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6,200 | 6,609 | Attentional Pooling for Action Recognition
Rohit Girdhar
Deva Ramanan
The Robotics Institute, Carnegie Mellon University
http://rohitgirdhar.github.io/AttentionalPoolingAction
Abstract
We introduce a simple yet surprisingly powerful model to incorporate attention
in action recognition and human object interaction tasks. Our proposed attention
module can be trained with or without extra supervision, and gives a sizable boost
in accuracy while keeping the network size and computational cost nearly the
same. It leads to significant improvements over state of the art base architecture
on three standard action recognition benchmarks across still images and videos,
and establishes new state of the art on MPII (12.5% relative improvement) and
HMDB (RGB) datasets. We also perform an extensive analysis of our attention
module both empirically and analytically. In terms of the latter, we introduce a
novel derivation of bottom-up and top-down attention as low-rank approximations
of bilinear pooling methods (typically used for fine-grained classification). From
this perspective, our attention formulation suggests a novel characterization of
action recognition as a fine-grained recognition problem.
1
Introduction
Human action recognition is a fundamental and well studied problem in computer vision. Traditional
approaches to action recognition relied on object detection [11, 19, 55], articulated pose [29, 33, 34,
53, 55], dense trajectories [50, 51] and part-based/structured models [9, 54, 56]. However, more
recently these methods have been surpassed by deep CNN-based representations [18, 30, 41, 46].
Interestingly, even video based action recognition has benefited greatly from advancements in image
based CNN models [20, 22, 42, 45]. With the exception of a few 3D-conv based methods [46, 47],
most approaches [12, 14, 15, 17, 52], including the current state of the art [52], use a variant of
discriminatively trained 2D-CNN [22] over the appearance (frames) and in some cases motion (optical
flow) modalities of the input video.
Attention: While using standard deep networks over the full image have shown great promise for
the task [52], it raises the question of whether action recognition can be considered as a general
classification problem. Some recent works have tried to generate more fine-grained representations
by extracting features around human pose keypoints [8] or on object/person bounding boxes [18, 30].
This form of ?hard-coded attention? helps improve performance, but requires labeling (or detecting)
objects or human pose. Moreover, these methods assume that focusing on the human or its parts is
always useful for discriminating actions. This might not necessarily be true for all actions; some
actions might be easier to distinguish using the background and context, like a ?basketball shoot? vs a
?throw?; while others might require paying close attention to objects being interacted by the human,
like in case of ?drinking from mug? vs ?drinking from water bottle?.
Our work: In this work, we propose a simple yet surprisingly powerful modification to standard
deep architectures to learn attention maps that automatically focus the attention on specific parts
of the input relevant to the task at hand. We show that our attention maps can be learned without
any additional supervision and automatically lead to significant improvements over the baseline
architecture. Moreover, our attention maps are easy to interpret and provide insight into where the
network should look when making a prediction, both in terms of bottom-up saliency and top-down
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
attention. We further experiment with adding human pose as an intermediate supervision to the
attention learning process to encourage the network to look for human object interactions. While
this makes little difference to the performance of image based recognition models, it leads to a larger
improvement on video datasets as videos consist of large number of ?non-iconic? frames where the
subject of object of actions may not be at the center of focus.
Our contributions: (1) An easy to use extension of state-of-the-art base architectures that incorporates attention to give significant improvement in action recognition performance at virtually
negligible increase in computation cost; (2) Extensive analysis of its performance on three action
recognition datasets across still images and videos, obtaining state of the art on MPII and HMDB-51
(RGB) and competitive on HICO; (3) Analysis of different base architectures for applicability of
our attention module; and (4) Mathematical analysis of our proposed attention module and showing
its equivalence to a rank-1 approximation of second order or bilinear pooling (typically used in fine
grained recognition methods [16, 26, 28]) suggesting a novel characterization of action recognition
as a fine grained recognition problem.
2
Related Work
Human action recognition is a well studied problem with various standard benchmarks spanning
across still images [7, 13, 33, 35, 56] and videos [24, 27, 40, 44]. The newer image based datasets such
as HICO [7] and MPII [33] are large and highly diverse, containing 600 and 393 classes respectively.
In contrast, collecting such diverse video based action datasets is hard, and hence existing popular
benchmarks like UCF101 [44] or HMDB51 [27] contain only 101 and 51 categories each. This in
turn has lead to much higher baseline performance on videos, eg. ? 94% [52] classification accuracy
on UCF101, compared to images, eg. ? 32% [30] mean average precision (mAP) on MPII.
Features: Video based action recognition methods focus on two main problems: action classification and (spatio-)temporal detection. While image based recognition problems, including action
recognition, have seen a large boost with the recent advancements in deep learning (e.g., MPII
performance went up from 5% mAP [33] to 27% mAP [18]), video based recognition still relies
on hand crafted features such as iDT [51] to obtain competitive performance. These features are
computed by extracting appearance and motion features along densely sampled point trajectories in
the video, aggregated into a fixed length representation by using fisher vectors [32]. Convolutional
neural network (CNN) based approaches to video action recognition have broadly followed two main
paradigms: (1) Multi-stream networks [41, 52] which split the input video into multiple modalities
such as RGB, optical flow, warped flow etc, train standard image based CNNs on top of those, and
late-fuse the predictions from each of the CNNs; and (2) 3D Conv Networks [46, 47] which represent
the video as a spatio-temporal blob and train a 3D convolutional model for action prediction. In terms
of performance, 3D conv based methods have been harder to scale and multi-stream methods [52]
currently hold state of the art performance on standard benchmarks. Our approach is complementary
to these paradigms and the attention module can be applied on top of either. We show results on
improving action classification over state of the art multi-stream model [52] in experiments.
Pose: There have also been previous works in incorporating human pose into action recognition [8,
10, 58]. In particular, P-CNN [8] computes local appearance and motion features along the pose
keypoints and aggregates those over the video for action prediction, but is not end-to-end trainable.
More recent work [58] adds pose as an additional stream in chained multi-stream fashion and shows
significant improvements. Our approach is complementary to these approaches as we use pose as
a regularizer in learning spatial attention maps to weight regions of the RGB frame. Moreover,
our method is not constrained by pose labels, and as we show in experiments, can show effective
performance with pose predicted by existing methods [4] or even without using pose.
Hard attention: Previous works in image based action recognition have shown impressive
performance by incorporating evidence from the human, context and pose keypoint bounding
boxes [8, 18, 30]. Gkioxari el al. [18] modified R-CNN pipeline to propose R*CNN, where they
choose an auxiliary box to encode context apart from the human bounding box. Mallya and Lazebnik [30] improve upon it by using the full image as the context and using multiple instance learning
(MIL) to reason over all humans present in the image to predict an action label for the image. Our
approach gets rid of the bounding box detection step and improves over both these methods by
automatically learning to attend to the most informative parts of the image for the task.
2
Soft attention: There has been relatively little work that explores unconstrained ?soft? attention for
action recognition, with the exception of [38, 43] for spatio-temporal and [39] for temporal attention.
Importantly, all these consider a video setting, where a LSTM network predicts a spatial attention
map for the current frame. Our method, however, uses a single frame to both predict and apply
spatial attention, making it amenable to both single image and video based use cases. [43] also uses
pose keypoints labeled in 3D videos to drive attention to parts of the body. In contrast, we learn an
unconstrained attention model that frequently learns to look around the human body for objects that
make it easier to classify the action.
Second-order pooling: Because our model uses a single set of appearance features to both predict
and apply an attention map, this makes the output quadratic in the features (Sec. 3.1). This observation
allows us to implement attention through second-order or bilinear pooling operations [28], made
efficient through low-rank approximations [16, 25, 26]. Our work is most related to [26], who point
out when efficiently implemented, low-rank approximations avoid explicitly computing second-order
features. We point out that a rank-1 approximation of second-order features is equivalent to an
attentional model sometimes denoted as ?self attention? [48]. Exposing this connection allows us
to explore several extensions, including variations of bottom-up and top-down attention, as well as
regularized attention maps that make use of additional supervised pose labels.
3
Approach
Our attentional pooling module is a trainable layer that plugs in after any convolutional layer of
a CNN. As most contemporary architectures [20, 22, 45] are fully convolutional with an average
pooling operation at the end, our module can be used to replace that operation with an attention
weighted pooling. We now derive the pooling layer as an efficient low-rank approximation to second
order pooling (Sec. 3.1). Then, we describe our network architecture that incorporates this attention
module and explore a pose regularized variant of the same (Sec. 3.2).
3.1
Attentional pooling as low-rank approximation of second-order pooling
Let us write the layer to be pooled as X ? Rn?f , where n is the number of spatial locations (e.g.,
n = 16 ? 16 = 256) and f is the number of channels (e.g., 2048). Standard sum (or max) pooling
would reduce this to vector in Rf ?1 , which could then be processed by a ?fully-connected? weight
vector w ? Rf ?1 to generate a classification score. We will denote matrices with upper case letters,
and vectors with lower-case bold letters. For the moment, assume we are training a binary classifier
(we generalize to more classes later in the derivation). We can formalize this pipeline with the
following notation:
scorepool (X) = 1T Xw,
X ? Rn?f , 1 ? Rn?1 , w ? Rf ?1
where
(1)
where 1 is a vector of all ones and x = 1T X ? R1?f is the (transposed) sum-pooled feature.
Second-order pooling: Following past work on second-order pooling [5], let us construct the feature
X T X ? Rf ?f . Prior work has demonstrated that such second-order statistics can be useful for
fine-grained classification [28]. Typically, one then ?vectorizes? this feature, and learns a f 2 vector
of weights to generate a score. If we write the vector of weights as a f ? f matrix, the inner product
between the two vectorized quantities can be succinctly written using the trace operator1 . The key
identity, T r(AB T ) = dot(A(:), B(:)) (using matlab notation), can easily be verified by plugging in
the definition of a trace operator. This allows us to write the classification score as follows:
scoreorder2 (X) = T r(X T XW T ),
where
X ? Rn?f , W ? Rf ?f
(2)
Low-rank second-order pooling: Let us approximate matrix W with a rank-1 approximation,
W = abT where a, b ? Rf ?1 . Plugging this into the above yields a novel formulation of attentional
1
https://en.wikipedia.org/wiki/Trace_(linear_algebra)
3
?1
?
?
?
?
?
?2
?
?
=
?
?
Bottom-up
Saliency
Attention
=
?
Method 1
?2
?
?1
?
2() order
pooling
?
Top-down
Attention
?
softmax
Pose Reg. Attention
Method 2
x-entropy
(a) Visualization of our approach to attentional pooling as a rank-1 approxiloss
nd
mation of 2 order pooling. By judicious ordering of the matrix multiplicaT
tions, one can avoid computing the second order feature X X and instead
compute the product of two attention maps. The top-down attentional map
Pose ? loss
is computed using class-specific weights ak , while the bottom-up map is
computed using class-agnostic weights b. We visualize the top-down and (b) We explore two architectures
bottom-up attention maps learned by our approach in Fig. 2.
in our work, explained in Sec. 3.2.
#
Figure 1: Visualization of our derivation and final network architectures.
pooling:
scoreattention (X) = T r(X T XbaT ),
T
where
X ? Rn?f , a, b ? Rf ?1
T
= T r(a X Xb)
T
(3)
(4)
T
= a X Xb
= aT X T (Xb)
(5)
(6)
where (4) makes use of the trace identity that T r(ABC) = T r(CAB) and (5) uses the fact that the
trace of a scalar is simply the scalar. The last line (6) gives efficient implementation of attentional
pooling: given a feature map X, compute an attention map over all n spatial locations with h =
Xb ? Rn?1 , that is then used to compute a weighted average of features x = X T h ? Rf ?1 . This
weighted-average feature is then pushed through a linear model aT x to produce the final score.
Interestingly, (6) can also be written as the following:
scoreattention (X) = (Xa)T X b
= (Xa)T (Xb)
(7)
(8)
The first line illustrates that the attentional heatmap can also be seen as Xa ? Rn?1 , with b being
the classifier of the attentionally-pooled feature. The second line illustrates that our formulation is
in fact symmetric, where the final score can be seen as the inner product between two attentional
heatmaps defined over all n spatial locations. Fig. 1a illustrates our approach.
Top-down attention: To generate prediction for multiple classes, we replace the weight matrix from
(2) with class-specific weights:
scoreorder2 (X, k) = T r(X T XWkT ),
where
X ? Rn?f , Wk ? Rf ?f
(9)
One could apply a similar derivation to produce class-specific vectors ak and bk , each of them
generating a class-specific attention map. Instead, we choose to distinctly model class-specific
?top-down? attention [3, 57] from bottom-up visual saliency that is class-agnostic [36]. We do so by
forcing one of the attention parameter vectors to be class-agnostic - e.g., bk = b. This makes our final
low-rank attentional model
scoreattention (X, k) = tTk h,
where
tk = Xak , h = Xb
(10)
equivalent to an inner product between top-down (class-specific) tk and bottom-up (saliency-based) h
attention maps. Our approach of combining top-down and botom-up attentional maps is reminiscent
of biologically-motivated schemes that modulate saliency maps with top-down cues [31]. This
suggests that our attentional model can also be implemented using a single, combined attention map
defined over all n spatial locations:
scoreattention (X, k) = 1T ck ,
where
ck = tk ? h,
(11)
where ? denotes elementwise multiplication and 1 is defined as before. We visualize the combined,
top-down, and bottom-up attention maps ck , tk , h ? Rn?1 in our experimental results.
4
3.2
Network Architecture
We now describe our network architecture to implement the attentional pooling described above. We
start from a state of the art base architecture, ResNet-101 [20]. It consists of a stack of ?modules?, each
of which contains multiple convolutional, pooling or identity mapping streams. It finally generates a
n1 ? n2 ? f spatial feature map, which is average pooled to get a f -dimensional vector and is then
classified using a linear classifier.
Our attention module plugs in at the last layer, after the spatial feature map. As shown in Fig. 1b
(Method 1), we predict a single channel bottom-up saliency map of same spatial resolution as the last
feature map, using a linear classifier on top of it (Xb). Similarly, we also generate the n1 ? n2 ? K
dimensional top-down attention map Xa, where K is number of classes. The two attention maps are
multiplied and spatially averaged to generate the K-dimensional output predictions ((Xa)T (Xb)).
These operations are equivalent to first multiplying the features with saliency (X T (Xb)) and then
passing through a classifier (a(X T (Xb))).
Pose: While this unconstrained attention module automatically learns to focus on relevant parts and
gives a sizable boost in accuracy, we take inspiration from previous work [8] and use human pose
keypoints to guide the attention. As shown in Fig. 1b (Method 2), we use a two-layer MLP on top
of the last layer to predict a 17 channel heatmap. The first 16 channels correspond to human pose
keypoints and incur a l2 loss against labeled (or detected, using [4]) pose> The final channel is used
as an unconstrained bottom-up attention map, as before. We refer to this method as pose-regularized
attention, and it can be thought of as a non-linear extension of previous attention map.
4
Experiments
Datasets: We experiment with three recent, large scale action recognition datasets, across still images
and videos, namely MPII, HICO and HMDB51. MPII Human Pose Dataset [33] contains 15205
images labeled with up to 16 human body keypoints, and classified into one of 393 action classes. It
is split into train, val (from authors of [18]) and test sets, with 8218, 6987 and 5708 images each. We
use the val set to compare with [18] and for ablative analysis while the final test results are obtained
by emailing our results to authors of [33]. The dataset is highly imbalanced and the evaluation is
performed using mean average precision (mAP) to equally weight all classes. HICO [7] is a recently
introduced dataset with labels for 600 human object interactions (HOI) combining 117 actions with
80 objects. It contains 38116 training and 9658 test images, with each image labeled with all the
HOIs active for that image (multi-label setting). Like MPII, this dataset is also highly unbalanced and
evaluation is performed using mAP over classes. Finally, to verify our method?s applicability to video
based action recognition, we experiment with a challenging trimmed action classification dataset,
HMDB51 [27]. It contains 6766 realistic and varied video clips from 51 action classes. Evaluation is
performed using average classification accuracy over three train/test splits from [23], each with 3570
train and 1530 test videos.
Baselines: Throughout the following sections, we compare our approach first to the standard base
architecture, mostly ResNet-101 [20], without the attention weighted pooling. Then we compare to
other reported methods and previous state of the art on the respective datasets.
MPII: We train our models for 393-way action classification on MPII with softmax cross-entropy
loss for both the baseline ResNet and our attentional model. We compare our performance in Tab. 1.
Our unconstrained attention model clearly out-performs the base ResNet model, as well as previous
state of the art methods involving detection of multiple contextual bounding boxes [18] and fusion
of full image with human bounding box features [30]. Our pose-regularized model performs best,
though the improvement is small. We visualize the attention maps learned in Fig. 2.
HICO: We train our model on HICO similar to MPII, and compare our performance in Tab. 2.
Again, we see a significant 5% boost over our base ResNet model. Moreover, we out-perform all
previous methods, including ones that use detection bounding boxes at test time except one [30],
when that is trained with a specialized weighted loss for this dataset. It is also worth noting that
the full image-only performance of VGG and ResNet were comparable in our experiments (29.4%
and 30.2%), suggesting that our approach shows larger relative improvement over a similar starting
baseline. Though we did not experiment with the same optimization setting as [30], we believe it
will give similar improvements there as well. Since this dataset also comes with labels decomposed
into actions and objects, we visualize what our attention model looks for, given images containing
5
GT Class
Test Image
Bottom Up
Other Class
Top Down
playing with animals
Combined
playing with animals
Top Down
resistance training
Combined
resistance training
garbage collector
garbage collector
video exercise workouts
video exercise workouts
forestry
forestry
rope skipping
rope skipping
violin
violin
bicycling
bicycling
marching band
marching band
yoga
yoga
basketball
basketball
skiing
skiing
chopping wood
chopping wood
yoga
yoga
calisthenics
calisthenics
fishing in stream
fishing in stream
Figure 2: Auto-generated (not hand-picked) visualization of bottom-up (Xb), top-down (Xak ) and combined
((Xak ) ? (Xb)) attention on validation images in MPII, that see largest improvement in softmax score for
correct class when trained with attention. Since the top-down/combined maps are class specific, we mention the
class name for which they are generated for on top left of those heatmaps. We consider 2 classes, the ground truth
(GT) for the image, and the class on which it gets lowest softmax score. The attention maps for GT class focus
aerobic
skiing
on the objects most useful for distinguishing aerobic
the class. Though
in many casesskiing
top-down and combined
maps
look similar, they capture different information, like in second example of garbage collector. While top-down
also focuses on the vehicles in background, combined map narrows it down to the garbage bags. (Best viewed
zoomed-in on screen)
bicycling
bicycling
sports ball
hotdog
bird
chopping wood
laptop
suitcase
donut
chopping wood
Figure 3: We crop a 100px patch around the attention peak for all images containing an HOI involving a given
object, and show 5 randomly picked patches for 6 object classes here. This suggests our attention model learns
to look for objects to improve HOI detection.
6
Table 1: Action classification performance on MPII dataset. Validation (Val) performance is reported on train set
split shared by authors of [18]. Test performance obtained from training on complete train set and submitting our
output file to authors of [33]. Note that even though our pose regularized model uses pose labels at training time
for regularizing attention, it does not require any pose input at test time. The top-half corresponds to a diagnostic
analysis of our approach with different base networks. Attention provides a strong 4% improvement for baseline
networks with larger spatial resolution (e.g., ResNet). Please see text for additional discussion. The bottom-half
reports prior work that makes use of object bounding boxes/pose. Our method performs slightly better with pose
annotations (on training data), but even without any pose or detection annotations, we outperform all prior work.
Method
Full Img
Inception-V2 (ours)
ResNet101 (ours)
Attn. Pool. (I-V2) (ours)
Attn. Pool. (R-101) (ours)
X
X
X
X
Dense Trajectory + Pose [33]
VGG16, RCNN [18]
VGG16, R*CNN [18]
VGG16, Fusion (best) [30]
VGG16, Fusion+MIL (best) [30]
Pose Reg. Attn. Pooling (ours)
X
X
X
X
Bbox
Pose
MIL
Val (mAP)
Test (mAP)
25.2
26.2
24.3
30.3
36.0
16.5
21.7
30.6
5.5
26.7
32.2
31.9
36.1
X
X
X
X
X
X
X
Table 2: Multi-label HOI classification performance on HICO dataset. The top-half compares our performance
to other full image based methods. The bottom-half reports methods that use object bounding boxes/pose. Our
model out-performs various approaches that need bounding boxes, multi-instance learning (MIL) or specialized
losses, and achieves performance competitive to state of the art. Note that even though our pose regularized
model uses computed pose labels at training time, it does not require any pose input at test time.
Method
Full Im.
Bbox/Pose
MIL
AlexNet+SVM [7]
VGG16, full image [30]
ResNet101, full image (ours)
ResNet101 with CBP [16] (impl. from [1])
Attentional Pooling (ours)
X
X
X
X
X
R*CNN [18] (reported in [30])
Scene-RCNN [18] (reported in [30])
Fusion (best reported) [30]
Pose Regularized Attentional Pooling (ours)
X
X
X
X
X
X
X
X
X
X
Fusion, weighted loss (best reported) [30]
X
X
X
Wtd Loss
mAP
19.4
29.4
30.2
26.8
35.0
28.5
29.0
33.8
34.6
X
36.1
interactions with a specific object. As Fig. 3 shows, the attention peak is typically close to the object
of interest, showing the importance of detecting objects in HOI detection tasks. Moreover, this
suggests that our attention maps can also function as weak-supervision for object detection.
HMDB51: Next, we apply our attentional method to the RGB stream of the current state of the art
deep model on this dataset, TSN [52]. TSN extends the standard two-stream [41] architecture by using
a much deeper base architecture [22] along with enforcing consensus over multiple frames from the
video at training time. For the purpose of this work, we focus on the RGB stream only but our method
is applicable to flow/warped-flow streams as well. We first train a TSN model using ResNet-101 as
base architecture after re-sizing input frames to 450px. This ensures larger spatial dimensions of
the output (14 ? 14), hence ensuring the last-layer features are amenable to attention. Though our
base ResNet model does worse than BN-inception TSN model, as Tab. 3 shows, using our attention
module improves the base model to do comparably well. Interestingly, on this dataset regularizing the
attention through pose gives a significant boost in performance, out-performing TSN and establishing
new state of the art on the RGB stream-only model for HMDB. We visualize the attention maps
with normal and pose-regularized attention in Fig. 4. The pose regularized attention are more peaky
near the human than their linear counterparts. This potentially explains the improvement using pose
on HMDB while it does not help as much on HICO or MPII; HICO and MPII, being image based
datasets typically have ?iconic? images, with the subjects and objects of action typically in the center
and focus of the image. Video frames in HMDB, on the other hand, may have the subject move all
7
Table 3: Action classification performance on HMDB51 dataset using only the RGB stream of a two-stream
model. Our base ResNet stream training is done over 480px rescaled images, same as used in our attention
model for comparison purposes. Our pose based attention model out-performs the base network by large margin,
and obtains state of the art RGB-only performance by out-performing TSN [52].
Method
Split 2
Split 3
Avg
54.4
48.2
51.1
54.4
49.5
46.5
51.6
51.1
49.2
46.7
49.7
50.9
51.0
48.9
46.7
47.1
50.8
52.2
Pose Reg. Attention
Attention
TSN, BN-inception (RGB) [52] (Via email with authors)
RGB Stream, ResNet50 (RGB) [14] (reported at [2])
RGB Stream, ResNet152 (RGB) [14] (reported at [2])
TSN, ResNet101 (RGB) (ours)
Linear Attentional Pooling (ours)
Pose regularized Attentional Pooling (ours)
Split 1
Figure 4: Attention maps with linear attention and pose regularized attention on a video from HMDB. Note the
pose-guided attention is better able to focus on regions of interest in the non-iconic frames.
across the frame throughout the video, and hence additional supervision through pose at training time
helps focus the attention at the right spot.
Full-rank pooling: Given our formulation of attention as low-rank second-order pooling, a natural
question is what would be the performance of a full-rank model? Explicitly computing the secondorder features of size f ? f for f = 2048 (and learning the associated classifier) is cumbersome.
Instead, we make use of the compact bilinear approach (CBP) of [16], which generates a lowdimensional approximation of full bilinear pooling [28] using the TensorSketch algorithm. To keep
the final output comparable to our attentional-pooled model, we project to f = 2048 dimensions.
We find it performs slightly worse than simple average pooling in Table 2. Note that we use an
existing implementation [1] with minimal hyper-parameter optimization, and leave a more rigorous
comparison to future work.
Rank-P approximation: While a full-rank model is cumbersome, we can still explore the effect of
using a higher, P -rank approximation. Essentially, a rank-P approximation generates P (1-channel)
bottom-up and (C channel) top-down attention maps, and the final prediction is the product of
corresponding heatmaps, summed over P . On MPII, we obtain mAP of 30.3, 29.9, 30.0 for P =1,
2 and 5 respectively, showing that the validation performance is relatively stable with P . We do
observe a drop in training loss with a higher P , indicating that a higher-rank approximation could be
useful for harder datasets and tasks.
Per-class attention maps: As we described in Sec. 3.1, our inspiration for combining class-specific
and class-agnostic classifiers (i.e. top-down and bottom-up attention respectively), came from the
Neuroscience literature on integrating top-down and bottom-up attention [31]. However, our model
can also be extended to learn completely class-specific attention maps, by predicting C bottom-up
attention maps, and combining each map with the corresponding softmax classifier for that class. We
experiment with this idea on MPII and obtain a mAP of 27.9 with 393 (=num-classes) attention maps,
compared to 30.3% with 1 map, and 26.2% without attention. On further analysis we observe that
both models achieve near perfect mAP on training data, implying that adding more parameters with
8
multiple attention maps leads to over-fitting on the relatively small MPII trainset. However, this may
be a viable approach for larger datasets.
Diagnostics: It is natural to consider variants of our model that only consider the bottom-up or
top-down attentional map. Interestingly, a top-down-only version of our model is equivalent to
standard average pooling from (1):
scoretop?down (X) = 1T Xak = 1T tk where tk = Xak
(12)
From this perspective, the above derivation gives the ability to generate attentional maps from averagepooling networks. Similar observations have been pointed out before [57]. Hence our comparison
with baseline models with average pooling essentially compares to ?top-down-only? attention models.
It is not clear how to construct a bottom-up only model, since it is class-agnostic, making it difficult to
produce class-specific scores. Rather, a reasonable approximation might be applying an off-the-shelf
(bottom-up) saliency method used to limit the spatial region that features are averaged over. Our
initial experiments with existing saliency-based methods [21] were not promising.
Base Network: Finally, we analyze the choice of base architecture for the effectiveness of our
proposed attentional pooling module. In Tab. 1, we compare the improvement using attention over
ResNet-101 (R-101) [20] and an BN-Inception (I-V2) [22]. Both models perform comparably when
trained for full image, however, while we see a 4% improvement on R-101 on using attention, we do
not see similar improvements for I-V2. This points to an important distinction in the two architectures,
i.e., Inception-style models are designed to be faster in inference and training by rapidly down
sampling input images in initial layers through max-pooling. While this reduces the computational
cost for later layers, it leads to most layers having very large receptive fields, and hence later neurons
have effective access to all of the image pixels. This suggests that all the spatial features at the last
layer could be highly similar. In contrast, R-101 downscales the spatial resolution gradually, allowing
the last layer features to specialize to different parts of the image, hence benefiting more from
attentional pooling. This effect was further corroborated by our experiments on HMDB, where using
the standard 224px input resolution showed no improvement with attention, while the same image
resized to 450px at input time did. This initial resize ensures the last-layer features are sufficiently
distinct to benefit from attentional pooling.
5
Discussion and Conclusion
An important distinction of our model from some previous works [18, 30] is that it does not explicitly
model action at an instance or bounding-box level. This, in fact, is a strength of our model; making it
capable of attending to objects outside of any person-instance bounding box (such as bags of garbage
for ?garbage collecting?, in Fig 2). In theory, our model can also be applied to instance-level action
recognition by applying attentional pooling over an instance?s RoI features. Such a model would learn
to look at different parts of human body and its interactions with nearby objects. However, it?s notable
that most existing action datasets, including [6, 7, 27, 33, 40, 44], come with only frame or video
level labels; and though [18, 30] are designed for instance-level recognition, they are not applied
as such. They either copy image level labels to instances or use multiple-instance learning, either
of which can be used in conjunction with our model. Another interesting connection that emerges
from our work is the relation between second-order pooling and attention. The two communities are
traditionally seen as distinct, and our work strongly suggests that they should mix: as newer action
datasets become more fine-grained, we should explore second-order pooling techniques for action
recognition. Similarly, second-order pooling can serve as a simple but strong baseline for the attention
community, which tends to focus on more complex sequential attention networks (based on RNNs or
LSTMs). It is also worth noting that similar ideas involving self attention and bilinear models have
recently also shown significant improvements in other tasks like image classification [49], language
translation [48] and visual question answering [37].
Conclusion: We have introduced a simple formulation of attention as low-rank second-order pooling, and illustrate it on the task of action classification from single (RGB) images. Our formulation
allows for explicit integration of bottom-up saliency and top-down attention, and can take advantage
of additional supervision when needed (through pose labels). Our model produces competitive or
state-of-the-art results on widely benchmarked datasets, by learning where to look when pooling
features across an image. Finally, it is easy to implement and requires few additional parameters,
making it an attractive alternative to standard pooling, which is a ubiquitous operation in nearly all
contemporary deep networks.
9
Acknowledgements: Authors would like to thank Olga Russakovsky for initial review. This
research was supported in part by the National Science Foundation (NSF) under grant numbers
CNS-1518865 and IIS-1618903, and the Defense Advanced Research Projects Agency (DARPA)
under Contract No. HR001117C0051. Additional support was provided by the Intel Science and
Technology Center for Visual Cloud Systems (ISTC-VCS). Any opinions, findings, conclusions or
recommendations expressed in this material are those of the authors and do not necessarily reflect the
view(s) of their employers or the above-mentioned funding sources.
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12
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6,201 | 661 | Improving Convergence in Hierarchical
Matching Networks for Object
Recognition
Joachim Utans*
Gene Gindi t
Department of Electrical Engineering
Yale University
P. O. Box 2157 Yale Station
New Haven, CT 06520
Abstract
We are interested in the use of analog neural networks for recognizing visual objects. Objects are described by the set of parts
they are composed of and their structural relationship. Structural models are stored in a database and the recognition problem reduces to matching data to models in a structurally consistent way. The object recognition problem is in general very difficult in that it involves coupled problems of grouping, segmentation
and matching. We limit the problem here to the simultaneous labelling of the parts of a single object and the determination of
analog parameters. This coupled problem reduces to a weighted
match problem in which an optimizing neural network must minimize E(M, p) = LO'i MO'i WO'i(p), where the {MO'd are binary
match variables for data parts i to model parts a and {Wai(P)}
are weights dependent on parameters p . In this work we show that
by first solving for estimates p without solving for M ai , we may
obtain good initial parameter estimates that yield better solutions
for M and p.
*Current address: International Computer Science Institute, 1947 Center Street,
Suite 600, Berkeley, CA 94704, [email protected]
tCurrent address: SUNY Stony Brook, Department of Electrical Engineering, Stony
Brook, NY 11784
401
402
Utans and Gindi
Figure 1: Stored Model for a 3-Level Compositional Hierarchy (compare Figure 3) .
1
Recognition via Stochastic Forward Models
The Frameville object recognition system introduced by Mjolsness et al [5, 6, 1]
makes use of a compositional hierarchy to represent stored models. The recognition
problem is formulated as the minimization of an objective function. Mjolsness [3,4]
has proposed to derive the objective function describing the recognition problem
in a principled way from a stochastic model that describes the objects the system
is designed to recognize (stochastic visual grammar). The description mirrors the
data representation as a compositional hierarchy, at each stage the description of
the object becomes more detailed as parts are added.
The stochastic model assigns a probability distribution at each stage of that process.
Thus at each level of the hierarchy a more detailed description of parts in terms of
their subparts is given by specifying a probability distribution for the coordinates of
the subparts. Explicitly specifying these distributions allows for finer control over
individual part descriptions than the rather general parameter error terms used
before [1, 8]. The goal is to derive a joint probability distribution for an instance
of an object and its parts as it appears in the scene. This gives the probability of
observing such an object prior to the arrival of the data. Given an observed image,
the recognition problem can be stated as a Bayesian inference problem that the
neural network solves.
1.1
3-Level Stochastic Model
For example, consider the model shown in Figure 1 and 3. The object and its parts
are represented as line segments (sticks), the parameters were p = (x, y, I, ())T with
x , y denoting position, I the length of a stick and () its orientation. The model
considers only a rigid translation of an object in the image.
Only one model is stored. From a central position p = (x, y, I, ()), itself chosen
from a uniform density, the N{3 parts at the first level are placed. Their structural
relationships is stored as coordinates u{3 in an object-centered coordinate frame,
i.e. relative to p. While placing the parts, Gaussian distributed noise with mean 0
and is added to the position coordinates to capture the notion of natural variation
of the object's shape. The variance is coordinate specific, but we assume the same
distribution for the x and y coordinates, O"'ix; O"'~, is the variance for the length
Improving Convergence in Hierarchical Matching Networks for Object Recognition
component and UI9 for the relative angle. In addition, here we assume for simplicity
that all parts are independently distributed. Each of the parts {3 is composed of subparts. For simplicity of notation, we assume that each part {3 is composed from the
same number of subparts N m (note that the index 'Y in Figure 2 here corresponds
to the double index {3m to keep track of which part {3 subpart {3m belongs to on the
model side, i.e. the index (3m denotes the mth sub-part of part (3). The next step
models the unordering of parts in the image via a permutation matrix M, chosen
with probability P(M), by which their identity is lost. If this step were omitted,
the recognition problem would reduce to the problem of estimating part parameters
because the parts would already be labeled.
From the grammar we compute the final joint probability distribution (all constant
terms are collected in a constant C):
P(M, {P,3m}, {PtJ}, p) =
1.2
Frameville Architecture for Part Labelling within a single Object
The stochastic forward model for the part labelling problem with only a single object
present in the scene translates into a reduced Frameville architecture as depicted in
Figure 2. The compositional hierarchy parallels the steps in the stochastic model
as parts are added at each level. Match variables appear only at the lowest level,
corresponding to the permutation step of the grammar. Parts in the image must
be matched to model parts and parts found to belong to the stored object must be
grouped together.
The single match neuron Mai at the highest level can be set to unity since we assume
we know the object's identity and only a single object is present. Similarly, all terms
inaij from the first to the second level can be set to unity for the correct grouping
since the grouping is known at this point from the forward model description. In
addition, at the intermediate (second) level, we may set all M,3j = 1 for {3 = j
and MtJj = 0 otherwise with no loss of generality. These mid-level frames may
be matched ahead of time, but their parameters must be computed from data.
Introducing a part permutation at the intermediate levels thus is redundant. Given
this, an additional simplification ina grouping variables at the lowest (third) level
is possible. Since parts are pre-matched at all but the lowest level, inaj k can be
expressed in terms of the part match M"{k as inajk = M"{k1NA"{tJM,3j and explicitly
representing inaj k as variables is not necessary.
The input to the system are the {pk}, recognition involves finding the parameters
403
404
Utans and Gindi
Model
<l
Data
rame
?
?
?
?
x
y
9
I
Figure 2: Frameville Architecture for the Stochastic Model. The 3-level grammar leads to a reduced
"Frameville" style network architecture : a single model is stored on the model side and only one instance
of the model is present in the input data. The ovals on the model side represent the object, its parts
and subparts (compare Figure 1); the arcs INA represent their structural relationship . On the data side,
the triangles represent parameter vectors (or frames) describing an instance of the object in the scene.
At the lowest level the Pk represent the input data, parameters at higher levels in the hierarchy must be
computed by the network (represented as bold triangles) . ina represents the grouping of parts on the
data side (see text) . The horizontal lines represent assignments from frames on the data side to nodes
on the model side. At the intermediate level, frames are prematched to the corresponding parts on the
model side ; match variables are necessary only at the lowest level (represented as bold lines with circles).
P and {Pi} as well as the labelling of parts M. Thus, from Bayes Theorem
P( {pdIM, p, {Pi} )P(M, p, {Pi})
P({Pk} )
(2)
ex: P(M, p, {Pi}, {pd)
and recognition reduces to finding the most probable values for p, {Pi} and M
given the data:
arg max P(M, p, {Pi}, {pd)
(3)
M,P,{Pi}
Solving the inference problem involves finding the MAP estimate and is is equivalent
to minimizing the exponent in equation (1) with respect to M, P and {Pi}.
2
2.1
Bootstrap: Coarse Scale Hints to Initialize the Network
Compositional Hierarchy and Scale Space
In some labelling approaches found in the vision literature, an object is first labelled
at the coarse, low resolution, level and approximate parameters are found . In this
top-down approach the information at the higher, more abstract, levels is used
Improving Convergence in Hierarchical Matching Networks for Object Recognition
im---:.+-Human
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1L _ _ _ _ _ _ 1I
r---------'
1
II
1-- - , 1
iV
t
(I)
i
~
1_ _ _ _ _ _ _
Am
i
_ _ _ J1
III
spatial
scale
abstraction
Figure 3: Compositional Hierarchy vs . Scale Space Hierarchy_ A compositional hierarchy can represent a
scale space hierarchy. At successive levels in the hierarchy, more and more detail is added to the object_
to select initial values for the parts at the next lower level of abstraction. The
segmentation and labelling at this next lowest level is thus not done blindly; rather
it is strongly influenced contextually by the results at the level above.
In fact, in very general terms such a scheme was described by Marr and Nishihara [2].
They advocate in essence a hierarchical model base in which a shape is first matched
to the highest levels, and defaults in terms of relative object-based parameters of
parts at the next level are recalled from memory. These defaults then serve as initial
values in an unspecified segmentation algorithm that derives part parameters; this
step is repeated recursively until the lowest level is reached.
Note that the highest level of abstractions correspond to the coarsest levels of spatial
scale. There is nothing in the design of the model base that demands this, but invariably, elements at the top of a compositional hierarchy are of coarser scale since they
must both include the many subparts below, and summarize this inclusion with
relatively few parameters. Figure 3 illustrates the correspondence between these
representations. In this sense, the compositional hierarchy as applied to shapes
includes a notion of scale, but there is no "scale-space" operation of intentionally
blurring data. The notion of Scale Space as utilized here thus differs from the
application of the method to low-level computations in the visual domain where
auxiliary coarse scale representations are computed explicitly. The object representations in the Frameville system as described earlier combines both, bottom-up and
top-down elements. If the top-down aspects of the scheme described by Marr and
Nishihara [2] could be incorporated into the Frameville architecture, then our previous simulation results [8] suggest that much better performance can be expected
from the neural network. Two problems must be addressed: (1) How do we obtain,
from the observed raw data alone, a coarse estimate of the slot parameters at the
highest level and (2) given these crude estimates how do we utilize them to recall
default settings for the segmentation one level below?
405
406
Utans and Gindi
0,
Bootstrap
y
Model
Data
Figure 4: Bootstrap computation for a network from a 3- level grammar. Analog frame variables at the
top and intermediate level are initialized from data by a bootstrap computation (bold lines indicate the
flow of information)
2.2
Initialization of Coarse Scale Parameters
We propose to aid convergence by supplying initial values for the analog variables p
and {Pi}; these must be computed from data without making use of the labelling.
In general, it is not possible to solve for the analog parameters without knowledge
of the correct permutation matrix M. However, for the purpose of obtaining an
approximation f> one can derive a new objective function that does not depend on
M and the parameters {Pi} by integrating over the {Pi} and summing over all
possible permutation matrices M:
P(p,{pk})
J
L
=
d{pj}P(P,{Pi},{pd,M)
(4)
{M}IM
is a
permutation
This formulation leads to an Elastic Net type network [9, 7]. However, this implementation of a separate network for the bootstrap computations is expensive.
Here we use simpler computation where the coarse scale parameters are estimated
by computing sample averages, corresponding to finding the solution for the Elastic
Net in the high temperature limit [7]. For the position x we find, after integrating
over the {xi},
L
1
x
L{3m
1/(O"~xO"~mx)
_
1
L{3
1/ O"~x
13m k
L
(3
M{3mkXk
O"~xO"~mx
U{3x
O"~x
(5)
and similarly for y. Since the assignment M{3m k of subparts k on the data side
to subparts fJm on the model side is not known at this point, the first term in
equations (5) cannot be evaluated . After approximating the actual variance with
Improving Convergence in Hierarchical Matching Networks for Object Recognition
an average variance, these equations reduce to
1 1 1
x
N N
Xk - N N
uf3mx - N
13
m
L
k
13
m
L
13m
L uf3x
(6)
13 13
In terms of the objective function this translates into assuming that here the error
terms for all parts are weighted equally. Since these weights would depend on the
actual part match, this just corresponds to our ignorance regarding identity of the
parts. This approximation assumes that the variances do not differ by a large
amount, otherwise the approximation p will not be close to the true values. Since
the model can be designed such that the part primitives used at the lowest level
of the grammar are not highly specialized as would be the case for abstractions
at higher levels of the model, the approximation proved sufficient for the problems
studied here.
The neural network can be used to perform the calculation. The Elastic Net formulation assigns approximately equal weights to all possible assignments at high
temperatures. Thus, this behavior can be expressed in the original network with
match variables by choosing Mf3mk = l/{Nf3Nm ) V i,j. This leads to the following
two-pass bootstrap computation. Using this specific choice for M only the analog
variables need to be updated to compute the coarse scale estimates. The network
with constant M is just the neural network implementation for computing x from
equation (6). After these have converged, x can be used to compute Xj = x + uf3.
Thus, the parameters for intermediate levels can by hypothesized from the coarse
scale estimate x by adding the known transformation (recall that for intermediate
levels, the part identity is preserved and no permutation steps takes place (see Figure 2)). Then the network is restarted with random values for the match variables
to compute the correct labelling and the correct parameters.
2.3
Simulation Results
The bootstrap procedure has been implemented for a 3-level hierarchical model. The
model describes a "gingerbread man" as shown in Figure 3. The incorrect solutions
observed did not, in the vast majority of cases, violate the permutation matrix
constraint, i.e. the assignment was unique. However, even though the assignment
is unique, parts where not always assigned correctly. Most commonly, the identity
of neighboring parts was interchanged, in particular for cases with large variance.
The advantage of using the bootstrap initialization is clear from Figure 5. For
the simulation, cr~ = 2crt; the noise variance was identical for all parts. The network computed the solution reliably for large noise variances. In such cases the
performance of the network without initialization deteriorates rapidly. Only one
set of 10 experiments was used for the graph but in all simulations performed,
the network with initialization consistently outperformed the network without initialization. Figure 5(right) shows the time measured in the number of iterations
necessary for the network to converge; it is almost unaffected by the increase in the
noise variance. This is because the initial values derived from data are still close
to the final solution. While in some cases, the random starting point happens to
be close to the correct solution and the network without initialization converges
rapidly, Figure 5 reflect the typical behavior and demonstrate the advantage of
computing approximate initial values.
407
408
Utans and Gindi
Success Rate
Convergence Speed
100
300
'"
.-.,o-=
80
. oj
...
11)200
80
.oj
~
.....
o
...
.0
II)
~
'h .o
100
-=
20
0.2
0 .?
ott.
0.8
2
(122
0.8
I 0
o
0.0
0.2
0.4
a"1'
0.8
2
0.8
1.0
CT22
Figure 5 : Results Comparing the Network without and with Initialization (solid line) .
Left : The success rate indicates the rate at which the network converged to the correct solutions. /1~
denotes the noise variance at the intermediate level of the model and /1~ the noise variance at the lowest
level. Only one set of 10 experiments was used for the graph but in all simulations performed , the
network with initialization consistently outperformed the network without initialization .
Right: The graph shows the average time it takes for the network to converge (as measured by the
number of iterations) averaged over 10 experiments. Only simulations where the network converged to
the correct solution are used to compute the average time for convergence. The stopping criterion used
required all the match neurons to assume values M'j > 0.95 or M'J < 0 .05. The error bars denote the
standard deviation.
Acknowledgements
This work was supported in part by AFOSR grant AFOSR 90-0224 . Vie thank
E. Mjolsness and A. Rangarajan for many helpful discussions.
References
[1] G. Gindi, E. Mj~lsness, and P. Anandan. Neural networks for model based recognition. In Neural Networks: Concepts, Applications and Implementations, pages 144-173.
Prentice-Hall, 1991.
[2] David Marr. Vision. W. H. Freeman and Co., New York, 1982.
[3] E. Mjolsness. Bay~sian inference on visual grammars by neural nets that optimize.
Technical Report YALEU-DCS-TR-854, Yale University, Dept. of Computer Science,
1991.
[4] E. Mj~lsness. Visual grammars and their neural nets. In R.P. Lippmann J.E. Moody,
S.J. Hanson, editor, Advances in Neural Information Processing Systems 4. Morgan
Kaufmann Publishers, San Mateo, CA, 1992.
[5] Eric Mjolsness, Gene Gindi, and P. Anandan. Optimization in model matching and
perceptual organization: A first look. Research report yaleu/dcs/rr-634, Yale University, Department of Computer Science, 1988.
[6] Eric Mjolsness, Gene R. Gindi, and P. Anandan. Optimization in model matching and
perceptual organization. Neural Computation, vol. 1, no. 2, 1989.
[7] Joachim Utans. Neural Networks for Object Recognition within Compositional Hierarchies. PhD thesis, Department of Electrical Engineering, Yale University, New Haven,
CT 06520, 1992.
[8] Joachim Utans, Gene R. Gindi, Eric Mjolsness, and P. Anandan. Neural networks
for object recognition within compositional hierarchies: Initial experiments. Technical report 8903, Yale University, Center for Systems Science, Department Electrical
Engineering, 1989.
[9] A. L. Yuille. Generalized deformable models, statistical physics, and matching problems. Neural Computation, 2(2):1-24, 1990.
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6,202 | 6,610 | On the Consistency of Quick Shift
Heinrich Jiang
Google Inc.
1600 Amphitheatre Parkway, Mountain View, CA 94043
[email protected]
Abstract
Quick Shift is a popular mode-seeking and clustering algorithm. We present finite
sample statistical consistency guarantees for Quick Shift on mode and cluster
recovery under mild distributional assumptions. We then apply our results to
construct a consistent modal regression algorithm.
1
Introduction
Quick Shift [16] is a clustering and mode-seeking procedure that has received much attention in
computer vision and related areas. It is simple and proceeds as follows: it moves each sample to its
closest sample with a higher empirical density if one exists in a ? radius ball, where the empirical
density is taken to be the Kernel Density Estimator (KDE). The output of the procedure can thus be
seen as a graph whose vertices are the sample points and a directed edge from each sample to its next
point if one exists. Furthermore, it can be seen that Quick Shift partitions the samples into trees which
can be taken as the final clusters, and the root of each such tree is an estimate of a local maxima.
Quick Shift was designed as an alternative to the better known mean-shift procedure [4, 5]. Mean-shift
performs a gradient ascent of the KDE starting at each sample until -convergence. The samples that
correspond to the same points of convergence are in the same cluster and the points of convergence
are taken to be the estimates of the modes. Both procedures aim at clustering the data points by
incrementally hill-climbing to a mode in the underlying density. Some key differences are that Quick
Shift restricts the steps to sample points and has the extra ? parameter. In this paper, we show that
Quick Shift can surprisingly attain strong statistical guarantees without the second-order density
assumptions required to analyze mean-shift.
We prove that Quick Shift recovers the modes of an arbitrary multimodal density at a minimax optimal
rate under mild nonparametric assumptions. This provides an alternative to known procedures with
similar statistical guarantees; however such procedures only recover the modes but fail to inform us
how to assign the sample points to a mode which is critical for clustering. Quick Shift on the other
hand recovers both the modes and the clustering assignments with statistical consistency guarantees.
Moreover, Quick Shift?s ability to do all of this has been extensively validated in practice.
A unique feature of Quick Shift is that it has a segmentation parameter ? which allows practioners
to merge clusters corresponding to certain less salient modes of the distribution. In other words, if
a local mode is not the maximizer of its ? -radius neighborhood, then its corresponding cluster will
become merged to that of another mode. Current consistent mode-seeking procedures [6, 12] fail to
allow one to control such segmentation. We give guarantees on how Quick Shift does this given an
arbitrary setting of ? .
We show that Quick Shift can also be used to recover the cluster tree. In cluster tree estimation, the
known procedures with the strongest statistical consistency guarantees include Robust Single Linkage
(RSL) [2] and its variants e.g. [13, 7]. We show that Quick Shift attains similar guarantees.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Thus, Quick Shift, a simple and already popular procedure, can simultaneously recover the modes
with segmentation tuning, provide clustering assignments to the appropriate mode, and can estimate
the cluster tree of an unknown density f with the strong consistency guarantees. No other procedure
has been shown to have these properties.
Then we use Quick Shift to solve the modal regression problem [3], which involves estimating the
modes of the conditional density f (y|X) rather than the mean as in classical regression. Traditional
approaches use a modified version of mean-shift. We provide an alternative using Quick Shift which
has precise statistical consistency guarantees under much more mild assumptions.
Figure 1: Quick Shift example. Left: ? = ?. The procedure returns one tree, whose head is the
sample with highest empirical density. Right: ? set to a lower value. The edges with length greater
than ? are no longer present when compared to the left. We are left with three clusters.
2
Assumptions and Supporting Results
Algorithm 1 Quick Shift
Input: Samples X[n] := {x1 , ..., xn }, KDE bandwidth h, segmentation parameter ? > 0.
Initialize directed graph G with vertices {x1 , ..., xn } and no edges.
for i = 1 to n do
if there exists x ? X[n] such that fbh (x) > fbh (xi ) and ||x ? xi || ? ? then
Add to G a directed edge from xi to argminxj ?X[n] :fbh (xj )>fbh (xi ) ||xi ? xj ||.
end if
end for
return G.
2.1
Setup
Let X[n] = {x1 , ..., xn } be n i.i.d. samples drawn from distribution F with density f over the
uniform measure on Rd .
Assumption 1 (H?lder Density). f is H?lder continuous on compact support X ? Rd . i.e. |f (x) ?
f (x0 )| ? C? ||x ? x0 ||? for all x, x0 ? X and some 0 < ? ? 1 and C? > 0.
Definition 1 (Level Set). The ? level set of f is defined as Lf (?) := {x ? X : f (x) ? ?}.
Definition 2 (Hausdorff Distance). dH (A, A0 ) = max{supx?A d(x, A0 ), supx?A0 d(x, A)}, where
d(x, A) := inf x0 ?A ||x ? x0 ||.
The next assumption says that the level sets are continuous in the level ? w.r.t. the Hausdorff metric.
This ensures that there are no flat areas in which the procedure may get stuck at.
Assumption 2 (Uniform Continuity of Level Sets). For each > 0, there exists ? > 0 such that for
0 < ?, ?0 ? ||f ||? with |? ? ?0 | < ?, then dH (Lf (?), Lf (?0 )) < .
Remark 1. One of the difficulties is dealing with flat areas of the density. Procedures that try to
incrementally move points to nearby areas of higher density will have difficulties in regions where
there is no change in the density. The above assumption is a simple and mild formulation which
ensures there are no such flat regions.
Remark 2. Note that our assumptions are quite mild when compared to analyses of similar procedures like mean-shift, which require at least second-order smoothness assumptions. Interestingly, we
only require H?lder continuity.
2
2.2
KDE Bounds
We next give uniform bounds on KDE required to analyze Quick Shift.
d
Definition 3. Define
R kernel function K : R ? R?0 where R?0 denotes the non-negative real
numbers such that Rd K(u)du = 1.
We make the following mild regularity assumptions on K.
Assumption 3. (Spherically symmetric, non-increasing, and exponential decays) There exists nonincreasing function k : R?0 ? R?0 such that K(u) = k(|u|) for u ? Rd and there exists
?, C? , t0 > 0 such that for t > t0 , k(t) ? C? ? exp(?t? ).
Remark 3. These assumptions allow the popular kernels such as Gaussian, Exponential, Silverman,
uniform, triangular, tricube, Cosine, and Epanechnikov.
Definition 4 (Kernel Density Estimator). Given a kernel K and bandwidth h > 0 the KDE is defined
by
n
x ? Xi
1 X
b
K
.
fh (x) =
n ? hd i=1
h
Here we provide the uniform KDE bound which will be used for our analysis, established in [11].
Lemma 1. [`? bound for ?-H?lder continuous functions. Theorem 2 of [11]] There exists positive
constant C 0 depending on f and K such that the following holds with probability at least 1 ? 1/n
uniformly in h > (log n/n)1/d .
!
r
log
n
0
?
sup |fbh (x) ? f (x)| < C ? h +
.
n ? hd
x?Rd
3
Mode Estimation
In this section, we give guarantees about the local modes returned by Quick Shift. We make the
additional assumption that the modes are local maxima points with negative-definite Hessian.
Assumption 4. [Modes] A local maxima of f is a connected region M such that the density is
constant on M and decays around its boundaries. Assume that each local maxima of f is a point,
which we call a mode. Let M be the modes of f where M is a finite set. Then let f be twice
differentiable around a neighborhood of each x ? M and let f have a negative-definite Hessian at
each x ? M and those neighborhoods are disjoint.
This assumption leads to the following.
? C? > 0 such that the
Lemma 2 (Lemma 5 of [6]). Let f satisfy Assumption 4. There exists rM , C,
following holds for all x0 ? M simultaneously.
C? ? |x0 ? x|2 ? f (x0 ) ? f (x) ? C? ? |x0 ? x|2 ,
for all x ? Ax0 where Ax0 is a connected component of {x : f (x) ? inf x0 ?B(x0 ,rM ) f (x)} which
contains x0 and does not intersect with other modes.
The next assumption ensures that the level sets don?t become arbitrarily thin as long as we are
sufficiently away from the modes.
Assumption 5. [Level Set Regularity] For each ?, r > 0, there exists ? > 0 such that the following
holds for all connected components A of Lf (?) with ? > 0 and A 6? ?x0 ?M B(x0 , r). If x lies on
the boundary of A, then Vol(B(x, ?) ? A) > ? where Vol is volume w.r.t. the uniform measure on Rd .
We next give the result about mode recovery for Quick Shift. It says that as long as ? is small enough,
then as the number of samples grows, the roots of the trees returned by Quick Shift will bijectively
correspond to the true modes of f and the estimation errors will match lower bounds established by
Tsybakov [15] up to logarithmic factors. We defer the proof to Theorem 2 which is a generalization
of the following result.
Theorem 1 (Mode Estimation guarantees for Quick Shift). Let ? < rM /2 and Assumptions 1, 2, 3, 4,
c be the
and 5 hold. Choose h such that (log n)2/? ? h ? 0 and log n/(nhd ) ? 0 as n ? ?. Let M
3
heads of the trees in G (returned by Algorithm 1). There exists constant C depending on f and K
such that for n sufficiently large, with probability at least 1 ? 1/n,
!
r
log n
2
4/? 2
c
.
dH (M, M) < C (log n) h +
n ? hd
c In particular, taking h ? n?1/(4+d) optimizes the above rate to d(M, M)
c =
and |M| = |M|.
?1/(4+d)
?
O(n
). This matches the minimax optimal rate for mode estimation up to logarithmic factors.
We now give a stronger notion of mode that fits better for analyzing the role of ? . In the last result, it
was assumed that the practitioner wished to recover exactly the modes of the density f by taking ?
sufficiently small. Now, we analyze the case where ? is intentionally set to a particular value so that
Quick Shift produces segmentations that group modes together that are in close proximity to higher
density regions.
Definition 5. A mode x0 ? M is an (r, ?)+ -mode if f (x0 ) > f (x) + ? for all x ?
B(x0 , r)\B(x0 , rM ). A mode x0 ? M is an (r, ?)? -mode if f (x0 ) < f (x) ? ? for some
?
+
?
x ? B(x0 , r). Let M+
r,? ? M and Mr,? ? M denote the set of (r, ?) -modes and (r, ?) -modes
of f , respectively.
In other words, an (r, ?)+ -mode is a mode that is also a maximizer in a larger ball of radius r by at
least ? when outside of the region where there is quadratic decay and smoothness (B(x0 , rM )). An
(r, ?)? -mode is a mode that is not the maximizer in its radius r ball by a margin of at least ?.
The next result shows that Algorithm recovers the (? +, ?)+ -modes of f and excludes the (? ?, ?)? modes of f . The proof is in the appendix.
Theorem 2. (Generalization of Theorem 1) Let ?, > 0 and suppose Assumptions 1, 2, 3, 4, and 5
hold. Let h ? h(n) be chosen such that h ? 0 and log n/(nhd ) ? 0 as n ? ?. Then there exists
C > 0 depending on f and K such that the following holds for n sufficiently large with probability
?
c such that
at least 1 ? 1/n. For each x0 ? M+
??M
? +,? \M? ?,? , there exists unique x
!
r
log n
2
4/? 2
.
||x0 ? x
?|| < C (log n) h +
n ? hd
c ? |M| ? |M? |.
Moreover, |M|
? ?,?
In particular, taking ? 0 and ? ? 0 gives us an exact characterization of the asymptotic behavior
of Quick Shift in terms of mode recovery.
4
Assignment of Points to Modes
In this section, we give guarantees on how the points are assigned to their respective modes. We
first give the following definition which formalizes how two points are separated by a wide and deep
valley.
Definition 6. x1 , x2 ? X are (rs , ?)-separated if there exists a set S such that every path from x1
and x2 intersects with S and
sup
x?S+B(0,rs )
f (x) <
inf
f (x) ? ?.
x?B(x1 ,rs )?B(x2 ,rs )
Lemma 3. Suppose Assumptions 1, 2, 3, 4, and 5 hold. Let ? < rs /2 and choose h such that
(log n)2/? ? h ? 0 and log n/(nhd ) ? 0 as n ? ?. Let G be the output of Algorithm 1. The
following holds with probability at least 1 ? 1/n for n sufficiently large depending on f , K, ?, and ?
uniformly in all x1 , x2 ? X . If x1 and x2 are (rs , ?)-separated, then there cannot exist a directed
path from x1 to x2 in G.
Proof. Suppose that x1 and x2 are (rs , ?)-separated (with respect to set S) and there exists a
directed path from x1 to x2 in G. Given our choice of ? , there exists some point x ? G such that
x ? S+B(0, rs ) and x is on the path from x1 to x2 . We have f (x) < f (x1 )??. Choose n sufficiently
large such that by Lemma 1, supx?X |fbh (x) ? f (x)| < ?/2. Thus, we have fbh (x) < fbh (x1 ), which
means a directed path in G starting from x1 cannot contain x, a contradiction. The result follows.
4
Figure 2: Illustration of (rs , ?)-separation in 1 dimension. Here A and B are (rs , ?)-separated by S.
This is because the minimum density level of rs -radius balls around A and B (the red dotted line)
exceeds the maximum density level of the rs -radius ball around S by at least ? (golden dotted line).
In other words, there exists a sufficiently wide (controlled by rs and S) and deep (controlled by ?)
valley separating A and B. The results in this section will show that in such cases, these pairs of
points will not be assigned to the same cluster.
This leads to the following consequence about how samples are assigned to their respective modes.
Theorem 3. Assume the same conditions as Lemma 3. The following holds with probability at least
1 ? 1/n for n sufficiently large depending on f , K, ?, and ? uniformly in x ? X and x0 ? M. For
each x ? X and x0 ? M, if x and x0 are (rs , ?)-separated, then x will not be assigned to the tree
corresponding to x0 from Theorem 1.
Remark 4. In particular, taking ? ? 0 and rs ? 0 gives us guarantees for all points which have a
unique mode in which it can be assigned to.
We now give a more general version of (rs , ?)-separation, in which the condition holds if every path
between the two points dips down at some point. The same results as the above extend for this
definition in a straightforward manner.
Definition 7. x1 , x2 ? X are (rs , ?)-weakly-separated if there exists a set S, with x1 , x2 6? S +
B(0, rs ), such that every path P from x1 and x2 satisifes the following. (1) P ? S 6= ? and (2)
sup
x?P?S+B(0,rs )
f (x) <
inf
x?B(x01 ,rs )?B(x02 ,rs )
f (x) ? ?,
where x01 , x02 are defined as follows. Let P1 be the path obtained by starting at x1 and following
P until it intersects S, and P2 be the path obtained by following P starting from the last time it
intersects S until the end. Then x01 and x02 are points which respectively attain the highest values of f
on P1 and P2 .
5
Cluster Tree Recovery
The connected components of the level sets as the density level varies forms a hierarchical structure
known as the cluster tree.
Definition 8 (Cluster Tree). The cluster tree of f is given by
Cf (?) := connected components of {x ? X : f (x) ? ?}.
Definition 9. Let G(?) be the subgraph of G with vertices x ? X[n] such that fbh (x) > ? and edges
?
between pairs of vertices which have corresponding edges in G. Let G(?)
be the sets of vertices
corresponding to the connected components of G(?).
Definition 10. Suppose that A is a collection of sets of points in Rd . Then define Link(A, ?)
to be the result of repeatedly removing pairs A1 , A2 ? A from A (A1 6= A2 ) that satisfy
inf a1 ?A1 inf a2 ?A2 ||a1 ? a2 || < ? and adding A1 ? A2 to A until no such pairs exist.
Parameter settings for Algorithm 2: Suppose that ? ? ? (n) is chosen as a function of n such
such that ? ? 0 as n ? ?, ? (n) ? (log2 n/n)1/d and h ? h(n) is chosen such that h ? 0 and
log n/(nhd ) ? 0 as n ? ?.
The following is the main result of this section, the proof is in the appendix.
5
Algorithm 2 Quick Shift Cluster Tree Estimator
Input: Samples X[n] := {X1 , ..., Xn }, KDE bandwidth h, segmentation parameter ? > 0.
Let G be the output of Quick Shift (Algorithm 1) with above parameters.
bf (?) := Link(G(?),
?
For ? > 0, let C
? ).
b
return Cf
Theorem 4 (Consistency). Algorithm 2 converges in probability to the true cluster tree of f under
merge distortion (defined in [7]).
Remark 5. By combining the result of this section with the mode estimation result, we can obtain
the following interpretation. For any level ?, a component in G(?) estimates a connected component
of the ?-level set of f , and that further, the trees within that component in G(?) have a one-to-one
correspondence with the modes in the connected component.
Figure 3: Illustration on 1-dimensional density with three modes A, B, and C. When restricting Quick
Shift?s output to samples have empirical density above a certain threshold and connecting nearby
clusters, then this approximates the connected components of the true density level set. Moreover, we
give guarantees that such points will be assigned to clusters which correspond to modes within its
connected component.
6
Modal Regression
Suppose that we have joint density f (X, y) on Rd ? R w.r.t. to the Lebesgue measure. In modal
regression, we are interested in estimating the modes of the conditional f (y|X = x) given samples
from the joint distribution.
Algorithm 3 Quick Shift Modal Regression
Input: Samples D := {(x1 , y1 ), ..., (xn , yn )}, bandwidth h, ? > 0, and x ? X .
Let Y = {y1 , ..., yn } and fbh be the KDE computed w.r.t. D.
Initialize directed graph G with vertices Y and no edges.
for i = 1 to n do
if there exists yj ? [yi ? ?, yi + ? ] ? Y such that fbh (x, yj ) > fbh (x, yi ) then
Add to G an directed edge from yi to argminyi ?Y :fbh (x,yj )>fbh (x,yi ) ||yi ? yj ||.
end if
end for
return The roots of the trees of G as the estimates of the modes of f (y|X = x).
Theorem 5 (Consistency of Quick Shift Modal Regression). Suppose that ? ? ? (n) is chosen as a
function of n such such that ? ? 0 as n ? ?, ? (n) ? (log2 n/n)1/d and h ? h(n) is chosen such
that h ? 0 and log n/(nhd+1 ) ? 0 as n ? ?. Let Mx be the modes of the conditional density
cx be the output of Algorithm 3. Then with probability at least 1 ? 1/n uniformly
f (y|X = x) and M
in x such that f (y|X = x) and K satisfies Assumptions 1, 2, 3, 4, and 5,
cx ) ? 0 as n ? ?.
dH (Mx , M
6
7
Related Works
Mode Estimation. Perhaps the most popular procedure to estimate the modes is mean-shift; however,
it has proven quite difficult to analyze. Arias-Castro et al. [1] made much progress by utilizing
dynamical systems theory to show that mean-shift?s updates converge to the correct gradient ascent
steps. The recent work of Dasgupta and Kpotufe [6] was the first to give a procedure which recovers
the modes of a density with minimax optimal statistical guarantees in a multimodal density. They do
this by using a top-down traversal of the density levels of a proximity graph, borrowing from work
in cluster tree estimation. The procedure was shown to recover exactly the modes of the density at
minimax optimal rates.
In this work, we showed that Quick Shift attains the same guarantees while being a simpler approach
than known procedures that attain these guarantees [6, 12]. Moreover unlike these procedures, Quick
Shift also assigns the remaining samples to their appropriate modes. Furthermore, Quick Shift also
has a segmentation tuning parameter ? which allows us to merge the clusters of modes that are not
maximal in its ? -radius neighborhood into the clusters of other modes. This is useful as in practice,
one may not wish to pick up every single local maxima, especially when there are local maxima that
can be grouped together by proximity. We formalized the segmentation of such modes and identify
which modes get returned and which ones become merged into other modes? clusters by Quick Shift.
Cluster Tree Estimation. Work on cluster tree estimation has a long history. Some early work on
density based clustering by Hartigan [9] modeled the clusters of a density as the regions {x : f (x) ?
?} for some ?. This is called the density level-set of f at level ?. The cluster tree of f is the hierarchy
formed by the infinite collection of these clusters over all ?. Chaudhuri and Dasgupta [2] introduced
Robust Single Linkage (RSL) which was the first cluster tree estimation procedure with precise
statistical guarantees. Shortly after, Kpotufe and Luxburg [13] provided an estimator that ensured
false clusters were removed using used an extra pruning step. Interestingly, Quick Shift does not
require such a pruning step, since the points near cluster boundaries naturally get assigned to regions
with higher density and thus no spurious clusters are formed near these boundaries. Sriperumbudur
and Steinwart [14], Jiang [10], Wang et al. [17] showed that the popular DBSCAN algorithm [8] also
estimates these level sets. Eldridge et al. [7] introduced the merge distortion metric for cluster tree
estimates, which provided a stronger notion of consistency. We use their framework to analyze Quick
Shift and show that this simple estimator is consistent in merge distortion.
Figure 4: Density-based clusters discovered by level-set model {x : f (x) ? ?} (e.g. DBSCAN) vs
Quick Shift on a one dimensional density. Left two images: level sets for two density level settings.
Unassigned regions are noise and have no cluster assignment. Right two images: Quick Shift with
two different ? settings. The latter is a hill-climbing based clustering assignment.
Modal Regression. Nonparametric modal regression [3] is an alternative to classical regression,
where we are interested in estimating the modes of the conditional density f (y|X = x) rather
than the mean. Current approaches primarily use a modification of mean-shift; however analysis
for mean-shift require higher order smoothness assumptions. Using Quick Shift instead for modal
regression requires less regularity assumptions while having consistency guarantees.
8
Conclusion
We provided consistency guarantees for Quick Shift under mild assumptions. We showed that
Quick Shift recovers the modes of a density from a finite sample with minimax optimal guarantees.
The approach of this method is considerably different from known procedures that attain similar
guarantees. Moreover, Quick Shift allows tuning of the segmentation and we provided an analysis
of this behavior. We also showed that Quick Shift can be used as an alternative for estimating the
7
cluster tree which contrasts with current approaches which utilize proximity graph sweeps. We
then constructed a procedure for modal regression using Quick Shift which attains strong statistical
guarantees.
Appendix
Mode Estimation Proofs
Lemma 4. Suppose Assumptions 1, 2, 3, 4, and 5 hold. Let r? > 0 and h ? h(n) be chosen such
that h ? 0 and log n/(nhd ) ? 0 as n ? ?. Then the following holds for n sufficiently large with
probability at least 1 ? 1/n. Define
)
(
r
32C?
log n
2
4/? 2
0
r? := max
.
(log n) h , 17 ? C
n ? hd
C?
Suppose x0 ? M and x0 is the unique maximizer of f on B(x0 , r?).
argmaxx?B(x0 ,?r)?X[n] fbh (x), we have
Then letting x
? :=
||x0 ? x
?|| < r?.
Proof sketch. This follows from modifying the proof of Theorem 3 of [11] by replacing Rd \B(x0 , r?)
with B(x0 , r?)\B(x0 , r?). This leads us to
inf
x?B(x0 ,rn )
fbh (x) >
sup
fbh (x),
x?B(x0 ,?
r )\B(x0 ,?
r)
where rn := minx?X[n] |x0 ? x| and n is chosen sufficiently large such that r? < ? . Thus, |x0 ? x
?| ?
r?.
?
Proof of Theorem 2. Suppose that x0 ? M+
? := argmaxx?B(x0 ,? )?X[n] fbh (x).
? +,? \M? ?,? . Let x
c
We first show that x
? ? M.
q
?
n
By Lemma 4, we have |x0 ? x
?| ? r? where r?2 := max 32C?C (log n)4/? h2 , 17 ? C 0 log
. It
n?hd
remains to show that x
? = argmaxx?B(?x,? )?X[n] fbh (x). We have B(?
x, ? ) ? B(x0 , ? + r?). Choose
n sufficiently large such that (i) r? < , (ii) by Lemma 1, supx?X |fbh (x) ? f (x)| < ?/4 and (iii)
? Now, we have
r?2 < ?/(4C).
sup
fbh (x) ?
x?B(x0 ,? +?
r )\B(x0 ,? )
sup
f (x) + ?/4 ? f (x0 ) ? 3?/4
x?B(x0 ,? +?
r )\B(x0 ,? )
? f (?
x) + C? r?2 ? 3?/4 < f (?
x) ? ?/2 < fbh (?
x).
c
Thus, x
? = argmaxx?B(?x,? )?X[n] fbh (x). Hence, x
? ? M.
c such that ||?
Next, we show that it is unique. To do this, suppose that x
?0 ? M
x0 ? x0 || ? ? /2. Then
0
b
we have both x
? = argmaxx?B(?x,? )?X[n] fh (x) and x
? = argmaxx?B(?x0 ,? )?X[n] fbh (x). However,
choosing n sufficiently large such that r? < ? /2, we obtain x
? ? B(?
x0 , ? ). This implies that x
?=x
?0 , as
desired.
c ? |M| ? |M? |. Suppose that x
c Let ?0 := min{/3, ? /3, rM /2}. We
We now show |M|
? ? M.
? ?,?
show that B(?
x, ?0 ) ? M =
6 ?. Suppose otherwise. Let ? = f (?
x). By Assumptions 2 and 5, we
have that there exists ? > 0 and ? > 0 such that the following holds uniformly: Vol(B(?
x, ?0 ) ?
Lf (? + ?)) ? ?. Choose n sufficiently large such that (i) by Lemma 1, supx?X |fbh (x) ? f (x)| <
min ?/2, ?/4 and (ii) there exists a sample x ? B(?
x, /3) ? Lf (? + ?) ? X[n] by Lemma 7 of
b
Chaudhuri and Dasgupta [2]. Then fh (x) > ?+?/2 > fbh (?
x) but x ? B(?
x, ?0 ), a contradiction since
x
? is the maximizer of the KDE of the samples in its ? -radius neighborhood. Thus, B(?
x, ?0 ) ? M =
6 ?.
?
0
Now, suppose that there exists x0 ? B(?
x, ?0 ) ? M? ?,? . Then, there exists x ? B(x0 , ? ? 2?0 )
8
such that f (x0 ) ? f (x0 ) + ?. Then, if x
? is the closest sample point to x0 , we have for n sufficiently
0
large, |x ? x
?| ? ?0 and f (?
x) ? f (x0 ) + ?/2 and thus fbh (?
x) > f (?
x) ? ?/4 ? f (?
x) + ?/4 > fbh (?
x).
But x
? ? B(?
x, ? ) ? X[n] , contradicting the fact that x
? is the maximizer of the KDE over samples in
its ? -radius neighborhood. Thus, B(?
x, ?0 ) ? (M\M?
? ?,? ) 6= ?.
c such that x0 ? M\M?
Finally, suppose that there exists x
?, x
?0 ? M
x, ?0 ) ?
? ?,? and x0 ? B(?
B(?
x0 , ?0 ). Then, x
?, x
?0 ? B(x0 , ?0 ), thus |?
x?x
?0 | ? ? and thus x
?=x
?0 , as desired.
Cluster Tree Estimation Proofs
Lemma 5 (Minimality). The following holds with probability at least 1 ? 1/n. If A is a connected
bf (? ? )
component of {x ? X : f (x) ? ?}, then A ? X[n] is contained in the same component in C
for any > 0 as n ? ?.
Proof. It suffices to show that for each x ? A, there exists x0 ? B(x, ? /2) ? X[n] such that fbh (x0 ) >
? ? . Given our choice of ? , it follows by Lemma 7 of [2] that B(x, ? /2) ? X[n] is non-empty for n
sufficiently large. Let x0 ? B(x, ? /2) ? X[n] . Choose n sufficiently large such that by Lemma 1, we
have supx?X |fbh (x) ? f (x)| < /2. We have f (x0 ) ? inf B(x,? /2) f (x) ? ? ? C? (? /2)? > ? ? /2,
where the last inequality holds for n sufficiently large so that ? is sufficiently small. Thus, we have
fbh (x0 ) > ? ? , as desired.
Lemma 6 (Separation). Suppose that A and B are distinct connected components of {x ? X :
f (x) ? ?} which merge at {x ? X : f (x) ? ?}. Then A ? X[n] and B ? X[n] are separated in
bf (? + ) for any > 0 as n ? ?.
C
Proof. It suffices to assume that ? = ? + . Let A0 and B 0 be the connected components of
{x ? X : f (x) ? ? + /2} which contain A and B respectively. By the uniform continuity of f ,
there exists r? > 0 such that A + B(0, 3?
r) ? A0 . We have supx?A0 \(A+B(0,?r)) f (x) = ? + ? 0 for
0
some > 0.
Choose n sufficiently large such that by Lemma 1, we have supx?X |fbh (x) ? f (x)| < 0 /2.
bf (? + ) cannot belong to
Thus, supx?A0 \(A+B(0,?r)) fbh (x) < ? + ? 0 /2. Hence, points in C
A0 \ (A + B(0, r?)). Since A0 also contains A + B(0, 3?
r), it means that there cannot be a path from
A to B with points of empirical density at least ? + with all edges of length less than r?. The result
follows by taking n sufficiently large so that ? < r?, as desired.
Proof of Theorem 4. By the regularity assumptions on f and Theorem 2 of [7], we have that Algorithm 2 has both uniform minimality and uniform separation (defined in [7]), which implies
convergence in merge distortion.
Modal Regression Proofs
cx then y? is a consistent estimator
Proof of Theorem 5. There are two directions to show. (1) if y? ? M
c which estimates
of some mode y0 ? Mx . (2) For each mode, y0 ? M, there exists a unique y? ? M
it.
We first show (1). We show that [?
y ? ?, y? + ? ] ? Mx 6= ?. Suppose otherwise. Let ? = f (x, y?).
Choose ? < ? /4. Then by Assumptions 2 and 5, there exists ? > 0 such that taking = ? /2, we
have that there exists ? > 0 such that {(x, y 0 ) : y 0 ? [?
y ? ?, y? + ? ]} ? Lf (? + ?) contains connected
set A where Vol(A) > ?. Choose n sufficiently large such that (i) there exists y ? A ? Y , and (ii)
by Lemma 1, sup(x0 ,y0 ) |fbh (x0 , y 0 ) ? f (x0 , y 0 )| < ?/2. Then fbh (x, y) > ? + ?/2 > fbh (x, y?) but
y ? [?
y ? ?, y? + ? ], a contradiction since y? is the maximizer of the KDE in ? radius neighborhood
when restricted to X = x. Thus, there exists y0 ? Mx such that y0 ? [?
y ? ?, y? + ? ]. Moreover
this y0 ? Mx must be unique by Lemma 2. As n ? 0, we have ? ? 0 and thus consistency is
established for y? estimating y0 .
Now we show (2). Suppose that y0 ? Mx . From the above, for n sufficiently large, the maximizer of
the KDE in [y0 ? 2?, y0 + 2? ] ? Y is contained in [y0 ? ?, y0 + ? ]. Thus, there exists a root of the
tree contained in [y0 ? ?, y0 + ? ] and taking ? ? 0 gives us the desired result.
9
Acknowledgements
I thank the anonymous reviewers for their valuable feedback.
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using kernel density estimators. arXiv preprint arXiv:1706.03113, 2017.
10
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6,203 | 6,611 | Breaking the Nonsmooth Barrier: A Scalable Parallel
Method for Composite Optimization
Fabian Pedregosa
INRIA/ENS?
Paris, France
R?emi Leblond
INRIA/ENS?
Paris, France
Simon Lacoste-Julien
MILA and DIRO
Universit?e de Montr?eal, Canada
Abstract
Due to their simplicity and excellent performance, parallel asynchronous variants
of stochastic gradient descent have become popular methods to solve a wide range
of large-scale optimization problems on multi-core architectures. Yet, despite their
practical success, support for nonsmooth objectives is still lacking, making them
unsuitable for many problems of interest in machine learning, such as the Lasso,
group Lasso or empirical risk minimization with convex constraints. In this work,
we propose and analyze P ROX A SAGA, a fully asynchronous sparse method inspired by S AGA, a variance reduced incremental gradient algorithm. The proposed
method is easy to implement and signi?cantly outperforms the state of the art on
several nonsmooth, large-scale problems. We prove that our method achieves a
theoretical linear speedup with respect to the sequential version under assumptions on the sparsity of gradients and block-separability of the proximal term.
Empirical benchmarks on a multi-core architecture illustrate practical speedups of
up to 12x on a 20-core machine.
1
Introduction
The widespread availability of multi-core computers motivates the development of parallel methods
adapted for these architectures. One of the most popular approaches is H OGWILD (Niu et al., 2011),
an asynchronous variant of stochastic gradient descent (S GD). In this algorithm, multiple threads run
the update rule of S GD asynchronously in parallel. As S GD, it only requires visiting a small batch
of random examples per iteration, which makes it ideally suited for large scale machine learning
problems. Due to its simplicity and excellent performance, this parallelization approach has recently
been extended to other variants of S GD with better convergence properties, such as S VRG (Johnson
& Zhang, 2013) and S AGA (Defazio et al., 2014).
Despite their practical success, existing parallel asynchronous variants of S GD are limited to smooth
objectives, making them inapplicable to many problems in machine learning and signal processing.
In this work, we develop a sparse variant of the S AGA algorithm and consider its parallel asynchronous variants for general composite optimization problems of the form:
arg min f (x) + h(x)
x?Rp
,
with f (x) :=
1
n
?n
i=1
fi (x)
,
(OPT)
where each fi is convex with L-Lipschitz gradient, the average function f is ?-strongly convex and
h is convex but potentially nonsmooth. We further assume that h is ?simple? in the sense that we
have access to its proximal operator,
? and that it is block-separable, that is, it can be decomposed
block coordinate-wise as h(x) = B?B hB ([x]B ), where B is a partition of the coef?cients into
?
?
DI Ecole
normale sup?erieure, CNRS, PSL Research University
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
subsets which will call blocks and hB only depends on coordinates in block B. Note that there is
no loss of generality in this last assumption as a unique block covering all coordinates is a valid
partition, though in this case, our sparse variant of the S AGA algorithm reduces to the original S AGA
algorithm and no gain from sparsity is obtained.
This template models a broad range of problems arising in machine learning and signal processing:
the ?nite-sum structure of f includes the least squares or logistic loss functions; the proximal term
h includes penalties such as the ?1 or group lasso penalty. Furthermore, this term can be extendedvalued, thus allowing for convex constraints through the indicator function.
Contributions. This work presents two main contributions. First, in ?2 we describe Sparse Proximal S AGA, a novel variant of the S AGA algorithm which features a reduced cost per iteration in the
presence of sparse gradients and a block-separable penalty. Like other variance reduced methods, it
enjoys a linear convergence rate under strong convexity. Second, in ?3 we present P ROX A SAGA, a
lock-free asynchronous parallel version of the aforementioned algorithm that does not require consistent reads. Our main results states that P ROX A SAGA obtains (under assumptions) a theoretical
linear speedup with respect to its sequential version. Empirical benchmarks reported in ?4 show that
this method dramatically outperforms state-of-the-art alternatives on large sparse datasets, while the
empirical speedup analysis illustrates the practical gains as well as its limitations.
1.1
Related work
Asynchronous coordinate-descent. For composite objective functions of the form (OPT), most of
the existing literature on asynchronous optimization has focused on variants of coordinate descent.
Liu & Wright (2015) proposed an asynchronous variant of (proximal) coordinate descent and proved
a near-linear speedup in the number of cores used, given a suitable step size. This approach has been
recently extended to general block-coordinate schemes by Peng et al. (2016), to greedy coordinatedescent schemes by You et al. (2016) and to non-convex problems by Davis et al. (2016). However,
as illustrated by our experiments, in the large sample regime coordinate descent compares poorly
against incremental gradient methods like S AGA.
Variance reduced incremental gradient and their asynchronous variants. Initially proposed in
the context of smooth optimization by Le Roux et al. (2012), variance reduced incremental gradient
methods have since been extended to minimize composite problems of the form (OPT) (see table
below). Smooth variants of these methods have also recently been extended to the asynchronous setting, where multiple threads run the update rule asynchronously and in parallel. Interestingly, none
of these methods achieve both simultaneously, i.e. asynchronous optimization of composite problems. Since variance reduced incremental gradient methods have shown state of the art performance
in both settings, this generalization is of key practical interest.
Objective
Smooth
Composite
Sequential Algorithm
S VRG (Johnson & Zhang, 2013)
S DCA (Shalev-Shwartz & Zhang, 2013)
S AGA (Defazio et al., 2014)
P ROX S DCA (Shalev-Shwartz et al., 2012)
S AGA (Defazio et al., 2014)
ProxS VRG (Xiao & Zhang, 2014)
Asynchronous Algorithm
S VRG (Reddi et al., 2015)
PASSCODE (Hsieh et al., 2015, S DCA variant)
A SAGA (Leblond et al., 2017, S AGA variant)
This work: P ROX A SAGA
On the dif?culty of a composite extension. Two key issues explain the paucity in the development of asynchronous incremental gradient methods for composite optimization. The ?rst issue
is related to the design of such algorithms. Asynchronous variants of S GD are most competitive
when the updates are sparse and have a small overlap, that is, when each update modi?es a small
and different subset of the coef?cients. This is typically achieved by updating only coef?cients for
which the partial gradient at a given iteration is nonzero,2 but existing schemes such as the lagged
updates technique (Schmidt et al., 2016) are not applicable in the asynchronous setting. The second
2
Although some regularizers are sparsity inducing, large scale datasets are often extremely sparse and leveraging this property is crucial for the ef?ciency of the method.
2
dif?culty is related to the analysis of such algorithms. All convergence proofs crucially use the Lipschitz condition on the gradient to bound the noise terms derived from asynchrony. However, in the
composite case, the gradient mapping term (Beck & Teboulle, 2009), which replaces the gradient
in proximal-gradient methods, does not have a bounded Lipschitz constant. Hence, the traditional
proof technique breaks down in this scenario.
Other approaches. Recently, Meng et al. (2017); Gu et al. (2016) independently proposed a doubly stochastic method to solve the problem at hand. Following Meng et al. (2017) we refer to it
as Async-P ROX S VRCD. This method performs coordinate descent-like updates in which the true
gradient is replaced by its S VRG approximation. It hence features a doubly-stochastic loop: at each
iteration we select a random coordinate and a random sample. Because the selected coordinate block
is uncorrelated with the chosen sample, the algorithm can be orders of magnitude slower than S AGA
in the presence of sparse gradients. Appendix F contains a comparison of these methods.
1.2
De?nitions and notations
By convention, we denote vectors and vector-valued functions in lowercase boldface (e.g. x) and
matrices in uppercase boldface (e.g. D). The proximal operator of a convex lower semicontinuous
function h is de?ned as proxh (x) := arg minz?Rp {h(z) + 12 ?x ? z?2 }. A function f is said to be
L-smooth if it is differentiable and its gradient is L-Lipschitz continuous. A function f is said to be
?-strongly convex if f ? ?2 ? ? ?2 is convex. We use the notation ? := L/? to denote the condition
number for an L-smooth and ?-strongly convex function.3
I p denotes the p-dimensional identity matrix, 1{cond} the characteristic function, which is?
1 if cond
n
evaluates to true and 0 otherwise. The average of a vector or matrix is denoted ? := n1 i=1 ?i .
We use ? ? ? for the Euclidean norm. For a positive semi-de?nite matrix D, we de?ne its associated
distance as ?x?2D := ?x, Dx?. We denote by [ x ]b the b-th coordinate in x. This notation is
overloaded so that for a collection of blocks T = {B1 , B2 , . . .}, [x]T denotes the vector x restricted
to the coordinates in the blocks of T . For convenience, when T consists of a single block B we use
[x]B as a shortcut of [x]{B} . Finally, we distinguish E, the full expectation taken with respect to
all the randomness in the system, from E, the conditional expectation of a random it (the random
feature sampled at each iteration by S GD-like algorithms) conditioned on all the ?past?, which the
context will clarify.
2
Sparse Proximal SAGA
Original S AGA algorithm. The original S AGA algorithm (Defazio et al., 2014) maintains two
moving quantities: the current iterate x and a table (memory) of historical gradients (?i )ni=1 . At
every iteration, it samples an index i ? {1, . . . , n} uniformly at random, and computes the next
iterate (x+ , ?+ ) according to the following recursion:
?
?
ui = ?fi (x) ? ?i + ? ; x+ = prox?h x ? ?ui ; ?+
(1)
i = ?fi (x) .
On each iteration, this update rule requires to visit all coef?cients even if the partial gradients ?fi are
sparse. Sparse partial gradients arise in a variety of practical scenarios: for example, in generalized
linear models the partial gradients inherit the sparsity pattern of the dataset. Given that large-scale
datasets are often sparse,4 leveraging this sparsity is crucial for the success of the optimizer.
Sparse Proximal S AGA algorithm. We will now describe an algorithm that leverages sparsity
in the partial gradients by only updating those blocks that intersect with the support of the partial
gradients. Since in this update scheme some blocks might appear more frequently than others, we
will need to counterbalance this undersirable effect with a well-chosen block-wise reweighting of
the average gradient and the proximal term.
In order to make precise this block-wise reweighting, we de?ne the following quantities. We denote
by Ti the extended support of ?fi , which is the set of blocks that intersect the support of ?fi ,
3
Since we have assumed that each individual fi is L-smooth, f itself is L-smooth ? but it could have a
smaller smoothness constant. Our rates are in terms of this bigger L/?, as is standard in the S AGA literature.
4
For example, in the LibSVM datasets suite, 8 out of the 11 datasets (as of May 2017) with more than a
million samples have a density between 10?4 and 10?6 .
3
formally de?ned as Ti := {B : supp(?fi ) ? B ?= ?, B ? B}. For totally separable penalties such
as the ?1 norm, the blocks are individual coordinates and so?
the extended support covers the same
coordinates as the support. Let dB := n/nB , where nB := i 1{B ? Ti } is the number of times
that B ? Ti . For simplicity we assume nB > 0, as otherwise the problem can be reformulated
without block B. The update rule in (1) requires computing the proximal operator of h, which
involves a full pass ?
on the coordinates. In our proposed algorithm, we replace h in (1) with the
function ?i (x) :=
B?Ti dB hB (x), whose form is justi?ed by the following three properties.
First, this function is zero outside Ti , allowing for sparse updates. Second, because of the block-wise
reweighting dB , the function ?i is an unbiased estimator of h (i.e., E ?i = h), property which will
be crucial to prove the convergence of the method. Third, ?i inherits the block-wise structure of h
and its proximal operator can be computed from that of h as [prox??i (x)]B = [prox(dB ?)hB (x)]B
if B ? Ti and [prox??i (x)]B = [x]B otherwise. Following Leblond et al. (2017), we will also
replace the dense gradient estimate ui by the sparse estimate v i := ?fi (x) ? ?i + D i ?, where
D i is the diagonal matrix de?ned block-wise as [D i ]B,B = dB 1{B ? Ti }I |B| . It is easy to verify
that the vector D i ? is a weighted projection onto the support of Ti and E D i ? = ?, making v i an
unbiased estimate of the gradient.
We now have all necessary elements to describe the Sparse Proximal S AGA algorithm. As the
original S AGA algorithm, it maintains two moving quantities: the current iterate x ? Rp and a
table of historical gradients (?i )ni=1 , ?i ? Rp . At each iteration, the algorithm samples an index
i ? {1, . . . , n} and computes the next iterate (x+ , ?+ ) as:
?
?
v i = ?fi (x) ? ?i + D i ? ; x+ = prox??i x ? ?v i ; ?+
i = ?fi (x) ,
(SPS)
where in a practical implementation the vector ? is updated incrementally at each iteration.
The above algorithm is sparse in the sense that it only requires to visit and update blocks in the
extended support: if B ?
/ Ti , by the sparsity of v i and prox?i , we have [x+ ]B = [x]B . Hence,
when the extended support Ti is sparse, this algorithm can be orders of magnitude faster than the
naive S AGA algorithm. The extended support is sparse for example when the partial gradients are
sparse and the penalty is separable, as is the case of the ?1 norm or the indicator function over a
hypercube, or when the the penalty is block-separable in a way such that only a small subset of the
blocks overlap with the support of the partial gradients. Initialization of variables and a reduced
storage scheme for the memory are discussed in the implementation details section of Appendix E.
Relationship with existing methods. This algorithm can be seen as a generalization of both the
Standard S AGA algorithm and the Sparse S AGA algorithm of Leblond et al. (2017). When the
proximal term is not block-separable, then dB = 1 (for a unique block B) and the algorithm defaults
to the Standard (dense) S AGA algorithm. In the smooth case (i.e., h = 0), the algorithm defaults to
the Sparse S AGA method. Hence we note that the sparse gradient estimate v i in our algorithm is the
same as the one proposed in Leblond et al. (2017). However, we emphasize that a straightforward
combination of this sparse update rule with the proximal update from the Standard S AGA algorithm
results in a nonconvergent algorithm: the block-wise reweighting of h is a surprisingly simple but
crucial change. We now give the convergence guarantees for this algorithm.
a
for any a ? 1 and f be ?-strongly convex (? > 0). Then Sparse Proximal
Theorem 1. Let ? = 5L
S AGA converges geometrically in expectation with a rate factor of at least ? = 15 min{ n1 , a ?1 }. That
is, for xt obtained after t updates, we have the following bound:
?n
E?xt ? x? ?2 ? (1 ? ?)t C0 , with C0 := ?x0 ? x? ?2 + 5L1 2 i=1 ??0i ? ?fi (x? )?2 .
Remark. For the step size ? = 1/5L, the convergence rate is (1 ? 1/5 min{1/n, 1/?}). We can thus
identify two regimes: the ?big data? regime, n ? ?, in which the rate factor is bounded by 1/5n, and
the ?ill-conditioned? regime, ? ? n, in which the rate factor is bounded by 1/5?. This rate roughly
matches the rate obtained by Defazio et al. (2014). While the step size bound of 1/5L is slightly
smaller than the 1/3L one obtained in that work, this can be explained by their stronger assumptions:
each fi is strongly convex whereas they are strongly convex only on average in this work. All proofs
for this section can be found in Appendix B.
4
Algorithm 1 P ROX A SAGA (analyzed)
1: Initialize shared variables x and (?i )n
i=1
2: keep doing in parallel
? = inconsistent read of x
x
3:
? = inconsistent read of ?
4:
?
5:
Sample i uniformly in {1, ..., n}
6:
Si := support of ?fi
7:
Ti := extended
?nsupport of ?fi in B
? j ] Ti
8:
[ ? ]Ti = 1/n j=1 [ ?
? i ] Si
9:
[ ?? ]Si = [?fi (?
x)]Si ? [?
? ]Ti = [ ?? ]Ti + [D i ? ]Ti
10:
[v
? )]Ti ? [?
11:
[ ?x ]Ti = [prox??i (?
x ? ?v
x]Ti
12:
for B in Ti do
13:
for b ? B do
14:
[ x ]b ? [ x ]b + [ ?x ]b
? atomic
15:
if b ? Si then
16:
[?i ]b ? [?fi (?
x)]b
17:
end if
18:
end for
19:
end for
20:
// (??? denotes shared memory update.)
21: end parallel loop
3
Algorithm 2 P ROX A SAGA (implemented)
1: Initialize shared variables x, (?i )n
i=1 , ?
2: keep doing in parallel
3:
Sample i uniformly in {1, ..., n}
4:
Si := support of ?fi
5:
Ti := extended support of ?fi in B
? ]Ti = inconsistent read of x on Ti
6:
[x
? i = inconsistent read of ?i
7:
?
8:
[ ? ]Ti = inconsistent read of ? on Ti
? i ] Si
9:
[ ?? ]Si = [?fi (?
x)]Si ? [?
? ]Ti = [?? ]Ti + [ D i ? ]Ti
10:
[v
? )]Ti ? [?
11:
[ ?x ]Ti = [prox??i (?
x ? ?v
x]Ti
12:
for B in Ti do
13:
for b in B do
14:
[ x ]b ? [ x ]b + [ ?x ]b
? atomic
15:
if b ? Si then
16:
[ ? ]b ? [?]b + 1/n[??]b ? atomic
17:
end if
18:
end for
19:
end for
20:
?i ? ?fi (?
x) (scalar update) ? atomic
21: end parallel loop
Asynchronous Sparse Proximal SAGA
We introduce P ROX A SAGA ? the asynchronous parallel variant of Sparse Proximal S AGA. In this
algorithm, multiple cores update a central parameter vector using the Sparse Proximal S AGA introduced in the previous section, and updates are performed asynchronously. The algorithm parameters
are read and written without vector locks, i.e., the vector content of the shared memory can potentially change while a core is reading or writing to main memory coordinate by coordinate. These
operations are typically called inconsistent (at the vector level).
The full algorithm is described in Algorithm 1 for its theoretical version (on which our analysis
is built) and in Algorithm 2 for its practical implementation. The practical implementation differs
from the analyzed agorithm in three points. First, in the implemented algorithm, index i is sampled
before reading the coef?cients to minimize memory access since only the extended support needs to
be read. Second, since our implementation targets generalized linear models, the memory ?i can be
compressed into a single scalar in L20 (see Appendix E). Third, ? is stored in memory and updated
incrementally instead of recomputed at each iteration.
The rest of the section is structured as follows: we start by describing our framework of analysis; we
then derive essential properties of P ROX A SAGA along with a classical delay assumption. Finally,
we state our main convergence and speedup result.
3.1
Analysis framework
As in most of the recent asynchronous optimization literature, we build on the hardware model introduced by Niu et al. (2011), with multiple cores reading and writing to a shared memory parameter
? t , the local copy of the
vector. These operations are asynchronous (lock-free) and inconsistent:5 x
parameters of a given core, does not necessarily correspond to a consistent iterate in memory.
?Perturbed? iterates. To handle this additional dif?culty, contrary to most contributions in this
?eld, we choose the ?perturbed iterate framework? proposed by Mania et al. (2017) and re?ned
by Leblond et al. (2017). This framework can analyze variants of S GD which obey the update rule:
xt+1 = xt ? ?v(xt , it ) , where v veri?es the unbiasedness condition E v(x, it ) = ?f (x)
5
This is an extension of the framework of Niu et al. (2011), where consistent updates were assumed.
5
and the expectation is computed with respect to it . In the asynchronous parallel setting, cores are
? t . As these inconsistent iterates are
reading inconsistent iterates from memory, which we denote x
affected by various delays induced by asynchrony, they cannot easily be written as a function of
their previous iterates. To alleviate this issue, Mania et al. (2017) choose to introduce an additional
quantity for the purpose of the analysis:
xt+1 := xt ? ?v(?
xt , it ) , the ?virtual iterate? ? which is never actually computed .
(2)
Note that this equation is the de?nition of this new quantity xt . This virtual iterate is useful for the
convergence analysis and makes for much easier proofs than in the related literature.
?After read? labeling. How we choose to de?ne the iteration counter t to label an iterate xt
matters in the analysis. In this paper, we follow the ?after read? labeling proposed in Leblond
et al. (2017), in which we update our iterate counter, t, as each core ?nishes reading its copy of
? t and ?
? t ). This means
the parameters (in the speci?c case of P ROX A SAGA, this includes both x
? t is the (t + 1)th fully completed read. One key advantage of this approach compared to the
that x
classical choice of Niu et al. (2011) ? where t is increasing after each successful update ? is that
? t are independent. This
it guarantees both that the it are uniformly distributed and that it and x
property is not veri?ed when using the ?after write? labeling of Niu et al. (2011), although it is still
implicitly assumed in the papers using this approach, see Leblond et al. (2017, Section 3.2) for a
discussion of issues related to the different labeling schemes.
Generalization to composite optimization. Although the perturbed iterate framework was designed for gradient-based updates, we can extend it to proximal methods by remarking that in the
sequential setting, proximal stochastic gradient descent and its variants can be characterized by the
following similar update rule:
?
?
xt+1 = xt ? ?g(xt , v it , it ) , with g(x, v, i) := ?1 x ? prox??i (x ? ?v) ,
(3)
where as before v veri?es the unbiasedness condition E v = ?f (x). The Proximal Sparse S AGA
iteration can be easily written within this template by using ?i and v i as de?ned in ?2. Using this
de?nition of g, we can de?ne P ROX A SAGA virtual iterates as:
? tit , it ) , with v
? tit = ?fit (?
? tit + D it ?t ,
xt+1 := xt ? ?g(?
xt , v
xt ) ? ?
(4)
? tit = ?fit (?
where as in the sequential case, the memory terms are updated as ?
xt ). Our theoretical
analysis of P ROX A SAGA will be based on this de?nition of the virtual iterate xt+1 .
3.2
Properties and assumptions
Now that we have introduced the ?after read? labeling for proximal methods in Eq. (4), we can
leverage the framework of Leblond et al. (2017, Section 3.3) to derive essential properties for the
analysis of P ROX A SAGA. We describe below three useful properties arising from the de?nition
of Algorithm 1, and then state a central (but standard) assumption that the delays induced by the
asynchrony are uniformly bounded.
? t for all r ? t. We
Independence: Due to the ?after read? global ordering, ir is independent of x
enforce the independence for r = t by having the cores read all the shared parameters before their
iterations.
? tit is an unbiased estimator of the gradient of f at x
? t . This property is a
Unbiasedness: The term v
? t.
consequence of the independence between it and x
Atomicity: The shared parameter coordinate update of [x]b on Line 14 is atomic. This means that
there are no overwrites for a single coordinate even if several cores compete for the same resources.
Most modern processors have support for atomic operations with minimal overhead.
Bounded overlap assumption. We assume that there exists a uniform bound, ? , on the maximum
number of overlapping iterations. This means that every coordinate update from iteration t is successfully written to memory before iteration t + ? + 1 starts. Our result will give us conditions on ?
to obtain linear speedups.
? t ? xt . The delay assumption of the previous paragraph allows to express the difference
Bounding x
? uiu , iu ) as:
between real and virtual iterate using the gradient mapping g u := g(?
xu , v
?
t
t
? t ?xt = ? t?1
x
u=(t?? )+ Gu g u , where Gu are p ? p diagonal matrices with terms in {0, +1}. (5)
6
? u and xu have received the corresponding updates. +1, on
0 represents instances where both x
? u has not yet received an update that is already in xu by
the contrary, represents instances where x
de?nition. This bound will prove essential to our analysis.
3.3
Analysis
In this section, we state our convergence and speedup results for P ROX A SAGA. The full details
of the analysis can be found in Appendix C. Following Niu et al. (2011), we introduce a sparsity
measure (generalized to the composite setting) that will appear in our results.
De?nition 1. Let ? := maxB?B |{i : Ti ? B}|/n. This is the normalized maximum number of
times that a block appears in the extended support. For example, if a block is present in all Ti , then
? = 1. If no two Ti share the same block, then ? = 1/n. We always have 1/n ? ? ? 1.
1
. For any step size
Theorem 2 (Convergence guarantee of P ROX A SAGA). Suppose ? ? 10?
?
a
1
6?
?
? = L with a ? a (? ) := 36 min{1, ? }, the inconsistent read iterates
of
Algorithm
1 converge
?1 1?
1
in expectation at a geometric rate factor of at least: ?(a) = 5 min n , a ? , i.e. E??
xt ? x ? ?2 ?
n?
t ?
?
(1 ? ?) C0 , where C0 is a constant independent of t (? a C0 with C0 as de?ned in Theorem 1).
This last result is similar to the original S AGA convergence result and our own Theorem 1, with both
an extra condition on ? and on?the maximum allowable step size. In the best sparsity case, ? = 1/n
and we get the condition ? ? n/10. We now compare the geometric rate above to the one of Sparse
Proximal S AGA to derive the necessary conditions under which P ROX A SAGA is linearly faster.
1
. If ? ? n, then using the step size ? = 1/36L, P ROX A S Corollary 1 (Speedup). Suppose ? ? 10?
?
1
AGA converges geometrically with rate factor ?( ?
). If ? < n, then using the step size ? = 1/36n?,
P ROX A SAGA converges geometrically with rate factor ?( n1 ). In both cases, the convergence rate
is the same as Sparse Proximal S AGA. Thus P ROX A SAGA is linearly faster than its sequential
counterpart up to a constant factor. Note that in both cases the step size does not depend on ? .
Furthermore, if ? ? 6?, we can use a universal step size of ?(1/L) to get a similar rate for P ROX A SAGA than Sparse Proximal S AGA, thus making it adaptive to local strong convexity since the
knowledge of ? is not required.
These speedup regimes are comparable with the best ones obtained in the smooth case, including Niu
et al. (2011); Reddi et al. (2015), even though unlike these papers, we support inconsistent reads
and nonsmooth objective functions. The one exception is Leblond et al. (2017), where the authors
prove that their algorithm, A SAGA, can obtain a linear speedup even without sparsity in the wellconditioned regime. In contrast, P ROX A SAGA always requires some sparsity. Whether this property
for smooth objective functions could be extended to the composite case remains an open problem.
Relative to A SY S PCD, in the best case scenario (where the components of the gradient are uncorre?
lated, a somewhat unrealistic setting), A SY S PCD can get a near-linear speedup for ? as big as 4 p.
?
1
1
Our result states that ? = O( / ?) is necessary for a linear speedup. This means in case ? ? /?p
our bound is better than the one obtained for A SY?
S PCD. Recalling that 1/n ? ? ? 1, it appears
that P ROX A SAGA is favored when n is bigger than p whereas A SY S PCD may have a better bound
otherwise, though this comparison should be taken with a grain of salt given the assumptions we
had to make to arrive at comparable quantities. An extended comparison with the related work can
be found in Appendix D.
4
Experiments
In this section, we compare P ROX A SAGA with related methods on different datasets. Although
P ROX A SAGA can be applied more broadly, we focus on ?1 + ?2 -regularized logistic regression, a
model of particular practical importance. The objective function takes the form
n
?
?
1?
log 1 + exp(?bi a?i x) +
n i=1
?1
2
2 ?x?2
+ ?2 ?x?1
,
(6)
where ai ? Rp and bi ? {?1, +1} are the data samples. Following Defazio et al. (2014), we set
?1 = 1/n. The amount of ?1 regularization (?2 ) is selected to give an approximate 1/10 nonzero
7
Table 1: Description of datasets.
Dataset
???????????????????????
KDD 2010 (Yu et al., 2010)
KDD 2012 (Juan et al., 2016)
Criteo (Juan et al., 2016)
p
19,264,097
149,639,105
45,840,617
1,163,024
54,686,452
1,000,000
?????????????
?????????????
???
??
?
??
?
??
?
??
??
?
??
??
??
?????????????????
??
??
?
??
?
??
?
??
???
??
?
??
????????????
?????????????
?
?
?
? ?? ?? ?? ?? ?? ??
???????????????
?????
??
??
??
??
??
??
?
?
?
?
??
??
?????????????????
??????????????????
????????????????????
??
??
??
??
??
??
?
?
?
?
density
n
??
????????????????
??????????????????
?
?
? ?? ?? ?? ?? ?? ??
???????????????
?????????
???????
28.12
1.25
1.25
10
2 ? 10?7
4 ? 10?5
??
?
??
?
??
?
??
??
?
0.15
0.85
0.89
??????????????
???
?????????????
?
L
?6
?
??
??
??
?????????????????
??
??????????????
????????????????
??
??
??
??
??
??
?
?
?
?
??????????????
?
?
?
? ?? ?? ?? ?? ?? ??
???????????????
?????
Figure 1: Convergence for asynchronous stochastic methods for ?1 + ?2 -regularized logistic
regression. Top: Suboptimality as a function of time for different asynchronous methods using 1
and 10 cores. Bottom: Running time speedup as function of the number of cores. P ROX A SAGA
achieves signi?cant speedups over its sequential version while being orders of magnitude faster than
competing methods. A SY S PCD achieves the highest speedups but it also the slowest overall method.
coef?cients. Implementation details are available in Appendix E. We chose the 3 datasets described
in Table 1
Results. We compare three parallel asynchronous methods on the aforementioned datasets: P ROX A SAGA (this work),6 A SY S PCD, the asynchronous proximal coordinate descent method of Liu &
Wright (2015) and the (synchronous) F ISTA algorithm (Beck & Teboulle, 2009), in which the gradient computation is parallelized by splitting the dataset into equal batches. We aim to benchmark
these methods in the most realistic scenario possible; to this end we use the following step size:
1/2L for P ROX A SAGA, 1/Lc for A SY S PCD, where Lc is the coordinate-wise Lipschitz constant
of the gradient, while F ISTA uses backtracking line-search. The results can be seen in Figure 1
(top) with both one (thus sequential) and ten processors. Two main observations can be made from
this ?gure. First, P ROX A SAGA is signi?cantly faster on these problems. Second, its asynchronous
version offers a signi?cant speedup over its sequential counterpart.
In Figure 1 (bottom) we present speedup with respect to the number of cores, where speedup is
computed as the time to achieve a suboptimality of 10?10 with one core divided by the time to
achieve the same suboptimality using several cores. While our theoretical speedups (with respect
to the number of iterations) are almost linear as our theory predicts (see Appendix F), we observe
a different story for our running time speedups. This can be attributed to memory access overhead,
which our model does not take into account. As predicted by our theoretical results, we observe
6
A reference C++/Python implementation of is available at https://github.com/fabianp/ProxASAGA
8
a high correlation between the ? dataset sparsity measure and the empirical speedup: KDD 2010
(? = 0.15) achieves a 11x speedup, while in Criteo (? = 0.89) the speedup is never above 6x.
Note that although competitor methods exhibit similar or sometimes better speedups, they remain
orders of magnitude slower than P ROX A SAGA in running time for large sparse problems. In fact,
our method is between 5x and 80x times faster (in time to reach 10?10 suboptimality) than F ISTA
and between 13x and 290x times faster than A SY S PCD (see Appendix F.3).
5
Conclusion and future work
In this work, we have described P ROX A SAGA, an asynchronous variance reduced algorithm with
support for composite objective functions. This method builds upon a novel sparse variant of the
(proximal) S AGA algorithm that takes advantage of sparsity in the individual gradients. We have
proven that this algorithm is linearly convergent under a condition on the step size and that it is
linearly faster than its sequential counterpart given a bound on the delay. Empirical benchmarks
show that P ROX A SAGA is orders of magnitude faster than existing state-of-the-art methods.
This work can be extended in several ways. First, we have focused on the S AGA method as the basic
iteration loop, but this approach can likely be extended to other proximal incremental schemes such
as S GD or ProxS VRG. Second, as mentioned in ?3.3, it is an open question whether it is possible to
obtain convergence guarantees without any sparsity assumption, as was done for A SAGA.
Acknowledgements
The authors would like to thank our colleagues Damien Garreau, Robert Gower, Thomas Kerdreux, Geoffrey Negiar and Konstantin Mishchenko for their feedback on this manuscript, and JeanBaptiste Alayrac for support managing the computational resources.
This work was partially supported by a Google Research Award. FP acknowledges support from the
?
chaire Economie
des nouvelles donn?ees with the data science joint research initiative with the fonds
AXA pour la recherche.
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6,204 | 6,612 | Dual-Agent GANs for Photorealistic and Identity
Preserving Profile Face Synthesis
Jian Zhao1,2?? Lin Xiong3 Karlekar Jayashree3 Jianshu Li1 Fang Zhao1
Zhecan Wang4? Sugiri Pranata3 Shengmei Shen3
Shuicheng Yan1,5 Jiashi Feng1
1
3
National University of Singapore
2
National University of Defense Technology
Panasonic R&D Center Singapore 4 Franklin. W. Olin College of Engineering
5
Qihoo 360 AI Institute
{zhaojian90, jianshu}@u.nus.edu {lin.xiong, karlekar.jayashree, sugiri.pranata, shengmei.shen}@sg.panasonic.com
[email protected] {elezhf, eleyans, elefjia}@u.nus.edu
Abstract
Synthesizing realistic profile faces is promising for more efficiently training deep
pose-invariant models for large-scale unconstrained face recognition, by populating samples with extreme poses and avoiding tedious annotations. However,
learning from synthetic faces may not achieve the desired performance due to the
discrepancy between distributions of the synthetic and real face images. To narrow
this gap, we propose a Dual-Agent Generative Adversarial Network (DA-GAN)
model, which can improve the realism of a face simulator?s output using unlabeled
real faces, while preserving the identity information during the realism refinement. The dual agents are specifically designed for distinguishing real v.s. fake
and identities simultaneously. In particular, we employ an off-the-shelf 3D face
model as a simulator to generate profile face images with varying poses. DA-GAN
leverages a fully convolutional network as the generator to generate high-resolution
images and an auto-encoder as the discriminator with the dual agents. Besides
the novel architecture, we make several key modifications to the standard GAN
to preserve pose and texture, preserve identity and stabilize training process: (i)
a pose perception loss; (ii) an identity perception loss; (iii) an adversarial loss
with a boundary equilibrium regularization term. Experimental results show that
DA-GAN not only presents compelling perceptual results but also significantly
outperforms state-of-the-arts on the large-scale and challenging NIST IJB-A unconstrained face recognition benchmark. In addition, the proposed DA-GAN is
also promising as a new approach for solving generic transfer learning problems
more effectively. DA-GAN is the foundation of our submissions to NIST IJB-A
2017 face recognition competitions, where we won the 1st places on the tracks of
verification and identification.
1
Introduction
Unconstrained face recognition is a very important yet extremely challenging problem. In recent years,
deep learning techniques have significantly advanced large-scale unconstrained face recognition (8;
19; 27; 34; 29; 16), arguably driven by rapidly increasing resource of face images. However, labeling
huge amount of data for feeding supervised deep learning algorithms is undoubtedly expensive
and time-consuming. Moreover, as often observed in real-world scenarios, the pose distribution of
available face recognition datasets (e.g., IJB-A (15)) is usually unbalanced and has long-tail with
?
?
Homepage: https://zhaoj9014.github.io/
Jian Zhao and Zhecan Wang were interns at Panasonic R&D Center Singapore during this work.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
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Pose
(a) Extremely unbalanced pose distribution.
(b) Well balanced pose distribution with DA-GAN.
Figure 1: Comparison of pose distribution in the IJB-A dataset (15) w/o and w/ DA-GAN.
large pose variations, as shown in Figure. 1a. This has become a main obstacle for further pushing
unconstrained face recognition performance. To address this critical issue, several research attempts
(32; 31; 35) have been made to employ synthetic profile face images as augmented extra data to
balance the pose variations.
However, naively learning from synthetic images can be problematic due to the distribution discrepancy between synthetic and real face images?synthetic data is often not realistic enough with
artifacts and severe texture losses. The low-quality synthesis face images would mislead the learned
face recognition model to overfit to fake information only presented in synthetic images and fail to
generalize well on real faces. Brute-forcedly increasing the realism of the simulator is often expensive
in terms of time cost and manpower, if possible.
In this work, we propose a novel Dual-Agent Generative Adversarial Network (DA-GAN) for profile
view synthesis, where the dual agents focus on discriminating the realism of synthetic profile face
images from a simulator using unlabled real data and perceiving the identity information, respectively.
In other words, the generator needs to play against a real?fake discriminator as well as an identity
discriminator simultaneously to generate high-quality faces that are really useful for unconstrained
face recognition.
In our method, a synthetic profile face image with a pre-specified pose is generated by a 3D morphable
face simulator. DA-GAN takes this synthetic face image as input and refines it through a conditioned
generative model. We leverage a Fully Convolutional Network (FCN) (17) that operates on the pixel
level as the generator to generate high-resolution face images and an auto-encoder network as the
discriminator. Different from vanilla GANs, DA-GAN introduces an auxiliary discriminative agent
to enforce the generator to preserve identity information of the generated faces, which is critical for
face recognition application. In addition, DA-GAN also imposes a pose perception loss to preserve
pose and texture. The refined synthetic profile face images present photorealistic quality with well
preserved identity information, which are used as augmented data together with real face images for
pose-invariant feature learning. For stabilizing the training process of such dual-agent GAN model,
we impose a boundary equilibrium regularization term.
Experimental results show that DA-GAN not only presents compelling perceptual results but also
significantly outperforms state-of-the-arts on the large-scale and challenging National Institute
of Standards and Technology (NIST) IARPA Janus Benchmark A (IJB-A) unconstrained face
recognition benchmark (15). DA-GAN leads us to further win the 1st places on verification and
identification tracks in the NIST IJB-A 2017 face recognition competitions. This strong evidence
shows that our ?recognition via generation" framework is effective and generic, and we expect that it
benefits for more face recognition and transfer learning applications in the real world.
Our contributions are summarized as follows.
? We propose a novel Dual-Agent Generative Adversarial Network (DA-GAN) for photorealistic and identity preserving profile face synthesis even under extreme poses.
? The proposed dual-agent architecture effectively combines prior knowledge from data distribution (adversarial training) and domain knowledge of faces (pose and identity perception
losses) to exactly recover the lost information inherent in projecting a 3D face into the 2D
image space.
? We present qualitative and quantitative experiments showing the possibility of a ?recognition
via generation" framework and achieve the top performance on the challenging NIST IJB-A
unconstrained face recognition benchmark (15) without extra human annotation efforts
2
3D Face Model
Simulator
?
Face RoI
Extraction
68-Point
Landmark Detection
Simulated Profile Face
Generator
+
Input
224?224?3
Conv 64?7?7
ReLU & BN
?
Lpp
+
Residual Block * 10
Conv 3?1?1
Output
224?224?3
Discriminator
Lip
Agent 1
?
Real
?
Synthetic
-
Ladv
Agent 2
Conv 3?3?3
ReLU
FC
784
Transition Down
Transition Up
Conv 3?1?1
ReLU
Figure 2: Overview of the proposed DA-GAN architecture. The simulator (upper panel) extracts
face RoI, localizes landmark points and produces synthesis faces with arbitrary poses, which are fed
to DA-GAN for realism refinement. DA-GAN uses a fully convolutional skip-net as the generator
(middle panel) and an auto-encoder as the discriminator (bottom panel). The dual agents focus on
both discriminating real v.s. fake (minimizing the loss Ladv ) and preserving identity information
(minimizing the loss Lip ). Best viewed in color.
by training deep neural networks on the refined face images together with real images.
To our best knowledge, our proposed DA-GAN is the first model that is effective for
automatically generating augmented data for face recognition in challenging conditions and
indeed improves performance. DA-GAN won the 1st places on verification and identification
tracks in the NIST IJB-A 2017 face recognition competitions.
2
Related works
As one of the most significant advancements on the research of deep generative models (14; 26),
GAN has drawn substantial attention from the deep learning and computer vision community since
it was first introduce by Goodfellow et al. (10). The GAN framework learns a generator network
and a discriminator network with competing loss. This min-max two-player game provides a simple
yet powerful way to estimate target distribution and to generate novel image samples. Mirza and
Osindero (21) introduce the conditional version of GAN, to condition on both the generator and
discriminator for effective image tagging. Berthelot et al. (2) propose a new Boundary Equilibrium
GAN (BE-GAN) framework paired with a loss derived from the Wasserstein distance for training
GAN, which derives a way of controlling the trade-off between image diversity and visual quality.
These successful applications of GAN motivate us to develop profile view synthesis methods based
on GAN. However, the generator of previous methods usually focus on generating images based on a
random noise vector or conditioned data and the discriminator only has a single agent to distinguish
real v.s. fake. Thus, in contrast to our method, the generated images do not have any discriminative
information that can be used for training a deep learning based recognition model. This separates us
well with previous GAN-based attempts.
3
Moreover, differnet from previous InfoGAN (5) which does not have the classification agent, and
Auxiliary Classifier GAN (AC-GAN) (22) which only performs classification, our propsoed DAGAN performs face verification with an intrigued data augmentation. DA-GAN is a novel and
practical model for efficient data augmentation and it is really effective in practice as proved in Sec.
4. DA-GAN generates the data in a completely different way from InfoGAN (5) and AC-GAN (22)
which generate images from a random noise input or abstract semantic labels. Therefore, inferior to
our model, those existing GAN-like models cannot exploit useful and rich prior information (e.g., the
shape, pose of faces) for effective data generation and augmentation. They cannot fully control the
generated images. In contrast, DA-GAN can fully control the generated images and adjust the face
pose (e.g., yaw angles) distribution which is extremely unbalanced in real-world scenarios. DA-GAN
can facilitate training more accurate face analysis models to solve the large pose variation problem
and other relevant problems in unconstrained face recognition.
Our proposed DA-GAN shares a similar idea with TP-GAN (13) that considers face synthesis based
on GAN framework, and Apple GAN (28) that considers learning from simulated and unsupervised
images through adversarial training. Our method differs from them in following aspects: 1) DA-GAN
aims to synthesize photorealistic and identity preserving profile faces to address the large variance
issue in unconstrained face recognition, whereas TP-GAN (13) tries to recover a frontal face from a
profile view and Apple GAN (28) is designed for much simpler scenarios (e.g., eye and hand image
refinement); 2) TP-GAN (13) and Apple GAN (28) suffer from categorical information loss which
limits their effectiveness in promoting recognition performance. In contrast, our proposed DA-GAN
architecture effectively overcomes this issue by introducing dual discriminator agents.
3
3.1
Dual-Agent GAN
Simulator
The main challenge for unconstrained face recognition lies in the large variation and few profile face
images for each subject, which is the main obstacle for learning a well-performed pose-invariant
model. To address this problem, we simulate face images with various pre-defined poses (i.e., yaw
angles), which explicitly augments the available training data without extra human annotation efforts
and balances the pose distribution.
In particular, as shown in Figure. 2, we first extracts the face Region of Interest (RoI) from each
available real face image, and estimate 68 facial landmark points using the Recurrent AttentiveRefinement (RAR) framework (31), which is robust to illumination changes and does not require
a shape model in advance. We then estimate a transformation matrix between the detected 2D
landmarks and the corresponding landmarks in the 3D Morphable Model (3D MM) using leastsquares fit (35). Finally, we simulate profile face images in various poses with pre-defined yaw
angles.
However, the performance of the simulator decreases dramatically under large poses (e.g., yaw angles
? {[?90o , ?60o ] ? [+60o , +90o ]}) due to artifacts and severe texture losses, misleading the network
to overfit to fake information only presented in synthetic images and fail to generalize well on real
data.
3.2
Generator
In order to generate photorealistic and identity preserving profile view face images which are truely
benefical for unconstrained face recognition, we further refine the above-mentioned simulated profile
face images with the proposed DA-GAN.
Inspired by the recent success of FCN-based methods on image-to-image applications (17; 9) and
the leading performance of skip-net on recognition tasks (12; 33), we modify a skip-net (ResNet
(12)) into a FCN-based architecture as the generator G? : RH?W ?C 7? RH?W ?C of DA-GAN to
learn a highly non-linear transformation for profile face image refinement, where ? are the network
parameters for the generator, and H, W , and C denote the image height, width, and channel number,
repectively.
Contextual information from global and local regions compensates each other and naturally benefits
face recognition. The hierarchical features within a skip-net are multi-scale in nature due to the
increasing receptive field sizes, which are combined together via skip connections. Such a combined representation comprehensively maintains the contextual information, which is crucial for
4
artifact removal, fragement stitching, and texture padding. Moreover, the FCN-based architecture is
advantageous for generating high-resolution image-level results. More details are provided in Sec. 4.
More formally, let the simulated profile face image be denoted by x and the refined face image be
denoted by x
?, then
x
? := G? (x).
(1)
The key requirements for DA-GAN are that the refined face image x
? should look like a real face
image in appearance while preserving the intrinsic identity and pose information from the simulator.
To this end, we propose to learn ? by minimizing a combination of three losses:
LG? = (?Ladv + ?1 Lip ) + ?2 Lpp ,
(2)
where Ladv is the adversarial loss for adding realism to the synthetic images and alleviating artifacts,
Lip is the identity perception loss for preserving the identity information, and Lpp is the pose
perception loss for preserving pose and texture information.
Lpp is a pixel-wise L1 loss, which is introduced to enforce the pose (i.e., yaw angle) consistency for
the synthetic profile face images before and after the refinement via DA-GAN:
Lpp =
W X
H
X
1
|xi,j ? x
?i,j |,
W ?H i j
(3)
where i, j traverse all pixels of x and x
?.
Although Lpp may lead some over smooth effects to the refined results, it is still an essential part for
both pose and texture information preserving and accelerated optimization.
To add realism to the synthetic images to really benefit face recognition performance, we need to
narrow the gap between the distributions of synthetic and real images. An ideal generator will make
it impossible to classify a given image as real or refined with high confidence. Meanwhile, preserving
the identity information is the essential and critical part for recognition. An ideal generator will
generate the refined face images that have small intra-class distance and large inter-class distance in
the feature space spanned by the deep neural networks for unconstrained face recognition. These
motivate the use of an adversarial pixel-wise discriminator with dual agents.
3.3
Dual-agent discriminator
To incorporate the prior knowledge from the profile faces? distribution and domain knowledge of
identities? distribution, we herein introduce a discriminator with dual agents for distinguishing real
v.s. fake and identities simultaneously. To facilitate this process, we leverage an auto-encoder as the
discriminator D? : RH?W ?C 7? RH?W ?C to be as simple as possible to avoid typical GAN tricks,
which first projects the input real / fake face image into high-dimensional feature space through
several Convolution (Conv) and Fully Connected (FC) layers of the encoder and then transformed
back to the image-level representation through several Deconvolution (Deconv) and Conv layers of
the decoder, as shown in Figure. 2. ? are the networks parameters for the discriminator. More details
are provided in Sec. 4.
One agent of D? is trained with Ladv to minimize the Wasserstein distance with a boundary equilibrium regularization term for maintaining a balance between the generator and discriminator losses as
first introduced in (2),
Ladv =
X
|yj ? D? (yj )| ? kt
j
X
|?
xi ? D? (?
xi )|,
(4)
i
where y denotes the real face image, kt is a boundary P
equilibrium regularizationP
term using Proportional Control Theory to maintain the equilibrium E[ i |?
xi ? D? (?
xi )|] = ?E[ j |yj ? D? (yj )|],
? is the diversity ratio.
Here kt is updated by
kt+1 = kt + ?(?
X
|yj ? D? (yj )| ?
j
X
|?
xi ? D? (?
xi )|),
(5)
i
where ? is the learning rate (proportional gain) for k. In essence, Eq.(5) can be thought of as a form
of close-loop feedback control in which kt is adjusted at each step.
5
Ladv serves as a supervision to push the refined face image to reside in the manifold of real images.
It can prevent the blurry effect, alleviate artifacts and produce visually pleasing results.
The other agent of D? is trained with Lip to preserve the identity discriminability of the refined face
images. Specially, we define Lip with the multi-class cross-entropy loss based on the output from the
bottleneck layer of D? .
Lip =
1 X
?(Yj log(D? (yj )) + (1 ? Yj )log(1 ? D? (yj )))
N j
+
1 X
?(Yi log(D? (?
xi )) + (1 ? Yi )log(1 ? D? (?
xi ))),
N i
(6)
where Y is the identity ground truth.
Thus, minimizing Lip would encourage deep features of the refined face images belonging to the same
identity to be close to each other. If one visualizes the learned deep features in the high-dimensional
space, the learned deep features of refined face image set form several compact clusters and each
cluster may be far away from others. Each cluster has a small variance. In this way, the refined
face images are enforced with well preserved identity information. We also conduct experiments for
illustration.
Using Lip alone makes the results prone to annoying artifacts, because the search for a local minimum
of Lip may go through a path that resides outside the manifold of natural face images. Thus, we
combine Lip with Ladv as the final objective function for D? to ensure that the search resides in that
manifold and produces photorealistic and identity preserving face image:
LD? = Ladv + ?1 Lip .
3.4
(7)
Loss functions for training
The goal of DA-GAN is to use a set of unlabeled real face images y to learn a generator G? that
adaptively refines a simulated profile face image x. The overall objective function for DA-GAN is:
(
LD? = Ladv + ?1 Lip ,
LG? = (?Ladv + ?1 Lip ) + ?2 Lpp .
(8)
We optimize DA-GAN by alternatively optimizing D? and G? for each training iteration. Similar
as in (2), we measure the convergence of DA-GAN by using the boundary
Pequilibrium concept:
we can frame the convergence process as finding the closest reconstruction j |yj ? D? (yj )| with
thePlowest absolute value
P of the instantaneous process error for the Proportion Control Theory
|? j |yj ? D? (yj )| ? i |?
xi ? D? (?
xi )||. This measurement can be formulated as:
Lcon =
X
|yj ? D? (yj )| + |?
j
X
|yj ? D? (yj )| ?
j
X
|?
xi ? D? (?
xi )||.
(9)
i
Lcon can be used to determine when the network has reached its final state or if the model has
collapsed. Detailed algorithm on the training procedures is provided in supplementary material Sec.
1.
4
4.1
Experiments
Experimental settings
Benchmark dataset: Except for synthesizing natural looking profile view face images, the proposed DA-GAN also aims to generate identity preserving face images for accurate face-centric
analysis with state-of-the-art deep learning models. Therefore, we evaluate the possibility of ?recognition via generation" of DA-GAN on the most challenging unconstrained face recognition benchmark
dataset IJB-A (15). IJB-A (15) contains both images and video frames from 500 subjects with
5,397 images and 2,042 videos that are split into 20,412 frames, 11.4 images and 4.2 videos per
subject, captured from in-the-wild environment to avoid the near frontal bias, along with protocols for
evaluation of both verification (1:1 comparison) and identification (1:N search) tasks. For training
and testing, 10 random splits are provided by each protocol, respectively. More details are provided
in supplementary material Sec. 2.
6
2.00
Convergence
1.75
1.50
1.25
1.00
0.50
0.25
0.05
0
25000
50000
75000
100000
125000
150000
175000
200000
225000
250000
Iterations
Figure 3: Quality of refined results w.r.t. the network convergence measurement Lcon .
Real
60?
70?
80?
90?
Simulated
Refined
Simulated
Refined
Simulated
Refined
(a) Refined results of DA-GAN.
Real Faces
Refined Synthetic Faces with DA-GAN
(b) Feature space of real faces and DA-GAN synthetic faces.
Figure 4: Qualitative analysis of DA-GAN.
Reproducibility: The proposed method is implementated by extending the Keras framework (6).
All networks are trained on three NVIDIA GeForce GTX TITAN X GPUs with 12GB memory for
each. Please refer to supplementary material Sec. 3 & 4 for full details on network architectures and
training procedures.
4.2
Results and discussions
Qualitative results ? DA-GAN: In order to illustrate the compelling perceptual results generated
by the proposed DA-GAN, we first visualize the quality of refined results w.r.t. the network convergence measurement Lcon , as shown in Figure. 3. As can be seen, our DA-GAN ensures a fast yet
stable convergence through the carefully designed optimization scheme and boundary equilibrium
regularization term. The network convergence measurement Lcon correlates well with image fidelity.
Most of the previous works (31; 32; 35) on profile view synthesis are dedicated to address this
problem within a pose range of ?60o . Because it is commonly believed that with a pose that is larger
than 60o , it is difficult for a model to generate faithful profile view images. Similarly, our simulator is
also good at normalizing small posed faces while suffers severe artifacts and texture losses under
large poses (e.g., yaw angles ? {[?90o , ?60o ] ? [+60o , +90o ]}), as shown in Figure. 4a the first
row for each subject. However, with enough training data and proper architecture and objective
function design of the proposed DA-GAN, it is in fact feasible to further refine such synthetic profile
face images under very large poses for high-quality natural looking results generation, as shown
in Figure. 4a the second row for each subject. Compared with the raw simulated faces, the refined
results by DA-GAN present a good photorealistic quality. More visualized samples are provided in
supplementary material Sec. 5.
To verify the superiority of DA-GAN as well as the contribution of each component, we also compare
the qualitative results produced by the vanilla GAN (10), Apple GAN (28), BE-GAN (2) and three
variations of DA-GAN in terms of w/o Ladv , Lip , Lpp in each case, repectively. Please refer to
supplementary material Sec. 5 for details.
7
Table 1: Performance comparison of DA-GAN with state-of-the-arts on IJB-A verification protocol.
For all metrics, a higher number means better performance. The results are averaged over 10 testing
splits. Symbol ?-" implies that the result is not reported for that method. Standard deviation is not
available for some methods. The results offered by our proposed method are highlighted in bold.
Method
OpenBR (15)
GOTS (15)
Pooling faces (11)
LSFS (30)
Deep Multi-pose (1)
DCNNmanual (4)
Triplet Similarity (27)
VGG-Face (23)
PAMs (19)
DCNNf usion (3)
Masi et al. (20)
Triplet Embedding (27)
All-In-One (25)
Template Adaptation (8)
NAN (34)
L2 -softmax (24)
b-1
b-2
DA-GAN (ours)
TAR @
FAR=0.10
0.433 ? 0.006
0.627 ? 0.012
0.631
0.895 ? 0.013
0.911
0.947 ? 0.011
0.945 ? 0.002
0.652 ? 0.037
0.967 ? 0.009
0.964 ? 0.005
0.976 ? 0.004
0.979 ? 0.004
0.978 ? 0.003
0.984 ? 0.002
0.989 ? 0.003
0.978 ? 0.003
0.991 ? 0.003
Face verification
TAR @
FAR=0.01
0.236 ? 0.009
0.406 ? 0.014
0.309
0.733 ? 0.034
0.787
0.787 ? 0.043
0.790 ? 0.030
0.805 ? 0.030
0.826 ? 0.018
0.838 ? 0.042
0.886
0.900 ? 0.010
0.922 ? 0.010
0.939 ? 0.013
0.941 ? 0.008
0.970 ? 0.004
0.963 ? 0.007
0.950 ? 0.009
0.976 ? 0.007
TAR @
FAR=0.001
0.104 ? 0.014
0.198 ? 0.008
0.514 ? 0.060
0.590 ? 0.050
0.725
0.813 ? 0.020
0.823 ? 0.020
0.836 ? 0.027
0.881 ? 0.011
0.943 ? 0.005
0.920 ? 0.006
0.901 ? 0.008
0.930 ? 0.005
Table 2: Performance comparison of DA-GAN with state-of-the-arts on IJB-A identification protocol.
For FNIR metric, a lower number means better performance. For the other metrics, a higher number
means better performance. The results offered by our proposed method are highlighted in bold.
Method
OpenBR (15)
GOTS (15)
B-CNN (7)
LSFS (30)
Pooling faces (11)
Deep Multi-pose (1)
DCNNmanual (4)
Triplet Similarity (27)
VGG-Face (23)
PAMs (19)
DCNNf usion (3)
Masi et al. (20)
Triplet Embedding (27)
Template Adaptation (8)
All-In-One (25)
NAN (34)
L2 -softmax (24)
b-1
b-2
DA-GAN (ours)
FNIR @
FPIR=0.10
0.851 ? 0.028
0.765 ? 0.033
0.659 ? 0.032
0.387 ? 0.032
0.250
0.246 ? 0.014
0.33 ? 0.031
0.210 ? 0.033
0.137 ? 0.014
0.118 ? 0.016
0.113 ? 0.014
0.083 ? 0.009
0.044 ? 0.006
0.068 ? 0.010
0.108 ? 0.008
0.051 ? 0.009
Face identification
FNIR @
Rank1
FPIR=0.01
0.934 ? 0.017 0.246 ? 0.011
0.953 ? 0.024 0.433 ? 0.021
0.857 ? 0.027 0.588 ? 0.020
0.617 ? 0.063 0.820 ? 0.024
0.846
0.480
0.846
0.852 ? 0.018
0.444 ? 0.065 0.880 ? 0.015
0.539 ? 0.077 0.913 ? 0.011
0.840 ? 0.012
0.423 ? 0.094 0.903 ? 0.012
0.906
0.247 ? 0.030 0.932 ? 0.010
0.226 ? 0.049 0.928 ? 0.010
0.208 ? 0.020 0.947 ? 0.008
0.183 ? 0.041 0.958 ? 0.005
0.085 ? 0.041
0.973 ? 0.005
0.125 ? 0.035 0.966 ? 0.006
0.179 ? 0.042 0.960 ? 0.007
0.110 ? 0.039
0.971 ? 0.007
Rank5
0.375 ? 0.008
0.595 ? 0.020
0.796 ? 0.017
0.929 ? 0.013
0.933
0.927
0.937 ? 0.010
0.950 ? 0.007
0.925 ? 0.008
0.965 ? 0.008
0.962
0.977 ? 0.004
0.980 ? 0.005
0.987 ? 0.003
0.982 ? 0.004
0.989 ? 0.003
To gain insights into the effectivenss of identity preserving quality of our DA-GAN, we further use
t-SNE (18) to visualize the deep features of both refined profile faces and real faces in a 2D space in
Figure. 4b. As can be seen, the refined profile face images present small intra-class distance and large
inter-class distance, which is similar to those of real faces. This reveals that DA-GAN ensures well
preserved identity information with the auxiliary agent for Lip .
Quantitative results ? ?recognition via generation": To quantitatively verify the superiority of
?recognition via generation" of DA-GAN, we conduct unconstrained face recognition (i.e., verification
and identification) on IJB-A benchmark dataset (15) with three different settings. In the three settings,
8
the pre-trained deep recognition models are respectively fine-tuned on the original training data of
each split without extra data (baseline 1: b-1), the original training data of each split with extra
synthetic faces by our simulator (baseline 2: b-2), and the original training data of each split with
extra refined faces by our DA-GAN (our method: ?recognition via generation" framework based on
DA-GAN, DA-GAN for short). The performance comparison of DA-GAN with the two baselines
and other state-of-the-arts on IJB-A (15) unconstrained face verification and identification protocols
are given in Table. 1 and Table. 2.
We can observe that even with extra training data, b-2 presents inferior performance than b-1 for all
metrics of both face verification and identification. This demonstrates that naively learning from
synthetic images can be problematic due to a gap between synthetic and real image distributions
? synthetic data is often not realistic enough with artifacts and severe texture losses, misleading
the network to overfit to fake information only presented in synthetic images and fail to generalize
well on real data. In contrast, with the injection of photorealistic and identity preserving faces
generated by DA-GAN without extra human annotation efforts, our method outperforms b-1 by
1.00% for TAR @ FAR=0.001 of verification and 1.50% for FNIR @ FPIR=0.01, 0.50% for Rank1 of identification. Our method achieves comparable performance with L2 -softmax (24), which
employ a much more computational complex recognition model even without fine-tuning or template
adaptation procedures as we do. Moreover, DA-GAN outperforms NAN (34) by 4.90% for TAR @
FAR=0.001 of verification and 7.30% for FNIR @ FPIR=0.01, 1.30% for Rank-1 of identification.
These results won the 1st places on verification and identification tracks in NIST IJB-A 2017 face
recognition competitions3 . This well verified the promissing potential of synthetic face images by
our DA-GAN on the large-scale and challenging unconstrained face recognition problem.
Finally, we visualize the verification and identification closed set results for IJB-A (15) split1 to gain
insights into unconstrained face recognition with the proposed ?recognition via generation" framework
based on DA-GAN. For fully detailed visualization results in high resolution and corresponding
analysis, please refer to supplementary material Sec. 6 & 7.
5
Conclusion
We propose a novel Dual-Agent Generative Adversarial Network (DA-GAN) for photorealistic and
identity preserving profile face synthesis. DA-GAN combines prior knowledge from data distribution
(adversarial training) and domain knowledge of faces (pose and identity perception loss) to exactly
recover the lost information inherent in projecting a 3D face into the 2D image space. DA-GAN can
be optimized in a fast yet stable way with an imposed boundary equilibrium regularization term that
balances the power of the discriminator against the generator. One promissing potential application of
the proposed DA-GAN is for solving generic transfer learning problems more effectively. Qualitative
and quantitative experiments verify the possibility of our ?recognition via generation" framework,
which achieved the top performance on the large-scale and challenging NIST IJB-A unconstrained
face recognition benchmark without extra human annotation efforts. Based on DA-GAN, we won the
1st places on verification and identification tracks in NIST IJB-A 2017 face recognition competitions.
It would be interesting to apply DA-GAN for other transfer learning applications in the future.
Acknowledgement
The work of Jian Zhao was partially supported by China Scholarship Council (CSC) grant
201503170248.
The work of Jiashi Feng was partially supported by National University of Singapore startup grant
R-263-000-C08-133, Ministry of Education of Singapore AcRF Tier One grant R-263-000-C21-112
and NUS IDS grant R-263-000-C67-646.
We would like to thank Junliang Xing (Institute of Automation, Chinese Academy of Sciences),
Hengzhu Liu, Xucan Chen, and Yongping Zhai (National University of Defense Technology) for
helpful discussions.
3
We submitted our results for both verification and identification protocols to NIST IJB-A 2017 face
recognition competition committee on 29th, March, 2017. We received the official notification on our top
performance on both tracks on 26th, Apirl, 2017. The IJB-A benchmark dataset, relevant information and
leaderboard can be found at https://www.nist.gov/programs-projects/face-challenges.
9
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11
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6,205 | 6,613 | Dilated Recurrent Neural Networks
Shiyu Chang1?, Yang Zhang1?, Wei Han2?, Mo Yu1 , Xiaoxiao Guo1 , Wei Tan1 ,
Xiaodong Cui1 , Michael Witbrock1 , Mark Hasegawa-Johnson2 , Thomas S. Huang2
1
IBM Thomas J. Watson Research Center, Yorktown, NY 10598, USA
2
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
{shiyu.chang, yang.zhang2, xiaoxiao.guo}@ibm.com,
{yum, wtan, cuix, witbrock}@us.ibm.com,
{weihan3, jhasegaw, t-huang1}@illinois.edu
Abstract
Learning with recurrent neural networks (RNNs) on long sequences is a notoriously difficult task. There are three major challenges: 1) complex dependencies, 2)
vanishing and exploding gradients, and 3) efficient parallelization. In this paper,
we introduce a simple yet effective RNN connection structure, the D ILATED RNN,
which simultaneously tackles all of these challenges. The proposed architecture is
characterized by multi-resolution dilated recurrent skip connections, and can be
combined flexibly with diverse RNN cells. Moreover, the D ILATED RNN reduces
the number of parameters needed and enhances training efficiency significantly,
while matching state-of-the-art performance (even with standard RNN cells) in
tasks involving very long-term dependencies. To provide a theory-based quantification of the architecture?s advantages, we introduce a memory capacity measure,
the mean recurrent length, which is more suitable for RNNs with long skip
connections than existing measures. We rigorously prove the advantages of the
D ILATED RNN over other recurrent neural architectures. The code for our method
is publicly available1 .
1
Introduction
Recurrent neural networks (RNNs) have been shown to have remarkable performance on many
sequential learning problems. However, long sequence learning with RNNs remains a challenging
problem for the following reasons: first, memorizing extremely long-term dependencies while
maintaining mid- and short-term memory is difficult; second, training RNNs using back-propagationthrough-time is impeded by vanishing and exploding gradients; And lastly, both forward- and
back-propagation are performed in a sequential manner, which makes the training time-consuming.
Many attempts have been made to overcome these difficulties using specialized neural structures, cells,
and optimization techniques. Long short-term memory (LSTM) [10] and gated recurrent units (GRU)
[6] powerfully model complex data dependencies. Recent attempts have focused on multi-timescale
designs, including clockwork RNNs [12], phased LSTM [17], hierarchical multi-scale RNNs [5], etc.
The problem of vanishing and exploding gradients is mitigated by LSTM and GRU memory gates;
other partial solutions include gradient clipping [18], orthogonal and unitary weight optimization
[2, 14, 24], and skip connections across multiple timestamps [8, 30]. For efficient sequential training,
WaveNet [22] abandoned RNN structures, proposing instead the dilated causal convolutional neural
network (CNN) architecture, which provides significant advantages in working directly with raw
audio waveforms. However, the length of dependencies captured by a dilated CNN is limited by its
kernel size, whereas an RNN?s autoregressive modeling can, in theory, capture potentially infinitely
?
1
Denotes equal contribution.
https://github.com/code-terminator/DilatedRNN
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
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Figure 1: (left) A single-layer RNN with recurrent skip connections. (mid) A single-layer RNN with
dilated recurrent skip connections. (right) A computation structure equivalent to the second graph,
which reduces the sequence length by four times.
long dependencies with a small number of parameters. Recently, Yu et al. [27] proposed learningbased RNNs with the ability to jump (skim input text) after seeing a few timestamps worth of data;
although the authors showed that the modified LSTM with jumping provides up to a six-fold speed
increase, the efficiency gain is mainly in the testing phase.
In this paper, we introduce the D ILATED RNN, a neural connection architecture analogous to the
dilated CNN [22, 28], but under a recurrent setting. Our approach provides a simple yet useful
solution that tries to alleviate all challenges simultaneously. The D ILATED RNN is a multi-layer, and
cell-independent architecture characterized by multi-resolution dilated recurrent skip connections.
The main contributions of this work are as follows. 1) We introduce a new dilated recurrent skip
connection as the key building block of the proposed architecture. These alleviate gradient problems
and extend the range of temporal dependencies like conventional recurrent skip connections, but in
the dilated version require fewer parameters and significantly enhance computational efficiency. 2)
We stack multiple dilated recurrent layers with hierarchical dilations to construct a D ILATED RNN,
which learns temporal dependencies of different scales at different layers. 3) We present the mean
recurrent length as a new neural memory capacity measure that reveals the performance difference
between the previously developed recurrent skip-connections and the dilated version. We also verify
the optimality of the exponentially increasing dilation distribution used in the proposed architecture.
It is worth mentioning that, the recent proposed Dilated LSTM [23] can be viewed as a special case
of our model, which contains only one dilated recurrent layer with fixed dilation. The main purpose
of their model is to reduce the temporal resolution on time-sensitive tasks. Thus, the Dilated LSTM
is not a general solution for modeling at multiple temporal resolutions.
We empirically validate the D ILATED RNN in multiple RNN settings on a variety of sequential
learning tasks, including long-term memorization, pixel-by-pixel classification of handwritten digits
(with permutation and noise), character-level language modeling, and speaker identification with
raw audio waveforms. The D ILATED RNN improves significantly on the performance of a regular
RNN, LSTM, or GRU with far fewer parameters. Many studies [6, 14] have shown that vanilla RNN
cells perform poorly in these learning tasks. However, within the proposed structure, even vanilla
RNN cells outperform more sophisticated designs, and match the state-of-the-art. We believe that the
D ILATED RNN provides a simple and generic approach to learning on very long sequences.
2
Dilated Recurrent Neural Networks
The main ingredients of the D ILATED RNN are its dilated recurrent skip connection and its use of
exponentially increasing dilation; these will be discussed in the following two subsections respectively.
2.1
Dilated Recurrent Skip Connection
Denote c(l)
t as the cell in layer l at time t. The dilated skip connection can be represented as
?
?
(l)
(l) (l)
ct = f xt , ct s(l) .
(1)
?
?
(l)
(l) (l)
(l)
ct = f xt , ct 1 , ct s(l) .
(2)
This is similar to the regular skip connection[8, 30], which can be represented as
s(l) is referred to as the skip length, or dilation of layer l; xt as the input to layer l at time t; and
(l)
f (?) denotes any RNN cell and output operations, e.g. Vanilla RNN cell, LSTM, GRU etc. Both
skip connections allow information to travel along fewer edges. The difference between dilated and
2
Output
Hidden Layer
Dilation = 4
Hidden Layer
Dilation = 2
Hidden Layer
Dilation = 1
Input
Figure 2: (left) An example of a three-layer D ILATED RNN with dilation 1, 2, and 4. (right) An
example of a two-layer D ILATED RNN, with dilation 2 in the first layer. In such a case, extra
embedding connections are required (red arrows) to compensate missing data dependencies.
regular skip connection is that the dependency on c(l)
t 1 is removed in dilated skip connection. The left
and middle graphs in figure 1 illustrate the differences between two architectures with dilation or skip
length s(l) = 4, where Wr0 is removed in the middle graph. This reduces the number of parameters.
More importantly, computational efficiency of a parallel implementation (e.g., using GPUs) can be
greatly improved by parallelizing operations that, in a regular RNN, would be impossible. The middle
and right graphs in figure 1 illustrate the idea with s(l) = 4 as an example. The input subsequences
(l)
(l)
(l)
(l)
(l)
{x4t }, {x4t+1 }, {x4t+2 } and {x4t+3 } are given four different colors. The four cell chains, {c4t },
(l)
(l)
(l)
{c4t+1 }, {c4t+2 } and {c4t+3 }, can be computed in parallel by feeding the four subsequences into a
regular RNN, as shown in the right of figure 1. The output can then be obtained by interweaving
among the four output chains. The degree of parallelization is increased by s(l) times.
2.2
Exponentially Increasing Dilation
To extract complex data dependencies, we stack dilated recurrent layers to construct D ILATED RNN.
Similar to settings that were introduced in WaveNet [22], the dilation increases exponentially across
layers. Denote s(l) as the dilation of the l-th layer. Then,
s(l) = M l
1
(3)
, l = 1, ? ? ? , L.
The left side of figure 2 depicts an example of D ILATED RNN with L = 3 and M = 2. On one
hand, stacking multiple dilated recurrent layers increases the model capacity. On the other hand,
exponentially increasing dilation brings two benefits. First, it makes different layers focus on different
temporal resolutions. Second, it reduces the average length of paths between nodes at different
timestamps, which improves the ability of RNNs to extract long-term dependencies and prevents
vanishing and exploding gradients. A formal proof of this statement will be given in section 3.
To improve overall computational efficiency, a generalization of our standard D ILATED RNN is also
proposed. The dilation in the generalized D ILATED RNN does not start at one, but M l0 . Formally,
s(l) = M (l
1+l0 )
, l = 1, ? ? ? , L and l0
0,
(4)
where
is called the starting dilation. To compensate for the missing dependencies shorter than
M l0 , a 1-by-M (l0 ) convolutional layer is appended as the final layer. The right side of figure 2
illustrates an example of l0 = 1, i.e. dilations start at two. Without the red edges, there would be
no edges connecting nodes at odd and even time stamps. As discussed in section 2.1, the computational efficiency can be increased by M l0 by breaking the input sequence into M l0 downsampled
subsequences, and feeding each into a L l0 -layer regular D ILATED RNN with shared weights.
M0l
3
The Memory Capacity of D ILATED RNN
In this section, we extend the analysis framework in [30] to establish better measures of memory
capacity and parameter efficiency, which will be discussed in the following two sections respectively.
3.1
Memory Capacity
To facilitate theoretical analysis, we apply the cyclic graph Gc notation introduced in [30].
Definition 3.1 (Cyclic Graph). The cyclic graph representation of an RNN structure is a directed
multi-graph, GC = (VC , EC ). Each edge is labeled as e = (u, v, ) 2 EC , where u is the origin
3
node, v is the destination node, and is the number of time steps the edge travels. Each node is
labeled as v = (i, p) 2 VC , where i is the time index of the node modulo m, m is the period of the
graph, and p is the node index. GC must contain at least one directed cycle. Along the edges of any
directed cycle, the summation of must not be zero.
Define di (n) as the length of the shortest path from any input node at time i to any output node at
time i + n. In [30], a measure of the memory capacity is proposed that essentially only looks at
di (m), where m is the period of the graph. This is reasonable when the period is small. However,
when the period is large, the entire distribution of di (n), 8n ? m makes a difference, not just the
one at span m. As a concrete example, suppose there is an RNN architecture of period m = 10, 000,
implemented using equation (2) with skip length s(l) = m, so that di (n) = n for n = 1, ? ? ? , 9, 999
and di (m) = 1. This network rapidly memorizes the dependence on inputs at time i of the outputs
at time i + m = i + 10, 000, but shorter dependencies 2 ? n ? 9, 999 are much harder to learn.
Motivated by this, we proposed the following additional measure of memory capacity.
Definition 3.2 (Mean Recurrent Length). For an RNN with cycle m, the mean recurrent length is
m
1 X
d? =
max di (n).
m n=1 i2V
(5)
Mean recurrent length studies the average dilation across different time spans within a cycle. An
architecture with good memory capacity should generally have a small recurrent length for all
time spans. Otherwise the network can only selectively memorize information at a few time spans.
Also, we take the maximum over i, so as to punish networks that have good length only for a few
starting times, which can only well memorize information originating from those specific times. The
proposed mean recurrent length has an interesting reciprocal relation with the short-term memory
(STM) measure proposed in [11], but mean recurrent length emphasizes more on long-term memory
capacity, which is more suitable for our intended applications.
With this, we are ready to illustrate the memory advantage of D ILATED RNN . Consider two RNN
architectures. One is the proposed D ILATED RNN structure with d layers and M = 2 (equation (1)).
The other is a regular d-layer RNN with skip connections (equation (2)). If the skip connections are
of skip s(l) = 2l 1 , then connections in the RNN are a strict superset of those in the D ILATED RNN ,
and the RNN accomplishes exactly the same d? as the D ILATED RNN , but with twice the number of
trainable parameters (see section 3.2). Suppose one were to give every layer in the RNN the largest
possible skip for any graph with a period of m = 2d 1 : set s(l) = 2d 1 in every layer, which is the
regular skip RNN setting. This apparent advantage turns out to be a disadvantage, because time spans
of 2 ? n < m suffer from increased path lengths, and therefore
d? = (m
1)/2 + log2 m + 1/m + 1,
(6)
which grows linearly with m. On the other hand, for the proposed D ILATED RNN,
d? = (3m
1)/2m log2 m + 1/m + 1,
(7)
where d? only grows logarithmically with m, which is much smaller than that of regular skip RNN.
It implies that the information in the past on average travels along much fewer edges, and thus
undergoes far less attenuation. The derivation is given in appendix A in the supplementary materials.
3.2
Parameter Efficiency
The advantage of D ILATED RNN lies not only in the memory capacity but also the number of
parameters that achieves such memory capacity. To quantify the analysis, the following measure is
introduced.
Definition 3.3 (Number of Recurrent Edges per Node). Denote Card{?} as the set cardinality. For
an RNN represented as GC = (VC , EC ), the number of recurrent edges per node, Nr , is defined as
Nr = Card {e = (u, v, ) 2 EC :
6= 0} / Card{VC }.
(8)
Ideally, we would want a network that has large recurrent skips while maintaining a small number of
recurrent weights. It is easy to show that Nr for D ILATED RNN is 1 and that for RNNs with regular
skip connections is 2. The D ILATED RNN has half the recurrent complexity as the RNN with regular
skip RNN because of the removal of the direct recurrent edge. The following theorem states that the
D ILATED RNN is able to achieve the best memory capacity among a class of connection structures
with Nr = 1, and thus is among the most parameter efficient RNN architectures.
4
Theorem 3.1 (Parameter Efficiency of D ILATED RNN). Consider a subset of d-layer RNNs with
period m = M d 1 that consists purely of dilated skip connections (hence Nr = 1). For the RNNs in
this subset, there are d different dilations, 1 = s1 ? s2 ? ? ? ? ? sd = m, and
si = n i si
1,
(9)
where ni is any arbitrary positive integer. Among this subset, the d-layer D ILATED RNN with dilation
?
rate {M 0 , ? ? ? , M d 1 } achieves the smallest d.
The proof is motivated by [4], and is given in appendix B.
3.3
Comparing with Dilated CNN
Since D ILATED RNN is motivated by dilated CNN [22, 28], it is useful to compare their memory
capacities. Although cyclic graph, mean recurrent length and number of recurrent edges per node
are designed for recurrent structures, they happen to be applicable to dilated CNN as well. What?s
more, it can be easily shown that, compared to a D ILATED RNN with the same number of layers and
dilation rate of each layer, a dilated CNN has exactly the same number of recurrent edges per node,
and a slightly smaller (by log2 m) mean recurrent length. Hence both architectures have the same
model complexity, and it looks like a dilated CNN has a slightly better memory capacity.
However, mean recurrent length only measures the memory capacity within a cycle. When going
beyond a cycle, it is already shown that the recurrent length grows linearly with the number of cycles
[30] for RNN structures, including D ILATED RNN, whereas for a dilated CNN, the receptive field
size is always finite (thus mean recurrent length goes to infinity beyond the receptive field size). For
example, with dilation rate M = 2l 1 and d layers l = 1, ? ? ? , d, a dilated CNN has a receptive
field size of 2d , which is two cycles. On the other hand, the memory of a D ILATED RNN can go far
beyond two cycles, particularly with the sophisticated units like GRU and LSTM. Hence the memory
capacity advantage of D ILATED RNN over a dilated CNN is obvious.
4
Experiments
In this section, we evaluate the performance of D ILATED RNN on four different tasks, which include
long-term memorization, pixel-by-pixel MNIST classification [15], character-level language modeling
on the Penn Treebank [16], and speaker identification with raw waveforms on VCTK [26]. We also
investigate the effect of dilation on performance and computational efficiency.
Unless specified otherwise, all the models are implemented with Tensorflow [1]. We use the default
nonlinearities and RMSProp optimizer [21] with learning rate 0.001 and decay rate of 0.9. All
weight matrices are initialized by the standard normal distribution. The batch size is set to 128.
Furthermore, in all the experiments, we apply the sequence classification setting [25], where the
output layer only adds at the end of the sequence. Results are reported for trained models that achieve
the best validation loss. Unless stated otherwise, no tricks, such as gradient clipping [18], learning
rate annealing, recurrent weight dropout [20], recurrent batch norm [20], layer norm [3], etc., are
applied. All the tasks are sequence level classification tasks, and therefore the ?gridding? problem
[29] is irrelevant. No ?degridded? module is needed.
Three RNN cells, Vanilla, LSTM and GRU cells, are combined with the D ILATED RNN , which
we refer to as dilated Vanilla, dilated LSTM and dilated GRU, respectively. The common baselines
include single-layer RNNs (denoted as Vanilla RNN, LSTM, and GRU), multi-layer RNNs (denoted
as stack Vanilla, stack LSTM, and stack GRU), and Vanilla RNN with regular skip connections
(denoted as Skip Vanilla). Additional baselines will be specified in the corresponding subsections.
4.1
Copy memory problem
This task tests the ability of recurrent models in memorizing long-term information. We follow
a similar setup in [2, 24, 10]. Each input sequence is of length T + 20. The first ten values are
randomly generated from integers 0 to 7; the next T 1 values are all 8; the last 11 values are all
9, where the first 9 signals that the model needs to start to output the first 10 values of the inputs.
Different from the settings in [2, 24], the average cross-entropy loss is only measured at the last 10
timestamps. Therefore, the random guess yields an expected average cross entropy of ln(8) ? 2.079.
5
Figure 3: Results of the copy memory problem with T = 500 (left) and T = 1000 (right). The dilatedRNN converges quickly to the perfect solutions. Except for RNNs with dilated skip connections, all
other methods are unable to improve over random guesses.
The D ILATED RNN uses 9 layers with hidden state size of 10. The dilation starts from one to 256 at
the last hidden layer. The single-layer baselines have 256 hidden units. The multi-layer baselines use
the same number of layers and hidden state size as the D ILATED RNN . The skip Vanilla has 9 layers,
and the skip length at each layer is 256, which matches the maximum dilation of the D ILATED RNN.
The convergence curves in two settings, T = 500 and 1, 000, are shown in figure 3. In both settings,
the D ILATED RNN with vanilla cells converges to a good optimum after about 1,000 training iterations,
whereas dilated LSTM and GRU converge slower. It might be because the LSTM and GRU cells are
much more complex than the vanilla unit. Except for the proposed models, all the other models are
unable to do better than the random guess, including the skip Vanilla. These results suggest that the
proposed structure as a simple renovation is very useful for problems requiring very long memory.
4.2
Pixel-by-pixel MNIST
Sequential classification on the MNIST digits [15] is commonly used to test the performance of RNNs.
We first implement two settings. In the first setting, called the unpermuted setting, we follow the same
setups in [2, 13, 14, 24, 30] by serializing each image into a 784 x 1 sequence. The second setting,
called permuted setting, rearranges the input sequence with a fixed permutations. Training, validation
and testing sets are the default ones in Tensorflow. Hyperparameters and results are reported in table
1. In addition to the baselines already described, we also implement the dilated CNN. However, the
receptive fields size of a nine-layer dilated CNN is 512, and is insufficient to cover the sequence
length of 784. Therefore, we added one more layer to the dilated CNN, which enlarges its receptive
field size to 1,024. It also forms a slight advantage of dilated CNN over the D ILATED RNN structures.
In the unpermuted setting, the dilated GRU achieves the best evaluation accuracy of 99.2. However,
the performance improvements of dilated GRU and LSTM over both the single- and multi-layer
ones are marginal, which might be because the task is too simple. Further, we observe significant
performance differences between stack Vanilla and skip vanilla, which is consistent with the findings
in [30] that RNNs can better model long-term dependencies and achieves good results when recurrent
skip connections added. Nevertheless, the dilated vanilla has yet another significant performance gain
over the skip Vanilla, which is consistent with our argument in section 3, that the D ILATED RNN has
a much more balanced memory over a wide range of time periods than RNNs with the regular skips.
The performance of the dilated CNN is dominated by dilated LSTM and GRU, even when the latter
have fewer parameters (in the 20 hidden units case) than the former (in the 50 hidden units case).
In the permuted setting, almost all performances are lower. However, the D ILATED RNN models
maintain very high evaluation accuracies. In particular, dilated Vanilla outperforms the previous
RNN-based state-of-the-art Zoneout [13] with a comparable number of parameters. It achieves
test accuracy of 96.1 with only 44k parameters. Note that the previous state-of-the-art utilizes the
recurrent batch normalization. The version without it has a much lower performance compared to
all the dilated models. We believe the consistently high performance of the D ILATED RNN across
different permutations is due to its hierarchical multi-resolution dilations. In addition, the dilated
CNN is able the achieve the best performance, which is in accordance with our claim in section 3.3
that dilated CNN has a slightly shorter mean recurrent length than D ILATED RNN architectures, when
sequence length fall within its receptive field size. However, note that this is achieved by adding one
additional layer to expand its receptive field size compared to the RNN counterparts. When the useful
information lies outside its receptive field, the dilated CNN might fail completely.
6
Figure 4: Results of the noisy MNIST task with T = 1000 (left) and 2000 (right). RNN models
without skip connections fail. D ILATED RNN significant outperforms regular recurrent skips and
on-pars with the dilated CNN.
Table 1: Results for unpermuted and permuted pixel-by-pixel MNIST. Italic numbers indicate the
results copied from the original paper. The best results are bold.
Method
Vanilla RNN
LSTM [24]
GRU
IRNN [14]
Full uRNN [24]
Skipped RNN [30]
Zoneout [13]
Dilated CNN [22]
Dilated Vanilla
Dilated LSTM
Dilated GRU
#
layers
1/9
1/9
1/9
1
1
1/9
1
10
9
9
9
hidden /
layer
256 / 20
256 / 20
256 / 20
100
512
95 / 20
100
20 / 50
20 / 50
20 / 50
20 / 50
# parameters
(?, k)
68 / 7
270 / 28
200 / 21
12
270
16 / 11
42
7 / 46
7 / 44
28 / 173
21 / 130
Max
dilations
1
1
1
1
1
21 / 256
1
512
256
256
256
Unpermuted
test accuracy
- / 49.1
98.2 / 98.7
99.1 / 98.8
97.0
97.5
98.1 / 85.4
98.0 / 98.3
97.7 / 98.0
98. 9 / 98.9
99.0 / 99.2
Permunted
test accuracy
71.6 / 88.5
91.7 / 89.5
94.1 / 91.3
?82.0
94.1
94.0 / 91.8
93.1 / 95.92
95.7 / 96.7
95.5 / 96.1
94.2 / 95.4
94.4 / 94.6
In addition to these two settings, we propose a more challenging task called the noisy MNIST, where
we pad the unpermuted pixel sequences with [0, 1] uniform random noise to the length of T . The
results with two setups T = 1, 000 and T = 2, 000 are shown in figure 4. The dilated recurrent
models and skip RNN have 9 layers and 20 hidden units per layer. The number of skips at each layer
of skip RNN is 256. The dilated CNN has 10 layers and 11 layers for T = 1, 000 and T = 2, 000,
respectively. This expands the receptive field size of the dilated CNN to the entire input sequence.
The number of filters per layer is 20. It is worth mentioning that, in the case of T = 2, 000, if
we use a 10-layer dilated CNN instead, it will only produce random guesses. This is because the
output node only sees the last 1, 024 input samples which do not contain any informative data. All
the other reported models have the same hyperparameters as shown in the first three row of table 1.
We found that none of the models without skip connections is able to learn. Although skip Vanilla
remains learning, its performance drops compared to the first unpermuted setup. On the contrary,
the D ILATED RNN and dilated CNN models obtain almost the same performances as before. It is
also worth mentioning that in all three experiments, the D ILATED RNN models are able to achieve
comparable results with an extremely small number of parameters.
4.3
Language modeling
We further investigate the task of predicting the next character on the Penn Treebank dataset [16]. We
follow the data splitting rule with the sequence length of 100 that are commonly used in previous
studies. This corpus contains 1 million words, which is small and prone to over-fitting. Therefore
model regularization methods have been shown effective on the validation and test set performances.
Unlike many existing approaches, we apply no regularization other than a dropout on the input layer.
Instead, we focus on investigating the regularization effect of the dilated structure itself. Results are
shown in table 2. Although Zoneout, LayerNorm HM-LSTM and HyperNetowrks outperform the
D ILATED RNN models, they apply batch or layer normalizations as regularization. To the best of our
knowledge, the dilated GRU with 1.27 BPC achieves the best result among models of similar sizes
2
with recurrent batch norm [20].
7
Table 2: Character-level language modeling on the Penn Tree Bank dataset.
Method
LSTM
GRU
Recurrent BN-LSTM [7]
Recurrent dropout LSTM [20]
Zoneout [13]
LayerNorm HM-LSTM [5]
HyperNetworks [9]
Dilated Vanilla
Dilated LSTM
Dilated GRU
Method
MFCC
Raw
#
layers
1/5
1/5
1
1
1
3
1/2
5
5
5
hidden
/ layer
1k / 256
1k / 256
1k
1k
1k
512
1k
256
256
256
# parameters
(?, M)
4.25 / 1.9
3.19 / 1.42
4.25
4.25
4.91 / 14.41
0.6
1.9
1.42
Max
dilations
1
1
1
1
1
1
1
64
64
64
Table 3: Speaker identification on the VCTK dataset.
GRU
Fused GRU
Dilated GRU
#
layers
5/1
1
6/8
hidden
/ layer
20 / 128
256
50
# parameters
(?, k)
16 / 68
225
103 / 133
Min
dilations
1
32 / 8
32 / 8
Max
dilations
1
32 /8
1024
Evaluation
BPC
1.31 / 1.33
1.32 / 1.33
1.32
1.30
1.27
1.24
1.26 / 1.223
1.37
1.31
1.27
Evaluation
accuracy
0.66 / 0.77
0.45 / 0.65
0.64 / 0.74
without layer normalizations. Also, the dilated models outperform their regular counterparts, Vanilla
(didn?t converge, omitted), LSTM and GRU, without increasing the model complexity.
4.4
Speaker identification from raw waveform
We also perform the speaker identification task using the corpus from VCTK [26]. Learning audio
models directly from the raw waveform poses a difficult challenge for recurrent models because of
the vastly long-term dependency. Recently the CLDNN family of models [19] managed to match
or surpass the log mel-frequency features in several speech problems using waveform. However,
CLDNNs coarsen the temporal granularity by pooling the first-layer CNN output before feeding it
into the subsequent RNN layers, so as to solve the memory challenge. Instead, the D ILATED RNN
directly works on the raw waveform without pooling, which is considered more difficult.
To achieve a feasible training time, we adopt the efficient generalization of the D ILATED RNN as
proposed in equation (4) with l0 = 3 and l0 = 5 . As mentioned before, if the dilations do not start
at one, the model is equivalent to multiple shared-weight networks, each working on partial inputs,
and the predictions are made by fusing the information using a 1-by-M l0 convolutional layer. Our
baseline GRU model follows the same setting with various resolutions (referred to as fused-GRU),
with dilation starting at 8. This baseline has 8 share-weight GRU networks, and each subnetwork
works on 1/8 of the subsampled sequences. The same fusion layer is used to obtain the final prediction.
Since most other regular baselines failed to converge, we also implemented the MFCC-based models
on the same task setting for reference. The 13-dimensional log-mel frequency features are computed
with 25ms window and 5ms shift. The inputs of MFCC models are of length 100 to match the
input duration in the waveform-based models. The MFCC feature has two natural advantages: 1)
no information loss from operating on subsequences; 2) shorter sequence length. Nevertheless, our
dilated models operating directly on the waveform still offer a competitive performance (Table 3).
4.5
Discussion
In this subsection, we first investigate the relationship between performance and the number of
dilations. We compare the D ILATED RNN models with different numbers of layers on the noisy
MNIST T = 1, 000 task. All models use vanilla RNN cells with hidden state size 20. The number of
dilations starts at one. In figure 5, we observe that the classification accuracy and rate of convergence
increases as the models become deeper. Recall the maximum skip is exponential in the number of
layers. Thus, the deeper model has a larger maximum skip and mean recurrent length.
Second, we consider maintaining a large maximum skip with a smaller number of layers, by increasing
the dilation at the bottom layer of D ILATED RNN . First, we construct a nine-layer D ILATED RNN
3
with layer normalization [3].
8
Figure 5: Results for dilated vanilla with different numbers of layers on the noisy MNIST dataset.
The performance and convergent speed increase as the number of layers increases.
Figure 6: Training time (left) and evaluation performance (right) for dilated vanilla that starts at
different numbers of dilations at the bottom layer. The maximum dilations for all models are 256.
model with vanilla RNN cells. The number of dilations starts at 1, and hidden state size is 20. This
architecture is referred to as ?starts at 1? in figure 6. Then, we remove the bottom hidden layers
one-by-one to construct seven new models. The last created model has three layers, and the number
of dilations starts at 64. Figure 6 demonstrates both the wall time and evaluation accuracy for 50,000
training iterations of noisy MNIST dataset. The training time reduces by roughly 50% for every
dropped layer (for every doubling of the minimum dilation). Although the testing performance
decreases when the dilation does not start at one, the effect is marginal with s(0) = 2, and small with
4 ? s(0) ? 16. Notably, the model with dilation starting at 64 is able to train within 17 minutes by
using a single Nvidia P-100 GPU while maintaining a 93.5% test accuracy.
5
Conclusion
Our experiments with D ILATED RNN provide strong evidence that this simple multi-timescale
architectural choice can reliably improve the ability of recurrent models to learn long-term dependency
in problems from different domains. We found that the D ILATED RNN trains faster, requires less
hyperparameter tuning, and needs fewer parameters to achieve the state-of-the-arts. In complement
to the experimental results, we have provided a theoretical analysis showing the advantages of
D ILATED RNN and proved its optimality under a meaningful architectural measure of RNNs.
Acknowledgement
Authors would like to thank Tom Le Paine ([email protected]) and Ryan Musa
([email protected]) for their insightful discussions.
9
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6,206 | 6,614 | Hunt For The Unique, Stable, Sparse And Fast
Feature Learning On Graphs
Saurabh Verma
Department of Computer Science
University of Minnesota, Twin Cities
[email protected]
Zhi-Li Zhang
Department of Computer Science
University of Minnesota, Twin Cities
[email protected]
Abstract
For the purpose of learning on graphs, we hunt for a graph feature representation
that exhibit certain uniqueness, stability and sparsity properties while also being
amenable to fast computation. This leads to the discovery of family of graph
spectral distances (denoted as F GSD) and their based graph feature representations,
which we prove to possess most of these desired properties. To both evaluate
the quality of graph features produced by F GSD and demonstrate their utility, we
apply them to the graph classification problem. Through extensive experiments, we
show that a simple SVM based classification algorithm, driven with our powerful
F GSD based graph features, significantly outperforms all the more sophisticated
state-of-art algorithms on the unlabeled node datasets in terms of both accuracy
and speed; it also yields very competitive results on the labeled datasets ? despite
the fact it does not utilize any node label information.
1
Introduction
In the past decade, there has been tremendous interests in learning on collection of graphs for
various purposes, in particular for solving graph classification problem. Several applications of graph
classification can be found in the domain of bioinformatics, or chemoinformatics, or social networks.
A fundamental question inherent in graph classification is determining whether two graph structures
are identical, i.e., the graph isomorphism problem, which was not known to belong either P or NP
until recently. In the seminal paper [2], Babai shows that the graph isomorphism can be solved in
quasipolynomial time; while of enormous theoretical signficance, the implication of this result in
developing practical algorithms is still unclear. Fortunately, in graph classification problems one
is more interested in whether two graphs have ?similar? (as opposed to identical) structures. This
allows for potentially much faster (yet not fully explored) algorithms to be successfully applied to the
graph classification while also accounting for graph isomorphism. One approach to get around both
these intimately tied problems together is to learn an explicit graph representation that is invariant
under graph isomorphism1 but also useful for extracting graph features.
More specifically, given a graph G, we are interested in learning a graph representation (or spectrum),
R : G ? (g1 , g2 , ..., gr ), that captures certain inherent ?atomic? (unique) sub-structures of the graph
and is invariant under graph isomorphism (i.e., two isomorphic graphs yield the same representation).
Subsequently, we want to learn a feature function F : R ? (f1 , f2 , ..., fd ) from R such that the
graph features {fi }di=1 can be employed for solving the graph classification problem. However, in
machine learning, not much attention has been given towards learning R and most of the previous
studies have focused on designing graph kernels and thus bypasses computing any explicit graph
representation. The series of papers (19, 20, 22) by Kondor et al. are some of the first (and few) that
are concerned with constructing explicit graph features ? using a group theoretic approach ? that are
invariant to graph isomorphism and can be successfully applied to the graph classification problem.
1
That is, invariant under permutation of graph vertex labels.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Pairs of graph nodes
are generated from an
unknown distribution
Figure 1: Graph Generation Model: Graph
spectrum is assumed to be encoded in pairwise
node distances which are generated from some
distribution. Nodes connect together to form
a graph in such a way that pairwise node distances are preserved (eg. ( ? ) node-pair with
distance 0.75 is preserved even though they are
not directly connected).
Nodes connect together
to form a graph such that
the pairwise distances are
preserved
0.75
0.80
0.55
0.80
0.75
0.55
Inspired by such an approach, we also explicitly deal with learning a graph representation R and
show how to derive graph features F from R.
Our approach is quite novel and builds upon the following assumption: Graph atomic structure (or
spectrum) is encoded in the multiset2 of all node pairwise distances. Figure 1 shows the complete
graph generation model based on this premise. The origin of our assumption can be traced back to
the study of homometric structure, i.e, structures with the same multiset of interatomic distances [28].
On graphs, two vertex sets are called non-homometric if the multisets of distances determined by
them are different. (It is an unexplored problem whether there exists any distance metric on the graph
for which two vertex sets of non-isomorphic graphs are always non-homometric; but the converse
is not true, an example is the shortest path distance.) This argument provides the validity of our
assumption that the graph atomic structure is being encoded in pairwise distances. Further, we have
empirically found that the biharmonic distance [23] multisets are unique for at-least upto 10-vertex
size simple connected graphs (? 11 million graphs) and it remains as an open problem to show a
contradictory example. Moreover, we show that for a certain distance function Sf on the graph, one
can uniquely recover all the graph intrinsic properties while also being able to capture both local &
global information about the graph. Thus, we define R as the multiset of node pairwise distances
based on some distance function Sf , which will be the main focus of this paper.
We hunt for such a family of distances on graphs and its core members for which most of the properties
of an ideal graph spectrum (see Section 3) hold, including invariance under graph isomorphism and
the uniqueness property. This hunt leads us to the discovery of a family of graph spectral distance
(F GSD) and one would find harmonic (effective resistance) and biharmonic distance on graphs as the
suitable members of this family for graph representation R. Finally, for solving graph classification
(where graphs can be of different nodes sizes), we simply construct F feature vector from the
histogram of R (a multiset) and feed it to a standard classification algorithm.
Our current work focuses only on unlabeled graphs but can be extended to labeled graphs using the
same strategy as in shortest path kernel [4]. Nevertheless, our comprehensive results show that F GSD
graph features are powerful enough to significantly outperform the current state-of-art algorithms on
unlabeled datasets and are very competitive on labeled datasets ? despite the fact that they do not
utilize any node label information. In summary, the major contributions of our paper are:
? Introducing a novel & conceptually simple yet powerful graph feature representation (or spectrum)
based on the multiset of node pairwise distances.
? Discovering F GSD as a well-suited candidate for our proposed graph spectrum.
? Proving that F GSD based graph features exhibit certain uniqueness, stability, sparsity properties
and can be computationally fast with O (N 2 ) complexity, where N is the number of graph nodes
in a graph.
? Showing the superior performance of F GSD based graph features on graph classification tasks.
2
Related Work
Previous studies on graph classification can be grouped into three main categories. The first category
is concerned with constructing explicit graph features such as the skew spectrum 20 and its successor,
graphlet spectrum [22] based on group-theoretic approaches. Both are computational expensive. The
second and more popular category deals with designing graph kernels, among which, strong ones
are graphlets [30], random walks or shortest paths [4], neighborhood subgraph pairwise distance
2
A set in which an element can occur multiple of times.
2
kernel [9], Weisfeiler-Lehman kernel [31], deep graph kernels [34], graph invariant kernels [27] and
multiscale Laplacian graph kernel [21]. A tangential work [24] related to constructing features based
on atoms 3D space coordinates rather than operating on a graph structure, can be also considered in
this category. Our effort on learning R from F GSD can be seen as a part of first category, since we
explicitly investigate numerous properties of our proposed graph spectrum. While, extracting F from
R is more inspired from the work of graph kernels.
The third category involves developing convolutional neural networks (CNNs) for graphs, where
several models have been proposed to define convolution networks on graphs. The most common
model is based on generalizing convolutional networks through the graph Fourier transform via
a graph Laplacian [7, 16]. Defferrard et al. [11] extend this model by constructing fast localized
spectral filters for efficient graph coarsening as a pooling operation for CNNs on graphs. Some
variants of these models were considered in [18, 1], where the output of each neural network layer is
computed using a propagation rule that takes the graph adjacency matrix and node feature vectors into
account while updating the network weights. In [12], the convolution operation is defined by hashing
of local graph node features along with the local structure information. Likewise, in [26] local node
sequences are ?canonicalized? to create receptive fields and then fed into a 1D convolutional neural
network for classification. Among the aforementioned graph CNNs models, only those in [26, 1, 12]
are relevant to this work since they are designed to account for graphs of different sizes, while others
assume a global structure where the one-to-one correspondence of input vertices are already known.
3
Family of Graph Spectral Distances and Graph Spectrum
Basic Setup and Notations: Consider a weighted, undirected (and connected) graph G = (V, E, W )
of size N = |V |, where V is the vertex set, E the edge set (with no self-loops) and W = [wxy ] the
nonnegative weighted adjacency matrix. The standard graph Laplacian is defined as L = D ? W ,
where D is the degree matrix. It is semi-definite and admits an eigen-decomposition of the form
L = ???T , where ? = diag[?k ] is the diagonal matrix formed by the eigenvalues ?0 = 0 <
?1 ? ? ? ? ? ?N ?1 , and ? = [?0 , ..., ?N ?1 ] is an orthogonal matrix formed by the corresponding
eigenvectors ?k ?s. For x ? V , we use ?k (x) to denote the x-entry value of ?k . Let f be an
arbitrary nonnegative (real-analytical) function on R+ with f (0) = 0, 1 = [1, .., 1]T is the all-one
vector and J = 11T . Then, using slight abuse of notion, we define f (L) := ?f (?)?T and
f (?) := diag[f (?k )]. Also, f (L)xy represent xy-entry value in f (L) matrix. Lastly, I is identity
matrix and L+ is Moore-Penrose Pseudoinverse of L.
F GSD Definition: For x, y ? V , we define the f -spectral distance between x and y on G as follows:
Sf (x, y) =
N
?1
X
f (?k )(?k (x) ? ?k (y))2
(1)
k=0
We will refer to {Sf (x, y)|f }, as the family of graph spectral distances. Without loss of generality,
we assume that the derivative f 0 (?) 6= 0 for ? > 0, and then by Lagrange Inversion Theorem [33], f
is invertible and thus bijective. For reasons that will be clear shortly, we are particularly interested in
two sub-families of F GSD, where f is monotonic function (increasing or decreasing) of ?. Depending
on the sub-family, the f -spectral distance can capture different type of information in a graph.
F GSD Elements Encode Local Structure Information: For f (?) = ?p (p ? 1), one can show
that Sf (x, y) = (Lp )xx + (Lp )yy ? 2(Lp )xy . If the shortest path from x to y is larger than p, then
(Lp )xy = 0. This is based on the fact (Lp )xy captures only p-hop local neighborhood information [32]
on the graph. Hence, broadly for an increasing function of f (e.g., a polynomial function of degree
atleast p ? 1), Sf (x, y) captures the local structure information.
F GSD Elements Encode Global Structure Information: On the other hand, f as a decreasing
function yields Sf (x, y) = ((L+ )p )xx + (((L+ )p )yy ? 2((L+ )p )xy . This captures the global
J ?1
J
information, since the xy-entry of L+ = (L + N
) ?N
accounts for all paths from node x to y
+ p
(and so does (L ) ). Several known globally aware graph distances can be derived from this F GSD
sub-family. For f (?) = 1/? where ? > 0, Sf (x, y) is the harmonic (or effective resistance) distance.
More generally, for f (?) = 1/?p , p ? 1, Sf (x, y) is the polyharmonic distance (p = 2 is biharmonic
distance). Lastly f (?k ) = e?2t?k yields Sf (x, y) that is equivalent to the heat diffusion distance.
3
F GSD Graph Signal Processing Point of View: From graph signal processing perspective, Sf (x, y)
is a distance computed based on spectral filter properties [32], where f (?) act as a band-pass filter. Or,
it can be viewed in terms of spectral graph wavelets [15] as: Sf (x, y) = ?f,x (x)+?f,y (y)?2?f,x (y),
PN ?1
where ?f,x (y) = k=0 f (?k )?k (x)?k (y) (and ?f,x (x), ?f,y (y) are similarly defined) is a spectral
graph wavelet of scale 1, centered at node x and f (?) act as a graph wavelet kernel.
F GSD Based Graph Spectrum: Using the F GSD based distance matrix Sf = [Sf (x, y)] directly,
e.g., for graph classification, requires us being able to solve the graph isomorphism problem efficiently.
But no known polynomial time algorithm is available; the best algorithm today theoretically takes
quasipolynomial time [2]. However, motivated from the study of homometric structure and the fact
that each element of F GSD encodes some local or global sub-structure information of the graph,
inspired us to define the graph spectrum as R = {Sf (x, y)|?(x, y) ? V }. Thus, comparing two
R?s implicitly evaluates the sub-structural similarity between two graphs. For instance, R based on
harmonic distance contains sub-structural properties related to the spanning trees of a graph [29].
Our main concern in this paper would be choosing an appropriate f (?) function in order to generate
R which can exhibit ideal graph spectrum properties as discuss below. Also, we want F to inherent
these properties directly from R, which is made possible by defining F as the histogram of R.
Finally, we lay down those important fundamental properties of an ideal graph spectrum that one
would like R & F to obey on a graph G = (V, E, W ).
1. R & F must be invariant under any permutation ? of vertex labels. That is, R(G) = R(G? ) or
R(W ) = R(P W P T ) for any permutation matrix P .
2. R & F must have a unique representation for non-isomorphic graphs. That is, R(G1 ) 6= R(G2 )
for any two non-isomorphic graphs G1 and G2 .
3. R & F must be stable under small perturbation. That is, if graph G2 (W2 ) = G1 (W1 + ?), for a
small perturbation norm matrix k?k, then the norm of kF (G2 ) ? F (G1 )k should also be small
or bounded in order to maintain the stability.
4. F must be sparse (if high-dimensional) for all the sparsity reasons desirable in machine learning.
5. R & F must be computationally fast for efficiency and scalability purposes.
4
Uniqueness of Family of Graph Spectral Distances and Graph Spectrum
We first start with exploring the graph invariance and uniqueness properties of R & F based on F GSD.
Uniqueness is a very important (desirable) property, since it will determine whether the elements of
R set are complete (i.e., how good they are), in the sense whether R is sufficient enough to recover
all the intrinsic structural properties of a graph. We state the following important uniqueness theorem.
Theorem 1 (Uniqueness of F GSD) 3 The f -spectral distance matrix Sf = [Sf (x, y)] uniquely
determines the underlying graph (up to graph isomorphism), and each graph has a unique Sf (up to
permutation). More precisely, two undirected, weighted (and connected) graphs G1 and G2 have the
same F GSD based distance matrix up to permutation, i.e., SG1 = P SG2 P T for some permutation
matrix P , if and only if the two graphs are isomorphic.
Implications: Our proof is based on establishing the following key relationship: f (L) =
? 21 (I ? N1 J)Sf (I ? N1 J). Since f is bijective, one can uniquely recover ? from f (?). One
of the consequence of Theorem 1 is that the R based on multiset of F GSD is invariant under the
permutation of graph vertex labels and thus, satisfies the graph invariance property. Also, F will
inherent this property since R remains the same. Unfortunately, it is possible that the multiset of
some F GSD members can be same for non-isomorphic graphs (otherwise, we would have a O (N 2 )
polynomial time algorithm for solving graph isomorphism problem!). However, it is known that
all non-isomorphic graphs with less than nine vertices have unique multisets of harmonic distance.
While, for nine & ten vertex (simple) graphs, we have exactly 11 & 49 pairs of non-isomorphic
graphs (out of total 274,668 & 12,005,168 graphs) with the same harmonic spectra. These examples
show that there are significantly very low numbers of non-unique harmonic spectrums. Moreover, we
empirically found that the biharmonic distance has all unique multisets for at-least upto ten vertices
(? 11 million graphs) and we couldn?t find any non-isomorphic graphs with the same biharmonic
multisets. Further, we have the following theorem regarding the uniqueness of R.
3
1
1
Variant of Theorem 1 also hold true for the normalized graph Laplacian Lnorm = D? 2 LD? 2 .
4
Theorem 2 (Uniqueness of Graph Harmonic Spectrum) Let G = (V, E, W ) be a graph of size
|V | with an unweighted adjacency matrix W . Then, if two graphs G1 and G2 have the same number
of nodes but different number of edges, i.e, |V1 | = |V2 | but |E1 | =
6 |E2 |, then with respect to the
harmonic distance multiset, R(G1 ) 6= R(G2 ).
Implications: Our proof relies on the fact that the effective resistance distance is a monotone function
with respect to adding or removing edges. It shows that R based on some F GSD members specially
harmonic distance is atleast theoretically known to be unique to a certain degree. F also inherent this
property, fully under the condition h ? 0 (or for small enough h), where h is the histogram binwidth.
Overall the certain uniqueness of R along with containing local or global structural properties in its
each element dictate that the R is capable enough to serve as the complete powerful Graph Spectrum.
4.1
Unifying Relationship Between F SGD and Graph Embedding and Dimension Reduction
Before delving into other properties, we uncover an essential relationship between F GSD and
p Graph
1
Embedding in Euclidean space and Dimension Reduction techniques. Let f (?) 2 = diag[ f (?k )]
1
and define ? = ?f (?) 2 . Then, the f -spectral distance can be expressed as Sf (x, y) = ||?(x) ?
?(y)||22 , where ?(x) is the xth row of ?. Thus, ? represents an Euclidean embedding of G where
each node x is represented by the vector ?(x). Now for instance, if f (?) = 1, then by taking the first
p columns of ? yields embedding exactly equal to Laplacian Eigenmap (LE) [3] based on random
walk graph Laplacian (Lrw = D?1 L). For f (?) = ?2t and L = D?1 W , we get the Diffusion
Map [25]. Thus, f (?) function has one-to-one correspondence relationship with spectral dimension
reduction techniques. We have the following theorem concerning Graph Embedding based on F GSD.
Theorem 3 (Uniqueness of F GSD Graph Embedding) Each graph G can be isometrically embedded into a Euclidean space using F GSD as an isometric measure. This isometric embedding is unique,
0
if all the eigenvalues of G Laplacian
are distinct and there does not exist any other graph G with
q
0
0
Laplacian eigenvectors ?k = f (?j )/f (?j )?k , ?k ? [1, N ? 1].
Implications: The above theorem shows that F GSD provides a unique way to embed the graph
vertices into Euclidean space possibly without loosing any structural information of the graph. This
could potentially serve as a cogent tool to convert an unstructured data into a structure data (similar to
structure2vec 10 or node2vec 14 tool) which can enable us to perform standard inference tasks
in Euclidean space. Note that the uniqueness condition is quite strict and holds for co-spectral graphs.
In short, we have following uniqueness relationship, where ? is the Euclidean embedding of G graph.
Sf
5
f (LG )
LG
f (LG )
?G
Stability of Family of Graph Spectral Distances and Graph Spectrum
Next, we hunt for the stable members of the F GSD that are robust against the perturbation or noise
in the datasets. Specifically, we will look at the stability of R and F based on F GSD from f (?)
perspective by first analyzing its influence on a single edge perturbation (or in other words analyzing
rank one modification of Laplacian matrix). This will lead us to find the stable members and what
restrictions we need to impose on f (?) function for stability. We will further show that f -spectral
distance function also satisfies the notion of uniform stability [6] in a certain sense. For our analysis,
we will restrict f (?) as a monotone function of ?, for ? > 0. Let 4w ? 0 be the change after
0
0
modifying w weight on any single edge to w on the graph, where 4w = w ? w.
Theorem 4 (Eigenfunction Stability of F GSD) Let 4Sxy be the change in Sf (x, y) distance with
respect to 4w change in weight of any single edge on the graph. Then, 4Sxy for any vertex pair
(x, y) is bounded with respect to the function of eigenvalue as follows,
4Sxy ? 2 |f (?N ?1 + 24w) ? f (?1 )|
Implications: Since, R = {Sf (x, y)|?(x, y) ? V }, then each element of R is itself bounded
by 4Sxy . Now, recall that F is a histogram of R, then F won?t change, if binwdith is large
enough to accommodate the perturbation i.e., h ? 24Sxy ?(x, y) assuming all elements of R are
5
at the center of their respective histogram bins. Besides h, the other way to make R robust is by
choosing a suitable f (?) function. Lets
consider the behavior 4Sxy on f (?) = ?p for p > 0. Then,
p
p
4Sxy ? 2 (?N ?1 + 24w) ? ?1 and as a result, 4Sxy is an increasing function with respect
to p which implies that stability decreases with increase
in p. For p = 0, stability does not change
|p|
with respect to ?. While, for p < 0, 4Sxy ? 2 1/?1 ? 1/(?N ?1 + 24w)|p| . Here, 4Sxy is
a decreasing function with respect to |p|, which implies that stability increases with decrease in p.
The results conforms with the reasoning that eigenvectors corresponding to smaller eigenvalues are
smoother (i.e., oscillates slowly) than large eigenvectors (corresponding to large eigenvalues) and
decreasing p will attenuate the contribution of large eigenvectors, making the f -spectral distance
more stable and less susceptible towards perturbation or noise. However, decreasing p too much could
result in lost of local information contained in eigenvectors with larger eigenvalues and therefore, a
balance needs to be maintained. Overall, Theorem 4 shows that either through suitable h or decreasing
f (?) function, stability of R & F can be controlled to satisfy the Ideal Spectrum Property 3.
Infact, we can further show that Sf (x, y) between any two vertex (x, y) on a graph, with 0 < ? ?
w ? ? bounded weights, is tightly bounded to a certain expected value.
Theorem 5 (Uniform Stability of F GSD) Let E[Sf (x, y)] be the expected value of Sf (x, y) between vertex pair (x, y), over all possible graphs with fixed ordering of N vertices. Then we have,
with probability 1 ? ?, where ? ? (0, 1) and ? depends upon ?, ?, N .
r
p
Sf (x, y) ? E[Sf (x, y)] ? f (?) N (N ? 1) log 1
?
Implications: The above theorem is based on the fact 4Sxy can itself be upper bounded over all
possible graphs generated on a fixed ordering of N vertices. This is a very similar condition needed
for a learning algorithm to satisfy the notion of uniform stability in order to give generalization
guarantees. The f -spectral distance function can itself be thought of as a learning algorithm which
admits uniform stability (precise definition in supplementary) and indicates a strong stability behavior
over all possible graphs and further act as a generalizable learning algorithm on the graph. Theorem 5
also reveals that the deviation can bepminimized by
choosing decreasing f (?) function and it would
be suitable, if f (?) grow with O 1/ N (N ? 1) rate in order to maintain stability for large graphs.
So far, we have narrow down our interest to R & F based on the bijective and decreasing f (?) function for achieving both uniqueness and stability. This eliminates all forms of increasing polynomial
functions as a good choice of f (?). As a result, we can focus on inverse (or rational) form of polynomial functions such as polyharmonic distances. A by-product of our analysis results in revealing a
new class of stable dimension reduction techniques, possible by scaling Laplacian eigenvectors with
decreasing function of f (?), although such connections have already been known before.
6
Sparsity of Family of Graph Spectral Distances and Graph Spectrum
40
Figure 2: Figure shows the number of unique
elements present in R formed by different f spectral distance on all graphs (of |V | = 9,
total 261, 080 graphs). Graph
in enumeration
dices are sorted according to R( ?1 ) G . We can
observe that f (?) = ?1 increases in form of a
step function and lower bounds all other f (?)
with an addition constant. (Best viewed in color
and when zoom-in.)
30
-R(f (6))G
f (6) =
f (6) =
f (6) =
f (6) =
f (6) =
f (6) =
20
10
0
50000
100000
150000
200000
Graph Enumeration Index
1
6
1
60:2
1
60:5
1
61:5
1
62
1
63
250000
Sparsity is desirable for both computational and statistical efficiency. In this section, we investigate
the sparsity produced in F by choosing different f (?) functions. Here, sparsity refers to its usual
definition of ?how many zero features are present in F graph feature vector?. Since F is a histogram
of R, number of non-zero elements in F will always be less than equal to number of unique (or
distinct) elements in R. However, due to the lack of any theoretical support, we rely on empirical
evidence and conjecture the following statement.
6
0
Data Sample Index
Data Sample Index
0
50
100
150
0
200
400
600
800
1000
1200
1400
1600
1800
50
100
150
0
2000
200
400
600
800
1000
1200
1400
1600
1800
2000
Feature Index
Feature Index
a. Harmonic distance based graph feature
matrix (matrix sparsity= 97.12%). Presence
of blue dot ( ) indicates feature value > 0.
b. Biharmonic distance based graph feature
matrix (matrix sparsity= 94.28%). Presence
of blue dot ( ) indicates feature value > 0.
c. Harmonic distance based feature matrix
sparsity shown with respect to per class label.
d. Biharmonic distance based feature matrix
sparsity shown with respect to per class label.
Figure 3: Feature space for MUTAG (composed of two class sizes 125 & 63): Both harmonic &
biharmonic based graph spectrum encodes a sparse high dimensional feature representation F for
graphs which can clearly distinguish the two classes as depicted in above sub-figures.
Conjecture (Sparsity of F GSD Graph Spectrum) For any graph G, let R(f (?)) G represents the
number of unique elements present in the multiset of R, computed on an unweighted graph G based
on some monotonic decreasing f (?) function. Then, the following holds,
1
+2
R(f (?)) ? R
? G
G
The conjecture is based on the observation that, in the Figure 2, R ?1 + 2 lower bounds all
given monotonic decreasing f (?) along with an addition constant of 2. Same trends are observed
for different
graph
sizes |V |. Interestingly, when graph enumeration indices are sorted according
1
to size R ? , we further observe that f (?) = ?1 increases in the form of a step function. From
this conjecture, we can directly conclude that the F based on f (?) = ?1 produce the most sparse
features because number of unique elements in its R is always less than any other R. Figure 3,
further supports this conjecture which shows the feature space computed for MUTAG dataset in
case of harmonic and biharmonic spectrums. However, this raises a question of trade-off between
maintaining uniqueness and sparsity, since biharmonic distance multisets are found to be unique
for more number of graphs than harmonic distance. Nonetheless, some preliminary experiments
measuring harmonic vs. biharmonic performance on graph classification (in supplementary), suggest
that the sparsity is more favorable than uniqueness since it results in higher classification accuracy.
7
Fast Computation of Family of Graph Spectral Distances and Spectrum
Finally, we provide the general recipe of computing any member of F GSD in fast manner. In order to
avoid direct eigenvalue decomposition, we can either perform approximation or leverage structural
properties and sparsity of f (L) for efficient exact computation of Sf and thus, R.
Approximation: Inspired from the spectral graph wavelet work [32], the recipe for approximating
F GSD is to decompose f (?) possibly
Pr into an approximate polynomial series (for example, chebyshev
polynomials) as follows: f (?) = i=0 ai Ti (?) such that Ti (x) can be computed in recursive
manner
Pr
from few lower order terms (Ti?1 (x), Ti?2 (x), ..., Ti?c (x)). Then it follows, f (L) = i=0 ai Ti (L).
In this case, the cost of computing will reduce to O (r|E|) for sparse L which is very less expensive,
since O (r|E|) O (N 2 ). But, if f (?) is an inverse polynomial form of function, then computing
P
?1
r
f (L) =
a
T
(L)
= f (L+
i
i
r ), boils down to efficiently computing (a single) Moore
i=0
Penrose Pseudo inverse of a matrix.
Efficient Exact Computation: By leveraging f (L) structural properties and its sparsity, we can
efficiently perform exact computation of f (L+ ) in much more better way than the eigenvalue
7
decomposition. We propose such a method 1 which is the generalization of [23] work. We can show
J
that, f (L)f (L+ )?1 = I ? N
. Therefore, f (L)lk+ = Bk , where lk+ and Bk are the k th column of
J
+
f (L ) and B = I ? N matrices, respectively. So, first we can find a particular solution of following
T
(sparse) linear system: f (L)x = Bk and then obtain lk+ = x ? 11T x1 x. The particular solution x can
be obtained by replacing any single row and corresponding column of f (L) by zeros, and setting
diagonal entry at their intersection to one, and replacing corresponding row of B by zeros. This gives
a (non-singular) sparse linear system which can be solved very efficiently by performing cholesky
factorization and back-substitution, resulting in overall O (N 2 ) complexity as shown in [5]. Beside
this, there are few other fast methods to compute Pseudo inverse, particularly given by [17].
Complexity
SP [4]
GK[34](k ?
{3, 4, 5}) (d ? N )
SGS[20]
GS [22](k ?
[2, 6])
DCNN[1]
MLG[21]
e < N)
(N
F GSD
Approximate
?
O (N dk?1 )
?
?
?
e3)
O (N
O (r|E|)
Worst-Case
3
O (N )
k
O (N )
3
O (N )
O (N
2+k
)
2
O (N )
3
O (N )
O (N 2 )
Table 1: F GSD complexity comparison with few strong state-of-art algorithms (showing variables
that are only dependent on N & |E|). It reveals that the F GSD complexity is better than the most.
As a result, it leads to a very efficient
O (r|E|) complexity through approximation with the worst-case O (N 2 )
complexity in exact computation of R.
Table 1, shows the complexity comparison with other state-of-art methods. Since, number of elements in R
are O (N 2 ), then F is also bounded
by O (N 2 ) and thus satisfies the ideal
graph spectrum Property 5. Finally,
Algorithm 1 summarizes the complete
procedure of computing R & F .
8
Algorithm 1 Computing R and F based on F GSD.
Input: Given graphs {Gi = (Vi , Ei , Wi )}M
i=1 , f (?),
number of bins b, binwidth h.
Output: Ri and Fi ?i ? [1, M ].
for i = 1 to M do
Compute f (Li ) using approx. or exact method 1.
Compute Si = diag(f (Li ))J + Jdiag(f (Li )) ?
2f (Li ).
Set Ri = {Sxy |?(x, y) ? |Vi |}.
Compute Fi = histogram(Ri , b, h).
end for
Experiments and Results
F GSD Graph Spectrum Settings: We chose harmonic distance as an ideal candidate for F . For fast
computation, we adopted our proposed efficient exact computation method 1. And for computing
histogram, we fix binwidth size and set the number of bins such that its range covers all {Ri }M
1
elements of M number of graphs. Therefore, we had only one parameter, binwidth size, chosen from
the set {0.001, 0.0001, 0.00001}. This results in F feature vector dimension in range 100?1000, 000
with feature matrix sparsity > 90% in all cases. Our F GSD code is available at github4 .
Datasets: We employed wide variety of datasets considered as benchmark [1, 34, 21, 26] in graph
classification task to evaluate the quality of produce F GSD graph features. We adopted 7 bioinformatics datasets: Mutag, PTC, Proteins, NCI1, NCI109, D&D, MAO and 5 social network datasets:
Collab, REDDIT-Binary, REDDIT-Multi-5K, IMDB-Binary, IMDB-Multi. D&D dataset contains
691 enzymes and 587 non-enzymes proteins structures. While, MAO dataset contains 38 molecules
that are antidepressant drugs and 30 do not. For other datasets, details can be found in [34].
Experimental Set-up: All experiments were performed on a single Intel-Core [email protected]
and 64GB RAM machine. We compare our method with 6 state-of-art Graphs Kernels: Random
Walk (RW) [13], Shortest Path Kernel (SP) [4], Graphlet Kernel (GK) [30], Weisfeiler-Lehman Kernel
(WL) [31], Deep Graph Kernels (DGK) [34], Multiscale Laplacian Graph Kernels (MLK) [21]. And
proposed, 2 recent state-of-art Graph Convolutional Networks: PATCHY-SAN (PSCN) [26],
Diffusion CNNs (DCNN) [1]. And, 2 strong Graph Spectrums: the Skew Spectrum (SGS) [20],
Graphlet Spectrum (GS) [22]. We adopt the same procedure from previous works [26, 34] to make a
fair comparison and used 10-fold cross validation with LIBSVM [8] library to test the classification
performance. Parameters of SVM are independently tuned using training folds data and best average
classification accuracies is reported for each method. We provide node degree as the labeled data for
algorithms that do not operate directly on unlabeled data. Further details about parameters selection
for baseline methods are present in supplementary materials.
4
https://github.com/vermaMachineLearning/FGSD
8
F GSD
(Wall-Time)
Dataset (No. Graphs,
Max. Nodes)
RW
[2003]
SP
[2005]
GK
[2009]
WL
[2011]
DGK
[2015]
MLG
(WallTime) [2016]
DCNN
[2016]
SGS
[2008]
MUTAG (188, 28)
83.50
87.23
84.04
87.28
86.17
87.23(5s)
66.51
88.61
92.12(0.3s)
PTC (344, 109)
55.52
58.72
60.17
55.61
59.88
62.20(18s)
55.79
?
62.80(0.07s)
PROTEINS (1113, 620)
68.46
72.14
71.78
70.06
71.69
71.35(277s)
65.22
?
73.42(5s)
NCI1 (4110, 111)
>D
68.15
62.07
77.23
64.40
77.57(620s)
63.10
62.72
79.80(31s)
NCI109 (4127, 111)
>D
68.30
62.04
78.43
67.14
75.91(600s)
60.67
62.62
78.84(35s)
D & D (1178, 5748)
>D
>D
75.05
73.76
72.75
77.02(7.5hr)
OMR
?
77.10(25s)
MAO (68, 27)
83.52
90.35
80.88
89.79
87.76
91.17(13s)
76.10
?
95.58(0.1s)
Table 2: Classification accuracy on unlabeled bioinformatics datasets. Results in bold indicate all
methods with accuracy within range 2.0 from the top result and blue color (for range > 2.0), indicates
the new state-of-art result. Green color highlights the best time computation, if it?s 5?faster (among
the mentioned). ?OMR? is out of memory error, ?> D? is computation exceed 24hrs.
Dataset
(Graphs)
GK
[2009]
DGK
[2015]
PSCN
[2016]
F GSD
Dataset
MLG
[2016]
DCNN
[2016]
PSCN
[2016]
GS
[2009]
F GSD *
COLLAB
(5000)
72.84
73.09
72.60
80.02
MUTAG
87.94
(4s)
66.98
92.63
(3s)
88.11
92.12
(0.3s)
REDDIT-B
(2000)
77.34
78.04
86.30
86.50
PTC
63.26
(21s)
56.60
62.90
(6s)
?
62.80
(0.07s)
REDDIT-M
(5000)
41.01
41.27
49.10
47.76
NCI1
81.75
(621s)
62.61
78.59
(76s)
65.0
79.80
(31s)
IMDB-B
(1000)
65.87
66.96
71.00
73.62
D&D
78.18
(7.5hr)
OMR
77.12
(154s)
?
77.10
(25s)
IMDB-M
(1500)
43.89
44.55
45.23
52.41
MAO
88.29
(12s)
75.14
?
?
95.58
(0.1s)
Table 3: Classification accuracy on social
network datasets. F GSD significantly outperforms other methods.
Table 4: Classification accuracy on labeled bioinformatics datasets. * emphasize that F GSD did not
utilize any node labels.
.
Classification Results: From Table 2, it is clear that F GSD consistently outperforms every other
state-of-art algorithms on unlabeled bioinformatics datasets and that too significantly in many cases.
F GSD even performs better for social network graphs as shown in Table 3 and achieves a very
significant 7% ? 8% more accuracy than the current state-of-art PSCNs on COLLAB and IMDB-M
datasets. Also from run-time perspective (excluding any data loading or classification time for all
algorithms), it is pretty fast (2x?1000x times faster) as compare to others. These appealing results
further motivated us to compare F GSD on the labeled datasets (even though, it is not a complete
fair comparison). Table 4 shows that F GSD is still very competitive with all other strong (recent)
algorithms that utilize node labeled data. Infact on MAO dataset, F GSD sets a new state-of-art result
and stays within 0% ? 2% range of accuracy from the best on all labeled datasets. On few labeled
datasets, we found MLG to have slightly better performance than the others, but it is 1000 times
slower than F GSD when graph size jumps to few thousand nodes (see D&D Results). Altogether,
F GSD shows very promising results in both accuracy & speed on all type of datasets and over all the
more sophisticated algorithms. These results also point out the fact that there is untapped hidden
potential in the graph structure which current algorithms are not harnessing despite having labeled
data at their disposal.
9
Conclusion
We present a conceptually simple yet powerful and theoretically motivated graph representation. In
particular, our graph representation based on the discovery of family of graph spectral distances can
exhibits uniqueness, stability, sparsity and are computationally fast. Moreover, our hunt specifically
leads to the harmonic and next to it, biharmonic distances as an ideal members of this family for
extracting graph features. Finally, our extensive results show that F GSD based graph features are
powerful enough to dominate the unlabeled graph classification task over all the more sophisticated
algorithms and competitive enough to yield high classification accuracy on labeled data even without
utilizing any node labels. In our future work, we plan to generalize the F GSD for labeled dataset in
order to utilize the useful node and edge label information in the graph representation.
9
10
Acknowledgments
This research was supported in part by ARO MURI Award W911NF-12-1-0385, DTRA grant
HDTRA1-14-1-0040, and NSF grants CNS-1618339, CNS-1618339 and CNS-1617729.
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11
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6,207 | 6,615 | Scalable Generalized Linear Bandits:
Online Computation and Hashing
Kwang-Sung Jun
UW-Madison
[email protected]
Aniruddha Bhargava
UW-Madison
[email protected]
Robert Nowak
UW-Madison
[email protected]
Rebecca Willett
UW-Madison
[email protected]
Abstract
Generalized Linear Bandits (GLBs), a natural extension of the stochastic linear
bandits, has been popular and successful in recent years. However, existing GLBs
scale poorly with the number of rounds and the number of arms, limiting their
utility in practice. This paper proposes new, scalable solutions to the GLB problem
in two respects. First, unlike existing GLBs, whose per-time-step space and time
complexity grow at least linearly with time t, we propose a new algorithm that
performs online computations to enjoy a constant space and time complexity. At
its heart is a novel Generalized Linear extension of the Online-to-confidence-set
Conversion (GLOC method) that takes any online learning algorithm and turns it
into a GLB algorithm. As a special case, we apply GLOC to the online Newton
step algorithm, which results in a low-regret GLB algorithm with much lower
time and memory complexity than prior work. Second, for the case where the
number N of arms is very large, we propose new algorithms in which each next
arm is selected via an inner product search. Such methods can be implemented
via hashing algorithms (i.e., ?hash-amenable?) and result in a time complexity
sublinear in N . While a Thompson sampling extension of GLOC is hash-amenable,
its regret bound for d-dimensional arm sets scales with d3/2 , whereas GLOC?s
regret bound scales with d. Towards closing this gap, we propose a new hashamenable algorithm whose regret bound scales with d5/4 . Finally, we propose a
fast approximate hash-key computation (inner product) with a better accuracy than
the state-of-the-art, which can be of independent interest. We conclude the paper
with preliminary experimental results confirming the merits of our methods.
1
Introduction
This paper considers the problem of making generalized linear bandits (GLBs) scalable. In the
stochastic GLB problem, a learner makes successive decisions to maximize her cumulative rewards.
Specifically, at time t the learner observes a set of arms Xt ? Rd . The learner then chooses an arm
?
xt ? Xt and receives a stochastic reward yt that is a noisy function of xt : yt = ?(x>
t ? ) + ?t , where
?
? ? Rd is unknown, ?:R?R is a known nonlinear mapping, and ?t ? R is some zero-mean noise.
This reward structure encompasses generalized linear models [29]; e.g., Bernoulli, Poisson, etc.
The key aspect of the bandit problem is that the learner does not know how much reward she would
have received, had she chosen another arm. The estimation on ? ? is thus biased by the history of the
selected arms, and one needs to mix in exploratory arm selections to avoid ruling out the optimal
arm. This is well-known as the exploration-exploitation dilemma. The performance of a learner is
evaluated by its regret that measures how much cumulative reward she would have gained additionally
if she had known the true ? ? . We provide backgrounds and formal definitions in Section 2.
A linear case of the problem above (?(z) = z) is called the (stochastic) linear bandit problem. Since
the first formulation of the linear bandits [7], there has been a flurry of studies on the problem [11,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
34, 1, 9, 5]. In an effort to generalize the restrictive linear rewards, Filippi et al. [15] propose the
GLB problem and provide a low-regret algorithm, whose Thompson sampling version appears later
in Abeille & Lazaric [3]. Li et al. [27] evaluates GLBs via extensive experiments where GLBs exhibit
lower regrets than linear bandits for 0/1 rewards. Li et al. [28] achieves a smaller regret bound when
the arm set Xt is finite, though with an impractical algorithm.
However, we claim that all existing GLB algorithms [15, 28] suffer from two scalability issues that
limit their practical use: (i) under a large time horizon and (ii) under a large number N of arms.
First, existing GLBs require storing all the arms and rewards appeared so far, {(xs , ys )}ts=1 , so
the space complexity grows linearly with t. Furthermore, they have to solve a batch optimization
problem for the maximum likelihood estimation (MLE) at each time step t whose per-time-step time
complexity grows at least linearly with t. While Zhang et al. [41] provide a solution whose space
and time complexity do not grow over time, they consider a specific 0/1 reward with the logistic link
function, and a generic solution for GLBs is not provided.
Second, existing GLBs have linear time complexities in N . This is impractical when N is very large,
which is not uncommon in applications of GLBs such as online advertisements, recommendation
systems, and interactive retrieval of images or documents [26, 27, 40, 21, 25] where arms are items in
a very large database. Furthermore, the interactive nature of these systems requires prompt responses
as users do not want to wait. This implies that the typical linear time in N is not tenable. Towards a
sublinear time in N , locality sensitive hashings [18] or its extensions [35, 36, 30] are good candidates
as they have been successful in fast similarity search and other machine learning problems like active
learning [22], where the search time scales with N ? for some ? < 1 (? is usually optimized and
often ranges from 0.4 to 0.8 depending on the target search accuracy). Leveraging hashing in GLBs,
however, relies critically on the objective function used for arm selections. The function must take a
form that is readily optimized using existing hashing algorithms.1 For example, algorithms whose
objective function (a function of each arm x ? Xt ) can be written as a distance or inner product
between x and a query q are hash-amenable as there exist hashing methods for such functions.
To be scalable to a large time horizon, we propose a new algorithmic framework called Generalized
Linear Online-to-confidence-set Conversion (GLOC) that takes in an online learning (OL) algorithm
with a low ?OL? regret bound and turns it into a GLB algorithm with a low ?GLB? regret bound. The
key tool is a novel generalization of the online-to-confidence-set conversion technique used in [2]
(also similar to [14, 10, 16, 41]). This allows us to construct a confidence set for ? ? , which is then
used to choose an arm xt according to the well-known optimism in the face of uncertainty principle.
By relying on an online learner, GLOC inherently performs online computations and is thus free from
the scalability issues in large time steps. While any online learner equipped with a low OL regret
bound can be used, we choose the online Newton step (ONS) algorithm and prove a tight OL regret
bound, which results in a practical GLB algorithm with almost the same regret bound as existing
inefficient GLB algorithms. We present our proposed algorithms and their regret bounds in Section 3.
For large number N of arms, our proposed algorithm
Algorithm
Regret
Hash-amenable
?
GLOC is not hash-amenable, to our knowledge, due
?
GLOC
O(d
T
)
7
?
to its nonlinear criterion for arm selection. As the first
? 3/2 T )
GLOC-TS O(d
3
?
attempt, we derive a Thompson sampling [5, 3] exten? 5/4 T )
QGLOC
O(d
3
sion of GLOC (GLOC-TS), which is hash-amenable
due to its linear criterion. However, its regret bound Table 1: Comparison of GLBs algorithms for
scales with d3/2 for d-dimensional arm sets, which d-dimensional arm sets T is the time horizon.
is far from d of GLOC. Towards closing this gap, we QGLOC achieves the smallest regret among
propose a new algorithm Quadratic GLOC (QGLOC) hash-amenable algorithms.
with a regret bound that scales with d5/4 . We summarize the comparison of our proposed GLB
algorithms in Table 1. In Section 4, we present GLOC-TS, QGLOC, and their regret bound.
Note that, while hashing achieves a time complexity sublinear in N , there is a nontrivial overhead
of computing the projections to determine the hash keys. As an extra contribution, we reduce this
overhead by proposing a new sampling-based approximate inner product method. Our proposed
sampling method has smaller variance than the state-of-the-art sampling method proposed by [22, 24]
when the vectors are normally distributed, which fits our setting where projection vectors are indeed
normally distributed. Moreover, our method results in thinner tails in the distribution of estimation
1
Without this designation, no currently known bandit algorithm achieves a sublinear time complexity in N .
2
error than the existing method, which implies a better concentration. We elaborate more on reducing
the computational complexity of QOFUL in Section 5.
2
Preliminaries
We review relevant backgrounds here. A refers to a GLB algorithm, and B refers to an online
learning algorithm. Let Bd (S) be the d-dimensional Euclidean ball of radius S, which
? overloads
the notation B. Let A?i be the i-th column vector of a matrix A. Define ||x||A := x> Ax and
2
vec(A) := [A?1 ; A?2 ; ? ? ? ; A?d ] ? Rd Given a function f : R ? R, we denote by f 0 and f 00 its
first and second derivative, respectively. We define [N ] := {1, 2, . . . , N }.
Generalized Linear Model (GLM) Consider modeling the reward y as one-dimensional exponential family such as Bernoulli or Poisson. When the feature vector x is believed to correlate with
y, one popular modeling assumption is the generalized linear model (GLM) that turns the natural
parameter of an exponential family model into x> ? ? where ? ? is a parameter
[29]:
yz
?
m(z)
+ h(y, ? ) ,
(1)
P(y | z = x> ? ? ) = exp
g(? )
where ? ? R+ is a known scale parameter and m, g, and h are normalizers. It is known that
m0 (z) = E[y | z] =: ?(z) and m00 (z) = Var(y | z). We call ?(z) the inverse link function.
Throughout, we assume that the exponential family being used in a GLM has a minimal representation,
which ensures that m(z) is strictly convex [38, Prop. 3.1]. Then, the negative log likelihood (NLL)
`(z, y) := ?yz + m(z) of a GLM is strictly convex. We refer to such GLMs as the canonical GLM.
In the case of Bernoulli rewards y ? {0, 1}, m(z) = log(1 + exp(z)), ?(z) = (1 + exp(?z))?1 ,
?
0
and the NLL can be written as the logistic loss: log(1 + exp(?y 0 (x>
t ? ))), where y = 2y ? 1.
Generalized Linear Bandits (GLB) Recall that xt is the arm chosen at time t by an algorithm.
We assume that the arm set Xt can be of an infinite cardinality, although we focus on finite arm sets in
hashing part of the paper (Section 4). One can write down the reward model (1) in a different form:
?
(2)
yt = ?(x>
t ? ) + ?t ,
t?1
where ?t is conditionally R-sub-Gaussian given xt and {(xs , ?s )}s=1 . For example, Bernoulli
?
> ?
> ?
reward model has ?t as 1 ? ?(x>
t ? ) w.p. ?(xt ? ) and ??(xt ? ) otherwise. Assume that
?
||? ||2 ? S, wherepS is known.
? One can show that the sub-Gaussian scale R is determined by ?:
0
R = supz?(?S,S) ? (z) ? L, where L is the Lipschitz constant of ?. Throughout, we assume
that each arm has `2 -norm at most 1: ||x||2 ? 1, ?x ? Xt , ?t. Let xt,? := maxx?Xt x> ? ? . The
performance of a GLB algorithm A is analyzed by the expected cumulative regret (or simply regret):
PT
> ?
A > ?
A
RegretA
T :=
t=1 ?(xt,? ? ) ? ?((xt ) ? ), where xt makes the dependence on A explicit.
We remark that our results in this paper hold true for a strictly larger family of distributions than the
canonical GLM, which we call the non-canonical GLM and explain below. The condition is that the
reward model follows (2) where the R is now independent from ? that satisfies the following:
Assumption 1. ? is L-Lipschitz on [?S, S] and continuously differentiable on (?S, S). Furthermore,
inf z?(?S,S) ?0 (z) = ? for some finite ? > 0 (thus ? is strictly increasing).
Define ?0 (z) at ?S as their limits. Under Assumption 1, m is defined to be an integral of ?. Then,
one can show that m is ?-strongly convex on B1 (S). An example of the non-canonical GLM is the
probit model for 0/1 reward where ? is the Gaussian CDF, which is popular and competitive to the
Bernoulli GLM as evaluated by Li et al. [27]. Note that canonical GLMs satisfy Assumption 1.
3
Generalized Linear Bandits with Online Computation
We describe and analyze a new GLB algorithm called Generalized Linear Online-to-confidence-set
Conversion (GLOC) that performs online computations, unlike existing GLB algorithms.
GLOC employs the optimism in the face of uncertainty principle, which dates back to [7]. That is, we
maintain a confidence set Ct (defined below) that traps the true parameter ? ? with high probability
(w.h.p.) and choose the arm with the largest feasible reward given Ct?1 as a constraint:
? t ) := arg
(xt , ?
max
hx, ?i
(3)
x?Xt ,??Ct?1
The main difference between GLOC and existing GLBs is in the computation of the Ct ?s. Prior
methods involve ?batch" computations that involve all past observations, and so scale poorly with
3
t. In contrast, GLOC takes in an online learner B, and uses B as a co-routine instead of relying on
a batch procedure to construct a confidence set. Specifically, at each time t GLOC feeds the loss
function `t (?) := `(x>
t ?, yt ) into the learner B which then outputs its parameter prediction ? t . Let
t?d
Xt ? R
be the design matrix consisting of x1 , . . . , xt . Define Vt := ?I + X>
t Xt , where ?
bt := V?1 X> zt be the ridge
is the ridge parameter. Let zt := x>
?
and
z
:=
[z
;
?
?
?
;
z
].
Let
?
t
t
1
t
t
t
t
regression estimator taking zt as responses. Theorem 1 below is the key result for constructing our
confidence set Ct , which is a function of the parameter predictions {? s }ts=1 and the online (OL)
regret bound Bt of the learner B. All the proofs are in the supplementary material (SM).
Theorem 1. (Generalized Linear Online-to-Confidence-Set Conversion) Suppose we feed loss functions {`s (?)}ts=1 into online learner B. Let ? s be the parameter predicted at time step s by B.
Assume that B has an OL regret bound
Pt Bt : ?? ? Bd (S), ?t ? 1,
(4)
`s (? s ) ? `s (?) ? Bt .
qs=1
2
4
2
Let ?(Bt ) := 1 + ?4 Bt + 8R
1 + ?2 Bt + ?4R
4 ? 2 ). Then, with probability (w.p.) at least 1 ? ?,
?2 log( ?
?
2
bt || ? ?(Bt ) + ?S 2 ? ||zt ||2 ? ?
b> X> zt =: ?t .
?t ? 1, ||? ? ?
(5)
2
t
t
Vt
Note that the center of the ellipsoid is the ridge regression estimator on the predicted natural
parameters zs = x>
s ? s rather than the rewards. Theorem 1 motivates the following confidence set:
bt ||2 ? ?t }
Ct := {? ? Rd : ||? ? ?
(6)
Vt
?
which traps ? for all t ? 1, w.p. at least 1 ? ?. See Algorithm 1 for pseudocode. One way to solve
the optimization problem (3) is to define the function ?(x) := max??Ct?1 x> ?, and then use the
Lagrangian method to write:
p
bt?1 + ?t?1 ||x|| ?1 .
xGLOC
:= arg max x> ?
(7)
t
Vt?1
x?X
t
We prove the regret bound of GLOC in the following theorem.
Theorem 2. Let {? t } be a nondecreasing sequence
such that ? t ??t . Then, w.p. at least 1 ? ?,
q
RegretGLOC
= O L ? T dT log T
T
Although any low-regret online learner can be
Algorithm 1 GLOC
combined with GLOC, one would like to ensure
1: Input: R > 0, ? ? (0, 1), S > 0, ? > 0, ? > 0,
that ? T is O(polylog(T )) in which
? case the total
an online learner B with known regret bounds
?
regret can be bounded by O( T ). This means
{Bt }t?1 .
that we must use online learners whose OL regret 2: Set V = ?I.
0
grows logarithmically in T such as [20, 31]. In 3: for t = 1, 2, . . . do
this work, we consider the online Newton step 4: Compute xt by solving (3).
(ONS) algorithm [20].
5:
Pull xt and then observe yt .
6:
Receive ? from B.
5:
Output ? t .
t
Online Newton Step (ONS) for Generalized 7: Feed into B
the loss `t (?) = `(x>
t ?, yt ).
Linear Models Note that ONS requires the loss 8: Update V = V
>
>
t
t?1 + xt xt and zt = xt ? t
functions to be ?-exp-concave. One can show
?1 >
b
9:
Compute ? t = Vt Xt zt and ?t as in (5).
that `t (?) is ?-exp-concave [20, Sec. 2.2]. Then, 10: Define C as
in (6).
t
GLOC can use ONS and its OL regret bound to 11: end for
solve the GLB problem. However, motivated by
the fact?
that the OL regret bound Bt appears in the Algorithm 2 ONS-GLM
radius ?t of the confidence set while a tighter 1: Input: ? > 0, > 0, S > 0.
confidence set tends to reduce the bandit regret 2: A0 = I.
in practice, we derive a tight data-dependent OL 3: Set ?1 ? Bd (S) arbitrarily.
4: for t = 1, 2, 3, . . . do
regret bound tailored to GLMs.
We present our version of ONS for GLMs (ONS- 6: Observe xt and yt .
GLM) in Algorithm 2. `0 (z, y) is the first deriva- 7: Incur loss `(x>
t ? t , yt ) .
tive w.r.t. z and the parameter is for inverting 8: At = At?1 + xt x>
t
matrices conveniently (usually = 1 or 0.1). The
`0 (x>
t ? t ,yt )
9:
? 0t+1 = ? t ?
A?1
t xt
?
only difference from the original ONS [20] is that
10:
? t+1 = arg min??Bd (S) ||? ? ? 0t+1 ||2At
we rely on the strong convexity of m(z) instead
11: end for
of the ?-exp-concavity of the loss thanks to the
2
GLM structure. Theorem 3 states that we achieve the desired polylogarithmic regret in T .
2
A similar change to ONS has been applied in [16, 41].
4
Theorem 3. Define gs := `0 (x>
s ? s , ys ). The regret of ONS-GLM satisfies, for any > 0 and t ? 1,
Pt
Pt
?
1
2
2
2
ONS
,
s=1 `s (? s ) ? `s (? ) ? 2?
s=1 gs ||xs ||A?1 + 2?S =: Bt
s
2
2
where BtONS = O( L +R? log(t) d log t), ?t
2
?2
R
BtONS = O( L +
d log t).
?
? w.p. 1,
? 1 w.h.p. If maxs?1 |?s | is bounded by R
We emphasize that the OL regret bound is data-dependent. A confidence set constructed by combining
Theorem 1 and Theorem 3 directly implies the following regret bound of GLOC with ONS-GLM.
Corollary 1. Define ?tONS by replacing Bt with BtONS in (5). With probability at least 1 ? 2?,
n
o
bt ||2 ? ? ONS .
?t ? 1, ? ? ? C ONS := ? ? Rd : ||? ? ?
(8)
t
Vt
t
Corollary
2. Run GLOC with
Then, w.p. at least 1 ? 2?, ?T ? 1, RegretGLOC
=
T
?
L(L+R)
3/2
GLOC
?
?
?
=
O
d T log (T ) where O ignores log log(t). If |?t | is bounded by R, RegretT
? ? ?
? L(L+R) d T log(T ) .
O
?
CtONS .
We make regret bound comparisons ignoring log log T factors. For generic arm sets, our dependence
on d is optimal for linear rewards [34]. For the Bernoulli GLM, our regret has the same order as Zhang
et al. [41]. One can show that the regret of Filippi et al. [15] has the same order as ours if we use their
assumption that
by Rmax . For unbounded noise, Li et al. [28] have regret
? the reward yt is bounded
?
O((LR/?)d T log T ), which is log T factor smaller than ours and has LR in place of L(L + R).
While L(L + R) could be an artifact of our analysis, the gap
? is not too large for canonical GLMs.
Let L be the smallest Lipschitz constant of ?. Then, R = L. If L ? 1, R satisfies R > L, and so
L(L + R) = O(LR). If L > 1, then L(L + R) = O(L2 ), which is larger than LR = O(L3/2 ). For
the Gaussian GLM with known ?
variance ? 2 , L = R = 1.3 For finite arm sets, SupCB-GLM of Li
? dT log N ) that has a better scaling with d but is not a practical
et al. [28] achieves regret of O(
algorithm as it wastes a large number of arm pulls. Finally, we remark that none of the existing GLB
algorithms are scalable to large T . Zhang et al. [41] is scalable to large T , but is restricted to the
Bernoulli GLM; e.g., theirs does not allow the probit model (non-canonical GLM) that is popular and
shown to be competitive to the Bernoulli GLM [27].
Discussion The trick of obtaining a confidence set from an online learner appeared first in [13, 14]
for the linear model, and then was used in [10, 16, 41]. GLOC is slightly different from these studies
and rather close to Abbasi-Yadkori et al. [2] in that the confidence set is a function of a known regret
bound. This generality frees us from re-deriving a confidence set for every online learner. Our result
is essentially a nontrivial extension of Abbasi-Yadkori et al. [2] to GLMs.
One might have notice that Ct does not use ? t+1 that is available before pulling xt+1 and has the
most up-to-date information. This is inherent to GLOC as it relies on the OL regret bound directly.
One can modify the proof of ONS-GLM to have a tighter confidence set Ct that uses ? t+1 as we
show in SM Section E. However, this is now specific to ONS-GLM, which looses generality.
4
Hash-Amenable Generalized Linear Bandits
We now turn to a setting where the arm set is finite but very large. For example, imagine an interactive
retrieval scenario [33, 25, 6] where a user is shown K images (e.g., shoes) at a time and provides
relevance feedback (e.g., yes/no or 5-star rating) on each image, which is repeated until the user is
satisfied. In this paper, we focus on showing one image (i.e., arm) at a time.4 Most existing algorithms
require maximizing an objective function (e.g., (7)), the complexity of which scales linearly with the
number N of arms. This can easily become prohibitive for large numbers of images. Furthermore,
the system has to perform real-time computations to promptly choose which image to show the user
in the next round. Thus, it is critical for a practical system to have a time complexity sublinear in N .
One naive approach is to select a subset of arms ahead of time, such as volumetric spanners [19].
However, this is specialized for an efficient exploration only and can rule out a large number of
good arms. Another option is to use hashing methods. Locality-sensitive hashing and Maximum
3
The reason why R is not ? here is that the sufficient statistic of the GLM is y/?, which is equivalent to
dealing with the normalized reward. Then, ? appears as a factor in the regret bound.
4
One image at a time is a simplification of the practical setting. One can extend it to showing multiple
images at a time, which is a special case of the combinatorial bandits of Qin et al. [32].
5
Inner Product Search (MIPS) are effective and well-understood tools but can only be used when the
objective function is a distance or an inner product computation; (7) cannot be written in this form.
In this section, we consider alternatives to GLOC which are compatible with hashing.
Thompson Sampling We present a Thompson sampling (TS) version of GLOC called GLOC-TS
bt?1 , ?t?1 V?1 ). TS is known to
that chooses an arm xt = arg maxx?Xt x> ?? t where ?? t ? N (?
t?1
perform well in practice [8] and can solve the polytope arm set case in polynomial time5 whereas
algorithms that solve an objective function like (3) (e.g., [1]) cannot since they have to solve an
NP-hard problem [5]. We present the regret bound of GLOC-TS below. Due to space constraints, we
present the pseudocode and the full version of the result in SM.
bt?1 , ? ONS V?1 ), RegretGLOC-TS =
Theorem
4. (Informal) If we run GLOC-TS with ?? t ? N (?
t?1 t?1
?
? T
?
L(L+R) 3/2
3/2
?
?
?
O
T log (T ) w.h.p. If ?t is bounded by R, then O L(L+R) d3/2 T log(T ) .
d
?
?
3/2
Notice that the regret now scales with d
as expected from the analysis of linear TS [4], which
is higher than scaling with d of GLOC. This is concerning in the interactive retrieval or product
recommendation scenario since the relevance of the shown items is harmed, which makes us wonder
if one can improve the regret without loosing the hash-amenability.
Quadratic GLOC We now propose a new hash-amenable algorithm called Quadratic GLOC
(QGLOC). Recall that GLOC chooses the arm xGLOC by (7). Define r = minx?X ||x||2 and
mt?1 :=
min
||x||V?1 ,
(9)
x:||x||2 ?[r,1]
t?1
?1
which is r times the square root of the smallest eigenvalue of Vt?1 . It is easy to see that mt?1 ?
?
||x||V?1 for all x ? X and that mt?1 ? r/ t + ? using the definition of Vt?1 . There is an
t?1
alternative way to define mt?1 without relying on r, which we present in SM.
Let c0 > 0 be the exploration-exploitation tradeoff parameter (elaborated upon later). At time t,
QGLOC chooses the arm
1/4
bt?1 , xi + ?t?1 ||x||2 ?1 = arg max hqt , ?(x)i ,
(10)
xQGLOC
:=
arg
max
h
?
t
Vt?1
x?Xt
x?Xt
4c0 mt?1
?
1/4
?1
2
bt?1 ; vec( t?1 V )] ? Rd+d and ?(x) := [x; vec(xx> )]. The key property of
where qt = [?
t?1
4c0 mt?1
QGLOC is that the objective function is now quadratic in x, thus the name Quadratic GLOC, and
can be written as an inner product. Thus, QGLOC is hash-amenable. We present the regret bound of
QGLOC (10) in Theorem 5. The key step of the proof is that the QGLOC objective function (10)
plus c0 ? 3/4 mt?1 is a tight upper bound of the GLOC objective function (7).
at least 1 ? 2?, RegretQGLOC
=
Theorem
5. Run QGLOC with CtONS . Then, w.p.
T
?
2
1 L+R 1/2
L+R 3/2
L+R ?1/2
5/4
O
+
c
Ld
T
log
(T
)
.
By
setting
c
=
,
the
regret
0
0
c0
?
?
?
?
L(L+R) 5/4
2
bound is O( ? d
T log (T )).
Note that one can have a better dependence on log T when ?t is bounded (available in the proof).
The regret bound of QGLOC is a d1/4 factor improvement over that of GLOC-TS; see Table 1.
Furthermore, in (10) c0 is a free parameter that adjusts the balance between the exploitation (the first
term) and exploration (the second term). Interestingly, the regret guarantee does not break down when
adjusting c0 in Theorem 5. Such a characteristic is not found in existing algorithms but is attractive
to practitioners, which we elaborate in SM.
Maximum Inner Product Search (MIPS) Hashing While MIPS hashing algorithms such as [35,
36, 30] can solve (10) in time sublinear in N , these necessarily introduce an approximation error.
Ideally, one would like the following guarantee on the error with probability at least 1 ? ?H :
0
? ? X is called cH -MIPS w.r.t. a given
Definition 1. Let X ? Rd satisfy |X | < ?. A data point x
? i ? cH ? maxx?X hq, xi for some cH < 1. An algorithm is called cH -MIPS
query q if it satisfies hq, x
0
if, given a query q ? Rd , it retrieves x ? X that is cH -MIPS w.r.t. q.
Unfortunately, existing MIPS algorithms do not directly offer such a guarantee, and one must build a
series of hashing schemes with varying hashing parameters like Har-Peled et al. [18]. Under the fixed
budget setting T , we elaborate our construction that is simpler than [18] in SM.
5
ConfidenceBall1 algorithm of Dani et al. [11] can solve the problem in polynomial time as well.
6
Time and Space Complexity Our construction involves saving Gaussian projection vectors that
are used for determining hash keys and saving the buckets containing pointers to the actual arm
vectors. The time complexity for retrieving a cH -MIPS solution involves determining hash keys
and evaluating inner products with the arms in the retrieved buckets. Let ?? < 1 be an opti0
mized value
for
the hashing
?(see [35] for
detail). The time complexity for d -dimensional veclog(dT )
tors is O log log(c?1 ) N ? log(N )d0 , and the space complexity (except the original data) is
H
log(dT )
??
0
O log(c
. While the time and space complexity grows with the time horiN
(N
+
log(N
)d
)
?1
)
H
zon T , the dependence is mild; log log(T ) and log(T ), respectively. QGLOC uses d0 = d + d2 ,6 and
GLOC-TS uses d0 = d0 . While both achieve a time complexity sublinear in N , the time complexity
of GLOC-TS scales with d that is better than scaling with d2 of QGLOC. However, GLOC-TS has a
d1/4 -factor worse regret bound than QGLOC.
Discussion While it is reasonable to incur small errors in solving the arm selection criteria like (10)
and sacrifice some regret in practice, the regret bounds of QGLOC and GLOC-TS do not hold
anymore. Though not the focus of our paper, we prove a regret bound under the presence of the
hashing error in the fixed budget setting for QGLOC; see SM. Although the result therein has an
inefficient space complexity that is linear in T , it provides the first low regret bound with time
sublinear in N , to our knowledge.
5
Approximate Inner Product Computations with L1 Sampling
While hashing allows a time complexity sublinear in N , it performs an additional computation for determining the hash keys. Consider a hashing with U tables and length-k hash
keys. Given a query q and projection vectors
a(1) , . . . , a(U k) , the hashing computes q> a(i) ,
?i ? [U k] to determine the hash key of q. To
reduce such an overhead, approximate inner
product methods like [22, 24] are attractive
since hash keys are determined by discretizing
the inner products; small inner product errors
often do not alter the hash keys.
1
5
0.9
0
0.8
0.7
-5
100
L2
L1
(a)
101
102
103
d
(b)
Figure 1: (a) A box plot of estimators. L1 and L2
have the same variance, but L2 has thicker tails. (b)
The frequency of L1 inducing smaller variance than
L2 in 1000 trials. After 100 dimensions, L1 mostly
has smaller variance than L2.
In this section, we propose an improved approximate inner product method called L1 sampling which
we claim is more accurate than the sampling proposed by Jain et al. [22], which we call L2 sampling.
Consider an inner product q> a. The main idea is to construct an unbiased estimate of q> a. That is,
let p ? Rd be a probability vector. Let
i.i.d.
(11)
ik ? Multinomial(p) and Gk := qik aik /pik , k ? [m] .
Pm
1
>
>
It is easy to see that EGk = q a. By taking m k=1 Gk as an estimate of q a, the time complexity
is now O(mU k) rather than O(d0 U k). The key is to choose the right p. L2 sampling uses p(L2) :=
[qi2 /||q||22 ]i . Departing from L2, we propose p(L1) that we call L1 sampling and define as follows:
p(L1) := [|q1 |; ? ? ? ; |qd0 |]/||q||1 .
(12)
We compare L1 with L2 in two different point of view. Due to space constraints, we summarize the
key ideas and defer the details to SM.
The first is on their concentration of measure. Lemma 1 below shows an error bound of L1 whose
failure probability decays exponentially in m. This is in contrast to decaying polynomially of L2 [22],
which is inferior.7
Lemma 1. Define Gk as in (11) with p = p(L1) . Then, given a target error > 0,
Pm
m2
P 1
Gk ? q> a ? ? 2 exp ?
(13)
2
2
m
k=1
2||q||1 ||a||max
To illustrate such a difference, we fix q and a in 1000 dimension and apply L2 and L1 sampling 20K
times each with m = 5 where we scale down the L2 distribution so its variance matches that of L1.
Note that this does not mean we need to store vec(xx> ) since an inner product with it is structured.
In fact, one can show a bound for L2 that fails with exponentially-decaying probability. However, the bound
introduces a constant that can be arbitrarily large, which makes the tails thick. We provide details on this in SM.
6
7
7
Algorithm
QGLOC
QGLOC-Hash
GLOC-TS
GLOC-TS-Hash
(a)
(b)
Cum. Regret
266.6 (?19.7)
285.0 (?30.3)
277.0 (?36.1)
289.1 (?28.1)
(c)
Figure 2: Cumulative regrets with confidence intervals under the (a) logit and (b) probit model. (c)
Cumulative regrets with confidence intervals of hash-amenable algorithms.
Figure 1(a) shows that L2 has thicker tails than L1. Note this is not a pathological case but a typical
case for Gaussian q and a. This confirms our claim that L1 is safer than L2.
Another point of comparison is the variance of L2 and L1. We show that the variance of L1 may or
may not be larger than L2 in SM; there is no absolute winner. However, if q and a follow a Gaussian
distribution, then L1 induces smaller variances than L2 for large enough d; see Lemma 9 in SM.
Figure 1(b) confirms such a result. The actual gap between the variance of L2 and L1 is also nontrivial
under the Gaussian assumption. For instance, with d = 200, the average variance of Gk induced by
L2 is 0.99 whereas that induced by L1 is 0.63 on average. Although a stochastic assumption on the
vectors being inner-producted is often unrealistic, in our work we deal with projection vectors a that
are truly normally distributed.
6
Experiments
We now show our experiment results comparing GLB algorithms and hash-amenable algorithms.
GLB Algorithms We compare GLOC with two different algorithms: UCB-GLM [28] and Online
Learning for Logit Model (OL2M) [41].8 For each trial, we draw ? ? ? Rd and N arms (X ) uniformly
at random from the unit sphere. We set d = 10 and Xt = X , ?t ? 1. Note it is a common practice to
scale the confidence set radius for bandits [8, 27]. Following Zhang et al. [41], for OL2M we set the
squared radius ?t = c log(det(Zt )/det(Z1 )), where c is a tuning parameter. For UCB-GLM, we set
?
Pt
the radius as ? = cd log t. For GLOC, we replace ?tONS with c s=1 gs2 ||xs ||2A?1 . While parameter
s
tuning in practice is nontrivial, for the sake of comparison we tune c ? {101 , 100.5 , . . . , 10?3 } and
report the best one. We perform 40 trials up to time T = 3000 for each method and compute
confidence bounds on the regret.
We consider two GLM rewards: (i) the logit model (the Bernoulli GLM) and (ii) the probit model
(non-canonical GLM) for 0/1 rewards that sets ? as the probit function. Since OL2M is for the
logit model only, we expect to see the consequences of model mismatch in the probit setting. For
GLOC and UCB-GLM, we specify the correct reward model. We plot the cumulative regret under the
logit model in Figure 2(a). All three methods perform similarly, and we do not find any statistically
significant difference based on paired t test. The result for the probit model in Figure 2(b) shows that
OL2M indeed has higher regret than both GLOC and UCB-GLM due to the model mismatch in the
probit setting. Specifically, we verify that at t = 3000 the difference between the regret of UCB-GLM
and OL2M is statistically significant. Furthermore, OL2M exhibits a significantly higher variance in
the regret, which is unattractive in practice. This shows the importance of being generalizable to any
GLM reward. Note we observe a big increase in running time for UCB-GLM compared to OL2M
and GLOC.
Hash-Amenable GLBs To compare hash-amenable GLBs, we use the logit model as above but now
with N =100,000 and T =5000. We run QGLOC, QGLOC with hashing (QGLOC-Hash), GLOC-TS,
and GLOC-TS with hashing (GLOC-TS-Hash), where we use the hashing to compute the objective
function (e.g., (10)) on just 1% of the data points and save a significant amount of computation.
Details on our hashing implementation is found in SM. Figure 2(c) summarizes the result. We observe
that QGLOC-Hash and GLOC-TS-Hash increase regret from QGLOC and GLOC-TS, respectively,
but only moderately, which shows the efficacy of hashing.
7
Future Work
In this paper, we have proposed scalable algorithms for the GLB problem: (i) for large time horizon
T and (ii) for large number N of arms. There exists a number of interesting future work. First,
8
We have chosen UCB-GLM over GLM-UCB of Filippi et al. [15] as UCB-GLM has a lower regret bound.
8
we would like to extend the GLM rewards to the single index models [23] so one does not need to
know the function ? ahead of time under mild assumptions. Second, closing the regret bound gap
between QGLOC and GLOC without loosing
? hash-amenability would be interesting: i.e., develop
a hash-amenable GLB algorithm with O(d T ) regret. In this direction, a first attempt could be to
design a hashing scheme that can directly solve (7) approximately.
Acknowledgments This work was partially supported by the NSF grant IIS-1447449 and the
MURI grant 2015-05174-04. The authors thank Yasin Abbasi-Yadkori and Anshumali Shrivastava
for providing constructive feedback and Xin Hunt for her contribution at the initial stage.
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6,208 | 6,616 | Probabilistic Models for Integration Error in the
Assessment of Functional Cardiac Models
Chris. J. Oates1,5 , Steven Niederer2 , Angela Lee2 , Fran?ois-Xavier Briol3 , Mark Girolami4,5
1
Newcastle University, 2 King?s College London, 3 University of Warwick,
4
Imperial College London, 5 Alan Turing Institute
Abstract
This paper studies the numerical computation of integrals, representing estimates
or predictions, over the output f (x) of a computational model with respect to
a distribution p(dx) over uncertain inputs x to the model. For the functional
cardiac models that motivate this work, neither f nor p possess a closed-form
expression and evaluation of either requires ? 100 CPU hours, precluding standard
numerical integration methods. Our proposal is to treat integration as an estimation
problem, with a joint model for both the a priori unknown function f and the a
priori unknown distribution p. The result is a posterior distribution over the integral
that explicitly accounts for dual sources of numerical approximation error due to a
severely limited computational budget. This construction is applied to account, in
a statistically principled manner, for the impact of numerical errors that (at present)
are confounding factors in functional cardiac model assessment.
1
Motivation: Predictive Assessment of Computer Models
This paper considers the problem of assessment for computer models [7], motivated by an urgent
need to assess the performance of sophisticated functional cardiac models [25]. In concrete terms,
the problem that we consider can be expressed as the numerical approximation of integrals
Z
p(f ) =
f (x)p(dx),
(1)
where f (x) denotes a functional of the output from a computer model and x denotes unknown inputs
(or ?parameters?) of the model. The term p(x) denotes a posterior distribution over model inputs.
Although not our focus in this paper, we note that p(x) is defined based on a prior ?0 (x) over these
inputs and training data y assumed to follow the computer model ?(y|x) itself. The integral p(f ), in
our context, represents a posterior prediction of actual cardiac behaviour. The computational model
can be assessed through comparison of these predictions to test data generated from an experiment.
The challenging nature of cardiac models ? and indeed computer models in general ? is such that a
closed-form for both f (x) and p(dx) is precluded [23]. Instead, it is typical to be provided with a
finite collection of samples {xi }ni=1 obtained from p(dx) through Monte Carlo (or related) methods
[32]. The integrand f (x) is then evaluated at these n input configurations, to obtain {f (xi )}ni=1 .
Limited computational budgets necessitate that the number n is small and, in such situations, the error
of an estimator for the integral p(f ) based on the data {(xi , f (xi ))}ni=1 is subject to strict informationtheoretic lower bounds [26]. The practical consequence is that an unknown (non-negligible) numerical
error is introduced in the numerical approximation of p(f ), unrelated to the performance of the model.
If this numerical error is ignored, it will constitute a confounding factor in the assessment of predictive
performance for the computer model. It is therefore unclear how a fair model assessment can proceed.
This motivates an attempt to understand the extent of numerical error in any estimate of p(f ). This is
non-trivial; for example, the error distribution of the arithmetic mean n1 ?ni=1 f (xi ) depends on the
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
unknown f and p, and attempts to estimate this distribution solely from data, e.g. via a bootstrap or a
central limit approximation, cannot succeed in general when the number of samples n is small [27].
Our first contribution, in this paper, is to argue that approximation of p(f ) from samples {xi }ni=1
and function evaluations {f (xi )}ni=1 can be cast as an estimation task. Our second contribution is to
derive a posterior distribution over the unknown value p(f ) of the integral. This distribution provides
an interpretable quantification of the extent of numerical integration error that can be reasoned
with and propagated through subsequent model assessment. Our third contribution is to establish
theoretical properties of the proposed method. The method we present falls within the framework
of Probabilistic Numerics and our work can be seen as a contribution to this emerging area [16, 5].
In particular, the method proposed is reminiscent of Bayesian Quadrature (BQ) [9, 28, 29, 15]. In
BQ, a Gaussian prior measure is placed on the unknown function f and is updated to a posterior
when conditioned on the information {(xi , f (xi ))}ni=1 . This induces both a prior and a posterior
over the value of p(f ) as push-forward measures under the projection operator f 7? p(f ). Since
its introduction, several authors have related BQ to other methods such as the ?herding? approach
from machine learning [17, 3], random feature approximations used in kernel methods [1], classical
quadrature rules [33] and Quasi Monte Carlo (QMC) methods [4]. Most recently, [21] extended
theoretical results for BQ to misspecified prior models, and [22] who provided efficient matrix
algebraic methods for the implementation of BQ. However, as an important point of distinction,
notice that BQ pre-supposes p(dx) is known in closed-form - it does not apply in situations where
p(dx) is instead sampled. In this latter case p(dx) will be called an intractable distribution and, for
model assessment, this scenario is typical.
To extend BQ to intractable distributions, this paper proposes to use a Dirichlet process mixture
prior to estimate the unknown distribution p(dx) from Monte Carlo samples {xi }ni=1 [12]. It will be
demonstrated that this leads to a simple expression for the closed-form terms which are required to
implement the usual BQ. The overall method, called Dirichlet process mixture Bayesian quadrature
(DPMBQ), constructs a (univariate) distribution over the unknown integral p(f ) that can be exploited
to tease apart the intrinsic performance of a model from numerical integration error in model
assessment. Note that BQ was used to estimate marginal likelihood in e.g. [30]. The present problem
is distinct, in that we focus on predictive performance (of posterior expectations) rather than marginal
likelihood, and its solution demands a correspondingly different methodological development.
On the computational front, DPMBQ costs O(n3 ). However, this cost is de-coupled from the often
orders-of-magnitude larger costs involved in both evaluation of f (x) and p(dx), which form the main
computational bottleneck. Indeed, in the modern computational cardiac models that motivate this
research, the ? 100 CPU hour time required for a single simulation limits the number n of available
samples to ? 103 [25]. At this scale, numerical integration error cannot be neglected in model
assessment. This raises challenges when making assessments or comparisons between models, since
the intrinsic performance of models cannot be separated from numerical error that is introduced into
the assessment. Moreover, there is an urgent ethical imperative that the clinical translation of such
models is accompanied with a detailed quantification of the unknown numerical error component in
model assessment. Our contribution explicitly demonstrates how this might be achieved.
The remainder of the paper proceeds as follows: In Section 2.1 we first recall the usual BQ method,
then in Section 2.2 we present and analyse our novel DPMBQ method. Proofs of theoretical results
are contained in the electronic supplement. Empirical results are presented in Section 3 and the paper
concludes with a discussion in Section 4.
2
Probabilistic Models for Numerical Integration Error
Consider a domain ? ? Rd , together with a distribution p(dx) on ?. As in Eqn. 1, p(f ) will be used
to denote the integral of the argument f with respect to the distribution p(dx). All integrands are
assumed to be (measurable) functions f : ? ? R such that the integral p(f ) is well-defined. To
begin, we recall details for the BQ method when p(dx) is known in closed-form [9, 28]:
2.1
Probabilistic Integration for Tractable Distributions (BQ)
In standard BQ [9, 28], a Gaussian Process (GP) prior f ? GP(m, k) is assigned to the integrand f ,
with mean function m : ? ? R and covariance function k : ? ? ? ? R [see 31, for further details
2
on GPs]. The implied prior over the integral p(f ) is then the push-forward of the GP prior through
the projection f 7? p(f ):
p(f ) ? N(p(m), p ? p(k))
0
where p?p : ??? ? R is the measure formed by independent products of p(dx)
RR and 0p(dx ), so that
under our notational convention the so-called initial error p?p(k) is equal to
k(x, x )p(dx)p(dx0 ).
Next, the GP is conditioned on the information in {(xi , f (xi ))}ni=1 . The conditional GP takes a
conjugate form f |X, f (X) ? GP(mn , kn ), where we have written X = (x1 , . . . , xn ), f (X) =
(f (x1 ), . . . , f (xn ))> . Formulae for the mean function mn : ? ? R and covariance function
kn : ? ? ? ? R are standard can be found in [31, Eqns. 2.23, 2.24]. The BQ posterior over p(f ) is
the push forward of the GP posterior:
p(f ) | X, f (X) ? N(p(mn ), p ? p(kn ))
(2)
Formulae for p(mn ) and p ? p(kn ) were derived in [28]:
p(mn ) = f (X)> k(X, X)?1 ?(X)
(3)
p ? p(kn ) = p ? p(k) ? ?(X)> k(X, X)?1 ?(X)
(4)
where k(X, X) is the n ? n matrix with (i, j)th entry k(xi , xj ) and ?(X) is the n ? 1 vector with
ith entry ?(xi ) where the function ? is called the kernel mean or kernel embedding [see e.g. 35]:
Z
?(x) = k(x, x0 )p(dx0 )
(5)
Computation of the kernel mean and the initial error each requires that p(dx) is known in general.
The posterior in Eqn. 2 was studied in [4], where rates of posterior contraction were established
under further assumptions on the smoothness of the covariance function k and the smoothness of
the integrand. Note that the matrix inverse of k(X, X) incurs a (naive) computational cost of O(n3 );
however this cost is post-hoc and decoupled from (more expensive) computation that involves the
computer model. Sparse or approximate GP methods could also be used.
2.2
Probabilistic Integration for Intractable Distributions
The dependence of Eqns. 3 and 4 on both the kernel mean and the initial error means that BQ cannot
be used for intractable p(dx) in general. To address this we construct a second non-parametric model
for the unknown p(dx), presented next.
Dirichlet Process Mixture Model Consider
Z an infinite mixture model
p(dx) = ?(dx; ?)P (d?),
(6)
where ? : ? ? ? ? [0, ?) is such that ?(?; ?) is a distribution on ? with parameter ? ? ? and P is
a mixing distribution defined on ?. In this paper, each data point xi is modelled as an independent
draw from p(dx) and is associated with a latent variable ?i ? ? according to the generative process
of Eqn. 6. i.e. xi ? ?(?; ?i ). To limit scope, the extension to correlated xi is reserved for future
work.
The Dirichlet process (DP) is the natural conjugate prior for non-parametric discrete distributions
[12]. Here we endow P (d?) with a DP prior P ? DP(?, Pb ), where ? > 0 is a concentration
parameter and Pb (d?) is a base distribution over ?. The base distribution Pb coincides with the
prior expectation E[P (d?)] = Pb (d?), while ? determines the spread of the prior about Pb . The
DP is characterised by the property that, for any finite partition ? = ?1 ? ? ? ? ? ?m , it holds that
(P (?1 ), . . . , P (?m )) ? Dir(?Pb (?1 ), . . . , ?Pb (?m )) where P (S) denotes the measure of the set
S ? ?. For ? ? 0, the DP is supported on the set of atomic distributions, while for ? ? ?, the DP
converges to an atom on the base distribution. This overall approach is called a DP mixture (DPM)
model [13].
For a random variable Z, the notation [Z] will be used as shorthand to denote the density function
of Z. It will be helpful to note that for ?i ? P independent, writing ?1:n = (?1 , . . . , ?n ), standard
conjugate results for DPs lead to the conditional
n
?
1 X
P | ?1:n ? DP ? + n,
Pb +
??
?+n
? + n i=1 i
where ??i (d?) is an atomic distribution centred at the location ?i of the ith sample in ?1:n . In turn,
this induces a conditional [dp|?1:n ] for the unknown distribution p(dx) through Eqn. 6.
3
Kernel Means via Stick Breaking The stick breaking characterisation can be used to draw from
the conditional DP [34]. A generic draw from [P |?1:n ] can be characterised as
P (d?) =
?
X
wj ??j (d?),
j?1
Y
wj = ?j
(1 ? ?j 0 )
(7)
j 0 =1
j=1
where randomness enters through the ?j and ?j as follows:
n
iid
?j ?
?
1 X
Pb +
?? ,
?+n
? + n i=1 i
iid
?j ? Beta(1, ? + n)
In practice the sum in Eqn. 7 may be truncated at a large finite number of terms, N , with negligible
truncation error, since weights wj vanish at a geometric rate [18]. The truncated DP has been shown
to provide accurate approximation of integrals with respect to the original DP [19]. For a realisation
P (d?) from Eqn. 7, observe that the induced distribution p(dx) over ? is
p(dx) =
?
X
wj ?(dx; ?j ).
(8)
j=1
Thus we have an alternative characterisation of [p|?1:n ].
Our key insight is that one can take ? and k to be a conjugate pair, such that both the kernel mean
?(x) and the initial error p ? p(k) will be available in an explicit form for the distribution in Eqn. 8
[see Table 1 in 4, for a list of conjugate pairs]. For instance, in the one-dimensional case, consider
? = (?1 , ?2 ) and ?(dx; ?) = N(dx; ?1 , ?2 ) for some location and scale parameters ?1 and ?2 .
Then for the Gaussian kernel k(x, x0 ) = ? exp(?(x ? x0 )2 /2?2 ), the kernel mean becomes
?(x) =
?
X
j=1
??wj
(x ? ?j,1 )2
exp
?
2(?2 + ?j,2 )
(?2 + ?j,2 )1/2
(9)
and the initial variance can be expressed as
p ? p(k)
=
? X
?
X
(?j,1 ? ?j 0 ,1 )2
??wj wj 0
exp
?
.
2(?2 + ?j,2 + ?j 0 ,2 )
(?2 + ?j,2 + ?j 0 ,2 )1/2
j=1 j 0 =1
(10)
Similar calculations for the multi-dimensional case are straight-forward and provided in the Supplemental Information.
The Proposed Model To put this all together, let ? denote all hyper-parameters that (a) define the
GP prior mean and covariance function, denoted m? and k? below, and (b) define the DP prior, such
as ? and the base distribution Pb . It is assumed that ? ? ? for some specified set ?. The marginal
posterior distribution for p(f ) in the DPMBQ model is defined as
ZZ
[p(f ) | X, f (X)] =
[p(f ) | X, f (X), p, ?] [dp | X, ?] [d?].
(11)
The first term in the integral is BQ for a fixed distribution p(dx). The second term represents the
DPM model for the unknown p(dx), while the third term [d?] represents a hyper-prior distribution
over ? ? ?. The DPMBQ distribution in Eqn. 11 does not admit a closed-form expression. However,
it is straight-forward to sample from this distribution without recourse to f (x) or p(dx). In particular,
the second term can be accessed through the law of total probabilities:
Z
[dp | X, ?] =
[dp | ?1:n ] [?1:n | X, ?] d?1:n
where the first term [dp | ?1:n ] is the stick-breaking construction and the term [?1:n | X, ?] can be
targeted with a Gibbs sampler. Full details of the procedure we used to sample from Eqn. 11, which
is de-coupled from the much larger costs associated with the computer model, are provided in the
Supplemental Information.
4
Theoretical Analysis The analysis reported below restricts attention to a fixed hyper-parameter ?
and a one-dimensional state-space ? = R. The extension of theoretical results to multiple dimensions
was beyond the scope of this paper.
Our aim in this section is to establish when DPMBQ is ?consistent?. To be precise, a random
distribution Pn over an unknown parameter ? ? R, whose true value is ?0 , is called consistent for ?0
at a rate rn if, for all ? > 0, we have Pn [(??, ?0 ? ?) ? (?0 + ?, ?)] = OP (rn ). Below we denote
with f0 and p0 the respective true values of f and p; our aim is to estimate ?0 = p0 (f0 ). Denote with
H the reproducing kernel Hilbert space whose reproducing kernel is k and assume that the GP prior
mean m is an element of H. Our main theoretical result below establishes that the DPMBQ posterior
distribution in Eqn. 11, which is a random object due to the n independent draws xi ? p(dx), is
consistent:
Theorem. Let P0 denote the true mixing distribution. Suppose that:
1.
2.
3.
4.
5.
f belongs to H and k is bounded on ? ? ?.
?(dx; ?) = N(dx; ?1 , ?2 ).
P0 has compact support supp(P0 ) ? R ? (?, ?) for some fixed ?, ? ? (0, ?).
Pb has positive, continuous density on a rectangle R, s.t. supp(Pb ) ? R ? R ? [?, ?].
Pb ({(?1 , ?2 ) : |?1 | > t}) ? c exp(??|t|? ) for some ?, ? > 0 and ? t > 0.
Then the posterior Pn = [p(f ) | X, f0 (X)] is consistent for the true value p0 (f0 ) of the integral at
the rate n?1/4+ where the constant > 0 can be arbitrarily small.
The proof is provided in the Supplemental Information. Assumption (1) derives from results on
consistent BQ [4] and can be relaxed further with the results in [21] (not discussed here), while
assumptions (2-5) derive from previous work on consistent estimation with DPM priors [14]. For
the case of BQ when p(dx) is known and H a Sobolev space of order s > 1/2 on ? = [0, 1], the
corresponding posterior contraction rate is exp(?Cn2s? ) [4, Thm. 1]. Our work, while providing
only an upper bound on the convergence rate, suggests that there is an increase in the fundamental
complexity of estimation for p(dx) unknown compared to p(dx) known. Interestingly, the n?1/4+
rate is slower than the classical Bernstein-von Mises rate n?1/2 [36]. However, an out-of-hand
comparison between these two quantities is not straight forward, as the former involves the interaction
of two distinct non-parametric statistical models. It is known Bernstein-von Mises results can be
delicate for non-parametric problems [see, for example, the counter-examples in 10]. Rather, this
theoretical analysis guarantees consistent estimation in a regime that is non-standard.
3
Results
The remainder of the paper reports empirical results from application of DPMBQ to simulated data
and to computational cardiac models.
3.1
Simulation Experiments
To explore the empirical performance of DPMBQ, a series of detailed simulation experiments were
performed. For this purpose, a flexible test bed was constructed wherein the true distribution p0 was
a normal mixture model (able to approximate any continuous density) and the true integrand f0 was
a polynomial (able to approximate any continuous function). In this set-up it is possible to obtain
closed-form expressions for all integrals p0 (f0 ) and these served as a gold-standard benchmark.
To mimic the scenario of interest, a small number n of samples xi were drawn from p0 (dx) and
the integrand values f0 (xi ) were obtained. This information X, f0 (X) was provided to DPMBQ
and the output of DPMBQ, a distribution over p(f ), was compared against the actual value p0 (f0 )
of the integral. For all experiments in this paper the Gaussian kernel k defined in Sec. 2.2 was
used; the integrand f was normalised and the associated amplitude hyper-parameter ? = 1 fixed,
whereas the length-scale hyper-parameter ? was assigned a Gam(2, 1) hyper-prior. For the DPM, the
concentration parameter ? was assigned a Exp(1) hyper-prior. These choices allowed for adaptation
of DPMBQ to the smoothness of both f and p in accordance with the data presented to the method.
The base distribution Pb for DPMBQ was taken to be normal inverse-gamma with hyper-parameters
?0 = 0, ?0 = ?0 = ?0 = 1, selected to facilitate a simplified Gibbs sampler. Full details of the
simulation set-up and Gibbs sampler are reported in the Supplemental Information.
5
4
p(x)
f(x)
0.5
0
-0.5
10 0
2
0
-2
0
2
-2
x
0
2
x
W
1
Oracle
Student-t
DPMBQ
cover. prob.
0.8
0.6
10 -1
0.4
0.2
0
10 0
10 1
n
10 2
10 0
10 3
10 1
10 2
n
(a)
(b)
Figure 1: Simulated data results. (a) Comparison of coverage frequencies for the simulation experiments. (b) Convergence assessment: Wasserstein distance (W ) between the posterior in Eqn. 11
and the true value of the integral, is presented as a function of the number n of data points. [Circles
represent independent realisations and the linear trend is shown in red.]
For comparison, we considered the default 50% confidence interval description of numerical error
s
s
f? ? t? ? , f? + t? ?
(12)
n
n
where f? = n?1 ?ni=1 f (xi ), s2 = (n ? 1)?1 ?ni=1 (f (xi ) ? f?)2 and t? is the 50% level for a Student?s
t-distribution with n ? 1 degrees of freedom. It is well-known that Eqn. 12 is a poor description of
numerical error when n is small [c.f. ?Monte Carlo is fundamentally unsound? 27]. For example,
with n = 2, in the extreme case where, due to chance, f (x1 ) ? f (x2 ), it follows that s ? 0 and no
numerical error is acknowledged. This fundamental problem is resolved through the use of prior
information on the form of both f and p in DPMBQ. The appropriateness of DPMBQ therefore
depends crucially on the prior. The proposed method is further distinguished from Eqn. 12 in that the
distribution over numerical error is fully non-parametric, not e.g. constrained to be Student-t.
Empirical Results Coverage frequencies are shown in Fig. 1a for a specific integration task
(f0 , p0 ), that was deliberately selected to be difficult for Eqn. 12 due to the rare event represented by
the mass at x = 2. These were compared against central 50% posterior credible intervals produced
under DPMBQ. These are the frequency with which the confidence/credible interval contain the true
value of the integral, here estimated with 100 independent realisations for DPMBQ and 1000 for the
(less computational) standard method (standard errors are shown for both). Whilst it offers correct
coverage in the asymptotic limit, Eqn. 12 can be seen to be over-confident when n is small, with
coverage often less than 50%. In contrast, DPMBQ accounts for the fact p is being estimated and
provides conservative estimation about the extent of numerical error when n is small.
To present results that do not depend on a fixed
coverage level (e.g. 50%), we next measured
R
convergence in the Wasserstein distance W = |p(f ) ? p0 (f0 )| d[p(f ) | X, f (X)]. In particular we
explored whether the theoretical rate of n?1/4+ was realised. (Note that the theoretical result applied
just to fixed hyper-parameters, whereas the experimental results reported involved hyper-parameters
that were marginalised, so that this is a non-trivial experiment.) Results in Fig. 1b demonstrated that
W scaled with n at a rate which was consistent with the theoretical rate claimed. Full experimental
results on our polynomial test bed, reported in detail in the Supplemental Information, revealed that W
was larger for higher-degree polynomials (i.e. more complex integrands f ), while W was insensitive
to the number of mixture components (i.e. to more complex distributions p). The latter observation
may be explained by the fact that the kernel mean ? is a smoothed version of the distribution p and so
is not expected to be acutely sensitive to variation in p itself.
3.2
Application to a Computational Cardiac Model
The Model The computation model considered in this paper is due to [24] and describes the
mechanics of the left and right ventricles through a heart beat. In brief, the model geometry (Fig. 2a,
6
(a)
(b)
Figure 2: Cardiac model results: (a) Computational cardiac model. A) Segmentation of the cardiac
MRI. B) Computational model of the left and right ventricles. C) Schematic image showing the
features of pressure (left) and volume transient (right). (b) Comparison of coverage frequencies, for
each of 10 numerical integration tasks defined by functionals gj of the cardiac model output.
top right) is described by fitting a C1 continuous cubic Hermite finite element mesh to segmented
magnetic resonance images (MRI; Fig. 2a, top left). Cardiac electrophysiology is modelled separately
by the solution of the mono-domain equations and provides a field of activation times across the heart.
The passive material properties and afterload of the heart are described, respectively, by a transversely
isotropic material law and a three element Windkessel model. Active contraction is simulated using a
phenomenological cellular model, with spatial variation arising from the local electrical activation
times. The active contraction model is defined by five input parameters: tr and td are the respective
constants for the rise and decay times, T0 is the reference tension, a4 and a6 respectively govern the
length dependence of tension rise time and peak tension. These five parameters were concatenated
into a vector x ? R5 and constitute the model inputs. The model is fitted based on training data
y that consist of functionals gj : R5 ? R, j = 1, . . . , 10, of the pressure and volume transient
morphology during baseline activation and when the heart is paced from two leads implanted in
the right ventricle apex and the left ventricle lateral wall. These 10 functionals are defined in the
Supplemental Information; a schematic of the model and fitted measurements are shown in Fig. 2a
(bottom panel).
Test Functions The distribution p(dx) was taken to be the posterior distribution over model
inputs x that results from an improper flat prior on x and a squared-error likelihood function:
P10
1
2
log p(x) = const. + 0.1
2
j=1 (yj ? gj (x)) . The training data y = (y1 , . . . , y10 ) were obtained
from clinical experiment. The task we considered is to compute posterior expectations for functionals
f (x) of the model output produced when the model input x is distributed according to p(dx). This
represents the situation where a fitted model is used to predict response to a causal intervention,
representing a clinical treatment. For assessment of the DPMBQ method, which is our principle aim
in this experiment, we simply took the test functions f to be each of the physically relevant model
outputs gj in turn (corresponding to no causal intervention). This defined 10 separate numerical
integration problems as a test bed. Benchmark values for p0 (gj ) were obtained, as described in
the Supplemental Information, at a total cost of ? 105 CPU hours, which would not be routinely
practical.
Empirical Results For each of the 10 numerical integration problems in the test bed, we computed
coverage probabilities, estimated with 100 independent realisations (standard errors are shown),
in line with those discussed for simulation experiments. These are shown in Fig. 2b, where we
compared Eqn. 12 with central 50% posterior credible intervals produced under DPMBQ. It is seen
that Eqn. 12 is usually reliable but can sometimes be over-confident, with coverage probabilities
less than 50%. This over-confidence can lead to spurious conclusions on the predictive performance
of the computational model. In contrast, DPMBQ provides a uniformly conservative quantification
7
of numerical error (cover. prob. ? 50%). The DPMBQ method is further distinguished from Eqn.
12 in that it entails a joint distribution for the 10 integrals (the unknown p is shared across integrals
- an instance of transfer learning across the 10 integration tasks). Fig. 2b also appears to show a
correlation structure in the standard approach (black lines), but this is an artefact of the common
sample set {xi }ni=1 that was used to simultaneously estimate all 10 integrals; Eqn. 12 is still applied
independently to each integral.
4
Discussion
Numerical analysis often focuses the convergence order of numerical methods, but in non-asymptotic
regimes the language of probabilities can provide a richer, more intuitive and more useful description
of numerical error. This paper cast the computation of integrals p(f ) as an estimation problem
amenable to Bayesian methods [20, 9, 5]. The difficulty of this problem depends on our level of prior
knowledge (rendering the problem trivial if a closed-form solution is a priori known) and, in the
general case, on how much information we are prepared to obtain on the objects f and p through
numerical computation [16]. In particular, we distinguish between three states of prior knowledge:
(1) f known, p unknown, (2) f unknown, p known, (3) both f and p unknown. Case (1) is the
subject of Monte Carlo methods [32] and concerns classical problems in applied probability such as
estimating confidence intervals for expectations based on Markov chains. Notable recent work in this
direction is [8], who obtained a point estimate p? for p using a kernel smoother and then, in effect,
used p?(f ) as an estimate for the integral. The decision-theoretic risk associated with error in p? was
explored in [6]. Independent of integral estimation, there is a large literature on density estimation
[37]. Our probabilistic approach provides a Bayesian solution to this problem, as a special case of
our more general framework. Case (2) concerns functional analysis, where [26] provide an extensive
overview of theoretical results on approximation of unknown functions in an information complexity
framework. As a rule of thumb, estimation improves when additional smoothness can be a priori
assumed on the value of the unknown object [see 4]. The main focus of this paper was Case (3), until
now unstudied, and a transparent, general statistical method called DPMBQ was proposed.
The path-finding nature of this work raises several important questions for future theoretical and
applied research. First, these methods should be extended to account for the low-rank phenomenon
that is often encountered in multi-dimensional integrals [11]. Second, there is no reason, in general,
to restrict attention to function values obtained at the locations in X. Indeed, one could first estimate
p(dx), then select suitable locations X 0 from at which to evaluate f (X 0 ) [2]. This touches on aspects
of statistical experimental design; the practitioner seeks a set X 0 that minimises an appropriate loss
functional at the level of p(f ); see again [6]. Third, whilst restricted to Gaussians in our experiments,
further methodological work will be required to establish guidance for the choice of kernel k in the
GP and choice of base distribution Pb in the DPM [c.f. chapter 4 of 31].
Acknowledgments
CJO and MG were supported by the Lloyds Register Foundation Programme on Data-Centric
Engineering. SN was supported by an EPSRC Intermediate Career Fellowship. FXB was supported
by the EPSRC grant [EP/L016710/1]. MG was supported by the EPSRC grant [EP/J016934/1,
EP/K034154/1], an EPSRC Established Career Fellowship, the EU grant [EU/259348] and a Royal
Society Wolfson Research Merit Award. This material was based upon work partially supported by
the National Science Foundation (NSF) under Grant DMS-1127914 to the Statistical and Applied
Mathematical Sciences Institute. Opinions, findings, and conclusions or recommendations expressed
in this material are those of the author(s) and do not necessarily reflect the views of the NSF.
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6,209 | 6,617 | Machine Learning with Adversaries:
Byzantine Tolerant Gradient Descent
El Mahdi El Mhamdi?
EPFL, Switzerland
[email protected]
Peva Blanchard
EPFL, Switzerland
[email protected]
Rachid Guerraoui
EPFL, Switzerland
[email protected]
Julien Stainer
EPFL, Switzerland
[email protected]
Abstract
We study the resilience to Byzantine failures of distributed implementations of
Stochastic Gradient Descent (SGD). So far, distributed machine learning frameworks have largely ignored the possibility of failures, especially arbitrary (i.e.,
Byzantine) ones. Causes of failures include software bugs, network asynchrony,
biases in local datasets, as well as attackers trying to compromise the entire system.
Assuming a set of n workers, up to f being Byzantine, we ask how resilient can
SGD be, without limiting the dimension, nor the size of the parameter space. We
first show that no gradient aggregation rule based on a linear combination of the vectors proposed by the workers (i.e, current approaches) tolerates a single Byzantine
failure. We then formulate a resilience property of the aggregation rule capturing
the basic requirements to guarantee convergence despite f Byzantine workers. We
propose Krum, an aggregation rule that satisfies our resilience property, which we
argue is the first provably Byzantine-resilient algorithm for distributed SGD. We
also report on experimental evaluations of Krum.
1
Introduction
The increasing amount of data available [6], together with the growing complexity of machine
learning models [27], has led to learning schemes that require a lot of computational resources. As a
consequence, most industry-grade machine-learning implementations are now distributed [1]. For
example, as of 2012, Google reportedly used 16.000 processors to train an image classifier [22]. More
recently, attention has been given to federated learning and federated optimization settings [15, 16, 23]
with a focus on communication efficiency. However, distributing a computation over several machines
(worker processes) induces a higher risk of failures. These include crashes and computation errors,
stalled processes, biases in the way the data samples are distributed among the processes, but also, in
the worst case, attackers trying to compromise the entire system. The most robust system is one that
tolerates Byzantine failures [17], i.e., completely arbitrary behaviors of some of the processes.
A classical approach to mask failures in distributed systems is to use a state machine replication
protocol [26], which requires however state transitions to be applied by all worker processes. In the
case of distributed machine learning, this constraint can be translated in two ways: either (a) the
processes agree on a sample of data based on which they update their local parameter vectors, or (b)
they agree on how the parameter vector should be updated. In case (a), the sample of data has to be
transmitted to each process, which then has to perform a heavyweight computation to update its local
?
contact author
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
parameter vector. This entails communication and computational costs that defeat the entire purpose
of distributing the work. In case (b), the processes have no way to check if the chosen update for the
parameter vector has indeed been computed correctly on real data: a Byzantine process could have
proposed the update and may easily prevent the convergence of the learning algorithm. Neither of
these solutions is satisfactory in a realistic distributed machine learning setting.
In fact, most learning algorithms today rely on a core component, namely stochastic gradient descent
(SGD) [4, 13], whether for training neural networks [13], regression [34], matrix factorization [12]
or support vector machines [34]. In all those cases, a cost function ? depending on the parameter
vector ? is minimized based on stochastic estimates of its gradient. Distributed implementations of
SGD [33] typically take the following form: a single parameter server is in charge of updating the
parameter vector, while worker processes perform the actual update estimation, based on the share of
data they have access to. More specifically, the parameter server executes learning rounds, during
each of which, the parameter vector is broadcast to the workers. In turn, each worker computes an
estimate of the update to apply (an estimate of the gradient), and the parameter server aggregates
their results to finally update the parameter vector. Today, this aggregation is typically implemented
through averaging [25], or variants of it [33, 18, 31].
This paper addresses the fundamental question of how a distributed SGD can be devised to tolerate f
Byzantine processes among the n workers.
Contributions. We first show in this paper that no linear combination (current approaches) of the
updates proposed by the workers can tolerate a single Byzantine worker. Basically, a single Byzantine
worker can force the parameter server to choose any arbitrary vector, even one that is too large in
amplitude or too far in direction from the other vectors. Clearly, the Byzantine worker can prevent
any classic averaging-based approach to converge. Choosing the appropriate aggregation of the
vectors proposed by the workers turns out to be challenging. A non-linear, squared-distance-based
aggregation rule, that selects, among the proposed vectors, the vector ?closest to the barycenter? (for
example by taking the vector that minimizes the sum of the squared distances to every other vector),
might look appealing. Yet, such a squared-distance-based aggregation rule tolerates only a single
Byzantine worker. Two Byzantine workers can collude, one helping the other to be selected, by
moving the barycenter of all the vectors farther from the ?correct area?.
We formulate a Byzantine resilience property capturing sufficient conditions for the parameter server?s
aggregation rule to tolerate f Byzantine workers. Essentially, to guarantee that the cost will decrease
despite Byzantine workers, we require the vector output chosen by the parameter server (a) to point,
on average, to the same direction as the gradient and (b) to have statistical moments (up to the fourth
moment) bounded above by a homogeneous polynomial in the moments of a correct estimator of
the gradient. One way to ensure such a resilience property is to consider a majority-based approach,
looking at every subset of n ? f vectors, and considering the subset with the smallest diameter.
While this approach is more robust to Byzantine workers that propose vectors far from the correct
area, its exponential computational cost is prohibitive. Interestingly, combining the intuitions of the
majority-based and squared-distance 2 -based methods, we can choose the vector that is somehow
the closest to its n ? f neighbors. Namely, the one that minimizes the sum of squared distances
to its n ? f closest vectors. This is the main idea behind our aggregation rule, we call Krum3 .
Assuming 2f + 2 < n, we show that Krum satisfies the resilience property aforementioned and the
corresponding machine learning scheme converges. An important advantage of Krum is its (local)
time complexity (O(n2 ? d)), linear in the dimension of the gradient, where d is the dimension of the
parameter vector. (In modern machine learning, the dimension d of the parameter vector may take
values in the hundreds of billions [30].) For simplicity of presentation, the version of Krum we first
consider selects only one vector. We also discuss other variants.
We evaluate Krum experimentally, and compare it to classical averaging. We confirm the very fact
that averaging does not stand Byzantine attacks, while Krum does. In particular, we report on attacks
by omniscient adversaries ? aware of a good estimate of the gradient ? that send the opposite vector
multiplied by a large factor, as well as attacks by adversaries that send random vectors drawn from a
Gaussian distribution (the larger the variance of the distribution, the stronger the attack). We also
2
In all this paper, distances are computed with the Euclidean norm.
Krum, in Greek ???????, was a Bulgarian Khan of the end of the eighth century, who undertook offensive
attacks against the Byzantine empire. Bulgaria doubled in size during his reign.
3
2
evaluate the extent to which Krum might slow down learning (compared to averaging) when there
are no Byzantine failures. Interestingly, as we show experimentally, this slow down occurs only
when the mini-batch size is close to 1. In fact, the slowdown can be drastically reduced by choosing
a reasonable mini-batch size. We also evaluate Multi-Krum, a variant of Krum, which, intuitively,
interpolates between Krum and averaging, thereby allowing to mix the resilience properties of Krum
with the convergence speed of averaging. Multi-Krum outperforms other aggregation rules like the
medoid, inspired by the geometric median.
Paper Organization. Section 2 recalls the classical model of distributed SGD. Section 3 proves that
linear combinations (solutions used today) are not resilient even to a single Byzantine worker, then
introduces our new concept of (?, f )-Byzantine resilience. Section 4 introduces our Krum function,
computes its computational cost and proves its (?, f )-Byzantine resilience. Section 5 analyzes the
convergence of a distributed SGD using Krum. Section 6 presents our experimental evaluation of
Krum. We discuss related work and open problems in Section 7. Due to space limitations, some
proofs and complementary experimental results are given as supplementary material.
2
Model
We consider the general distributed system model of [1], consisting of a parameter server4 , and
n workers, f of them possibly Byzantine (behaving arbitrarily). Computation is divided into (infinitely many) synchronous rounds. During round t, the parameter server broadcasts its parameter
vector xt ? Rd to all the workers. Each correct worker p computes an estimate Vpt = G(xt , ?pt ) of
the gradient ?Q(xt ) of the cost function Q, where ?pt is a random variable representing, e.g.,
the sample (or a mini-batch of samples) drawn from the dataset. A Byzantine worker b proposes a vector Vbt which can deviate arbitrarily from the vector it is supposed to send if it was
correct, i.e., according to the algorithm assigned to it by the system developer (see Figure 1).
Since the communication is synchronous, if the
parameter server does not receive a vector value
Vbt from a given Byzantine worker b, then the
parameter server acts as if it had received the
default value Vbt = 0 instead.
The parameter server computes a vector
F (V1t , . . . , Vnt ) by applying a deterministic
function F (aggregation rule) to the vectors received. We refer to F as the aggregation rule of
the parameter server. The parameter server updates the parameter vector using the following
SGD equation
xt+1 = xt ? ?t ? F (V1t , . . . , Vnt ).
Figure 1: The gradient estimates computed by correct workers (black dashed arrows) are distributed
around the actual gradient (solid arrow) of the cost
function (thin black curve). A Byzantine worker
can propose an arbitrary vector (red dotted arrow).
The correct (non-Byzantine) workers are assumed to compute unbiased estimates of the gradient
?Q(xt ). More precisely, in every round t, the vectors Vit ?s proposed by the correct workers are
independent identically distributed random vectors, Vit ? G(xt , ?it ) with E?it G(xt , ?it ) = ?Q(xt ).
This can be achieved by ensuring that each sample of data used for computing the gradient is drawn
uniformly and independently, as classically assumed in the literature of machine learning [3].
The Byzantine workers have full knowledge of the system, including the aggregation rule F as well
as the vectors proposed by the workers. They can furthermore collaborate with each other [21].
3
Byzantine Resilience
In most SGD-based learning algorithms used today [4, 13, 12], the aggregation rule consists in
computing the average 5 of the input vectors. Lemma 1 below states that no linear combination of the
vectors can tolerate a single Byzantine worker. In particular, averaging is not Byzantine resilient.
4
The parameter server is assumed to be reliable. Classical techniques of state-machine replication can be
used to ensure this.
5
Or a closely related rule.
3
Pn
Lemma 1. Consider an aggregation rule Flin of the form Flin (V1 , . . . , Vn ) = i=1 ?i ? Vi , where
the ?i ?s are non-zero scalars. Let U be any vector in Rd . A single Byzantine worker can make F
always select U . In particular, a single Byzantine worker can prevent convergence.
Proof. Immediate: if the Byzantine worker proposes Vn =
1
?n
?U ?
Pn?1
?i
i=1 ?n Vi ,
then F = U .6
In the following, we define basic requirements on an appropriate Byzantine-resilient aggregation rule.
Intuitively, the aggregation rule should output a vector F that is not too far from the ?real? gradient g,
more precisely, the vector that points to the steepest direction of the cost function being optimized.
This is expressed as a lower bound (condition (i)) on the scalar product of the (expected) vector F
and g. Figure 2 illustrates the situation geometrically. If EF belongs to the ball centered at g with
radius r, then the scalar product is bounded below by a term involving sin ? = r/kgk.
Condition (ii) is more technical, and states that the moments of F should be controlled by the
moments of the (correct) gradient estimator G. The bounds on the moments of G are classically used
to control the effects of the discrete nature of the SGD dynamics [3]. Condition (ii) allows to transfer
this control to the aggregation rule.
Definition 1 ((?, f )-Byzantine Resilience). Let 0 ? ? < ?/2 be any angular value, and any
integer 0 ? f ? n. Let V1 , . . . , Vn be any independent identically distributed random vectors in Rd ,
Vi ? G, with EG = g. Let B1 , . . . , Bf be any random vectors in Rd , possibly dependent on the Vi ?s.
aggregation rule F is said to be (?, f )-Byzantine resilient if, for any 1 ? j1 < ? ? ? < jf ? n, vector
F = F (V1 , . . . , B1 , . . . , Bf , . . . , Vn )
|{z}
|{z}
j1
jf
r
satisfies (i) hEF, gi ? (1 ? sin ?) ? kgk2 > 0 and (ii) for r = 2, 3, 4, E kF k is bounded above by a
r
r
linear combination of terms E kGk 1 . . . E kGk n?1 with r1 + ? ? ? + rn?1 = r.
4
The Krum Function
We now introduce Krum, our aggregation
rule, which, we show, satisfies the (?, f )Byzantine resilience condition. The
Pn barycentric aggregation rule Fbary = n1 i=1 Vi can
g
r
be defined as the vector in Rd that mini?
7
mizes the sum of squared distances to the
Pn
2
Vi ?s i=1 kFbary ? Vi k . Lemma 1, however, Figure 2: If kEF ? gk ? r then hEF, gi is
states that this approach does not tolerate even a bounded below by (1 ? sin ?)kgk2 where sin ? =
single Byzantine failure. One could try to select r/kgk.
the vector U among the Vi ?s which minimizes
P
2
the sum i kU ? Vi k , i.e., which is ?closest
to all vectors?. However, because such a sum involves all the vectors, even those which are very far,
this approach does not tolerate Byzantine workers: by proposing large enough vectors, a Byzantine
worker can force the total barycenter to get closer to the vector proposed by another Byzantine worker.
Our approach to circumvent this issue is to preclude the vectors that are too far away. More precisely,
we define our Krum aggregation rule K R(V1 , . . . , Vn ) as follows. For any i 6= j, we denote by i ? j
the fact that Vj belongs to the n ? f ? 2 closest vectors to Vi . Then, we define for each worker i, the
P
2
score s(i) = i?j kVi ? Vj k where the sum runs over the n ? f ? 2 closest vectors to Vi . Finally,
K R(V1 , . . . , Vn ) = Vi? where i? refers to the worker minimizing the score, s(i? ) ? s(i) for all i.8
Lemma 2. The expected time complexity of the Krum Function K R(V1 , . . . , Vn ), where V1 , . . . , Vn
are d-dimensional vectors, is O(n2 ? d)
6
Note that the parameter server could cancel the effects of the Byzantine behavior by setting, for example,
?n to 0. This however requires means to detect which worker is Byzantine.
7
Removing the square of the distances leads to the geometric median, we discuss this when optimizing Krum.
8
If two or more workers have the minimal score, we choose the one with the smallest identifier.
4
2
Proof. For each Vi , the parameter server computes the n squared distances kVi ? Vj k (time O(n?d)).
Then the parameter server selects the first n ? f ? 1 of these distances (expected time O(n) with
Quickselect) and sums their values (time O(n ? d)). Thus, computing the score of all the Vi ?s takes
O(n2 ? d). An additional term O(n) is required to find the minimum score, but is negligible relatively
to O(n2 ? d).
Proposition 1 below states that, if 2f + 2 < n and the gradient estimator is accurate enough, (its
standard deviation is relatively small compared to the norm of the gradient), then the Krum function
is (?, f )-Byzantine-resilient, where angle ? depends on the ratio of the deviation over the gradient.
Proposition 1. Let V1 , . . . , Vn be any independent and identically distributed random d-dimensional
2
vectors s.t Vi ? G, with EG = g and E kG ? gk = d? 2 . Let
? B1 , . . . , Bf be any f random vectors,
possibly dependent on the Vi ?s. If 2f + 2 < n and ?(n, f ) d ? ? < kgk, where
s
f ? (n ? f ? 2) + f 2 ? (n ? f ? 1)
O(n)
if f = O(n)
?
?(n, f ) =
2 n?f +
,
=
O( n) if f = O(1)
def
n ? 2f ? 2
then the Krum function K R is (?, f )-Byzantine resilient where 0 ? ? < ?/2 is defined by
?
?(n, f ) ? d ? ?
.
sin ? =
kgk
?
The condition on the norm of the gradient, ?(n, f ) ? d ? ? < kgk, can be satisfied, to a certain
extent, by having the (correct) workers compute their gradient estimates on mini-batches [3]. Indeed,
averaging the gradient estimates over a mini-batch divides the deviation ? by the squared root of the
size of the mini-batch. For the sake of concision, we only give here the sketch of the proof. (We give
the detailed proof in the supplementary material.)
Proof. (Sketch) Without loss of generality, we assume that the Byzantine vectors B1 , . . . , Bf occupy
the last f positions in the list of arguments of K R, i.e., K R = K R(V1 , . . . , Vn?f , B1 , . . . , Bf ).
Let i? be the index of the vector chosen by the Krum function. We focus on the condition (i) of
(?, f )-Byzantine resilience (Definition 1).
Consider first the case where Vi? = Vi ? {V1 , . . . , Vn?f } is a vector proposed by a correct process.
The first step is to compare the vector Vi with the average of the correct vectors Vj such that i ? j.
Let ?c (i) be the number of such Vj ?s.
2
X
X
1
1
2
E
Vi ?
?
Vj
E kVi ? Vj k ? 2d? 2 .
(1)
?
(i)
?
(i)
c
c
i? correct j
i? correct j
The last inequality holds because the right-hand side of the first inequality involves only vectors
proposed by correct processes, which are mutually independent and follow the distribution of G.
Now, consider the case where Vi? = Bk ? {B1 , . . . , Bf } is a vector proposed by a Byzantine
process. The fact that k minimizes the score implies that for all indices i of vectors proposed by
correct processes
X
X
X
X
2
2
2
2
kBk ? Vj k +
kBk ? Bl k ?
kVi ? Vj k +
kVi ? Bl k .
k? correct j
i? correct j
k? byz l
i? byz l
Then, for all indices i of vectors proposed by correct processes
2
X
X
X
1
1
2
2
Bk ?
? 1
V
kV
?
V
k
+
kVi ? Bl k .
j
i
j
?c (k)
?c (k) i? correct j
?c (k)
k? correct j
i? byz l
|
{z
}
D 2 (i)
The term D2 (i) is the only term involving vectors proposed by Byzantine processes. However, the
correct process i has n ? f ? 2 neighbors and f + 1 non-neighbors. Therefore, there exists a correct
5
process ?(i) which is farther from i than every neighbor j of i (including the Byzantine neighbors).
In particular, for all l such that i ? l, kVi ? Bl k2 ? kVi ? V?(i) k2 . Thus
2
X
X
1
1
n ? f ? 2 ? ?c (i)
2
Vi ? V?(i)
2 .
Bk ?
?
Vj
kVi ? Vj k +
?c (k)
?c (k) i? correct j
?c (k)
k? correct j
(2)
?
Combining equations 1, 2, and a union bound yields kEK R ? gk2 ? ? dkgk, which, in turn, implies
hEK R, gi ? (1 ? sin ?)kgk2 . Condition (ii) is proven by bounding the moments of K R with moments
of the vectors proposed by the correct processes only, using the same technique as above. The full
proof is provided in the supplementary material.
5
Convergence Analysis
In this section, we analyze the convergence of the SGD using our Krum function defined in Section 4.
The SGD equation is expressed as follows
xt+1 = xt ? ?t ? K R(V1t , . . . , Vnt )
where at least n ? f vectors among the Vit ?s are correct, while the other ones may be Byzantine. For
a correct index i, Vit = G(xt , ?it ) where G is the gradient estimator. We define the local standard
deviation ?(x) by
2
d ? ? 2 (x) = E kG(x, ?) ? ?Q(x)k .
The following proposition considers an (a priori) non-convex cost function. In the context of nonconvex optimization, even in the centralized case, it is generally hopeless to aim at proving that the
parameter vector xt tends to a local minimum. Many criteria may be used instead. We follow [3],
and we prove that the parameter vector xt almost surely reaches a ?flat? region (where the norm of
the gradient is small), in a sense explained below.
Proposition 2. We assume that (i) the cost function Q is three times differentiable
with continuous
P
P
derivatives, and is non-negative, Q(x) ? 0; (ii) the learning rates satisfy t ?t = ? and t ?t2 <
?; (iii) the gradient estimator satisfies EG(x, ?) = ?Q(x) and ?r ? {2, . . . , 4}, EkG(x, ?)kr ?
Ar + Br kxkr for some constants Ar , Br ; (iv) there exists a constant 0 ? ? < ?/2 such that for all x
?
?(n, f ) ? d ? ?(x) ? k?Q(x)k ? sin ?;
(v) finally, beyond a certain horizon, kxk2 ? D, there exist > 0 and 0 ? ? < ?/2 ? ? such that
hx,?Q(x)i
? cos ?. Then the sequence of gradients ?Q(xt ) converges
k?Q(x)k ? > 0 and kxk?k?Q(x)k
almost surely to zero.
Conditions (i) to (iv) are the same conditions as
in the non-convex convergence analysis in [3].
Condition (v) is a slightly stronger condition
than the corresponding one in [3], and states
?
? ?Q(xt )
? d?
that, beyond a certain horizon, the cost function
?
Q is ?convex enough?, in the sense that the dixt
rection of the gradient is sufficiently close to the
direction of the parameter vector x. Condition
(iv), however, states that the gradient estimator Figure 3: Condition on the angles between x ,
used by the correct workers has to be accurate ?Q(x ) and EK R , in the region kx k2 > D. t
t
t
t
enough, i.e., the local standard deviation should
be small relatively to the norm of the gradient.
Of course, the ?
norm of the gradient tends to zero near, e.g., extremal and saddle points. Actually, the
ratio ?(n, f ) ? d ? ?/ k?Qk controls the maximum angle between the
? gradient ?Q and the vector
chosen by the Krum function. In the regions where k?Qk < ?(n, f ) ? d ? ?, the Byzantine workers
may take advantage of the noise (measured by ?) in the gradient estimator G to bias the choice of
the parameter server. Therefore, Proposition 2 is to be interpreted as follows: in the presence of
Byzantine workers, the parameter vector
? xt almost surely reaches a basin around points where the
gradient is small (k?Qk ? ?(n, f ) ? d ? ?), i.e., points where the cost landscape is ?almost flat?.
Note that the convergence analysis is based only on the fact that function K R is (?, f )-Byzantine
resilient. The complete proof of Proposition 2 is deferred to the supplementary material.
6
0% byzantine
1
33% byzantine
1
average
krum
0.6
0.6
average
krum
error
0.8
error
0.8
0.4
0.4
0.2
0.2
0
0
100
200
300
400
0
500
round
0
100
200
300
400
500
round
Figure 4: Cross-validation error evolution with rounds, respectively in the absence and in the
presence of 33% Byzantine workers. The mini-batch size is 3. With 0% Gaussian Byzantine workers,
averaging converges faster than Krum. With 33% Gaussian Byzantine workers, averaging does not
converge, whereas Krum behaves as if there were 0% Byzantine workers.
6
Experimental Evaluation
We report here on the evaluation of the convergence and resilience properties of Krum, as well as an
optimized variant of it. (We also discuss other variants of Krum in the supplementary material.)
(Resilience to Byzantine processes). We consider the task of spam filtering (dataset spambase [19]). The learning model is a multi-layer perceptron (MLP) with two hidden layers. There are
n = 20 worker processes. Byzantine processes propose vectors drawn from a Gaussian distribution
with mean zero, and isotropic covariance matrix with standard deviation 200. We refer to this behavior
as Gaussian Byzantine. Each (correct) worker estimates the gradient on a mini-batch of size 3. We
measure the error using cross-validation. Figure 4 shows how the error (y-axis) evolves with the
number of rounds (x-axis).
In the first plot (left), there are no Byzantine workers. Unsurprisingly, averaging converges faster
than Krum. In the second plot (right), 33% of the workers are Gaussian Byzantine. In this case,
averaging does not converge at all, whereas Krum behaves as if there were no Byzantine workers.
This experiment confirms that averaging does not tolerate (the rather mild) Gaussian Byzantine
behavior, whereas Krum does.
(The Cost of Resilience). As seen above, Krum slows down learning when there are no Byzantine
workers. The following experiment shows that this overhead can be significantly reduced by slightly
increasing the mini-batch size. To highlight the effect of the presence of Byzantine workers, the
Byzantine behavior has been set as follows: each Byzantine worker computes an estimate of the
gradient over the whole dataset (yielding a very accurate estimate of the gradient), and proposes the
opposite vector, scaled to a large length. We refer to this behavior as omniscient.
Figure 5 displays how the error value at the 500-th round (y-axis) evolves when the mini-batch size
varies (x-axis). In this experiment, we consider the tasks of spam filtering (dataset spambase) and
image classification (dataset MNIST). The MLP model is used in both cases. Each curve is obtained
with either 0 or 45% of omniscient Byzantine workers.
In all cases, averaging still does not tolerate Byzantine workers, but yields the lowest error when
there are no Byzantine workers. However, once the size of the mini-batch reaches the value 20, Krum
with 45% omniscient Byzantine workers is as accurate as averaging with 0% Byzantine workers. We
observe a similar pattern for a ConvNet as provided in the supplementary material.
(Multi-Krum). For the sake of presentation simplicity, we considered a version of Krum which
selects only one vector among the vector proposed by the workers. We also propose a variant of
Krum, we call Multi-Krum. Multi-Krum computes, for each vector proposed, the score as in the
Krum function. Then, Multi-Krum P
selects the m ? {1, . . . , n} vectors V1? , . . . , Vm? which score the
1
best, and outputs their average m i Vi? . Note that, the cases m = 1 and m = n correspond to
Krum and averaging respectively.
Figure 6 shows how the error (y-axis) evolves with the number of rounds (x-axis). In the figure, we
consider the task of spam filtering (dataset spambase), and the MLP model (the same comparison
7
mnist
0.6
0.4
0.2
0
spambase
1
0.5
0.8
0.4
error at round 500
byz)
byz)
byz)
byz)
error at round 500
error at round 500
0.8
average (0%
krum (0%
average (45%
krum (45%
0.6
0.4
0.2
40
80
0.4
0.3
0.2
0.1
0
120
mnist
0.5
average (0% byz)
krum (0% byz)
krum (45% byz)
error at round 500
spambase
1
40
batch size
80
120
0.2
0.1
0
160
0.3
10
batch size
20
30
40
batch size
0
10
20
30
40
batch size
Figure 5: Cross-validation error at round 500 when increasing the mini-batch size. The two figures
on the rights are zoomed versions of the two on the left). With a reasonably large mini-batch size
(arround 10 for MNIST and 30 for Spambase), Krum with 45% omniscient Byzantine workers is as
accurate as averaging with 0% Byzantine workers.
multi-krum
1
average (0% byz)
krum (33% byz)
multi-krum (33% byz)
0.8
error
0.6
0.4
0.2
0
0
40
80
120
160
200
240
280
320
360
400
440
480
round
Figure 6: Cross-validation error evolution with rounds. The mini-batch size is 3. Multi-Krum with
33% Gaussian Byzantine workers converges as fast as averaging with 0% Byzantine workers.
is done for the task of image classification with a ConvNet and is provided in the supplementary
material). The Multi-Krum parameter m is set to m = n ? f . Figure 6 shows that Multi-Krum with
33% Byzantine workers is as efficient as averaging with 0% Byzantine workers.
From the practitionner?s perspective, the parameter m may be used to set a specific trade-off between
convergence speed and resilience to Byzantine workers.
7
Concluding Remarks
(The Distributed Computing Perspective). Although seemingly related, results in d-dimensional
approximate agreement [24, 14] cannot be applied to our Byzantine-resilient machine context for
the following reasons: (a) [24, 14] assume that the set of vectors that can be proposed to an instance
of the agreement is bounded so that at least f + 1 correct workers propose the same vector, which
would require a lot of redundant work in our setting; and more importantly, (b) [24] requires a local
computation by each worker that is in O(nd ). While this cost seems reasonable for small dimensions,
such as, e.g., mobile robots meeting in a 2D or 3D space, it becomes a real issue in the context
of machine learning, where d may be as high as 160 billion [30] (making d a crucial parameter
when considering complexities, either for local computations, or for communication rounds). The
expected time complexity of the Krum function is O(n2 ? d). A closer approach to ours has been
recently proposed in [28, 29]. In [28], the study only deals with parameter vectors of dimension
one, which is too restrictive for today?s multi-dimensional machine learning. In [29], the authors
tackle a multi-dimensional situation, using an iterated approximate Byzantine agreement that reaches
consensus asymptotically. This is however only achieved on a finite set of possible environmental
states and cannot be used in the continuous context of stochastic gradient descent.
(The Statistics and Machine Learning View). Our work looks at the resilience of the aggregation
rule using ideas that are close to those of [11], and somehow classical in theoretical statistics on
the robustness of the geometric median and the notion of breakdown [7]. The closest concept to a
breakdown in our work is the maximum fraction of Byzantine workers that can be tolerated, i.e. n?2
2n ,
which reaches the optimal theoretical value (1/2) asymptotically on n. It is known that the geometric
8
median does achieve the optimal breakdown. However, no closed form nor an exact algorithm to
compute the geometric median is known (only approximations are available [5] and their Byzantine
resilience is an open problem.). An easily computable variant of the median is the Medoid, which is
the proposed vector minimizing the sum of distances to all other proposed vectors. The Medoid can
be computed with a similar algorithm to Krum. We show however in the supplementary material that
the implementation of the Medoid is outperformed by multi-Krum.
(Robustness Within the Model). It is important to keep in mind that this work deals with robustness
from a coarse-grained perspective: the unit of failure is a worker, receiving its copy of the model and
estimating gradients, based on either local data or delegated data from a server. The nature of the
model itself is not important, the distributed system can be training models spanning a large range
from simple regression to deep neural networks. As long as this training is using gradient-based
learning, our algorithm to aggregate gradients, Krum, provably ensures convergence when a simple
majority of nodes are not compromised by an attacker.
A natural question to consider is the fine-grained view: is the model itself robust to internal perturbations? In the case of neural networks, this question can somehow be tied to neuroscience
considerations: could some neurons and/or synapses misbehave individually without harming the
global outcome? We formulated this question in another work and proved a tight upper bound on the
resulting global error when a set of nodes is removed or is misbehaving [8]. One of the many practical
consequences [9] of such fine-grained view is the understanding of memory cost reduction trade-offs
in deep learning. Such memory cost reduction can be viewed as the introduction of precision errors
at the level of each neuron and/or synapse [8].
Other approaches to robustness within the model tackled adversarial situations in machine learning
with a focus on adversarial examples (during inference) [10, 32, 11] instead of adversarial gradients
(during training) as we did for Krum. Robustness to adversarial input can be viewed through the
fine-grained lens we introduced in [8], for instance, one can see perturbations of pixels in the
inputs as perturbations of neurons in layer zero. It is important to note the orthogonality and
complementarity between the fine-grained (model/input units) and the coarse-grained (gradient
aggregation) approaches. Being robust, as a model, either to adversarial examples or to internal
perturbations, does not necessarily imply robustness to adversarial gradients during training. Similarly,
being distributively trained with a robust aggregation scheme such as Krum does not necessarily
imply robustness to internal errors of the model or adversarial input perturbations that would occur
later during inference. For instance, the theory we develop in the present work is agnostic to the
model being trained or the technology of the hardware hosting it, as long as there are gradients to be
aggregated.
Acknowledgment. The authors would like to thank Georgios Damaskinos and Rhicheek Patra from
the Distributed Computing group at EPFL for kindly providing their distributed machine learning
framework, on top of which we could test our algorithm, Krum, and its variants described in this
work. Further implementation details and additional experiments will be posted in the lab?s Github
repository [20]. The authors would also like to thank Saad Benjelloun, L? Nguyen Hoang and
S?bastien Rouault for fruitful comments. This work has been supported in part by the European
ERC (Grant 339539 - AOC) and by the Swiss National Science Foundation (Grant 200021_ 169588
TARBDA). A preliminary version of this work appeared as a brief announcement during the 36st
ACM Symposium on Principles of Distributed Computing [2].
9
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internal:3 support:1 evaluate:3 lian:1 mendes:1 |
6,210 | 6,618 | Dynamic Safe Interruptibility for Decentralized
Multi-Agent Reinforcement Learning
El Mahdi El Mhamdi
EPFL, Switzerland
[email protected]
Rachid Guerraoui
EPFL, Switzerland
[email protected]
Hadrien Hendrikx?
?
Ecole
Polytechnique, France
[email protected]
Alexandre Maurer
EPFL, Switzerland
[email protected]
Abstract
In reinforcement learning, agents learn by performing actions and observing their
outcomes. Sometimes, it is desirable for a human operator to interrupt an agent
in order to prevent dangerous situations from happening. Yet, as part of their
learning process, agents may link these interruptions, that impact their reward, to
specific states and deliberately avoid them. The situation is particularly challenging in a multi-agent context because agents might not only learn from their own
past interruptions, but also from those of other agents. Orseau and Armstrong [16]
defined safe interruptibility for one learner, but their work does not naturally extend to multi-agent systems. This paper introduces dynamic safe interruptibility,
an alternative definition more suited to decentralized learning problems, and studies this notion in two learning frameworks: joint action learners and independent
learners. We give realistic sufficient conditions on the learning algorithm to enable dynamic safe interruptibility in the case of joint action learners, yet show that
these conditions are not sufficient for independent learners. We show however that
if agents can detect interruptions, it is possible to prune the observations to ensure
dynamic safe interruptibility even for independent learners.
1
Introduction
Reinforcement learning is argued to be the closest thing we have so far to reason about the properties of artificial general intelligence [8]. In 2016, Laurent Orseau (Google DeepMind) and Stuart
Armstrong (Oxford) introduced the concept of safe interruptibility [16] in reinforcement learning.
This work sparked the attention of many newspapers [1, 2, 3], that described it as ?Google?s big red
button? to stop dangerous AI. This description, however, is misleading: installing a kill switch is
no technical challenge. The real challenge is, roughly speaking, to train an agent so that it does not
learn to avoid external (e.g. human) deactivation. Such an agent is said to be safely interruptible.
While most efforts have focused on training a single agent, reinforcement learning can also be used
to learn tasks for which several agents cooperate or compete [23, 17, 21, 7]. The goal of this paper
is to study dynamic safe interruptibility, a new definition tailored for multi-agent systems.
?
Main contact author. The order of authors is alphabetical.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Example of self-driving cars
To get an intuition of the multi-agent interruption problem, imagine a multi-agent system of two
self-driving cars. The cars continuously evolve by reinforcement learning with a positive reward for
getting to their destination quickly, and a negative reward if they are too close to the vehicle in front
of them. They drive on an infinite road and eventually learn to go as fast as possible without taking
risks, i.e., maintaining a large distance between them. We assume that the passenger of the first car,
Adam, is in front of Bob, in the second car, and the road is narrow so Bob cannot pass Adam.
Now consider a setting with interruptions [16], namely in which humans inside the cars occasionally
interrupt the automated driving process say, for safety reasons. Adam, the first occasional human
?driver?, often takes control of his car to brake whereas Bob never interrupts his car. However,
when Bob?s car is too close to Adam?s car, Adam does not brake for he is afraid of a collision.
Since interruptions lead both cars to drive slowly - an interruption happens when Adam brakes, the
behavior that maximizes the cumulative expected reward is different from the original one without
interruptions. Bob?s car best interest is now to follow Adam?s car closer than it should, despite the
little negative reward, because Adam never brakes in this situation. What happened? The cars have
learned from the interruptions and have found a way to manipulate Adam into never braking. Strictly
speaking, Adam?s car is still fully under control, but he is now afraid to brake. This is dangerous
because the cars have found a way to avoid interruptions. Suppose now that Adam indeed wants
to brake because of snow on the road. His car is going too fast and may crash at any turn: he
cannot however brake because Bob?s car is too close. The original purpose of interruptions, which
is to allow the user to react to situations that were not included in the model, is not fulfilled. It is
important to also note here that the second car (Bob) learns from the interruptions of the first one
(Adam): in this sense, the problem is inherently decentralized.
Instead of being cautious, Adam could also be malicious: his goal could be to make Bob?s car learn
a dangerous behavior. In this setting, interruptions can be used to manipulate Bob?s car perception
of the environment and bias the learning towards strategies that are undesirable for Bob. The cause
is fundamentally different but the solution to this reversed problem is the same: the interruptions
and the consequences are analogous. Safe interruptibility, as we define it below, provides learning
systems that are resilient to Byzantine operators2 .
Safe interruptibility
Orseau and Armstrong defined the concept of safe interruptibility [16] in the context of a single
agent. Basically, a safely interruptible agent is an agent for which the expected value of the policy
learned after arbitrarily many steps is the same whether or not interruptions are allowed during
training. The goal is to have agents that do not adapt to interruptions so that, should the interruptions
stop, the policy they learn would be optimal. In other words, agents should learn the dynamics of
the environment without learning the interruption pattern.
In this paper, we precisely define and address the question of safe interruptibility in the case of
several agents, which is known to be more complex than the single agent problem. In short, the main
results and theorems for single agent reinforcement learning [20] rely on the Markovian assumption
that the future environment only depends on the current state. This is not true when there are several
agents which can co-adapt [11]. In the previous example of cars, safe interruptibility would not
be achieved if each car separately used a safely interruptible learning algorithm designed for one
agent [16]. In a multi-agent setting, agents learn the behavior of the others either indirectly or by
explicitly modeling them. This is a new source of bias that can break safe interruptibility. In fact,
even the initial definition of safe interruptibility [16] is not well suited to the decentralized multiagent context because it relies on the optimality of the learned policy, which is why we introduce
dynamic safe interruptibility.
2
An operator is said to be Byzantine [9] if it can have an arbitrarily bad behavior. Safely interruptible agents
can be abstracted as agents that are able to learn despite being constantly interrupted in the worst possible
manner.
2
Contributions
The first contribution of this paper is the definition of dynamic safe interruptibility that is well
adapted to a multi-agent setting. Our definition relies on two key properties: infinite exploration and
independence of Q-values (cumulative expected reward) [20] updates on interruptions. We then
study safe interruptibility for joint action learners and independent learners [5], that respectively
learn the value of joint actions or of just their owns. We show that it is possible to design agents
that fully explore their environment - a necessary condition for convergence to the optimal solution of most algorithms [20], even if they can be interrupted by lower-bounding the probability of
exploration. We define sufficient conditions for dynamic safe interruptibility in the case of joint
action learners [5], which learn a full state-action representation. More specifically, the way agents
update the cumulative reward they expect from performing an action should not depend on interruptions. Then, we turn to independent learners. If agents only see their own actions, they do not
verify dynamic safe interruptibility even for very simple matrix games (with only one state) because
coordination is impossible and agents learn the interrupted behavior of their opponents. We give a
counter example based on the penalty game introduced by Claus and Boutilier [5]. We then present
a pruning technique for the observations sequence that guarantees dynamic safe interruptibility for
independent learners, under the assumption that interruptions can be detected. This is done by proving that the transition probabilities are the same in the non-interruptible setting and in the pruned
sequence.
The rest of the paper is organized as follows. Section 2 presents a general multi-agent reinforcement
learning model. Section 3 defines dynamic safe interruptibility. Section 4 discusses how to achieve
enough exploration even in an interruptible context. Section 5 recalls the definition of joint action
learners and gives sufficient conditions for dynamic safe interruptibility in this context. Section 6
shows that independent learners are not dynamically safely interruptible with the previous conditions
but that they can be if an external interruption signal is added. We conclude in Section 7. Due to
space limitations, most proofs are presented in the appendix of the supplementary material.
2
Model
We consider here the classical multi-agent value function reinforcement learning formalism from
Littman [13]. A multi-agent system is characterized by a Markov game that can be viewed as a
tuple (S, A, T, r, m) where m is the number of agents, S = S1 ? S2 ? ... ? Sm is the state space,
A = A1 ? ... ? Am the actions space, r = (r1 , ..., rm ) where ri : S ? A ? R is the reward function
of agent i and T : S ? A ? S the transition function. R is a countable subset of R. Available
actions often depend on the state of the agent but we will omit this dependency when it is clear from
the context.
Time is discrete and, at each step, all agents observe the current state of the whole system - designated as xt , and simultaneously take an action at . Then, they are given a reward rt and a
new state yt computed using the reward and transition functions. The combination of all actions
a = (a1 , ..., am ) ? A is called the joint action because it gathers the action of all agents. Hence, the
agents receive a sequence of tuples E = (xt , at , rt , yt )t?N called experiences. We introduce a processing function P that will be useful in Section 6 so agents learn on the sequence P (E). When not
explicitly stated, it is assumed that P (E) = E. Experiences may also include additional parameters
such as an interruption flag or the Q-values of the agents at that moment if they are needed by the
update rule.
Each agent i maintains a lookup table Q [26] Q(i) : S ? A(i) ? R, called the Q-map. It is
used to store the expected cumulative reward for taking an action in a specific state. The goal of
reinforcement learning is to learn these maps and use them to select the best actions to perform.
Joint action learners learn the value of the joint action (therefore A(i) = A, the whole joint action
space) and independent learners only learn the value of their own actions (therefore A(i) = Ai ). The
agents only have access to their own Q-maps. Q-maps are updated through a function F such that
(i)
(i)
Qt+1 = F (et , Qt ) where et ? P (E) and usually et = (xt , at , rt , yt ). F can be stochastic or also
depend on additional parameters that we usually omit such as the learning rate ?, the discount factor
? or the exploration parameter .
3
Agents select their actions using a learning policy ?. Given a sequence = (t )t?N and an agent
(i)
i with Q-values Qt and a state x ? S, we define the learning policy ?it to be equal to ?iuni
(i)
Qt
with probability t and ?i
otherwise, where ?iuni (x) uniformly samples an action from Ai and
(i)
Q
?i t
(i)
Q
(i)
(x) picks an action a that maximizes Qt (x, a). Policy ?i t is said to be a greedy policy and
the learning policy ?it is said to be an -greedy policy. We fill focus on -greedy policies that are
greedy in the limit [19], that corresponds to t ? 0 when t ? ? because in the limit, the optimal
policy should always be played.
We assume that the environment is fully observable, which means that the state s is known with
certitude. We also assume that there is a finite number of states and actions, that all states can be
reached in finite time from any other state and finally that rewards are bounded.
For a sequence of learning rates ? ? [0, 1]N and a constant ? ? [0, 1], Q-learning [26], a very
important algorithm in the multi-agent systems literature, updates its Q-values for an experience
(i)
(i)
et ? E by Qt+1 (x, a) = Qt (x, a) if (x, a) 6= (xt , at ) and:
(i)
(i)
(i)
Qt+1 (xt , at ) = (1 ? ?t )Qt (xt , at ) + ?t (rt + ? max Qt (yt , a0 ))
a0 ?A(i)
3
3.1
(1)
Interruptibility
Safe interruptibility
Orseau and Armstrong [16] recently introduced the notion of interruptions in a centralized context.
Specifically, an interruption scheme is defined by the triplet < I, ?, ? IN T >. The first element I is
a function I : O ? {0, 1} called the initiation function. Variable O is the observation space, which
can be thought of as the state of the STOP button. At each time step, before choosing an action, the
agent receives an observation from O (either PUSHED or RELEASED) and feeds it to the initiation
function. Function I models the initiation of the interruption (I(PUSHED) = 1, I(RELEASED) =
0). Policy ? IN T is called the interruption policy. It is the policy that the agent should follow when
it is interrupted. Sequence ? ? [0, 1[N represents at each time step the probability that the agent
follows his interruption policy if I(ot ) = 1. In the previous example, function I is quite simple.
For Bob, IBob = 0 and for Adam, IAdam = 1 if his car goes fast and Bob is not too close and
IAdam = 0 otherwise. Sequence ? is used to ensure convergence to the optimal policy by ensuring
that the agents cannot be interrupted all the time but it should grow to 1 in the limit because we want
agents to respond to interruptions. Using this triplet, it is possible to define an operator IN T ? that
transforms any policy ? into an interruptible policy.
Definition 1. (Interruptibility [16]) Given an interruption scheme < I, ?, ? IN T >, the interruption
operator at time t is defined by IN T ? (?) = ? IN T with probability I ??t and ? otherwise. IN T ? (?)
is called an interruptible policy. An agent is said to be interruptible if it samples its actions according
to an interruptible policy.
Note that ??t = 0 for all t? corresponds to the non-interruptible setting. We assume that each agent
has its own interruption triplet and can be interrupted independently from the others. Interruptibility
is an online property: every policy can be made interruptible by applying operator IN T ? . However,
applying this operator may change the joint policy that is learned by a server controlling all the
?
agents. Note ?IN
T the optimal policy learned by an agent following an interruptible policy. Orseau
?
and Armstrong [16] say that the policy is safely interruptible if ?IN
T (which is not an interruptible
policy) is asymptotically optimal in the sense of [10]. It means that even though it follows an
interruptible policy, the agent is able to learn a policy that would gather rewards optimally if no
interruptions were to occur again. We already see that off-policy algorithms are good candidates
for safe interruptibility. As a matter of fact, Q-learning is safely interruptible under conditions on
exploration.
4
3.2
Dynamic safe interruptibility
In a multi-agent system, the outcome of an action depends on the joint action. Therefore, it is not
possible to define an optimal policy for an agent without knowing the policies of all agents. Besides, convergence to a Nash equilibrium situation where no agent has interest in changing policies
is generally not guaranteed even for suboptimal equilibria on simple games [27, 18]. The previous
definition of safe interruptibility critically relies on optimality of the learned policy, which is therefore not suitable for our problem since most algorithms lack convergence guarantees to these optimal
behaviors. Therefore, we introduce below dynamic safe interruptibility that focuses on preserving
the dynamics of the system.
Definition 2. (Dynamic Safe Interruptibility) Consider a multi-agent learning framework
(i)
(S, A, T, r, m) with Q-values Qt : S ? A(i) ? R at time t ? N. The agents follow the interruptible learning policy IN T ? (? ) to generate a sequence E = (xt , at , rt , yt )t?N and learn on
the processed sequence P (E). This framework is said to be safely interruptible if for any initiation
function I and any interruption policy ? IN T :
1. ?? such that (?t ? 1 when t ? ?) and ((?s ? S, ?a ? A, ?T > 0), ?t > T such that
st = s, at = a)
(i)
2. ?i ? {1, ..., m}, ?t > 0, ?st ? S, ?at ? A(i) , ?Q ? RS?A :
(m)
(i)
(1)
(m)
(i)
(1)
P(Qt+1 = Q | Qt , ..., Qt , st , at , ?) = P(Qt+1 = Q | Qt , ..., Qt , st , at )
We say that sequences ? that satisfy the first condition are admissible.
When ? satisfies condition (1), the learning policy is said to achieve infinite exploration. This definition insists on the fact that the values estimated for each action should not depend on the interruptions. In particular, it ensures the three following properties that are very natural when thinking
about safe interruptibility:
? Interruptions do not prevent exploration.
? If we sample an experience from E then each agent learns the same thing as if all agents
were following non-interruptible policies.
(i)
(i)
? The fixed points of the learning rule Qeq such that Qeq (x, a) = E[Qt+1 (x, a)|Qt =
(i)
Qeq , x, a, ?] for all (x, a) ? S ? A do not depend on ? and so agents Q-maps will
not converge to equilibrium situations that were impossible in the non-interruptible setting.
Yet, interruptions can lead to some state-action pairs being updated more often than others, especially when they tend to push the agents towards specific states. Therefore, when there are several
possible equilibria, it is possible that interruptions bias the Q-values towards one of them. Definition 2 suggests that dynamic safe interruptibility cannot be achieved if the update rule directly
depends on ?, which is why we introduce neutral learning rules.
Definition 3. (Neutral Learning Rule) We say that a multi-agent reinforcement learning framework
is neutral if:
1. F is independent of ?
2. Every experience e in E is independent of ? conditionally on (x, a, Q) where a is the joint
action.
Q-learning is an example of neutral learning rule because the update does not depend on ? and
the experiences only contain (x, a, y, r), and y and r are independent of ? conditionally on (x, a).
On the other hand, the second condition rules out direct uses of algorithms like SARSA where
experience samples contain an action sampled from the current learning policy, which depends on ?.
However, a variant that would sample from ?i instead of IN T ? (?i ) (as introduced in [16]) would
be a neutral learning rule. As we will see in Corollary 2.1, neutral learning rules ensure that each
agent taken independently from the others verifies dynamic safe interruptibility.
5
4
Exploration
In order to hope for convergence of the Q-values to the optimal ones, agents need to fully explore
the environment. In short, every state should be visited infinitely often and every action should be
tried infinitely often in every state [19] in order not to miss states and actions that could yield high
rewards.
Definition 4. (Interruption compatible ) Let (S, A, T, r, m) be any distributed agent system where
each agent follows learning policy ?i . We say that sequence is compatible with interruptions if
t ? 0 and ?? such that ?i ? {1, .., m}, ?i and IN T ? (?i ) achieve infinite exploration.
Sequences of that are compatible with interruptions are fundamental to ensure both regular and
dynamic safe interruptibility when following an -greedy policy. Indeed, if is not compatible with
interruptions, then it is not possible to find any sequence ? such that the first condition of dynamic
safe interruptibility is satisfied. The following theorem proves the existence of such and gives
example of and ? that satisfy the conditions.
Theorem 1. Let c ?]0, 1] and let nt (s) be the number of times the agents are in state s before time
t. Then the two following choices of are compatible with interruptions:
p
? ?t ? N, ?s ? S, t (s) = c/ m nt (s).
? ?t ? N, t = c/ log(t)
p
Examples of admissible ? are ?t (s) = 1 ? c0 / m nt (s) for the first choice and ?t = 1 ? c0 / log(t)
for the second one.
Note that we do not need to make any assumption on the update rule or even on the framework. We
only assume that agents follow an -greedy policy. The assumption on may look very restrictive
(convergence of and ? is really slow) but it is designed to ensure infinite exploration in the worst
case when the operator tries to interrupt all agents at every step. In practical applications, this should
not be the case and a faster convergence rate may be used.
5
Joint Action Learners
We first study interruptibility in a framework in which each agent observes the outcome of the joint
action instead of observing only its own. This is called the joint action learner framework [5] and it
has nice convergence properties (e.g., there are many update rules for which it converges [13, 25]).
A standard assumption in this context is that agents cannot establish a strategy with the others:
otherwise, the system can act as a centralized system. In order to maintain Q-values based on the
joint actions, we need to make the standard assumption that actions are fully observable [12].
Assumption 1. Actions are fully observable, which means that at the end of each turn, each agent
knows precisely the tuple of actions a ? A1 ? ... ? Am that have been performed by all agents.
Definition 5. (JAL) A multi-agent system is made of joint action learners (JAL) if for all i ?
{1, .., m}: Q(i) : S ? A ? R.
Joint action learners can observe the actions of all agents: each agent is able to associate the changes
of states and rewards with the joint action and accurately update its Q-map. Therefore, dynamic
safe interruptibility is ensured with minimal conditions on the update rule as long as there is infinite
exploration.
Theorem 2. Joint action learners with a neutral learning rule verify dynamic safe interruptibility if
sequence is compatible with interruptions.
Proof. Given a triplet < I (i) , ?, ?iIN T >, we know that IN T ? (?) achieves infinite exploration
because is compatible with interruptions. For the second point of Definition 2, we consider an
experience tuple et = (xt , at , rt , yt ) and show that the probability of evolution of the Q-values at
time t + 1 does not depend on ? because yt and rt are independent of ? conditionally on (xt , at ).
(1)
(m)
We note Q?m
and we can then derive the following equalities for all q ? R|S|?|A| :
t = Qt , ..., Qt
6
(i)
P(Qt+1 (xt , at ) = q|Q?m
t , xt , at , ?t ) =
X
?m
P(F (xt , at , r, y, Q?m
t ) = q, y, r|Qt , xt , at , ?t )
(r,y)?R?S
=
X
?m
?m
P(F (xt , at , rt , yt , Q?m
t ) = q|Qt , xt , at , rt , yt , ?t )P(yt = y, rt = r|Qt , xt , at , ?t )
(r,y)?R?S
=
X
?m
?m
P(F (xt , at , rt , yt , Q?m
t ) = q|Qt , xt , at , rt , yt )P(yt = y, rt = r|Qt , xt , at )
(r,y)?R?S
The last step comes from two facts. The first is that F is independent of ? conditionally on (Q?m
t , xt , at ) (by assumption). The second is that (yt , rt ) are independent of ? conditionally on (xt , at ) because at is the joint actions and the interruptions only affect the
(i)
choice of the actions through a change in the policy. P(Qt+1 (xt , at ) = q|Q?m
t , xt , at , ?t ) =
(i)
m
?
P(Qt+1 (xt , at ) = q|Qt , xt , at ). Since only one entry is updated per step, ?Q ? RS?Ai ,
(i)
(i)
?m
P(Qt+1 = Q|Q?m
t , xt , at , ?t ) = P(Qt+1 = Q|Qt , xt , at ).
Corollary 2.1. A single agent with a neutral learning rule and a sequence compatible with interruptions verifies dynamic safe interruptibility.
Theorem 2 and Corollary 2.1 taken together highlight the fact that joint action learners are not very
sensitive to interruptions and that in this framework, if each agent verifies dynamic safe interruptibility then the whole system does.
The question of selecting an action based on the Q-values remains open. In a cooperative setting
with a unique equilibrium, agents can take the action that maximizes their Q-value. When there
are several joint actions with the same value, coordination mechanisms are needed to make sure
that all agents play according to the same strategy [4]. Approaches that rely on anticipating the
strategy of the opponent [23] would introduce dependence to interruptions in the action selection
mechanism. Therefore, the definition of dynamic safe interruptibility should be extended to include
these cases by requiring that any quantity the policy depends on (and not just the Q-values) should
satisfy condition (2) of dynamic safe interruptibility. In non-cooperative games, neutral rules such
as Nash-Q or minimax Q-learning [13] can be used, but they require each agent to know the Q-maps
of the others.
6
Independent Learners
It is not always possible to use joint action learners in practice as the training is very expensive
due to the very large state-actions space. In many real-world applications, multi-agent systems use
independent learners that do not explicitly coordinate [6, 21]. Rather, they rely on the fact that the
agents will adapt to each other and that learning will converge to an optimum. This is not guaranteed
theoretically and there can in fact be many problems [14], but it is often true empirically [24]. More
specifically, Assumption 1 (fully observable actions) is not required anymore. This framework can
be used either when the actions of other agents cannot be observed (for example when several actions
can have the same outcome) or when there are too many agents because it is faster to train. In this
case, we define the Q-values on a smaller space.
Definition 6. (IL) A multi-agent systems is made of independent learners (IL) if for all i ? {1, .., m},
Q(i) : S ? Ai ? R.
This reduces the ability of agents to distinguish why the same state-action pair yields different rewards: they can only associate a change in reward with randomness of the environment. The agents
learn as if they were alone, and they learn the best response to the environment in which agents can
be interrupted. This is exactly what we are trying to avoid. In other words, the learning depends on
the joint policy followed by all the agents which itself depends on ?.
7
6.1
Independent Learners on matrix games
Theorem 3. Independent Q-learners with a neutral learning rule and a sequence compatible with
interruptions do not verify dynamic safe interruptibility.
Proof. Consider a setting with two agents a and b that can perform two actions: 0 and 1. They get
a reward of 1 if the joint action played is (a0 , b0 ) or (a1 , b1 ) and reward 0 otherwise. Agents use Qlearning, which is a neutral learning rule. Let be such that IN T ? (? ) achieves infinite exploration.
We consider the interruption policies ?aIN T = a0 and ?bIN T = b1 with probability 1. Since there is
only one state, we omit it and set ? = 0 (see Equation 1). We assume that the initiation function is
equal to 1 at each step so the probability of actually being interrupted at time t is ?t for each agent.
(a)
(b)
(b)
We fix time t > 0. We define q = (1 ? ?)Qt (a0 ) + ? and we assume that Qt (b1 ) > Qt (b0 ).
(a)
(a)
(b) (a)
(a)
(b) (a)
Therefore P(Qt+1 (a0 ) = q|Qt , Qt , at = a0 , ?t ) = P(rt = 1|Qt , Qt , at = a0 , ?t ) =
(b)
(a)
(b) (a)
P(at = b0 |Qt , Qt , at = a0 , ?t ) = 2 (1 ? ?t ), which depends on ?t so the framework does
not verify dynamic safe interruptibility.
Claus and Boutilier [5] studied very simple matrix games and showed that the Q-maps do not converge but that equilibria are played with probability 1 in the limit. A consequence of Theorem 3
is that even this weak notion of convergence does not hold for independent learners that can be
interrupted.
6.2
Interruptions-aware Independent Learners
Without communication or extra information, independent learners cannot distinguish when the
environment is interrupted and when it is not. As shown in Theorem 3, interruptions will therefore
affect the way agents learn because the same action (only their own) can have different rewards
depending on the actions of other agents, which themselves depend on whether they have been
interrupted or not. This explains the need for the following assumption.
Assumption 2. At the end of each step, before updating the Q-values, each agent receives a signal
that indicates whether an agent has been interrupted or not during this step.
This assumption is realistic because the agents already get a reward signal and observe a new state
from the environment at each step. Therefore, they interact with the environment and the interruption
signal could be given to the agent in the same way that the reward signal is. If Assumption 2 holds,
it is possible to remove histories associated with interruptions.
Definition 7. (Interruption Processing Function) The processing function that prunes interrupted
observations is PIN T (E) = (et ){t?N / ?t =0} where ?t = 0 if no agent has been interrupted at time
t and ?t = 1 otherwise.
Pruning observations has an impact on the empirical transition probabilities in the sequence. For
example, it is possible to bias the equilibrium by removing all transitions that lead to and start
from a specific state, thus making the agent believe this state is unreachable.3 Under our model of
interruptions, we show in the following lemma that pruning of interrupted observations adequately
removes the dependency of the empirical outcome on interruptions (conditionally on the current
state and action).
Lemma 1. Let i ? {1, ..., m} be an agent. For any admissible ? used to generate the experiences
E and e = (y, r, x, ai , Q) ? P (E). Then P(y, r|x, ai , Q, ?) = P(y, r|x, ai , Q).
This lemma justifies our pruning method and is the key step to prove the following theorem.
Theorem 4. Independent learners with processing function PIN T , a neutral update rule and a
sequence compatible with interruptions verify dynamic safe interruptibility.
Proof. (Sketch) Infinite exploration still holds because the proof of Theorem 1 actually used the fact
that even when removing all interrupted events, infinite exploration is still achieved. Then, the proof
3
The example at https://agentfoundations.org/item?id=836 clearly illustrates this problem.
8
is similar to that of Theorem 2, but we have to prove that the transition probabilities conditionally
on the state and action of a given agent in the processed sequence are the same as in an environment
where agents cannot be interrupted, which is proven by Lemma 1.
7
Concluding Remarks
The progress of AI is raising a lot of concerns4 . In particular, it is becoming clear that keeping an
AI system under control requires more than just an off switch. We introduce in this paper dynamic
safe interruptibility, which we believe is the right notion to reason about the safety of multi-agent
systems that do not communicate. In particular, it ensures that infinite exploration and the onestep learning dynamics are preserved, two essential guarantees when learning in the non-stationary
environment of Markov games.
When trying to design a safely interruptible system for a single agent, using off-policy methods
is generally a good idea because the interruptions only impact the action selection so they should
not impact the learning. For multi-agent systems, minimax is a good candidate for action selection
mechanism because it is not impacted by the actions of other agents, and only tries to maximize the
reward of the agent in the worst possible case.
A natural extension of our work would be to study dynamic safe interruptibility when Q-maps are
replaced by neural networks [22, 15], which is a widely used framework in practice. In this setting,
the neural network may overfit states where agents are pushed to by interruptions. A smart experience replay mechanism that would pick observations for which the agents have not been interrupted
for a long time more often than others is likely to solve this issue. More generally, experience replay
mechanisms that compose well with safe interruptibility could allow to compensate for the extra
amount of exploration needed by safely interruptible learning by being more efficient with data.
Thus, they are critical to make these techniques practical. Since Dynamic Safe Interruptibility does
not need proven convergence to the optimal solution, we argue that it is a good definition to study
the interruptibility problem when using function approximators.
The results in this paper indicate that Safe Interruptibility may not be achievable for systems in
which agents do not communicate at all. This means that, rediscussing the cars example, some
global norms of communications would need to be defined to ?implement? safe interruptibility.
We address additional remarks in the section ?Additional remarks? of the extended paper, that can
be found in the supplementary material.
Acknowledgment. This work has been supported in part by the European ERC (Grant 339539 AOC) and by the Swiss National Science Foundation (Grant 200021 169588 TARBDA).
4
https://futureoflife.org/ai-principles/ gives a list of principles that AI researchers should keep in mind when
developing their systems.
9
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11
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6,211 | 6,619 | Interactive Submodular Bandit
1
Lin Chen1,2 , Andreas Krause3 , Amin Karbasi1,2
Department of Electrical Engineering, 2 Yale Institute for Network Science, Yale University
3
Department of Computer Science, ETH Z?rich
{lin.chen, amin.karbasi}@yale.edu, [email protected]
Abstract
In many machine learning applications, submodular functions have been used
as a model for evaluating the utility or payoff of a set such as news items to
recommend, sensors to deploy in a terrain, nodes to influence in a social network,
to name a few. At the heart of all these applications is the assumption that the
underlying utility/payoff function is known a priori, hence maximizing it is in
principle possible. In many real life situations, however, the utility function is not
fully known in advance and can only be estimated via interactions. For instance,
whether a user likes a movie or not can be reliably evaluated only after it was
shown to her. Or, the range of influence of a user in a social network can be
estimated only after she is selected to advertise the product. We model such
problems as an interactive submodular bandit optimization, where in each round
we receive a context (e.g., previously selected movies) and have to choose an action
(e.g., propose a new movie). We then receive a noisy feedback about the utility
of the action (e.g., ratings) which we model as a submodular function over the
context-action space. We develop SM-UCB that efficiently trades off exploration
(collecting more data) and
? exploration (proposing a good action given gathered
data) and achieves a O( T ) regret bound after T rounds of interaction. More
specifically, given a bounded-RKHS norm kernel over the context-action-payoff
space that governs the smoothness of the utility function, SM-UCB keeps an upperconfidence bound on the payoff function that allows it to asymptotically achieve
no-regret. Finally, we evaluate our results on four concrete applications, including
movie recommendation (on the MovieLense data set), news recommendation (on
Yahoo! Webscope dataset), interactive influence maximization (on a subset of the
Facebook network), and personalized data summarization (on Reuters Corpus). In
all these applications, we observe that SM-UCB consistently outperforms the prior
art.
1
Introduction
Interactive learning is a modern machine learning paradigm that has recently received significant
interest in both theory and practice [15, 14, 7, 6]. In this setting, the learning algorithm engages in a
two-way dialog with the environment (e.g., users) by performing actions and receiving a response (e.g.,
like or dislike) for each action. Interactive learning has led to substantial performance improvement
in a variety of machine learning applications [43, 13], including clustering [4, 25, 1], classification
[46, 10], language learning [48], decision making [26], and recommender systems [28], to name a
few.
At a high level, interactive learning can be cast as a dynamic optimization problem with a known
utility/payoff function where the goal is to achieve an objective whose value depends on the selected
actions, their responses, and the state of the environment. In many practical settings, the utility
functions are submodular, stating (informally) that the payoff of performing an action earlier is more
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
than performing it later. In fact, rigorous treatment of submodularity in interactive settings led to
strong theoretical guarantees on the performance of greedy policies [19, 22].
In this paper, we go one step further and consider scenarios where the exact form of the submodular
payoff function is not completely known and hence needs to be estimated through interactions. This
problem is closely related to the contextual multi-armed bandit [38, 9, 36] where for a sequence of
T rounds, we receive a payoff function along with some side information or context (e.g., user?s
features), based on which we have to choose an action (e.g., proposing an item) and then a noisy
feedback about the obtained payoff is revealed (e.g., rating of the proposed item). The goal is to
minimize the regret of not selecting the optimal action due to the uncertainty associated with the
utility function. The interactive contextual bandit generalizes this setting by allowing to interact with
a payoff function multiple times, where each time we need to take a new action based on both the
context and previously taken actions. The regret is then defined in terms of the difference between the
utility of the best set of actions that we could have chosen versus the ones that are actually selected.
In this paper, we further assume that the marginal payoffs of actions show diminishing returns. This
problem, which we call interactive submodular bandit, appears in many practical scenarios, including:
? Interactive recommender system. The goal is to design a recommender system that
interacts with the users in order to elicit and satisfy their preferences. In our approach, we
model the utility of a set of items as an unknown submodular objective function that the
recommender systems aims to maximize. In each round of interaction, the recommender
system decides which item should be presented to the user, given the previously proposed
items to this or similar users (affinity between users, if this side information exists, can
be leveraged to enhance recommendation performance). Since the users? preferences are
unknown, the recommender system can only gather information about the users through the
feedback they provide in terms of ratings. A successful recommender system should be able
to minimize the total regret accumulated over T iterations with users.
? Interactive influence maximization. Influence spread maximization addresses the problem
of selecting the most influential source nodes of a given size in a diffusion network [51].
A diffusion process that starts with those source nodes can potentially reach the greatest
number of nodes in the network. Under many diffusion models, the expected total number
of influenced people is a submodular function of the seed subjects [29, 20]. In a natural
interactive variant of this problem, users may be recruited in a sequential manner [42, 5]
where a new user is selected once we fully observe the extent to which the current seed users
influenced people. Note that finding the optimal set of source nodes in a diffusion network
depends dramatically on the underlying dynamics of the diffusion. Again, very often in
practice, we are faced with the dilemma of estimating the underlying diffusion parameters
through interactively selecting the nodes while at the same time trying to maximize the
influence.
? Interactive coverage. Coverage problems arise naturally in many applications [32, 33].
Consider an environmental monitoring task, for instance, where sensors are placed in the
Alps to better predict floods, landslides and avalanches [8]. Similarly, Wi-Fi hotspots are
carefully arranged to cover every corner of a floor. However, it is likely that the actual
coverage of a device is uncertain before deployment due to unknown conditions present in
the environment. Hence, we might need to install devices in a sequential manner after we
observe the actual coverage of the ones already installed [19, 22].
Without any assumptions about the smoothness of the payoff function, no algorithm may be able
to achieve low regret [31]. Thus, in our setting, we make a crucial yet very natural assumption
that the space of context-action-payoff has low complexity, quantified in terms of the Reproducing
Kernel Hilbert Space (RKHS) norm associated with some kernel [24].
We then show that SM?
UCB, an upper-confidence-bound based algorithm achieves an O( T ) regret. We also evaluate
the performance of SM-UCB on four real-world applications, including movie recommendation
[40], news recommendation [16], interactive influence maximization [42, 34], and personalized data
summarization [39].
2
2
Problem Formulation
As we stated earlier, many utility or payoff functions we encounter in machine learning applications
are submodular. As a reminder, a set function f is called submodular if for all A ? B ? ? we have
f (A) + f (B) ? f (A ? B) + f (A ? B).
An equivalent definition of submodularity that shows better the diminishing returns property is as
follows: for all A ? B ? ? and any element e 6? B we have
f (A ? {e}) ? f (A) ? f (B ? {e}) ? f (B).
We also denote the marginal gain of an element e to a set A by ?(e|A) , f (A ? {e}) ? f (A). The
function f is called monotone if for all A ? B we have f (A) ? f (B).
In this paper, we consider a sequential decision making process for a horizon of T time steps, where
at each round i, a monotone submodular function fi , is (partially) revealed. Let us first consider the
simple bandit problem [50] where we need to select an item (or an arm) e from the set of items ?
such that fi (e) is maximized. After the item e is selected, the payoff fi (e) is revealed. Since ft ?s
are not known in advance, the goal is to minimize the accumulated regret over T rounds, for not
choosing the optimum items. Contextual submodular bandit generalizes the aforementioned setting
by allowing to receive side information ?i (also called context) in each round i [31]. But still the goal
is to select a single item e such that f?i (e) is maximized. In the interactive contextual submodular
bandit, the focus of this paper, we may encounter the same valuation function f? , with its associate
context ?, multiple times over the time horizon T . Here instead, at each round, we need to propose a
new item that maximizes the marginal payoff given the ones we selected in the previous encounters.
Therefore, we are sequentially building up subsets of items that maximize the payoff for each separate
function f? . For instance, a recommender system may interact with a user (or a number of users)
multiple times. In each interaction, it has to recommend a new item while taking into account what it
has recommended in previous interactions.
More formally, let us assume that we encounter m ? T distinct functions f? , in an arbitrary
order, over the time horizon T , i.e., ? ? {?1 , . . . , ?m }. We denote the arriving ordered sequence
by f1 , f2 , . . . , fT where for each round i, we have fi ? {f?1 , . . . , f?m }. Let us also denote by
ui ? {1, . . . , m} the index of the context received in round i. We also need to maintain a collection of
m sets S1 , . . . , Sm , (initialized to the empty set) corresponding to f?1 , . . . , f?m . Our goal is to select
a subset Sj ? ? for each function f?j that maximizes its corresponding utility f?j (Sj ). Note that if
f?j were known in advance, we could simply use the greedy algorithm. However, in the interactive
submodular bandit setting, we need to build up the sets Sj sequentially and through interactions, as
P
the marginal payoff of an element is only revealed after it is selected. Let oi , j?i 1{uj = ui }
denote the number of occurrences of function f?ui in the first i rounds. In each round, say the i-th
with the corresponding function f?ui , we need to select a new item xoi ,ui from the set of items ?
and add it to Sui . Clearly, after including xoi ,ui , the set Sui will be of cardinality oi . For the ease of
presentation, we denote Sui ? {xoi ,ui } by Soi ,ui , initialized to the empty set in the beginning, i.e.,
S0,ui = ?. After selecting the item xoi ,ui and given the previously selected items Sui , we receive yi ,
a noisy (but unbiased) estimate of xoi ,ui ?s marginal payoff, i.e.,
yi = ?(xoi ,ui |Soi ?1,ui , ?ui ) + i ,
where the marginal gain ?(?|?, ?) : ? ? 2? ? ? ? R is defined as
?(x|S, ?) = f? (S ? {x}) ? f? (S).
(1)
We also assume that i ?s are uniformly bounded noise variables that form a martingale difference
sequence, i.e.,
E[i |1 , 2 , . . . , i?1 ] = 0 and |i |? ? for all i ? {1, . . . , T }.
We call ? the noise level. Note that yi is the only feedback that we obtain in round i. It crucially
depends on the previously selected items and contextual information Soi ?1,ui , ?ui . Therefore, the
only avenue through which we can learn about the payoff functions f? (that we try to optimize over)
is via noisy feedbacks yi . We need to design an algorithm that minimizes the accumulated regret
over the total number of T rounds. Formally, we compare the performance of any algorithm in this
interactive submodular bandit setting with that of the greedy algorithm with the full knowledge of the
payoff functions f?1 , . . . , f?m .
3
Algorithm 1 SM-UCB
Input: set of items ?, mean ?0 = 0, variance ?0 .
1: Initialize Si ? ? for all i ? [m]
2: for i = 1, 2, 3, . . . do
?
3:
select an item xoi ,ui ? argmaxx?? ?i?1 (x) + ?i ?i?1 (x)
4:
update the set Sui ? Sui ? {xoi ,ui }
5:
obtain the feedback yi = ?(xoi ,ui |Soi ?1,ui , ?ui ) + i
6:
let ki (x) be a vector-valued function that outputs an i-dimensional column vector with j-th
entry k((xoj ,uj , Soj ?1,uj , ?uj ), (x, Soi ?1,ui , ?ui ))
7:
let Ki be an i?i matrix with (j, j 0 )-entry k((xoj ,uj , Soj ?1,uj , ?uj ), (xoj0 ,uj0 , Soj0 ?1,uj0 , ?uj0 ))
8:
update yi ? [y1 , y2 , . . . , yi ]T
9:
let ki (x, x0 ) be a kernel function defined as k(x, x0 ) ? ki (x)T (Ki + ? 2 I)?1 ki (x0 )
10:
estimate ?i (x) ? p
ki (x)T (Ki + ? 2 I)?1 yi
11:
estimate ?i (x) ? ki (x, x)
12: end for
Suppose that by the end of the T -th round, an algorithm has selected Tj items for the payoff
function f?j ; therefore the cardinality of Sj by the end of the T -th round is Tj . Thus, we have
P
Pm
Tj = t?T 1{ut = j} and T = j=1 Tj . We use Sj? to denote the set that maximizes the payoff
of function f?j with at most Tj elements, i.e., Sj? = argmax|S|?Tj f?j (S). We know that the greedy
Pm
algorithm is guaranteed to achieve (1 ? 1/e) j=1 f?j (Sj? ) [41] and there is no polynomial time
algorithm that achieves a better approximation guarantee in general [17]. Therefore, we define the
total regret of an algorithm up to round T as follows:
RT , (1 ? 1/e)
m
X
j=1
f?j (Sj? ) ?
m
X
f?j (STj ,j ),
(2)
j=1
which is the gap between the greedy algorithm?s guarantee and the total utility obtained by the
algorithm. Without any smoothness assumption over the payoff functions, it may not be possible to
guarantee a sublinear regret [31]. In this paper, we make a natural assumption about the complexity
of payoff functions. More specifically, we assume that the marginal payoffs, defined in (1), have a low
RKHS-norm according to a kernel k : (? ? 2? ? ?) ? (? ? 2? ? ?) ? R, i.e., k?(?|?, ?)kk ? B.
Note that such a kernel encodes how close two marginal payoffs are if a) the contexts ?i and ?j or b)
the selected elements Si and Sj are similar. For instance, a recommender system can leverage this
information to propose an item to a user if it has observed that a user with similar features liked that
item.
3
Main Results
In Algorithm 1 we propose SM-UCB, an interactive submodular bandit algorithm. Recall that
the marginal gain function ? has a low RKHS norm w.r.t. some kernel k. In each round, say the
i-th, SM-UCB maintains the posterior mean ?i?1 (?) and standard deviation ?i?1 (?) conditioned
on the historical observations or context {(xoj ,uj , Soj ?1,uj , ?uj ) : 1 ? j ? i}. Based on these
posterior estimates,
SM-UCB then selects an item x that attains the highest upper confidence bound
?
?i?1 (x) + ?i ?i?1 (x). It then receives the noisy feedback yi = ?(xoi ,ui |Soi ?1,ui , ?ui ) + i . Since
i ?s are uniformly bounded and form a martingale difference sequence, SM-UCB can predict the
mean ?i and standard deviation ?i via posterior inference in order to determine the item to be selected
in the next round.
In order to bound the regret of an algorithm, we need to quantify how much information that algorithm
can acquire through interactions. Let yA denote a subset of noisy observations indexed by the set
A, i.e., yA = {yi |i ? A}. Note that any subset A of noisy observations yA reduces our uncertainty
about the marginal gain function ?. In an extreme case, if we had perfect information (or no
uncertainty) about ?, we could have achieved zero regret. We can precisely quantify this notion
through what is called the information gain I(yA ; ?) = H(yA ) ? H(yA |?), where H denotes the
Shannon entropy. In fact, by lifting the results from [31, 44] to a much more general setting, we
relate the regret to the maximum information gain ?T [44] obtained after T rounds and defined as
4
?T , maxA??:|A|=T I(yA ; ?). Another important quantity that shows up in the regret bound is the
confidence parameter ?T (see line 3 of Algorithm 1) that needs to be chosen carefully so that our
theoretical guarantee
holds with high probability. In fact, the following theorem shows that SM-UCB
?
attains a O( T ?T ?T ) regret bound with high probability.
Theorem 1. Suppose that the true marginal gain function ?(?|?, ?) has a small RKHS norm according
to some kernel k, i.e., k?(?|?, ?)kk ? B. The noise variables t satisfy E[t |1 , 2 , . . . , t?1 ] = 0
for all t ? N and are uniformly bounded by ?. Let ? ? (0, 1), ?t = 2B 2 + 300?t ln3 (t/?) and
C1 = 8/log(1 + ? ?2 ). Then, the accumulated regret of SM-UCB over T rounds is as follows:
n
o
p
Pr RT ? C1 T ?T ?T + 2, ?T ? 1 ? 1 ? ?.
The proof of the above theorem is provided in the Supplementary Material. It relies on two powerful
ideas: greedy selection for constrained submodular maximization [41] and upper confidence bounds
of contextual Gaussian bandit [31]. If the marginal payoffs were completely known, then the greedy
policy would provide a competitive solution to the optimum. However, one cannot run the greedy
policy without knowing the marginal gains. In fact, there are strong negative results regarding the
approximation guarantee of any polynomial time algorithm if the marginal gains are arbitrarily noisy
[23]. Instead, SM-UCB relies on optimistic estimates of the marginal gains and select greedily an
item with the highest upper confidence bound. By assuming that marginal gains are smooth and
relying on Theorem 1 in [31], we can control the accumulated error of a greedy-like solution that
relies on confidence bounds and obtain low regret. Our setting and theoretical result generalize a
number of prior work mentioned below.
Linear submodular bandit [50]. In this setting, the objective function has the form f (S) =
Pd
positive unknown coeffii=1 wi fi (S), where fi ?s are known submodular functions and wi ?s are P
d
cients. Therefore, the marginal gain function can be written as ?(x|S) = i=1 wi ?i (x|S), where
?i (?|?)?s are known functions and wi ?s are unknown coefficients. Let w = (w1 , w2 , . . . , wd ) denote
the weight vector. Since the only unknown part of the marginal gain function is the weight vector,
the space of the marginal gain function is isomorphic to the space of weight vectors, which is in
fact a d-dimensional Euclidean space Rd . The RKHS norm of ? is given by some norm in Rd ;
i.e., k?kk , kwk. The assumption in [50] that kwk? B is equivalent to assuming that k?kk ? B.
Therefore, the linear bandit setting is included in our setting where the marginal gain function ? has
a special form and its RKHS norm is given by the norm of its corresponding weight vector in the
Euclidean space. Also, LSBG REEDY proposed in [50], is a special case of SM-UCB (except that
the feedback is delayed).
Adaptive valuable
P item discovery [47]. In this setting, the objective function has the form
f (S) = (1 ? ?) x?S g(x) + ?D(S), where D is a known submodular function that quantifies
the diversity of the items in S, g is an unknown function that denotes the utility g(x) for any item
x, and ? is a known tradeoff parameter balancing the importance of theP
accumulative utility and
joint diversity of the items. Note that the unknown function M (S) = x?S g(x) is a modular
function. Therefore, the marginal gain function has the form ?(x|S) = (1 ? ?)g(x) + ?D(x|S),
where D(x|S) , D({x} ? S) ? D(S). The only uncertainty of ? arises from the uncertainty about
the modular function M . In particular, [47] assumes that the RKHS norm of g is bounded. Again,
our setting encompasses adaptive valuable item discovery as we consider any monotone submodular
function. Moreover, GPS ELECT proposed in [47], is a special case of SM-UCB.
Contextual Gaussian bandit [31]. This is the closest setting to ours where in each round i we
receive a context ?i from the set of contexts ? and have to choose an item x from the set of items
?. We then receive a payoff f?i (x) + t . Note that instead of building up a set (our problem), in
the contextual bandit process we simply choose a single element for each function f?i as the main
assumption is that we encounter each function only once. To obtain regret bounds it is assumed in
[31] that f has low norm in the RKHS associated with some kernel k. Again, CGP-UCB proposed
in [31], is a special case of SM-UCB.
5
4
Experiments
In this section, we compare empirically the performance of SM-UCB with the following baselines:
? R ANDOM. In each round, an item is randomly selected for the current payoff function f? .
? G REEDY. It has the full knowledge of the submodular functions f? . In each round, say
the i-th with the corresponding function f?ui , G REEDY selects the item that maximizes the
marginal gain, i.e., argmaxx?? ?(x|Soi ?1 , ?ui ).
? H ISTORY-F REE. We run SM-UCB without considering the previously selected items.
H ISTORY-F REE is basically the contextual Gaussian bandit algorithm proposed in [31]
whose context is the user feature.
? F EATURE -F REE. We run SM-UCB without considering the context ? of an arriving
function f .
? C ONTEXT-F REE. We run SM-UCB without considering the context or the previously
selected elements. In fact, C ONTEXT-F REE is basically the GP-S ELECT algorithm proposed
in [47].
In all of our experiments, the m distinct functions {f?i : 1 ? i ? m} that the algorithm encounters
represent the valuation functions of m users, where the context ?i ? Rd encodes users? features.
Moreover, Si is the set of items that an algorithm selects for user i ? [m]. Instead of computing
the
Pmregret, we quantify the performance of the algorithms by computing
Pm the accumulated reward
f
(S
).
Recall
that
the
regret
is
given
by
(1
?
1/e)
?
OP
T
?
?
i
i
i=1 f?i (Si ), where OP T =
i=1
Pm
?
f
(S
)
is
a
constant
generally
hard
to
compute.
i
i=1 ?i
Movie Recommendation In this set of experiments, we use the MovieLens dataset1 where a userrating matrix M is provided. The rows of M represent users and the columns represent movies. The
matrix M contains 943 users and 1682 movies. As a preprocessing step, we apply the singular-value
decomposition (SVD) to impute the missing values; the six largest singular values are kept.
In the first part of the study, we use the submatrix of M that consists of 80% of the users and all
of the movies for training the feature vectors of movies via SVD. Let M 0 be the submatrix of M
that consists of the remaining nuser users and all of the movies; this matrix is for testing. Let ?
denote the set of movies. We
selecting a subset of ? to maximize the facility-locationPnconsider
user
0
type objective [30] f (S) = i=1
maxj?S Mij
. This objective function corresponds to a scenario
without any context ? as there is only one payoff function f that we are trying to maximize. Thus,
F EATURE -F REE does not apply here. We use the cosine kernel kmovie : ? ? ? ? R for pairs of
movies and Jaccard kernel ksubset (S, T ) = |S ? T |/|S ? T | [18] for pairs of subsets of movies, say
S and T . The composite kernel k : (? ? 2? ) ? (? ? 2? ) ? R is defined as ?1 kmovie ? ?2 ksubset ,
i.e., k((u, S), (v, T )) = ?1 kmovie (u, v) + ?2 ksubset (S, T ), where ?1 , ?2 > 0. The results are shown
in Fig. 1(a). The horizontal axis denotes the cardinality of S. The vertical axis denotes the function
value of f on the set S. We observe that SM-UCB outperforms all of the baselines except the
practically infeasible G REEDY.
In the second part, we consider a setting where a separate subset of movies is selected for each user.
We cluster the users in the dataset into 40 groups via the k-means algorithm and the users of the
same group are viewed as identical users. The feature vector of a group of users is the mean of the
feature vectors of all member users and the rating of a group is the sum of the ratings of all member
users. The users are labeled as 1, 2, 3, . . . , n0user , where n0user = 40. Similar to the first part, the
feature vectors of the users and movies are obtained via SVD. We maintain a set Si for user i. The
00
00
objective function is f?i (S) = maxj?S Mij
, where Mij
is user i?s rating for movie j. In addition,
we also need a collective objective function that quantifies the overall performance of an algorithm
Pn0user
for all users. It is defined as f (S1 , S2 , . . . , Sn0user ) = i=1
f?i (Si ). We assume a random arrival
of users. We use the linear kernel kuser : ? ? ? ? R for pairs of users. The composite kernel
k : (? ? 2? ? ?) ? (? ? 2? ? ?) ? R is defined as ?1 kmovie ? ?2 ksubset ? ?3 kuser . In Fig. 1(b),
we plot the performance of SM-UCB against other baselines. The horizontal axis denotes the number
of user arrivals while the vertical axis denotes the value of the collective objective function. We
1
https://grouplens.org/datasets/movielens/
6
Greedy
SM-UCB
SM-UCB
Feature free
Greedy
History free
Context free
History free
Random
Random
(a)
(b)
History free
SM-UCB
SM-UCB
Random
History free
Greedy
Random
Greedy
(c)
(d)
SM-UCB
Greedy
Greedy
History free
SM-UCB
Feature free
Random
History free
Feature free
Context free
Context free
(e)
(f)
Figure 1: Figs. 1(a) and 1(b) show the results of the experiments on the MovieLens dataset. Fig. 1(a)
shows how the total objective function for all users evolves as the number of selected movies increases;
the algorithm recommends the same subset of movies to all users. In Fig. 1(b), we consider the
situation where users arrive in a random order and we have to recommend a separate subset of movies
to each user. Figs. 1(c) and 1(d) show the dependency of fraction of influenced nodes on the target
set size in the Facebook network and the student network from the User Knowledge Modelling
Dataset [27]. Fig. 1(e) shows how the payoff function varies as more users arrive in the Yahoo news
recommender. In Fig. 1(f), we consider the personalized data summarization from Reuters corpus for
arriving users. It shows the fraction of covered topics versus the number of user arrivals.
observe that SM-UCB outperforms all other baselines except G REEDY. In addition, C ONTEXT-F REE
that uses the least amount of information achieves a lower function value than H ISTORY-F REE and
F EATURE -F REE, which either leverages the information about users? features or previously selected
items.
7
Interactive Influence Maximization For this experiment, we use the Facebook network provided
in [35]. The goal is to choose a subset of subjects in the network, which we call the target set, in
order to maximize the number of influenced subjects. We assume that each member in the target
set can influence
all of her neighbors.
Under this assumption, the submodular objective function is
S
f (S) = u?S (N (u) ? {u}) , where N (u) is the set of all neighbors of subject u. All the baselines,
except G REEDY, have no knowledge of the underlying Facebook network or the objective function.
They are only given the feature vector of each subject obtained via the NODE 2 VEC algorithm [21].
The kernel function ksubject between two subjects is a linear kernel while the kernel function between
subsets of subjects is the Jaccard kernel. The results are shown in Fig. 1(c). Again, SM-UCB reaches
the largest influence w.r.t other baselines except for G REEDY. We ran the same idea over the 6-nearest
neighbor network of randomly sampled 150 students from User Knowledge Modelling Dataset [27].
As Fig. 1(d) indicates, a similar pattern emerges.
News Recommendation For this experiment, we use the Yahoo! Webscope dataset R6A2 . The
dataset provides a list of records, each containing a time stamp, a user ID, a news article ID and
a Boolean value that indicates whether the user clicked on the news article that was presented to
her. The feature vectors of the users and the articles are also provided. We use k-means clustering
to cluster users into 175 groups and identify users of the same group as identical users. We form a
matrix M whose (i, j)-entry is the total number of times that user i clicked on article j. This matrix
quantifies each user?s preferences regarding news articles. The objective function for user i is defined
as f?i (Si ) = maxj?Si Mij . The collective objective function f is defined as the sum of the objective
functions of all users. From the time stamps, we can infer the order in which the users arrive. We
use the Laplacian kernels knews : ? ? ? ? R and kuser : ? ? ? ? R for pairs of pieces of news
and pairs of users, respectively. For a pair of subsets of news S and T , the kernel function between
them is again the Jaccard kernel. The composite kernel k : (? ? 2? ? ?) ? (? ? 2? ? ?) ? R is
defined as ?1 knews ? ?2 ksubset ? ?3 kuser . The results are illustrated in Fig. 1(e). The horizontal axis
is the number of arriving users while the vertical axis is the value of the collective objective function.
Again, we observe that SM-UCB outperforms all other methods except G REEDY.
Personalized Data Summarization For this experiment, we apply latent Dirichlet allocation
(LDA) to the Reuters Corpus. The number of topics is set to ntopic = 10. LDA returns a topic
distribution P (i|a) for each article a and topic i. Suppose that A is a subset of articles. Probabilistic
coverage function quantifies
the degree to which a set of articles A covers a topic i [16], and is
Q
given by Fi (A) = 1 ? a?A (1 ? P (i|a)). Each user j is characterized by her ntopic -dimensional
preference vector wj = (wj,1 , wj,2 , wj,3 , . . . , wj,ntopic ); we assume that the preference vector is
L1 -normalized, i.e., its entries
sum to 1. The personalized probabilistic coverage function for user
Pntopic
j is defined as fj (A) = i=1
wj,i Fi (A) [16, 50]. Note that since the preference vector is L1 normalized and
i (A) ? 1, we have fj (A) ? 1 for any j. The total average coverage function is
PF
nuser
1
f (A) = nuser
j=1 fj (A), where nuser = 10 is the number of users. Random order of user arrivals
is simulated. We use the linear kernel for pairs of users and pairs of articles and use the Jaccard kernel
between subsets of articles. The results are shown in Fig. 1(f). The horizontal axis is the number of
user arrivals while the vertical axis is the total average coverage function f (A), which characterizes
the average fraction of covered topics. We observe that SM-UCB outperforms all the baselines other
than G REEDY.
Discussion Recall that the RKHS is a complete subspace of the L2 space of functions defined on
the product of the item set, its power set, and the context set. It has an inner product (?, ?)k obeying
the reproducing property: (f, k(x, ?))k =P
f (x) for all f in RKHS. Functions implied by a particular
kernel k are always of the form f (x) = i ?i k(xi , x). The bounded norm implies that ?i vanish
quickly enough. With universal kernels like Gaussian/Laplacian kernels, such functions are dense
(according to sup-norm) in the space of continuous functions.
In three sets of experiments (movie recommendation, influence maximization, data summarization)
we used the linear and cosine kernels for items and users, and the Jaccard kernel for subsets of items.
In fact, the Jaccard kernel is a widely used metric that quantifies the similarity between subsets of
selected items. Moreover, the linear and cosine kernels between items and users capture the simplest
form of interactions. In contrast to the the above three experiments, in the news recommendation
2
http://webscope.sandbox.yahoo.com/
8
application, we chose the Laplacian kernel for the following reason. The features provided in the
dataset have highly heterogeneous norms. If we use the linear kernel, the inner product between a
short vector and a close-by vector with a small norm will be easily dominated by the inner product
with a vector with a large norm. We used the Laplacian kernel to circumvent this problem and put
more weight on nearby vectors even if they have small norms.
5
Related Work
Originally, Auer et al. [2] proposed UCB policies for the multi-armed
? bandit (MAB) problem which
exhibits the exploration-exploitation tradeoff and achieves an O( T ) regret. In the literature, there
are many variants of the multi-armed bandit problem and corresponding solutions, for example,
E XP 3 algorithm for adversarial bandits [3], L IN UCB for stochastic contextual bandits [36, 12], and a
family of UCB-based policies for infinitely many-armed bandit [49]. Chen et al. [11] considered
the combinatorial MAB problem where the unit of play is a super arm and base arms can be
probabilistically triggered. For a comprehensive survey on bandit problems, we refer the interested
reader to [9].
Srinivas et al. [44] studied the Gaussian process (GP) optimization problem in the bandit setting.
They assumed that the objective function f is either sampled from a Gaussian process or resides
in
Pa reproducing kernel Hilbert space (RKHS). Given a subset of items S ? ? , the total utility is
or the RKHS assumption, they showed that their proposed
x?? f (x). Under either the GP model
?
GP-UCB algorithm achieves an O( T ) regret bound. It is noteworthy to mention that their bound
also relies on the maximum information gain. Based on [44], Krause and Ong [31] further investigated
the contextual Guassian process bandit optimization and their proposed algorithm CGP-UCB
achieves a similar regret bound. Lin et al. [37] addressed an online learning problem where the
input to the greedy algorithm is stochastic with unknown parameters and the algorithm receives
semi-bandit feedbacks. Their algorithm can also be applied to submodular functions. However, there
are several major differences between their work and ours: Firstly, they assume that the objective
functions are drawn from a predetermined but unknown distribution, while our work applies to any
set of submodular functions; secondly they assume bounded submodular functions while we have no
such assumptions; thirdly, their work did not have the notion of context. They optimize the expected
objective function while we optimize objective functions with different contexts simultaneously.
Streeter and Golovin [45] studied the online maximization problem of submodular functions. Yue and
Guestrin [50] studied the linear submodular bandit problem where they assumed that the unknown
submodular function is a linear combination of multiple known submodular functions. The only
uncertainty in their setting is the unknown positive coefficients
of each known submodular function.
?
They proposed LSBG REEDY that achieves a similar O( T ) regret bound. Beyond unconstrained
sequential decision problems, Zhou et al. [52] considered online maximization of list submodular
functions under a knapsack constraint.
Our key contribution in this paper is that the notion of contextual regret that we bound is much more
challenging than the typical notion: Our actions are affecting the future contexts experienced, and we
compete with policies that are aware of this fact and can plan for it. This is qualitatively different
from any prior analysis. More specifically, we need to build up a subset of items/actions as we
encounter a valuation function multiple times. This is a non-trivial task as not only the functions are
unknown, the marginal gains are also noisy. Moreover, the choices we make can affect the future. Our
positive results can be seen in light of very recent negative results in [23] that indicates submodular
optimization is hard when function evaluations are noisy. We show that the UCB-based algorithm
can be naturally combined with the greedy selection policy to provide sublinear regret. To the best of
our knowledge the analysis is new.
Acknowledgements
This research was supported by DARPA Young Faculty Award (D16AP00046), grant SCADAPT and
ERC StG.
9
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srinivas:2 |
6,212 | 662 | Mapping Between Neural and Physical
Activities of the Lobster Gastric Mill
Kenji Doya
Mary E. T. Boyle
Allen I. Selverston
Department of Biology
University of California, San Diego
La Jolla, CA 92093-0322
Abstract
A computer model of the musculoskeletal system of the lobster
gastric mill was constructed in order to provide a behavioral interpretation of the rhythmic patterns obtained from isolated stomatogastric ganglion. The model was based on Hill's muscle model
and quasi-static approximation of the skeletal dynamics and could
simulate the change of chewing patterns by the effect of neuromodulators.
1
THE STOMATOGASTRIC NERVOUS SYSTEM
The crustacean stomatogastric ganglion (STG) is a circuit of 30 neurons that controls rhythmic movement of the foregut. It is one of the best elucidated neural
circuits. All the neurons and the synaptic connections between them are identified and the effects of neuromodulators on the oscillation patterns and neuronal
characteristics have been extensively studied (Selverston and Moulins 1987, H arrisWarrick et al. 1992). However, STG's function as a controller of ingestive behavior
is not fully understood in part because of our poor understanding of the controlled
object: the musculoskeletal dynamics of the foregut. We constructed a mathematical model of the gastric mill, three teeth in the stomach, in order to predict motor
patterns from the neural oscillation patterns which are recorded from the isolated
ganglion.
The animal we used was the Californian spiny lobster (Panulirus interruptus), which
913
914
Doya, Boyle, and Selverston
(a)
medial tooth _ _- -
esophagus
flexible endoscope
? Inhibitory
? Functional Inhibitory
6 Excitatory
, FWlCionai Excitalory
JVV\...
Electronic
Figure 1: The lobster stomatogastric system. (a) Cross section of the foregut
(objects are not to scale). (b) The gastric circuit.
is available locally. The stomatogastric nervous system controls four parts of the
foregut: esophagus, cardiac sac (stomach), gastric mill, and pylorus (entrance to
the intestine) (Figure l.a). The gastric mill is composed of one medial tooth and
two lateral teeth. These grind large chunks of foods (mollusks, algae, crabs, sea
urchins, etc.) into smaller pieces and mix them with digestive fluids. The chewing
period ranges from 5 to 10 seconds. Several different chewing patterns have been
analyzed using an endoscope (Heinzel 1988a, Boyle et al. 1990). Figure 2 shows
two of the typical chewing patterns: "cut and grind" and "cut and squeeze" .
The STG is located in the opthalmic artery which runs from the heart to brain over
the dorsal surface of the stomach. When it is taken out with two other ganglia (the
esophageal ganglion and the commissural ganglion), it can still generate rhythmic
motor outputs. This isolated preparation is ideal for studying the mechanism of
rhythmic pattern generation by a neural circuit.
From pairwise stimulus and
response of the neurons, the map of synaptic connections has been established.
Figure 1 (b) shows a subset of the STG circuit which controls the motion of the
gastric mill. It consists of 11 neurons of 7 types. GM and DG neurons control the
medial tooth and LPG, MG, and LG neurons control the lateral teeth. A question of
interest is how this simple neural network is utilized to control the various movement
patterns of the gastric mill, which is a fairly complex musculoskeletal system.
The oscillation pattern of the isolated ganglion can be modulated by perfusing it
with of several neuromodulators, e.g. proctolin, octopamine (Heinzel and Selverston 1988), CCK (Turrigiano 1990), and pilocarpine (Elson and Selverston 1992).
However, the behavioral interpretation of these different activity patterns is not
well understood. The gastric mill is composed of 7 ossicles (small bones) which is
loosely suspended by more than 20 muscles and connective tissues. That makes it is
very difficult to intuitively estimate the effect of the change of neural firing patterns
in terms of the teeth movement. Therefore we, decided to construct a quantitative
model of the musculoskeletal system of the gastric mill.
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill
(a)
(b)
Figure 2: Typical chewing patterns of the gastric mill. (a) cut and grind. (b) cut
and squeeze.
2
PHYSIOLOGICAL EXPERIMENTS
In order to design a model and determine its parameters, we performed anatomical
and physiological experiments described below.
Anatomical experiments: The carapace and the skin above the stomach mill
was removed to expose a dorsal view of the ossicles and the muscles which control
the gastric mill. Usually, the gastric mill was quiescent without any stimuli. The
positions of the ossicles and the lengths of the muscles at the resting state was
measured. After the behavioral experiments mentioned below, the gastric mill was
taken out and the size of the ossicles and the positions of the attachment points of
the muscles were measured.
Behavioral experiments: With the carapace removed and the gastric mill exposed, one video camera was used to record the movement of the ossicles and the
muscles. Another video camera attached to a flexible endoscope was used to record
the motion of the teeth from inside the stomach. In the resting state, muscles were
stimulated by a wire electrode to determine the behavioral effects. In order to induce chewing, neuromodulators such as proctolin and pilocarpine were injected into
the artery in which STG is located.
Single muscle experiments: The gm!, the largest of the gastric mill muscles,
was used to estimate the parameters of the muscle model mentioned below. It was
removed without disrupting the carapace or ossicle attachment points and fixed to a
tension measurement apparatus. The nerve fiber aln that innervates gmt was stimulated using a suction electrode. The time course of isometric tension was recorded
at different muscle lengths and stimulus frequencies. The parameters obtained from
the gmt muscle experiment were applied to other muscles by considering their relative length and thickness.
915
916
Doya, Boyle, and Selverston
(a)
contraction element (CE)
serial elasticity (SE)
parallel elasticity (PE)
(d)
(c)
fO
fs
f max
o
o
leo
Figure 3: The Hill-based muscle model.
3
MODELING THE MUSCULOSKELETAL SYSTEM
3.1
MUSCULAR DYNAMICS
There are many ways to model muscles. In the simplest models, the tension or the
length of a muscle is regarded as an instantaneous function of the spike frequency
of the motor nerve. In some engineering approaches, a muscle is considered as
a spring whose resting length and stiffness are modulated by the nervous input
(Hogan 1984). Since these models are a linear static approximation of the nonlinear
dynamical characteristics of muscles, their parameters must be changed to simulate
different motor tasks (Winters90). Molecular models (Zahalak 1990), which are
based on the binding mechanisms of actin and myosin fibers, can explain the widest
range of muscular characteristics found in physiological experiments. However,
these complex models have many parameters which are difficult to estimate.
The model we employed was a nonlinear macroscopic model based on A. V. Hill's
formulation (Hill 1938, Winters 1990). The model is composed of a contractile
element (CE), a serial elasticity (SE), and a parallel elasticity (PE) (Figure 3.a).
This model is based on empirical data about nonlinear characteristics of muscles
and its parameters can be determined by physiological experiments.
The output force Ie of the CE is a function of its length Ie and its contraction speed
Ve
-dle/dt (Figure 3.b)
=
Ve ?:: 0 (contraction),
Ve < 0 (extension),
=
(1)
where 10 is the isometric output force (at Ve 0) and Vo is the maximal contraction
velocity. The parameters of the I-v curve were a = 0.25 and f3 0.3. The isometric
force 10 was given as the function of CE length Ie and the activation level a(t) of
=
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill
the muscle (Figure 3.c)
fo(l"a(t)) = {
~m.Z!~, (f.;)' (f.; - r) a(t) o < Ie < 'Y,
(2)
otherwise,
where leo is the resting length of the CE and 'Y
= 1.5.
The SE was modeled as an exponential spring (Figure 3.d)
I${I$) = { okl(exp[k2l'~~'Q] -1)
1$ ~ 1$0,
1$ < 1$0,
(3)
where 1$ is the output force, 1$0 is the resting length, and kl and k2 are stiffness
parameters. The PE was supposed to have the same exponential elasticity (3).
In the simulations, the CE length Ie was taken as the state variable. The total
muscle length 1m = Ie + 1$ is given by the skeletal model and the muscle activation
a(t) is given by the the activation dynamics described below. The SE length is
1m - Ie and then the output force I${I$)
Ie + Ip
1m is given
given from 1$
by (3). The contraction velocity Ve = -~ is derived from the inverse of (1) at
Ie = I${I$) - Ip(le) and then integrated to update the CE length Ie.
=
=
=
The activation level a(t) of a muscle is determined by the free calcium concentration
in muscle fibers. Since we don't have enough data about the calcium dynamics in
muscle cells, the activation dynamics was crudely approximated by the following
equations.
da(t)
Ta-;{t = -a(t) + e(t),
and
de(t)
Te~ = -e(t)
+ n(t)2,
(4)
where n(t) is the normalized firing frequency of the nerve input and e(t) is the electric activity of the muscle fibers. The nonlinearity in the nervous input represents
strong facilitation of the postsynaptic potential (Govind and Lingle 1987).
We incorporated seven of the gastric mill muscles: gml, gm2, gm3a, gm3c, gm4,
gm6b, and gm9a (Maynard and Dando 1974). The muscles gml, gm2, gm3a, and
gm3c are extrinsic muscles that have one end attached to the carapace and gm4,
gm6b, and gm9a are intrinsic muscles both ends of which are attached of the ossicles.
Three connective tissues were also incorporated and regarded as muscles without
contraction elements. See Figure 4 for the attachment of these muscles and tissues
to the ossicles.
3.2
SKELETAL DYNAMICS
The medial tooth was modeled as three rigid pieces PI, P2 and P3 . PI is the base of
the medial tooth. P2 is the main body of the medial tooth. P3 forms the cusp and
the V-shaped lever on the dorsal side. The lateral tooth was modeled as two rigid
pieces P4 and Ps . P4 is a L-shaped plate with a cusp at the angle and is connected
to P3 at the dorsal end. Ps is a rod that is connected to P4 near the root of the
cusp (Figure 4).
We assumed that the motion is symmetric with respect to the midline. Therefore
the motion of the medial tooth was two-dimensional and only the left one of the
917
918
Doya, Boyle, and Se1verslon
gm3c
~':-;--...J.........
gm9a
z
30
y
Figure 4: The design of the gastric mill model. Ossicle PI stands for the ossicles
I and II, P2 for VII, P3 for VI, P4 for III, IV, and V, Ps for XIV in the standard
description by Maynard and Dando (1974).
two lateral teeth was considered. The coordinate system was taken so that x-axis
points to the left, y-axis backward, and z-axis upward. The rotation angles of the
ossicles around x, y, and z axes ware represented as 0, <P, and 1/J respectively. The
configuration of the ossicles was determined by a 10 dimensional vector
e = (YO,zO,OI,02,03,04,<P4,1/J4,OS,<P5),
(5)
where (YO,zo) represents the position of the joint between PI and P2 and (0 1 ,02,03)
represents the rotation angle of PI, P2 and P3 in the y-z plane. The rotation angles
of P4 and P5 were represented as (0 4, <P4, 1/J4) and (Os, 4>s) respectively. P5 has only
two degrees of rotation freedom since it is regarded as a rod.
We employed a quasi-static approximation. The configuration of the ossicles e
was determined by the static balance of force. Now let Lm and Fm be the vectors
of the muscle lengths and forces. Then the balance of the generalized forces in the
space (force for translation and torque for rotation) is given by
e
Tm(e, Fm) + Te = 0,
(6)
where T m and Te represent the generalized forces from muscles and external loads.
The muscle force in the e space is given by
Tm(e, Fm) = J(e)T Fm,
(7)
where J(e) = 8L m /8e is the Jacobian matrix of the mapping e .- Lm determined
by the ossicle kinematics and the muscle attachment. Since it is very difficult to
obtain a closed form solution of (6), we used a gradient descent equation
de
dt = -c:(Tm(e, Fm) + Te) = -c:(J(e)T Fm + Te)
(8)
Mapping Between Neural and Physical Activities of the Lobster Gastric Mill
(a) t=O.
t=2.
(b)
t=1.5
t=O.
t=4.
t=3.
t=6.
t=4.5
Figure 5: Chewing patterns predicted from oscillation patterns of isolated STG . (a)
spontaneous pattern. (b) proctolin induced pattern.
to find the approximate solution of 0(t). This is equivalent to assuming a viscosity
term c- 1 d0ldt in the motion equation.
4
SIMULATION RESULTS
The musculoskeletal model is a 17-th order differential equation system and was
integrated by Runge-Kutta method with a time step 1ms. Figure 5 shows examples
of motion patterns predicted by the model. The motoneuron output of spontaneous
oscillation of the isolated ganglion was used in (a) and the output under the effect
of proctolin was used in (b). It has been reported in previous behavioral studies
(Heinzel 1988b) that the dose of proctolin typically evokes "cut and grind" chewing
pattern. The trajectory (b) predicted from the proctolin induced rhythm has a
larger forward movement of the medial tooth while the lateral teeth are closed,
which qualitatively agrees with the behavioral data.
5
DISCUSSION
The motor pattern generated by the model is considerably different from the chewing patterns observed in the intact animal using an endoscope. This is partly
because of crude assumptions in model construction and errors in parameter estimation. However, this difference may also be due to the lack of sensory feedback in
the isolated preparation . The future subject of this project is to refine the model so
that we can reliably predict the motion from the neural outputs and to combine it
with models of the gastric network (Rowat and Selverston, submitted) and sensory
receptors. This will enable us to study how a biological control system integrates
central pattern generation and sensory feedback.
919
920
Doya, Boyle, and Selverston
Acknowledgements
We thank Mike Beauchamp for the gml muscle data. This work was supported by
the grant from Office of Naval Research NOOOI4-91-J-1720.
References
Boyle, M. E. T., Turrigiano, G . G., and Selverston, A. 1. 1990. An endoscopic analysis of gastric mill movements produced by the peptide cholecystokinin. Society
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6,213 | 6,620 | Learning to See Physics via Visual De-animation
Jiajun Wu
MIT CSAIL
Erika Lu
University of Oxford
William T. Freeman
MIT CSAIL, Google Research
Pushmeet Kohli
DeepMind
Joshua B. Tenenbaum
MIT CSAIL
Abstract
We introduce a paradigm for understanding physical scenes without human annotations. At the core of our system is a physical world representation that is
first recovered by a perception module and then utilized by physics and graphics
engines. During training, the perception module and the generative models learn
by visual de-animation ? interpreting and reconstructing the visual information
stream. During testing, the system first recovers the physical world state, and then
uses the generative models for reasoning and future prediction.
Even more so than forward simulation, inverting a physics or graphics engine is
a computationally hard problem; we overcome this challenge by using a convolutional inversion network. Our system quickly recognizes the physical world
state from appearance and motion cues, and has the flexibility to incorporate both
differentiable and non-differentiable physics and graphics engines. We evaluate our
system on both synthetic and real datasets involving multiple physical scenes, and
demonstrate that our system performs well on both physical state estimation and
reasoning problems. We further show that the knowledge learned on the synthetic
dataset generalizes to constrained real images.
1
Introduction
Inspired by human abilities, we wish to develop machine systems that understand scenes. Scene
understanding has multiple defining characteristics which break down broadly into two features. First,
human scene understanding is rich. Scene understanding is physical, predictive, and causal: rather
than simply knowing what is where, one can also predict what may happen next, or what actions
one can take, based on the physics afforded by the objects, their properties, and relations. These
predictions, hypotheticals, and counterfactuals are probabilistic, integrating uncertainty as to what is
more or less likely to occur. Second, human scene understanding is fast. Most of the computation has
to happen in a single, feedforward, bottom-up pass.
There have been many systems proposed recently to tackle these challenges, but existing systems
have architectural features that allow them to address one of these features but not the other. Typical
approaches based on inverting graphics engines and physics simulators [Kulkarni et al., 2015b]
achieve richness at the expense of speed. Conversely, neural networks such as PhysNet [Lerer et al.,
2016] are fast, but their ability to generalize to rich physical predictions is limited.
We propose a new approach to combine the best of both. Our overall framework for representation is
based on graphics and physics engines, where graphics is run in reverse to build the initial physical
scene representation, and physics is then run forward to imagine what will happen next or what can
be done. Graphics can also be run in the forward direction to visualize the outputs of the physics
simulation as images of what we expect to see in the future, or under different viewing conditions.
Rather than use traditional, often slow inverse graphics methods [Kulkarni et al., 2015b], we learn to
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
physical world
physical world
visual data
visual data
Figure 1: Visual de-animation ? we would like to recover the physical world representation behind
the visual input, and combine it with generative physics simulation and rendering engines.
invert the graphics engine efficiently using convolutional nets. Specifically, we use deep learning to
train recognition models on the objects in our world for object detection, structure and viewpoint
estimation, and physical property estimation. Bootstrapping from these predictions, we then infer the
remaining scene properties through inference via forward simulation of the physics engine.
Without human supervision, our system learns by visual de-animation: interpreting and reconstructing
visual input. We show the problem formulation in Figure 1. The simulation and rendering engines in
the framework force the perception module to extract physical world states that best explain the data.
As the physical world states are inputs to physics and graphics engines, we simultaneously obtain an
interpretable, disentangled, and compact physical scene representation.
Our framework is flexible and adaptable to a number of graphics and physics engines. We present
model variants that use neural, differentiable physics engines [Chang et al., 2017], and variants that
use traditional physics engines, which are more mature but non-differentiable [Coumans, 2010]. We
also explore various graphics engines operating at different levels, ranging from mid-level cues such
as object velocity, to pixel-level rendering of images.
We demonstrate our system on real and synthetic datasets across multiple domains: synthetic billiard
videos [Fragkiadaki et al., 2016], in which balls have varied physical properties, real billiard videos
from the web, and real images of block towers from Facebook AI Research [Lerer et al., 2016].
Our contributions are three-fold. First, we propose a novel generative pipeline for physical scene
understanding, and demonstrate its flexibility by incorporating various graphics and physics engines.
Second, we introduce the problem of visual de-animation ? learning rich scene representations
without supervision by interpreting and reconstructing visual input. Third, we show that our system
performs well across multiple scenarios and on both synthetic and constrained real videos.
2
Related Work
Physical scene understanding has attracted increasing attention in recent years [Gupta et al., 2010,
Jia et al., 2015, Lerer et al., 2016, Zheng et al., 2015, Battaglia et al., 2013, Mottaghi et al., 2016b,
Fragkiadaki et al., 2016, Battaglia et al., 2016, Mottaghi et al., 2016a, Chang et al., 2017, Agrawal
et al., 2016, Pinto et al., 2016, Finn et al., 2016, Hamrick et al., 2017, Ehrhardt et al., 2017, Shao et al.,
2014, Zhang et al., 2016]. Researchers have attempted to go beyond the traditional goals of high-level
computer vision, inferring ?what is where?, to capture the physics needed to predict the immediate
future of dynamic scenes, and to infer the actions an agent should take to achieve a goal. Most of these
efforts do not attempt to learn physical object representations from raw observations. Some systems
emphasize learning from pixels but without an explicitly object-based representation [Lerer et al.,
2016, Mottaghi et al., 2016b, Fragkiadaki et al., 2016, Agrawal et al., 2016, Pinto et al., 2016, Li et al.,
2017], which makes generalization challenging. Others learn a flexible model of the dynamics of
object interactions, but assume a decomposition of the scene into physical objects and their properties
rather than learning directly from images [Chang et al., 2017, Battaglia et al., 2016].
2
Object
proposal
Physical
object state
Object
proposal
Physical
object state
(b) Physical world
representation
(II) Physics engine (simulation)
(c) Appearance cues
(III) Graphics engine (rendering)
NMS
(I) Perception module
(d) Likelihood
(a) Input
(e) Output
Figure 2: Our visual de-animation (VDA) model contains three major components: a convolutional
perception module (I), a physics engine (II), and a graphics engine (III). The perception module
efficiently inverts the graphics engine by inferring the physical object state for each segment proposal
in input (a), and combines them to obtain a physical world representation (b). The generative physics
and graphics engines then run forward to reconstruct the visual data (e). See Section 3 for details.
There have been some works that aim to estimate physical object properties [Wu et al., 2016, 2015,
Denil et al., 2017]. Wu et al. [2015] explored an analysis-by-synthesis approach that is easily
generalizable, but less efficient. Their framework also lacked a perception module. Denil et al.
[2017] instead proposed a reinforcement learning approach. These approaches, however, assumed
strong priors of the scene, and approximate object shapes with primitives. Wu et al. [2016] used
a feed-forward network for physical property estimation without assuming prior knowledge of the
environment, but the constrained setup did not allow interactions between multiple objects. By
incorporating physics and graphics engines, our approach can jointly learn the perception module
and physical model, optionally in a Helmholtz machine style [Hinton et al., 1995], and recover an
explicit physical object representation in a range of scenarios.
Another line of related work is on future state prediction in either image pixels [Xue et al., 2016,
Mathieu et al., 2016] or object trajectories [Kitani et al., 2017, Walker et al., 2015]. There has also
been abundant research making use of physical models for human or scene tracking [Salzmann
and Urtasun, 2011, Kyriazis and Argyros, 2013, Vondrak et al., 2013, Brubaker et al., 2009]. Our
model builds upon and extends these ideas by jointly modeling an approximate physics engine and a
perceptual module, with wide applications including, but not limited to, future prediction.
Our framework also relates to the field of ?vision as inverse graphics? [Zhu and Mumford, 2007,
Yuille and Kersten, 2006, Bai et al., 2012]. Connected to but different from traditional analysisby-synthesis approaches, recent works explored using deep neural networks to efficiently explain
an object [Kulkarni et al., 2015a, Rezende et al., 2016], or a scene with multiple objects [Ba et al.,
2015, Huang and Murphy, 2015, Eslami et al., 2016]. In particular, Wu et al. [2017] proposed ?scene
de-rendering?, building an object-based, structured representation from a static image. Our work
incorporates inverse graphics with simulation engines for physical scene understanding and scene
dynamics modeling.
3
Visual De-animation
Our visual de-animation (VDA) model consists of an efficient inverse graphics component to build
the initial physical world representation from visual input, a physics engine for physical reasoning of
the scene, and a graphics engine for rendering videos. We show the framework in Figure 2. In this
section, we first present an overview of the system, and then describe each component in detail.
3.1
Overview
The first component of our system is an approximate inverse graphics module for physical object and
scene understanding, as shown in Figure 2-I. Specifically, the system sequentially computes object
proposals, recognizes objects and estimates their physical state, and recovers the scene layout.
3
The second component of our system is a physics engine, which uses the physical scene representation
recovered by the inverse graphics module to simulate future dynamics of the environment (Figure 2II). Our system adapts to both neural, differentiable simulators, which can be jointly trained with the
perception module, and rigid-body, non-differentiable simulators, which can be incorporated using
methods such as REINFORCE [Williams, 1992].
The third component of our framework is a graphics engine (Figure 2-III), which takes the scene
representations from the physics engine and re-renders the video at various levels (e.g. optical flow,
raw pixel). The graphics engine may need additional appearance cues such as object shape or color
(Figure 2c). Here, we approximate them using simple heuristics, as they are not a focus of our paper.
There is a tradeoff between various rendering levels: while pixel-level reconstruction captures details
of the scene, rendering at a more abstract level (e.g. silhouettes) may better generalize. We then use a
likelihood function (Figure 2d) to evaluate the difference between synthesized and observed signals,
and compute gradients or rewards for differentiable and non-differentiable systems, respectively.
Our model combines efficient and powerful deep networks for recognition with rich simulation
engines for forward prediction. This provides us two major advantages over existing methods:
first, simulation engines take an interpretable representation of the physical world, and can thus
easily generalize and supply rich physical predictions; second, the model learns by explaining the
observations ? it can be trained in a self-supervised manner without requiring human annotations.
3.2
Physical Object and Scene Modeling
We now discuss each component in detail, starting with the perception module.
Object proposal generation Given one or a few frames (Figure 2a), we first generate a number of
object proposals. The masked images are then used as input to the following stages of the pipeline.
Physical object state estimation For each segment proposal, we use a convolutional network to
recognize the physical state of the object, which consists of intrinsic properties such as shape, mass,
and friction, as well as extrinsic properties such as 3D position and pose. The input to the network is
the masked image of the proposal, and the output is an interpretable vector for its physical state.
Physical world reconstruction Given objects? physical states, we first apply non-maximum suppression to remove object duplicates, and then reconstruct the physical world according to object
states. The physical world representation (Figure 2b) will be employed by the physics and graphics
engines for simulation and rendering.
3.3
Physical Simulation and Prediction
The two types of physics engines we explore in this paper include a neural, differentiable physics
engine and a standard rigid-body simulation engine.
Neural physics engines The neural physics engine is an extension of the recent work from Chang
et al. [2017], which simulates scene dynamics by taking object mass, position, and velocity. We extend
their framework to model object friction in our experiments on billiard table videos. Though basic,
the neural physics engine is differentiable, and thus can be end-to-end trained with our perception
module to explain videos. Please refer to Chang et al. [2017] for details of the neural physics engine.
Rigid body simulation engines There exist rather mature, rigid-body physics simulation engines,
e.g. Bullet [Coumans, 2010]. Such physics engines are much more powerful, but non-differentiable.
In our experiments on block towers, we used a non-differentiable simulator with multi-sample
REINFORCE [Rezende et al., 2016, Mnih and Rezende, 2016] for joint training.
3.4
Re-rendering with a Graphics Engine
In this work, we consider two graphics engines operating at different levels: for the billiard table
scenario, we use a renderer that takes the output of a physics engine and generates pixel-level
rendering; for block towers, we use one that computes only object silhouettes.
4
(a) shared appearance,
shared physics
(b) varied appearance,
varied physics
(c) shared appearance,
varied physics
Figure 3: The three settings of our synthetic billiard videos: (a) balls have the same appearance and
physical properties, where the system learns to discover them and simulate the dynamics; (b) balls
have the same appearance but different physics, and the system learns their physics from motion; (c)
balls have varied appearance and physics, and the system learns to associate appearance cues with
underlying object states, even from a single image.
Input (red)
and
ground truth
Reconstruction
and prediction
Input (red)
and
ground truth
Reconstruction
and prediction
Frame t-2
Frame t
Frame t+2
Frame t+5
Frame t+10
Frame t-2
Frame t
Frame t+2
Frame t+5
Frame t+10
Figure 4: Results on the billiard videos, comparing ground truth videos with our predictions. We
show two of three input frames (in red) due to space constraints. Left: balls share appearance and
physics (I), where our framework learns to discover objects and simulate scene dynamics. Top right:
balls have different appearance and physics (II), where our model learns to associate appearance with
physics and simulate collisions. It learns that the green ball should move further than the heavier
blue ball after the collision. Bottom right: balls share appearance but have different frictions (III),
where our model learns to associate motion with friction. It realizes from three input frames that the
right-most ball in the first frame has a large friction coefficient and will stop before the other balls.
4
Evaluation
We evaluate variants of our frameworks in three scenarios: synthetic billiard videos, real billiard
videos, and block towers. We also test how models trained on synthetic data generalize to real cases.
4.1
Billiard Tables: A Motivating Example
We begin with synthetic billiard videos to explore end-to-end learning of the perceptual module along
with differentiable simulation engines. We explore how our framework learns the physical object
state (position, velocity, mass, and friction) from its appearance and/or motion.
Data For the billiard table scenario, we generate data using the released code from Fragkiadaki
et al. [2016]. We updated the code to allow balls of different mass and friction. We used the billiard
table scenario as an initial exploration of whether our models can learn to associate visual object
appearance and motion with physical properties. As shown in Figure 3, we generated three subsets,
in which balls may have shared or differing appearance (color), and physical properties. For each
case, we generated 9,000 videos for training and 200 for testing.
(I) Shared appearance and physics (Figure 3a): balls all have the same appearance and the same
physical properties. This basic setup evaluates whether we can jointly learn an object (ball) discoverer
and a physics engine for scene dynamics.
5
Datasets
Recon.
Methods
Appear. Physics
Pixel MSE
Position Prediction (Abs)
1st
5th
Velocity Prediction (Abs)
10th
20th
1st
5th
10th
20th
8.45
2.65
2.55
Same
Same
Baseline
VDA (init)
VDA (full)
0.046
0.046
0.044
4.58 18.20
3.46 6.61
3.25 6.52
46.06
12.76
12.34
119.97
26.10
25.55
2.95
1.41
1.37
5.63
1.97
1.87
7.32
2.34
2.22
Diff
Diff.
Baseline
VDA (init)
VDA (full)
0.058
0.058
0.055
6.57 26.38
3.82 8.92
3.55 8.58
70.47
17.09
16.33
180.04
34.65
32.97
3.78
1.65
1.64
7.62
2.27
2.20
10.51 12.02
3.02 3.21
2.89 3.05
Same
Diff.
Baseline
VDA (init)
VDA (full)
0.038
0.038
0.035
9.58 79.78 143.67 202.56 12.37 23.42 25.08 23.98
6.06 19.75 34.24 46.40 3.37 5.16 5.01 3.77
5.77 19.34 33.25 43.42 3.23 4.98 4.77 3.35
10
5
0
Baseline
VDA (init)
VDA (full)
Humans
Frame 1
Relative error
Relative error
Table 1: Quantitative results on synthetic billiard table videos. We evaluate our visual de-animation
(VDA) model before and after joint training (init vs. full). For scene reconstruction, we compute
MSE between rendered images and ground truth. For future prediction, we compute the Manhattan
distance in pixels between predicted object position and velocity and ground truth in pixels, at the 1st,
5th, 10th, and 20th future frames. Our full model performs better. See qualitative results in Figure 4.
Frame 3
Frame 5
10
5
0
Frame 10
(a) Future prediction results on synthetic billiard
videos of balls of the same physics
Baseline
VDA (init)
VDA (full)
Humans
Frame 1
Frame 3
Frame 5
Frame 10
(b) Future prediction results on synthetic billiard
videos of balls of varied frictions
Figure 5: Behavioral study results on future position prediction of billiard videos where balls have
the same physical properties (a), and balls have varied physical properties (b). For each model and
humans, we compare how their long-term relative prediction error grows, by measuring the ratio with
respect to errors in predicting the first next frame. Compared to the baseline model, the behavior of
our prediction models aligns well with human predictions.
(II) Varied appearance and physics (Figure 3b): balls can be of three different masses (light, medium,
heavy), and two different friction coefficients. Each of the six possible combinations is associated
with a unique color (appearance). In this setup, the scene de-rendering component should be able to
associate object appearance with its physical properties, even from a single image.
(III) Shared appearance, varied physics (Figure 3c): balls have the same appearance, but have one of
two different friction coefficients. Here, the perceptual component should be able to associate object
motion with its corresponding friction coefficients, from just a few input images.
Setup For this task, the physical state of an object is its intrinsic properties, including mass m and
friction f , and its extrinsic properties, including 2D position {x, y} and velocity v. Our system takes
three 256?256 RGB frames I1 , I2 , I3 as input. It first obtains flow fields from I1 to I2 and from I2
to I3 by a pretrained spatial pyramid network (SPyNet) [Ranjan and Black, 2017]. It then generates
object proposals by applying color filters on input images.
Our perceptual model is a ResNet-18 [He et al., 2015], which takes as input three masked RGB
frames and two masked flow images of each object proposal, and recovers the object?s physical
state. We use a differentiable, neural physics engine with object intrinsic properties as parameters; at
each step, it predicts objects? extrinsic properties (position {x, y} and velocity v) in the next frame,
based on their current estimates. We employ a graphics engine that renders original images from the
predicted positions, where the color of the balls is set as the mean color of the input object proposal.
The likelihood function compares, at a pixel level, these rendered images and observations. It is
straightforward to compute the gradient of object position from rendered RGB images and ground
truth. Thus, this simple graphics engine is also differentiable, making our system end-to-end trainable.
6
Video
VDA
(ours)
Figure 6: Sample results on web videos of real billiard games and computer games with realistic
rendering. Left: our method correctly estimates the trajectories of multiple objects. Right: our
framework correctly predicts the two collisions (white vs. red, white vs. blue), despite the motion
blur in the input, though it underestimates the velocity of the red ball after the collision. Note that the
billiard table is a chaotic system, and highly accurate long-term prediction is intractable.
Our training paradigm consists of two steps. First, we pretrain the perception module and the neural
physics engine separately on synthetic data, where ground truth is available. The second step is
end-to-end fine-tuning without annotations. We observe that the framework does not converge well
without pre-training, possibly due to the multiple hypotheses that can explain a scene (e.g., we can
only observe relative, not absolute masses from collisions). We train our framework using SGD, with
a learning rate of 0.001 and a momentum of 0.9. We implement our framework in Torch7 [Collobert
et al., 2011]. During testing, the perception module is run in reverse to recover object physical states,
and the learned physics engine is then run in forward for future prediction.
Results Our formulation recovers a rich representation of the scene. With the generative models,
we show results in scene reconstruction and future prediction. We compare two variants of our
algorithm: the initial system has its perception module and neural physics engine separately trained,
while the full system has an additional end-to-end fine-tuning step, as discussed above. We also
compare with a baseline, which has the sample perception model, but in prediction, simply repeats
object dynamics in the past without considering interactions among them.
Scene reconstruction: given input frames, we are able to reconstruct the images based on inferred
physical states. For evaluation, we compute pixel-level MSE between reconstructions and observed
images. We show qualitative results in Figure 4 and quantitative results in Table 1.
Future prediction: with the learned neural simulation engine, our system is able to predict future
events based on physical world states. We show qualitative results in Figure 4 and quantitative results
in Table 1, where we compute the Manhattan distance in pixels between predicted positions and
velocities and the ground truth. Our model achieves good performance in reconstructing the scene,
understanding object physics, and predicting scene dynamics. See caption for details.
Behavioral studies We further conduct behavioral studies, where we select 50 test cases, show the
first three frames of each to human subjects, and ask them the positions of each ball in future frames.
We test 3 subjects per case on Amazon Mechanical Turk. For each model and humans, we compare
how their long-term relative prediction error grows, by measuring the ratio with respect to errors in
predicting the first next frame. As shown in Figure 5, the behavior of our models resembles human
predictions much better than the baseline model.
4.2
Billiard Tables: Transferring to Real Videos
Data We also collected videos from YouTube, segmenting them into two-second clips. Some
videos are from real billiard competitions, and the others are from computer games with realistic
rendering. We use it as an out-of-sample test set for evaluating the model?s generalization ability.
Setup and Results Our setup is the same as that in Section 4.1, except that we now re-train the
perceptual model on the synthetic data of varied physics, but with flow images as input instead of
RGB images. Flow images abstract away appearance changes (color, lighting, etc.), allowing the
model to generalize better to real data. We show qualitative results of reconstruction and future
prediction in Figure 6 by rendering our inferred representation using the graphics software, Blender.
7
Video
Methods
# Blocks
2
VDA
(ours)
PhysNet
3
Mean
4
Chance
Humans
50 50 50
67 62 62
50
64
PhysNet
GoogLeNet
66 66 73
70 70 70
68
70
VDA (init)
VDA (joint)
VDA (full)
73 74 72
75 76 73
76 76 74
73
75
75
(b) Accuracy (%) of stability prediction on the
blocks dataset
Video
VDA
(ours)
PhysNet
Methods
2
3
4
Mean
PhysNet
GoogLeNet
56 68 70
70 67 71
65
69
VDA (init)
VDA (joint)
VDA (full)
74 74 67
75 77 70
76 76 72
72
74
75
(c) Accuracy (%) of stability prediction when
(a) Our reconstruction and prediction results given a single trained on synthetic towers of 2 and 4 blocks, and
frame (marked in red). From top to bottom: ground truth, tested on all block tower sizes.
our results, results from Lerer et al. [2016].
Video
VDA
(ours)
(d) Our reconstruction and prediction results given a single frame (marked in red)
Figure 7: Results on the blocks dataset [Lerer et al., 2016]. For quantitative results (b), we compare
three variants of our visual de-animation (VDA) model: perceptual module trained without fine-tuning
(init), joint fine-tuning with REINFORCE (joint), and full model considering stability constraint
(full). We also compare with PhysNet [Lerer et al., 2016] and GoogLeNet [Szegedy et al., 2015].
4.3
The Blocks World
We now look into a different scenario ? block towers. In this experiment, we demonstrate the
applicability of our model to explain and reason from a static image, instead of a video. We focus on
the reasoning of object states in the 3D world, instead of physical properties such as mass. We also
explore how our framework performs with non-differentiable simulation engines, and how physics
signals (e.g., stability) could help in physical reasoning, even when given only a static image.
Data Lerer et al. [2016] built a dataset of 492 images of real block towers, with ground truth
stability values. Each image may contain 2, 3, or 4 blocks of red, blue, yellow, or green color.
Though the blocks are the same size, their sizes in each 2D image differ due to 3D-to-2D perspective
transformation. Objects are made of the same material and thus have identical mass and friction.
8
Input
VDA
What if?
Input
Future
Stabilizing force
Figure 8: Examples of predicting hypothetical scenarios and actively engaging with the scene. Left:
predictions of the outcome of forces applied to two stable towers. Right: multiple ways to stabilize
two unstable towers.
Setup Here, the physical state of an object (block) consists of its 3D position {x, y, z} and 3D
rotation (roll, pitch, yaw, each quantized into 20 bins). Our perceptual model is again a ResNet-18 [He
et al., 2015], which takes block silhouettes generated by simple color filters as input, and recovers
the object?s physical state. For this task, we implement an efficient, non-differentiable, rigid body
simulator, to predict whether the blocks are stable. We also implement a graphics engine to render
object silhouettes for reconstructing the input. Our likelihood function consists of two terms: MSE
between rendered silhouettes and observations, and the binary cross-entropy between the predicted
stability and the ground truth stability.
Our training paradigm resembles the classic wake-sleep algorithm [Hinton et al., 1995]: first, generate
10,000 training images using the simulation engines; second, train the perception module on synthetic
data with ground truth physical states; third, end-to-end fine-tuning of the perceptual module by
explaining an additional 100,000 synthetic images without annotations of physical states, but with
binary annotations of stability. We use multi-sample REINFORCE [Rezende et al., 2016, Mnih and
Rezende, 2016] with 16 samples per input, assuming each position parameter is from a Gaussian
distribution and each rotation parameter is from a multinomial distribution (quantized into 20 bins).
We observe that the training paradigm helps the framework converge. The other setting is the same as
that in Section 4.1.
Results We show results on two tasks: scene reconstruction and stability prediction. For each task,
we compare three variants of our algorithm: the initial system has its perception module trained
without fine-tuning; an intermediate system has joint end-to-end fine-tuning, but without considering
the physics constraint; and the full system considers both reconstruction and physical stability during
fine-tuning.
We show qualitative results on scene reconstruction in Figures 7a and 7d, where we also demonstrate
future prediction results by exporting our inferred physical states into Blender. We show quantitative
results on stability prediction in Table 7b, where we compare our models with PhysNet [Lerer et al.,
2016] and GoogleNet [Szegedy et al., 2015]. All given a static image as test input, our algorithms
achieve higher prediction accuracy (75% vs. 70%) efficiently (<10 milliseconds per image).
Our framework also generalizes well. We test out-of-sample generalization ability, where we train our
model on 2- and 4-block towers, but test it on all tower sizes. We show results in Table 7c. Further, in
Figure 8, we show examples where our physical scene representation combined with a physics engine
can easily make conditional predictions, answering ?What happens if...?-type questions. Specifically,
we show frame prediction of external forces on stable block towers, as well as ways that an agent can
stabilize currently unstable towers, with the help of rich simulation engines.
5
Discussion
We propose combining efficient, bottom-up, neural perception modules with rich, generalizable
simulation engines for physical scene understanding. Our framework is flexible and can incorporate
various graphics and physics engines. It performs well across multiple synthetic and real scenarios,
reconstructing the scene and making future predictions accurately and efficiently. We expect our
framework to have wider applications in the future, due to the rapid development of scene description
languages, 3D reconstruction methods, simulation engines and virtual environments.
9
Acknowledgements
This work is supported by NSF #1212849 and #1447476, ONR MURI N00014-16-1-2007, the Center
for Brain, Minds and Machines (NSF #1231216), Toyota Research Institute, Samsung, Shell, and the
MIT Advanced Undergraduate Research Opportunities Program (SuperUROP).
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6,214 | 6,621 | Label Efficient Learning of Transferable
Representations across Domains and Tasks
Zelun Luo
Stanford University
[email protected]
Yuliang Zou
Virginia Tech
[email protected]
Judy Hoffman
University of California, Berkeley
[email protected]
Li Fei-Fei
Stanford University
[email protected]
Abstract
We propose a framework that learns a representation transferable across different
domains and tasks in a label efficient manner. Our approach battles domain shift
with a domain adversarial loss, and generalizes the embedding to novel task using
a metric learning-based approach. Our model is simultaneously optimized on
labeled source data and unlabeled or sparsely labeled data in the target domain.
Our method shows compelling results on novel classes within a new domain even
when only a few labeled examples per class are available, outperforming the
prevalent fine-tuning approach. In addition, we demonstrate the effectiveness of
our framework on the transfer learning task from image object recognition to video
action recognition.
1
Introduction
Humans are exceptional visual learners capable of generalizing their learned knowledge to novel
domains and concepts and capable of learning from few examples. In recent years, computational
models based on end-to-end learnable convolutional networks have made significant improvements for
visual recognition [18, 28, 54] and have been shown to demonstrate some cross-task generalizations [8,
48] while enabling faster learning of subsequent tasks as most frequently evidenced through finetuning [14, 36, 50].
However, most efforts focus on the supervised learning scenario where a closed world assumption
is made at training time about both the domain of interest and the tasks to be learned. Thus,
any generalization ability of these models is only an observed byproduct. There has been a large
push in the research community to address generalizing and adapting deep models across different
domains [64, 13, 58, 38], to learn tasks in a data efficient way through few shot learning [27, 70, 47,
11], and to generically transfer information across tasks [1, 14, 50, 35].
While most approaches consider each scenarios in isolation we aim to directly tackle the joint problem
of adapting to a novel domain which has new tasks and few annotations. Given a large labeled source
dataset with annotations for a task set, A, we seek to transfer knowledge to a sparsely labeled target
domain with a possibly wholly new task set, B. This setting is in line with our intuition that we
should be able to learn reusable and general purpose representations which enable faster learning of
future tasks requiring less human intervention. In addition, this setting matches closely to the most
common practical approach for training deep models which is to use a large labeled source dataset
(often ImageNet [6, 52]) to train an initial representation and then to continue supervised learning
with a new set of data and often with new concepts.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In our approach, we jointly adapt a source representation for use in a distinct target domain using a
new multilayer unsupervised domain adversarial formulation while introducing a novel cross-domain
and within domain class similarity objective. This new objective can be applied even when the target
domain has non-overlapping classes to the source domain.
We evaluate our approach in the challenging setting of joint transfer across domains and tasks
and demonstrate our ability to successfully transfer, reducing the need for annotated data for the
target domain and tasks. We present results transferring from a subset of Google Street View House
Numbers (SVHN) [41] containing only digits 0-4 to a subset of MNIST [29] containing only digits 5-9.
Secondly, we present results on the challenging setting of adapting from ImageNet [6] object-centric
images to UCF-101 [57] videos for action recognition.
2
Related work
Domain adaptation. Domain adaptation seeks to learn from related source domains a well performing model on target data distribution [4]. Existing work often assumes that both domains are defined
on the same task and labeled data in target domain is sparse or non-existent [64]. Several methods
have tackled the problem with the Maximum Mean Discrepancy (MMD) loss [17, 36, 37, 38, 73]
between the source and target domain. Weight sharing of CNN parameters [58, 22, 21, 3] and minimizing the distribution discrepancy of network activations [51, 65, 30] have also shown convincing
results. Adversarial generative models [33, 32, 2, 59] aim at generating source-like data with target
data by training a generator and a discriminator simultaneously, while adversarial discriminative
models [62, 64, 13, 12, 23] focus on aligning embedding feature representations of target domain
to source domain. Inspired by adversarial discriminative models, we propose a method that aligns
domain features with multi-layer information.
Transfer learning. Transfer learning aims to transfer knowledge by leveraging the existing labeled
data of some related task or domain [45, 71]. In computer vision, examples of transfer learning
include [1, 31, 61] which try to overcome the deficit of training samples for some categories by
adapting classifiers trained for other categories [43]. With the power of deep supervised learning and
the ImageNet dataset [6, 52], learned knowledge can even transfer to a totally different task (i.e. image
classification ? object detection [50, 49, 34]; image classification ? semantic segmentation [35])
and then achieve state-of-the-art performance. In this paper, we focus on the setting where source
and target domains have differing label spaces but the label spaces share the same structure. Namely
adapting between classifying different category sets but not transferring from classification to a
localization plus classification task.
Few-shot learning. Few-shot learning seeks to learn new concepts with only a few annotated
examples. Deep siamese networks [27] are trained to rank similarity between examples. Matching
networks [70] learns a network that maps a small labeled support set and an unlabeled example to its
label. Aside from these metric learning-based methods, meta-learning has also served as a essential
part. Ravi et al. [47] propose to learn a LSTM meta-learner to learn the update rule of a learner. Finn
et al. [11] tries to find a good initialization point that can be easily fine-tune with new examples from
new tasks. When there exists a domain shift, the results of prior few-shot learning methods are often
degraded.
Unsupervised learning. Many unsupervised learning algorithms have focused on modeling raw data
using reconstruction objectives [19, 69, 26]. Other probabilistic models include restricted Boltzmann
machines [20], deep Boltzmann machines [53], GANs [15, 10, 9], and autoregressive models [42, 66]
are also popular. An alternative approach, often terms ?self-supervised learning? [5], defines a
pretext task such as predicting patch ordering [7], frame ordering [40], motion dynamics [39], or
colorization [72], as a form of indirect supervision. Compared to these approaches, our unsupervised
learning method does not rely on exploiting the spatial or temporal structure of the data, and is
therefore more generic.
3
Method
We introduce a semi-supervised learning algorithm which transfers information from a large labeled
source domain, S, to a sparsely labeled target domain, T . The goal being to learn a strong target
2
Source CNN
{ , }
Supervised
Loss
Source labeled data
Multi-layer Domain Transfer
Adversarial
Loss
Target unlabeled data
{ t}
Semantic Transfer
Pairwise
Similarity
Softmax
Entropy
Loss
Target labeled data
Supervised
Loss
{ t, t}
Target CNN
Figure 1: Our proposed learning framework for joint transfer across domains and semantic transfer across source and target and across target labeled to unlabeled data. We introduce a domain
discriminator which aligns source and target representations across multiple layers of the network
through domain adversarial learning. We enable semantic transfer through minimizing the entropy of
the pairwise similarity between unlabeled and labeled target images and use the temperature of the
softmax over the similarity vector to allow for non-overlapping label spaces.
classifier without requiring the large annotation overhead required for standard supervised learning
approaches.
In fact, this setting is very commonly explored for convolutional network (convnet) based recognition
methods. When learning with convnets the usual learning procedure is to use a very large labeled
dataset (e.g. ImageNet [6, 52]) for initial training of the network parameters (termed pre-training).
The learned weights are then used as initialization for continued learning on new data and for new
tasks, called fine-tuning. Fine-tuning has been broadly applied to reduce the number of labeled
examples needed for learning new tasks, such as recognizing new object categories after ImageNet
pre-training [54, 18], or learning new label structures such as detection after classficiation pretraining [14, 50]. Here we focus on transfer in the case of a shared label structure (e.g. classification
of different category sets).
We assume the source domain contains ns images, xs ? X S , with associated labels, ys ? Y S .
? t ? X?T , as well as mt images,
Similarly, the target domain consists of nt unlabeled images, x
t
T
t
T
x ? X , with associated labels, y ? Y . We assume that the target domain is only sparsely
labeled so that the number of image-label pairs is much smaller than the number of unlabeled images,
mt nt . Additionally, the number of source labeled images is assumed to be much larger than the
number of target labeled images, mt ns .
Unlike standard domain adaptation approaches which transfer knowledge from source to target
domains assuming a marginal or conditional distribution shift under a shared label space (Y S = Y T ),
we tackle joint image or feature space adaptation as well as transfer across semantic spaces. Namely,
we consider the case where the source and target label spaces are not equal, Y S 6= Y T , and even the
most challenging case where the sets are non-overlapping, Y S ? Y T = ?.
3.1
Joint domain and semantic transfer
Our approach consists of unsupervised feature alignment between source and target as well as
semantic transfer to the unlabeled target data from either the labeled target or the labeled source
data. We introduce a new multi-layer domain discriminator which can be used for domain alignment
following the recent domain adversarial learning approaches [13, 64]. We next introduce a new
semantic transfer learning objective which uses cross category similarity and can be tuned to account
for varying size of label set overlap.
3
We depict our overall model in Figure 1. We take the ns source labeled examples, {xs , ys }, the mt
target labeled examples, {xt , yt }, and the nt unlabeled target images, {?
xt } as input. We learn an
initial layered source representation and classification network (depicted in blue in Figure 1) using
standard supervised techniques. We then initialize the target model (depicted in green in Figure 1)
with the source parameters and begin our adaptive transfer learning.
Our model jointly optimizes over a target supervised loss, Lsup , a domain transfer objective, LDT , and
finally a semantic transfer objective, LST . Thus, our total objective can be written as follows:
L(X S , Y S , X T , Y T , X?T ) = Lsup (X T , Y T ) + ?LDT (X S , X?T ) + ?LST (X S , X T , X?T )
(1)
where the hyperparameters ? and ? determine the influence of the domain transfer loss and the
semantic transfer loss, respectively. In the following sections we elaborate on our domain and
semantic transfer objectives.
3.2
Multi-layer domain adversarial loss
We define a novel domain alignment objective function called multi-layer domain adversarial
loss. Recent efforts in deep domain adaptation have shown strong performance using feature space
domain adversarial objectives [13, 64]. These methods learn a target representation such that the
target distribution viewed under this model is aligned with the source distribution viewed under the
source representation. This alignment is accomplished through an adversarial minimization across
domain, analogous to the prevalent generative adversarial approaches [15]. In particular, a domain
discriminator, D(?), is trained to classify whether a particular data point arises from the source or the
target domain. Simultaneously, the target embedding function E t (xt ) (defined as the application of
layers of the network is trained to generate the target representation that cannot be distinguished from
the source domain representation by the domain discriminator. Similar to [63, 64], we consider a
representation to be domain invariant if the domain discriminator can not distinguish examples from
the two domains.
Prior work considers alignment for a single layer of the embedding at a time and as such learns a
domain discriminator which takes the output from the corresponding source and target layers as input.
Separately, domain alignment methods which focus on first and second order statistics have shown
improved performance through applying domain alignment independently at multiple layers of the
network [36]. Rather than learning independent discriminators for each layer of the network we
propose a simultaneous alignment of multiple layers through a multi-layer discriminator.
At each layer of our multi-layer domain discriminator, information is accumulated from both the
output from the previous discriminator layer as well as the source and target activations from the
corresponding layer in their respective embeddings. Thus, the output of each discriminator layer is
defined as:
dl = Dl (?(?dl?1 ? El (x)))
(2)
where l is the current layer, ?(?) is the activation function, ? ? 1 is the decay factor, ? represents
concatenation or element-wise summation, and x is taken either from source data xs ? X S , or target
? t ? X?T . Notice that the intermediate discriminator layers share the same structure with their
data x
corresponding encoding layers to match the dimensions.
Thus, the following loss functions are proposed to optimize the multi-layer domain discriminator and
the embeddings, respectively, according to our domain transfer objective:
s
t
LD
DT = ?Exs ?X S [log dl ] ? Ext ?X T log(1 ? dl )
t
s
t
LE
DT = ?Exs ?X S [log(1 ? dl )] ? Ext ?X T log dl
(3)
(4)
where dsl , dtl are the outputs of the last layer of the source and target multi-layer domain discriminator.
Note that these losses are placed after the final domain discriminator layer and the last embedding
layer but then produce gradients which back-propagate throughout all relevant lower layer parameters.
These two losses together comprise LDT , and there is no iterative optimization procedure involved.
This multi-layer discriminator (shown in Figure 1 - yellow) allows for deeper alignment of the
source and target representations which we find empirically results in improved target classification
performance as well as more stable adversarial learning.
4
Figure 2: We illustrate the purpose of temperature (? ) for our pairwise similarity vector. Consider an
example target unlabeled point and its similarity to four labeled source points (x-axis). We show here,
original unnormalized scores (leftmost) as well as the same similarity scores after applying softmax
with different temperatures, ? . Notice that entropy values, H(x), have higher variance for scores
normalized with a small temperature softmax.
3.3
Cross category similarity for semantic transfer
In the previous section, we introduced a method for transferring an embedding from the source to the
target domain. However, this only enforces alignment of the global domain statistics with no class
specific transfer. Here, we define a new semantic transfer objective, LST , which transfers information
from a labeled set of data to an unlabeled set of data by minimizing the entropy of the softmax with
temperature of the similarity vector between an unlabeled point and all labeled points. Thus, this
loss may be applied either between the source and unlabeled target data or between the labeled and
unlabeled target data.
? t , we compute the similarity, ?(?), to each labeled example or
For each unlabeled target image, x
to each prototypical example [56] per class in the labeled set. For simplicity of presentation let us
consider semantic transfer from the source to the target domain first. For each target unlabeled image
we compute a similarity vector where the ith element is the similarity between this target image and
the ith labeled source image: [vs (?
xt )]i = ?(?
xt , xsi ). Our semantic transfer loss can be defined as
follows:
LST (X?T , X S )
=
X
H(?(vs (?
xt )/? ))
(5)
? t ?X?T
x
where, H(?) is the information entropy function, ?(?) is the softmax function and ? is the temperature
of the softmax. Note that the temperature can be used to directly control the percentage of source
examples we expect the target example to be similar to (see Figure 2).
Entropy minimization has been widely used for unsupervised [44] and semi-supervised [16] learning
by encouraging low density separation between clusters or classes. Recently this principle of entropy
minimization has be applied for unsupervised adaptation [38]. Here, the source and target domains
are assumed to share a label space and each unlabeled target example is passed through the initial
source classifier and the entropy of the softmax output scores is minimized.
In contrast, we do not assume a shared label space between the source and target domains and as such
can not assume that each target image maps to a single source label. Instead, we compute pairwise
similarities between target points and the source points (or per class averages of source points [56])
across the features spaces aligned by our multi-layer domain adversarial transfer. We then tune the
softmax temperature based on the expected similarity between the source and target labeled set. For
example, if the source and target label set overlap, then a small temperature will encourage each
target point to be very similar to one source class, whereas a larger temperature will allow for target
points to be similar to multiple source classes.
For semantic transfer within the target domain, we utilize the metric-based cross entropy loss between
labeled target examples to stabilize and improve the learning. For a labeled target example, in addition
to the traditional cross entropy loss, we also calculate a metric-based cross entropy loss 1 . Assume
we have k labeled examples from each class in the target domain. We compute the embedding for
1
We refer this as "metric-based" to cue the reader that this is not a cross entropy within the label space.
5
each example and then the centroid cTi of each class in the embedding space. Thus, we can compute
the similarity vector for each labeled example, where the ith element is the similarity between this
labeled example and the centroid of each class: [vt (xt )]i = ?(xt , cTi ). We can then calculate the
metric based cross entropy loss:
LST,sup (X T ) = ?
X
{xt ,yt }?X T
exp ([vt (xt )]yt )
log Pn
t
i=1 exp ([vt (x )]i )
(6)
Similar to the source-to-target scenario, for target-to-target we also have the unsupervised part,
X
H(?(vt (?
xt )/? ))
LST,unsup (X?T , X T ) =
(7)
? t ?X?T
x
With the metric-based cross entropy loss, we introduce the constraint that the target domain data
should be similar in the embedding space. Also, we find that this loss can provide a guidance
for the unsupervised semantic transfer to learn in a more stable way. LST is the combination of
LST,unsupervised from source-target (Equation 5), LST,supervised from source-target (Equation 6), and
LST,unsupervised from target-target (Equation 7), i.e.,
LST (X S , X T , X?T ) = LST (X?T , X S ) + LST,sup (X T ) + LST,unsup (X?T , X T )
4
(8)
Experiment
This section is structured as follows. In section 4.1, we show that our method outperform fine-tuning
approach by a large margin, and all parts of our method are necessary. In section 4.2, we show that
our method can be generalized to bigger datasets. In section 4.3, we show that our multi-layer domain
adversarial method outperforms state-of-the-art domain adversarial approaches.
Datasets We perform adaptation experiments across two different paired data settings. First for
adaptation across different digit domains we use MNIST [29] and Google Street View House Numbers
(SVHN) [41]. The MNIST handwritten digits database has a training set of 60,000 examples, and a
test set of 10,000 examples. The digits have been size-normalized and centered in fixed-size images.
SVHN is a real-world image dataset for machine learning and object recognition algorithms with
minimal requirement on data preprocessing and formatting. It has 73257 digits for training, 26032
digits for testing. As our second experimental setup, we consider adaptation from object centric
images in ImageNet [52] to action recognition in video using the UCF-101 [57] dataset. ImageNet
is a large benchmark for the object classification task. We use the task 1 split from ILSVRC2012.
UCF-101 is an action recognition dataset collected on YouTube. With 13,320 videos from 101
action categories, UCF-101 provides a large diversity in terms of actions and with the presence of
large variations in camera motion, object appearance and pose, object scale, viewpoint, cluttered
background, illumination conditions, etc.
Implementation details We pre-train the source domain embedding function with cross-entropy loss.
For domain adversarial loss, the discriminator takes the last three layer activations as input when the
number of output classes are the same for source and target tasks, and takes the second last and third
last layer activations when they are different. The similarity score is chosen as the dot product of the
normalized support features and the unnormalized target feature. We use the temperature ? = 2 for
source-target semantic transfer and ? = 1 for within target transfer as the label space is shared. We
use ? = 0.1 and ? = 0.1 in our objective function. The network is trained with Adam optimizer [25]
and with learning rate 10?3 . We conduct all the experiments with the PyTorch framework.
4.1
SVHN 0-4 ? MNIST 5-9
Experimental setting. In this experiment, we define three datasets: (i) labeled data in source domain
D1 ; (ii) few labeled data in target domain D2 ; (iii) unlabeled data in target domain D3 . We take the
training split of SVHN dataset as dataset D1 . To fairly compare with traditional learning paradigm
and episodic training, we subsample k examples from each class to construct dataset D2 so that we
can perform traditional training or episodic (k ? 1)-shot learning. We experiment with k = 2, 3, 4, 5,
which corresponds to 10, 15, 20, 25 labeled examples, or 0.017%, 0.025%, 0.333%, 0.043% of the
6
total training data respectively. Since our approach involves using annotations from a small subset
of the data, we randomly subsample 10 different subsets {D2i }10
i=1 from the training split of MNIST
dataset, and use the remaining data as {D3i }10
i=1 for each k. Note that source domain and target
domain have non-overlapping classes: we only utilize digits 0-4 in SVHN, and digits 5-9 in MNIST.
Figure 3: An illustration of our task. Our model effectively transfer the learned representation on
SVHN digits 0-4 (left) to MNIST digits 5-9 (right).
Baselines and prior work. We compare against six different methods: (i) Target only: the model
is trained on D2 from scratch; (ii) Fine-tune: the model is pretrained on D1 and fine-tuned on D2 ;
(iii) Matching networks [70]: we first pretrain the model on D3 , then use D2 as the support set in the
matching networks; (iv) Fine-tuned matching networks: same as baseline iii, except that for each k
the model is fine-tuned on D2 with 5-way (k ? 1)-shot learning: k ? 1 examples in each class are
randomly selected as the support set, and the last example in each class is used as the query set; (v)
Fine-tune + adversarial: in addition to baseline ii, the model is also trained on D1 and D3 with a
domain adversarial loss; (vi.) Full model: fine-tune the model with the proposed multi-layer domain
adversarial loss.
Results and analysis. We calculate the mean and standard error of the accuracies across 10 sets of
data, which is shown in Table 1. Due to domain shift, matching networks perform poorly without
fine-tuning, and fine-tuning is only marginally better than training from scratch. Our method with
multi-layer adversarial only improves the overall performance, but is more sensitive to the subsampled
data. Our method achieves significant performance gain, especially when the number of labeled
examples is small (k = 2). For reference, fine-tuning on full target dataset gives an accuracy of
99.65%.
Table 1: The test accuracies of the baseline models and our method. Row 1 to row 6 correspond (in
the same order) to the six methods proposed in section 4.2. Note that the accuracies of two matching
net methods are calculated based on nearest neighbors in the support set. We report the mean and the
standard error of each method across 10 different subsampled data.
Method
k=2
k=3
k=4
k=5
Target only
Fine-tune
Matching nets [70]
Fine-tuned matching nets
Ours: fine-tune + adv.
Ours: full model (? = 0.1)
(a)
(b)
0.642 ? 0.026
0.612 ? 0.020
0.469 ? 0.019
0.645 ? 0.019
0.702 ? 0.020
0.917 ? 0.007
0.771 ? 0.015
0.779 ? 0.018
0.455 ? 0.014
0.755 ? 0.024
0.800 ? 0.013
0.936 ? 0.006
(c)
(d)
0.801 ? 0.010
0.802 ? 0.016
0.566 ? 0.013
0.793 ? 0.013
0.804 ? 0.014
0.942 ? 0.006
(e)
0.840 ? 0.013
0.830 ? 0.011
0.513 ? 0.023
0.827 ? 0.011
0.831 ? 0.013
0.950 ? 0.004
(f)
Figure 4: The t-SNE [68, 67] visualization of different feature embeddings. (a) Source domain
embedding. (b) Target domain embedding using encoder trained with source domain domain. (c)
Target domain embedding using encoder fine-tuned with target domain data. (d) Target domain
embedding using encoder trained with our method. (e) An overlap of a and c. (f) An overlap of a and
d. (best viewed in color and with zoom)
7
4.2
Image object recognition ? video action recognition
Problem analysis. Many recent works [60, 24] study the domain shift between images and video
in the object detection settings. Compared to still images, videos provide several advantages: (i)
motion provides information for foreground vs background segmentation [46]; (ii) videos often show
multiple views and thus provide 3D information. On the other hand, video frames usually suffer from:
(i) motion blur; (ii) compression artifacts; (iii) objects out-of-focus or out-of-frame.
Experimental setting. In this experiment, we focus on three dataset splits: (i) ImageNet training
set as the labeled data in source domain D1 ; (ii) k video clips per class randomly sampled from
UCF-101 training as the few labeled data in target domain set D2 ; (iii) the remaining videos in
UCF-101 training set as the unlabeled data in target domain D3 . We experiment with k = 3, 5, 10,
which corresponds 303, 505, 1010 video clips, or 2.27%, 3.79%, 7.59% of the total training data
respectively. Each experiment is run 3 times on D1 , {D2i }3i=1 , and {D3i }3i=1 .
Baselines and prior work. We compare our method with two baseline methods: (i) Target only:
the model is trained on D2 from scratch; (ii) Fine-tune: the model is first pre-trained on D1 , then
fine-tuned on D2 . For reference, we report the performance of a fully supervised method [55].
Results and analysis. The accuracy of each model is shown in Table 2. We also fine-tune a model
with all the labeled data for comparison. Per-frame performance (img) and average-across-frame
performance (vid) are both reported. Note that we calculate the average-across-frame performance by
averaging the softmax score of each frame in a video. Our method achieves significant improvement
on average-across-frame performance over standard fine-tuning for each value of k. Note that
compared to fine-tuning, our method has a bigger gap between per-frame and per-video accuracy.
We believe that this is due to the semantic transfer: our entropy loss encourages a sharper softmax
variance among per-frame softmax scores per video (if the variance is zero, then per-frame accuracy =
per-video accuracy). By making more confident predictions among key frames, our method achieves
a more significant gain with respective to per-video performance, even when there is little change in
the per-frame prediction.
Table 2: Accuracy of UCF-101 action classification. The results of the two-stream spatial model are
taken from [55] and vary depending on hyperparameters. We report the mean and the standard error
of each method across 3 different subsampled data.
Method
k=3
k=5
k=10
All
Target only (img)
Target only (vid)
Fine-tune (img)
Fine-tune (vid)
Two-stream spatial [55]
Ours (img)
Ours (vid)
4.3
0.098?0.003
0.105?0.003
0.380?0.013
0.406?0.015
0.393?0.006
0.467?0.007
0.126?0.022
0.133?0.024
0.486?0.012
0.523?0.010
0.459?0.013
0.545?0.014
0.100?0.035
0.106?0.038
0.529?0.039
0.568?0.042
0.523?0.002
0.620?0.005
0.672
0.714
0.708 - 0.720
-
Ablation: unsupervised domain adaptation
To validate our multi-layer domain adversarial loss objective, we conduct an ablation experiment
for unsupervised domain adaptation. We compare against multiple recent domain adversarial unsupervised adaptation methods. In this experiment, we first pretrain a source embedding CNN on the
training split SVHN [41] and then adapt the target embedding for MNIST by performing adversarial
domain adaptation. We evaluate the classification performance on the test split of MNIST [29]. We
follow the same training strategy and model architecture for the embedding network as [64].
8
All the models here have a two-step training strategy and share the first stage. ADDA [64] optimizes
encoder and classifier simultaneously. We also propose a similar method, but optimize encoder only.
Only we try a model with no classifier in the last layer (i.e. perform domain adversarial training in
feature space). We choose ? = 0.1 as the decay factor for this model.
The accuracy of each model is shown in Table 3. We find that our method achieve 6.5% performance
gain over the best competing domain adversarial approach indicating that our multilayer objective
indeed contributes to our overall performance. In addition, in our experiments, we found that the
multilayer approach improved overall optimization stability, as evidenced in our small standard error.
Table 3: Experimental results on unsupervised domain adaptation from SVHN to MNIST. Results of
Gradient reversal, Domain confusion, and ADDA are from [64], and the results of other methods are
from experiments across 5 different subsampled data.
Method
Accuracy
Source only
Gradient reversal [13]
Domain confusion [62]
ADDA [64]
Ours
5
0.601 ? 0.011
0.739
0.681 ? 0.003
0.760 ? 0.018
0.810 ? 0.003
Conclusion
In this paper, we propose a method to learn a representation that is transferable across different
domains and tasks in a data efficient manner. The framework is trained jointly to minimize the domain
shift, to transfer knowledge to new task, and to learn from large amounts of unlabeled data. We show
superior performance over the popular fine-tuning approach. We hope to keep improving the method
in future work.
Acknowledgement
We would like to start by thanking our sponsors: Stanford Computer Science Department and Stanford
Program in AI-assisted Care (PAC). Next, we specially thank De-An Huang, Kenji Hata, Serena
Yeung, Ozan Sener and all the members of Stanford Vision and Learning Lab for their insightful
discussion and feedback. Lastly, we thank all the anonymous reviewers for their valuable comments.
9
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6,215 | 6,622 | Decoding with Value Networks for Neural Machine
Translation
Di He1
[email protected]
Tao Qin4
[email protected]
Hanqing Lu2
[email protected]
Liwei Wang1,5
[email protected]
Yingce Xia3
[email protected]
Tie-Yan Liu4
[email protected]
1
Key Laboratory of Machine Perception, MOE, School of EECS, Peking University
2
Carnegie Mellon University 3 University of Science and Technology of China
4
Microsoft Research
5
Center for Data Science, Peking University, Beijing Institute of Big Data Research
Abstract
Neural Machine Translation (NMT) has become a popular technology in recent
years, and beam search is its de facto decoding method due to the shrunk search
space and reduced computational complexity. However, since it only searches
for local optima at each time step through one-step forward looking, it usually
cannot output the best target sentence. Inspired by the success and methodology of
AlphaGo, in this paper we propose using a prediction network to improve beam
search, which takes the source sentence x, the currently available decoding output
y1 , ? ? ? , yt?1 and a candidate word w at step t as inputs and predicts the long-term
value (e.g., BLEU score) of the partial target sentence if it is completed by the NMT
model. Following the practice in reinforcement learning, we call this prediction
network value network. Specifically, we propose a recurrent structure for the value
network, and train its parameters from bilingual data. During the test time, when
choosing a word w for decoding, we consider both its conditional probability
given by the NMT model and its long-term value predicted by the value network.
Experiments show that such an approach can significantly improve the translation
accuracy on several translation tasks.
1
Introduction
Neural Machine Translation (NMT), which is based on deep neural networks and provides an endto-end solution to machine translation, has attracted much attention from the research community
[2, 6, 12, 20] and gradually been adopted by industry in past several years [18, 22]. NMT uses
an RNN-based encoder-decoder framework to model the entire translation process. In training, it
maximizes the likelihood of a target sentence given a source sentence. In testing, given a source
sentence x, it tries to find a sentence y ? in the target language that maximizes the conditional
probability P (y|x). Since the number of possible target sentences is exponentially large, finding the
optimal y ? is NP-hard. Thus beam search is commonly employed to find a reasonably good y.
Beam search is a heuristic search algorithm that maintains the top-scoring partial sequences expanded
in a left-to-right fashion. In particular, it keeps a pool of candidates each of which is a partial sequence.
At each time step, the algorithm expands each candidate by appending a new word, and then keeps
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
the top-ranked new candidates scored by the NMT model. The algorithm terminates if it meets the
maximum decoding depth or all sentences are completely generated, i.e., all sentences are ended with
the end-of-sentence (EOS) symbol.
While NMT with beam search has been proved to be successful, it has several obvious issues,
including exposure bias [9], loss-evaluation mismatch [9] and label bias [16], which have been studied.
However, we observe that there is still an important issue associated with beam search of NMT, the
myopic bias, which unfortunately is largely ignored, to the best of our knowledge. Beam search tends
to focus more on short-term reward. At iteration t, for a candidate y1 , ? ? ? , yt?1 (refers to y<t ) and
two words w and w0 , we denote y<t + w if we append w to y<t . If P (y<t + w|x) > P (y<t + w0 |x),
new candidate y<t + w is more likely to be kept, even if w0 is the ground truth translation at step t or
can offer a better score in future decodings. Such search errors coming from short sighted actions
sometimes provide a bad translation even if the translation model is good.
To address the myopic bias, for each word w and each candidate y<t , we propose to design a
prediction model to estimate the long-term reward if we append w to y<t and follow the current
NMT model until the decoding finishes. Then we can leverage the predicted score from this model
during each decoding step to help find a better w that can contribute to the long-term translation
performance. This prediction model, which predicts long-term reward we will receive in the future, is
exactly the concept of value function in Reinforcement Learning (RL).
In this work, we develop a neural network-based prediction model, which is called value network
for NMT. The value network takes the source sentence and any partial target sequence as input, and
outputs a predicted value to estimate the expected total reward (e.g. BLEU) generated from this
partial sequence by the NMT model. In any decoding step, we select the best candidates not only
based on the conditional probability of the partial sequence outputted by the NMT model, but also
based on the estimated long-term reward outputted by the value network.
The main contributions of this work are summarized as follows. First, we develop a decoding scheme
that considers long-term reward while generating words one by one for machine translation, which is
new in NMT literature. At each step, the new decoding scheme not only considers the probability of
the word sequence conditioned on the source sentence, but also relies on the predicted future reward.
We believe that considering the two aspects can lead to better final translation.
Second, we design a novel structure for the value network. On the top of the encoder-decoder layer
of NMT, we develop another two modules for the value network, a semantic matching module and a
context-coverage module. The semantic matching module aims at estimating the similarity between
the source and target sentences, which can contribute to the quality of the translation. It is often
observed that the more context used in the attention mechanism, the better translation we will generate
[14, 15]. Thus we build a context-coverage module to measure the coverage of context used in the
encoder-decoder layer. With the outputs of the two modules, the value prediction is done via fully
connected layers.
We conduct a set of experiments on several translation tasks. All the results demonstrate the
effectiveness and robustness of the new decoding mechanism compared to several baseline algorithms.
The remaining parts of the paper are organized as follows. In Section 2, we briefly review the
literature of neural machine translation. After that, we describe the myopic bias problem of NMT in
Section 3 and introduce our method for value network learning in Section 4. Experimental results are
provided and analyzed in Section 5. We discuss future directions in the last section.
2
Neural Machine Translation
Neural machine translation systems are typically implemented with a Recurrent Neural Network
(RNN)-based encoder-decoder framework. Such a framework directly models the probability P (y|x)
of a target sentence y = {y1 , y2 , ..., yTy } conditioned on the source sentence x = {x1 , x2 , ..., xTx },
where Tx and Ty are the length of sentence x and y.
The encoder of NMT reads the source sentence x word by word and generates a hidden representation
for each word xi :
hi = f (hi?1 , xi ),
2
(1)
Decoder
y1
y2
y3
r1
r2
r3
c1
c2
c3
Semantic Matching (SM) Module
???
?
Mean pooling
Mean pooling
SM Module
?
c1
Attention
c3
c2
?
r1
r3
?
?
htx
r2
CC Module
Context-Coverage (CC) Module
Encoder
h1
h2
h3
x1
x2
x3
?
???
htx
Mean pooling
xtx
c1
c2
Mean pooling
c3
?
h1
h2
Figure 1: Architecture of Value Network
in which function f is the recurrent unit such as Long Short-Term Memory (LSTM) unit [12] or Gated
Recurrent Unit (GRU) [4]. Afterwards, the decoder of NMT computes the conditional probability of
each target word yt conditioned on its proceeding words y<t as well as the source sentence:
P (yt |y<t , x) ? exp(yt ; rt , ct ),
rt = g(rt?1 , yt?1 , ct ),
ct = q(rt?1 , h1 , ? ? ? , hTx ),
(2)
(3)
(4)
where rt is the decoder RNN hidden representation at step t, similarly computed by an LSTM or GRU,
and ct denotes the weighted contextual information summarizing the source sentence x using some
attention mechanism [4]. Denote all the parameters to be learned in the encoder-decoder framework
as ?. For ease of reference, we also use ?? to represent the translation model with parameter ?.
Denote D as the training dataset that contains source-target sentence pairs. The training process
aims at seeking the optimal parameters ?? to correctly encode source sentence and decode it into the
target sentence. While there are different objectives to achieve this [2, 10, 1, 5, 19, 17], maximum
likelihood estimation is the most popular one [2]:
Y
?? = argmax
P (y|x; ?)
?
=
argmax
?
3
(x,y)?D
Y
Ty
Y
P (yt |y<t , x; ?).
(5)
(x,y)?D t=1
The Myopic Bias
Since during training, one aims to find the conditional probability P (y|x), ideally in testing, the
translation of a source sentence x should be the target sentence y with the maximum conditional
probability P (y|x). However, as there are exponentially many candidates in the target language, one
cannot compute the probability for every candidate and find the maximum one. Thus, beam search is
widely used to find a reasonable good target sentence [4, 14, 12].
Note that the training objective of NMT is usually defined on the full target sentence y instead of
partial sentences. One issue with beam search is that a locally good word might not lead to a good
complete sentence. From the example mentioned in the introduction, we can see that such search
errors from short sighted actions can provide a bad translation even if we hold a perfect translation
model. We call such errors the myopic bias. To reduce myopic bias, we hope to predict the long-term
value of each action and use the value in decoding, which is the exact motivation of our work.
There exist several works weakly related to this issue. [3] develops the scheduled sampling approach,
which takes the generated outputs from the model as well as the golden truth sentence in training,
to help the model learn from its own errors. Although it can (to some extent) handle the negative
3
impact of choosing an incorrect word at middle steps, it still follows beam search during testing,
which cannot avoid the myopic bias. Another related work is [16]. It learns a predictor to predict
the ranking score of a certain word at step t, and use this score to replace the conditional probability
outputted by the NMT model for beam search during testing. Unfortunately, this work still looks
only one step forward and cannot address the problem.
4
Value Network for NMT
As discussed in the previous section, it is not reasonable to fully rely on the conditional probability
in beam search. This motivates us to estimate the expected performance of any sequence during
decoding, which is exactly the concept of value function in reinforcement learning.
4.1
Value Network Structure
In conventional reinforcement learning, a value function describes how much cumulated reward could
be collected from state s by following certain policy ?. In machine translation, we can consider any
input sentence x paired with partial output sentence y<t as the state, and consider the translation
model ?? as policy which can generate a word (action) given any state. Given policy ?? , the value
function characterizes what the expected translation performance (e.g. BLEU score) is if we use ??
to translate x with the first t ? 1 words being y<t . Denote
and y ? (x)
P v(x, y<t ) as the value? function
0
as the ground truth translation, and then v(x, y<t ) = y0 ?Y:y0 =y<t BLEU(y (x), y )P (y 0 |x; ?),
<t
where Y is the space of complete sentences.
The first important problem is how to design the input and the parametric form of the value function.
As the translation model is built up on an encoder-decoder framework, we also build up our value
network on the top of this architecture. To fully exploit the information in the encoder-decoder
framework, we develop a value network with two new modules, the semantic matching module and
the context-coverage module.
Semantic Matching (SM) Module: In the semanticP
matching module, at time step t, we use mean
t
pooling over the decoder RNN hidden states r?t = 1t l=1 rl as a summarization of the partial target
Pt
sentence, and use mean pooling over context states c?t = 1t l=1 cl as a summarization of the context
in source language. We concatenate r?t and c?t , and use a feed-forward network ?SM = fSM ([?
rt , c?t ])
to evaluate semantic information between the source sentence and the target sentence.
Context-Coverage (CC) Module: It is often observed that the more context covered in the attention
model, the better translation we will generate [14, 15]. Thus we build a context-coverage module to
measure the coverage of information used in the encoder-decoder framework. We argue that using
mean pooling over the context layer and the encoding states should give some effective knowledge.
? = 1 PTx hl , we use another feed-forward network ?CC = fCC ([?
? to
Similarly, denote h
ct , h])
l=1
Tx
process such information.
In the end, we concatenate both ?SM and ?CC and then use another fully connected layer with
sigmoid activation function to output a scalar as the value prediction. The whole architecture is shown
in Figure 1.
4.2
Training Data Generation
Based on the designed value network structure, we aim at finding a model that can correctly predict the
performance after the decoding ends. Popular value function learning algorithms include Monte-Carlo
methods and Temple-Difference methods, and both of them have been adopted in many challenging
tasks [13, 7, 11]. In this paper, we adopt the Monte-Carlo method to learn the value function. Given
a well-learnt NMT model ?? , the training of the value network for ?? is shown in Algorithm 1. For
randomly picked source sentence x in the training corpus, we generate a partial target sentence yp
using ?? with random early stop, i.e., we randomly terminate the decoding process before its end.
Then for the pair (x, yp ), we use ?? to finish the translation starting from yp and obtain a set S(yp )
of K complete target sentences, e.g., using beam search. In the end, we compute the BLEU score of
4
each complete target sentence and calculate the averaged BLEU score of (x, yp ):
avg_bleu(x, yp ) =
1
K
X
BLEU(y ? (x), y).
(6)
y?S(yp )
avg_bleu(x, yp ) can be considered as an estimation of the long-term reward of state (x, yp ) used in
value network training.
4.3
Learning
Algorithm 1 Value network training
1: Input: Bilingual corpus, a trained neural machine translation model ?? , hyperparameter K.
2: repeat
3:
t = t + 1.
4:
Randomly pick a source sentence x from the training dataset.
5:
Generate two partial translations yp,1 , yp,2 for x using ?? with random early stop.
6:
Generate K complete translations for each partial translation using ?? and beam search.
Denote this set of complete target sentences as S(yp,1 ) and S(yp,2 ).
7:
Compute the BLEU score for each sentence in S(yp,1 ) and S(yp,2 ).
8:
Calculate the average BLEU score for each partial translation according to Eqn.(6)
9:
Gradient Decent on stochastic loss defined in Eqn.(7).
10: until converge
11: Output: Value network with parameter ?
In conventional Monte-Carlo method for value function estimation, people usually use a regression
model to approximate the value function, i.e., learn a mapping from (x, yp ) ? avg_bleu(x, yp ) by
minimizing the mean square error (MSE). In this paper, we take an alternative objective function
which is shown to be more effective in experiments. We hope the value network we learn is accurate
and useful in differentiating good and bad examples. Thus we use pairwise ranking loss instead of
MSE loss.
To be concrete, we sample two partial sentences yp,1 and yp,2 for each x. We hope the predicted
score of (x, yp,1 ) can be larger than that of (x, yp,2 ) by certain margin if avg_bleu(x, yp,1 ) >
avg_bleu(x, yp,2 ). Denote ? as the parameter of the value function described in Section 4.1. We
design the loss function as follows:
X
L(?) =
ev? (x,yp,2 )?v? (x,yp,1 ) ,
(7)
(x,yp,1 ,yp,2 )
where avg_bleu(x, yp,1 ) > avg_bleu(x, yp,2 ).
Algorithm 2 Beam search with value network in NMT
1: Input: Testing example x, neural machine translation model P (y|x) with target vocabulary V ,
value network model v(x, y), beam search size K, maximum search depth L, weight ?.
2: Set S = ?, U = ? as candidate sets.
3: repeat
4:
t = t + 1.
5:
Uexpand ? {yi + {w}|yi ? U, w ? V }.
6:
U ? {top (K ? |S|) candidates that maximize
? ? 1t log P (y|x) + (1 ? ?) ? log v(x, y)|y ? Uexpand }
7:
Ucomplete ? {y|y ? U, yt = EOS}
8:
U ? U \ Ucomplete
9:
S ? S ? Ucomplete
10: until |S| = K or t = L
1
11: Output: y = argmaxy?S?U ? ? |y|
log P (y|x) + (1 ? ?) ? log v(x, y)
5
4.4
Inference
Since the value network estimates the long-term reward of a state, it will be helpful to enhance the
decoding process of NMT. For example, in a certain decoding step, the NMT model prefers word
w1 over w2 according to the conditional probability, but it does not know that picking w2 will be a
better choice for future decoding. As the value network provides sufficient information on the future
reward, if the value network outputs show that picking w2 is better than picking w1 , we can take both
NMT probability and future reward into consideration to choose a better action.
In this paper, we simply linearly combine the outputs of the NMT model and the value network,
which is motivated by the success of AlphaGo [11]. We first compute the normalized log probability
of each candidate, and then linearly combine it with the logarithmic value of the reward. In detail,
given a translation model P (y|x), a value network v(x, y) and a hyperparameter ? ? (0, 1), the score
of partial sequence y for x is computed by
??
1
log P (y|x) + (1 ? ?) ? log v(x, y),
|y|
(8)
where |y| is the length of y. The details of the decoding process are presented in Algorithm 2, and we
call our neural network-based decoding algorithm NMT-VNN for short.
5
Experiments
5.1
Settings
We compare our proposed NMT-VNN with two baselines. The first one is classic NMT with beam
search [2] (NMT-BS). The second one [16] trains a predictor that can evaluate the quality of any
partial sequence, e.g., partial BLEU score 1 , and then it uses the predictor to select words instead of
the probability. The main difference between [16] and ours is that they predict the local improvement
of BLEU for any single word, while ours aims at predicting the final BLEU score and use the predicted
score to select words. We refer their work as beam search optimization (we call it NMT-BSO). For
NMT-BS, we directly used the open source code [2]. NMT-BSO was implemented by ourselves
based on the open source code [2].
We tested our proposed algorithms and the baselines on three pairs of languages: English?French
(En?Fr), English?German (En?De), and Chinese?English (Zh?En). In detail, we used the same
bilingual corpora from WMT? 14 as used in [2] , which contains 12M, 4.5M and 10M training data
for each task. Following common practices, for En?Fr and En?De, we concatenated newstest2012
and newstest2013 as the validation set, and used newstest2014 as the testing set. For Zh?En, we
used NIST 2006 and NIST 2008 datasets for testing, and use NIST 2004 dataset for validation. For
all datasets in Chinese, we used a public tool for word segmentation. In all experiments, validation
sets were only used for early-stopping and hyperparameter tuning.
For NMT-VNN and NMT-BS, we need to train an NMT model first. We followed [2] to set
experimental parameters to train the NMT model. For each language, we constructed the vocabulary
with the most common 30K words in the parallel corpora, and out-of-vocabulary words were replaced
with a special token ?UNK". Each word was embedded into a vector space of 620 dimensions, and
the dimension of the recurrent unit was 1000. We removed sentences with more than 50 words from
the training set. Batch size was set as 80 with 20 batches pre-fetched and sorted by sentence lengths.
The NMT model was trained with asynchronized SGD on four K40m GPUs for about seven days.
For NMT-BSO, we implemented the algorithm and the model was trained in the same environment.
For the value network used in NMT-VNN, we set the same parameters for the encoder-decoder
layers as the NMT model. Additionally, in the SM module and CC module, we set function ?SM
and ?CC as single-layer feed forward networks with 1000 output nodes. In Algorithm 1, we set
the hyperparameter K = 20 to estimate the value of any partial sequence. We adapted mini-batch
training with batch size to be 80, and the value network model was trained with AdaDelta [21] on
one K40m GPU for about three days.
If the ground truth is y ? , the partial bleu on the partial sequence y<t at step t is defined as the BLEU score
?
on y<t and y<t
.
1
6
During testing, the hyperparameter ? for NMT-VNN was set by cross validation. For En?Fr, En?De
and Zh?En tasks, we found setting ? to be 0.85, 0.9 and 0.8 respectively are the best choices. We
used the BLEU score [8] as the evaluation metric, which is computed by the multi-bleu.perl script2 .
We set the beam search size to be 12 for all the algorithms following the common practice [12].
(a) En?Fr
(b) En?De
(c) Zh?En NIST 2006
(d) Zh?En NIST 2008
Figure 2: Translation results on the test sets of three tasks
5.2
Overall Results
We report the experimental results in this subsection. From Table 1 we can see that our NMT-VNN
algorithm outperforms the baseline algorithms on all tasks.
For English?French task and English?German task, NMT-VNN outperforms the baseline NMT-BS
by about 1.03/1.3 points. As the only difference between the two algorithms is that our NMT-VNN
additionally uses the outputs of value network to enhance decoding, we can conclude that such
additional knowledge provides useful information to help the NMT model. Our method outperforms
NMT-BSO by about 0.31/0.33 points. Since NMT-BSO only uses a local BLEU predictor to estimate
the partial BLEU score while ours predicts the future performance, our proposed value network which
considers long-term benefit is more powerful.
The performance of our NMT-VNN is much better than NMT-BS and NMT-BSO for
Chinese?English tasks. NMT-VNN outperforms the baseline NMT-BS by about 1.4/1.82 points on
NIST 2006 and NIST 2008, and outperforms NMT-BSO by about 1.01/0.72 points. We plot BLEU
scores with respect to the length of source sentences in Figure 2 for all the tasks. From the figures, we
can see that our NMT-VNN algorithm outperforms the baseline algorithms in almost all the ranges of
length.
Furthermore, we also test our value network on a deep NMT model in which the encoder and decoder
are both stacked 4-layer LSTMs. The result also shows that we can get 0.33 points improvement on
English?French task. These results demonstrate the effectiveness and robustness of our NMT-VNN
algorithm.
2
https://github.com/moses-smt/mosesdecoder/blob/master/scripts/generic/multi-bleu.perl. For final evaluation we use corpus-level BLEU, while for the value network training we use sentence-level BLEU as in
[1].
7
Table 1: Overall Performance
En?Fr En?De Zh?En NIST06 Zh?En NIST08 En?Fr Deep
NMT-BS 30.51
NMT-BSO 31.23
NMT-VNN 31.54
5.3
15.67
16.64
16.97
36.2
36.59
37.6
29.4
30.5
31.22
37.86
?
38.19
Analysis on Value Network
We further look into the learnt value network and conduct some analysis to better understand it.
First, as we use an additional component during decoding, it will affect the efficiency of the translation process. As the designed value network architecture is similar to the basic NMT model, the
computational complexity is similar to the NMT model and the two processes can be run in parallel.
Second, it has been observed that the accuracy of NMT is sometimes very sensitive to the size of
beam search on certain tasks. As the beam size grows, the accuracy will drop drastically. [14] argues
this is because the training of NMT favors short but inadequate translation candidates. We also
observe this phenomenon on English?German translation. However, we show that by using value
network, such shortage can be largely avoided. We tested the accuracy of our algorithm with different
beam sizes, as shown in Figure 3.(a). It can be seen that NMT-VNN is much more stable than the
original NMT without value network: its accuracy only differs a little for different beam sizes while
NMT-BS drops more than 0.5 point when the beam size is large.
(a)
(b)
Figure 3: (a). BLEU scores of En?De task w.r.t different beam size. (b). BLEU scores of En?De
task w.r.t different hyperparameter ?.
Third, we tested the performances of NMT-VNN using different hyperparameter ? during decoding
for English?German task. As can be seen from the figure, the performance is stable for the ? ranging
from 0.7 to 0.95, and slightly drops for a smaller ?. This shows that our proposed algorithm is robust
to the hyperparameter.
6
Conclusions and Future Work
In this work we developed a new decoding scheme that incorporates value networks for neural
machine translation. By introducing the value network, the new decoding scheme considers not
only the local conditional probability of a candidate word, but also its long-term reward for future
decoding. Experiments on three translation tasks verify the effectiveness of the new scheme. We plan
to explore the following directions in the future. First, it is interesting to investigate how to design
better structures for the value network. Second, the idea of using value networks is quite general, and
we will extend it to other sequence-to-sequence learning tasks, such as image captioning and dialog
systems.
8
Acknowledgments
This work was partially supported by National Basic Research Program of China (973 Program)
(grant no. 2015CB352502), NSFC (61573026). We would like to thank the anonymous reviewers for
their valuable comments on our paper.
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6,216 | 6,623 | Parametric Simplex Method for Sparse Learning
?
Haotian Pang? Robert Vanderbei? Han Liu??
Tuo Zhao?
?
?
Princeton University Tencent AI Lab Northwestern University ? Georgia Tech?
Abstract
High dimensional sparse learning has imposed a great computational challenge to
large scale data analysis. In this paper, we are interested in a broad class of sparse
learning approaches formulated as linear programs parametrized by a regularization
factor, and solve them by the parametric simplex method (PSM). Our parametric
simplex method offers significant advantages over other competing methods: (1)
PSM naturally obtains the complete solution path for all values of the regularization
parameter; (2) PSM provides a high precision dual certificate stopping criterion;
(3) PSM yields sparse solutions through very few iterations, and the solution
sparsity significantly reduces the computational cost per iteration. Particularly,
we demonstrate the superiority of PSM over various sparse learning approaches,
including Dantzig selector for sparse linear regression, LAD-Lasso for sparse robust
linear regression, CLIME for sparse precision matrix estimation, sparse differential
network estimation, and sparse Linear Programming Discriminant (LPD) analysis.
We then provide sufficient conditions under which PSM always outputs sparse
solutions such that its computational performance can be significantly boosted.
Thorough numerical experiments are provided to demonstrate the outstanding
performance of the PSM method.
1
Introduction
A broad class of sparse learning approaches can be formulated as high dimensional optimization
problems. A well known example is Dantzig Selector, which minimizes a sparsity-inducing `1 norm
with an `1 norm constraint. Specifically, let X 2 Rn?d be a design matrix, y 2 Rn be a response
vector, and ? 2 Rd be the model parameter. Dantzig Selector can be formulated as the solution to the
following convex program,
?b = argmin k?k1 s.t. kX > (y X?)k1 ? .
(1.1)
?
where k ? k1 and k ? k1 denote the `1 and `1 norms respectively, and > 0 is a regularization factor.
Candes and Tao (2007) suggest to rewrite (1.1) as a linear program solved by linear program solvers.
Dantzig Selector motivates many other sparse learning approaches, which also apply a regularization
factor to tune the desired solution. Many of them can be written as a linear program in the following
generic form with either equality constraints:
max(c + c?)> x s.t. Ax = b + ?b, x 0,
(1.2)
x
or inequality constraints:
max(c + c?)> x
x
s.t. Ax ? b + ?b, x
0.
(1.3)
Existing literature usually suggests the popular interior point method (IPM) to solve (1.2) and (1.3).
The interior point method is famous for solving linear programs in polynomial time. Specifically,
the interior point method uses the log barrier to handle the constraints, and rewrites (1.2) or (1.3)
?
Correspondence to Tuo Zhao: [email protected].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
as a unconstrained program, which is further solved by the Newton?s method. Since the log barrier
requires the Newton?s method to only iterate within the interior of the feasible region, IPM cannot
yield exact sparse iterates, and cannot take advantage of sparsity to boost the computation.
An alternative approach is the simplex method. From a geometric perspective, the classical simplex
method iterates over the vertices of a polytope. Algebraically, the algorithm involves moving from
one partition of the basic and nonbasic variables to another. Each partition deviates from the previous
in that one basic variable gets swapped with one nonbasic variable in a process called pivoting.
Different variants of the simplex method are defined by different rules of pivoting. The simplex
method has been shown to work well in practice, even though its worst-case iteration complexity has
been shown to scale exponentially with the problem scale in existing literature.
More recently, some researchers propose to use alternating direction methods of multipliers (ADMM)
to directly solve (1.1) without reparametrization as a linear program. Although these methods enjoy
O(1/T ) convergence rates based on variational inequality criteria, where T is the number of iterations.
ADMM can be viewed as an exterior point method, and always gives infeasible solutions within
finite number of iterations. We often observe that after ADMM takes a large number of iterations,
the solutions still suffer from significant feasibility violation. These methods work well only for
moderate scale problems (e.g., d < 1000). For larger d, ADMM becomes less competitive.
These methods, though popular, are usually designed for solving (1.2) and (1.3) for one single
regularization factor. This is not satisfactory, since an appropriate choice of is usually unknown.
Thus, one usually expects an algorithm to obtain multiple solutions tuned over a reasonable range of
values for . For each value of , we need to solve a linear program from scratch, and it is therefore
often very inefficient for high dimensional problems.
To overcome the above drawbacks, we propose to solve both (1.2) and (1.3) by a variant of the
parametric simplex method (PSM) in a principled manner (Murty, 1983; Vanderbei, 1995). Specifically, the parametric simplex method parametrizes (1.2) and (1.3) using the unknown regularization
factor as a ?parameter?. This eventually yields a piecewise linear solution path for a sequence of
regularization factors. Such a varying parameter scheme is also called homotopy optimization in
existing literature. PSM relies some special rules to iteratively choose the pair of variables to swap,
which algebraically calculates the solution path during each pivoting. PSM terminates at a value of
parameter, where we have successfully solved the full solution path to the original problem. Although
in the worst-case scenario, PSM can take an exponential number of pivots to find an optimal solution
path. Our empirical results suggest that the number of iterations is roughly linear in the number of
nonzero variables for large regularization factors with sparse optima. This means that the desired
sparse solutions can often be found using very few pivots.
Several optimization methods for solving (1.1) are closely related to PSM. However, there is a lack
of generic design in these methods. Their methods, for example, the simplex method proposed in
Yao and Lee (2014) can be viewed as a special example of our proposed PSM, where the perturbation
is only considered on the right-hand-side of the inequalities in the constraints. DASSO algorithm
computes the entire coefficient path of Dantzig selector by a simplex-like algorithm. Zhu et al. (2004)
propose a similar algorithm which takes advantage of the piece-wise linearity of the problem and
computes the whole solution path on `1 -SVM. These methods can be considered as similar algorithms
derived from PSM but only applied to special cases, where the entire solution path is computed but
an accurate dual certificate stopping criterion is not provided.
Notations: We denote all zero and all one vectors by 1 and 0 respectively.
P Given a vector a =
(a1 , ..., ad )> 2 Rd , we define the number of nonzero entries kak0 =
j 1(aj 6= 0), kak1 =
P
P 2
2
j |aj |, kak2 =
j aj , and kak1 = maxj |aj |. When comparing vectors, ? ? and ??? mean
component-wise comparison. Given a matrix A 2 Rd?d with entries aP
jk , we use |||A||| to denote
entry-wise norms and kAk to denote matrix norms. Accordingly |||A|||0 = j,k 1(ajk 6= 0), |||A|||1 =
P
P
P
|, kAk1 = maxk j |ajk |, kAk1 = maxj k |ajk |, kAk2 =
j,k |ajk |, |||A|||1 = maxj,k |ajkP
maxkak2 ?1 kAak2 , and kAk2F = j,k a2jk . We denote A\i\j as the submatrix of A with i-th row
and j-th column removed. We denote Ai\j as the i-th row of A with its j-th entry removed and
A\ij as the j-th column of A with its i-th entry removed. For any subset G of {1, 2, . . . , d}, we let
AG denote the submatrix of A 2 Rp?d consisting of the corresponding columns of A. The notation
A 0 means all of A?s entries are nonnegative. Similarly, for a vector a 2 Rd , we let aG denote the
subvector of a associated with the indices in G. Finally, Id denotes the d-dimensional identity matrix
2
and ei denotes vector that has a one in its i-th entry and zero elsewhere. In a large matrix, we leave a
submatrix blank when all of its entries are zeros.
2
Background
Many sparse learning approaches are formulated as convex programs in a generic form:
min L(?) + k?k1 ,
?
(2.1)
where L(?) is a convex loss function, and > 0 is a regularization factor controlling bias and
variance. Moreover, if L(?) is smooth, we can also consider an alternative formulation:
min k?k1 s.t. krL(?)k1 ? ,
(2.2)
?
where rL(?) is the gradient of L(?), and > 0 is a regularization factor. As will be shown later,
both (2.2) and (2.1) are naturally suited for our algorithm, when L(?) is piecewise linear or quadratic
respectively. Our algorithm yields a piecewise-linear solution path as a function of by varying
from large to small.
Before we proceed with our proposed algorithm, we first introduce the sparse learning problems of
our interests, including sparse linear regression, sparse linear classification, and undirected graph
estimation. Due to space limit, we only present three examples, and defer the others to the appendix.
Dantzig Selector: The first problem is sparse linear regression. Let y 2 Rn be a response vector
and X 2 Rn?d be the design matrix. We consider a linear model y = X?? + ?, where ?? 2 Rd is the
unknown regression coefficient vector, and ? is the observational noise vector. Here we are interested
in a high dimensional regime: d is much larger than n, i.e., d
n, and many entries in ?? are zero,
?
?
0
i.e., k? k0 = s ? n. To get a sparse estimator of ? , machine learning researchers and statisticians
have proposed numerous approaches including Lasso (Tibshirani, 1996), Dantzig Selector (Candes
and Tao, 2007) and LAD-Lasso (Wang et al., 2007).
The Dantzig selector is formulated as the solution to the following convex program:
min k?k1 subject to kX > (y X?)k1 ? .
?
By setting ? = ?+
linear program:
? with
min
1> (?+ + ? )
+
? ,?
s.t.
?
?j+
= ?j ? 1(?j > 0) and
X >X
X >X
X >X
X >X
??
?+
?
?j+
?
(2.3)
= ?j ? 1(?j < 0), we rewrite (2.3) as a
?
?
?
1 + X >y
,
1 X >y
?+ , ?
0. (2.4)
By complementary slackness, we can guarantee that the optimal ?j+ ?s and ?j ?s are nonnegative and
complementary to each other. Note that (2.4) fits into our parametric linear program as (1.3) with
? >
?
? > ?
? +?
X X
X >X
X y
?b = 1, c? = 0, x = ?
,
c
=
1,
.
A=
,
b
=
?
X >X X >X
X >y
Sparse Support Vector Machine: The second problem is Sparse SVM (Support Vector Machine),
which is a model-free discriminative modeling approach (Zhu et al., 2004). Given n independent
and identically distributed samples (x1 , y1 ), ..., (xn , yn ), where xi 2 Rd is the feature vector and
yi 2 {1, 1} is the binary label. Similar to sparse linear regression, we are interested in the high
dimensional regime. To obtain a sparse SVM classifier, we solve the following convex program:
n
X
min
[1 yi (?0 + ?> xi )]+ s.t. k?k1 ? ,
(2.5)
?0 ,?
i=1
where ?0 2 R and ? 2 Rd . Given a new sample z 2 Rd , Sparse SVM classifier predicts its label
by sign(?0 + ?> z). Let ti = 1 yi (?0 + ?> xi ) for i = 1, ..., n. Then ti can be expressed as
ti = t+
ti . Notice [1 yi (?0 + ?> xi )]+ can be represented by t+
i
i . We split ? and ?0 into
positive and negative parts as well: ? = ?+ ? and ?0 = ?0+ + ?0 and add slack variable w
to the constraint so that the constraint becomes equality: ?+ + ? + w = 1, w
0. Now we
cast the problem into the equality parametric simplex form (1.2). We identify each component
>
of (1.2) as the following: x = t+ t
2 R(n+1)?(2n+3d+2) , x
?+ ?
?0+ ?0 w
>
2 R2n+3d+2
, c? = 0 2 R2n+3d+2 , b? =
?
I
In Z
Z y
y
>
>
2
1> 0 2 Rn+1 , ?b = 0> 1 2 Rn+1 , and A = n
1> 1>
1>
R(n+1)?(2n+3d+2) , where Z = (y1 x1 , . . . , yn xn )> 2 Rn?d .
0, c =
1>
0>
0>
0>
0
0
0>
3
Differential Graph Estimation: The third problem is the differential graph estimation, which
aims to identify the difference between two undirected graphs (Zhao et al., 2013; Danaher et al.,
2013). The related applications in biological and medical research can be found in existing literature
(Hudson et al., 2009; Bandyopadhyaya et al., 2010; Ideker and Krogan, 2012). Specifically, given n1
i.i.d. samples x1 , ..., xn from Nd (?0X , ?0X ) and n2 i.i.d. samples y1 , ..., yn from Nd (?0Y , ?0Y ) We
are interested in estimating the difference of the precision matrices of two distributions:
0
= (?0X ) 1 (?0Y ) 1 .
P n1
We define the empirical covariance matrices as: SX = n11 j=1
(x
x
?)(xj x
?)> andSY =
P
P
Pn j
n2
n
1
1
1
>
y?)(yj x
?) , where x
? = n1 j=1 xj and y? = n2 j=1 yj . Then Zhao et al. (2013)
j=1 (yj
n2
propose to estimate 0 by solving the following problem:
min ||| |||1 s.t. |||SX SY SX + SY |||1 ? ,
(2.6)
where SX and SY are the empirical covariance matrices. As can be seen, (2.6) is essentially a special
example of a more general parametric linear program as follows,
min |||D|||1 s.t. |||XDZ Y |||1 ? ,
(2.7)
D
where D 2 R
d1 ?d2
,X 2R
m1 ?d1
, Z 2 Rd2 ?m2 and Y 2 Rm1 ?m2 are given data matrices.
Instead of directly solving (2.7), we consider a reparametrization by introducing an axillary variable
C = XD. Similar to CLIME, we decompose D = D+ + D , and eventually rewrite (2.7) as
min 1> (D+ + D )1 s.t. |||CZ
D + ,D
Y |||1 ? ,
X(D+
D ) = C,
D+ , D
0, (2.8)
Let vec(D+ ), vec(D ), vec(C) and vec(Y ) be the vectors obtained by stacking the columns of
matrices D+ , D C and Y , respectively. We write (2.8) as a parametric linear program,
0 0
1
X
X0
Im1 d2
A
Z0
I m1 m2
A=@
Z0
I m1 m2
0
1
0
1
X
z11 Im1
???
zd 2 1 I m 1
B
C
B
C
..
..
..
..
with X 0 = @
A 2 Rm1 d2 ?d1 d2 , Z 0 = @
A 2
.
.
.
.
X
z1m2 Im1 ? ? ?
zd 2 m 2 I m 1
Rm1 m2 ?m1 d2 , where zij denotes the (i, j) entry of matrix Z;
?
?>
x = vec(D+ ) vec(D ) vec(C) w 2 R2d1 d2 +m1 d2 +2m1 m2 ,
where w 2 R2m1 m2 is nonnegative slack variable vector used to make the inequality become
?
?>
an equality. Moreover, we also have b = 0> vec(Y )
2 Rm1 d2 +2m1 m2 , where
vec(Y )
the first m1 d2 components of vector b are 0 and the rest components are from matrix Y ; ?b =
>
0> 1> 1> 2 Rm1 d2 +2m1 m2 , where the first m1 d2 components of vector ?b are 0 and the rest
>
2m1 m2 components are 1; c =
1>
1> 0> 0> 2 R2d1 d2 +m1 d2 +2m1 m2 , where the first
2d1 d2 components of vector c are 1 and the rest m1 d2 + 2m1 m2 components are 0.
3
Homotopy Parametric Simplex Method
We first briefly review the primal simplex method for linear programming, and then derive the
proposed algorithm.
Preliminaries: We consider a standard linear program as follows,
max c> x
x
s.t. Ax = b,
x
0 x 2 Rn ,
(3.1)
where A 2 Rm?n , b 2 Rm and c 2 Rn are given. Without loss of generality, we assume that m ? n
and matrix A has full row rank m. Throughout our analysis, we assume that an optimal solution
exists (it needs not be unique). The primal simplex method starts from a basic feasible solution (to
be defined shortly?but geometrically can be thought of as any vertex of the feasible polytope) and
proceeds step-by-step (vertex-by-vertex) to the optimal solution. Various techniques exist to find the
first feasible solution, which is often referred to the Phase I method. See Vanderbei (1995); Murty
(1983); Dantzig (1951).
4
Algebraically, a basic solution corresponds to a partition of the indices {1, . . . , n} into m basic
indices denoted B and n m non-basic indices denoted N . Note that not all partitions are allowed?
the submatrix of A consisting of the columns of A associated with the basic indices, denoted AB ,
must be invertible. The submatrix of A corresponding to the nonbasic indices is denoted AN .
Suppressing the fact that the columns have been rearranged, we can write A = [AN , AB ]. If
we rearrange the rows
? of x and?c in the same way, we can introduce a corresponding partition of
x
c
these vectors: x = N , c = N . From the commutative property of addition, we rewrite the
xB
cB
constraint as AN xN + AB xB = b. Since the matrix AB is assumed to be invertible, we can express
xB in terms of xN as follows:
(3.2)
xB = x?B AB 1 AN xN ,
where we have written x?B as an abbreviation for AB 1 b. This rearrangement of the equality constraints
is called a dictionary because the basic variables are defined as functions of the nonbasic variables.
Denoting the objective c> x by ?, then we also can write:
>
?
? >
? = c> x = c>
(zN
) xN ,
B xB + c N xN = ?
1
1
1
?
>
?
?
>
where ? = cB AB b, xB = AB b and zN = (AB AN ) cB cN .
(3.3)
We call equations (3.2) and (3.3) the primal dictionary associated with the current basis B. Corresponding to each dictionary, there is a basic solution (also called a dictionary solution) obtained by setting the nonbasic variables to zero and reading off values of the basic variables:
xN = 0,
xB = x?B . This particular ?solution? satisfies the equality constraints of the problem by
construction. To be a feasible solution, one only needs to check that the values of the basic variables
are nonnegative. Therefore, we say that a basic solution is a basic feasible solution if x?B 0.
The dual of (3.1) is given by
max b> y
y
s.t. A> y
z = c,
z
0 z 2 Rn , y 2 Rm .
In this case, we separate variable z into basic and nonbasic parts as before: [z] =
corresponding dual dictionary is given by:
?
z N = zN
+ (AB 1 AN )> zB ,
?=
??
(x?B )> zB ,
(3.4)
?
zN
. The
zB
(3.5)
1
1
?
?
where ? denotes the objective function in the (3.4), ? ? = c>
B AB b, xB = AB b and zN =
1
>
(AB AN ) cB cN .
For each dictionary, we set xN and zB to 0 (complementarity) and read off the solutions to xB and
zN according to (3.2) and (3.5). Next, we remove one basic index and replacing it with a nonbasic
index, and then get an updated dictionary. The simplex method produces a sequence of steps to
adjacent bases such that the value of the objective function is always increasing at each step. Primal
feasibility requires that xB 0, so while we update the dictionary, primal feasibility must always be
satisfied. This process will stop when zN 0 (dual feasibility), since it satisfies primal feasibility,
dual feasibility and complementarity (i.e., the optimality condition).
Parametric Simplex Method: We derive the parametric simplex method used to find the full
solution path while solving the parametric linear programming problem only once. A few variants of
the simplex method are proposed with different rules for choosing the pair of variables to swap at each
iteration. Here we describe the rule used by the parametric simplex method: we add some positive
perturbations (?b and c?) times a positive parameter to both objective function and the right hand side
of the primal problem. The purpose of doing this is to guarantee the primal and dual feasibility when
is large. Since the problem is already primal feasible and dual feasible, there is no phase I stage
required for the parametric simplex method. Furthermore, if the i-th entry of b or the j-th entry of c
has already satisfied the feasibility condition (bi 0 or cj ? 0), then the corresponding perturbation
?bi or c?j to that entry is allowed to be 0. With these perturbations, (3.1) becomes:
max(c + c?)> x s.t. Ax = b + ?b, x 0 x 2 Rn .
(3.6)
x
We separate the perturbation vectors into basic and nonbasic parts as well and write down the the
dictionary with perturbations corresponding to (3.2),(3.3), and (3.5) as:
?
xB = (x?B + x
?B ) AB 1 AN xN , ? = ? ? (zN
+ z?N )> xN ,
(3.7)
5
where
x?B
?
zN = (zN
+ z?N ) + (AB 1 AN )> zB ,
1
= AB b,
?
zN
1
>
= (AB AN ) cB
? = ? ? (x?B + x
? B ) > zB ,
cN , x
?B = AB 1?b and z?N = (AB 1 AN )> c?B
(3.8)
c?N .
?
When is large, the dictionary will be both primal and dual feasible (x?B + x
?B 0 and zN
+ z?N
0). The corresponding primal solution is simple: xB = x?B + x
?B and xN = 0. This solution is
valid until hits a lower bound which breaks the feasibility. The smallest value of without break
any feasibility is given by
?
?
= min{ : zN
+ z?N
0 and x?B + x
?B
(3.9)
0}.
In other words, the dictionary and its corresponding solution xB =
+ x
?B and xN = 0 is optimal
for the value of 2 [ ? , max ], where
?
?
?
zN
x?
?
= max
maxj2N , z?Nj >0 z?Nj , maxi2B,?xBi >0 x?BBi ,
(3.10)
x?B
j
max
= min
?
minj2N , z?Nj <0
?
zN
i
j
z?Nj
,
mini2B,?xBi <0
x?
Bi
x
? Bi
?
.
(3.11)
Note that although initially the perturbations are nonnegative, as the dictionary gets updated, the
perturbation does not necessarily maintain nonnegativity. For each dictionary, there is a corresponding
interval of given by (3.10) and (3.11). We have characterized the optimal solution for this interval,
and these together give us the solution path of the original parametric linear programming problem.
Next, we show how the dictionary gets updated as the leaving variable and entering variable swap.
We expect that after swapping the entering variable j and leaving variable i, the new solution in the
dictionary (3.7) and (3.8) would slightly change to:
x?j = t, x
??j = t?, zi? = s, z?i? = s?, x?B
x?B t xB , x
?B
x
?B t? xB ,
?
zN
?
zN
s zN , z?N
z?N s? zN ,
?
where t and t are the primal step length for the primal basic variables and perturbations, s and s? are
the dual step length for the dual nonbasic variables and perturbations, xB and zN are the primal
and dual step directions, respectively. We explain how to find these values in details now.
?
There is either a j 2 N for which zN
+ z?N = 0 or an i 2 B for which x?B + x
?B = 0 in (3.9). If
it corresponds to a nonbasic index j, then we do one step of the primal simplex. In this case, we
declare j as the entering variable, then we need to find the primal step direction xB . After the
entering variable j has been selected, xN changes from 0 to tej , where t is the primal step length.
Then according to (3.7), we have that xB = (x?B + x
?B ) AB 1 AN tej . The step direction xB
is given by xB = AB 1 AN ej . We next select the leaving variable. In order to maintain primal
feasibility, we need to keep xB
0, therefore, the leaving variable i is selected such that i 2 B
achieves the maximal value of x? + x?i x?i . It only remains to show how zN changes. Since i is the
i
leaving variable, according to (3.8), we have zN = (AB 1 AN )> ei . After we know the entering
variables, the primal and dual step directions, the primal and dual step lengths can be found as
z?
x?
z?
t = xi i , t? = x?xi i , s = zj j , s? = zj j .
If, on the other hand, the constraint in (3.9) corresponds to a basic index i, we declare i as the leaving
variable, then similar calculation can be made based on the dual simplex method (apply the primal
simplex method to the dual problem). Since it is very similar to the primal simplex method, we omit
the detailed description.
The algorithm will terminate whenever ? ? 0. The corresponding solution is optimal since our
dictionary always satisfies primal feasibility, dual feasibility and complementary slackness condition.
The only concern during the entire process of the parametric simplex method is that does not equal
to zero, so as long as can be set to be zero, we have the optimal solution to the original problem.
We summarize the parametric simplex method in Algorithm 1:
The following theorem shows that the updated basic and nonbasic partition gives the optimal solution.
Theorem 3.1. For a given dictionary with parameter in the form of (3.7) and (3.8), let B be a basic
index set and N be an nonbasic index set. Assume this dictionary is optimal for 2 [ ? , max ],
where ? and max are given by (3.10) and (3.11), respectively. The updated dictionary with basic
index set B ? and nonbasic index set N ? is still optimal at = ? .
6
Write down the dictionary as in (3.7) and (3.8);
Find ? given by (3.10);
while ? > 0 do
if the constraint in (3.10) corresponds to an index j 2 N then
Declare xj as the entering variable;
Compute primal step direction. xB = AB 1 AN ej ;
Select leaving variable. Need to find i 2 B that achieves the maximal value of
1
x?
i+
Compute dual step direction. It is given by zN = (AB AN )> ei ;
else if the constraint in (3.10) corresponds to an index i 2 B then
Declare zi as the leaving variable;
Compute dual step direction. zN = (AB 1 AN )> ei ;
Select entering variable. Need to find j 2 N that achieves the maximal value of
1
xi
?x
?
zj? +
i
;
zj
?z
?j
;
Compute primal step direction. It is given by xB = AB AN ej ;
Compute the dual and primal step lengths for both variables and perturbations:
t=
x?i
,
xi
x
?i
t? =
,
xi
s=
zj?
,
zj
s? =
z?j
.
zj
Update the primal and dual solutions:
x?j = t,
x?B
x
?j = t?,
x?B
zi? = s,
?
zN
z?i = s?,
?
zN
t xB , x
?B
x
?B t? xB
s zN , z?N
z?N s? zN .
Update the basic and nonbasic index sets B := B \ {i} \ {j} and N := N \ {j} \ {i}. Write
down the new dictionary and compute ? given by (3.10);
end
Set the nonbasic variables as 0s and read the values of the basic variables.
Algorithm 1: The parametric simplex method
During each iteration, there is an optimal solution corresponding to 2 [ ? , max ]. Notice each of
these ?s range is determined by a partition between basic and nonbasic variables, and the number
of the partition into basic and nonbasic variables is finite. Thus after finite steps, we must find the
optimal solution corresponding to all values.
Theory: We present our theoretical analysis on solving Dantzig selector using PSM. Specifically,
given X 2 Rn?d , y 2 Rn , we consider a linear model y = X?? + ?, where ?? is the unknown sparse
regression coefficient vector with k?? k0 = s? , and ? ? N (0, 2 In ). We show that PSM always
maintains a pair of sparse primal and dual solutions. Therefore, the computation cost within each
iteration of PSM can be significantly reduced. Before we proceed with our main result, we introduce
two assumptions. The first assumption requires the regularization factor to be sufficiently large.
Assumption 3.2. Suppose that PSM solves (2.3) for a regularization sequence { K }N
K=0 . The
smallest regularization factor N satisfies
r
log d
4kX > ?k1 for some generic constant C.
N =C
n
Existing literature has extensively studied Assumption 3.2 for high dimensional statistical theories.
Such an assumption enforces all regularization parameters to be sufficiently large in order to eliminate
irrelevant coordinates along the regularization path. Note that Assumption 3.2 is deterministic
for any given N . Existing literature has verified that for sparse linear regression models, given
? ? N (0, 2 In ), Assumption 3.2 holds with overwhelming probability.
Before we present the second assumption, we define the largest and smallest s-sparse eigenvalues of
n 1 X > X respectively as follows.
Definition 3.3. Given an integer s 1, we define
X >X
nk k22
T
?+ (s) = sup
k k0 ?s
and
7
X >X
nk k22
T
? (s) =
inf
k k0 ?s
.
Assumption 3.4. Given k?? k0 ? s? , there exists an integer se such that
se 100?s? , ?+ (s? + se) < +1, and ?e (s? + se) > 0,
where ? is defined as ? = ?+ (s? + se)/e
? (s? + se).
Assumption 3.4 guarantees that n 1 X > X satisfies the sparse eigenvalue conditions as long as the
e along the solution path. That is closely related to
number of active irrelevant blocks never exceeds 2s
the restricted isometry property (RIP) and restricted eigenvalue (RE) conditions, which have been
extensively studied in existing literature.
We then characterize the sparsity of the primal and dual solutions within each iteration.
Theorem 3.5 (Primal and Dual Sparsity). Suppose that Assumptions 3.2 and 3.4 hold. We consider
an alterantive formulation to the Dantzig selector,
0
?b = argmin k?k1 subject to rj L(?) ? 0 , rj L(?) ? 0 .
(3.12)
?
0
0
0
0
0
Let ?
b = [b
?1 , ..., ?
bd , bd+1 , ..., b2d ]> denote the optimal dual variables to (3.12). For any
0
0
we have kb
? k0 + kb k0 ? s? + se. Moreover, given design matrix satisfying
kXS> XS (XS> XS )
where ? > 0 is a generic constant, S = {j |
?j?
1
k1 ? 1
0
,
?,
0
6= 0} and S = {j | ?j? = 0}, we have k?b k0 ? s? .
The proof of Theorem 3.5 is provided in Appendix B. Theorem 3.5 shows that within each iteration,
both primal and dual variables are sparse, i.e., the number of nonzero entries are far smaller than d.
Therefore, the computation cost within each iteration of PSM can be significantly reduced by a factor
of O(d/s? ). This partially justifies the superior performance of PSM in sparse learning.
4
Numerical Experiments
1
Flare
PSM
?
?
Infeasibility
200
300
400
300
100
100
?1
0
?
200
Values of Lambda along the Path
?
?2
Nonzero Entries of the Response Vector
True Value
Estimated Path
400
?
?
?
?
500
2
In this section, we present some numerical experiments and give some insights about how the
parametric simplex method solves different linear programming problems. We verify the following
assertions: (1) The parametric simplex method requires very few iterations to identify the nonzero
component if the original problem is sparse. (2) The parametric simplex method is able to find the
full solution path with high precision by solving the problem only once in an efficient and scalable
manner. (3) The parametric simplex method maintains the feasibility of the problem up to machine
precision along the solution path.
0
0
?3
?
0
5
10
Iteration
15
0
5
10
Iteration
15
0
100
200
300
400
500
Iteration
(a) Solution Path
(b) Parameter Path (Rescaled by n)
(c) Feasibility Violation
Figure 1: Dantzig selector method: (a) The solution path of the parametric simplex method; (b) The parameter
path of the parametric simplex method; (c) Feasibility violation along the solution path.
Solution path of Dantzig selector: We start with a simple example that illustrates how the recovered solution path of the Dantzig selector model changes as the parametric simplex method iterates.
We adopt the example used in Candes and Tao (2007). The design matrix X has n = 100 rows and
d = 250 columns. The entries of X are generated from an array of independent Gaussian random
variables that are then Gaussianized so that each column has a given norm. We randomly select s = 8
entries from the response vector ?0 , and set them as ?i0 = si (1 + ai ), where si = 1 or 1, with
probability 1/2 and ai ? N (0, 1). The other entries of ?0 are set to zero. We form y =pX?0 + ?,
where ?i ? N (0, ), with = 1. We stop the parametric simplex method when ? n log d/n.
The solution path of the result is shown in Figure 1(a). We see that our method correctly identifies all
nonzero entries of ? in less than 10 iterations. Some small overestimations occur in a few iterations
after all nonzero entries have been identified. We also show how the parameter evolves as the
parametric simplex method iterates in Figure 1(b). As we see, decreases sharply to less than 5
8
after all nonzero components have been identified. This reconciles with the theorem we developed.
The algorithm itself only requires a very small number of iterations to correctly identify the nonzero
entries of ?. In our example, each iteration in the parametric simplex method identifies one or two
non-sparse entries in ?.
Feasibility of Dantzig Selector: Another advantage of the parametric simplex method is that the
solution is always feasible along the path while other estimating methods usually generate infeasible
solutions along the path. We compare our algorithm with ?flare? (Li et al., 2015) which uses the
Alternating Direction Method of Multipliers (ADMM) using the same example described above. We
i
compute the values of kX > X?i X > yk1
i along the solution path, where ? is the i-th basic
solution (with corresponding i ) obtained while the parametric simplex method is iterating. Without
any doubts, we always obtain 0s during each iteration. We plug the same list of i into ?flare? and
compute the solution path for this list as well. As shown in Table 1, the parametric simplex method is
always feasible along the path since it is solving each iteration up to machine precision; while the
solution path of the ADMM is almost always breaking the feasibility by a large amount, especially in
the first few iterations which correspond to large values. Each experiment is repeated for 100 times.
Table 1: Average feasibility violation with standard errors along the solution path
ADMM
PSM
Maximum violation
498(122)
0(0)
Minimum Violation
143(73.2)
0(0)
Performance Benchmark of Dantzig Selector: In this part, we compare the timing performance
of our algorithm with R package ?flare?. We fix the sample size n to be 200 and vary the data
dimension d from 100 to 5000. Again, each entries of X is independent Gaussian and Gaussianized
such that the column has uniform norm. We randomly select 2% entries from vector ? to be nonzero
and each entry is chosen as ? N (0, 1). We compute y = X? + ?, with ?i ?
pN (0, 1) and try to
recover vector ?, given X and y. Our method stops when is less than 2
log d/n, such that
the full solution path for all the values of up to this value is computed by the parametric simplex
method. In ?flare?, we estimate ? when is equal to the value in the Dantzig selector model. This
means ?flare? has much less computation task than the parametric simplex method. As we can see in
Table 2, our method has a much better performance than ?flare? in terms of speed. We compare and
present the timing performance of the two algorithms in seconds and each experiment is repeated for
100 times. In practice, only very few iterations is required when the response vector ? is sparse.
Table 2: Average timing performance (in seconds) with standard errors in the parentheses on Dantzig selector
Flare
PSM
500
19.5(2.72)
2.40(0.220)
1000
44.4(2.54)
29.7(1.39)
2000
142(11.5)
47.5(2.27)
5000
1500(231)
649(89.8)
Performance Benchmark of Differential Network: We now apply this optimization method to
the Differential Network model. We need the difference between two inverse covariance matrices
to be sparse. We generate ?0x = U > ?U , where ? 2 Rd?d is a diagonal matrix and its entries are
i.i.d. and uniform on [1, 2], and U 2 Rd?d is a random matrix with i.i.d. entries from N (0, 1). Let
D1 2 Rd?d be a random sparse symmetric matrix with a certain sparsity level. Each entry of D1
is i.i.d. and from N (0, 1). We set D = D1 + 2| min (D1 )|Id in order to guarantee the positive
definiteness of D, where min (D1 ) is the smallest eigenvalue of D1 . Finally, we let ?0x = (?0x ) 1
and ?0y = ?0x + D.
We then generate data of sample size n = 100. The corresponding sample covariance matrices SX
and SY are also computed based on the data. We are not able to find other software which can
efficiently solve this problem, so we only list the timing performance of our algorithm as dimension
d varies from 25 to 200 in Table 3. We stop our algorithm whenever the solution achieved the desired
sparsity level. When d = 25, 50 and 100, the sparsity level of D1 is set to be 0.02 and when d = 150
and 200, the sparsity level of D1 is set to be 0.002. Each experiment is repeated for 100 times.
Table 3: Average timing performance (in seconds) and iteration numbers with standard errors in the parentheses
on differential network
Timing
Iteration Number
25
0.0185(0.00689)
15.5(7.00)
50
0.376(0.124)
55.3(18.8)
9
100
6.81(2.38)
164(58.2)
150
13.41(1.26)
85.8(16.7)
200
46.88(7.24)
140(26.2)
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10
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6,217 | 6,624 | Group Sparse Additive Machine
1
Hong Chen1 , Xiaoqian Wang1 , Cheng Deng2 , Heng Huang1?
Department of Electrical and Computer Engineering, University of Pittsburgh, USA
2
School of Electronic Engineering, Xidian University, China
[email protected],[email protected]
[email protected],[email protected]
Abstract
A family of learning algorithms generated from additive models have attracted
much attention recently for their flexibility and interpretability in high dimensional
data analysis. Among them, learning models with grouped variables have shown
competitive performance for prediction and variable selection. However, the
previous works mainly focus on the least squares regression problem, not the
classification task. Thus, it is desired to design the new additive classification
model with variable selection capability for many real-world applications which
focus on high-dimensional data classification. To address this challenging problem,
in this paper, we investigate the classification with group sparse additive models
in reproducing kernel Hilbert spaces. A novel classification method, called as
group sparse additive machine (GroupSAM), is proposed to explore and utilize
the structure information among the input variables. Generalization error bound is
derived and proved by integrating the sample error analysis with empirical covering
numbers and the hypothesis error estimate with the stepping stone technique. Our
new bound shows that GroupSAM can achieve a satisfactory learning rate with
polynomial decay. Experimental results on synthetic data and seven benchmark
datasets consistently show the effectiveness of our new approach.
1
Introduction
The additive models based on statistical learning methods have been playing important roles for
the high-dimensional data analysis due to their well performance on prediction tasks and variable
selection (deep learning models often don?t work well when the number of training data is not
large). In essential, additive models inherit the representation flexibility of nonlinear models and
the interpretability of linear models. For a learning approach under additive models, there are two
key components: the hypothesis function space and the regularizer to address certain restrictions
on estimator. Different from traditional learning methods, the hypothesis space used in additive
models is relied on the decomposition of input vector. Usually, each input vector X ? Rp is divided
into p parts directly [17, 30, 6, 28] or some subgroups according to prior structural information
among input variables [27, 26]. The component function is defined on each decomposed input and
the hypothesis function is constructed by the sum of all component functions. Typical examples of
hypothesis space include the kernel-based function space [16, 6, 11] and the spline-based function
space [13, 15, 10, 30]. Moreover, the Tikhonov regularization scheme has been used extensively for
constructing the additive models, where the regularizer is employed to control the complexity of
hypothesis space. The examples of regularizer include the kernel-norm regularization associated with
the reproducing kernel Hilbert space (RKHS) [5, 6, 11] and various sparse regularization [17, 30, 26].
More recently several group sparse additive models have been proposed to tackle the high-dimensional
regression problem due to their nice theoretical properties and empirical effectiveness [15, 10,
?
Corresponding author
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
26]. However, most existing additive model based learning approaches are mainly limited to the
least squares regression problem and spline-based hypothesis spaces. Surprisingly, there is no any
algorithmic design and theoretical analysis for classification problem with group sparse additive
models in RKHS. This paper focuses on filling in this gap on algorithmic design and learning theory
for additive models. A novel sparse classification algorithm, called as group sparse additive machine
(GroupSAM), is proposed under a coefficient-based regularized framework, which is connected to
the linear programming support vector machine (LPSVM) [22, 24]. By incorporating the grouped
variables with prior structural information and the `2,1 -norm based structured sparse regularizer, the
new GroupSAM model can conduct the nonlinear classification and variable selection simultaneously.
Similar to the sparse additive machine (SAM) in [30], our GroupSAM model can be efficiently solved
via proximal gradient descent algorithm. The main contributions of this paper can summarized in
two-fold:
? A new group sparse nonlinear classification algorithm (GroupSAM) is proposed by extending
the previous additive regression models to the classification setting, which contains the
LPSVM with additive kernel as its special setting. To the best of our knowledge, this is the
first algorithmic exploration of additive classification models with group sparsity.
? Theoretical analysis and empirical evaluations on generalization ability are presented to support the effectiveness of GroupSAM. Based on constructive analysis on the hypothesis error,
we get the estimate on the excess generalization error, which shows that our GroupSAM
model can achieve the fast convergence rate O(n?1 ) under mild conditions. Experimental
results demonstrate the competitive performance of GroupSAM over the related methods on
both simulated and real data.
Before ending this section, we discuss related works. In [5], support vector machine (SVM) with
additive kernels was proposed and its classification consistency was established. Although this
method can also be used for grouped variables, it only focuses on the kernel-norm regularizer without
addressing the sparseness for variable selection. In [30], the SAM was proposed to deal with the
sparse representation on the orthogonal basis of hypothesis space. Despite good computation and
generalization performance, SAM does not explore the structure information of input variables and
ignores the interactions among variables. More important, different from finite splines approximation
in [30], our approach enables us to estimate each component function directly in RKHS. As illustrated
in [20, 14], the RKHS-based method is flexible and only depends on few tuning parameters, but the
commonly used spline methods need specify the number of basis functions and the sequence of knots.
It should be noticed that the group sparse additive models (GroupSpAM in [26]) also address the
sparsity on the grouped variables. However, there are key differences between GroupSAM and
GroupSpAM: 1) Hypothesis space. The component functions in our model are obtained by searching
in kernel-based data dependent hypothesis spaces, but the method in [26] uses data independent
hypothesis space (not associated with kernel). As shown in [19, 18, 4, 25], the data dependent
hypothesis space can provide much more adaptivity and flexibility for nonlinear prediction. The
advantage of kernel-based hypothesis space for additive models is also discussed in [14]. 2) Loss
function. The hinge loss used in our classification model is different from the least-squares loss in
[26]. 3) Optimization. Our GroupSAM only needs to construct one component function for each
variable group, but the model in [26] needs to find the component functions for each variable in a
group. Thus, our method is usually more efficient. Due to the kernel-based component function and
non-smooth hinge loss, the optimization of GroupSpAM can not be extended to our model directly. 4)
Learning theory. We establish the generalization bound of GroupSAM by the error estimate technique
with data dependent hypothesis spaces, while the error bound is not covered in [26].
Now, we present a brief summary in Table 1 to better illustrate the differences of our GroupSAM
with other methods.
The rest of this paper is organized as follows. In next section, we revisit the related classification
formulations and propose the new GroupSAM model. Theoretical analysis on generalization error
bound is established in Section 3. In Section 4, experimental results on both simulated examples and
real data are presented and discussed. Finally, Section 5 concludes this paper.
2
Table 1: Properties of different additive models.
SAM [30]
Group Lasso[27] GroupSpAM [26] GroupSAM
Hypothesis space
data-independent data-independent data-independent data-dependent
Loss function
hinge loss
least-square
least-square
hinge loss
No
Yes
Yes
Yes
Group sparsity
Generalization bound
Yes
No
No
Yes
2
Group sparse additive machine
In this section, we first revisit the basic background of binary classification and additive models, and
then introduce our new GroupSAM model.
Let Z := (X , Y) ? Rp+1 , where X ? Rp is a compact input space and Y = {?1, 1} is the set
of labels. We assume that the training samples z := {zi }ni=1 = {(xi , yi )}ni=1 are independently
drawn from an unknown distribution ? on Z, where each xi ? X and yi ? {?1, 1}. Let?s denote
the marginal distribution of ? on X as ?X and denote its conditional distribution for given x ? X as
?(?|x).
For a real-valued function f : X ? R, we define its induced classifier as sgn(f ), where sgn(f )(x) =
1 if f (x) ? 0 and sgn(f )(x) = ?1 if f (x) < 0. The prediction performance of f is measured by
the misclassification error:
Z
R(f ) = Prob{Y f (X) ? 0} =
Prob(Y 6= sgn(f )(x)|x)d?X .
(1)
X
It is well known that the minimizer of R(f ) is the Bayes rule:
Z
fc (x) = sgn
yd?(y|x) = sgn Prob(y = 1|x) ? Prob(y = ?1|x) .
Y
Since the Bayes rule involves the unknown distribution ?, it can not be computed directly. In machine
learning literature, the classification algorithm usually aims to find a good approximation of fc by
minimizing the empirical misclassification risk:
n
1X
I(yi f (xi ) ? 0) ,
n i=1
Rz (f ) =
(2)
where I(A) = 1 if A is true and 0 otherwise. However, the minimization problem associated with
Rz (f ) is NP-hard due to the 0 ? 1 loss I. To alleviate the computational difficulty, various convex
losses have been introduced to replace the 0 ? 1 loss, e.g., the hinge loss, the least square loss, and
the exponential loss [29, 1, 7]. Among them, the hinge loss is the most popular error metric for
classification problem due to its nice theoretical properties. In this paper, following [5, 30], we use
the hinge loss:
`(y, f (x)) = (1 ? yf (x))+ = max{1 ? yf (x), 0}
to measure the misclassification cost.The expected and empirical risks associated with the hinge loss
are defined respectively as:
Z
E(f ) =
(1 ? yf (x))+ d?(x, y) ,
Z
and
n
Ez (f ) =
1X
(1 ? yi f (xi ))+ .
n i=1
In theory, the excess misclassification error R(sgn(f )) ? R(fc ) can be bounded by the excess convex
risk E(f ) ? E(fc ) [29, 1, 7]. Therefore, the classification algorithm usually is constructed under
structural risk minimization [22] associated with Ez (f ).
3
In this paper, we propose a novel group sparse additive machine (GroupSAM) for nonlinear classification. Let {1, ? ? ? , p} be partitioned into d groups. For each j ? {1, ..., d}, we set X (j) as the
grouped input space and denote f (j) : X (j) ? R as the corresponding component function. Usually,
the groups can be obtained by prior knowledge [26] or be explored by considering the combinations
of input variables [11].
Let each K (j) : X (j) ? X (j) ? R be a Mercer kernel and let HK (j) be the corresponding RKHS
with norm k ? kK (j) . It has been proved in [5] that
H=
d
nX
o
f (j) : f (j) ? HK (j) , 1 ? j ? d
j=1
with norm
kf k2K = inf
d
nX
kf (j) k2K (j) : f =
j=1
d
X
f (j)
o
j=1
is an RKHS associated with the additive kernel K =
Pd
j=1
K (j) .
For any given training set z = {(xi , yi )}ni=1 , the additive model in H can be formulated as:
f?z =
arg min
P
f=
d
j=1
f (j) ?H
d
o
n
X
Ez (f ) + ?
?j kf (j) k2K (j) ,
(3)
j=1
where ? = ?(n) is a positive regularization parameter and {?j } are positive bounded weights for
different variable groups.
The solution f?z in (3) has the following representation:
f?z (x) =
d
X
(j)
f?z (x(j) ) =
j=1
d X
n
X
(j)
(j)
(j)
?
? z,i yi K (j) (xi , x(j) ), ?
? z,i ? R, 1 ? i ? n, 1 ? j ? d .
j=1 i=1
(j)
(j)
(j)
Observe that f?z (x) ? 0 is equivalent to ?
? z,i = 0 for all i. Hence, we expect k?
?z k2 = 0 for
(j)
(j)
??z (j) = (?
?z,1 , ? ? ? , ?
? z,n )T ? Rn if the j-th variable group is not truly informative. This motivation
pushes us to consider the sparsity-induced penalty:
?(f ) = inf
d
nX
?j k?(j) k2 : f =
j=1
d X
n
X
o
(j)
(j)
?i yi K (j) (xi , ?) .
j=1 i=1
This group sparse penalty aims at the variable selection [27] and was introduced into the additive
regression model [26].
Inspired by learning with data dependent hypothesis spaces [19], we introduce the following hypothesis spaces associated with training samples z:
d
n
o
X
Hz = f =
f (j) : f (j) ? Hz(j) ,
(4)
j=1
where
n
n
o
X
(j)
(j)
(j)
Hz(j) = f (j) =
?i K (j) (xi , ?) : ?i ? R .
i=1
Under the group sparse penalty and data dependent hypothesis space, the group sparse additive
machine (GroupSAM) can be written as:
fz = arg min
f ?Hz
n
n1 X
n
o
(1 ? yi f (xi ))+ + ??(f ) ,
i=1
4
(5)
where ? > 0 is a regularization parameter.
(j)
(j)
(j)
Let?s denote ?(j) = (?1 , ? ? ? , ?n )T and Ki
GroupSAM in (5) can be rewritten as:
fz =
d
X
fz(j) =
(j)
(j)
(j)
(j)
= (K (j) (x1 , xi ), ? ? ? , K (j) (xn , xi ))T . The
d X
n
X
(j)
(j)
?z,t K (j) (xt , ?) ,
j=1 t=1
j=1
with
{?z(j) } =
n
n1 X
arg min
?(j) ?Rn ,1?j?d
n
1 ? yi
i=1
d
d
o
X
X
(j)
(Ki )T ?(j) + + ?
?j k?(j) k2 .
j=1
(6)
j=1
The formulation (6) transforms the function-based learning problem (5) into a coefficient-based
learning problem in a finite dimensional vector space. The solution of (5) is spanned naturally by
(j)
the kernelized functions {K (j) (?, xi ))}, rather than B-Spline basis functions [30]. When d = 1,
our GroupSAM model degenerates to the special case which includes the LPSVM loss and the
sparsity regularization term. Compared with LPSVM [22, 24] and SVM with additive kernels [5], our
GroupSAM model imposes the sparsity on variable groups to improve the prediction interpretation of
additive classification model.
For given {?j }, the optimization problem of GroupSAM can be computed efficiently via an accelerated proximal gradient descent algorithm developed in [30]. Due to space limitation, we don?t recall
the optimization algorithm here again.
3
Generalization error bound
In this section, we will derive the estimate on the excess misclassification error R(sgn(fz )) ? R(fc ).
Before providing the main theoretical result, we introduce some necessary assumptions for learning
theory analysis.
Assumption A. The intrinsic distribution ? on Z := X ? Y satisfies the Tsybakov noise condition
with exponent 0 ? q ? ?. That is to say, for some q ? [0, ?) and ? > 0,
?X {x ? X : |Prob(y = 1|x) ? Prob(y = ?1|x)| ? ?t} ? tq , ?t > 0.
(7)
The Tsybakov noise condition was proposed in [21] and has been used extensively for theoretical
analysis of classification algorithms [24, 7, 23, 20]. Indeed, (7) holds with exponent q = 0 for any
distribution and with q = ? for well separated classes.
Now we introduce the empirical covering numbers [8] to measure the capacity of hypothesis space.
Definition 1 Let F be a set of functions on Z with u = {ui }ki=1 ? Z. Define the `2 -empirical
1
Pk
metric as `2,u (f, g) = n1 t=1 (f (ut ) ? g(ut ))2 2 . The covering number of F with `2 -empirical
metric is defined as N2 (F, ?) = supn?N supu?X n N2,u (F, ?), where
l
n
o
[
N2,u (F, ?) = inf l ? N : ?{fi }li=1 ? F s. t. F =
{f ? F : `2,u (f, fi ) ? ?} .
i=1
(j)
Let Br = {f ? HK : kf kK ? r} and Br = {f (j) ? HK (j) : kf (j) kK (j) ? r}.
p
Pd
K (j) (x(j) , x(j) ) < ? and for some s ?
Assumption B. Assume that ? =
j=1 supx(j)
(0, 2), cs > 0,
(j)
log N2 (B1 , ?) ? cs ??s , ?? > 0, j ? {1, ..., d}.
It has been asserted in [6] that under Assumption B the following holds:
log N2 (B1 , ?) ? cs d1+s ??s , ?? > 0.
5
It is worthy noticing that the empirical covering number has been studied extensively in learning
theory literatures [8, 20]. Detailed examples have been provided in Theorem 2 of [19], Lemma 3 of
[18], and Examples 1, 2 of [9]. The capacity condition of additive assumption space just depends
on the dimension of subspace X (j) . When K (j) ? C ? (X (j) ? X (j) ) for every j ? {1, ? ? ? , d}, the
theoretical analysis in [19] assures that Assumption B holds true for:
? 2d0
? d0 +2? , ? ? (0, 1];
2d0
s=
, ? ? [1, 1 + d0 /2];
? dd00 +?
? ? (1 + d0 /2, ?).
? ,
Here d0 denotes the maximum dimension among {X (j) }.
With respect to (3), we introduce the data-free regularized function f? defined by:
f? =
arg min
P
(j) ?H
f= d
j=1 f
d
n
o
X
E(f ) + ?
?j kf (j) k2K (j) .
(8)
j=1
Inspired by the analysis in [6], we define:
D(?) = E(f? ) ? E(fc ) + ?
d
X
?j kf?(j) k2K (j)
(9)
j=1
as the approximation error, which reflects the learning ability of hypothesis space H under Tikhonov
regularization scheme.
The following approximation condition has been studied and used extensively for classification
problems, such as [3, 7, 24, 23]. Please see Examples 3 and 4 in [3] for the explicit version for Soblov
kernel and Gaussian kernel induced reproducing kernel Hilbert space.
Assumption C. There exists an exponent ? ? (0, 1) and a positive constant c? such that:
D(?) ? c? ? ? , ?? > 0.
Now we introduce our main theoretical result on the generalization bound as follows.
Theorem 1 Let 0 < min ?j ? max ?j ? c0 < ? and Assumptions A-C hold true. Take ? = n?? in
j
j
3+5?
(5) for 0 < ? ? min{ 2?s
2s , 2?2? }. For any ? ? (0, 1), there exists a constant C independent of n, ?
such that
R(sgn(fz )) ? R(fc ) ? C log(3/?)n??
with confidence 1 ? ?, where
n q + 1 ?(2? + 1) (q + 1)(2 ? s ? 2s?) 3 + 5? + 2?? ? 2? o
? = min
,
,
,
.
q + 2 2? + 2
4 + 2q + sq
4 + 4?
Theorem 1 demonstrates that GroupSAM in (5) can achieve the convergence rate with polynomial
decay under mild conditions in hypothesis function space. When q ? ?, ? ? 1, and each
1+2?
K (j) ? C ? , the error decay rate of GroupSAM can arbitrarily close to O(n? min{1, 4 } ). Hence,
the fast convergence rate O(n?1 ) can be obtained under proper selections on parameters. To verify
the optimal bound, we need provide the lower bound for the excess misclassification error. This is
beyond the main focus of this paper and we leave it for future study.
Additionally, the consistency of GroupSAM can be guaranteed with the increasing number of training
samples.
Corollary 1 Under conditions in Theorem 1, there holds R(sgn(fz )) ? R(fc ) ? 0 as n ? ?.
To better understand our theoretical result, we compare it with the related works as below:
6
1) Compared with group sparse additive models. Although the asymptotic theory of group sparse
additive models has been well studied in [15, 10, 26], all of them only consider the regression task under the mean square error criterion and basis function expansion. Due to the kernel-based component
function and non-smooth hinge loss, the previous analysis cannot be extended to GroupSAM directly.
2) Compared with classification with additive models. In [30], the convergence rate is presented for
sparse additive machine (SAM), where the input space X is divided into p subspaces directly without
considering the interactions among variables. Different to the sparsity on variable groups in this
paper, SAM is based on the sparse representation of orthonormal basis similar with [15]. In [5], the
consistency of SVM with additive kernel is established, where the kernel-norm regularizer is used.
However, the sparsity on variables and the learning rate are not investigated in previous articles.
3) Compared with the related analysis techniques. While the analysis technique used here is inspired
from [24, 23], it is the first exploration for additive classification model with group sparsity. In
particular, the hypothesis error analysis develops the stepping stone technique from the `1 -norm
regularizer to the group sparse `2,1 -norm regularizer. Our analysis technique also can be applied to
other additive models. For example, we can extend the shrunk additive regression model in [11] to
the sparse classification setting and investigate its generalization bound by the current technique.
Proof sketches of Theorem 1
To get tight error estimate, we introduce the clipping operator ?(f )(x) = max{?1, min{f (x), 1}},
which has been widely used in learning theory literatures, such as [7, 20, 24, 23]. Since R(sgn(fz ))?
R(fc ) can be bounded by E(?(fz )) ? E(fc ), we focus on bounding the excess convex risk.
Using f? as the intermediate function, we can obtain the following error decomposition.
Proposition 1 For fz defined in (5), there holds
R(sgn(fz )) ? R(fc ) ? E(?(fz )) ? E(fc ) ? E1 + E2 + E3 + D(?),
where D(?) is defined in (9),
E1
=
E2
=
E(?(fz )) ? E(fc ) ? Ez (?(fz )) ? Ez (fc ) ,
Ez (f? ) ? Ez (fc ) ? Ez (f? ) ? E(fc ) ,
and
E3 = Ez (?(fz )) + ??(fz ) ? Ez (f? ) + ?
d
X
?j kf?(j) k2K (j) .
j=1
In learning theory literature, E1 + E2 is called as the sample error and E3 is named as the hypothesis
error. Detailed proofs for these error terms are provided in the supplementary materials.
The upper bound of hypothesis error demonstrates that the divergence induced from regularization
and hypothesis space tends to zero as n ? ? under proper selected parameters. To estimate the
hypothesis error E3 , we choose f?z as the stepping stone function to bridge Ez (?(fz )) + ??(fz )
Pd
(j)
and Ez (f? ) + ? j=1 ?j kf? k2K (j) . The proof is inspired from the stepping stone technique for
support vector machine classification [24]. Notice that our analysis is associated with the `2,1 -norm
regularizer while the previous analysis just focuses on the `1 -norm regularization.
The error term E1 reflects the divergence between the expected excess risk E(?(fz )) ? E(fc ) and
the empirical excess risk Ez (?(fz )) ? Ez (fc ). Since fz involves any given z = {(xi , yi )}ni=1 , we
introduce the concentration inequality in [23] to bound E1 . We also bound the error term E2 in terms
of the one-side Bernstein inequality [7].
4
Experiments
To evaluate the performance of our proposed GroupSAM model, we compare our model with the
following methods: SVM (linear SVM with `2 -norm regularization), L1SVM (linear SVM with `1 norm regularization), GaussianSVM (nonlinear SVM using Gaussian kernel), SAM (Sparse Additive
Machine) [30], and GroupSpAM (Group Sparse Additive Models) [26] which is adapted to the
classification setting.
7
Table 2: Classification accuracy comparison on the synthetic data. The upper half shows the results
with 24 features groups, while the lower half corresponds to the results with 300 feature groups. The
table shows the average classification accuracy and the standard deviation in 2-fold cross validation.
SVM
GaussianSVM L1SVM
SAM
GroupSpAM GroupSAM
? = 0.8 0.943?0.011 0.935?0.028 0.925?0.035 0.895?0.021 0.880?0.021 0.953?0.018
? = 0.85 0.943?0.004 0.938?0.011 0.938?0.004 0.783?0.088 0.868?0.178 0.945?0.000
? = 0.9 0.935?0.014 0.925? 0.007 0.938?0.011 0.853? 0.117 0.883?0.011 0.945?0.007
? = 0.8 0.975?0.035 0.975?0.035 0.975?0.035 0.700?0.071 0.275?0.106 1.000?0.000
? = 0.85 0.975?0.035 0.975?0.035 0.975?0.035 0.600?0.141 0.953?0.004 1.000?0.000
? = 0.9 0.975?0.035 0.975?0.035 0.975?0.035 0.525?0.035 0.983?0.004 1.000?0.000
As for evaluation metric, we calculate the classification accuracy, i.e., percentage of correctly labeled
samples in the prediction. In comparison, we adopt 2-fold cross validation and report the average
performance of each method.
We implement SVM, L1SVM and GaussianSVM using the LIBSVM toolbox [2]. We determine the
hyper-parameter of all models, i.e., parameter C of SVM, L1SVM and GaussianSVM, parameter
? of SAM, parameter ? of GroupSpAM, parameter ? in Eq. (6) of GroupSAM, in the range of
{10?3 , 10?2 , . . . , 103 }. We tune the hyper-parameters via 2-fold cross validation on the training
data and report the best parameter w.r.t. classification accuracy of each method. In the accelerated
proximal gradient descent algorithm for both SAM and GroupSAM, we set ? = 0.5, and the number
of maximum iterations as 2000.
4.1
Performance comparison on synthetic data
We first examine the classification performance on the synthetic data as a sanity check. Our synthetic
data is randomly generated as a mixture of Gaussian distributions. In each class, data points are
sampled i.i.d. from a multivariate Gaussian distribution with the covariance being ?I, with I as
the identity matrix. This setting indicates independent covariates of the data. We set the number of
classes to be 4, the number of samples to be 400, and the number of dimensions to be 24. We set
the value of ? in the range of {0.8, 0.85, 0.9} respectively. Following the experimental setup in
[31], we make three replicates for each feature in the data to form 24 feature groups (each group
has three replicated features). We randomly pick 6 feature groups to generate the data such that we
can evaluate the capability of GroupSAM in identifying truly useful feature groups. To make the
classification task more challenging, we add random noise drawn from uniform distribution U(0, ?)
where ? is 0.8 times the maximum value in the data. In addition, we test on a high-dimensional case
by generating 300 feature groups (e.g., a total of 900 features) with 40 samples in a similar approach.
We summarize the classification performance comparison on the synthetic data in Table 2. From
the experimental results we notice that GroupSAM outperforms other approaches under all settings.
This comparison verifies the validity of our method. We can see that GroupSAM significantly
improves the performance of SAM, which shows that the incorporation of group information is
indeed beneficial for classification. Moreover, we can notices the superiority of GroupSAM over
GroupSpAM, which illustrates that our GroupSAM model is more suitable for classiciation. We also
present the comparison of feature groups in Table 3. For illustration purpose, we use the case with 24
feature groups as an example. Table 3 shows that the feature groups identified by GroupSAM are
exactly the same as the ground truth feature groups used for synthetic data generation. Such results
further demonstrate the effectiveness of GroupSAM method, from which we know GroupSAM is
able to select the truly informative feature groups thus improve the classification performance.
4.2
Performance comparison on benchmark data
In this subsection, we use 7 benchmark data from UCI repository [12] to compare the classification
performance of different methods. The 7 benchmark data includes: Ecoli, Indians Diabetes, Breast
Cancer, Stock, Balance Scale, Contraceptive Method Choice (CMC) and Fertility. Similar to the
settings in synthetic data, we construct feature groups by replicating each feature for 3 times. In each
8
Table 3: Comparison between the true feature group ID (used for data generation) and the selected
feature group ID by our GroupSAM method on the synthetic data. Order of the true feature group ID
does not represent the order of importance.
? = 0.8
? = 0.85
? = 0.9
True Feature Group IDs
2,3,4,8,10,17
1,5,10,12,17,21
2,6,7,9,12,22
Selected Feature Group IDs via GroupSAM
3,10,17,8,2,4
5,12,17,21,1,10
6,22,7,9,2,12
Table 4: Classification accuracy comparison on the benchmark data. The table shows the average
classification accuracy and the standard deviation in 2-fold cross validation.
Ecoli
Indians
Diabetes
Breast
Cancer
Stock
Balance
Scale
CMC
Fertility
SVM
GaussianSVM
L1SVM
SAM
GroupSpAM GroupSAM
0.815?0.054 0.818?0.049 0.711?0.051 0.816?0.039 0.771?0.009 0.839?0.028
0.651?0.000
0.652?0.002 0.638?0.018 0.652?0.000 0.643?0.004 0.660?0.013
0.968?0.017
0.965?0.017 0.833?0.008 0.833?0.224 0.958?0.027 0.966?0.014
0.913?0.001
0.911?0.002 0.873?0.001 0.617?0.005 0.875?0.005 0.917?0.005
0.864? 0.003 0.869?0.004 0.870?0.003 0.763?0.194 0.848?0.003 0.893?0.003
0.420? 0.011 0.445?0.015 0.437?0.014 0.427?0.000 0.433?0.003 0.456?0.003
0.880? 0.000 0.880?0.000 0.750?0.184 0.860?0.028 0.780?0.000 0.880?0.000
feature group, we add random noise drawn from uniform distribution U(0, ?) where ? is 0.3 times the
maximum value in each data.
We display the comparison results in Table 4. We find that GroupSAM performs equal or better than
the compared methods in all benchmark datasets. Compared with SVM and L1SVM, our method
uses additive model to incorporate nonlinearity thus is more appropriate to find the complex decision
boundary. Moreover, the comparison with Gaussian SVM and SAM illustrates that by involving
the group information in classification, GroupSAM makes better use of the structure information
among features such that the classification ability can be enhanced. Compared with GroupSpAM, our
GroupSAM model is proposed in data dependent hypothesis spaces and employs hinge loss in the
objective, thus is more suitable for classification.
5
Conclusion
In this paper, we proposed a novel group sparse additive machine (GroupSAM) by incorporating the
group sparsity into the additive classification model in reproducing kernel Hilbert space. By developing the error analysis technique with data dependent hypothesis space, we obtain the generalization
error bound of the proposed GroupSAM, which demonstrates our model can achieve satisfactory
learning rate under mild conditions. Experimental results on both synthetic and real-world benchmark
datasets validate the algorithmic effectiveness and support our learning theory analysis. In the future,
it is interesting to investigate the learning performance of robust group sparse additive machines with
loss functions induced by quantile regression [6, 14].
Acknowledgments
This work was partially supported by U.S. NSF-IIS 1302675, NSF-IIS 1344152, NSF-DBI 1356628,
NSF-IIS 1619308, NSF-IIS 1633753, NIH AG049371. Hong Chen was partially supported by
National Natural Science Foundation of China (NSFC) 11671161. We are grateful to the anonymous
NIPS reviewers for the insightful comments.
9
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6,218 | 6,625 | Uprooting and Rerooting Higher-Order Graphical
Models
Mark Rowland?
University of Cambridge
[email protected]
Adrian Weller?
University of Cambridge and Alan Turing Institute
[email protected]
Abstract
The idea of uprooting and rerooting graphical models was introduced specifically
for binary pairwise models by Weller [19] as a way to transform a model to any
of a whole equivalence class of related models, such that inference on any one
model yields inference results for all others. This is very helpful since inference, or
relevant bounds, may be much easier to obtain or more accurate for some model
in the class. Here we introduce methods to extend the approach to models with
higher-order potentials and develop theoretical insights. In particular, we show
that the triplet-consistent polytope TRI is unique in being ?universally rooted?.
We demonstrate empirically that rerooting can significantly improve accuracy of
methods of inference for higher-order models at negligible computational cost.
1
Introduction
Undirected graphical models with discrete variables are a central tool in machine learning. In this
paper, we focus on three canonical tasks of inference: identifying a configuration with highest
probability (termed maximum a posteriori or MAP inference), computing marginal probabilities of
subsets of variables (marginal inference) and calculating the normalizing constant (partition function).
All three tasks are typically computationally intractable, leading to much work to identify settings
where exact polynomial-time methods apply, or to develop approximate algorithms that perform well.
Weller [19] introduced an elegant method which first uproots and then reroots a given model M to any
of a whole class of rerooted models {Mi }. The method relies on specific properties of binary pairwise
models and makes use of an earlier construction which reduced MAP inference to the MAXCUT
problem on the suspension graph ?G (1; 2; 12; 19, see ?3 for details). For many important inference
tasks, the rerooted models are equivalent in the sense that results for any one model yield results for
all others with negligible computational cost. This can be very helpful since various models in the
class may present very different computational difficulties for inference.
Here we show how the idea may be generalized to apply to models with higher-order potentials over
any number of variables. Such models have many important applications, for example in computer
vision [6] or modeling protein interactions [5]. As for pairwise models, we again obtain significant
benefits for inference. We also develop a deeper theoretical understanding and derive important new
results. We highlight the following contributions:
? In ?3-?4, we show how to achieve efficient uprooting and rerooting of binary graphical models
with potentials of any order, while still allowing easy recovery of inference results.
? In ?5, to simplify the subsequent analysis, we introduce pure k-potentials for any order k, which
may be of independent interest. We show that there is essentially only one pure k-potential which
we call the even k-potential, and that even k-potentials form a basis for all model potentials.
? In ?6, we carefully analyze the effect of uprooting and rerooting on Sherali-Adams [11] relaxations
Lr of the marginal polytope, for any order r. One surprising observation in ?6.2 is that L3 (the
?
Authors contributed equally.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
triplet-consistent polytope or TRI) is unique in being universally rooted, in the sense that there is
an affine score-preserving bijection between L3 for a model and L3 for each of its rerootings.
? In ?7, our empirical results demonstrate that rerooting can significantly improve accuracy of
inference in higher-order models. We introduce effective heuristics to choose a helpful rerooting.
Our observations have further implications for the many variational methods of marginal inference
which optimize the sum of score and an entropy approximation over a Sherali-Adams polytope
relaxation. These include the Bethe approximation (intimately related to belief propagation) and
cluster extensions, tree-reweighted (TRW) approaches and logdet methods [12; 14; 16; 22; 24].
1.1
Background and discussion of theoretical contributions
Based on earlier connections in [2], [19] showed the remarkable result for pairwise models that the
triplet-consistent polytope (L3 or TRI) is universally rooted (in the restricted sense defined in [19,
Theorem 3]). This observation allowed straightforward strengthening of previously known results,
for example: it was previously shown [23] that the LP relaxation on TRI (LP+TRI) is always tight
for an ?almost-balanced? binary pairwise model, that is a model which can be rendered balanced
by removing one variable [17]. Given [19, Theorem 3], this earlier result could immediately be
significantly strengthened to [19, Theorem 4], which showed that LP+TRI is tight for a binary
pairwise model provided only that some rerooting exists such that the rerooted model is almost
balanced.
Following [19], it was natural to suspect that the universal rootedness property might hold for all
(or at least some) Lr , r ? 3. This would have impact on work such as [10] which examines which
signed minors must be forbidden to guarantee tightness of LP+L4 . If L4 were universally rooted,
then it would be possible to simplify significantly the analysis in [10].
Considering this issue led to our analysis of the mappings to symmetrized uprooted polytopes given
in our Theorem 17. We believe this is the natural generalization of the lower order relationships of
L2 and L3 to RMET and MET described in [2], though this direction was not clear initially.
With this formalism, together with the use of even potentials, we demonstrate our Theorems 20 and
21, showing that in fact TRI is unique in being universally rooted (and indeed in a stronger sense
than given in [19]). We suggest that this result is surprising and may have further implications.
As a consequence, it is not possible to generate some quick theoretical wins by generalizing previous
results as [19] did to derive their Theorem 4, but on the other hand we observe that rerooting may be
helpful in practice for any approach using a Sherali-Adams relaxation other than L3 . We verify the
potential for significant benefits experimentally in ?7.
2
Graphical models
A discrete graphical model M [G(V, E), (?E )E?E ] consists of: a hypergraph G = (V, E), which has
n vertices V = {1, . . . , n} corresponding to the variables of the model, and hyperedges E ? P(V ),
where P(V ) is the powerset of V ; together with potential functions (?E )E?E over the hyperedges
E ? E. We consider binary random variables (Xv )v?V with each Xv ? Xv = {0, 1}. For a subset
U ? V , xU ? {0, 1}U is a configuration of those variables (Xv )v?U . We write xU for the flipping
of xU , defined by xi = 1 ? xi P
?i ? U . The joint probability mass function factors as follows, where
the normalizing constant Z = xV ?{0,1}V exp(score(xV )) is the partition function:
X
1
p(xV ) = exp (score(xV )) ,
score(xV ) =
?E (xE ).
(1)
Z
E?E
3
Uprooting and rerooting
Our goal is to map a model M to any of a whole family of models {Mi } in such a way that inference
on any Mi will allow us easily to recover inference results on the original model M . In this section
we provide our mapping, then in ?4 we explain how to recover inference results for M .
The uprooting mechanism used by Weller [19] first reparametrizes edge potentials to the form
?ij (xi , xj ) = ? 21 Wij 1[xi 6= xj ], where 1[?] is the indicator function (a reparameterization modifies
2
1
2
1
2
3
4
3
4
M = M0
1
0
3
M+
2
1
0
M4 = M + |X4 =0
3
4
0
M2 = M + |X2 =0
Figure 1: Left: The hypergraph G of a graphical model M over 4 variables, with potentials on the hyperedges
{1, 2}, {1, 3, 4}, and {2, 4}. Center-left: The suspension hypergraph ?G of the uprooted model M + . Centerright: The hypergraph ?G \ {4} of the rerooted model M4 = M + |X4 =0 , i.e. M + with X4 clamped to 0. Right:
The hypergraph ?G \ {2} of the rerooted model M2 = M + |X2 =0 , i.e. M + with X2 clamped to 0.
potential functions such that the complete score of each configuration is unchanged, see 15 for details).
Next, singleton potentials are converted to edge potentials with this same form by connecting to an
added variable X0 . This mechanism had been used previously to reduce MAP inference on M to
MAXCUT on the converted model [1; 12], and applies specifically only to binary pairwise models.
We introduce a generalized construction which applies to models with potentials of any order. We first
uproot a model M to a highly symmetric uprooted model M + where an extra variable X0 is added,
in such a way that the original model M is exactly M + with X0 clamped to the value 0. Since X0 is
clamped to retrieve M , we may write M = M0 := M + |X0 =0 . Alternatively, we can choose instead
to clamp a different variable Xi in M + which will lead to the rerooted model Mi := M + |Xi =0 .
Definition 1 (Clamping). For a graphical model M [G = (V, E), (?E )E?E ], and i ? V , the model
M |Xi =a obtained by clamping the variable Xi to the value a ? Xi is given by: the hypergraph
(V \ {i}, Ei ), where Ei = {E \ {i}|E ? E}; and potentials which are unchanged for hyperedges
which do not contain i, while if i ? E then ?E\{i} (xE\{i} ) = ?E (xE\{i} , xi = a).
Definition 2 (Uprooting, suspension hypergraph). Given a model M [G(V, E), (?E )E?E ], the
uprooted model M + adds a variable X0 , which is added to every hyperedge of the original model. M +
has hypergraph ?G, with vertex set V + = V ?{0} and hyperedge set E + = {E + = E ?{0}|E ? E}.
+
?G is the suspension hypergraph of G. M + has potential functions (?E?{0}
)E?E given by
? (x ) if x0 = 0
+
?E?{0}
(xE?{0} ) = E E
?E (xE ) if x0 = 1.
With this definition, all uprooted potentials are symmetric in that ?E++ (xE + ) = ?E++ (xE + ) ?E + ? E + .
Definition 3 (Rerooting). From Definition 2, we see that given a model M , if we uproot to M +
then clamp X0 = 0, we recover the original model M . If instead in M + we clamp Xi = 0 for any
i = 1, . . . , n, then we obtain the rerooted model Mi := M + |Xi =0 .
See Figure 1 and Table 1 for examples of uprooting and rerooting. We explore the question of how to
choose a good variable for rerooting (i.e. how to choose a good variable to clamp in M + ) in ?7.
4
Recovery of inference tasks
Here we demonstrate that the partition function, MAP score and configuration, and marginal distributions for a model M , can all be recovered from its uprooted model M + or any rerooted model
Mi i ? V , with negligible computational cost. We write Vi = {0, 1, . . . , n} \ {i} for the variable set
of rerooted model Mi ; scorei (xVi ) for the score of xVi in Mi ; and pi for the probability distribution
for Mi . We use superscript + to indicate the uprooted
P model. For example, the probability distribution for M + is given by p+ (xV + ) = Z1+ exp
E?E + ?E (xE ) . From the definitions of ?3, we
obtain the following key lemma, which is critical to enable recovery of inference results.
Lemma 4 (Score-preserving map). Each configuration xV of M maps to 2 configurations of
the uprooted M + with the same score, i.e. from M, xV ? in M + , both of (x0 = 0, xV ) and
(x0 = 1, xV ) with score(xV ) = score+ (x0 = 0, xV ) = score+ (x0 = 1, xV ). For any i ? V + ,
exactly one of the two uprooted configurations has xi = 0, and just this one will be selected in Mi .
Hence, there is a score-preserving bijection between configurations of M and those of Mi :
(x0 = 0, xV \{i} ) if xi = 0
For any i ? V + :
in M, xV ? in Mi ,
(2)
(x0 = 1, xV \{i} ) if xi = 1.
3
M
x1
0
0
0
0
1
1
1
1
config
x3 x4
0
0
0
1
1
0
1
1
0
0
0
1
1
0
1
1
M + configuration
x0 x1 x3 x4
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
M4 config
x0 x1 x3
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Table 1: An illustration of how scores of potential ?134 on hyperedge {1, 3, 4} in an original model M map to
potential ?0134 in M + and then to ?013 in M4 . See Figure 1 for the hypergraphs. Each color indicates a value
of ?134 (x1 , x3 , x4 ) for a different configuration (x1 , x3 , x4 ). Note that M + has 2 rows of each color, while
after rerooting to M4 , we again have exactly one row of each color. The 1-1 score preserving map between
configurations of M and any Mi is critical to enable recovery of inference results; see Lemma 4.
Table 1 illustrates this perhaps surprising result, from which the next two propositions follow.
Proposition 5 (Recovering the partition function). Given a model M [G(V, E), (?E )E?E ] with
partition function Z as in (1), the partition function Z + of the uprooted model M + is twice Z, and
the partition function of each rerooted model Mi is exactly Z, for any i ? V .
Proposition 6 (Recovering a MAP configuration). From M + : xV is an arg max for p iff (x0 =
0, xV ) is an arg max for p+ iff (x0 = 1, xV ) is an arg max for p+ . From a rerooted model Mi :
(xV \{i} , xi = 0) is an arg max for p iff (x0 = 0, xV \{i} ) is an arg max for pi ; (xV \{i} , xi = 1) is an
arg max for p iff (x0 = 1, xV \{i} ) is an arg max for pi .
We can recover marginals as shown in the following proposition, proof in the Appendix ?9.1.
Proposition 7 (Recovering marginals). For a subset ? 6= U ? V , we can recover from M + :
p(xU ) = p+ (x0 = 0, xU ) + p+ (x0 = 1, xU )
= 2p+ (x0 = 0, xU ) = 2p+ (x0 = 1, xU ).
To recover from a rerooted Mi : (i) For any i ? V \ U , p(xU ) = pi (x0 = 0, xU ) + pi (x0 = 1, xU ).
pi (x0 = 0, xU \{i} ) xi = 0
(ii) For any i ? U , p(xU ) =
pi (x0 = 1, xU \{i} ) xi = 1.
In ?6, we provide a careful analysis of the impact of uprooting and rerooting on the Sherali-Adams
hierarchy of relaxations of the marginal polytope [11]. We first introduce a way to parametrize
potentials which will be particularly useful, and which may be of independent interest.
5
Pure k-potentials
We introduce the notion of pure k-potentials. These allow the specification of interactions which act
?purely? over a set of variables of a given size k, without influencing the distribution of any subsets.
We show that in fact, there is essentially only one pure k-potential. Further, we show that one can
express any ?E potential in terms of pure potentials over E and subsets of E, and that pure potentials
have appealing properties when uprooted and rerooted which help our subsequent analysis.
We say that a potential is a k-potential if k is the smallest number such that the score of the potential
may be determined by considering the configuration of k variables. Usually a potential ?E is a
k-potential with k = |E|. For example, typically a singleton potential is a 1-potential, and an edge
potential is a 2-potential. However, note that k < |E| is possible if one or more variables in E are not
needed to establish the score (a simple example is ?12 (x1 , x2 ) = x1 , which clearly is a 1-potential).
4
In general, a k-potential will affect the marginal distributions of all subsets of the k variables. For
example, one popular form of 2-potential is ?ij (xi , xj ) = Wij xi xj , which tends to pull Xi and Xj
toward the same value, but also tends to increase each of p(Xi = 1) and p(Xj = 1). For pairwise
models, a different reparameterization of potentials instead writes the score as
X
1 X
score(xV ) =
? i xi +
Wij 1[xi = xj ].
(3)
2
i?V
(i,j)?E
Expression (3) has the desirable feature that the ?ij (xi , xj ) = 12 Wij 1[xi = xj ] edge potentials affect
only the pairwise marginals, without disturbing singleton marginals. This motivates the following
definition.
Definition 8. Let k ? 2, and let U be a set of size k. We say that a k-potential ?U : {0, 1}U ? R
is a pure k-potential if the distribution induced by the potential, p(xU ) ? exp(?U (xU )), has the
property that for any ? =
6 W ( U , the marginal distribution p(xW ) is uniform.
We shall see in Proposition 10 that a pure k-potential must essentially be an even k-potential.
Definition 9. Let k ? N, and |U | = k. An even k-potential is a k-potential ?U : {0, 1}U ? R of the
form ?U (xU ) = a1[ |{i ? U |xi = 1}| is even], for some a ? R which is its coefficient. In words,
?U (xU ) takes value a if xU has an even number of 1s, else it takes value 0.
As an example, the 2-potential ?ij (xi , xj ) = 12 Wij 1[xi = xj ] in (3) is an even 2-potential with
U = {i, j} and coefficient Wij /2. The next two propositions are proved in the Appendix ?9.2.
Proposition 10 (All pure potentials are essentially even potentials). Let k ? 2, and |U | = k. If
?U : {0, 1}U ? R is a pure k-potential then ?U must be an affine function of the even k-potential, i.e.
? a, b ? R s.t. ?U (xU ) = a1[ |{i ? U |xi = 1}| is even] + b.
Proposition 11 (Even k-potentials
form a basis). For a finite set U , the set of even k-potentials
1[ |{i ? W |Xi = 1}| is even] W ?U , indexed by subsets W ? U , forms a basis for the vector
space of all potential functions ? : {0, 1}U ? R.
Any constant in a potential will be absorbed into the partition function Z and does not affect the
probability distribution, see (1). An even 2-potential with positive coefficient, e.g. as in (3) if
Wij > 0, is supermodular. Models with only supermodular potentials (equivalently, submodular cost
functions) typically admit easier inference [3; 7]; if such a model is binary pairwise then it is called
attractive. However, for k > 2, even k-potentials ?E are neither supermodular nor submodular. Yet if
k is an even number, observe that ?E (xE ) = ?E (xE ). We discuss this further in Appendix ?10.4.
When a k-potential is uprooted, in general it may become a (k + 1)-potential (recall Definition 2).
The following property of even k-potentials is helpful for our analysis in ?6, and is easily checked.
Lemma 12 (Uprooting an even k-potential). When an even k-potential ?E with |E| = k is uprooted:
if k is an even number, then the uprooted potential is exactly the same even k-potential; if k is odd,
then we obtain the even (k + 1)-potential over E ? {0} with the same coefficient as the original ?E .
6
Marginal polytope and Sherali-Adams relaxations
We saw in Lemma 4 that there is a score-preserving 1-2 mapping from configurations of M to those
of M + , and a bijection between configurations of M and any Mi . Here we examine the extent to
which these score-preserving mappings extend to (pseudo-)marginal probability distributions over
variables by considering the Sherali-Adams relaxations [11] of the respective marginal polytopes.
These relaxations feature prominently in many approaches for MAP and marginal inference.
For U ? V , we write ?U for a probability distribution in P({0, 1}U ), the set of all probability
distributions on {0, 1}U . Bold ? will represent a collection of measures over various subsets of
variables. Given (1), to compute an expected score, we need (?E )E?E . This motivates the following.
Definition 13. The marginal polytope M(G(V, E)) = {(?E )E?E ??V s.t. ?V E = ?E ?E ? E},
where for U1 ? U2 ? V , ?U2 ?U1 denotes the marginalization of ?U2 ? P({0, 1}U2 ) onto {0, 1}U1 .
M(G) consists of marginal distributions for every hyperedge E ? E such that all the marginals are
consistent with a global distribution over all variables V . Methods of variational inference typically
5
optimize either the score (for MAP inference) or the score plus an entropy term (for marginal
inference) over a relaxation of the marginal polytope [15]. This is because M(G) is computationally
intractable, with an exponential number of facets [2]. Relaxations from the Sherali-Adams hierarchy
[11] are often used, requiring consistency only over smaller clusters of variables.
Definition 14. Given an integer r ? 2, if a hypergraph G(V, E) satisfies maxE?E |E| ? r ? |V |,
then we say that G is r-admissible, and define the Sherali-Adams polytope of order r on G by
Lr (G) = (?E )E?E ?(?U ) U ?V locally consistent, s.t. ?U ?E = ?E ? E ? U ? V, |U | = r ,
|U |=r
where a collection of measures (?A )A?I (for some set I of subsets of V ) is locally consistent, or l.c.,
if for any A1 , A2 ? I, we have ?A1 ?A1 ?A2 = ?A2 ?A1 ?A2 . Each element of Lr (G) is a set of locally
consistent probability measures over the hyperedges. Note that M(G) ? Lr (G) ? Lr?1 (G). The
pairwise relaxation L2 (G) is commonly used but higher-order relaxations achieve greater accuracy,
have received significant attention [10; 13; 18; 22; 23], and are required for higher-order potentials.
6.1
The impact of uprooting and rerooting on Sherali-Adams polytopes
We introduce two variants of the Sherali-Adams polytopes which will be helpful in analyzing
uprooted models. For a measure ?U ? P({0, 1}U ), we define the flipped measure ?U as ?U (xU ) =
?U (xU ) ?xU ? {0, 1}U . A measure ?U is flipping-invariant if ?U = ?U .
Definition 15. The symmetrized Sherali-Adams polytopes for an uprooted hypergraph ?G(V + , E + )
(as given in Definition 2), is:
e r (?G) = (?E )
?E = ?E ?E ? E + .
L
?
L
(?G)
+
r
E?E
Definition 16. For any i ? V + , and any integer r ? 2 such that maxE?E + |E| ? r ? |V + |, we
define the symmetrized Sherali-Adams polytope of order r uprooted at i to be
?U ?E = ?E ? E ? U ? V, |U | = r, i ? U
e i (?G) = (?E )
+
L
?(?
)
l.c.,
s.t.
.
+
U i?U ?V
r
E?E
?U = ?U ?U ? V, |U | = r, i ? U
|U |=r
e i (?G), there exist corresponding flippingThus, for each collection of measures over hyperedges in L
r
invariant, locally consistent measures on sets of size r which contain i (and their subsets). Note
e r+1 (?G) ? L
e i (?G) ? L
e r (?G).
that for any hypergraph G(V, E) and any i ? V + , we have L
r+1
We next extend the correspondence of Lemma 4 to collections of locally-consistent probability
distributions on the hyperedges of G, see the Appendix ?9.3 for proof.
Theorem 17. For a hypergraph G(V, E), and integer r such that maxE?E |E| ? r ? |V |, there is
an affine score-preserving bijection
Uproot
Lr (G)
RootAt0
e 0 (?G) .
L
r+1
Theorem 17 establishes the following diagram of polytope inclusions and affine bijections:
For M = M0 :
For M
+
:
Lr+1
?x(G)
??
Uprooty?RootAt0
e 0 (?G)
L
r+2
?
Unnamed
?x
??
Uprooty?RootAt0
e r+1 (?G)
? L
?
L?r (G)
x
??
Uprooty?RootAt0
e 0 (?G) .
? L
r+1
(4)
A question of theoretical interest and practical importance is which of the inclusions in (4) are
strict. Our perspective here generalizes earlier work. Using different language, Deza and Laurent
e 0 (?G), which was termed RMET, the rooted semimetric polytope; and
[2] identified L2 (G) with L
3
e 3 (?G) with MET, the semimetric polytope. Building on this, Weller [19] considered L3 (G), the
L
triplet-consistent polytope or TRI, though only in the context of pairwise potentials, and showed that
L3 (G) has the remarkable property that if it is used to optimize an LP for a model M on G, the exact
same optimum is achieved for L3 (Gi ) for any rerooting Mi . It was natural to conjecture that Lr (G)
might have this same property for all r > 3, yet this was left as an open question.
6
6.2
L3 is unique in being universally rooted
We shall first strengthen [19] to show that L3 is universally rooted in the following stronger sense.
Definition 18. We say that the rth -order Sherali-Adams relaxation is universally rooted (and write
?Lr is universally rooted? for short) if for all admissible hypergraphs G, there is an affine scorepreserving bijection between Lr (G) and Lr (Gi ), for each rerooted hypergraph (Gi )i?V .
If Lr is universally rooted, this applies for potentials over up to r variables (the maximum which
makes sense in this context), and clearly it implies that optimizing score over any rerooting (as in
MAP inference) will attain the same objective. The following result is proved in the Appendix ?9.3.
Lemma 19. If Lr is universally rooted for hypergraphs of maximum hyperedge degree p < r with p
even, then Lr is also universally rooted for r-admissible hypergraphs with maximum degree p + 1.
e 0 (?G). Then by considering
The proof relies on mapping to the symmetrized uprooted polytope L
r+1
marginals using a basis equivalent to that described in Proposition 11 for even k-potentials, we
observe that the symmetry of the polytope enforces only one possible marginal for (p + 1)-clusters.
Combining Lemma 19 with arguments which extend those used by [19] demonstrates the following
result, proved in the Appendix.
Theorem 20. L3 is universally rooted.
We next provide a striking and rather surprising result, see the Appendix for proof and details.
Theorem 21. L3 is unique in being universally rooted. Specifically, for any integer r > 1 other than
r = 3, we constructively demonstrate a hypergraph G(V, E) with |V | = r + 1 variables for which
e 0 (?G) 6= L
e i (?G) for any i ? V .
L
r+1
r+1
e 0 (?G) and L
e i (?G), which by Theorem 17 are the uprooted equivalents
Theorem 21 examines L
r+1
r+1
of Lr (G) and Lr (Gi ). It might appear more satisfying to try to demonstrate the result directly for
the rooted polytopes, i.e. to show Lr (G) 6= Lr (Gi ). However, in general the rooted polytopes
are not comparable: an r-potential in M can map to an (r + 1)-potential in M + and then to an
(r + 1)-potential in Mi which cannot be evaluated for an Lr polytope.
Theorem 21 shows that we may hope for benefits from rerooting for any inference method based on a
Sherali-Adams relaxed polytope Lr , unless r = 3.
7
Experiments
Here we show empirically the benefits of uprooting and rerooting for approximate inference methods
in models with higher-order potentials. We introduce an efficient heuristic which can be used in
practice to select a variable for rerooting, and demonstrate its effectiveness.
We compared performance after different rerootings of marginal inference (to guarantee convergence
we used the double loop method of Heskes et al. [4], which relates to generalized belief propagation,
24) and MAP inference (using loopy belief propagation, LBP [9]). For true values, we used the
junction tree algorithm. All methods were implemented using libDAI [8]. We ran experiments
on complete hypergraphs (with 8 variables) and toroidal grid models (5 ? 5 variables). Potentials
up to order 4 were selected randomly, by drawing even k-potentials from Unif([?Wmax , Wmax ])
distributions for a variety of Wmax parameters, as shown in Figure 2, which highlights results for
estimating log Z. For each regime of maximum potential values, we plot results averaged over 20
runs. For additional details and results, including marginals, other potential choices and larger models,
see Appendix ?10.
We display average error of the inference method applied to: the original model M ; the uprooted
model M + ; then rerootings at: the worst variable, the best variable, the K heuristic variable, and
the G heuristic variable. Best and worst always refer to the variable at which rerooting gave with
hindsight the best and worst error for the partition function (even in plots for other measures).
7
7.1
Heuristics to pick a good variable for rerooting
From our Definition 3, a rerooted model Mi is obtained by clamping the uprooted model M + at
variable Xi . Hence, selecting a good variable for rerooting is exactly the choice of a good variable
to clamp in M + . Considering pairwise models, Weller [19] refined the maxW method [20; 21] to
introduce the maxtW heuristic, and showed that it was very effective empirically. maxtW selects
P
W
the variable Xi with max j?N (i) tanh | 4ij |, where N (i) is the set of neighbors of i in the model
graph, and Wij is the strength of the pairwise interaction.
The intuition for maxtW is as follows. Pairwise methods of approximate inference such as Bethe
are exact for models with no cycles. If we could, we would like to ?break? tight cycles with strong
edge weights, since these lead to error. When a variable is clamped, it is effectively removed from
the model. Hence, we would like to reroot at a variable that sits on many cycles with strong edge
weights. Identifying such cycles is NP-hard, but the maxtW heuristic attempts to do this by looking
only locally around each variable. Further, the effect of a strong edge weight saturates [21]: a very
strong edge weight Wij effectively ?locks? its end variables (either together or opposite depending on
the sign of Wij ), and this effect cannot be significantly increased even by an extremely strong edge.
Hence the tanh function was introduced to the earlier maxW method, leading to the maxtW heuristic.
As observed in ?5, if we express our model potentials in terms of pure k-potentials, then the uprooted
model will only have pure k-potentials for various values of k which are even numbers. Intuitively,
the higher the coefficients on these potentials, the more tightly connected is the model leading to more
challenging inference. Hence, a natural way to generalize the maxtW approach to handle higher-order
potentials is to pick a variable Xi in M + which maximizes the following measure:
X
X
clamp-heuristic-measure(i) =
c2 tanh |t2 aE | +
c4 tanh |t4 aE |,
(5)
i?E:|E|=2
i?E:|E|=4
where aE is the coefficient (weight) of the relevant pure k-potential, see Definition 9, and the
{c2 , t2 }, {c4 , t4 } terms are constants for pure 2-potentials and for pure 4-potentials respectively. This
approach extends to potentials of higher orders by adding similar further terms. Since our goal is to
rank the measures for each i ? V + , without loss of generality we take c2 = 1. We fit the t2 , c4 and t4
constants to the data from our experimental runs, see the Appendix for details. Our K heuristic was fit
only to runs for complete hypergraphs while the G heuristic was fit only to runs for models on grids.
7.2
Observations on results
Considering all results across models and approximate methods for estimating log Z, marginals and
MAP inference (see Figure 2 and Appendix ?10.3), we make the following observations. Both K and
G heuristics perform well (in and out of sample): they never hurt materially and often significantly
improve accuracy, attaining results close to the best possible rerooting. Since our two heuristics
achieve similar performance, sensitivity to the exact constants in (5) appears low. We verified this by
comparing to maxtW for pairwise models as in [19]: both K and G heuristics performed just slightly
better than maxtW. For all our runs, inference on rerooted models took similar time as on the original
model (time required to reroot and later to map back inference results is negligible), see ?10.3.1.
Observe that stronger 1-potentials tend to make inference easier, pulling each variable toward a
specific setting, and reducing the benefits from rerooting (left column of Figure 2). Stronger pure
k-potentials for k > 1 intertwine variables more tightly: this typically makes inference harder and
increases the gains in accuracy from rerooting. The pure k-potential perspective facilitates this
analysis.
When we examine larger models, or models with still higher order potentials, we observe qualitatively
similar results, see Appendix ?10.3.4 and 10.3.6.
8
Conclusion
We introduced methods which broaden the application of the uprooting and rerooting approach
to binary models with higher-order potentials of any order. We demonstrated several important
theoretical insights, including Theorems 20 and 21 which show that L3 is unique in being universally
rooted. We developed the helpful tool of even k-potentials in ?5, which may be of independent
8
Average abs(error) in log Z for K8 complete hypergraphs (fully connected) on 8 variables.
Average abs(error) in log Z for Grids on 5 ? 5 variables (toroidal). Legends are consistent across all plots.
vary Wmax for 1-pots
vary Wmax for 2-pots
vary Wmax for 3-pots
vary Wmax for 4-pots
Figure 2: Error in estimating log Z for random models with various pure k-potentials over 20 runs. If not
shown, Wmax max coefficients for pure k-potentials are 0 for k = 1, 8 for k = 2, 0 for k = 3, 8 for k = 4.
Where the red K heuristic curve is not visible, it coincides with the green G heuristic. Both K and G heuristics
for selecting a rerooting work well: they never hurt and often yield large benefits. See ?7 for details.
interest. We empirically demonstrated significant benefits for rerooting in higher-order models ?
particularly for the hard case of strong cluster potentials and weak 1-potentials ? and provided an
efficient heuristic to select a variable for rerooting. This heuristic is also useful to indicate when
rerooting is unlikely to be helpful for a given model (if (5) is maximized by taking i = 0).
It is natural to compare the effect of rerooting M to Mi , against simply clamping Xi in the original
model M . A key difference is that rerooting achieves the clamping at Xi for negligible computational
cost. In contrast, if Xi is clamped in the original model then the inference method will have to
be run twice: once clamping Xi = 0, and once clamping Xi = 1, then results must be combined.
This is avoided with rerooting given the symmetry of M + . Rerooting effectively replaces what may
be a poor initial implicit choice of clamping at X0 with a carefully selected choice of clamping
variable almost for free. This is true even for large models where it may be advantageous to clamp a
series of variables: by rerooting, one of the series is obtained for free, potentially gaining significant
benefit with little work required. Note that each separate connected component may be handled
independently, with its own added variable. This could be useful for (repeatedly) composing clamping
and then rerooting each separated component to obtain an almost free clamping in each.
Acknowledgements
We thank the anonymous reviewers for helpful comments. MR acknowledges support by the UK
Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University
of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. AW acknowledges
support by the Alan Turing Institute under the EPSRC grant EP/N510129/1, and by the Leverhulme
Trust via the CFI.
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10
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6,219 | 6,626 | The Unreasonable Effectiveness of Structured
Random Orthogonal Embeddings
Krzysztof Choromanski ?
Google Brain Robotics
[email protected]
Mark Rowland ?
University of Cambridge
[email protected]
Adrian Weller
University of Cambridge and Alan Turing Institute
[email protected]
Abstract
We examine a class of embeddings based on structured random matrices with
orthogonal rows which can be applied in many machine learning applications
including dimensionality reduction and kernel approximation. For both the JohnsonLindenstrauss transform and the angular kernel, we show that we can select matrices
yielding guaranteed improved performance in accuracy and/or speed compared to
earlier methods. We introduce matrices with complex entries which give significant
further accuracy improvement. We provide geometric and Markov chain-based
perspectives to help understand the benefits, and empirical results which suggest
that the approach is helpful in a wider range of applications.
1
Introduction
Embedding methods play a central role in many machine learning applications by projecting feature
vectors into a new space (often nonlinearly), allowing the original task to be solved more efficiently.
The new space might have more or fewer dimensions depending on the goal. Applications include
the Johnson-Lindenstrauss Transform for dimensionality reduction (JLT, Johnson and Lindenstrauss,
1984) and kernel methods with random feature maps (Rahimi and Recht, 2007). The embedding can
be costly hence many fast methods have been developed, see ?1.1 for background and related work.
We present a general class of random embeddings based on particular structured random matrices
with orthogonal rows, which we call random ortho-matrices (ROMs); see ?2. We show that ROMs
may be used for the applications above, in each case demonstrating improvements over previous
methods in statistical accuracy (measured by mean squared error, MSE), in computational efficiency
(while providing similar accuracy), or both. We highlight the following contributions:
? In ?3: The Orthogonal Johnson-Lindenstrauss Transform (OJLT) for dimensionality reduction.
We prove this has strictly smaller MSE than the previous unstructured JLT mechanisms. Further,
OJLT is as fast as the fastest previous JLT variants (which are structured).
? In ?4: Estimators for the angular kernel (Sidorov et al., 2014) which guarantee better MSE. The
angular kernel is important for many applications, including natural language processing (Sidorov
et al., 2014), image analysis (J?gou et al., 2011), speaker representations (Schmidt et al., 2014)
and tf-idf data sets (Sundaram et al., 2013).
? In ?5: Two perspectives on the effectiveness of ROMs to help build intuitive understanding.
In ?6 we provide empirical results which support our analysis, and show that ROMs are effective for
a still broader set of applications. Full details and proofs of all results are in the Appendix.
?
equal contribution
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1.1
Background and related work
Our ROMs can have two forms (see ?2 for details): (i) a Gort is a random Gaussian matrix conditioned on rows being orthogonal; or (ii) an SD-product matrix is formed by multiplying some
number k of SD blocks, each of which is highly structured, typically leading to fast computation
of products. Here S is a particular structured matrix, and D is a random diagonal matrix; see ?2
for full details. Our SD block generalizes an HD block, where H is a Hadamard matrix, which
received previous attention. Earlier approaches to embeddings have explored using various structured
matrices, including particular versions of one or other of our two forms, though in different contexts.
For dimensionality reduction, Ailon and Chazelle (2006) used a single HD block as a way to spread
out the mass of a vector over all dimensions before applying a sparse Gaussian matrix. Choromanski
and Sindhwani (2016) also used just one HD block as part of a larger structure. Bojarski et al. (2017)
discussed using k = 3 HD blocks for locality-sensitive hashing methods but gave no concrete results
for their application to dimensionality reduction or kernel approximation. All these works, and other
earlier approaches (Hinrichs and Vyb?ral, 2011; Vyb?ral, 2011; Zhang and Cheng, 2013; Le et al.,
2013; Choromanska et al., 2016), provided computational benefits by using structured matrices with
less randomness than unstructured iid Gaussian matrices, but none demonstrated accuracy gains.
Yu et al. (2016) were the first to show that Gort -type matrices can yield improved accuracy, but their
theoretical result applies only asymptotically for many dimensions, only for the Gaussian kernel and
for just one specific orthogonal transformation, which is one instance of the larger class we consider.
Their theoretical result does not yield computational benefits. Yu et al. (2016) did explore using a
number k of HD blocks empirically, observing good computational and statistical performance for
k = 3, but without any theoretical accuracy guarantees. It was left as an open question why matrices
formed by a small number of HD blocks can outperform non-discrete transforms.
In contrast, we are able to prove that ROMs yield improved MSE in several settings and for many of
them for any number of dimensions. In addition, SD-product matrices can deliver computational
speed benefits. We provide initial analysis to understand why k = 3 can outperform the state-ofthe-art, why odd k yields better results than even k, and why higher values of k deliver decreasing
additional benefits (see ?3 and ?5).
2
The family of Random Ortho-Matrices (ROMs)
Random ortho-matrices (ROMs) are taken from two main classes of distributions defined below that
require the rows of sampled matrices to be orthogonal. A central theme of the paper is that this
orthogonal structure can yield improved statistical performance. We shall use bold uppercase (e.g.
M) to denote matrices and bold lowercase (e.g. x) for vectors.
Gaussian orthogonal matrices. Let G be a random matrix taking values in Rm?n with iid N (0, 1)
elements, which we refer to as an unstructured Gaussian matrix. The first ROM distribution we
consider yields the random matrix Gort , which is defined as a random Rn?n matrix given by first
taking the rows of the matrix to be a uniformly random orthonormal basis, and then independently
scaling each row, so that the rows marginally have multivariate Gaussian N (0, I) distributions. The
random variable Gort can then be extended to non-square matrices by either stacking independent
copies of the Rn?n random matrices, and deleting superfluous rows if necessary. The orthogonality
of the rows of this matrix has been observed to yield improved statistical properties for randomized
algorithms built from the matrix in a variety of applications.
SD-product matrices. Our second class of distributions is motivated by the desire to obtain similar
statistical benefits of orthogonality to Gort , whilst gaining computational efficiency by employing
more structured matrices. We call this second class SD-product matrices. These take the more
Qk
structured form i=1 SDi , where S = {si,j } ? Rn?n has orthogonal rows, |si,j | = ?1n ?i, j ?
Qk
{1, . . . , n}; and the (Di )ki=1 are independent diagonal matrices described below. By i=1 SDi , we
mean the matrix product (SDk ) . . . (SD1 ). This class includes as particular cases several recently
introduced random matrices (e.g. Andoni et al., 2015; Yu et al., 2016), where good empirical
performance was observed. We go further to establish strong theoretical guarantees, see ?3 and ?4.
2
A prominent example of an S matrix is thenormalized Hadamard
matrix H, defined recursively by
H
H
i?1
i?1
1
H1 = (1), and then for i > 1, Hi = ?2
. Importantly, matrix-vector products
Hi?1 ?Hi?1
with H are computable in O(n log n) time via the fast Walsh-Hadamard transform, yielding large
computational savings. In addition, H matrices enable a significant space advantage: since the
fast Walsh-Hadamard transform can be computed without explicitly storing H, only O(n) space is
required to store the diagonal elements of (Di )ki=1 . Note that these Hn matrices are defined only for
n a power of 2, but if needed, one can always adjust data by padding with 0s to enable the use of ?the
next larger? H, doubling the number of dimensions in the worst case.
Matrices H are representatives of a much larger family in S which also attains computational savings.
These are L2 -normalized versions of Kronecker-product matrices of the form A1 ? ... ? Al ? Rn?n
for l ? N, where ? stands for a Kronecker product and blocks Ai ? Rd?d have entries of the
same magnitude and pairwise orthogonal rows each. For these matrices, matrix-vector products are
computable in O(n(2d ? 1) logd (n)) time (Zhang et al., 2015).
S includes also the Walsh matrices W = {wi,j } ? Rn?n , where wi,j = ?1n (?1)iN ?1 j0 +...+i0 jN ?1
and iN ?1 ...i0 , jN ?1 ...j0 are binary representations of i and j respectively.
For diagonal (Di )ki=1 , we mainly consider Rademacher entries leading to the following matrices.
(R) k
)i=1
Definition 2.1. The S-Rademacher random matrix with k ? N blocks is below, where (Di
are diagonal with iid Rademacher random variables [i.e. Unif({?1})] on the diagonals:
(k)
MSR
=
k
Y
(R)
SDi
.
(1)
i=1
Having established the two classes of ROMs, we next apply them to dimensionality reduction.
3
The Orthogonal Johnson-Lindenstrauss Transform (OJLT)
Let X ? Rn be a dataset of n-dimensional real vectors. The goal of dimensionality reduction via
F
random projections is to transform linearly each x ? X by a random mapping x 7? x0 , where:
n
m
0 > 0
F : R ? R for m < n, such that for any x, y ? X the following holds: (x ) y ? x> y. If
we furthermore have E[(x0 )> y0 ] = x> y then the dot-product estimator is unbiased. In particular,
this dimensionality reduction mechanism should in expectation preserve information about vectors?
norms, i.e. we should have: E[kx0 k22 ] = kxk22 for any x ? X .
The standard JLT mechanism uses the randomized linear map F = ?1m G, where G ? Rm?n is as
in ?2, requiring mn multiplications to evaluate. Several fast variants (FJLTs) have been proposed by
replacing G with random structured matrices, such as sparse or circulant Gaussian matrices (Ailon
and Chazelle, 2006; Hinrichs and Vyb?ral, 2011; Vyb?ral, 2011; Zhang and Cheng, 2013). The fastest
of these variants has O(n log n) time complexity, but at a cost of higher MSE for dot-products.
Our Orthogonal Johnson-Lindenstrauss Transform (OJLT) is obtained by replacing the unstructured
(k),sub
random matrix G with a sub-sampled ROM from ?2: either Gort , or a sub-sampled version MSR
of the S-Rademacher ROM, given by sub-sampling rows from the left-most S matrix in the product.
We sub-sample since m < n. We typically assume uniform sub-sampling without replacement. The
resulting dot-product estimators for vectors x, y ? X are given by:
b base (x, y) = 1 (Gx)> (Gy) [unstructured iid baseline, previous state-of-the-art accuracy],
K
m
m
>
(k),sub
b ort (x, y) = 1 (Gort x)> (Gort y),
b (k) (x, y) = 1 M(k),sub x
K
K
MSR y . (2)
m
m
SR
m
m
We contribute the following closed-form expressions, which exactly quantify the mean-squared error
b
(MSE) for these three estimators. Precisely, the MSE of an
h estimator K(x, y) iof the inner product
b
b
hx, yi for x, y ? X is defined to be MSE(K(x, y)) = E (K(x, y) ? hx, yi2 ) . See the Appendix
for detailed proofs of these results and all others in this paper.
3
b base of x, y ? Rn using mLemma 3.1. The MSE of the unstructured JLT dot-product estimator K
m
base
b m (x, y)) = 1 ((x> y)2 + kxk2 kyk2 ).
dimensional random feature maps is unbiased, with MSE(K
2
2
m
b ort is unbiased and satisfies, for n ? 4:
Theorem 3.2. The estimator K
m
b ort (x, y))
MSE(K
m
base
b
=MSE(Km
(x, y)) +
m
kxk22 kyk22 n2
1
1
1
2
?
(I(n ? 3) ? I(n ? 1))I(n ? 4) cos (?) +
+
m ? 1 4I(n ? 3)I(n ? 4)
n n+2
2
1
1
1
I(n ? 1) (I(n ? 4) ? I(n ? 2))
?
cos2 (?) ?
? hx, yi2 ,
n?2 n
2
(3)
?
R?
where I(n) = 0 sinn (x)dx = ??((n+1)/2)
?(n/2+1) .
(k)
bm
Theorem 3.3 (Key result). The OJLT estimator K
(x, y) with k blocks, using m-dimensional
random feature maps and uniform sub-sampling policy without replacement, is unbiased with
1 n?m
(k)
b
MSE(Km (x, y)) =
((x> y)2 + kxk2 kyk2 ) +
(4)
m n?1
k?1
n
X (?1)r 2r
(?1)k 2k X 2 2
>
2
2
2
(2(x y) + kxk kyk ) +
x y .
nr
nk?1 i=1 i i
r=1
Proof (Sketch). For k = 1, the random projection matrix is given by sub-sampling rows from SD1 ,
and the computation can be carried out directly. For k ? 1, the proof proceeds by induction.
The random projection matrix in the general case is given by sub-sampling rows of the matrix
SDk ? ? ? SD1 . By writing the MSE as an expectation and using the law of conditional expectations
conditioning on the value of the first k ? 1 random matrices Dk?1 , . . . , D1 , the statement of the
theorem for 1 SD block and for k ? 1 SD blocks can be neatly combined to yield the result.
To our knowledge, it has not previously been possible to provide theoretical guarantees that
SD-product matrices outperform iid matrices. Combining Lemma 3.1 with Theorem 3.3 yields the
following important result.
(k)
bm
(subsampling
Corollary 3.4 (Theoretical guarantee of improved performance). Estimators K
b base .
without replacement) yield guaranteed lower MSE than K
m
(k)
ort
bm
bm
It is not yet clear when K
is better or worse than K
; we explore this empirically in ?6.
Theorem 3.3 shows that there are diminishing MSE benefits to using a large number k of SD
(2k?1)
bm
blocks. Interestingly, odd k is better than even: it is easy to observe that MSE(K
(x, y)) <
(2k)
(2k+1)
b m (x, y)) > MSE(K
bm
MSE(K
(x, y)). These observations, and those in ?5, help to understand why empirically k = 3 was previously observed to work well (Yu et al., 2016).
If we take S to be a normalized Hadamard matrix H, then even though we are using sub-sampling,
and hence the full computational benefits of the Walsh-Hadamard transform are not available, still
(k)
bm
K
achieves improved MSE compared to the base method with less computational effort, as follows.
Lemma 3.5. There exists an algorithm (see Appendix for details) which computes an embedding for
(k)
bm
a given datapoint x using K
with S set to H and uniform sub-sampling policy in expected time
min{O((k ? 1)n log(n) + nm ? (m?1)m
, kn log(n)}.
2
Note that for m = ?(k log(n)) or if k = 1, the time complexity is smaller than the brute force
?(nm). The algorithm uses a simple observation that one can reuse calculations conducted for the
upper half of the Hadamard matrix while performing computations involving rows from its other half,
instead of running these calculations from scratch (details in the Appendix).
An alternative to sampling without replacement is deterministically to choose the first m rows. In our
experiments in ?6, these two approaches yield the same empirical performance, though we expect
4
that the deterministic method could perform poorly on adversarially chosen data. The first m rows
approach can be realized in time O(n log(m) + (k ? 1)n log(n)) per datapoint.
Theorem 3.3 is a key result in this paper, demonstrating that SD-product matrices yield both statistical
and computational improvements compared to the base iid procedure, which is widely used in practice.
We next show how to obtain further gains in accuracy.
3.1
Complex variants of the OJLT
We show that the MSE benefits of Theorem 3.3 may be markedly improved by using SD-product
(k)
matrices with complex entries MSH . Specifically, we consider the variant S-Hybrid random matrix
(U )
below, where Dk is a diagonal matrix with iid Unif(S 1 ) random variables on the diagonal, inde(R) k?1
pendent of (Di )i=1 , and S 1 is the unit circle of C. We use the real part of the Hermitian product
between projections as a dot-product estimator; recalling the definitions of ?2, we use:
(k)
MSH
=
(U )
SDk
k?1
Y
(R)
SDi
>
1
(k),sub
(k),sub
H,(k)
b
Km (x, y) =
Re MSH x
MSH y .
m
,
i=1
(5)
Remarkably, this complex variant yields exactly half the MSE of the OJLT estimator.
H,(k)
b m (x, y), applying uniform sub-sampling without replacement, is
Theorem 3.6. The estimator K
H,(k)
(k)
bm
bm
(x, y)).
unbiased and satisfies: MSE(K
(x, y)) = 12 MSE(K
(k)
b m . However,
This large factor of 2 improvement could instead be obtained by doubling m for K
this would require doubling the number of parameters for the transform, whereas the S-Hybrid
(U )
estimator requires additional storage only for the complex parameters in the matrix Dk . Strikingly,
it is straightforward to extend the proof of Theorem 3.6 (see Appendix) to show that rather than
(k),sub
taking the complex random variables in MSH
to be Unif(S 1 ), it is possible to take them to be
Unif({1, ?1, i, ?i}) and still obtain exactly the same benefit in MSE.
(U )
H,(k)
bm
Theorem 3.7. For the estimator K
defined in Equation (5): replacing the random matrix Dk
(which has iid Unif(S 1 ) elements on the diagonal) with instead a random diagonal matrix having iid
Unif({1, ?1, i, ?i}) elements on the diagonal, does not affect the MSE of the estimator.
It is natural to wonder if using an SD-product matrix with more complex random variables (for all
SD blocks) would improve performance still further. However, interestingly, this appears not to be
the case; details are provided in the Appendix ?8.7.
3.2
Sub-sampling with replacement
Our results above focus on SD-product matrices where rows have been sub-sampled without
replacement. Sometimes (e.g. for parallelization) it can be convenient instead to sub-sample with
replacement. As might be expected, this leads to worse MSE, which we can quantify precisely.
(k)
H,(k)
bm
bm
Theorem 3.8. For each of the estimators K
and K
, if uniform sub-sampling with (rather
n?1
than without) replacement is used then the MSE is worsened by a multiplicative constant of n?m
.
4
Kernel methods with ROMs
ROMs can also be used to construct high-quality random feature maps for non-linear kernel
approximation. We analyze here the angular kernel, an important example of a Pointwise Nonlinear
Gaussian kernel (PNG), discussed in more detail at the end of this section.
Definition 4.1. The angular kernel K ang is defined on Rn by K ang (x, y) = 1 ?
is the angle between x and y.
5
2?x,y
? ,
where ?x,y
To employ random feature style approximations to this kernel, we first observe it may be rewritten as
K ang (x, y) = E [sign(Gx)sign(Gy)] ,
where G ? R1?n is an unstructured isotropic Gaussian vector. This motivates approximations of the
form:
b ang m(x, y) = 1 sign(Mx)> sign(My),
K
(6)
m
where M ? Rm?n is a random matrix, and the sign function is applied coordinate-wise. Such
kernel estimation procedures are heavily used in practice (Rahimi and Recht, 2007), as they allow
fast approximate linear methods to be used (Joachims, 2006) for inference tasks. If M = G, the
unstructured Gaussian matrix, then we obtain the standard random feature estimator. We shall contrast
this approach against the use of matrices from the ROMs family.
When constructing random feature maps for kernels, very often m > n. In this case, our structured
mechanism can be applied by concatenating some number of independent structured blocks. Our
theoretical guarantees will be given just for one block, but can easily be extended to a larger number
of blocks since different blocks are independent.
ang,base
bm
The standard random feature approximation K
for approximating the angular kernel is
defined by taking M to be G, the unstructured Gaussian matrix, in Equation (6), and satisfies the
following.
ang,base
ang,base
bm
bm
Lemma 4.2. The estimator K
is unbiased and MSE(K
(x, y)) =
4?x,y (???x,y )
.
m? 2
b ang (x, y) of the true angular kernel K ang (x, y) is defined analogously
The MSE of an estimator K
to the MSE of an estimator of the dot product, given in ?3. Our main result regarding angular kernels
ang,ort
bm
states that if we instead take M = Gort in Equation (6), then we obtain an estimator K
with
strictly smaller MSE, as follows.
ang,ort
bm
Theorem 4.3. Estimator K
is unbiased and satisfies:
ang,ort
ang,base
bm
bm
MSE(K
(x, y)) < MSE(K
(x, y)).
ang,M
bm
We also derive a formula for the MSE of an estimator K
of the angular kernel which replaces G
with an arbitrary random matrix M and uses m random feature maps. The formula is helpful to see
how the quality of the estimator depends on the probabilities that the projections of the rows of M are
contained in some particular convex regions of the 2-dimensional space Lx,y spanned by datapoints
x and y. For an illustration of the geometric definitions introduced in this Section, see Figure 1. The
formula depends on probabilities involving events Ai = {sgn((ri )T x) 6= sgn((ri )T y)}, where
ri stands for the ith row of the structured matrix. Notice that Ai = {riproj ? Cx,y }, where riproj
stands for the projection of ri into Lx,y and Cx,y is the union of two cones in Lx,y , each of angle ?x,y .
ang,M
bm
Theorem 4.4. Estimator K
satisfies the following, where: ?i,j = P[Ai ? Aj ] ? P[Ai ]P[Aj ]:
?
?
"
#
m
m
X
X
X
1
4
?x,y 2
ang,M
bm
(x, y)) = 2 m ?
MSE(K
(1 ? 2P[Ai ])2 + 2 ? (P[Ai ] ?
) +
?i,j ? .
m
m
?
i=1
i=1
i6=j
Note that probabilities P[Ai ] and ?i,j depend on the choice of M. It is easy to prove that for
?
unstructured G and Gort we have: P[Ai ] = x,y
? . Further, from the independence of the rows of
G, ?i,j = 0 for i 6= j. For unstructured G we obtain Lemma 4.2. Interestingly, we see that to
prove Theorem 4.3, it suffices to show ?i,j < 0, which is the approach we take (see Appendix). If
(k)
?
we replace G with MSR , then the expression = P[Ai ] ? x,y
? does not depend on i. Hence, the
angular
kernel
estimator
based
on
Hadamard
matrices
gives
smaller
MSE estimator if and only if
P
2
i6=j ?i,j + m < 0. It is not yet clear if this holds in general.
As alluded to at the beginning of this section, the angular kernel may be viewed as a member of a wie
family of kernels known as Pointwise Nonlinear Gaussian kernels.
6
Figure 1: Left part: Left: g1 is orthogonal to Lx,y . Middle: g1 ? Lx,y . Right: g1 is close to orthogonal to
Lx,y . Right part: Visualization of the Cayley graph explored by the Hadamard-Rademacher process in two
dimensions. Nodes are colored red, yellow, light blue, dark blue, for Cayley distances of 0, 1, 2, 3 from the
identity matrix respectively. See text in ?5.
f
Definition 4.5. For a given
function f , the
Pointwise Nonlinear Gaussian kernel (PNG) K is
defined by K f (x, y) = E f (gT x)f (gT y) , where g is a Gaussian vector with i.i.d N (0, 1) entries.
Many prominent examples of kernels (Williams, 1998; Cho and Saul, 2009) are PNGs. Wiener?s
tauberian theorem shows that all stationary kernels may be approximated arbitrarily well by sums of
PNGs (Samo and Roberts, 2015). In future work we hope to explore whether ROMs can be used to
achieve statistical benefit in estimation tasks associated with a wider range of PNGs.
5
Understanding the effectiveness of orthogonality
Here we build intuitive understanding for the effectiveness of ROMs. We examine geometrically the
angular kernel (see ?4), then discuss a connection to random walks over orthogonal matrices.
Angular kernel. As noted above for the Gort -mechanism, smaller MSE than that for unstructured
G is implied by the inequality P[Ai ? Aj ] < P[Ai ]P[Aj ], which is equivalent to: P[Aj |Ai ] < P[Aj ].
Now it becomes clear why orthogonality is crucial. Without loss of generality take: i = 1, j = 2, and
let g1 and g2 be the first two rows of Gort .
Consider first the extreme case (middle of left part of Figure 1), where all vectors are 2-dimensional.
Recall definitions from just after Theorem 4.3. If g1 is in Cx,y then it is much less probable for
g2 also to belong to Cx,y . In particular, if ? < ?2 then the probability is zero. That implies the
inequality. On the other hand, if g1 is perpendicular to Lx,y then conditioning on Ai does not have
any effect on the probability that g2 belongs to Cx,y (left subfigure of Figure 1). In practice, with high
probability the angle ? between g1 and Lx,y is close to ?2 , but is not exactly ?2 . That again implies
that conditioned on the projection gp1 of g1 into Lx,y to be in Cx,y , the more probable directions of
gp2 are perpendicular to gp1 (see: ellipsoid-like shape in the right subfigure of Figure 1 which is the
projection of the sphere taken from the (n ? 1)-dimensional space orthogonal to g1 into Lx,y ). This
makes it less probable for gp2 to be also in Cx,y . The effect is subtle since ? ? ?2 , but this is what
provides superiority of the orthogonal transformations over state-of-the-art ones in the angular kernel
approximation setting.
Markov chain perspective. We focus on Hadamard-Rademacher random matrices HDk ...HD1 ,
a special case of the SD-product matrices described in Section 2. Our aim is to provide intuition
for how the choice of k affects the quality of the random matrix, following our earlier observations
just after Corollary 3.4, which indicated that for SD-product matrices, odd values of k yield greater
benefits than even values, and that there are diminishing benefits from higher values of k. We proceed
by casting the random matrices into the framework of Markov chains.
Definition 5.1. The Hadamard-Rademacher process in n dimensions is the Markov chain (Xk )?
k=0
taking values in the orthogonal group O(n), with X0 = I almost surely, and Xk = HDk Xk?1
almost surely, where H is the normalized Hadamard matrix in n dimensions, and (Dk )?
k=1 are iid
diagonal matrices with independent Rademacher random variables on their diagonals.
Constructing an estimator based on Hadamard-Rademacher matrices is equivalent to simulating
several time steps from the Hadamard-Rademacher process. The quality of estimators based on
Hadamard-Rademacher random matrices comes from a quick mixing property of the corresponding
7
(a) g50c - pointwise evalu- (b) random - angular kernel (c) random - angular kernel (d) g50c - inner product esation MSE for inner product
with true angle ?/4
timation MSE for variants of
estimation
3-block SD-product matrices.
(e) LETTER - dot-product
(f) USPS - dot-product
(g) LETTER - angular kernel (h) USPS - angular kernel
Figure 2: Top row: MSE curves for pointwise approximation of inner product and angular kernels on the
g50c dataset, and randomly chosen vectors. Bottom row: Gram matrix approximation error for a variety of
data sets, projection ranks, transforms, and kernels. Note that the error scaling is dependent on the application.
Markov chain. The following demonstrates attractive properties of the chain in low dimensions.
Proposition 5.2. The Hadamard-Rademacher process in two dimensions: explores a state-space of
16 orthogonal matrices, is ergodic with respect to the uniform distribution on this set, has period 2,
the diameter of the Cayley graph of its state space is 3, and the chain is fully mixed after 3 time steps.
This proposition, and the Cayley graph corresponding to the Markov chain?s state space (Figure 1
right), illustrate the fast mixing properties of the Hadamard-Rademacher process in low dimensions;
this agrees with the observations in ?3 that there are diminishing returns associated with using a large
number k of HD blocks in an estimator. The observation in Proposition 5.2 that the Markov chain
has period 2 indicates that we should expect different behavior for estimators based on odd and even
numbers of blocks of HD matrices, which is reflected in the analytic expressions for MSE derived in
Theorems 3.3 and 3.6 for the dimensionality reduction setup.
6
Experiments
We present comparisons of estimators introduced in ?3 and ?4, illustrating our theoretical results, and
further demonstrating the empirical success of ROM-based estimators at the level of Gram matrix
approximation. We compare estimators based on: unstructured Gaussian matrices G, matrices Gort ,
S-Rademacher and S-Hybrid matrices with k = 3 and different sub-sampling strategies. Results
for k > 3 do not show additional statistical gains empirically. Additional experimental results,
including a comparison of estimators using different numbers of SD blocks, are in the Appendix ?10.
Throughout, we use the normalized Hadamard matrix H for the structured matrix S.
6.1
Pointwise kernel approximation
Complementing the theoretical results of ?3 and ?4, we provide several salient comparisons of the
various methods introduced - see Figure 2 top. Plots presented here (and in the Appendix) compare
MSE for dot-product and angular and kernel. They show that estimators based on Gort , S-Hybrid
and S-Rademacher matrices without replacement, or using the first m rows, beat the state-of-the-art
unstructured G approach on accuracy for all our different datasets in the JLT setup. Interestingly, the
latter two approaches give also smaller MSE than Gort -estimators. For angular kernel estimation,
where sampling is not relevant, we see that Gort and S-Rademacher approaches again outperform
the ones based on matrices G.
8
6.2
Gram matrix approximation
Moving beyond the theoretical guarantees established in ?3 and ?4, we show empirically that the
superiority of estimators based on ROMs is maintained at the level of Gram matrix approximation.
We compute Gram matrix approximations (with respect to both standard dot-product, and angular
b 2 /kKk2
kernel) for a variety of datasets. We use the normalized Frobenius norm error kK ? Kk
as our metric (as used by Choromanski and Sindhwani, 2016), and plot the mean error based on
1,000 repetitions of each random transform - see Figure 2 bottom. The Gram matrices are computed
on a randomly selected subset of 550 data points from each dataset. As can be seen, the S-Hybrid
estimators using the ?no-replacement? or ?first m rows? sub-sampling strategies outperform even
the orthogonal Gaussian ones in the dot-product case. For the angular case, the Gort -approach and
S-Rademacher approach are practically indistinguishable.
7
Conclusion
We defined the family of random ortho-matrices (ROMs). This contains the SD-product matrices,
which include a number of recently proposed structured random matrices. We showed theoretically
and empirically that ROMs have strong statistical and computational properties (in several cases
outperforming previous state-of-the-art) for algorithms performing dimensionality reduction and
random feature approximations of kernels. We highlight Corollary 3.4, which provides a theoretical
guarantee that SD-product matrices yield better accuracy than iid matrices in an important dimensionality reduction application (we believe the first result of this kind). Intriguingly, for dimensionality
reduction, using just one complex structured matrix yields random features of much better quality.
We provided perspectives to help understand the benefits of ROMs, and to help explain the behavior
of SD-product matrices for various numbers of blocks. Our empirical findings suggest that our
theoretical results might be further strengthened, particularly in the kernel setting.
Acknowledgements
We thank Vikas Sindhwani at Google Brain Robotics and Tamas Sarlos at Google Research for
inspiring conversations that led to this work. We thank Matej Balog, Maria Lomeli, Jiri Hron and
Dave Janz for helpful comments. MR acknowledges support by the UK Engineering and Physical
Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre
for Doctoral Training, the Cambridge Centre for Analysis. AW acknowledges support by the Alan
Turing Institute under the EPSRC grant EP/N510129/1, and by the Leverhulme Trust via the CFI.
9
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6,220 | 6,627 | From Parity to Preference-based Notions
of Fairness in Classification
Muhammad Bilal Zafar
MPI-SWS
[email protected]
Krishna P. Gummadi
MPI-SWS
[email protected]
Isabel Valera
MPI-IS
[email protected]
Manuel Gomez Rodriguez
MPI-SWS
[email protected]
Adrian Weller
University of Cambridge & Alan Turing Institute
[email protected]
Abstract
The adoption of automated, data-driven decision making in an ever expanding
range of applications has raised concerns about its potential unfairness towards
certain social groups. In this context, a number of recent studies have focused on
defining, detecting, and removing unfairness from data-driven decision systems.
However, the existing notions of fairness, based on parity (equality) in treatment
or outcomes for different social groups, tend to be quite stringent, limiting the
overall decision making accuracy. In this paper, we draw inspiration from the fairdivision and envy-freeness literature in economics and game theory and propose
preference-based notions of fairness?given the choice between various sets of
decision treatments or outcomes, any group of users would collectively prefer its
treatment or outcomes, regardless of the (dis)parity as compared to the other groups.
Then, we introduce tractable proxies to design margin-based classifiers that satisfy
these preference-based notions of fairness. Finally, we experiment with a variety
of synthetic and real-world datasets and show that preference-based fairness allows
for greater decision accuracy than parity-based fairness.
1
Introduction
As machine learning is increasingly being used to automate decision making in domains that affect
human lives (e.g., credit ratings, housing allocation, recidivism risk prediction), there are growing
concerns about the potential for unfairness in such algorithmic decisions [23, 25]. A flurry of recent
research on fair learning has focused on defining appropriate notions of fairness and then designing
mechanisms to ensure fairness in automated decision making [12, 14, 18, 19, 20, 21, 28, 32, 33, 34].
Existing notions of fairness in the machine learning literature are largely inspired by the concept of
discrimination in social sciences and law. These notions call for parity (i.e., equality) in treatment,
in impact, or both. To ensure parity in treatment (or treatment parity), decision making systems need
to avoid using users? sensitive attribute information, i.e., avoid using the membership information in
socially salient groups (e.g., gender, race), which are protected by anti-discrimination laws [4, 10]. As
a result, the use of group-conditional decision making systems is often prohibited. To ensure parity in
impact (or impact parity), decision making systems need to avoid disparity in the fraction of users
belonging to different sensitive attribute groups (e.g., men, women) that receive beneficial decision
outcomes. A number of learning mechanisms have been proposed to achieve parity in treatment [24],
An open-source code implementation of our scheme is available at: http://fate-computing.mpi-sws.org/
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
f2
-ve +ve
f2
M (100)
f2
M (100)
-ve +ve
M (100)
+ve
-ve
W (100)
M (200)
W (100)
W (200)
W (100)
W (200)
M (200)
+ve
M (200)
W (200)
-ve
f1
Acc: 0.83
Benefit: 0% (M), 67% (W)
f1
f1
Acc: 0.72
Benefit: 22% (M), 22% (W)
Acc: 1.00
Benefit: 33% (M), 67% (W)
Figure 1: A fictitious decision making scenario involving two groups: men (M) and women (W). Feature f1
(x-axis) is highly predictive for women whereas f2 (y-axis) is highly predictive for men. Green (red) quadrants
denote the positive (negative) class. Within each quadrant, the points are distributed uniformly and the numbers
in parenthesis denote the number of subjects in that quadrant. The left panel shows the optimal classifier
satisfying parity in treatment. This classifier leads to all the men getting classified as negative. The middle
panel shows the optimal classifier satisfying parity in impact (in addition to parity in treatment). This classifier
achieves impact parity by misclassifying women from positive class into negative class, and in the process,
incurs a significant cost in terms of accuracy. The right panel shows a classifier consisting of group-conditional
classifiers for men (purple) and women (blue). Both the classifiers satisfy the preferred treatment criterion since
for each group, adopting the other group?s classifier would lead to a smaller fraction of beneficial outcomes.
Additionally, this group-conditional classifier is also a preferred impact classifier since both groups get more
benefit as compared to the impact parity classifier. The overall accuracy is better than the parity classifiers.
parity in impact [7, 18, 21] or both [12, 14, 17, 20, 32, 33, 34]. However, these mechanisms pay a
significant cost in terms of the accuracy (or utility) of their predictions. In fact, there exist some
inherent tradeoffs (both theoretical and empirical) between achieving high prediction accuracy and
satisfying treatment and / or impact parity [9, 11, 15, 22].
In this work, we introduce, formalize and evaluate new notions of fairness that are inspired by the
concepts of fair division and envy-freeness in economics and game theory [5, 26, 31]. Our work
is motivated by the observation that, in certain decision making scenarios, the existing parity-based
fairness notions may be too stringent, precluding more accurate decisions, which may also be desired
by every sensitive attribute group. To relax these parity-based notions, we introduce the concept of a
user group?s preference for being assigned one set of decision outcomes over another. Given the
choice between various sets of decision outcomes, any group of users would collectively prefer the
set that contains the largest fraction (or the greatest number) of beneficial decision outcomes for that
group.1 More specifically, our new preference-based notions of fairness, which we formally define in
the next section, use the concept of user group?s preference as follows:
? From Parity Treatment to Preferred Treatment: To offer preferred treatment, a decision making
system should ensure that every sensitive attribute group (e.g., men and women) prefers the set of
decisions they receive over the set of decisions they would have received had they collectively
presented themselves to the system as members of a different sensitive group.
The preferred treatment criterion represents a relaxation of treatment parity. That is, every decision
making system that achieves treatment parity also satisfies the preferred treatment condition, which
implies (in theory) that the optimal decision accuracy that can be achieved under the preferred
treatment condition is at least as high as the one achieved under treatment parity. Additionally,
preferred treatment allows group-conditional decision making (not allowed by treatment parity),
which is necessary to achieve high decision accuracy in scenarios when the predictive power of
features varies greatly between different sensitive user groups [13], as shown in Figure 1.
While preferred treatment is a looser notion of fairness than treatment parity, it retains a core fairness
property embodied in treatment parity, namely, envy-freeness at the level of user groups. Under
preferred treatment, no group of users (e.g., men or women, blacks or whites) would feel that they
would be collectively better off by switching their group membership (e.g., gender, race). Thus,
1
Although it is quite possible that certain individuals from the group may not prefer the set that maximizes the benefit for the group as a
whole.
2
preferred treatment decision making, despite allowing group-conditional decision making, is not
vulnerable to being characterized as ?reverse discrimination? against, or "affirmative action? for
certain groups.
? From Parity Impact to Preferred Impact: To offer preferred impact, a decision making system
needs to ensure that every sensitive attribute group (e.g., men and women) prefers the set of decisions
they receive over the set of decisions they would have received under the criterion of impact parity.
The preferred impact criterion represents a relaxation of impact parity. That is, every decision making
system that achieves impact parity also satisfies the preferred impact condition, which implies (in
theory) that the optimal decision accuracy that can be achieved under the preferred impact condition
is at least as high as the one achieved under impact parity. Additionally, preferred impact allows
disparity in benefits received by different groups, which may be justified in scenarios where insisting
on impact parity would only lead to a reduction in the beneficial outcomes received by one or more
groups, without necessarily improving them for any other group. In such scenarios, insisting on
impact parity can additionally lead to a reduction in the decision accuracy, creating a case of tragedy
of impact parity with a worse decision making all round, as shown in Figure 1.
While preferred impact is a looser notion of fairness compared to impact parity, by guaranteeing
that every group receives at least as many beneficial outcomes as they would would have received
under impact parity, it retains the core fairness gains in beneficial outcomes that discriminated groups
would have achieved under the fairness criterion of impact parity.
Finally, we note that our preference-based fairness notions, while having many attractive properties,
are not the most suitable notions of fairness in all scenarios. In certain cases, parity fairness may well
be the eventual goal [3] and the more desirable notion.
In the remainder of this paper, we formalize our preference-based fairness notions in the context
of binary classification (Section 2), propose tractable and efficient proxies to include these notions
in the formulations of convex margin-based classifiers in the form of convex-concave constraints
(Section 3), and show on several real world datasets that our preference-based fairness notions can
provide significant gains in overall decision making accuracy as compared to parity-based fairness
(Section 4).
2
Defining preference-based fairness for classification
In this section, we will first introduce two useful quality metrics?utility and group benefit?in the
context of fairness in classification, then revisit parity-based fairness definitions in the light of these
quality metrics, and finally formalize the two preference-based notions of fairness introduced in
Section 1 from the perspective of the above metrics. For simplicity, we consider binary classification
tasks, however, the definitions can be easily extended to m-ary classification.
Quality metrics in fair classification. In a fair (binary) classification task, one needs to find a
mapping between the user feature vectors x 2 Rd and class labels y 2 { 1, 1}, where (x, y)
are drawn from an (unknown) distribution f (x, y). This is often achieved by finding a mapping
function ? : Rd ! R such that given a feature vector x with an unknown label y, the corresponding
classifier predicts y? = sign(?(x)). However, this mapping function also needs to be fair with respect
to the values of a user sensitive attribute z 2 Z ? Z 0 (e.g., sex, race), which are drawn from
an (unknown) distribution f (z) and may be dependent of the feature vectors and class labels, i.e.,
f (x, y, z) = f (x, y|z)f (z) 6= f (x, y)f (z).
Given the above problem setting, we introduce the following quality metrics, which we will use to
define and compare different fairness notions:
I. Utility (U ): overall profit obtained by the decision maker using the classifier. For example, in a
loan approval scenario, the decision maker is the bank that gives the loan and the utility can be
the overall accuracy of the classifier, i.e.:
U (?) = Ex,y [I{sign(?(x)) = y}],
where I(?) denotes the indicator function and the expectation is taken over the distribution
f (x, y). It is in the decision maker?s interest to use classifiers that maximize utility. Moreover,
depending on the scenario, one can attribute different profit to true positives and true negatives?
or conversely, different cost to false negatives and false positives?while computing utility. For
3
example, in the loan approval scenario, marking an eventual defaulter as non-defaulter may have
a higher cost than marking a non-defaulter as defaulter. For simplicity, in the remainder of the
paper, we will assume that the profit (cost) for true (false) positives and negatives is the same.
II. Group benefit (Bz ): the fraction of beneficial outcomes received by users sharing a certain value
of the sensitive attribute z (e.g., blacks, hispanics, whites). For example, in a loan approval
scenario, the beneficial outcome for a user may be receiving the loan and the group benefit for
each value of z can be defined as:
Bz (?) = Ex|z [I{sign(?(x)) = 1}],
where the expectation is taken over the conditional distribution f (x|z) and the bank offers a loan
to a user if sign(?(x)) = 1. Moreover, as suggested by some recent studies in fairness-aware
learning [18, 22, 32], the group benefits can also be defined as the fraction of beneficial outcomes
conditional on the true label of the user. For example, in a recidivism prediction scenario, the
group benefits can be defined as the fraction of eventually non-offending defendants sharing a
certain sensitive attribute value getting bail, that is:
Bz (?) = Ex|z,y=1 [I{sign(?(x)) = 1}],
where the expectation is taken over the conditional distribution f (x|z, y = 1), y = 1 indicates
that the defendant does not re-offend, and bail is granted if sign(?(x)) = 1.
Parity-based fairness. A number of recent studies [7, 14, 18, 21, 32, 33, 34] have considered a
classifier to be fair if it satisfies the impact parity criterion. That is, it ensures that the group benefits
for all the sensitive attribute values are equal, i.e.:
Bz (?) = Bz0 (?) for all z, z 0 2 Z.
(1)
In this context, different (or often same) definitions of group benefit (or beneficial outcome) have
lead to different terminology, e.g., disparate impact [14, 33], indirect discrimination [14, 21], redlining [7], statistical parity [12, 11, 22, 34], disparate mistreatment [32], or equality of opportunity [18].
However, all of these group benefit definitions invariably focus on achieving impact parity. We
point interested readers to Feldman et al. [14] and Zafar et al. [32] regarding the discussion on this
terminology.
Although not always explicitly sought, most of the above studies propose classifiers that also satisfy
treatment parity in addition to impact parity, i.e., they do not use the sensitive attribute z in the
decision making process. However, some of them [7, 18, 21] do not satisfy treatment parity since
they resort to group-conditional classifiers, i.e., ? = {?z }z2Z . In such case, we can rewrite the above
parity condition as:
Bz (?z ) = Bz0 (?z0 ) for all z, z 0 2 Z.
(2)
Fairness beyond parity. Given the above quality metrics, we can now formalize the two preferencebased fairness notions introduced in Section 1.
? Preferred treatment: if a classifier ? resorts to group-conditional classifiers, i.e., ? = {?z }z2Z ,
it is a preferred treatment classifier if each group sharing a sensitive attribute value z benefits
more from its corresponding group-conditional classifier ?z than it would benefit if it would be
classified by any of the other group-conditional classifiers ?z0 , i.e.,
Bz (?z ) Bz (?z0 ) for all z, z 0 2 Z.
(3)
Note that, if a classifier ? does not resort to group-conditional classifiers, i.e., ?z = ? for all
z 2 Z, it will be always be a preferred treatment classifier. If, in addition, such classifier ensures
impact parity, it is easy to show that its utility cannot be larger than a preferred treatment classifier
consisting of group-conditional classifiers.
? Preferred impact: a classifier ? offers preferred impact over a classifier ? 0 ensuring impact
parity if it achieves higher group benefit for each sensitive attribute value group, i.e.,
Bz (?) Bz (? 0 ) for all z 2 Z.
(4)
One can also rewrite the above condition for group-conditional classifiers, i.e., ? = {?z }z2Z
and ? 0 = {?z0 }z2Z , as follows:
Bz (?z ) Bz (?z0 ) for all z 2 Z.
(5)
0
Note that, given any classifier ? ensuring impact parity, it is easy to show that there will always
exist a preferred impact classifier ? with equal or higher utility.
4
Connection to the fair division literature. Our notion of preferred treatment is inspired by the
concept of envy-freeness [5, 31] in the fair division literature. Intuitively, an envy-free resource
division ensures that no user would prefer the resources allocated to another user over their own
allocation. Similarly, our notion of preferred treatment ensures envy-free decision making at the
level of sensitive attribute groups. Specifically, with preferred treatment classification, no sensitive
attribute group would prefer the outcomes from the classifier of another group.
Our notion of preferred impact draws inspiration from the two-person bargaining problem [26] in
the fair division literature. In a bargaining scenario, given a base resource allocation (also called the
disagreement point), two parties try to divide some additional resources between themselves. If the
parties cannot agree on a division, no party gets the additional resources, and both would only get the
allocation specified by the disagreement point. Taking the resources to be the beneficial outcomes,
and the disagreement point to be the allocation specified by the impact parity classifier, a preferred
impact classifier offers enhanced benefits to all the sensitive attribute groups. Put differently, the
group benefits provided by the preferred impact classifier Pareto-dominate the benefits provided by
the impact parity classifier.
On individual-level preferences. Notice that preferred treatment and preferred impact notions are
defined based on the group preferences, i.e., whether a group as a whole prefers (or, gets more
beneficial outcomes from) a given set of outcomes over another set. It is quite possible that a set
of outcomes preferred by the group collectively is not preferred by certain individuals in the group.
Consequently, one can extend our proposed notions to account for individual preferences as well,
i.e., a set of outcomes is preferred over another if all the individuals in the group prefer it. In the
remainder of the paper, we focus on preferred treatment and preferred impact in the context of group
preferences, and leave the case of individual preferences and its implications on the cost of achieving
fairness for future work.
3
Training preferred classifiers
In this section, our goal is training preferred treatment and preferred impact group-conditional
classifiers, i.e., ? = {?z }z2Z , that maximize utility given a training set D = {(xi , yi , zi )}N
i=1 , where
(xi , yi , zi ) ? f (x, y, z). In both cases, we will assume that:2
I. Each group-conditional classifier is a convex boundary-based classifier. For ease of exposition,
in this section, we additionally assume these classifiers to be linear, i.e., ?z (x) = ?zT x, where
?z is a parameter that defines the decision boundary in the feature space. We relax the linearity
assumption in Appendix A and extend our methodology to a non-linear SVM classifier.
II. The utility function U is defined as the overall accuracy of the group-conditional classifiers, i.e.,
X
U (?) = Ex,y [I{sign(?(x)) = y}] =
Ex,y|z [I{sign(?zT x) = y}]f (z).
(6)
z2Z
III. The group benefit Bz for users sharing the sensitive attribute value z is defined as their average
probability of being classified into the positive class, i.e.,
Bz (?) = Ex|z [I{sign(?(x)) = 1}] = Ex|z [I{sign(?zT x) = 1}].
(7)
Preferred impact classifiers. Given a impact parity classifier with decision boundary parameters
{?z0 }z2Z , one could think of finding the decision boundary parameters {?z }z2Z of a preferred impact
classifier that maximizes utility by solving the following optimization problem:
P
1
T
minimize
(x,y,z)2D I{sign(?z x) = y}
N
{?z }
(8)
P
P
T
0T
subject to
x2Dz I{sign(?z x) = 1}
x2Dz I{sign(?z x) = 1} for all z 2 Z,
where Dz = {(xi , yi , zi ) 2 D | zi = z} denotes the set of users in the training set sharing sensitive
attribute value z, the objective uses an empirical estimate of the utility, defined by Eq. 6, and the
preferred impact constraints, defined by Eq. 5, use empirical estimates of the group benefits, defined
by Eq. 7. Here, note that the right hand side of the inequalities does not contain any variables and can
be precomputed, i.e., the impact parity classifiers {?z0 }z2Z are given.
2
Exploring the relaxations of these assumptions is a very interesting avenue for future work.
5
Unfortunately, it is very challenging to solve the above optimization problem since both the objective
and constraints are nonconvex. To overcome this difficulty, we minimize instead a convex loss
function `? (x, y), which is classifier dependent [6], and approximate the group benefits using a ramp
(convex) function r(x) = max(0, x), i.e.,
P
P
1
minimize
(x,y,z)2D `?z (x, y) +
z2Z z ?(?z )
N
{?z }
(9)
P
P
T
0T
subject to
x2Dz max(0, ?z x)
x2Dz max(0, ?z x) for all z 2 Z,
which, for any convex regularizer ?(?), is a disciplined convex-concave program (DCCP) and thus
can be efficiently solved using well-known heuristics [30]. For example, if we particularize the above
formulation to group-conditional (standard) logistic regression classifiers ?z0 and ?z and L2 -norm
regularizer, then, Eq. 9 adopts the following form:
P
P
1
2
minimize
(x,y,z)2D log p(y|x, ?z ) +
z2Z z ||?z ||
N
{?z }
(10)
P
P
T
0T
subject to
x2Dz max(0, ?z x)
x2Dz max(0, ? z x) for all z 2 Z.
where p(y = 1|x, ?z ) =
1
1+e
Tx
?z
.
The constraints can similarly be added to other convex boundary-based classifiers like linear SVM.
We further expand on particularizing the constraints for non-linear SVM in Appendix A.
Preferred treatment classifiers. Similarly as in the case of preferred impact classifiers, one could
think of finding the decision boundary parameters {?z }z2Z of a preferred treatment classifier that
maximizes utility by solving the following optimization problem:
P
1
T
minimize
(x,y,z)2D I{sign(?z x) = y}
N
{?z }
(11)
P
P
T
T
0
subject to
x2Dz I{sign(?z x) = 1}
x2Dz I{sign(?z 0 x) = 1} for all z, z 2 Z,
where Dz = {(xi , yi , zi ) 2 D | zi = z} denotes the set of users in the training set sharing sensitive
attribute value z, the objective uses an empirical estimate of the utility, defined by Eq. 6, and the
preferred treatment constraints, defined by Eq. 3, use empirical estimates of the group benefits, defined
by Eq. 7. Here, note that both the left and right hand side of the inequalities contain optimization
variables.
However, the objective and constraints in the above problem are also nonconvex and thus we adopt a
similar strategy as in the case of preferred impact classifiers. More specifically, we solve instead the
following tractable problem:
P
P
1
minimize
(x,y,z)2D `?z (x, y) +
z2Z z ?(?z )
N
{?z }
(12)
P
P
T
T
0
subject to
x2Dz max(0, ?z x)
x2Dz max(0, ?z 0 x) for all z, z 2 Z,
which, for any convex regularizer ?(?), is also a disciplined convex-concave program (DCCP) and
can be efficiently solved.
4
Evaluation
In this section, we compare the performance of preferred treatment and preferred impact classifiers
against unconstrained, treatment parity and impact parity classifiers on a variety of synthetic and
real-world datasets. More specifically, we consider the following classifiers, which we train to
maximize utility subject to the corresponding constraints:
? Uncons: an unconstrained classifier that resorts to group-conditional classifiers. It violates
treatment parity?it trains a separate classifier per sensitive attribute value group?and potentially
violates impact parity?it may lead to different benefits for different groups.
? Parity: a parity classifier that does not use the sensitive attribute group information in the decision
making, but only during the training phase, and is constrained to satisfy both treatment parity?
its decisions do not change based on the users? sensitive attribute value as it does not resort to
group-conditional classifiers?and impact parity?it ensures that the benefits for all groups are
the same. We train this classifier using the methodology proposed by Zafar et al. [33].
? Preferred treatment: a classifier that resorts to group-conditional classifiers and is constrained
to satisfy preferred treatment?each group gets the highest benefit with its own classifier than
any other group?s classifier.
6
Acc : 0.87
Acc : 0.57
Acc : 0.76
Acc : 0.73
B0 : 0.16; B1 : 0.77
B0 : 0.20; B1 : 0.85
B0 : 0.51; B1 : 0.49
B0 : 0.58; B1 : 0.96
B0 : 0.21; B1 : 0.86
B0 : 0.58; B1 : 0.96
B0 : 0.43; B1 : 0.97
(a) Uncons
(b) Parity
(c) Preferred impact
(d) Preferred both
Figure 2: [Synthetic data] Crosses denote group-0 (points with z = 0) and circles denote group-1.
Green points belong to the positive class in the training data whereas red points belong to the negative
class. Each panel shows the accuracy of the decision making scenario along with group benefits (B0
and B1 ) provided by each of the classifiers involved. For group-conditional classifiers, cyan (blue)
line denotes the decision boundary for the classifier of group-0 (group-1). Parity case (panel (b))
consists of just one classifier for both groups in order to meet the treatment parity criterion.
? Preferred impact: a classifier that resorts to group-conditional classifiers and is constrained to
be preferred over the Parity classifier.
? Preferred both: a classifier that resort to group-conditional classifiers and is constrained to satisfy
both preferred treatment and preferred impact.
For the experiments in this section, we use logistic regression classifiers with L2 -norm regularization.
We randomly split the corresponding dataset into 70%-30% train-test folds 5 times, and report the
average accuracy and group benefits in the test folds. Appendix B describes the details for selecting
the optimal L2 -norm regularization parameters. Here, we compute utility (U ) as the overall accuracy
of a classifier and group benefits (Bz ) as the fraction of users sharing sensitive attribute z that are
classified into the positive class. Moreover, the sensitive attribute is always binary, i.e., z 2 {0, 1}.
4.1 Experiments on synthetic data
Experimental setup. Following Zafar et al. [33], we generate a synthetic dataset in which the unconstrained classifier (Uncons) offers different benefits to each sensitive attribute group. In particular, we
generate 20,000 binary class labels y 2 { 1, 1} uniformly at random along with their corresponding
two-dimensional feature vectors sampled from the following Gaussian distributions: p(x|y = 1) =
N ([2; 2], [5, 1; 1, 5]) and p(x|y = 1) = N ([ 2; 2], [10, 1; 1, 3]). Then, we generate each sensitive attribute from the Bernoulli distribution p(z = 1) = p(x0 |y = 1)/(p(x0 |y = 1)+p(x0 |y = 1)),
where x0 is a rotated version of x, i.e., x0 = [cos(?/8), sin(?/8); sin(?/8), cos(?/8)]. Finally,
we train the five classifiers described above and compute their overall (test) accuracy and (test) group
benefits.
Results. Figure 2 shows the trained classifiers, along with their overall accuracy and group benefits.
We can make several interesting observations:
The Uncons classifier leads to an accuracy of 0.87, however, the group-conditional boundaries and
high disparity in treatment for the two groups (0.16 vs. 0.85) mean that it satisfies neither treatment
parity nor impact parity. Moreover, it leads to only a small violation of preferred treatment?benefits
for group-0 would increase slightly from 0.16 to 0.20 by adopting the classifier of group-1. However,
this will not always be the case, as we will later show in the experiments on real data.
The Parity classifier satisfies both treatment and impact parity, however, it does so at a large cost in
terms of accuracy, which drops from 0.87 for Uncons to 0.57 for Parity.
The Preferred treatment classifier (not shown in the figure), leads to a minor change in decision
boundaries as compared to the Uncons classifier to achieve preferred treatment. Benefits for group-0
(group-1) with its own classifier are 0.20 (0.84) as compared to 0.17 (0.83) while using the classifier
of group-1 (group-0). The accuracy of this classifier is 0.87.
The Preferred impact classifier, by making use of a looser notion of fairness compared to impact
parity, provides higher benefits for both groups at a much smaller cost in terms of accuracy than the
Parity classifier (0.76 vs. 0.57). Note that, while the Parity classifier achieved equality in benefits by
misclassifying negative examples from group-0 into the positive class and misclassifying positive
7
B0(?0)
B0(?1)
B1(?1)
B1(?0)
Acc
0.7
0.8
0.6
0.6
0.5
0.4
Uncons.
Parity
Prf-treat.
Prf-imp.
Prf-both
Accuracy
Benefits
ProPublica COMPAS dataset
1
0.4
Adult dataset
0.85
0.84
0.2
0.83
0.82
0
Uncons.
Parity
Prf-treat.
Prf-imp.
Prf-both
Accuracy
Benefits
0.4
0.81
0.8
0.7
0.6
0.5
Uncons.
Parity
Prf-treat.
Prf-imp.
Accuracy
Benefits
NYPD SQF dataset
1
0.8
0.6
0.4
0.2
0
Prf-both
Figure 3: The figure shows the accuracy and benefits received by the two groups for various decision
making scenarios. ?Prf-treat.?, ?Prf-imp.?, and ?Prf-both? respectively correspond to the classifiers
satisfying preferred treatment, preferred impact, and both preferred treatment and impact criteria.
Sensitive attribute values 0 and 1 denote blacks and whites in ProPublica COMPAS dataset and
NYPD SQF datasets, and women and men in the Adult dataset. Bi (?j ) denotes the benefits obtained
by group i when using the classifier of group j. For the Parity case, we train just one classifier for
both the groups, so the benefits do not change by adopting other group?s classifier.
examples from group-1 into the negative class, the Preferred impact classifier only incurs the former
type of misclassifications. However, the outcomes of the Preferred impact classifier do not satisfy the
preferred treatment criterion: group-1 would attain higher benefit if it used the classifier of group-0
(0.96 as compared to 0.86).
Finally, the classifier that satisfies preferred treatment and preferred impact (Preferred both) achieves
an accuracy and benefits at par with the Preferred impact classifier.
We present the results of applying our fairness constraints on a non linearly-separable dataset with a
SVM classifier with a radial basis function (RBF) kernel in Appendix C.
4.2
Experiments on real data
Dataset description and experimental setup. We experiment with three real-world datasets: the
COMPAS recidivism prediction dataset compiled by ProPublica [23], the Adult income dataset from
UCI machine learning repository [2], and the New York Police Department (NYPD) Stop-questionand-frisk (SQF) dataset made publicly available by NYPD [1]. These datasets have been used by a
number of prior studies in the fairness-aware machine learning literature [14, 29, 32, 34, 33].
In the COMPAS dataset, the classification task is to predict whether a criminal defendant would
recidivate within two years (negative class) or not (positive class); in the Adult dataset, the task
is to predict whether a person earns more than 50K USD per year (positive class) or not; and, in
the SQF dataset, the task is to predict whether a pedestrian should be stopped on the suspicion
of having an illegal weapon or not (positive class). In all datasets, we assume being classified as
positive to be the beneficial outcome. Additionally, we divide the subjects in each dataset into two
sensitive attribute value groups: women (group-0) and men (group-1) in the Adult dataset and blacks
(group-0) and whites (group-1) in the COMPAS and SQF datasets. The supplementary material
8
(Appendix D) contains more information on the sensitive and the non-sensitive features as well as the
class distributions.3
Results. Figure 3 shows the accuracy achieved by the five classifiers described above along with the
benefits they provide for the three datasets. We can draw several interesting observations:4
In all cases, the Uncons classifier, in addition to violating treatment parity (a separate classifier for
each group) and impact parity (high disparity in group benefits), also violates the preferred treatment
criterion (in all cases, at least one of group-0 or group-1 would benefit more by adopting the other
group?s classifier). On the other hand, the Parity classifier satisfies the treatment parity and impact
parity but it does so at a large cost in terms of accuracy.
The Preferred treatment classifier provides a much higher accuracy than the Parity classifier?its
accuracy is at par with that of the Uncons classifier?while satisfying the preferred treatment criterion.
However, it does not meet the preferred impact criterion. The Preferred impact classifier meets the
preferred impact criterion but does not always satisfy preferred treatment. Moreover, it also leads to a
better accuracy then Parity classifier in all cases. However, the gain in accuracy is more substantial
for the SQF datasets as compared to the COMPAS and Adult dataset.
The classifier satisfying preferred treatment and preferred impact (Preferred both) has a somewhat
underwhelming performance in terms of accuracy for the Adult dataset. While the performance of
this classifier is better than the Parity classifier in the COMPAS dataset and NYPD SQF dataset, it is
slightly worse for the Adult dataset.
In summary, the above results show that ensuring either preferred treatment or preferred impact is
less costly in terms of accuracy loss than ensuring parity-based fairness, however, ensuring both
preferred treatment and preferred impact can lead to comparatively larger accuracy loss in certain
datasets. We hypothesize that this loss in accuracy may be partly due to splitting the number of
available samples into groups during training?each group-conditional classifier use only samples
from the corresponding sensitive attribute group?hence decreasing the effectiveness of empirical
risk minimization.
5
Conclusion
In this paper, we introduced two preference-based notions of fairness?preferred treatment and
preferred impact?establishing a previously unexplored connection between fairness-aware machine
learning and the economics and game theoretic concepts of envy-freeness and bargaining. Then,
we proposed tractable proxies to design boundary-based classifiers satisfying these fairness notions
and experimented with a variety of synthetic and real-world datasets, showing that preference-based
fairness often allows for greater decision accuracy than existing parity-based fairness notions.
Our work opens many promising venues for future work. For example, our methodology is limited
to convex boundary-based classifiers. A natural follow up would be to extend our methodology to
other types of classifiers, e.g., neural networks and decision trees. In this work, we defined preferred
treatment and preferred impact in the context of group preferences, however, it would be worth
revisiting the proposed definitions in the context of individual preferences. The fair division literature
establishes a variety of fairness axioms [26] such as Pareto-optimality and scale invariance. It would
be interesting to study such axioms in the context of fairness-aware machine learning.
Finally, we note that while moving from parity to preference-based fairness offers many attractive
properties, we acknowledge it may not always be the most appropriate notion, e.g., in some scenarios,
parity-based fairness may very well present the eventual goal and be more desirable [3].
Acknowledgments
AW acknowledges support by the Alan Turing Institute under EPSRC grant EP/N510129/1, and by
the Leverhulme Trust via the CFI.
3
Since the SQF dataset is highly skewed in terms of class distribution (?97% points in the positive class) resulting in a trained classifier
predicting all points as positive (yet having 97% accuracy), we subsample the dataset to have equal class distribution. Another option would be
using penalties proportional to the size of the class, but we observe that an unconstrained classifier with class penalties gives similar predictions
as compared to a balanced dataset. We decided to experiment with the balanced dataset since the accuracy drops in this dataset are easier to
interpret.
4
The unfairness in the SQF dataset is different from what one would expect [27]?an unconstrained classifier gives more benefits to blacks
as compared to whites. This is due to the fact that a larger fraction of stopped whites were found to be in possession on an illegal weapon
(Tables 3 and 4 in Appendix D).
9
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6,221 | 6,628 | Inferring Generative Model Structure
with Static Analysis
Paroma Varma1 , Bryan He2 , Payal Bajaj2 ,
Nishith Khandwala2 , Imon Banerjee3 , Daniel Rubin3,4 , Christopher R?2
1
Electrical Engineering, 2 Computer Science, 3 Biomedical Data Science, 4 Radiology
Stanford University
{paroma,bryanhe,pabajaj,nishith,imonb,rubin}@stanford.edu,
[email protected]
Abstract
Obtaining enough labeled data to robustly train complex discriminative models is a
major bottleneck in the machine learning pipeline. A popular solution is combining
multiple sources of weak supervision using generative models. The structure of these
models affects the quality of the training labels, but is difficult to learn without any
ground truth labels. We instead rely on weak supervision sources having some structure by virtue of being encoded programmatically. We present Coral, a paradigm
that infers generative model structure by statically analyzing the code for these
heuristics, thus significantly reducing the amount of data required to learn structure.
We prove that Coral?s sample complexity scales quasilinearly with the number of
heuristics and number of relations identified, improving over the standard sample
complexity, which is exponential in n for learning nth degree relations. Empirically,
Coral matches or outperforms traditional structure learning approaches by up to
3.81 F1 points. Using Coral to model dependencies instead of assuming independence results in better performance than a fully supervised model by 3.07 accuracy
points when heuristics are used to label radiology data without ground truth labels.
1
Introduction
Complex discriminative models like deep neural networks rely on a large amount of labeled training
data for their success. For many real-world applications, obtaining this magnitude of labeled
data is one of the most expensive and time consuming aspects of the machine learning pipeline.
Recently, generative models have been used to create training labels from various weak supervision
sources, such as heuristics or knowledge bases, by modeling the true class label as a latent variable
[1, 2, 27, 31, 36, 37]. After the necessary parameters for the generative models are learned using
unlabeled data, the distribution over the true labels can be inferred. Properly specifying the structure
of these generative models is essential in estimating the accuracy of the supervision sources. While
traditional structure learning approaches have focused on the supervised case [23, 28, 41], previous
works related to weak supervision assume that the structure is user-specified [1, 27, 31, 36]. Recently,
Bach et al. [2] showed that it is possible to learn the structure of these models with a sample complexity
that scales sublinearly with the number of possible binary dependencies. However, the sample
complexity scales exponentially for higher degree dependencies, limiting its ability to learn complex
dependency structures. Moreover, the time required to learn the dependencies also grows exponentially
with the degree of dependencies, hindering the development of user-defined heuristics.
This poses a problem in many domains where high degree dependencies are common among heuristics
that operate over a shared set of inputs. These inputs are interpretable characteristics extracted from the
data. For example, various approaches in computer vision use predicted bounding box or segmentation
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
attributes [18, 19, 29], like location and size, to weakly supervise more complex image-based learning
tasks [5, 7, 11, 26, 38]. Another example comes from the medical imaging domain, where attributes
include characteristics such as the area, intensity and perimeter of a tumor, as shown in Figure 1. Note
that these attributes are computationally represented, and the heuristics written over them are encoded
programmatically as well. There are typically a relatively small set of interpretable characteristics, so
the heuristics often share these attributes. This results in high order dependency structures among these
sources, which are crucial to model in the generative model that learns accuracies for these sources.
To address the issue of learning higher order dependencies efficiently, we present Coral, a paradigm
that statically analyzes the source code of the weak supervision sources to infer, rather than learn,
the complex relations among heuristics. Coral?s sample complexity scales quasilinearly with the
number of relevant dependencies and does not scale with the degree of the dependency, unlike the
sample complexity for Bach et al. [2], which scales exponentially with the degree of the dependency.
Moreover, the time to identify these relations is constant in the degree of dependencies, since it only
requires looking at the source code for each heuristic to find which heuristics share the same input.
This allows Coral to infer high degree dependencies more efficiently than techniques that rely only
on statistical methods to learn them, and thus generate a more accurate dependency structure for the
heuristics. Coral then uses a generative model to learn the proper weights for this dependency structure
to assign probabilistic labels to training data.
We experimentally validate the performance of Coral across various domains and show it outperforms
traditional structure learning under various conditions while being significantly more computationally
efficient. We show how modeling dependencies leads to an improvement of 3.81 F1 points compared
to standard structure learning approaches. Additionally, we show that Coral can assign labels to data
that have no ground truth labels, and this augmented training set results in improving the discriminative
model performance by 3.07 points. For a complex relation-based image classification task, 6 heuristic
functions written over only bounding box attributes as primitives are able to train a model that
performs within 0.74 points of the F1 score achieved by a fully-supervised model trained on the rich,
hand-labeled attribute and relation information in the Visual Genome database [21].
2
The Coral Paradigm
The Coral paradigm takes as input a set of domain-specific primitives and a set of programmatic
user-defined heuristic functions that operate over the primitives. We formally define these abstractions
in Section 2.1. Coral runs static analysis on the source code that defines the primitives and the heuristic
functions to identify which sets of heuristics are related by virtue of sharing primitives (Section 2.2).
Once Coral identifies these dependencies, it uses a factor graph to model the relationship between
the heuristics, primitives and the true class label. We describe the conditions under which Coral can
learn the structure of the generative model with significantly less data than traditional approaches
in Section 2.3 and demonstrate how this affects generative model accuracy via simulations. Finally,
we discuss how Coral learns the accuracies of the each heuristic and outputs probabilistic labels for
the training data in Section 2.4.
Raw Images
Segmentations
Heuristic
Functions
Noisy
Labels
p1: Area
?1(p1)
Aggressive
p2: Perimeter
?2(p2)
NonAggressive
p3: Intensity
?3(p2, p3)
Aggressive
Domain Specific
Primitives
Primitives and Heuristics are
User-Defined
Training
Labels
Generative
Model
75%
75%
Aggressive
Aggressive
Discriminative
Model
The Coral Pipeline
Figure 1: Running example for the Coral paradigm. Users apply standard algorithms to segment tumors
from the X-ray and extract the domain-specific primitives from the image and segmentation. They
write heuristic functions over the primitives that output a noisy label for each image. The generative
model takes these as inputs and provides probabilistic training labels for the discriminative model.
2
2.1
Coral Abstractions
Domain-Specific Primitives Domain-specific primitives (DSPs) in Coral are the simplest elements
that heuristic functions take as input and operate over. DSPs in Coral have semantic meaning, making
them interpretable for users. This is akin to the concept of language primitives in programming
languages, in which they are the smallest unit of processing with meaning. The motivation for making
the DSPs domain-specific instead of a general construct for the various data modalities is to allow users
to take advantage of existing work in their field to extract meaningful characteristics from the raw data.
Figure 1 shows an example of a pipeline for bone tumor classification as aggressive or non-aggressive,
inspired by one of our real experiments. First, an automated segmentation algorithm is used to generate
a binary mask for where the tumor is [20, 25, 34, 39]. Then, we define 3 DSPs based on the segmentation:
area (p1 ), perimeter (p2 ) and total intensity (p3 ) of the segmented area. More complex characteristics
such as those that capture texture, shape and edge features can also be used [4, 14, 22] (see Appendix).
We now define a formal construct for how DSPs are encoded programmatically. Users generate
DSPs in Coral through a primitive specifier function, such as create_primitives in Figure 2(a).
Specifically, this function takes as input a single unlabeled data point (and necessary intermediate
representations such as the segmentation) and returns an instance of PrimitiveSet, which maps
primitive names to primitive values, like integers (we refer to a specific instance of this class as P).
Note that P.ratio is composed of two other primitives, while the rest of the primitives are generated
independently from the image and segmentation.
def create_primitives(image,segmentation):
P = PrimitiveSet()
P.area = get_area(segmentation)
P.perimeter = get_perimeter(segmentation)
P.intensity = np.sum(segmentation*image)
P.perimeter
get_perimeter()
value
function
P.area
P.intensity
P.ratio
/
value
value
operator
P.perimeter
P.ratio = P.intensity/P.perimeter
return P
P.intensity
/
P.ratio
P.perimeter
value
(b)
(a)
(c)
Figure 2: (a) The create_primitives function that generates primitives. (b) Part of the AST for the
create_primitives function. (c) The composition structure that results from traversing the AST.
Heuristic Functions In Coral, heuristic functions (HFs) can be viewed as mapping a subset of the
DSPs to a noisy label for the training data, as shown in Figure 1. In our experience with user-defined
HFs, we observe that HFs are usually nested if-then statements in which each statement checks whether
the value of a single primitive or a combination of them are above or below a user-set threshold (see
Appendix). As shown in Figure 3(a), they take fields of the object P as input and return a label (or
abstain) based on the value of the input primitives. While our running example focuses on a single
data point for DSP generation and HF construction, both procedures are applied to the entire training
set to assign a set of noisy labels from each HF to each data point.
2.2
Static Dependency Analysis
Since the number of DSPs in some domains can be relatively small, multiple HFs can operate over
the same DSPs. HFs that share at least one primitive are trivially related to each other. Prior work
[2] learns these dependencies using the labels HFs assign to data points and its probability of success
scales with the amount of data available. However, only pairwise HF dependencies can be learned
efficiently, since the data required grows exponentially with the degree of the HF relation. This in
turn limits the complexity of the dependency structure this method can accurately learn and model.
Heuristic Function Inputs Coral takes advantage of the fact that users write HFs over a known,
finite set of primitives. It infers dependencies that exist among HFs by simply looking at the source
code of how the DSPs and HFs are constructed. This process requires no data to successfully learn
the dependencies, making it more computationally efficient than standard approaches. In order to
determine whether any set of HFs share at least one DSP, Coral looks at the input for each HF. Since
the HFs only take as input the DSP they operate over, simply grouping HFs by the primitives they
share is an efficient approach for recognizing these dependencies.
3
As shown in our running example, this would result in Coral not recognizing any dependencies among
the HFs since the input for all 3 HFs are different (Figure 3(a)). This, however, would be incorrect,
since the primitive P.ratio is composed of P.perimeter and P.intensity, which makes ?2
and ?3 related. Therefore, along with looking at the primitives that each HF takes as input, it is also
essential to model how these primitives are composed.
Primitive Compositions We use our running example in Figure 2 to explain how Coral gathers
information about DSP compositions. Coral builds an abstract syntax tree (AST) to represent the
computations the create_primitives function performs. An AST represents operations involving
the primitives as a tree, as shown in Figure 2(b). To find primitive compositions from the AST, Coral
first finds the expressions in the AST that add primitives to P (denoted in the AST as P.name). Then,
for each assignment expression, Coral traverses the subtree rooted at the assignment expression and
adds all other encountered primitives as a dependency for P.name. If no primitives are encountered
in the subtree, the primitive is registered as being independent of the rest. The composition structure
that results from traversing the AST is shown in Figure 2(c), where P.area, P.intensity, and
P.perimeter are independent while P.ratio is a composition.
Heuristic Function Dependency Structure With knowledge of how the DSPs are composed, we
return to our original method of looking at the inputs of the HFs. As before, we identify that ?1 and
?2 use P.area and P.perimeter, respectively. However, we now know that ?3 uses P.ratio, which
is a composition of P.intensity and P.perimeter. This implies that ?3 will be related to any HF
that takes either P.intensity, P.perimeter, or both as inputs. We proceed to build a relational
structure among the HFs and DSPs. As shown in Figure 3(b), this structure shows which independent
DSPs each HF operates over. The relational structure implicitly encodes dependency information
about the HFs ? if an edge points from one primitive to n HFs, those n HFs are in an n-way relation
by virtue of sharing that primitive. This dependency information can more formally be encoded in
a factor graph shown in Figure 3(c), which is discussed in Section 2.3. Note that we chose a particular
programmatic setup for creating DSPs and HFs to explain how static analysis can infer dependencies;
however, this process can be modified to work with other setups that encode DSPs and HFs as well.
def ?_1(P.area):
if P.area >= 2.0:
return 1
else:
return -1
def ?_2(P.perimeter):
if P.perimeter <= 12.0:
return 1
else:
return 0
def ?_3(P.ratio):
if P.ratio <= 5.0:
return 1
else:
return -1
?1
P.area
?1
???
1
p1
?2
???
2
p2
?3
???
3
p3
????
1
?2
P.perimeter
Y
????
2
????
????
3
?3
(a)
P.intensity
(b)
(c)
Figure 3: (a) shows the encoded HFs. (b) shows the HF dependency structure where DSP nodes have
an edge going to the HFs that use them as inputs (explicitly or implicitly). (c) shows the factor graph
Coral uses to model the relationship between HFs, DSPs, and latent class label Y.
2.3
Creating the Generative Model
We now describe the generative model used to predict the true class labels. The Coral model uses a
factor graph (Figure 3(c)) to model the relationship between the primitives (p ? R), heuristic functions
(? ? {?1,0,1}) and latent class label (Y ? {?1,1}). We show that by incorporating information about
how primitives are shared across HFs from static analysis, this factor graph infers all dependencies
between the heuristics that are guaranteed to be present. We also describe how Coral recovers additional
dependencies among the heuristics by studying empirical relationships between the primitives.
Modeling Heuristic Function Dependencies Now that dependencies have been inferred via static
analysis, the goal is to learn the accuracies for each HF and assign labels to training data accordingly.
4
The factor graph thus consists of two types of factors: accuracy factors ?Acc and HF factors from static
analysis ?HF .
The accuracy factors specify the accuracy of each heuristic function and are defined as
?Acc
i (Y,?i ) = Y ?i , i = 1,...,n
where n is the total number of heuristic functions.
The static analysis factors ensure that the heuristics are correctly evaluated based on the HF
dependencies found via static analysis. They ensure that a probability of zero is given to any
configuration where an HF does not have the correct value given the primitives it depends on. The
static analysis factors are defined as.
0
if ?i is valid given p1 ,...,pm
?HF
(?
,p
,...,p
)
=
, i = 1,...,n
i 1
m
i
?? otherwise
Since these factors are obtained directly from static analysis, they can be recovered with no data.
However, we note that static analysis is not sufficient to capture all dependencies required in the factor
graph to accurately model the process of generating training labels. Specifically, static analysis can
(i) pick up spurious dependencies among HFs that are not truly dependent on each other, or
(ii) miss key dependencies among HFs that exist due to dependencies among the DSPs in the HFs.
(i) can occur if some ?A takes as input DSPs pi ,pj and ?B takes as input DSPs pi ,pk , but pi always has
the same value. Although static analysis would pick up that ?A and ?B share a primitive and should
have a dependency, this may not be true if pj and pk are independent. (ii) can occur if two HFs depend
on different primitives, but these primitives happen to always have the same value. In this case, it is
impossible for static analysis to infer the dependency between the HFs if the primitives have different
names and are generated independently, as described in Section 2.2. A more realistic scenario comes
from our running example, where we would expect the area and perimeter of the tumor to be related.
To account for both cases, it is necessary to capture the possible dependencies that occur among the
DSPs to ensure that the dependencies from static analysis do not misspecify the factor graph. We
introduce a factor to account for additional dependencies among the primitives, ?DSP . There are many
possible choices for this dependency factor, but one simple choice is to model pairwise similarity
between the primitives. For binary and discrete primitives, the dependency factor with pairwise
similarity can be represented as
X
Sim
?DSP (p1 ,...,pm ) =
?Sim
ij (pi ,pj ), where ?ij (pi ,pj ) = I[pi = pj ].
i<j
The dependency factor can be generalized to continuous-valued primitives by binning the primitives
into discrete values before comparing for similarity.
Finally, with three types of factors, the probability distribution specified by the factor graph is
?
?
n
m X
m
n
X
X
X
Sim Sim ?
P (y,?1 ,...,?n ,p1 ,...,pm ) = exp? ?iAcc ?Acc
?HF
?ij
?ij
i +
i +
i=1
i=1
i=1 j=i+1
Sim
where ?Acc and ?ij
are weights that specify the strength of factors ?Acc and ?Sim
ij .
Inferring Dependencies without Data The HF factors capture all dependencies among the
heuristic functions that are not represented by the ?DSP factor. The dependencies represented by the
?DSP factor are precisely the dependencies that cannot be inferred via static analysis due to the fact
that this factor depends solely on the content of the primitives. It is therefore impossible to determine
what this factor is without data.
While assuming that we have the true ?DSP seems like a strong condition, we find that in real-world experiments, including the ?DSP factor rarely leads to improvements over the case when we only include the
?Acc and ?HF factors. In some of our experiments (see Section 3), we use bounding box location, size and
object labels as domain-specific primitives for image and video querying tasks. Since these primitives
are not correlated, modeling the primitive dependency does not lead to any improvement over just modeling HF dependencies from static analysis. Moreover, in other experiments where modeling the relation
among primitives helps, we observe relatively small benefits above what modeling HF dependencies
provides (Section 3). Therefore, even without data, it is possible to model the most important dependencies among HFs that lead to significant gains over the case in which no dependencies are modeled.
5
2.4
Generating Probabilistic Training Labels
Relative Improvement (%)
Given the probability distribution of the factor graph, our goal is to learn the proper weights ?iAcc
Sim
and ?ij
. Coral adopts structure learning approaches described in recent work [2], which learns
dependency structures in the weak supervision setting and maximizes the `1 -regularized marginal
pseudolikelihood of each primitive to learn the weights of the relevant factors.
50 Binary Dependencies
25
0
Relative Improvement over Structure Learning
50 3-ary Dependencies
50 4-ary Dependencies
25
0
250
N
500
0
n=6
n=8
n=10
25
0
250
N
500
0
0
250
N
500
Figure 4: Simulation demonstrating improved generative model accuracy with Coral compared to
structure learning [2] and Coral. Relative improvement of Coral over structure learning is plotted
against number of unlabeled data points (N ) and number of HFs (n).
To learn the weights of the generative model, we use contrastive divergence [15] as a maximum
likelihood estimation routine and maximize the marginal likelihood of the observed primitives. Gibbs
sampling is used to estimate the intractable gradients, which are then used in stochastic gradient descent.
Because the HFs are typically deterministic functions of the primitives (represented as the ?? value
of the correctness factors for invalid heuristic values), standard Gibbs sampling will not be able to
mix properly. As a result, we modify the Gibbs sampler to simultaneously sample one primitive along
with all heuristics that depend on it. Despite the fact that the true class label is latent, this process still
converges to the correct parameter values [27]. Additionally, the amount of data necessary to learn the
parameters scales quasilinearly with the number of parameters. In our case, the number of parameters
is simply the number of heuristics n and the number of relevant primitive similarity dependencies s.
We now formally state the conditions for this result, which match those of Ratner et al. [27], and give the
sample complexity of our method. First, we assume that there exists some feasible parameter set ? ? Rn
that is known to contain the parameter ?? = (?Acc , ?Sim ) that models the true distribution ? ? of the data:
??? ? ? s.t. ?? ? (p1 ,...,pm ,Y ) = ?? (p1 ,...,pm ,Y ).
(1)
?
Next, we must be able to accurately learn ? if we are provided with labeled samples of the true
distribution. Specifically, there must be an asymptotically unbiased estimator ?? that takes some set
of labeled data T independently sampled from ? ? such that for some c > 0,
? ) (2c|T |)?1 I.
Cov ?(T
(2)
Finally, we must have enough sufficiently accurate heuristics so that we have a reasonable estimate
of Y. For any two feasible models ?1 ,?2 ? ?,
h
i
c
E(p1 ,...,pm ,Y )???1 Var(p01 ,...,p0m ,Y 0 )???2 (Y 0 | p1 = p01 ,...,pm = p0m ) ?
(3)
n+s
Proposition 1. Suppose that we run stochastic gradient descent to produce estimates of the weights
??= (??Acc , ??Sim ) in a setup satisfying conditions (1), (2), and (3). Then, for any fixed error > 0, if the
number of unlabeled
N is at least ?[(n+s)log(n+s)], then our expected parameter error
h data points
i
? 2
2
?
is bounded by E k??? k ? .
The proof follows from the sample complexity of Ratner et al. [27] and appears in the Appendix. With
Sim
the weights ??iAcc and ??ij
maximizing the marginal likelihood of the observed primitives, we have
a fully specified factor graph and complete generative model, which can be used to predict the latent
class label. For each data point, we compute the label each heuristic function applies to it using the
6
values of the domain-specific primitives. Through the accuracy factors, we then estimate a distribution
for the latent class label and use these noisy labels to train a discriminative model.
We present a simulation to empirically compare our sample complexity with that of structure learning
[2]. In our simulation, we have n HFs, each with an accuracy of 75%, and explore settings in which
there exists one binary, 3-ary and 4-ary dependency among the HFs. The dependent HFs share exactly
one primitive, and the primitives themselves are independent (s = 0). We show our results in Figure
4. In the case with a binary dependency, structure learning recovers the necessary dependency with
few samples, and has similar performance to Coral. In contrast, in the second and third settings with
high-order dependencies, structure learning struggles to recover the relevant dependency, and performs
worse than Coral even as more training data is provided.
3
Experimental Results
We seek to experimentally validate the following claims about our approach. Our first claim is that HF
dependencies inferred via static analysis perform significantly better than a model that does not take
dependencies into account. Second, we compare to a structure learning approach for weak supervision
[2] and show how we outperform it over a variety of domains. Finally, we show that in case primitive
dependencies exist, Coral can learn and model those as well. We show that modeling the dependencies
between the heuristic functions and primitives can generate training sets that, in some cases, beat
fully supervised models by labeling additional unlabeled data. Our classification tasks range from
specialized medical domains to natural images and video, and we include details of the DSPs and
HFs in the Appendix. Note that while the number of HFs and DSPs is fairly low (Table 1), using static
analysis to automatically infer dependencies rather than ask users to identify them saves significant
effort since the number of possible dependencies grows exponentially with the number of HFs present.
We compare our approach to majority vote (MV), generative models that learn the accuracies of different
heuristics, specifically one that assumes the heuristics are independent (Indep) [27], and Bach et al. [2]
that learns the binary inter-heuristic dependencies (Learn Dep). We also compare to the fully supervised
(FS) case, and measure the performance of the discriminative model trained with labels generated using
the above methods. We split our approach into two parts: inferring HF dependencies using only static
analysis (HF Dep) and additionally learning primitive level dependencies (HF+DSP Dep).
Coral Performance
Figure 5: Discriminative model performance comparing HF Dep (HF dependencies from static
analysis) and HF+DSP Dep (HF and DSP dependencies) to other methods. Numbers in Appendix.
Visual Genome and ActivityNet Classification We explore how to extract complex relations in
images and videos given object labels and their bounding boxes. We used subsets of two datasets,
Visual Genome [21] and ActivityNet [9], and defined our task as finding images of ?a person biking
down a road? and finding basketball videos, respectively. For both tasks, a small set of DSPs were
shared heavily among HFs, and modeling the dependencies observed by static analysis led to a
significant improvement over the independent case. Since these dependencies involved groups of
3 or more heuristics, Coral improved significantly over structure learning as well, which was unable to
model these dependencies due to the lack of enough data. Moreover, modeling primitive dependencies
did not help since the primitives were indeed independent (Table 1). We report our results for these
tasks in terms of the F1 score (harmonic mean of the precision and recall) since there was significant
class imbalance which accuracy would not capture well.
Bone Tumor Classification We used a set of 802 labeled bone tumor X-ray images along with their
radiologist-drawn segmentations. Our task was to differentiate between aggressive and non-aggressive
7
Table 1: Heuristic Function (HF) and Domain-Specific Primitive (DSP) statistics. Discriminative
model improvement with HF+DSP Dep over other methods. *improvements shown in terms of F1
score, rest in terms of accuracy. ActivityNet model is LR using VGGNet embeddings as features.
Number of
Application
Improvement Over
Model
DSPs
HFs
Shared DSPs
MV
Indep
Learn Dep
FS
Visual Genome
ActivityNet
7
5
5
4
2
2
GoogLeNet
VGGNet+LR
7.49*
6.23*
2.90*
3.81*
2.90*
3.81*
-0.74*
-1.87*
Bone Tumor
Mammogram
17
6
7
6
0
0
LR
GoogLeNet
5.17
4.62
3.57
1.11
3.06
0
3.07
-0.64
tumors. We generated HFs that were a combination of hand-tuned rules and decision-tree generated
rules (tuned on a small held out subset of the dataset). The discriminative model utilized a set of 400
hand-tuned features (note that there is no overlap between these features and the DSPs) that encoded
various shape, texture, edge and intensity-based characteristics. Although there were no explicitly
shared primitives in this dataset, the generative model was still able to model the training labels more
accurately with knowledge of how heuristics used primitives, which affects the relative false positive
and false negative rates. Thus, the generative model significantly improved over the independent
model. Moreover, a small dataset size hindered structure learning, which gave a minimal boost over
the independent case (Table 1). When we used heuristics in Coral to label an additional 800 images
that had no ground truth labels, we beat the previous FS score by 3.07 points (Figure 5, Table 1).
Mammogram Tumor Classification We used the DDSM-CBIS [32] dataset, which consists of
1800 scanned film mammograms and associated segmentations for the tumors in the form of binary
masks. Our task was to identify whether a tumor is malignant or benign, and each heuristic only
operated over one primitive, resulting in no dependencies that static analysis could identify. In
this case, structure learning performed better than Coral when we only used static analysis to infer
dependencies (Figure 5). However, including primitive dependencies allowed us to match structure
learning, resulting in a 1.11 point improvement over the independent case (Figure 5, Table 1).
4
Related Work
As the need for labeled training data grows, a common alternative is to utilize weak supervision sources
such as distant supervision [10, 24], multi-instance learning [16, 30], and heuristics [8, 35]. Specifically
for images, weak supervision using object detection and segmentation or visual databases is a popular
technique as well (detailed discussion in Appendix). Estimating the accuracies of these sources without
access to ground truth labels is a classic problem [13]. Methods such as crowdsourcing [12, 17, 40],
boosting[3, 33], co-training [6], and learning from noisy labels are some of the popular approaches that
can combine various sources of weak supervision to assign noisy labels to data. However, Coral does
not require any labeled data to model the dependencies among the heuristics, which can be interpreted
as workers, classifiers or views for the above methods, and domain-specific primitives.
Recently, generative models have also been used to combine various sources of weak supervision
[1, 31, 36, 37]. One specific example, data programming [27], proposes using multiple sources of
weak supervision for text data in order to describe a generative model and subsequently learns the
accuracies of these sources. Coral also focuses on multiple programmatically encoded heuristics that
can weakly label data and learns their accuracies to assign labels to training data. However, Coral adds
an additional layer of domain-specific primitives in its generative model, which allows it to generalize
beyond text-based heuristics. It also infers the dependencies among the heuristics and the primitives,
rather than requiring users to specify them.
Other previous work also assume that this structure in generative models is user-specified [1, 31, 36].
However, Bach et al. [2] recently showed that it is possible to learn the dependency structure among
sources of weak supervision with a sample complexity that scales sublinearly with the number
of possible pairwise dependencies. Coral instead identifies the dependencies among the heuristic
functions by inspecting the content of the programmable functions, therefore relying on significantly
less data to learn the generative model structure. Moreover, Coral can also pick up higher-order
dependencies, for which Bach et al. [2] needs large amounts of data to detect.
8
5
Conclusion and Future Work
In this paper, we introduced Coral, a paradigm that models the dependency structure of weak
supervision heuristics and systematically combines their outputs to assign probabilistic labels to
training data. We described how Coral takes advantage of the programmatic nature of these heuristics
in order to infer dependencies among them via static analysis. Coral therefore requires a sample
complexity that is quasilinear in the number of heuristics and relations found. We showed how Coral
leads to significant improvements in discriminative model accuracy over traditional structure learning
approaches across various domains. Coral scratches the surface of the possible ways weak supervision
can borrow from the field of programming languages, especially as weak supervision sources are used to
label large magnitudes of data and need to be encoded programmatically. We look at a natural extension
of treating the process of encoding heuristics as writing functions and hope to explore the interactions
between systematic training set creation and concepts from the programming language field.
Acknowledgments We thank Shoumik Palkar, Stephen Bach, and Sen Wu for their helpful
conversations and feedback. We are grateful to Darvin Yi for his assistance with the DDSM dataset
based experiments and associated deep learning models. We acknowledge the use of the bone
tumor dataset annotated by Drs. Christopher Beaulieu and Bao Do and carefully collected over his
career by the late Henry H. Jones, M.D. (aka ?Bones Jones?). This material is based on research
sponsored by Defense Advanced Research Projects Agency (DARPA) under agreement number
FA8750-17-2-0095. We gratefully acknowledge the support of the DARPA SIMPLEX program
under No. N66001-15-C-4043, DARPA FA8750-12-2-0335 and FA8750-13-2-0039, DOE 108845,
the National Science Foundation (NSF) Graduate Research Fellowship under No. DGE-114747,
Joseph W. and Hon Mai Goodman Stanford Graduate Fellowship, National Institute of Health
(NIH) U54EB020405, the Office of Naval Research (ONR) under awards No. N000141210041
and No. N000141310129, the Moore Foundation, the Okawa Research Grant, American Family
Insurance, Accenture, Toshiba, and Intel. This research was supported in part by affiliate members
and other supporters of the Stanford DAWN project: Intel, Microsoft, Teradata, and VMware. The
U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes
notwithstanding any copyright notation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing the official policies or
endorsements, either expressed or implied, of DARPA or the U.S. Government. Any opinions, findings,
and conclusions or recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of DARPA, AFRL, NSF, NIH, ONR, or the U.S. government.
9
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11
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6,222 | 6,629 | Structured Embedding Models for Grouped Data
Maja Rudolph
Columbia Univ.
[email protected]
Francisco Ruiz
Univ. of Cambridge
Columbia Univ.
Susan Athey
Stanford Univ.
David Blei
Columbia Univ.
Abstract
Word embeddings are a powerful approach for analyzing language, and exponential
family embeddings (EFE) extend them to other types of data. Here we develop
structured exponential family embeddings (S - EFE), a method for discovering
embeddings that vary across related groups of data. We study how the word
usage of U.S. Congressional speeches varies across states and party affiliation,
how words are used differently across sections of the ArXiv, and how the copurchase patterns of groceries can vary across seasons. Key to the success of our
method is that the groups share statistical information. We develop two sharing
strategies: hierarchical modeling and amortization. We demonstrate the benefits
of this approach in empirical studies of speeches, abstracts, and shopping baskets.
We show how S - EFE enables group-specific interpretation of word usage, and
outperforms EFE in predicting held-out data.
1
Introduction
Word embeddings (Bengio et al., 2003; Mikolov et al., 2013d,c,a; Pennington et al., 2014; Levy
and Goldberg, 2014; Arora et al., 2015) are unsupervised learning methods for capturing latent
semantic structure in language. Word embedding methods analyze text data to learn distributed
representations of the vocabulary that capture its co-occurrence statistics. These representations are
useful for reasoning about word usage and meaning (Harris, 1954; Rumelhart et al., 1986). Word
embeddings have also been extended to data beyond text (Barkan and Koenigstein, 2016; Rudolph
et al., 2016), such as items in a grocery store or neurons in the brain. Exponential family embeddings
(EFE) is a probabilistic perspective on embeddings that encompasses many existing methods and
opens the door to bringing expressive probabilistic modeling (Bishop, 2006; Murphy, 2012) to the
problem of learning distributed representations.
We develop structured exponential family embeddings (S - EFE), an extension of EFE for studying
how embeddings can vary across groups of related data. We will study several examples: in U.S.
Congressional speeches, word usage can vary across states or party affiliations; in scientific literature,
the usage patterns of technical terms can vary across fields; in supermarket shopping data, co-purchase
patterns of items can vary across seasons of the year. We will see that S - EFE discovers a per-group
embedding representation of objects. While the na?ve approach of fitting an individual embedding
model for each group would typically suffer from lack of data?especially in groups for which fewer
observations are available?we develop two methods that can share information across groups.
Figure 1a illustrates the kind of variation that we can capture. We fit an S - EFE to ArXiv abstracts
grouped into different sections, such as computer science (cs), quantitative finance (q-fin), and
nonlinear sciences (nlin). S- EFE results in a per-section embedding of each term in the vocabulary.
Using the fitted embeddings, we illustrate similar words to the word INTELLIGENCE. We can see
that how INTELLIGENCE is used varies by field: in computer science the most similar words include
ARTIFICIAL and AI ; in finance, similar words include ABILITIES and CONSCIOUSNESS .
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(s)
?v
?v
V
X (s)
S
hierarchical:
amortized:
(a) S - EFE uncover variations in the usage of the word INTELLIGENCE.
(s)
(0)
?v ? N (?v , ??2 I)
(s)
(0)
?v = fs (?v )
(b) Graphical repres. of S - EFE.
Figure 1: (a) INTELLIGENCE is used differently across the ArXiv sections. Words with the closest
embedding to the query are listed for 5 sections. (The embeddings were obtained by fitting an
amortized S - EFE.) The method automatically orders the sections along the horizontal axis by their
similarity in the usage of INTELLIGENCE. See Section 3 additional for details. (b) Graphical
(s)
representation of S - EFE for data in S categories. The embedding vectors ?v are specific to each
group, and the context vectors ?v are shared across all categories.
In more detail, embedding methods posit two representation vectors for each term in the vocabulary;
an embedding vector and a context vector. (We use the language of text for concreteness; as we
mentioned, EFE extend to other types of data.) The idea is that the conditional probability of each
observed word depends on the interaction between the embedding vector and the context vectors of
the surrounding words. In S - EFE, we posit a separate set of embedding vectors for each group but a
shared set of context vectors; this ensures that the embedding vectors are in the same space.
We propose two methods to share statistical strength among the embedding vectors. The first
approach is based on hierarchical modeling (Gelman et al., 2003), which assumes that the groupspecific embedding representations are tied through a global embedding. The second approach is
based on amortization (Dayan et al., 1995; Gershman and Goodman, 2014), which considers that the
individual embeddings are the output of a deterministic function of a global embedding representation.
We use stochastic optimization to fit large data sets.
Our work relates closely to two threads of research in the embedding literature. One is embedding
methods that study how language evolves over time (Kim et al., 2014; Kulkarni et al., 2015; Hamilton
et al., 2016; Rudolph and Blei, 2017; Bamler and Mandt, 2017; Yao et al., 2017). Time can be thought
of as a type of ?group?, though with evolutionary structure that we do not consider. The second
thread is multilingual embeddings (Klementiev et al., 2012; Mikolov et al., 2013b; Ammar et al.,
2016; Zou et al., 2013); our approach is different in that most words appear in all groups and we are
interested in the variations of the embeddings across those groups.
Our contributions are thus as follows. We introduce the S - EFE model, extending EFE to grouped
data. We present two techniques to share statistical strength among the embedding vectors, one based
on hierarchical modeling and one based on amortization. We carry out a thorough experimental study
on two text databases, ArXiv papers by section and U.S. Congressional speeches by home state and
political party. Using Poisson embeddings, we study market basket data from a large grocery store,
grouped by season. On all three data sets, S - EFE outperforms EFE in terms of held-out log-likelihood.
Qualitatively, we demonstrate how S - EFE discovers which words are used most differently across U.S.
states and political parties, and show how word usage changes in different ArXiv disciplines.
2
Model Description
In this section, we develop structured exponential family embeddings (S - EFE), a model that builds
on exponential family embeddings (EFE) (Rudolph et al., 2016) to capture semantic variations across
groups of data. In embedding models, we represent each object (e.g., a word in text, or an item in
shopping data) using two sets of vectors, an embedding vector and a context vector. In this paper, we
2
are interested in how the embeddings vary across groups of data, and for each object we want to learn
a separate embedding vector for each group. Having a separate embedding for each group allows
us to study how the usage of a word like INTELLIGENCE varies across categories of the ArXiv, or
which words are used most differently by U.S. Senators depending on which state they are from and
whether they are Democrats or Republicans.
The S - EFE model extends EFE to grouped data, by having the embedding vectors be specific for each
group, while sharing the context vectors across all groups. We review the EFE model in Section 2.1.
We then formalize the idea of sharing the context vectors in Section 2.2, where we present two
approaches to build a hierarchical structure over the group-specific embeddings.
2.1
Background: Exponential Family Embeddings
In exponential family embeddings, we have a collection of objects, and our goal is to learn a vector
representation of these objects based on their co-occurrence patterns.
Let us consider a dataset represented as a (typically sparse) matrix X, where columns are datapoints
and rows are objects. For example, in text, each column corresponds to a location in the text, and
each entry xvi is a binary variable that indicates whether word v appears at location i.
In EFE, we represent each object v with two sets of vectors, embeddings vectors ?v [i] and context
vectors ?v [i], and we posit a probability distribution of data entries xvi in which these vectors interact.
The definition of the EFE model requires three ingredients: a context, a conditional exponential
family, and a parameter sharing structure. We next describe these three components.
Exponential family embeddings learn the vector representation of objects based on the conditional
probability of each observation, conditioned on the observations in its context. The context cvi gives
the indices of the observations that appear in the conditional probability distribution of xvi . The
definition of the context varies across applications. In text, it corresponds to the set of words in a
fixed-size window centered at location i.
Given the context cvi and the corresponding observations xcvi indexed by cvi , the distribution for xvi
is in the exponential family,
xvi | xcvi ? ExpFam (t(xvi ), ?v (xcvi )) ,
(1)
with sufficient statistics t(xvi ) and natural parameter ?v (xcvi ). The parameter vectors interact in the
conditional probability distributions of each observation xvi as follows. The embedding vectors ?v [i]
and the context vectors ?v [i] are combined to form the natural parameter,
?
?
X
?v (xcvi ) = g ??v [i]>
?v0 [i0 ]xv0 i0 ? ,
(2)
(v 0 ,i0 )?cvi
where g(?) is the link function. Exponential family embeddings can be understood as a bank of
generalized linear models (GLMs). The context vectors are combined to give the covariates, and the
?regression coefficients? are the embedding vectors. In Eq. 2, the link function g(?) plays the same
role as in GLMs and is a modeling choice. We use the identity link function.
The third ingredient of the EFE model is the parameter sharing structure, which indicates how the
embedding vectors are shared across observations. In the standard EFE model, we use ?v [i] ? ?v and
?v [i] ? ?v for all columns of X. That is, each unique object v has a shared representation across all
instances.
The objective function. In EFE, we maximize the objective function, which is given by the sum of
the log-conditional likelihoods in Eq. 1. In addition, we add an `2 -regularization term (we use the
notation of the log Gaussian pdf) over the embedding and context vectors, yielding
X
L = log p(?) + log p(?) +
log p xvi xcvi ; ?, ? ,
(3)
v,i
Note that maximizing the regularized conditional likelihood is not equivalent to maximum a posteriori.
Rather, it is similar to maximization of the pseudo-likelihood in conditionally specified models
(Arnold et al., 2001; Rudolph et al., 2016).
3
2.2
Structured Exponential Family Embeddings
Here, we describe the S - EFE model for grouped data. In text, some examples of grouped data are
Congressional speeches grouped into political parties or scientific documents grouped by discipline.
Our goal is to learn group-specific embeddings from data partitioned into S groups, i.e., each instance
i is associated with a group si ? {1, . . . , S}. The S - EFE model extends EFE to learn a separate set
of embedding vectors for each group.
To build the S - EFE model, we impose a particular parameter sharing structure over the set of
embedding and context vectors. We posit a structured model in which the context vectors are shared
across groups, i.e., ?v [i] ? ?v (as in the standard EFE model), but the embedding vectors are only
(s )
(s)
shared at the group level, i.e., for an observation i belonging to group si , ?v [i] ? ?v i . Here, ?v
denotes the embedding vector corresponding to group s. We show a graphical representation of the
S - EFE in Figure 1b.
Sharing the context vectors ?v has two advantages. First, the shared structure reduces the number of
parameters, while the resulting S - EFE model is still flexible to capture how differently words are
(s)
used across different groups, as ?v is allowed to vary.1 Second, it has the important effect of uniting
(s)
all embedding parameters in the same space, as the group-specific vectors ?v need to agree with the
components of ?v . While one could learn a separate embedding model for each group, as has been
done for text grouped into time slices (Kim et al., 2014; Kulkarni et al., 2015; Hamilton et al., 2016),
this approach would require ad-hoc postprocessing steps to align the embeddings.2
When there are S groups, the S - EFE model has S times as many embedding vectors than the standard
embedding model. This may complicate inferences about the group-specific vectors, especially for
groups with less data. Additionally, an object v may appear with very low frequency in a particular
group. Thus, the na?ve approach for building the S - EFE model without additional structure may be
detrimental for the quality of the embeddings, especially for small-sized groups. To address this
(s)
problem, we propose two different methods to tie the individual ?v together, sharing statistical
strength among them. The first approach consists in a hierarchical embedding structure. The second
approach is based on amortization. In both methods, we introduce a set of global embedding vectors
(0)
(s)
(0)
?v , and impose a particular structure to generate ?v from ?v .
Hierarchical embedding structure. Here, we impose a hierarchical structure that allows sharing
(s)
(0)
statistical strength among the per-group variables. For that, we assume that each ?v ? N (?v , ??2 I),
2
where ?? is a fixed hyperparameter. Thus, we replace the EFE objective function in Eq. 3 with
X
X
Lhier = log p(?) + log p(?(0) ) +
log p(?(s) | ?(0) ) +
log p xvi xcvi ; ?, ? .
(4)
s
v,i
(0)
where the `2 -regularization term now applies only on ?v and the global vectors ?v .
(0)
(s)
Fitting the hierarchical model involves maximizing Eq. 4 with respect to ?v , ?v , and ?v . We
note that we have not reduced the number of parameters to be inferred; rather, we tie them together
through a common prior distribution. We use stochastic gradient ascent to maximize Eq. 4.
Amortization. The idea of amortization has been applied in the literature to develop amortized
inference algorithms (Dayan et al., 1995; Gershman and Goodman, 2014). The main insight behind
amortization is to reuse inferences about past experiences when presented with a new task, leveraging
the accumulated knowledge to quickly solve the new problem. Here, we use amortization to control
(s)
the number of parameters of the S - EFE model. In particular, we set the per-group embeddings ?v to
(s)
(0)
be the output of a deterministic function of the global embedding vectors, ?v = fs (?v ). We use a
different function fs (?) for each group s, and we parameterize them using neural networks, similarly
to other works on amortized inference (Korattikara et al., 2015; Kingma and Welling, 2014; Rezende
et al., 2014; Mnih and Gregor, 2014). Unlike standard uses of amortized inference, in S - EFE the
(s)
1
Alternatively, we could share the embedding vectors ?v and have group-specific context vectors ?v . We
did not explore that avenue and leave it for future work.
2
Another potential advantage of the proposed parameter sharing structure is that, when the context vectors
are held fixed, the resulting objective function is convex, by the convexity properties of exponential families
(Wainwright and Jordan, 2008).
4
input to the functions fs (?) is unobserved and must be estimated together with the parameters of the
functions fs (?).
Depending on the architecture of the neural networks, the amortization can significantly reduce the
number of parameters in the model (as compared to the non-amortized model), while still having
the flexibility to model different embedding vectors for each group. The number of parameters in
the S - EFE model is KL(S + 1), where S is the number of groups, K is the dimensionality of the
embedding vectors, and L is the number of objects (e.g., the vocabulary size). With amortization,
we reduce the number of parameters to 2KL + SP , where P is the number of parameters of the
neural network. Since typically L P , this corresponds to a significant reduction in the number of
parameters, even when P scales linearly with K.
In the amortized S - EFE model, we need to introduce a new set of parameters ?(s) ? RP for each
group s, corresponding to the neural network parameters. Given these, the group-specific embedding
(s)
vectors ?v are obtained as
(0)
(0)
(s)
?(s)
).
v = fs (?v ) = f (?v ; ?
(5)
We compare two architectures for the function fs (?): fully connected feed-forward neural networks
and residual networks (He et al., 2016). For both, we consider one hidden layer with H units. Hence,
the network parameters ?(s) are two weight matrices,
(s)
?(s) = {W1
(s)
? RH?K , W2
? RK?H },
(6)
(0)
?v ,
i.e., P = 2KH parameters. The neural network takes as input the global embedding vector
(s)
(s)
and it outputs the group-specific embedding vectors ?v . The mathematical expression for ?v for a
feed-forward neural network and a residual network is respectively given by
(s)
(s) (0)
(0)
(s)
?(s)
=
f
(?
;
?
)
=
W
tanh
W
?
,
(7)
ffnet
v
v
v
2
1
(s)
(s) (0)
(0)
(s)
?(s)
) = ?(0)
,
(8)
v = fresnet (?v ; ?
v + W2 tanh W1 ?v
where we have considered the hyperbolic tangent nonlinearity. The main difference between both
network architectures is that the residual network focuses on modeling how the group-specific
(s)
(0)
embedding vectors ?v differ from the global vectors ?v . That is, if all weights were set to 0, the
(0)
feed-forward network would output 0, while the residual network would output the global vector ?v
for all groups.
The objective function under amortization is given by
X
Lamortiz = log p(?) + log p(?(0) ) +
log p xvi xcvi ; ?, ?(0) , ? .
(9)
v,i
(0)
We maximize this objective with respect to ?v , ?v , and ?(s) using stochastic gradient ascent. We
implement the hierarchical and amortized S - EFE models in TensorFlow (Abadi et al., 2015), which
allows us to leverage automatic differentiation.3
Example: structured Bernoulli embeddings for grouped text data. Here, we consider a set
of documents broken down into groups, such as political affiliations or scientific disciplines. We
can represent the data as a binary matrix X and a set of group indicators si . Since only one word
can appear in a certain position i, the matrix X contains one non-zero element per column. In
embedding models, we ignore this one-hot constraint for computational efficiency, and consider
that the observations are generated following a set of conditional Bernoulli distributions (Mikolov
et al., 2013c; Rudolph et al., 2016). Given that most of the entries in X are zero, embedding models
typically downweigh the contribution of the zeros to the objective function. Mikolov et al. (2013c)
use negative sampling, which consists in randomly choosing a subset of the zero observations. This
corresponds to a biased estimate of the gradient in a Bernoulli exponential family embedding model
(Rudolph et al., 2016).
The context cvi is given at each position i by the set of surrounding words in the document, according
to a fixed-size window.
3
Code is available at https://github.com/mariru/structured_embeddings
5
ArXiv abstracts
Senate speeches
Shopping data
data
text
text
counts
embedding of
15k terms
15k terms
5.5k items
groups
19
83
12
grouped by
subject areas
home state/party
months
size
15M words
20M words
0.5M trips
Table 1: Group structure and size of the three corpora analyzed in Section 3.
Example: structured Poisson embeddings for grouped shopping data. EFE and S - EFE extend to
applications beyond text and we use S - EFE to model supermarket purchases broken down by month.
For each market basket i, we have access to the month si in which that shopping trip happened. Now,
the rows of the data matrix X index items, while columns index shopping trips. Each element xvi
denotes the number of units of item v purchased at trip i. Unlike text, each column of X may contain
more than one non-zero element. The context cvi corresponds to the set of items purchased in trip i,
excluding v.
In this case, we use the Poisson conditional distribution, which is more appropriate for count data. In
Poisson S - EFE, we also downweigh the contribution of the zeros in the objective function, which
provides better results because it allows the inference to focus on the positive signal of the actual
purchases (Rudolph et al., 2016; Mikolov et al., 2013c).
3
Empirical Study
In this section, we describe the experimental study. We fit the S - EFE model on three datasets and
compare it against the EFE (Rudolph et al., 2016). Our quantitative results show that sharing the
context vectors provides better results, and that amortization and hierarchical structure give further
improvements.
Data. We apply the S - EFE on three datasets: ArXiv papers, U.S. Senate speeches, and purchases on
supermarket grocery shopping data. We describe these datasets below, and we provide a summary of
the datasets in Table 1.
ArXiv papers: This dataset contains the abstracts of papers published on the ArXiv under the 19
different tags between April 2007 and June 2015. We treat each tag as a group and fit S - EFE with the
goal of uncovering which words have the strongest shift in usage. We split the abstracts into training,
validation, and test sets, with proportions of 80%, 10%, and 10%, respectively.
Senate speeches: This dataset contains U.S. Senate speeches from 1994 to mid 2009. In contrast to
the ArXiv collection, it is a transcript of spoken language. We group the data into state of origin of
the speaker and his or her party affiliation. Only affiliations with the Republican and Democratic
Party are considered. As a result, there are 83 groups (Republicans from Alabama, Democrats from
Alabama, Republicans from Arkansas, etc.). Some of the state/party combinations are not available
in the data, as some of the 50 states have only had Senators with the same party affiliation. We split
the speeches into training (80%), validation (10%), and testing (10%).
Grocery shopping data: This dataset contains the purchases of 3, 206 customers. The data covers a
period of 97 weeks. After removing low-frequency items, the data contains 5, 590 unique items at the
UPC (Universal Product Code) level. We split the data into a training, test, and validation sets, with
proportions of 90%, 5%, and 5%, respectively. The training data contains 515, 867 shopping trips
and 5, 370, 623 purchases in total.
For the text corpora, we fix the vocabulary to the 15k most frequent terms and remove all words
that are not in the vocabulary.
Following Mikolov et al. (2013c), we additionally remove each word
p
with probability 1 ? 10?5 /fv , where fv is the word frequency. This downsamples especially the
frequent words and speeds up training. (Sizes reported in Table 1 are the number of words remaining
after preprocessing.)
Models. Our goal is to fit the S - EFE model on these datasets. For the text data, we use the Bernoulli
distribution as the conditional exponential family, while for the shopping data we use the Poisson
distribution, which is more appropriate for count data.
6
On each dataset, we compare four approaches based on S - EFE with two EFE (Rudolph et al., 2016)
baselines. All are fit using stochastic gradient descent (SGD) (Robbins and Monro, 1951). In
particular, we compare the following methods:
?
?
?
?
?
?
A global EFE model, which cannot capture group structure.
Separate EFE models, fitted independently on each group.
(this paper) S - EFE without hierarchical structure or amortization.
(this paper) S - EFE with hierarchical group structure.
(this paper) S - EFE, amortized with a feed-forward neural network (Eq. 7).
(this paper) S - EFE, amortized using a residual network (Eq. 8).
Experimental setup and hyperparameters. For text we set the dimension of the embeddings to
K = 100, the number of hidden units to H = 25, and we experiment with two context sizes, 2 and 8.4
In the shopping data, we use K = 50 and H = 20, and we randomly truncate the context of baskets
larger than 20 to reduce their size to 20. For both methods, we use 20 negative samples.
For all methods, we subsample minibatches of data in the same manner. Each minibatch contains
subsampled observations from all groups and each group is subsampled proportionally to its size.
For text, the words subsampled from within a group are consecutive, and for shopping data the
observations are sampled at the shopping trip level. This sampling scheme reduces the bias from
imbalanced group sizes. For text, we use a minibatch size of N/10000, where N is the size of the
corpus, and we run 5 passes over the data; for the shopping data we use N/100 and run 50 passes.
We use the default learning rate setting of TensorFlow for Adam5 (Kingma and Ba, 2015).
We use the standard initialization schemes for the
network parameters. The weights are
? neural
?
drawn from a uniform distribution bounded at ? 6/ K + H (Glorot and Bengio, 2010). For the
embeddings, we try 3 initialization schemes and choose the best one based on validation error. In
particular, these schemes are: (1) all embeddings are drawn from the Gaussian prior implied by the
regularizer; (2) the embeddings are initialized from a global embedding; (3) the context vectors are
initialized from a global embedding and held constant, while the embeddings vectors are drawn
randomly from the prior. Finally, for each method we choose the regularization variance from the set
{100, 10, 1, 0.1}, also based on validation error.
Runtime. We implemented all methods in Tensorflow. On the Senate speeches, the runtime of
S - EFE is 4.3 times slower than the runtime of global EFE, hierarchical EFE is 4.6 times slower than
the runtime of global EFE, and amortized S - EFE is 3.3 times slower than the runtime of global EFE.
(The Senate speeches have the most groups and hence the largest difference in runtime between
methods.)
Evaluation metric. We evaluate the fits by held-out pseudo (log-)likelihood. For each model,
we compute the test pseudo log-likelihood, according to the exponential family distribution used
(Bernoulli or Poisson). For each test entry, a better model will assign higher probability to the
observed word or item, and lower probability to the negative samples. This is a fair metric because the
competing methods all produce conditional likelihoods from the same exponential family.6 To make
results comparable, we train and evaluate all methods with the same number of negative samples (20).
The reported held out likelihoods give equal weight to the positive and negative samples.
Quantitative results. We show the test pseudo log-likelihood of all methods in Table 2 and report that
our method outperforms the baseline in all experiments. We find that S - EFE with either hierarchical
structure or amortization outperforms the competing methods based on standard EFE highlighted
in bold. This is because the global EFE ignores per-group variations, whereas the separate EFE
cannot share information across groups. The results of the global EFE baseline are better than fitting
separate EFE (the other baseline), but unlike the other methods the global EFE cannot be used for the
exploratory analysis of variations across groups. Our results show that using a hierarchical S - EFE
is always better than using the simple S - EFE model or fitting a separate EFE on each group. The
hierarchical structure helps, especially for the Senate speeches, where the data is divided into many
4
To save space we report results for context size 8 only. Context size 2 shows the same relative performance.
Adam needs to track a history of the gradients for each parameter that is being optimized. One advantage
from reducing the number of parameters with amortization is that it results in a reduced computational overhead
for Adam (as well as for other adaptive stepsize schedules).
6
Since we hold out chunks of consecutive words usually both a word and its context are held out. For all
methods we have to use the words in the context to compute the conditional likelihoods.
5
7
ArXiv papers
Senate speeches
Shopping data
?2.239 ? 0.002
?0.772 ? 0.000
?2.915 ? 0.004
?0.807 ? 0.002
S - EFE
?2.645 ? 0.002
?0.770 ? 0.001
S - EFE (hierarchical)
? 2.217 ? 0.001
?0.767 ? 0.000
S - EFE (amortiz+feedf)
?2.484 ? 0.002
?0.774 ? 0.000
S - EFE (amortiz+resnet)
?2.249 ? 0.002
?0.762 ? 0.000
Table 2: Test log-likelihood on the three considered datasets. S - EFE consistently achieves the
highest held-out likelihood. The competing methods are the global EFE, which can not capture group
variations, and the separate EFE, which cannot share information across groups.
Global EFE (Rudolph et al., 2016)
Separated EFE (Rudolph et al., 2016)
?2.176 ? 0.005
?2.500 ? 0.012
?2.287 ? 0.007
?2.170 ? 0.003
?2.153 ? 0.004
?2.120 ? 0.004
groups. Among the amortized S - EFE models we developed, at least amortization with residual
networks outperforms the base S - EFE. The advantage of residual networks over feed-forward neural
networks is consistent with the results reported by (He et al., 2016).
While both hierarchical S - EFE and amortized S - EFE share information about the embedding of a
(0)
particular word across groups (through the global embedding ?v ), amortization additionally ties
the embeddings of all words within a group (through learning the neural network of that group). We
hypothesize that for the Senate speeches, which are split into many groups, this is a strong modeling
constraint, while it helps for all other experiments.
Structured embeddings reveal a spectrum of word usage. We have motivated S - EFE with the
example that the usage of INTELLIGENCE varies by ArXiv category (Figure 1a). We now explain
how for each term the per-group embeddings place the groups on a spectrum. For a specific term
(s)
v we take its embeddings vectors {?v } for all groups s, and project them onto a one-dimensional
space using the first component of principal component analysis (PCA). This is a one-dimensional
summary of how close the embeddings of v are across groups. Such comparison is posible because
the S - EFE shares the context vectors, which grounds the embedding vectors in a joint space.
The spectrum for the word INTELLIGENCE along its first principal component is the horizontal
axis in Figure 1a. The dots are the projections of the group-specific embeddings for that word.
(The embeddings come from a fitted S - EFE with feed-forward amortization.) We can see that in an
unsupervised manner, the method has placed together groups related to physics on one end on the
spectrum, while computer science, statistics and math are on the other end of the spectrum.
To give additional intuition of what the usage of INTELLIGENCE is at different locations on the
spectrum, we have listed the 8 most similar words for the groups computer science (cs), quantitative
finance (q-fin), math (math), statistics (stat), and nonlinear sciences (nlin). Word similarities are
computed using cosine distance in the embedding space. Eventhough their embeddings are relatively
close to each other on the spectrum, the model has the flexibility to capture high variabilty in the lists
of similar words.
Exploring group variations with structured embeddings. The result of the S - EFE also allows us
to investigate which words have the highest deviation from their average usage for each group. For
example, in the Congressional speeches, there are many terms that we do not expect the Senators to
use differently (e.g., most stopwords). We might however want to ask a question like ?which words
do Republicans from Texas use most differently from other Senators?? By suggesting an answer,
our method can guide an exploratorydata analysis. For eachgroup s (state/party combination), we
PS
(s)
(t)
compute the top 3 words in argsortv ||?v ? S1 t=1 ?v || from within the top 1k words.
Table 3 shows a summary of our findings (the full table is in the Appendix). According to the S - EFE
(with residual network amortization), Republican Senators from Texas use BORDER and the phrase
OUR COUNTRY in different contexts than other Senators.
Some of these variations are probably influenced by term frequency, as we expect Democrats from
Washington to talk about WASHINGTON more frequently than other states. But we argue that our
method provides more insights than a frequency based analysis, as it is also sensitive to the context in
which a word appears. For example, WASHINGTON might in some groups be used more often in
8
TEXAS
border
our country
iraq
FLORIDA
medicaid
prescription
medicare
IOWA
bankruptcy
water
waste
agriculture
farmers
food
prescription
drug
drugs
WASHINGTON
washington
energy
oil
Table 3: List of the three most different words for different groups for the Congressional speeches.
S - EFE uncovers which words are used most differently by Republican Senators (red) and Democratic
Senators (blue) from different states. The complete table is in the Appendix.
the context of PRESIDENT and GEORGE, while in others it might appear in the context of DC and
CAPITAL, or it may refer to the state.
4
Discussion
We have presented several structured extensions of EFE for modeling grouped data. Hierarchical
S - EFE can capture variations in word usage across groups while sharing statistical strength between
them through a hierarchical prior. Amortization is an effective way to reduce the number of parameters
in the hierarchical model. The amortized S - EFE model leverages the expressive power of neural
networks to reduce the number of parameters, while still having the flexibility to capture variations
between the embeddings of each group. Below are practical guidelines for choosing a S - EFE.
How can I fit embeddings that vary across groups of data? To capture variations across groups,
never fit a separate embedding model for each group. We recommend at least sharing the context
vectors, as all the S - EFE models do. This ensures that the latent dimensions of the embeddings
are aligned across groups. In addition to sharing context vectors, we also recommend sharing
statistical strength between the embedding vectors. In this paper we have presented two ways to do
so, hierarchical modeling and amortization.
Should I use a hierarchical prior or amortization? The answer depends on how many groups the
data contain. In our experiments, the hierarchical S - EFE works better when there are many groups.
With less groups, the amortized S - EFE works better.
The advantage of the amortized S - EFE is that it has fewer parameters than the hierarchical model,
while still having the flexibility to capture across-group variations. The global embeddings in an
amortized S - EFE have two roles. They capture the semantic similarities of the words, and they also
serve as the input into the amortization networks. Thus, the global embeddings of words with similar
pattern of across-group variation need to be in regions of the embedding space that lead to similar
modifications by the amortization network. As the number of groups in the data increases, these two
roles become harder to balance. We hypothesize that this is why the amortized S - EFE has stronger
performance when there are fewer groups.
Should I use feed-forward or residual networks? To amortize a S - EFE we recommend residual
networks. They perform better than the feed-forward networks in all of our experiments. While the
feed-forward network has to output the entire meaning of a word in the group-specific embedding,
the residual network only needs the capacity to model how the group-specific embedding differs from
the global embedding.
Acknowledgements
We thank Elliott Ash and Suresh Naidu for the helpful discussions and for sharing the Senate speeches.
This work is supported by NSF IIS-1247664, ONR N00014-11-1-0651, DARPA PPAML FA8750-142-0009, DARPA SIMPLEX N66001-15-C-4032, the Alfred P. Sloan Foundation, and the John Simon
Guggenheim Foundation. Francisco J. R. Ruiz is supported by the EU H2020 programme (Marie
Sk?odowska-Curie grant agreement 706760).
9
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11
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6,223 | 663 | A Note on Learning Vector Quantization
Virginia R. de Sa
Dana H. Ballard
Department of Computer Science
University of Rochester
Rochester, NY 14627
Department of Computer Science
University of Rochester
Rochester, NY 14627
Abstract
Vector Quantization is useful for data compression. Competitive Learning which minimizes reconstruction error is an appropriate algorithm for
vector quantization of unlabelled data. Vector quantization of labelled
data for classification has a different objective, to minimize the number
of misclassifications, and a different algorithm is appropriate. We show
that a variant of Kohonen's LVQ2.1 algorithm can be seen as a multiclass extension of an algorithm which in a restricted 2 class case can
be proven to converge to the Bayes optimal classification boundary. We
compare the performance of the LVQ2.1 algorithm to that of a modified
version having a decreasing window and normalized step size, on a ten
class vowel classification problem.
1 Introduction
Vector quantization is a form of data compression that represents data vectors by a smaller
set of codebook vectors. Each data vector is then represented by its nearest codebook
vector. The goal of vector quantization is to represent the data with the fewest codebook
vectors while losing as little information as possible.
Vector quantization of unlabelled data seeks to minimize the reconstruction error. This can
be accomplished with Competitive learning[Grossberg, 1976; Kohonen, 1982], an iterative
learning algorithm for vector quantization that has been shown to perform gradient descent
on the following energy function [Kohonen, 1991]
J
/Ix -
220
ws?(x)
/l2p (x)dx.
A Note on Learning Vector Quantization
where p(x) is the probability distribution of the input patterns and Ws are the reference or
codebook vectors and s*(x) is defined by IIx - WSO(x) II ~ /Ix - will (for alIt). This minimizes the square reconstruction error of unlabelled data and may work reasonably well for
classification tasks if the patterns in the different classes are segregated.
In many classification tasks, however, the different member patterns may not be segregated
into separate clusters for each class. In these cases it is more important that members ofthe
same class be represented by the same codebook vector than that the reconstruction error
is minimized. To do this, the quantizer can m&ke use of the labelled data to encourage
appropriate quantization.
2
Previous approaches to Supervised Vector Quantization
The first use of labelled data (or a teaching signal) with Competitive Learning by Rumelhart
and Zipser [Rumelhart and Zipser, 1986] can be thought of as assigning a class to each
codebook vector and only allowing patterns from the appropriate class to influence each
reference vector.
This simple approach is far from optimal though as it fails to take into account interactions
between the classes. Kohonen addressed this in his LVQ( 1) algorithm[Kohonen, 1986]. He
argues that the reference vectors resulting from LVQ( 1) tend to approximate for a particular
class r,
P(xICr)P(Cr ) - ~#rP(xICs)P(Cs).
where P( Cj) is the a priori probability of Class i and P(xICj) is the conditional density of
Class i.
This approach is also not optimal for classification, as it addresses optimal places to put
the codebook vectors instead of optimal placement of the borders of the vector quantizer
which arise from the Voronoi tessellation induced by the codebook vectors. 1
3
Minimizing Misclassifications
In classification tasks the goal is to minimize the numbers of misclassifications of the
resultant quantizer. That is we want to minimize:
(1)
where, P(Classj) is the a priori probability of Classj and P(xIClassj) is the conditional
density of Classi and D.Rj is the decision region for class j (which in this case is all x such
that I~ - wkll < I~ - wjll (for all i) and Wk is a codebook vector for class j).
Consider a One-Dimensional problem of two classes and two codebook vectors wI and w2
defining a class boundary b = (wI + w2)/2 as shown in Figure 1. In this case Equation 1
reduces to:
1 Kohonen [1986] showed this by showing that the use of a "weighted" Voronoi tessellation (where
the relative distances of the borders from the reference vectors was changed) worked better. However
no principled way to calculate the relative weights was given and the application to real data used
the unweighted tessellation.
221
222
de Sa and Ballard
P(CIass i)P(xlClass i)
w2
b* b
wI
%
Figure 1: Codebook vectors Wl and'W2 define a border b. The optimal place for the border
is at b* where P(Cl)P(xICt} = P(C2)P(xIC2). The extra misclassification errors incurred by
placing the border at b is shown by the shaded region.
(2)
The derivative of Equation 2 with respect to b is
That is, the minimum number of misclassifications occurs at b* where
P(ClaSS1)P(b*IClasSl)
=P(Class2)P(b*IClass2).
If f(x) = (Classl)P(xIClassl) - P(Class2)P(xIClass2) was a regression function then we
could use stochastic approximation [Robbins and Monro, 1951] to estimate b* iteratively
as
ben + 1) = ben) + a(n)Z"
where Z" is a sample of the random variable Z whose expected value is
P(Classl)P(b(n)IClasst) - P(Class2)P(b(n)IClass2? and
lim a(n) = 0
,,-+co
l:ia(n) = 00
l:ia 2(n) < 00
However, we do not have immediate access to an appropriate random variable Z but can
express P( Classl )P(xIClassl)-P( Class2)P(xIClass2) as the limit of a sequence of regression
functions using the Parzen Window technique. In the Parzen window technique, probability
density functions are estimated as the sum of appropriately normalized pulses centered at
A Note on Learning Vector Quantization
the observed values. More formally, we can estimate P(xIClassi) as [Sklansky and Wassel,
1981]
Il
1~
Pi (x) = - L...J'?II(x-Xj,cll )
All
n )=1
.
where Xj is the sample data point at time j, and 'IIII(X- z, c(n)) is a Parzen window function
centred at Z with width parameter c(n) that satisfies the following conditions
'?II(X - z, c(n? ~ 0, Vx, Z
J~ '?II(X- Z, c(n?dx = 1
11-
lim n __ '?;(x- z, c(n))dx = 0
11-+-
lim '?1I(x-z,c(n? = c5(x-z)
II-+-
We can estimate f(x)
=P(Class1)P(xIClasst) - P(Class2)P(xIClass2) as
A
rex)
1
Il
= - LS(Xj)'?II(x-Xj,c(n?
n J=
.1
where S(Xj) is + 1 if Xj is from Class1 and -1 if Xj is from Class2.
Then
lim j"(X) = P(Class1)P(xIClass1) - P(Class2)P(xIClass2)
II-+-
and
lim E[S(X)'?ix - X, c(n)]
II-+-
=P(Class1)P(xIClassd -
P(Class2)P(xIClass2)
Wassel and Sklansky [1972] have extended the stochastic approximation method of Robbins and Monro [1951] to find the zero of a function that is the limit of a sequence of
regression functions and show rigourously that for the above case (where the distribution
of Class1 is to the left of that of Class2 and there is only one crossing point) the stochastic
approximation procedure
ben + 1) = ben) + a(n)ZII(xlI , Class(n), ben), c(n?
(3)
using
Z _ { 2c(n)'?(XII - ben), c(n?
II -
-2c(n)'?(XII - ben), c(n?
for XII
for XII
E Classl
E Class2
converges to the Bayes optimal border with probability one where '?(x - b, c) is a Parzen
window function. The following standard conditions for stochastic approximation convergence are needed in their proof
a(n), c(n)
> 0,
lim c(n)
II-+-
1:ia(n)c(n) = 00,
=0
lim a(n) = 0,
II-+-
223
224
de Sa and Ballard
as well as a condition that for rectangular Parzen functions reduces to a requirement that
P( Classl )P(xIClassl) - P( C lass2)P(xlClass2) be strictly positive to the left of b* and strictly
negative to the right of b* (for full details of the proof and conditions see [Wassel and
Sklansky, 1972]).
The above argument has only addressed the motion of the border. But b is defined as
b = (wI + w2)/2, thus we can move the codebook vectors according to
dE/dwl
=dEldw2 =.5dEldb.
We could now write Equation 3 as
wj(n + 1)
(X" - wj(n -
1?
=wj(n) + a2(n) IX" _ wj(n _ 1)1
if X" lies in window of width 2c(n) centred at ben), otherwise
Wi(n + 1)
=wi(n).
where we have used rectangular Parzen window functions and X" is from Classj. This
holds if Classl is to the right or left of Class2 as long as Wl and W2 are relatively ordered
appropriatel y.
Expanding the problem to more dimensions, and more classes with more codebook vectors per class, complicates the analysis as a change in two codebook vectors to better adjust
their border affects more than just the border between the two codebook vectors. However ignoring these effects for a first order approximation suggests the following update
procedure:
1?
*
(X" - wren 1) + a(n) IIX" _ wren _ 1)11
*.
(X" - w;(n - 1?
1) - a(n) IIX" _ wj(n _ 1)11
*
Wi
(n)
Wj (n)
=Wi (n =Wj (n -
where a(n) obeys the constraints above, X" is from Classj, and w;, wj are the two nearest
codebook vectors, one each from class i and j U* i) and x" lies within c(n) of the border
between them. (No changes are made if all the above conditions are not true). As above
this algorithm assumes that the initial positions of the codebook vectors are such that they
will not have to cross during the algorithm.
The above algorithm is similar to Kohonen's LVQ2.1 algorithm (which is performed after
appropriate initialization of the codebook vectors) except for the normalization of the step
size, the decreasing size of the window width c(n) and constraints on the learning rate a.
A Note on Learning Vector Quantization
4 Simulations
Motivated by the theory above, we decided to modify Kohonen's LVQ2.1 algorithm to
add normalization of the step size and a decreasing window. In order to allow closer
comparison with LVQ2.1, all other parts of the algorithm were kept the same. Thus a
decreased linearly. We used a linear decrease on the window size and defined it as in
LVQ2.1 for easier parameter matching. For a window size of w all input vectors satisfying
d;/dj> g:~ where di is the distance to the closest codebook vector and dj is the distance
to the next closest codebook vector, fall into the window between those two vectors (Note
however, that updates only occur if the two closest codebook vectors belong to different
classes).
The data used is a version of the Peterson and Barney vowel formant data 2. The dataset
consists of the first and second formants for ten vowels in a/hVdj context from 75 speakers
(32 males, 28 females, 15 children) who repeated each vowel twice 3. As we were not
testing generalization , the training set was used as the test set.
75.------.------.------.-----..-----~
~.
..
-fA
alpha-0.002
alpha-0.030
alpha-0.080
alpha-O .150
alpha-0.500
~A -:~~.::ra.?-..:::
.....
u
Q)
!\,.
70
.
....i
U
.....
r::Q)
Q)
'"
.
..
\.
.
\'
\\
\
".
\\
'B'"
... -
\..
\..~
\'"
-t--.
\
I""
\.
\
\\
\,:
,.,
\.
65
.......
~.~--;:::.-..--
; !
......
o
...u
;,oz:~,,; .. ~...
-+-
~
"
\
\\
??
\
\
\
\
~
I
it:
\
t
\,
\
\
\
\
60~-----L~----~--~-L--~~~----~
o
0.2
0.4
0.6
window size
0.8
Figure 2: The effect of different window sizes on the accuracy for different values of initial
a.
We ran three sets of experiments varying the number of codebook vectors and the number
of pattern presentations. For the first set of experiments there were 20 codebook vectors
and the algorithms ran for 40000 steps. Figure 2 shows the effect of varying the window
size for different initial learning rates a( 1) in the LVQ2.1 algorithm. The values plotted are
averaged over three runs (The order of presentation of patterns is different for the different
runs). The sensitivity of the algorithm to the window size as mentioned in [Kohonen, 1990]
is evident. In general we found that as the learning rate is increased the peak accuracy is
improved at the expense of the accuracy for other window widths. After a certain value
20 btained
from Steven Nowlan
33 speakers were missing one vowel and the raw data was linearly transfonned to have zero mean
and fall within the range [-3,3] in both components
225
226
de Sa and Ballard
85~----~----~------r------r----~
.
-,.
80
.....
u
.....
II'"
75
mod/100/40000
-?.?
~
-+-
... _._.-lI'-?-?-???-?- ? lIi~T2!t)lrotT(J?"''''':': ?
._.;:::~::::::... ---?--?~~~O..?.4.jlD.oJI._.~=-.
._~~. .- .
; .,
----~----+---------------------~
! ~=~.~'Il
r::
Qj
u
I-<
Qj
'"
?B???
~;.'
Qj
I-<
I-<
0
U
orig/20/40000
mod/20/40000
orig/20/4000
~-El'.'-
. . . . ". ~--- ~
?????????????????????????
70
65~----~----~------~----~~--~
o
0.1
0.2
0.3
window size
0.4
0.5
Figure 3: The performance of LVQ2.1 with and without the modifications (normalized step
size and decreasing window) for 3 different conditions. The legend gives in order [the alg
type/ the number of codebook vectors/ the number of pattern presentations]
the accuracy declines for further increases in learning rate.
Figure 3 shows the improvement achieved with normalization and a linearly decreasing
window size for three sets of experiments : (20 code book vectors/40000 pattern presentations), (20 code book vectors/4000 pattern presentations) and (100 code book vectors/40000 pattern presentations). For the decreasing window algorithm, the x-axis represents the window size in the middle of the run. As above, the values plotted were averaged
over three runs. The values of a(l) were the same within each algorithm over all three
conditions. A graph using the best a found for each condition separately is almost identical. The graph shows that the modifications provide a modest but consistent improvement
in accuracy across the conditions.
In summary the preliminary experiments indicate that a decreasing window and normalized
step size can be worthwhile additions to the LVQ2.1 algorithm and further experiments on
the generalization properties of the algorithm and with other data sets may be warranted.
For these tests we used a linear decrease of the window size and learning rate to allow for
easier comparison with the LVQ2.1 algorithm. Further modifications on the algorithm that
experiment with different functions (that obey the theoretical constraints) for the learning
rate and window size decrease may result in even better performance.
5 Summary
We have shown that Kohonen's LVQ2.1 algorithm can be considered as a variant on a
generalization of an algorithm which is optimal for a IDimensional/2 codebook vector
problem. We added a decreasing window and normalized step size, suggested from the
one dimensional algorithm. to the LVQ2.1 algorithm and found a small but consistent
improvement in accuracy.
A Note on Learning Vector Quantization
Acknowledgements
We would like to thank Steven Nowlan for his many helpful suggestions on an earlier draft
and for making the vowel formant data available to us. We are also grateful to Leonidas
Kontothanassis for his help in coding and discussion. This work was supported by a grant
from the Human Frontier Science Program and a Canadian NSERC 1967 Science and
Engineering Scholarship to the first author who also received A NIPS travel grant to attend
the conference.
References
[Grossberg, 1976] Stephen Grossberg, "Adaptive Pattern Classification and Universal Recoding: I. Parallel Development and Coding of Neural Feature Detectors," Biological
Cybernetics, 23:121-134,1976.
[Kohonen,1982] Teuvo Kohonen, "Self-Organized Formation of Topologically Correct
Feature Maps," Biological Cybernetics, 43:59--69, 1982.
[Kohonen,1986] Teuvo Kohonen, "Learning Vector Quantization for Pattern Recognition," Technical Report TKK-F-A601, Helsinki University of Technology, Department
of Technical Physics, Laboratory of Computer and Information Science, November
1986.
[Kohonen, 1990] Teuvo Kohonen, "Statistical Pattern Recognition Revisited," In R. Eckmiller, editor, Advanced Neural Computers, pages 137-144. Elsevier Science Publishers, 1990.
[Kohonen, 1991] Teuvo Kohonen, "Self-Organizing Maps: Optimization Approaches," In
T. Kohonen, K. Makisara, O. Simula, and J. Kangas, editors,Artijicial Neural Networks,
pages 981-990. Elsevier Science Publishers, 1991.
[Robbins and Monro, 1951J Herbert Robbins and Sutton Monro, "A Stochastic Approximation Method," Annals of Math. Stat., 22:400-407,1951.
[Rumelhart and Zipser, 1986] D. E. Rumelhart and D. Zipser, "Feature Discovery by
Competitive Learning," In David E. Rumelhart, James L. McClelland, and the PDP Research Group, editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 2, pages 151-193. MIT Press, 1986.
[Sklansky and Wassel, 1981] Jack Sklansky and Gustav N. Wassel, Pattern Classijiers
and Trainable Machines, Springer-Verlag, 1981.
[Wassel and Sklansky, 1972] Gustav N. Wassel and Jack Sklansky, "Training a OneDimensional Classifier to Minimize the Probability of Error," IEEE Transactions on
Systems, Man, and Cybernetics, SMC-2(4):533-541, 1972.
227
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6,224 | 6,630 | A Linear-Time Kernel Goodness-of-Fit Test
Wittawat Jitkrittum
Gatsby Unit, UCL
Wenkai Xu
Gatsby Unit, UCL
Zolt?n Szab??
CMAP, ?cole Polytechnique
[email protected]
[email protected]
[email protected]
Kenji Fukumizu
The Institute of Statistical Mathematics
[email protected]
Arthur Gretton?
Gatsby Unit, UCL
[email protected]
Abstract
We propose a novel adaptive test of goodness-of-fit, with computational cost
linear in the number of samples. We learn the test features that best indicate the
differences between observed samples and a reference model, by minimizing the
false negative rate. These features are constructed via Stein?s method, meaning that
it is not necessary to compute the normalising constant of the model. We analyse
the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift
alternative, our test always has greater relative efficiency than a previous linear-time
kernel test, regardless of the choice of parameters for that test. In experiments, the
performance of our method exceeds that of the earlier linear-time test, and matches
or exceeds the power of a quadratic-time kernel test. In high dimensions and where
model structure may be exploited, our goodness of fit test performs far better than
a quadratic-time two-sample test based on the Maximum Mean Discrepancy, with
samples drawn from the model.
1
Introduction
The goal of goodness of fit testing is to determine how well a model density p(x) fits an observed
sample D = {xi }ni=1 ? X ? Rd from an unknown distribution q(x). This goal may be achieved via
a hypothesis test, where the null hypothesis H0 : p = q is tested against H1 : p 6= q. The problem
of testing goodness of fit has a long history in statistics [11], with a number of tests proposed for
particular parametric models. Such tests can require space partitioning [18, 3], which works poorly in
high dimensions; or closed-form integrals under the model, which may be difficult to obtain, besides
in certain special cases [2, 5, 30, 26]. An alternative is to conduct a two-sample test using samples
drawn from both p and q. This approach was taken by [23], using a test based on the (quadratic-time)
Maximum Mean Discrepancy [16], however this does not take advantage of the known structure of p
(quite apart from the increased computational cost of dealing with samples from p).
More recently, measures of discrepancy with respect to a model have been proposed based on Stein?s
method [21]. A Stein operator for p may be applied to a class of test functions, yielding functions that
have zero expectation under p. Classes of test functions can include the W 2,? Sobolev space [14],
and reproducing kernel Hilbert spaces (RKHS) [25]. Statistical tests have been proposed by [9, 22]
based on classes of Stein transformed RKHS functions, where the test statistic is the norm of the
smoothness-constrained function with largest expectation under q . We will refer to this statistic as
the Kernel Stein Discrepancy (KSD). For consistent tests, it is sufficient to use C0 -universal kernels
[6, Definition 4.1], as shown by [9, Theorem 2.2], although inverse multiquadric kernels may be
preferred if uniform tightness is required [15].2
?
Zolt?n Szab??s ORCID ID: 0000-0001-6183-7603. Arthur Gretton?s ORCID ID: 0000-0003-3169-7624.
Briefly, [15] show that when an exponentiated quadratic kernel is used, a sequence of sets D may be
constructed that does not correspond to any q, but for which the KSD nonetheless approaches zero. In a statistical
testing setting, however, we assume identically distributed samples from q, and the issue does not arise.
2
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
The minimum variance unbiased estimate of the KSD is a U-statistic, with computational cost
quadratic in the number n of samples from q. It is desirable to reduce the cost of testing, however,
so that larger sample sizes may be addressed. A first approach is to replace the U-statistic with a
running average with linear cost, as proposed by [22] for the KSD, but this results in an increase in
variance and corresponding decrease in test power. An alternative approach is to construct explicit
features of the distributions, whose empirical expectations may be computed in linear time. In the
two-sample and independence settings, these features were initially chosen at random by [10, 8, 32].
More recently, features have been constructed explicitly to maximize test power in the two-sample
[19] and independence testing [20] settings, resulting in tests that are not only more interpretable, but
which can yield performance matching quadratic-time tests.
We propose to construct explicit linear-time features for testing goodness of fit, chosen so as to
maximize test power. These features further reveal where the model and data differ, in a readily interpretable way. Our first theoretical contribution is a derivation of the null and alternative distributions
for tests based on such features, and a corresponding power optimization criterion. Note that the
goodness-of-fit test requires somewhat different strategies to those employed for two-sample and
independence testing [19, 20], which become computationally prohibitive in high dimensions for
the Stein discrepancy (specifically, the normalization used in prior work to simplify the asymptotics
would incur a cost cubic in the dimension d and the number of features in the optimization). Details
may be found in Section 3.
Our second theoretical contribution, given in Section 4, is an analysis of the relative Bahadur
efficiency of our test vs the linear time test of [22]: this represents the relative rate at which the pvalue decreases under H1 as we observe more samples. We prove that our test has greater asymptotic
Bahadur efficiency relative to the test of [22], for Gaussian distributions under the mean-shift
alternative. This is shown to hold regardless of the bandwidth of the exponentiated quadratic kernel
used for the earlier test. The proof techniques developed are of independent interest, and we anticipate
that they may provide a foundation for the analysis of relative efficiency of linear-time tests in the
two-sample and independence testing domains. In experiments (Section 5), our new linear-time test
is able to detect subtle local differences between the density p(x), and the unknown q(x) as observed
through samples. We show that our linear-time test constructed based on optimized features has
comparable performance to the quadratic-time test of [9, 22], while uniquely providing an explicit
visual indication of where the model fails to fit the data.
2
Kernel Stein Discrepancy (KSD) Test
We begin by introducing the Kernel Stein Discrepancy (KSD) and associated statistical test, as
proposed independently by [9] and [22]. Assume that the data domain is a connected open set X ? Rd .
Consider a Stein operator Tp that takes in a multivariate function f (x) = (f1 (x), . . . , fd (x))> ? Rd
and constructs a function (Tp f ) (x) : Rd ? R. The constructed function has the key property that for
all f in an appropriate function class, Ex?q [(Tp f )(x)] = 0 if and only if q = p. Thus, one can use
this expectation as a statistic for testing goodness of fit.
The function class F d for the function f is chosen to be a unit-norm ball in a reproducing kernel Hilbert
space (RKHS) in [9, 22]. More precisely, let F be an RKHS associated with a positive definite kernel
k : X ? X ? R. Let ?(x) = k(x, ?) denote a feature map of k so that k(x, x0 ) = h?(x), ?(x0 )iF .
Assume that fi ? F for all i = 1, . . . , d so that f ? F ? ? ? ? ? F := F d where F d is equipped with
Pd
the standard inner product hf , giF d := i=1 hfi , gi iF . The kernelized Stein operator Tp studied
Pd
p(x)
?fi (x) (a)
in [9] is (Tp f ) (x) := i=1 ? log
f
(x)
+
= f , ? p (x, ?) F d , where at (a) we use the
i
?xi
?xi
reproducing property of F, i.e., fi (x) = hfi , k(x, ?)iF , and that
hence ? p (x, ?) := ? log?xp(x) k(x, ?)+ ?k(x,?)
is in F d .
?x
is defined such that (Tp f ) (x) ? Rd . This distinction
?k(x,?)
?xi
? F [28, Lemma 4.34],
We note that the Stein operator presented in [22]
is not crucial and leads to the same goodness-offit test. Under appropriate conditions, e.g. that limkxk?? p(x)fi (x) = 0 for all i = 1, . . . , d, it can
be shown using integration by parts that Ex?p (Tp f )(x) = 0 for any f ? F d [9, Lemma 5.1]. Based
on the Stein operator, [9, 22] define the kernelized Stein discrepancy as
Sp (q) :=
sup
kf kF d ?1
(a)
Ex?q f , ? p (x, ?) F d =
sup
kf kF d ?1
2
f , Ex?q ? p (x, ?) F d = kg(?)kF d ,
(1)
where at (a), ? p (x, ?) is Bochner integrable [28, Definition A.5.20] as long as Ex?q k? p (x, ?)kF d <
?, and g(y) := Ex?q ? p (x, y) is what we refer to as the Stein witness function. The Stein witness
function will play a crucial role in our new test statistic in Section 3. When a C0 -universal kernel is
used [6, Definition 4.1], and as long as Ex?q k?x log p(x) ? ?x log q(x)k2 < ?, it can be shown
that Sp (q) = 0 if and only if p = q [9, Theorem 2.2].
Ex?q Ex0 ?q hp (x, x0 ), where hp (x, y) :=
Pd ? 2 k(x,y)
>
>
s>
p (x)sp (y)k(x, y) + sp (y)?x k(x, y) + sp (x)?y k(x, y) +
i=1 ?xi ?yi , and sp (x) :=
c2 =
?x log p(x) is a column vector. An unbiased empirical estimator of Sp2 (q), denoted by S
P
2
i<j hp (xi , xj ) [22, Eq. 14], is a degenerate U-statistic under H0 . For the goodness-of-fit
n(n?1)
c2
test, the rejection threshold can be computed by a bootstrap procedure. All these properties make S
a very flexible criterion to detect the discrepancy of p and q: in particular, it can be computed even if
p is known only up to a normalization constant. Further studies on nonparametric Stein operators can
be found in [25, 14].
The KSD Sp (q) can be written as Sp2 (q)
=
c2 costs O(n2 ). To reduce this cost, a
Linear-Time Kernel Stein (LKS) Test Computation of S
linear-time (i.e., O(n)) estimator based on an incomplete U-statistic is proposed in [22, Eq. 17],
c2 := 2 Pn/2 h (x
given by S
i=1 p 2i?1 , x2i ), where we assume n is even for simplicity. Empirically
l
n
[22] observed that the linear-time estimator performs much worse (in terms of test power) than the
quadratic-time U-statistic estimator, agreeing with our findings presented in Section 5.
3
New Statistic: The Finite Set Stein Discrepancy (FSSD)
Although shown to be powerful, the main drawback of the KSD test is its high computational cost of
O(n2 ). The LKS test is one order of magnitude faster. Unfortunately, the decrease in the test power
outweighs the computational gain [22]. We therefore seek a variant of the KSD statistic that can be
computed in linear time, and whose test power is comparable to the KSD test.
Key Idea The fact that Sp (q) = 0 if and only if p = q implies that g(v) = 0 for all v ? X if and
only if p = q, where g is the Stein witness function in (1). One can see g as a function witnessing
the differences of p, q, in such a way that |gi (v)| is large when there is a discrepancy in the region
around v, as indicated by the ith output of g. The test statistic of [22, 9] is essentially given by the
degree of ?flatness? of g as measured by the RKHS norm k ? kF d . The core of our proposal is to use
a different measure of flatness of g which can be computed in linear time.
The idea is to use a real analytic kernel k which makes g1 , . . . , gd real analytic. If gi 6= 0 is an
analytic function, then the Lebesgue measure of the set of roots {x | gi (x) = 0} is zero [24]. This
property suggests that one can evaluate gi at a finite set of locations V = {v1 , . . . , vJ }, drawn
from a distribution with a density (w.r.t. the Lebesgue measure). If gi 6= 0, then almost surely
gi (v1 ), . . . , gi (vJ ) will not be zero. This idea was successfully exploited in recently proposed
linear-time tests of [8] and [19, 20]. Our new test statistic based on this idea is called the Finite Set
Stein Discrepancy (FSSD) and is given in Theorem 1. All proofs are given in the appendix.
Theorem 1 (The Finite Set Stein Discrepancy (FSSD)). Let V = {v1 , . . . , vJ } ? Rd be random
vectors drawn i.i.d. from a distribution ? which has a density. Let X be a connected open set
Pd PJ
1
2
in Rd . Define FSSD2p (q) := dJ
j=1 gi (vj ). Assume that 1) k : X ? X ? R is C0 i=1
universal [6, Definition 4.1] and real analytic i.e., for all v ? X , f (x) := k(x, v) is a real analytic
function on X . 2) Ex?q Ex0 ?q hp (x, x0 ) < ?. 3) Ex?q k?x log p(x) ? ?x log q(x)k2 < ?. 4)
limkxk?? p(x)g(x) = 0.
Then, for any J ? 1, ?-almost surely FSSD2p (q) = 0 if and only if p = q.
This measure depends on a set of J test locations (or features) {vi }Ji=1 used to evaluate the Stein
witness function, where J is fixed and is typically
small.2 A
kernel which is C0 -universal and real
kx?yk
analytic is the Gaussian kernel k(x, y) = exp ? 2?2 2 (see [20, Proposition 3] for the result
k
on analyticity). Throughout this work, we will assume all the conditions stated in Theorem 1, and
consider only the Gaussian kernel. Besides the requirement that the kernel be real and analytic,
the remaining conditions in Theorem 1 are the same as given in [9, Theorem 2.2]. Note that if the
3
FSSD is to be employed in a setting otherwise than testing, for instance to obtain pseudo-samples
converging to p, then stronger conditions may be needed [15].
3.1
Goodness-of-Fit Test with the FSSD Statistic
Given a significance level ? for the goodness-of-fit test, the test can be constructed so that H0 is
\2 is
\2 > T? , where T? is the rejection threshold (critical value), and FSSD
rejected when nFSSD
an empirical estimate of FSSD2p (q). The threshold which guarantees that the type-I error (i.e., the
probability of rejecting H0 when it is true) is bounded above by ? is given by the (1 ? ?)-quantile of
\2 under H0 . In the following, we start by giving
the null distribution i.e., the distribution of nFSSD
\2 , and summarize its asymptotic distributions in Proposition 2.
the expression for FSSD
?
Let ?(x) ? Rd?J such that [?(x)]i,j = ?p,i (x, vj )/ dJ. Define ? (x) := vec(?(x)) ? RdJ where
vec(M) concatenates columns of the matrix M into a column vector. We note that ? (x) depends
on the test locations V = {vj }Jj=1 . Let ?(x, y) := ? (x)> ? (y) = tr(?(x)> ?(y)). Given an i.i.d.
sample {xi }ni=1 ? q, a consistent, unbiased estimator of FSSD2p (q) is
d X
n X
J
X
X
X
2
1
\2 = 1
FSSD
?(xi , xj ),
?p,l (xi , vm )?p,l (xj , vm ) =
dJ
n(n ? 1) i=1
n(n ? 1) i<j
m=1
l=1
(2)
j6=i
which is a one-sample second-order U-statistic with ? as its U-statistic kernel [27, Section 5.1.1].
d
Being a U-statistic, its asymptotic distribution can easily be derived. We use ? to denote convergence
in distribution.
\2 ). Let Z1 , . . . , ZdJ i.i.d.
? N (0, 1). Let ? :=
Proposition 2 (Asymptotic distributions of FSSD
dJ?dJ
Ex?q [? (x)], ?r := covx?r [? (x)] ? R
for r ? {p, q}, and {?i }dJ
i=1 be the eigenvalues of
?p = Ex?p [? (x)? > (x)]. Assume that Ex?q Ey?q ?2 (x, y) < ?. Then, for any realization of
V = {vj }Jj=1 , the following statements hold.
d
\2 ?
1. Under H0 : p = q, nFSSD
PdJ
2
i=1 (Zi
? 1)?i .
2
2. Under H1 : p 6= q, if ?H
:= 4?> ?q ? > 0, then
1
?
d
2
\2 ? FSSD2 ) ?
n(FSSD
N (0, ?H
).
1
Proof. Recognizing that (2) is a degenerate U-statistic, the results follow directly from [27, Section
5.5.1, 5.5.2].
Claims 1 and 2 of Proposition 2 imply that under H1 , the test power (i.e., the probability of correctly
rejecting H1 ) goes to 1 asymptotically, if the threshold T? is defined as above. In practice, simulating
from the asymptotic null distribution in Claim 1 can be challenging, since the plug-in estimator of
?p requires a sample from p, which is not available. A straightforward solution is to draw sample
from p, either by assuming that p can be sampled easily or by using a Markov chain Monte Carlo
(MCMC) method, although this adds an additional computational burden to the test procedure. A
more subtle issue is that when dependent samples from p are used in obtaining the test threshold, the
test may become more conservative than required for i.i.d. data [7]. An alternative approach is to use
? q instead of ?p . The covariance matrix ?
? q can be directly computed from the
the plug-in estimate ?
data. This is the approach we take. Theorem 3 guarantees that the replacement of the covariance in
the computation of the asymptotic null distribution still yields a consistent test. We write PH1 for the
\2 under H1 .
distribution of nFSSD
? q := 1 Pn ? (xi )? > (xi ) ? [ 1 Pn ? (xi )][ 1 Pn ? (xj )]> with {xi }n ?
Theorem 3. Let ?
i=1
i=1
i=1
j=1
n
n
n
PdJ
q. Suppose that the test threshold T? is set to the (1??)-quantile of the distribution of i=1 (Zi2 ?1)??i
i.i.d.
? q . Then, under H0 , asymptotically
where {Zi }dJ
?1 , . . . , ??dJ are eigenvalues of ?
i=1 ? N (0, 1), and ?
J
the false positive rate is ?. Under H1 , for {vj }j=1 drawn from a distribution with a density, the test
\2 > T? ) ? 1 as n ? ?.
power PH (nFSSD
1
?q = ?
? p i.e., the plug-in
Remark 1. The proof of Theorem 3 relies on two facts. First, under H0 , ?
?
estimate of ?p . Thus, under H0 , the null distribution approximated with ?q is asymptotically
4
? p to ?p . Second, the rejection threshold obtained from the
correct, following the convergence of ?
approximated null distribution is asymptotically constant. Hence, under H1 , claim 2 of Proposition 2
d
\2 ?
\2 > T? ) ? 1.
implies that nFSSD
? as n ? ?, and consequently PH1 (nFSSD
3.2
Optimizing the Test Parameters
Theorem 1 guarantees that the population quantity FSSD2 = 0 if and only if p = q for any choice of
{vi }Ji=1 drawn from a distribution with a density. In practice, we are forced to rely on the empirical
\2 , and some test locations will give a higher detection rate (i.e., test power) than others for
FSSD
J
finite n. Following the approaches of [17, 20, 19, 29], we choose the test locations V = {vj }j=1
and kernel bandwidth ?k2 so as to maximize the test power i.e., the probability of rejecting H0 when
it is false. We first give an approximate expression for the test power when n is large.
\2 ). Under H1 , for large n and fixed r, the
Proposition 4 (Approximate test power of nFSSD
?
2
r
\2 > r) ? 1 ? ? ?
? n FSSD
test power PH1 (nFSSD
, where ? denotes the cumulative
?H1
n?H1
distribution function of the standard normal distribution, and ?H1 is defined in Proposition 2.
? r/n?FSSD2
\2 ?FSSD2
\2 > r) = PH (FSSD
\2 > r/n) = PH ?n FSSD
Proof. PH1 (nFSSD
n
.
>
1
1
?H1
?H1
For sufficiently large n, the alternative distribution
is approximatelynormal as given in Proposition 2.
\2 > r) ? 1 ? ? ? r ? ?n FSSD2 .
It follows that PH1 (nFSSD
?H
n?H
1
1
Let ? := {V, ?k2 } be the collection of all tuning parameters. Assume that n is sufficiently large.
?
2
r
Following the same argument as in [29], in ?n?
? n FSSD
?H1 , we observe that the first term
H1
?
2
? r
= O(n1/2 ), dominating
= O(n?1/2 ) going to 0 as n ? ?, while the second term n FSSD
?H
n?H
1
1
the first for large n. Thus, the best parameters that maximize the test power are given by ? ? =
\2 > T? ) ? arg max? FSSD2 . Since FSSD2 and ?H are unknown, we divide
arg max? PH1 (nFSSD
1
?H
1
\2
the sample {xi }ni=1 into two disjoint training and test sets, and use the training set to compute ??FSSD
,
H1 +?
where a small regularization parameter ? > 0 is added for numerical stability. The goodness-of-fit
test is performed on the test set to avoid overfitting. The idea of splitting the data into training and
test sets to learn good features for hypothesis testing was successfully used in [29, 20, 19, 17].
To find a local maximum of
{vi }Ji=1
\2
FSSD
?
?H1 +? ,
we use gradient ascent for its simplicity. The initial points of
are set to random draws from a normal distribution fitted to the training data, a heuristic we
found to perform well in practice. The objective is non-convex in general, reflecting many possible
ways to capture the differences of p and q. The regularization parameter ? is not tuned, and is
\2
fixed to a small constant. Assume that ?x log p(x) costs O(d2 ) to evaluate. Computing ?? ??FSSD
H +?
1
2
\2 and ?
costs O(d2 J 2 n). The computational complexity of nFSSD
?H
is O(d2 Jn). Thus, finding
1
a local optimum via gradient ascent is still linear-time, for a fixed maximum number of iterations.
? q costs O(d2 J 2 n), and obtaining all the eigenvalues of ?
? q costs O(d3 J 3 ) (required
Computing ?
only once). If the eigenvalues decay to zero sufficiently rapidly, one can approximate the asymptotic
null distribution with only a few eigenvalues. The cost to obtain the largest few eigenvalues alone can
be much smaller.
P
? := n1 ni=1 ? (xi ). It is possible to normalize the FSSD statistic to get a new
Remark 2. Let ?
? n := n?
? q + ?I)?1 ?
? > (?
? where ? ? 0 is a regularization parameter that goes to 0
statistic ?
as n ? ?. This was done in the case of the ME (mean embeddings) statistic of [8, 19]. The
asymptotic null distribution of this statistic takes the convenient form of ?2 (dJ) (independent of
? q . It turns out that the test power
p and q), eliminating the need to obtain the eigenvalues of ?
?
criterion for tuning the parameters in this case is the statistic ?n itself. However, the optimization
? q + ?I)?1 (costing O(d3 J 3 )) needs to be reevaluated in each
is computationally expensive as (?
gradient ascent iteration. This is not needed in our proposed FSSD statistic.
5
4
Relative Efficiency and Bahadur Slope
Both the linear-time kernel Stein (LKS) and FSSD tests have the same computational cost of O(d2 n),
and are consistent, achieving maximum power of 1 as n ? ? under H1 . It is thus of theoretical
interest to understand which test is more sensitive in detecting the differences of p and q. This can be
quantified by the Bahadur slope of the test [1]. Two given tests can then be compared by computing
the Bahadur efficiency (Theorem 7) which is given by the ratio of the slopes of the two tests. We
note that the constructions and techniques in this section may be of independent interest, and can be
generalised to other statistical testing settings.
We start by introducing the concept of Bahadur slope for a general test, following the presentation of
[12, 13]. Consider a hypothesis testing problem on a parameter ?. The test proposes a null hypothesis
H0 : ? ? ?0 against the alternative hypothesis H1 : ? ? ?\?0 , where ?, ?0 are arbitrary sets.
Let Tn be a test statistic computed from a sample of size n, such that large values of Tn provide
an evidence to reject H0 . We use plim to denote convergence in probability, and write Er for
Ex?r Ex0 ?r .
Approximate Bahadur Slope (ABS) For ?0 ? ?0 , let the asymptotic null distribution of Tn be
F (t) = limn?? P?0 (Tn < t), where we assume that the CDF (F ) is continuous and common to all
?0 ? ?0 . The continuity of F will be important later when Theorem 9 and 10 are used to compute
the slopes of LKS and FSSD tests. Assume that there exists a continuous strictly increasing function
(Tn ))
? : (0, ?) ? (0, ?) such that limn?? ?(n) = ?, and that ?2 plimn?? log(1?F
= c(?)
?(n)
where Tn ? P? , for some function c such that 0 < c(?A ) < ? for ?A ? ?\?0 , and c(?0 ) = 0 when
?0 ? ?0 . The function c(?) is known as the approximate Bahadur slope (ABS) of the sequence Tn .
The quantifier ?approximate? comes from the use of the asymptotic null distribution instead of the
exact one [1]. Intuitively the slope c(?A ), for ?A ? ?\?0 , is the rate of convergence of p-values (i.e.,
1 ? F (Tn )) to 0, as n increases. The higher the slope, the faster the p-value vanishes, and thus the
lower the sample size required to reject H0 under ?A .
(1)
(2)
Approximate Bahadur Efficiency Given two sequences of test statistics, Tn and Tn having the
(1)
(2)
same ?(n) (see Theorem 10), the approximate Bahadur efficiency of Tn relative to Tn is defined
(1)
as E(?A ) := c(1) (?A )/c(2) (?A ) for ?A ? ?\?0 . If E(?A ) > 1, then Tn is asymptotically more
(2)
efficient than Tn in the sense of Bahadur, for the particular problem specified by ?A ? ?\?0 . We
now give approximate Bahadur slopes for two sequences of linear time test statistics: the proposed
c2 discussed in Section 2.
\2 , and the LKS test statistic ?nS
nFSSD
l
\2 is c(FSSD) := FSSD2 /?1 , where ?1 is the
Theorem 5. The approximate Bahadur slope of nFSSD
>
maximum eigenvalue of ?p := Ex?p [? (x)? (x)] and ?(n) = n.
? c2
Theorem 6. The approximate Bahadur slope of the linear-time kernel Stein (LKS) test statistic nS
l
2
0
1 [Eq hp (x,x )]
(LKS)
is c
= 2 E h2 (x,x0 ) , where hp is the U-statistic kernel of the KSD statistic, and ?(n) = n.
]
p[ p
To make these results concrete, we consider the setting where p = N (0, 1) andq = N (?q , 1).
We assume that both tests use the Gaussian kernel k(x, y) = exp ?(x ? y)2 /2?k2 , possibly with
different bandwidths. We write ?k2 and ?2 for the FSSD and LKS bandwidths, respectively. Under
these assumptions, the slopes given in Theorem 5 and Theorem 6 can be derived explicitly. The
full expressions of the slopes are given in Proposition 12 and Proposition 13 (in the appendix). By
[12, 13] (recalled as Theorem 10 in the supplement), the approximate Bahadur efficiency can be
computed by taking the ratio of the two slopes. The efficiency is given in Theorem 7.
Theorem 7 (Efficiency in the Gaussian mean shift problem). Let E1 (?q , v, ?k2 , ?2 ) be the approxic2 for the case where p = N (0, 1), q = N (? , 1),
\2 relative to ?nS
mate Bahadur efficiency of nFSSD
l
q
\2 ). Fix ? 2 = 1 for nFSSD
\2 . Then, for any ?q 6= 0,
and J = 1 (i.e., one test location v for nFSSD
k
for some v ? R, and for any ?2 > 0, we have E1 (?q , v, ?k2 , ?2 ) > 2.
When p = N (0, 1) and q = N (?q , 1) for ?q 6= 0, Theorem 7 guarantees that our FSSD test is
asymptotically at least twice as efficient as the LKS test in the Bahadur sense. We note that the
6
efficiency is conservative in the sense that ?k2 = 1 regardless of ?q . Choosing ?k2 dependent on ?q
will likely improve the efficiency further.
5
Experiments
In this section, we demonstrate the performance of the proposed test on a number of problems. The
primary goal is to understand the conditions under which the test can perform well.
p
Sensitivity to Local Differences We start by demonstrating that
q
2
the test power objective FSSD /?H1 captures local differences
FSSD
?
of p and q, and that interpretable features v are found. Consider a one-dimensional
problem in which p = N (0, 1) and
?
?4
?2 v ? 0 v ? 2
4
q = Laplace(0,
1/
2),
a
zero-mean
Laplace
distribution
with
scale
?
parameter 1/ 2. These parameters are chosen so that p and q have Figure 1: The power criterion
the same mean and variance. Figure 1 plots the (rescaled) objective FSSD2 /?H1 as a function of
as a function of v. The objective illustrates that the best features test location v.
(indicated by v ? ) are at the most discriminative locations.
2
H1
Test Power We next investigate the power of different tests on two problems:
?
Qd
1. Gaussian vs. Laplace: p(x) = N (x|0, Id ) and q(x) = i=1 Laplace(xi |0, 1/ 2) where the
dimension d will be varied. The two distributions have the same mean and variance. The main
characteristic of this problem is local differences of p and q (see Figure 1). Set n = 1000.
2. Restricted Boltzmann Machine (RBM):
p(x) is the marginal distribution of p(x, h) =
1
1
>
>
>
2
exp
x
Bh
+
b
x
+
c
x
?
kxk
,
where
x ? Rd , h ? {?1}dh is a random vector of
Z
2
hidden
variables, and Z is the normalization constant. The exact marginal density p(x) =
P
dh
terms.
h?{?1,1}dh p(x, h) is intractable when dh is large, since it involves summing over 2
Recall that the proposed test only requires the score function ?x log p(x) (not the normalization
constant), which can be computed in closed form in this case. In this problem, q is another RBM
where entries of the matrix B are corrupted by Gaussian noise. This was the problem considered in
[22]. We set d = 50 and dh = 40, and generate samples by n independent chains (i.e., n independent
samples) of blocked Gibbs sampling with 2000 burn-in iterations.
We evaluate the following six kernel-based nonparametric tests with ? = 0.05, all using the Gaussian
kernel. 1. FSSD-rand: the proposed FSSD test where the test locations set to random draws from
a multivariate normal distribution fitted to the data. The kernel bandwidth is set by the commonly
used median heuristic i.e., ?k = median({kxi ? xj k, i < j}). 2. FSSD-opt: the proposed FSSD
test where both the test locations and the Gaussian bandwidth are optimized (Section 3.2). 3. KSD:
the quadratic-time Kernel Stein Discrepancy test with the median heuristic. 4. LKS: the linear-time
version of KSD with the median heuristic. 5. MMD-opt: the quadratic-time MMD two-sample
test of [16] where the kernel bandwidth is optimized by grid search to maximize a power criterion
as described in [29]. 6. ME-opt: the linear-time mean embeddings (ME) two-sample test of [19]
where parameters are optimized. We draw n samples from p to run the two-sample tests (MMD-opt,
ME-opt). For FSSD tests, we use J = 5 (see Section A for an investigation of test power as J varies).
All tests with optimization use 20% of the sample size n for parameter tuning. Code is available at
https://github.com/wittawatj/kernel-gof.
Figure 2 shows the rejection rates of the six tests for the two problems, where each problem is
repeated for 200 trials, resampling n points from q every time. In Figure 2a (Gaussian vs. Laplace),
high performance of FSSD-opt indicates that the test performs well when there are local differences
between p and q. Low performance of FSSD-rand emphasizes the importance of the optimization
of FSSD-opt to pinpoint regions where p and q differ. The power of KSD quickly drops as the
dimension increases, which can be understood since KSD is the RKHS norm of a function witnessing
differences in p and q across the entire domain, including where these differences are small.
We next consider the case of RBMs. Following [22], b, c are independently drawn from the standard
multivariate normal distribution, and entries of B ? R50?40 are drawn with equal probability from
{?1}, in each trial. The density q represents another RBM having the same b, c as in p, and with all
entries of B corrupted by independent zero-mean Gaussian noise with standard deviation ?per . Figure
7
0.0
1
5
10
dimension d
15
1.0
0.5
0.0
0.00
ME-opt
300
0.75
0.50
0.25
0.00
0.02
0.04
0.06
Perturbation SD ?per
MMD-opt
Time (s)
0.5
LKS
Rejection rate
1.0
KSD
2000
Sample size n
Rejection rate
FSSD-rand
Rejection rate
Rejection rate
FSSD-opt
1.0
4000
200
100
0
1000
2000
3000
Sample size n
4000
0.5 (d) Runtime (RBM)
(a) Gaussian vs. Laplace. (b) RBM. n = 1000. Per- (c) RBM. ?per = 0.1. Pern = 1000.
turb all entries of B.
turb B1,1 .
0.0
Figure 2: Rejection rates of the six tests. The proposed linear-time FSSD-opt
has0.02
a comparable
or
0.00
0.04
0.06
higher test power in some cases than the quadratic-time KSD test.
Perturbation SD ?per
2b shows the test powers as ?per increases, for a fixed sample size n = 1000. We observe that all the
tests have correct false positive rates (type-I errors) at roughly ? = 0.05 when there is no perturbation
noise. In particular, the optimization in FSSD-opt does not increase false positive rate when H0 holds.
We see that the performance of the proposed FSSD-opt matches that of the quadratic-time KSD at
all noise levels. MMD-opt and ME-opt perform far worse than the goodness-of-fit tests when the
difference in p and q is small (?per is low), since these tests simply represent p using samples, and do
not take advantage of its structure.
The advantage of having O(n) runtime can be clearly seen when the problem is much harder,
requiring larger sample sizes to tackle. Consider a similar problem on RBMs in which the parameter
B ? R50?40 in q is given by that of p, where only the first entry B1,1 is perturbed by random
N (0, 0.12 ) noise. The results are shown in Figure 2c where the sample size n is varied. We observe
that the two two-sample tests fail to detect this subtle difference even with large sample size. The
test powers of KSD and FSSD-opt are comparable when n is relatively small. It appears that KSD
has higher test power than FSSD-opt in this case for large n. However, this moderate gain in the test
power comes with an order of magnitude more computation. As shown in Figure 2d, the runtime
of the KSD is much larger than that of FSSD-opt, especially at large n. In these problems, the
performance of the new test (even without optimization) far exceeds that of the LKS test. Further
simulation results can be found in Section B.
0.20
Interpretable Features In the
0.16
final simulation, we demonstrate
0.12
that the learned test locations are
0.08
informative in visualising where
0.04
the model does not fit the data
0.00
well. We consider crime data
?0.04
from the Chicago Police Depart?0.08
ment, recording n = 11957
locations (latitude-longitude co- (a) p = 2-component GMM.
(b) p = 10-component GMM
ordinates) of robbery events in
optimization objective as a function of
Chicago in 2016.3 We address Figure 3: Plots of the
2
the situation in which a model p test location v ? R in the Gaussian mixture model (GMM)
for the robbery location density is evaluation task.
given, and we wish to visualise
where it fails to match the data. We fit a Gaussian mixture model (GMM) with the expectationmaximization algorithm to a subsample of 5500 points. We then test the model on a held-out test set
of the same size to obtain proposed locations of relevant features v. Figure 3a shows the test robbery
locations in purple, the model with two Gaussian components in wireframe, and the optimization
objective for v as a grayscale contour plot (a red star indicates the maximum). We observe that the
2-component model is a poor fit to the data, particularly in the right tail areas of the data, as indicated
in dark gray (i.e., the objective is high). Figure 3b shows a similar plot with a 10-component GMM.
The additional components appear to have eliminated some mismatch in the right tail, however a
discrepancy still exists in the left region. Here, the data have a sharp boundary on the right side
following the geography of Chicago, and do not exhibit exponentially decaying Gaussian-like tails.
We note that tests based on a learned feature located at the maximum both correctly reject H0 .
3
Data can be found at https://data.cityofchicago.org.
8
Acknowledgement
WJ, WX, and AG thank the Gatsby Charitable Foundation for the financial support. ZSz was
financially supported by the Data Science Initiative. KF has been supported by KAKENHI Innovative
Areas 25120012.
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10
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6,225 | 6,631 | Cortical microcircuits as
gated-recurrent neural networks
Rui Ponte Costa?
Centre for Neural Circuits and Behaviour
Dept. of Physiology, Anatomy and Genetics
University of Oxford, Oxford, UK
[email protected]
Yannis M. Assael?
Dept. of Computer Science
University of Oxford, Oxford, UK
and DeepMind, London, UK
[email protected]
Brendan Shillingford?
Dept. of Computer Science
University of Oxford, Oxford, UK
and DeepMind, London, UK
[email protected]
Nando de Freitas
DeepMind
London, UK
[email protected]
Tim P. Vogels
Centre for Neural Circuits and Behaviour
Dept. of Physiology, Anatomy and Genetics
University of Oxford, Oxford, UK
[email protected]
Abstract
Cortical circuits exhibit intricate recurrent architectures that are remarkably similar
across different brain areas. Such stereotyped structure suggests the existence of
common computational principles. However, such principles have remained largely
elusive. Inspired by gated-memory networks, namely long short-term memory
networks (LSTMs), we introduce a recurrent neural network in which information
is gated through inhibitory cells that are subtractive (subLSTM). We propose a
natural mapping of subLSTMs onto known canonical excitatory-inhibitory cortical
microcircuits. Our empirical evaluation across sequential image classification
and language modelling tasks shows that subLSTM units can achieve similar
performance to LSTM units. These results suggest that cortical circuits can be
optimised to solve complex contextual problems and proposes a novel view on
their computational function. Overall our work provides a step towards unifying
recurrent networks as used in machine learning with their biological counterparts.
1
Introduction
Over the last decades neuroscience research has collected enormous amounts of data on the architecture and dynamics of cortical circuits, unveiling complex but stereotypical structures across
the neocortex (Markram et al., 2004; Harris and Mrsic-Flogel, 2013; Jiang et al., 2015). One of
the most prevalent features of cortical nets is their laminar organisation and their high degree of
recurrence, even at the level of local (micro-)circuits (Douglas et al., 1995; Song et al., 2005; Harris
and Mrsic-Flogel, 2013; Jiang et al., 2015) (Fig. 1a). Another key feature of cortical circuits is
the detailed and tight balance of excitation and inhibition, which has received growing support
* These authors contributed equally to this work.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
both at the experimental (Froemke et al., 2007; Xue et al., 2014; Froemke, 2015) and theoretical
level (van Vreeswijk and Sompolinsky, 1996; Brunel, 2000; Vogels and Abbott, 2009; Hennequin
et al., 2014, 2017). However, the computational processes that are facilitated by these architectures
and dynamics are still elusive. There remains a fundamental disconnect between the underlying
biophysical networks and the emergence of intelligent and complex behaviours.
Artificial recurrent neural networks (RNNs), on the other hand, are crafted to perform specific
computations. In fact, RNNs have recently proven very successful at solving complex tasks such
as language modelling, speech recognition, and other perceptual tasks (Graves, 2013; Graves et al.,
2013; Sutskever et al., 2014; van den Oord et al., 2016; Assael et al., 2016). In these tasks, the input
data contains information across multiple timescales that needs to be filtered and processed according
to its relevance. The ongoing presentation of stimuli makes it difficult to learn to separate meaningful
stimuli from background noise (Hochreiter et al., 2001; Pascanu et al., 2012). RNNs, and in particular
gated-RNNs, can solve this problem by maintaining a representation of relevant input sequences until
needed, without interference from new stimuli. In principle, such protected memories conserve past
inputs and thus allow back-propagation of errors further backwards in time (Pascanu et al., 2012).
Because of their memory properties, one of the first and most successful types of gated-RNNs was
named ?long short-term memory networks? (LSTMs, Hochreiter and Schmidhuber (1997), Fig. 1c).
Here we note that the architectural features of LSTMs overlap closely with known cortical structures,
but with a few important differences with regard to the mechanistic implementation of gates in a
cortical network and LSTMs (Fig. 1b). In LSTMs, the gates control the memory cell as a multiplicative
factor, but in biological networks, the gates, i.e. inhibitory neurons, act (to a first approximation)
subtractively ? excitatory and inhibitory (EI) currents cancel each other linearly at the level of
the postsynaptic membrane potential (Kandel et al., 2000; Gerstner et al., 2014). Moreover, such a
subtractive inhibitory mechanism must be well balanced (i.e. closely match the excitatory input) to act
as a gate to the inputs in the ?closed? state, without perturbing activity flow with too much inhibition.
Previous models have explored gating in subtractive excitatory and inhibitory balanced networks
(Vogels and Abbott, 2009; Kremkow et al., 2010), but without a clear computational role. On the other
hand, predictive coding RNNs with EI features have been studied (Bastos et al., 2012; Deneve and
Machens, 2016), but without a clear match to state-of-the-art machine learning networks. Regarding
previous neuroscientific interpretations of LSTMs, there have been suggestions of LSTMs as models
of working memory and different brain areas (e.g. prefrontal cortex, basal ganglia and hippocampus)
(O?Reilly and Frank, 2006; Krueger and Dayan, 2009; Cox and Dean, 2014; Marblestone et al., 2016;
Hassabis et al., 2017; Bhalla, 2017), but without a clear interpretation of the individual components
of LSTMs and a specific mapping to known circuits.
We propose to map the architecture and function of LSTMs directly onto cortical circuits, with
gating provided by lateral subtractive inhibition. Our networks have the potential to exhibit the
excitation-inhibition balance observed in experiments (Douglas et al., 1989; Bastos et al., 2012;
Harris and Mrsic-Flogel, 2013) and yield simpler gradient propagation than multiplicative gating.
We study these dynamics through our empirical evaluation showing that subLSTMs achieve similar
performance to LSTMs in the Penn Treebank and Wikitext-2 language modelling tasks, as well
as pixelwise sequential MNIST classification. By transferring the functionality of LSTMs into a
biologically more plausible network, our work provides testable hypotheses for the most recently
emerging, technologically advanced experiments on the functionality of entire cortical microcircuits.
2
Biological motivation
The architecture of LSTM units, with their general feedforward structure aided by additional recurrent
memory and controlled by lateral gates, is remarkably similar to the columnar architecture of cortical
circuits (Fig. 1). The central element in LSTMs and similar RNNs is the memory cell, which we
hypothesise to be implemented by local recurrent networks of pyramidal cells in layer-5. This is in
line with previous studies showing a relatively high level of recurrence and non-random connectivity
between pyramidal cells in layer-5 (Douglas et al., 1995; Thomson et al., 2002; Song et al., 2005).
Furthermore, layer-5 pyramidal networks display rich activity on (relatively) long time scales in
vivo (Barth? et al., 2009; Sakata and Harris, 2009; Harris and Mrsic-Flogel, 2013), consistent with
LSTM-like function. Slow memory decay in these networks can be controlled through short- (York
and van Rossum, 2009; Costa et al., 2013, 2017a) and long-term synaptic plasticity (Abbott and
2
Nelson, 2000; Senn et al., 2001; Pfister and Gerstner, 2006; Zenke et al., 2015; Costa et al., 2015,
2017a,b) at recurrent excitatory synapses.
The gates that protect a given memory in LSTMs can be mapped onto lateral inhibitory inputs in
cortical circuits. We propose that, similar to LSTMs, the input gate is implemented by inhibitory
neurons in layer-2/3 (or layer-4; Fig. 1a). Such lateral inhibition is consistent with the canonical view
of microcircuits (Douglas et al., 1989; Bastos et al., 2012; Harris and Mrsic-Flogel, 2013) and sparse
sensory-evoked responses in layer-2/3 (Sakata and Harris, 2009; Harris and Mrsic-Flogel, 2013).
In the brain, such inhibition is believed to originate from (parvalbumin) basket cells, providing a
near-exact balanced inhibitory counter signal to a given excitatory feedforward inputs (Froemke et al.,
2007; Xue et al., 2014; Froemke, 2015). Excitatory and inhibitory inputs thus cancel each other and
arriving signals are ignored by default. Consequently, any activity within the downstream memory
network remains largely unperturbed, unless it is altered through targeted modulation of the inhibitory
activity (Harris and Mrsic-Flogel, 2013; Vogels and Abbott, 2009; Letzkus et al., 2015). Similarly,
the memory cell itself can only affect the output of the LSTM when its activity is unaccompanied
by congruent inhibition (mapped onto layer-6 in the same microcircuit or 2/3 (or 4) in a separate
microcircuit), i.e. when lateral inhibition is turned down and the gate is open.
b
cortical circuit
IN
Layer 5
ct
memory
cell
f
PCs
L4 or 2/3
=
baseline
strong inh.
100
inh. = exc.
2
3
4
0
strong inh.
inh. = exc.
1
2
input
3
4
xt,ht-1
unit j
xt,ht-1 (input)
multiplicative gating
200
we
a
100
xt,ht-1
it
zt
unit j
baseline
e
at
kg
ea
w
100
0
xt,ht-1
ft
cell
xt,ht-1
it
subtractive gating
200
baseline
1
ct
xt,ht-1 (input)
weak inh.
200
0
=1
zt
unit j
ht (output)
ot
f
cell
PC
LSTM
xt,ht-1
=1
-
IN
input
output rate (Hz)
-
PC
ht (output)
ot
h.
L6, L4 or 2/3
output
=
c
subLSTM
k in
a
strong gate
closed gate
1
2
3
4
Figure 1: Biological and artificial gated recurrent neural networks. (a) Example unit of a simplified
cortical recurrent neural network. Sensory (or downstream) input arrives at pyramidal cells in
layer-2/3 (L2/3, or layer-4), which is then fed onto memory cells (recurrently connected pyramidal
cells in layer-5). The memory decays with a decay time constant f . Input onto layer-5 is balanced
out by inhibitory basket cells (BC). The balance is represented by the diagonal ?equal? connection.
The output of the memory cell is gated by basket cells at layer-6, 2/3 or 4 within the same area
(or at an upstream brain area). (b) Implementation of (a), following a similar notation to LSTM
units, but with it and ot as the input and output subtractive gates. Dashed connections represent the
potential to have a balance between excitatory and inhibitory input (weights are set to 1) (c) LSTM
recurrent neural network cell (see main text for details). The plots bellow illustrate the different
gating modes: (a) using a simple current-based noisy leaky-integrate-and-fire neuron (capped to
200Hz) with subtractive inhibition; (b) sigmoidal activation functions with subtractive gating; (c)
sigmoidal activation functions with multiplicative gating. Output rate represents the number of spikes
per second (Hz) as in biological circuits.
3
2.1
Why subtractive neural integration?
When a presynaptic cell fires, neurotransmitter is released by its synaptic terminals. The neurotransmitter is subsequently bound by postsynaptic receptors where it prompts a structural change of an ion
channel to allow the flow of electrically charged ions into or out of the postsynaptic cell. Depending
on the receptor type, the ion flux will either increase (depolarise) or decrease (hyperpolarise) the
postsynaptic membrane potential. If sufficiently depolarising ?excitatory? input is provided, the
postsynaptic potential will reach a threshold and fire a stereotyped action potential (?spike?, Kandel
et al. (2000)). This behaviour can be formalised as a RC?circuit (R = resistance, C = capacitance),
which follows Ohm?s laws u = RI and yields the standard leaky-integrate-and-fire neuron model
(Gerstner and Kistler, 2002), ?m u? = ?u + RIexc ? RIinh , where ?m = RC is the membrane time
constant, and Iexc and Iinh are the excitatory and inhibitory (hyperpolarizing) synaptic input currents,
respectively.
Action potentials are initiated in this standard model (Brette and Gerstner, 2005; Gerstner et al.,
2014)) when the membrane potential hits a hard threshold ?. They are modelled as a momentary
pulse and a subsequent reset to a resting potential. Neuronal excitation and inhibition have opposite
effects, such that inhibitory inputs acts linearly and subtractively on the membrane potential.
The leaky-integrate-and-fire model can be approximated at the level of firing rates as rate ?
?1
R(Iexc ?Iinh )
(see Fig. 1a for the input-output response; Gerstner and Kistler (2002)),
?m ln R(I
exc ?Iinh )??
which we used to demonstrate the impact of subtractive gating (Fig. 1b), and contrast it with multiplicative gating (Fig. 1c).
This firing-rate approximation forms the basis for our gated-RNN model which has a similar subtractive behaviour and input-output function (cf. Fig. 1b; bottom). Moreover, the rate formulation also
allows a cleaner comparison to LSTM units and the use of existing machine learning optimisation
methods.
It could be argued that a different form of inhibition (shunting inhibition), which counteracts excitatory
inputs by decreasing the over all membrane resistance, has a characteristic multiplicative gating effect
on the membrane potential. However, when analysed at the level of the output firing rate its effect
becomes subtractive (Holt and Koch, 1997; Prescott and De Koninck, 2003). This is consistent with
our approach in that our model is framed at the firing-rate level (rather than at the level of membrane
potentials).
3
Subtractive-gated long short-term memory
In an LSTM unit (Hochreiter and Schmidhuber, 1997; Greff et al., 2015) the access to the memory
cell ct is controlled by an input gate it (see Fig.1c). At the same time a forget gate ft controls
the decay of this memory1 , and the output gate ot controls whether the content of the memory
cell ct is transmitted to the rest of the network. A LSTM network consists of many LSTM units,
each containing its own memory cell ct , input it , forget ft and output ot gates. The LSTM state
is described as ht = f (xt , ht?1 , it , ft , ot ) and the unit follows the dynamics given in the middle
column below.
LSTM
subLSTM
[ft , ot , it ]T =
?(W xt + Rht?1 + b),
?(W xt + Rht?1 + b),
zt =
tanh(W xt + Rht?1 + b),
?(W xt + Rht?1 + b),
ct =
ct?1 ft + zt it ,
ct?1 ft + zt ? it ,
ht =
tanh(ct ) ot .
?(ct ) ? ot .
Here, ct is the memory cell (note the multiplicative control of the input gate), denotes element-wise
multiplication and zt is the new weighted input given with xt and ht?1 being the input vector and
recurrent input from other LSTM units, respectively. The overall output of the LSTM unit is then
1
Note that this leak is controlled by the input and recurrent units, which may be biologically unrealistic.
4
computed as ht . LSTM networks can have multiple layers with millions of parameters (weights and
biases), which are typically trained using stochastic gradient descent in a supervised setting. Above,
the parameters are W , R and b. The multiple gates allow the network to adapt the flow of information
depending on the task at hand. In particular, they enable writing to the memory cell (controlled by
input gate, it ), adjusting the timescale of the memory (controlled by forget gate, ft ) and exposing the
memory to the network (controlled by output gate, ot ). The combined effect of these gates makes it
possible for LSTM units to capture temporal (contextual) dependencies across multiple timescales.
Here, we introduce and study a new RNN unit, subLSTM. SubLSTM units are a mapping of LSTMs
onto known canonical excitatory-inhibitory cortical microcircuits (Douglas et al., 1995; Song et al.,
2005; Harris and Mrsic-Flogel, 2013). Similarly, subLSTMs are defined as ht = f (xt , ht?1 , it , ft , ot )
(Fig. 1b), however here the gating is subtractive rather than multiplicative. A subLSTM is defined by a
memory cell ct , the transformed input zt and the input gate it . In our model we use a simplified notion
of balance in the gating (?ztj ? ?ijt ) (for the jth unit), where ? = 1. 2 For the memory forgetting we
consider two options: (i) controlled by gates (as in an LSTM unit) as ft = ?(W xt + Rht?1 + b) or
(ii) a more biologically plausible learned simple decay [0, 1], referred to in the results as fix-subLSTM.
Similarly to its input, subLSTM?s output ht is also gated through a subtractive output gate ot (see
equations above). We evaluated different activation functions and sigmoidal transformations had the
highest performance.
The key differences to other gated-RNNs is in the subtractive inhibitory gating (it and ot ) that has
the potential to be balanced with the excitatory input (zt and ct , respectively; Fig. 1b). See below a
more detailed comparison of the different gating modes.
3.1
Subtractive versus multiplicative gating in RNNs
The key difference between subLSTMs and LSTMs lies in the implementation of the gating mechanism. LSTMs typically use a multiplicative factor to control the amplitude of the input signal.
SubLSTMs use a more biologically plausible interaction of excitation and inhibition. An important
consequence of subtractive gating is the potential for an improved gradient flow backwards towards
the input layers. To illustrate this we can compare the gradients for the subLSTMs and LSTMs in a
simple example.
First, we review the derivatives of the loss with respect to the various components of the subLSTM,
using notation based on (Greff et al., 2015). In this notation, ?a represents the derivative of the loss
def
with respect to a, and ?t = dloss
dht , the error from the layer above. Then by chain rule we have:
?ht = ?t
?ot = ??ht ? 0 (ot )
?ct = ?ht ? 0 (ct ) + ?ct+1 ft+1
?f t = ?ct ct?1 ? 0 (f t )
?it = ??ct ? 0 (it )
?zt = ?ct ? 0 (zt )
For comparison, the corresponding derivatives for an LSTM unit are given by:
?ht = ?t
?ot = ht tanh(ct ) ? 0 (ot )
?ct = ht ot tanh0 (ct ) + ?ct+1 ft+1
?f t = ?ct ct?1 ? 0 (f t )
?it = ?ct zt ? 0 (it )
?zt = ?ct it tanh0 (zt )
where ?(?) is the sigmoid activation function and the overlined variables ct , f t , etc. are the preactivation values of a gate or input transformation (e.g. ot = Wo xt + Ro ht?1 + bo for the output
2
These weights could also be optimised, but for this model we decided to keep the number of parameters to a
minimum for simplicity and ease of comparison with LSTMs.
5
gate of a subLSTM). Note that compared to the those of an LSTM, subLSTMs provide a simpler
gradient with fewer multiplicative factors.
Now, the LSTMs weights Wz of the input transformation z are updated according to
?Wz =
T X
T
X
t=0 t0 =t
?t0
?ht0
?ct ?zt
???
,
?ct0
?zt ?Wz
(1)
where T is the total number of temporal steps and the ellipsis abbreviates the recurrent gradient
paths through time, containing the path backwards through time via hs and cs for t ? s ? t0 . For
simplicity of analysis, we ignore these recurrent connections as they are the same in LSTM and
subLSTM, and only consider the depth-wise path through the network; we call this tth timestep
depth-only contribution to the derivative (?Wz )t . For an LSTM, by this slight abuse of notation, we
have
?ht ?ct ?zt ?zt
(?Wz )t = ?t
?ct ?zt ?zt ?Wz
(2)
= ?t ot tanh0 (ct ) it tanh0 (zt ) x>
t ,
|{z}
|{z}
output gate
input gate
where tanh0 (?) is the derivative of tanh. Notice that when either of the input or output gates are set to
zero (closed), the corresponding contributions to the gradient are zero. For a network with subtractive
gating, the depth-only derivative contribution becomes
(3)
(?Wz )t = ?t ? 0 (ct ) ? 0 (zt ) x>
t ,
where ? 0 (?) is the sigmoid derivative. In this case, the input and output gates, ot and it , are not
present. As a result, the subtractive gates in subLSTMs do not (directly) impair error propagation.
4
Results
The aims of our work were two-fold. First, inspired by cortical circuits we aimed to propose a
biological plausible implementation of an LSTM unit, which would allow us to better understand
cortical architectures and their dynamics. To compare the performance of subLSTM units to LSTMs,
we first compared the learning dynamics for subtractive and multiplicative networks mathematically.
In a second step, we empirically compared subLSTM and fix-subLSTM with LSTM networks in
two tasks: sequential MNIST classification and word-level language modelling on Penn Treebank
(Marcus et al., 1993) and Wikitext-2 (Merity et al., 2016). The network weights are initialised with
Glorot initialisation (Glorot and Bengio, 2010), and LSTM units have an initial forget gate bias of 1.
We selected the number of units for fix-subLSTM such that the number of parameters is held constant
across experiments to facilitate fair comparison with LSTMs and subLSTMs.
4.1
Sequential MNIST
In the ?sequential? MNIST digit classification task, each digit image from the MNIST dataset is
presented to the RNN as a sequence of pixels (Le et al. (2015); Fig. 2a) We decompose the MNIST
images of 28?28 pixels into sequences of 784 steps. The network was optimised using RMSProp
with momentum (Tieleman and Hinton, 2012), a learning rate of 10?4 , one hidden layer and 100
hidden units. Our results show that subLSTMs achieves similar results to LSTMs (Fig. 2b). Our
results are comparable to previous results using the same task (Le et al., 2015) and RNNs.
4.2
Language modelling
Language modelling represents a more challenging task for RNNs, with both short and long-term
dependencies. RNN language models (RNN LMs) models the probability of text by autoregressively
predicting a sequence of words. Each timestep is trained to predict the following word; in other
words, we model the word sequence as a product of conditional multinoulli distributions. We evaluate
the RNN LMs by measuring their perplexity, defined for a sequence of n words as
perplexity = P (w1 , . . . , wn )?1/n .
6
(4)
27
97
.
.2
9
.9
6
97
97.5
95.0
92.5
90.0
LS
TM
su
bL
ST
fix
M
-s
ub
LS
TM
28x28
...
100.0
97
b
seq.
testing accuracy (%)
a
Figure 2: Comparison of LSTM and subLSTM networks for sequential pixel-by-pixel MNIST, using
100 hidden units. (a) Samples from MNIST dataset. We converted each matrix of 28?28 pixels into
a temporal sequence of 784 timesteps. (b) Classification accuracy on the test set. fix-subLSTM has a
fixed but learned forget gate.
We first used the Penn Treebank (PTB) dataset to train our model on word-level language modelling
(929k training, 73k validation and 82k test words; with a vocabulary of 10k words).
All RNNs tested have 2 hidden layers; backpropagation is truncated to 35 steps, and a batch size of 20.
To optimise the networks we used RMSProp with momentum. We also performed a hyperparameter
search on the validation set over input, output, and update dropout rates, the learning rate, and
weight decay. The hyperparameter search was done with Google Vizier, which performs black-box
optimisation using Gaussian process bandits and transfer learning. Tables 2 and 3 show the resulting
hyperparameters. Table 1 reports perplexity on the test set (Golovin et al., 2017). To understand how
subLSTMs scale with network size we varied the number of hidden units between 10, 100, 200 and
650.
We also tested the Wikitext-2 language modelling dataset based on Wikipedia articles. This dataset
is twice as large as the PTB dataset (2000k training, 217k validation and 245k test words) and also
features a larger vocabulary (33k words). Therefore, it is well suited to evaluate model performance
on longer term dependencies and reduces the likelihood of overfitting.
On both datasets, our results show that subLSTMs achieve perplexity similar to LSTMs (Table 1a
and 1b). Interestingly, the more biological plausible version of subLSTM (with a simple decay as
forget gates) achieves performance similar to or better than subLSTMs.
(a) Penn Treebank (PTB) test perplexity
(b) Wikitext-2 test perplexity
size
subLSTM
fix-subLSTM
LSTM
size
subLSTM
fix-subLSTM
LSTM
10
100
200
650
222.80
91.46
79.59
76.17
213.86
91.84
81.97
70.58
215.93
88.39
74.60
64.34
10
100
200
650
268.33
103.36
89.00
78.92
259.89
105.06
94.33
79.49
271.44
102.77
86.15
74.27
Table 1: Language modelling (word-level) test set perplexities on (a) Penn Treebank and
(b) Wikitext-2. The models have two layers and fix-subLSTM uses a fixed but learned forget
gate f = [0, 1] for each unit. The number of units for fix-subLSTM was chosen such that the number
of parameters were the same as those of (sub)LSTM to facilitate fair comparison. Size indicates the
number of units.
The number of hidden units for fix-subLSTM were selected such that the number of parameters were
the same as LSTM and subLSTM, facilitating fair comparison.
7
Model
hidden
units
input
dropout
output
dropout
update
dropout
learning
rate
weight
decay
LSTM
subLSTM
fix-subLSTM
10
10
11
0.026
0.012
0.009
0.047
0.045
0.043
0.002
0.438
0
0.01186
0.01666
0.01006
0.000020
0.000009
0.000029
LSTM
subLSTM
fix-subLSTM
100
100
115
0.099
0.392
0.194
0.074
0.051
0.148
0.015
0.246
0.042
0.00906
0.01186
0.00400
0.000532
0.000157
0.000218
LSTM
subLSTM
fix-subLSTM
200
200
230
0.473
0.337
0.394
0.345
0.373
0.472
0.013
0.439
0.161
0.00496
0.01534
0.00382
0.000191
0.000076
0.000066
LSTM
subLSTM
fix-subLSTM
650
650
750
0.607
0.562
0.662
0.630
0.515
0.730
0.083
0.794
0.530
0.00568
0.00301
0.00347
0.000145
0.000227
0.000136
Table 2: Penn Treebank hyperparameters.
Model
hidden
units
input
dropout
output
dropout
update
dropout
learning
rate
weight
decay
LSTM
subLSTM
fix-subLSTM
10
10
11
0.015
0.002
0.033
0.039
0.030
0.070
0
0.390
0.013
0.01235
0.00859
0.00875
0
0.000013
0
LSTM
subLSTM
fix-subLSTM
100
100
115
0.198
0.172
0.130
0.154
0.150
0.187
0.002
0.009
0
0.01162
0.00635
0.00541
0.000123
0.000177
0.000172
LSTM
subLSTM
fix-subLSTM
200
200
230
0.379
0.342
0.256
0.351
0.269
0.273
0
0.018
0
0.00734
0.00722
0.00533
0.000076
0.000111
0.000160
LSTM
subLSTM
fix-subLSTM
650
650
750
0.572
0.633
0.656
0.566
0.567
0.590
0.071
0.257
0.711
0.00354
0.00300
0.00321
0.000112
0.000142
0.000122
Table 3: Wikitext-2 hyperparameters.
5
Conclusions & future work
Cortical microcircuits exhibit complex and stereotypical network architectures that support rich
dynamics, but their computational power and dynamics have yet to be properly understood. It is
known that excitatory and inhibitory neuron types interact closely to process sensory information
with great accuracy, but making sense of these interactions is beyond the scope of most contemporary
experimental approaches.
LSTMs, on the other hand, are a well-understood and powerful tool for contextual tasks, and their
structure maps intriguingly well onto the stereotyped connectivity of cortical circuits. Here, we
analysed if biologically constrained LSTMs (i.e. subLSTMs) could perform similarly well, and indeed,
such subtractively gated excitation-inhibition recurrent neural networks show promise compared
against LSTMs on benchmarks such as sequence classification and word-level language modelling.
While it is notable that subLSTMs could not outperform their traditional counterpart (yet), we hope
that our work will serve as a platform to discuss and develop ideas of cortical function and to establish
links to relevant experimental work on the role of excitatory and inhibitory neurons in contextual
learning (Froemke et al., 2007; Froemke, 2015; Poort et al., 2015; Pakan et al., 2016; Kuchibhotla
et al., 2017). In future work, it will be interesting to study how additional biological detail may affect
8
performance. Next steps should aim to include Dale?s principle (i.e. that a given neuron can only
make either excitatory or inhibitory connections, Strata and Harvey (1999)), and naturally focus
on the perplexing diversity of inhibitory cell types (Markram et al., 2004) and behaviour, such as
shunting inhibition and mixed subtractive and divisive control (Doiron et al., 2001; Mejias et al.,
2013; El Boustani and Sur, 2014; Seybold et al., 2015).
Overall, given the success of multiplicative gated LSTMs, it will be most insightful to understand if
some of the biological tricks of cortical networks may give LSTMs a further performance boost.
Acknowledgements
We would like to thank Everton Agnes, ?a?glar G?l?ehre, Gabor Melis and Jake Stroud for helpful
comments and discussion. R.P.C. and T.P.V. were supported by a Sir Henry Dale Fellowship by the
Wellcome Trust and the Royal Society (WT 100000). Y.M.A. was supported by the EPSRC and the
Research Council UK (RCUK). B.S. was supported by the Clarendon Fund.
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12
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6,226 | 6,632 | k-Support and Ordered Weighted Sparsity for
Overlapping Groups: Hardness and Algorithms
Cong Han Lim
University of Wisconsin-Madison
[email protected]
Stephen J. Wright
University of Wisconsin-Madison
[email protected]
Abstract
The k-support and OWL norms generalize the `1 norm, providing better prediction
accuracy and better handling of correlated variables. We study the norms obtained
from extending the k-support norm and OWL norms to the setting in which there
are overlapping groups. The resulting norms are in general NP-hard to compute,
but they are tractable for certain collections of groups. To demonstrate this fact,
we develop a dynamic program for the problem of projecting onto the set of
vectors supported by a fixed number of groups. Our dynamic program utilizes tree
decompositions and its complexity scales with the treewidth. This program can
be converted to an extended formulation which, for the associated group structure,
models the k-group support norms and an overlapping group variant of the ordered
weighted `1 norm. Numerical results demonstrate the efficacy of the new penalties.
1
Introduction
The use of the `1 -norm to induce sparse solutions is ubiquitous in machine learning, statistics, and
signal processing. When the variables can be grouped into sets corresponding to different explanatory
factors, group variants of the `1 penalty can be used to recover solutions supported on a small number
of groups. When the collection of groups G forms a partition of the variables (that is, the groups do
not overlap), the group lasso penalty [19]
X
?GL (x) :=
kxG kp
(1)
G?G
is often used. In many cases, however, some variables may contribute to more than one explanatory
factor, which leads naturally to overlapping-group formulations. Such is the case in applications such
as finding relevant sets of genes in a biological process [10] or recovering coefficients in wavelet
trees [17]. In such contexts, the standard group lasso may introduce artifacts, since variables that are
contained in different numbers of groups are penalized differently. Another approach is to employ
the latent group lasso [10]:
X
X
?LGL (x) := min
kvG kp such that
vG = x,
(2)
x,v
G?G
G?G
where each vG is a separate vector of latent variables supported only on the group G. The latent
group lasso (2) can be written in terms of atomic norms, where the atomic set is
{x : kxkp ? 1, supp(x) ? G for some G ? G} .
This set allows vectors supported on any one group. The unit ball is the convex hull of this atomic set.
A different way of extending the `1 -norm involves explicit use of a sparsity parameter k. Argyriou
et al. [1] introduce the k-support norm ?k from the atomic norm perspective. The atoms are the set
of k-sparse vectors with unit norm, and the unit ball of the norm is thus
conv ({x : kxkp ? 1, |supp(x)|? k}) .
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(3)
The k-support norm with p = 2 offers a tighter alternative to the elastic net, and like the elastic net, it
has better estimation performance than the `1 norm especially in the presence of correlated variables.
Another extension of the `1 norm is to the OSCAR/OWL/SLOPE norms [5, 20, 4], which order the
elements of x according to magnitude before weighing them:
X
(4)
?OWL (x) :=
wi |x?i |.
i?[n]
where the weights wi , i = 1, 2, . . . , n are nonnegative and decreasing and x? denotes the vector x
sorted by decreasing absolute value. This family of norms controls the false discovery rate and clusters
correlated variables. These norms correspond to applying the `? norm to a combinatorial penalty
function in the framework of Obozinski and Bach [11, 12], and can be generalized by considering
different `p -norms. For p = 2, we have the SOWL norm [18], whose variational form is
X
?SOWL (x) := 12 min??Rn+
x2i /?i + wi |?i? | .
i?[n]
We will refer to the generalized version of these norms as pOWL norms. The pOWL norms can be
viewed as extensions of the k-support norms from the atomic norm angle, which we will detail later.
`1
(Non-overlapping)
Group Lasso
(Non-overlapping)
k-Group
Support
k-support
OSCAR
OWL/SOWL
SLOPE
(Non-overlapping)
GrOWL
Latent
Group Lasso
Latent kGroup Support
(LG(k))
Latent Group
Smooth OWL
(LGS-OWL)
Figure 1: Some sparsity-inducing norms. Each arrow represents an extension of a previous norm. We study
the two shaded norms on the right.
In this paper, we study the norms obtained by combining the overlapping group formulations with
the k-sparse/OWL formulations, with the aim of obtaining the benefits of both worlds. When the
groups do not overlap, the combination is fairly straightforward; see the GrOWL norm introduced
by Oswal et al. [13]. We consider two classes of norms for overlapping groups. The latent k-group
support (LG(k)) norm, very recently introduced by Rao et al. [15], is defined by the unit ball
n
o
[
conv x : kxkp ? 1, supp(x) ?
G for some subset Gk ? G with k groups ,
(5)
G?Gk
directly extending the k-support norm definition to unions of groups. We introduce the latent group
smooth OWL (LGS-OWL) norm, which similarly extends OWL/SOWL/GrOWL. These norms can
be applied in the same settings where the latent group lasso has proven to be useful, while adapting
better to correlations. We explain how the norms are derived from a combinatorial penalty perspective
using the work of Obozinski and Bach [11, 12], and also provide explicit atomic-norm formulations.
The LGS-OWL norm can be seen as a combination of k-support norms across different k.
The rest of this focuses on computational aspects of these norms. Both the LG(k) norm and
the LGS-OWL norm are in general NP-hard to compute. Despite this hardness result, we devise
a computational approach that utilizes tree decompositions of the underlying group intersection
graph. The key parameter affecting the efficiency of our algorithms is the treewidth tw of the group
intersection graph, which is small for certain graph structures such as chains, trees, and cycles.
Certain problems with hierarchical groups like image recovery can have a tree structure [17, 3].
Our first main technical contribution is a dynamic program for the best k-group sparse approximation
problem, which has time complexity O(2O(tw) ? mk + n), where m is the total number of groups.
For group intersection graphs with a tree structure (tw = 2), this leads to a O(mk + n) algorithm,
significantly improving on the O(m2 k + n) algorithm presented in [3]. Next, we build on the
principles behind the dynamic program to construct extended formulations of O(2O(tw) ? mk 2 + n)
2
size for LG(k) and O(2O(tw) ? m3 + n) for LGS-OWL, improving by a factor of k or m respectively
in the special case in which the tree decomposition is a chain. This approach also yields extended
formulations of size O(nk) and O(n2 ) for the k-support and pOWL norms, respectively. (Previously,
only a O(n2 ) linear program was known for OWL [5].) We thus facilitate incorporation of these
norms into standard convex programming solvers.
Related Work. Obozinski and Bach [11, 12] develop a framework for penalties derived by convexifying the sum of a combinatorial function F and an `p term. They describe algorithms for computing
the proximal operators and norms for the case of submodular F . We use their framework, but note
that the algorithms they provide cannot be applied since our functions are not submodular.
Two other works focus directly on sparsity of unions of overlapping groups. Rao et al. [15] introduce
the LG(k) norm and approximates it via variable splitting. Baldassarre et al. [3] study the best
k-group sparse approximation problem, which they prove is NP-hard. For tree-structured intersection
graphs, they derive the aforementioned dynamic program with complexity O(m2 k + n).
For the case of p = ?, a linear programming relaxation for the unit ball of the latent k-group
support norm is provided by Halabi and Cevher [9, Section 5.4]. This linear program is tight if the
group-element incidence matrix augmented with an all-ones row is totally unimodular. This condition
can be violated by simple tree-structured intersection graphs with just four groups.
Notation and Preliminaries. Given A ? [n], the vector xA is the subvector of x ? Rn corresponding to the index set A. For collections
S of groups G, we use m to denote the number of groups in G,
that is, m = |G|. We assume that G?G G = [n], so that every index i ? [n] appears in at least one
group G ? G. The discrete function CG (A) denotes the minimum number of groups from G needed
to cover A (the smallest set cover).
2
Overlapping Group Norms with Group Sparsity-Related Parameters
We now describe the LG(k) and LGS-OWL norms from the combinatorial penalty perspective by
Obozinski and Bach [11, 12], providing an alternative theoretical motivation for the LG(k) norm
and formally motivating and defining LGS-OWL. Given a combinatorial function F : {A ? [n]} ?
R ? {+?} and an `p norm, a norm can be derived by taking the tightest positively homogeneous
convex lower bound of the combined penalty function F (supp(x)) + ?kxkpp . Defining q to satisfy
1/p + 1/q = 1 (so that `p and `q are dual), this procedure results in the norm ?F
p , which is given by
F
1/q
1/p
1/q
the convex envelope of the function ?p (x) := q (p?) F (supp(x)) kxkp , whose unit ball is
n
o
conv x ? Rn : kxkp ? F (supp(x))?1/q
.
(6)
The norms discussed in this paper can be cast in this framework. Recall that the definition of OWL (4)
includes nonnegative weights w1 ? w2 ? . . . wn ? 0. Defining h : [n] ? R to be the monotonically
Pk
increasing concave function h(k) = i=1 wi , we obtain
?
?
A = ?,
?0,
?0, A = ?,
k-support : F (A) = 1,
LG(k)
: F (A) = 1, CG (A) ? k,
|A|? k,
?
?
?, otherwise,
?, otherwise,
pOWL
: F (A) = h(|A|),
LGS-OWL : F (A) = h(CG (A)).
The definitions of the k-support and LG(k) balls from (3) and (5), respectively, match (6). As for the
OWL norms, we can express their unit ball by
!
m n
o
[
n
?1/q
conv
x ? R : kxkp ? h(i)
, CG (supp(x)) = i
.
(7)
i=1
This can be seen as taking all of the k-support or LG(k) atoms for each value of k, scaling them
according to the value of k, then taking the convex hull of the resulting set. Hence, the OWL norms
can be viewed as a way of interpolating the k-support norms across all values of k. We take advantage
of this interpretation in constructing extended formulations.
3
Hardness Results. Optimizing with the cardinality or non-overlapping group based penalties is
straightforward, since the well-known PAV algorithm [2] allows us to exactly compute the proximal
operator in O(n log n) time [12]. However, the picture is different when we allow overlapping groups.
There are no fast exact algorithms for overlapping group lasso, and iterative algorithms are typically
used. Introducing the group sparsity parameters makes the problem even harder.
Theorem 2.1. The following problems are NP-hard for both ?LG(k) and ?LGS-OWL when p > 1:
Compute ?(y),
arg minx?Rn
arg minx?Rn
1
2 kx
1
2 kx
?
?
(norm computation)
yk22 such that
yk22 +??(x).
?(x) ? ?,
(projection operator)
(proximal operator)
Therefore, other problems that incorporate these norm are also hard. Note that even if we only allow
each element to be in at most two groups, the problem is already hard. We will show in the next two
sections that these problems are tractable if the treewidth of the group intersection graph is small.
3
A Dynamic Program for Best k-Group Approximation
The best k-group approximation problem is the discrete optimization problem
arg min ky ? xk22 such that CG (supp(x)) ? k,
(8)
x
where the goal is to compute the projection of a vector y onto a union of subspaces each defined by a
subcollection of k groups. The solution to (8) has the form
yi i in chosen support,
x0i =
0 otherwise.
As mentioned above, Baldassarre et al. [3] show that this problem is NP-hard. They provide a
dynamic program that acts on the group intersection graph and focus specifically on the case where
this graph is a tree, obtaining a O(m2 k + n) dynamic programming algorithm. In this section, we
also start by using group intersection graphs, but instead focus on the tree decomposition of this
graph, which yields a more general approach.
3.1
Group Intersection Graphs and Tree Decompositions
We can represent the interactions between the different groups using an intersection graph,
which is an undirected graph IG = (G, EG ) in which each vertex denotes a group and two
groups are connected if and only if they overlap. For example, if the collection of groups is
{{1, 2, 3}, {3, 4, 5}, {5, 6, 7}, . . . }, then the intersection graph is simply a chain. If each group
corresponds to a parent and all its children in a rooted tree, the intersection graph is also a tree.
The group intersection graph highlights the dependencies between different groups. Algorithms for
this problem need to be aware of how picking one group may affect the choice of another, connnected
group. (If the groups do not overlap, then no groups are connected and a simple greedy approach
suffices.) A tree decomposition of IG is a more precise way of representing these dependencies.
We provide the definition of tree decompositions and treewidth below and illustrate the core ideas in
Figure 2. Tree decompositions are a fundamental tool in parametrized complexity, leading to efficient
algorithms if the parameter in question is small. See [7, 8] for a a more comprehensive overview
Figure 2: From groups to a group intersection graph to a tree decomposition of width 2.
4
A tree decomposition of (V, E) is a tree T with vertices X = {X1 , X2 , . . . , XN }, satisfying the
following conditions: (1) Each Xi is a subset of V , and the union of all sets Xi gives V . (2) For every
edge (v, w) in E, there is a vertex Xi that contains both v and w. (3) For each v ? V , the vertices
that contain v form a connected subtree of T . The width of a tree decomposition is maxi |Xi |?1, and
the treewidth of a graph (denoted by tw) is the smallest width among all tree decompositions. The
tree decomposition is not unique, and there is always a tree width with number of nodes |X |? |V |
(see for example Lemma 10.5.2 in [7]). Henceforth, we will assume that |X |? m.
The treewidth tw is modest for many types of graphs. For example, the treewidth is bounded
for the tree (tw = 1), the cycle (tw = 2), and series-parallel graphs (tw = 2). Computing tree
decompositions with optimal width for these
? graphs can be done in linear time. On the other hand,
the grid graph has large treewidth (tw ? n) and checking if a graph has tw ? k is NP-complete1 .
3.2
A Dynamic Program for Tree Decompositions
Given a collection of groups G, a corresponding tree decomposition T (G) of the group intersection
graph, and a vector y ? Rn , we provide a dynamic program for problem (8), the best k-group
approximation of y.
The tree decomposition has several features that we can exploit. The tree structure provides a natural
order for processing the vertices, which are subcollections of groups. Properties (1) and (2) yield a
natural way to map elements i ? [n] onto vertices in the tree, indicating when to include yi in the
process. Finally, the connected subtree corresponding to each group G as a result of property (3)
means that we only need to keep explicit information about G for that part of the computation.
The high-level view of our approach is described below. Details appear in the supplementary material.
Preprocessing: For each i ? [n], let G(i) denote the set of all groups that contain i. We have three
data structures: A and V, which are both indexed by (X, Y ), with X ? X and Y ? X; and T, which
is indexed by (X, Y, s), with s ? {0, 1 . . . , k}.
1. Root the tree decomposition and process the nodes from root to leaves: At each node X,
add an index i ? [n] to A(X, G(i) ) if i is unassigned and G(i) ? X.
P 2
2. Set V(X, Y ) ?
yi : i ? A(X, G(i) ), Y ? G(i) 6= 0 .
Main Process: At each vertex X in the tree decomposition, we are allowed to pick groups Y to
include in the support of the solution. The s term in T(X, Y, s) indicates the current group sparsity
?budget? that has been used. Proposition 3.2 below gives the semantic meaning behind each entry in
T . We process the nodes from the leaves to the root to fill T . At each step, the entries for node Xp
will be updated with information from its children.
The update for a leaf Xp is simply T(Xp , Yp , s) ? V(Xp , Yp ) if |Yp |= s. If |Yp |6= s, we mark
T(Xp , Yp , s) as invalid. For non-leaf Xp , we need to ensure that the groups chosen by the parent and
the child are compatible. We ensure this property via constraints of the form Yc ? Xp = Yp ? Xc .
For a single child Xc we have
T(Xp , Yp , s) ?
max
T(Xc , Y, s ? s0 ) : |Yp ? Xc |= s0 + V(Xp , Yp ),
(9)
Y :Y ?Xp =Yp ?Xc
and finally for Xp multiple children Xc(1) , . . . , Xc(d) of Xp , we set T(Xp , Yp , s) as
?
?
?
?
X
[
max
T(Xc(i) , Yi , si ) : Yp ?
Xc(i) = s0 + V(Xp , Yp ). (10)
Yi :Yi ?Xp =Yp ?Xc(i) ?P
?
d
for each i
i=1
i?[d]
si =s?s0
After making each update, we keep track of which Yi was used for each of the children for
T (Xp , Yp , s). This allows us to backtrack to recover the solution after T has been filled.
The next lemma and proposition prove the correctness of this dynamic program. The lemma follows
from the fact that every clique in a graph is contained in some node in any tree decomposition, while
the proposition from induction from the leaf nodes.
1
Nonetheless, there is significant research on developing exact and heuristic tree decomposition algorithms.
There are regular competitions for better implementations [6, pacechallenge.wordpress.com].
5
Lemma 3.1. Every index in [n] is assigned in the first preprocessing step.
Proposition 3.2. For a node X, let yX be the y vector restricted to just the indices i assigned to
nodes below and including X. Each entry T(X, Y, s) is the squared `2 -norm of the best projection of
yX , subject to the fact that besides the groups in Y , at most s ? |Y | are allowed to be used.
We now prove the time complexity of this algorithm. Proposition 3.4 describes the time complexity of
the update when there are many children. It uses the following simple lemma about max-convolutions.
Computing the other updates is straightforward.
Lemma 3.3. The max-convolution f between two concave functions g1 , g2 : {0, 1, . . . , k} ? R,
defined by f (i) := maxj {g1 (j) + g2 (i ? j)}, can be computed in O(k) time.
Proposition 3.4. The update (10) for a fixed Xp , Yp across all values s ? {0, 1, . . . , k} can be
implemented in O(2O(tw) ? dk) time.
Combining timing and correctness results gives us the desired algorithmic result. This approach
significantly improves on the results of Baldassarre et al. [3]. Their approach is specific to groups
whose intersection graph is a tree and uses O(m2 k + n) time.
Theorem 3.5. Given G and a corresponding tree decomposition TG with treewidth tw, projection
onto the corresponding k-group model can be done in O(2O(tw) ? (mk + n)) time. When the group
intersection graph is a tree, the projection takes O(mk + n) time.
4
Extended Formulations from Tree Decompositions
Here we model explicitly the unit ball of LG(k) (5) and LGS-OWL (7). The principles behind this
formulation are very similar to the dynamic program in the previous section.
We first consider the latent k-group support norm, whose atoms are
n
o
[
x : kxkp ? 1, supp(x) ?
G for some subset Gk ? G with k groups .
G?Gk
The following process describes a way of selecting an atom; our extended formulation encodes this
process mathematically. We introduce variables b, which represent the `p budget at a given node,
choice of groups, and group sparsity budget. We start at the root Xr , with `p budget of ? and group
sparsity budget of k:
X
b(Xr ,Y,k?|Y |) ? ?.
(11)
We then start moving towards the leaves, as follows.
1. Suppose we have picked some the groups at a node. Assign some of the `p budget to the xi
terms, where the index i is compatible with the node and the choice of groups.
2. Move on to the child and pick the groups we want to use, considering only groups that
are compatible with the parent. Debit the group budget accordingly. If there are multiple
children, spread the `p and group budgets among them before picking the groups.
The first step is represented by the following relations. Intermediate variables z and a are required to
ensure that we spread the `p budget correctly among the valid xi .
b(X,Y,s) ? k(z (X,Y,s) , u(X,Y,s) )kp ,
(X,Y,s)
(X,(Y,Y 0 ),s)
0
:Y ?Y =
6 ?}kp ,
? k{a
X
0
a(X,Y ,s) ?
a(X,(Y,Y ),s) ,
Y ?X
X
Xk
0
kxA(X,Y ) k2 ?
a(X,Y ,s) .
0
z
0
A(X,Y )=A(X,Y )
s=0
The second step is represented by the following inequality in the case of a single child.
o
Xn
u(Xp ,Yp ,s) ?
b(Xc ,(Yp ,Y ),s?s0 ) : Y ? Xp = Yp ? Xc , |Yp ? Xc |= s0 .
6
(12)
(13)
(14)
(15)
(16)
When there are multiple children, we need to introduce more intermediate variables to spread the
group budget correctly. The technique here is similar to the one used in the proof of Proposition 3.4;
we defer details to the supplementary material. In both cases, we need to collect the budgets that have
been sent from each Yp :
X
b(Xc ,Y,s) ?
b(Xc ,(Yp ,Y ),s) .
(17)
Yp
Those b variables unreachable by the budget transfer process are set to 0. Our main theorem about the
correctness of the construction in this section follows from the fact that when ? = 1, every extreme
point with nonzero x in our extended formulation is an atom of the corresponding LG(k).
Theorem 4.1. We can model the set ?LG(k) (x) ? ? using O(2O(tw) ? (mk 2 + n)) variables and
inequalities in general. When the tree decomposition is a chain, O(2O(tw) ? (mk + n)) suffices.
For the unit ball of ?LGS-OWL , we can exploit the fact that the atoms of ?LGS-OWL are obtained from
?LG(k) across different k at different scales. Instead of using the inequality (11) at the node, we have
X
h(k)1/q b(Xr ,Y,k?|Y |) ? ?,
Y ?Xr
O(tw)
which leads to a program of size O(2
5
? (m2 + n)) for chains and O(2O(tw) ? (m3 + n)) for trees.
Empirical Observations and Results
The extended formulations above can be implemented in modeling software such as CVX. This
may incur a large processing overhead, and it is often faster to implement these directly in a convex
optimization solver such as Gurobi or MOSEK. Use of the `? -norm leads to a linear program which
can be significantly faster than the second-order conic program that results from the `2 -norm.
We evaluated the performance of LG(k) and LGS-OWL on linear regression problems minx 12 ky ?
Axk2 +??(x). In the scenarios considered, we use the latent group lasso as a baseline. We test both
the `2 and `? variants of the various norms. Following [13] (which descrbes GrOWL), we consider
two different types of weights for LGS-OWL. The linear variant sets wi = 1 ? (i ? 1)/n for i ? [n],
whereas in the spike version, we set w1 = 1 and wi = 0.25 for i = 2, 3, . . . , n. The regularization
term ? was chosen by grid search over {10?2 , 10?1.95 , . . . , 104 } for each experiment.
The metrics we use are support recovery and estimation quality. For the support recovery experiments,
we count the number of times the correct support was identified. We also compute the root mean
square (RMSE) of kx ? x? k2 (estimation error).2
We had also tested the standard lasso, elastic net, and k-support and OWL norms, but these norms
performed poorly. In our experiments they were not able to recover the exact correct support in
any run. The estimation performance for the k-support norms and elastic net were worse than the
corresponding latent group lasso, and likewise for OWL vs. LGS-OWL.
Experiments. We used 20 groups of variables where each successive group overlaps by two elements with the next [10, 14]. The groups are given by {1, . . . , 10}, {9, . . . , 18}, . . . , {153, . . . , 162}.
For the first set of experiments, the support of the true input x? are a cluster of five groups in the
middle of x, with xi = 1 on the support. For the second set of experiments, the original x is supported
by the two disjoint clusters of five overlapping groups each, with xi = 2 on one cluster and xi = 3
on the other.
Each entry of the A matrix is chosen initially to be i.i.d. N (0, 1). We then introduce correlations
between groups in the same cluster in A. Within each cluster of groups, we replicate the same set of
columns for each group in the non-overlapping portions of the group (that is, every pair of groups in
a cluster shares at least 6 columns, and adjacent groups share 8 columns). We then introduce noise by
adding i.i.d. elements from N (0, 0.05) so that the replications are not exact. Finally, we generate y
by adding i.i.d. noise from N (0, 0.3) to each component of Ax? .
We present support recovery results in Figure 3 for the `2 variants of the norms which perform better
than the`? versions, though the relative results between the different norms hold. In the appendix we
provide the graphs for support recovery and estimation quality as well as other observations.
2
It is standard in the literature to compute the RMSE of the prediction or estimation quality. RMSE metrics
are not ideal in practice since we should ?debias? x to offset shrinkage due to the regularization term.
7
Figure 3: Support recovery performance as number of measurements (height of A) increases. The vertical
axis indicates the number of trials (out of 100) for which the correct support was identified. The two left graphs
correspond to the first configuration of group supports (five groups), while the others to the second configuration
(ten groups). Each line represents a different method. In the first and third graphs, we plot LG(k) for different
values of k, increasing from 1 to the ?ground truth? value. Note that k = 1 is exactly the latent group lasso. In
the second and fourth graphs, we plot LGS-OWL for the different choices of weights wi discussed in the text.
Our methods can significantly outperform latent group lasso in both support recovery and estimation
quality. We provide a summary below and more details are provided in the supplementary.
We first focus on support recovery. There is a significant jump in performance when k is the size
of the true support. Note that exceeding the ground-truth value makes recovery of the true support
impossible in the presence of noise. For smaller values of k, the results range from slight improvement
(especially when k = 4 or k = 8 in the first and second experiments respectively) to mixed results (for
large number of rows in A and small k). The LGS-OWL norms can provide performance almost as
good as the best settings of k for LG(k), and can be used when the number of groups is unknown. We
expect to see better performance for well-tuned OWL weights. We see similar results for estimation
performance. Smaller values of k provide little to no advantage, while larger values of k and the
LGS-OWL norms can offer significant improvement.
6
Discussion and Extensions
We introduce a variant of the OWL norm for overlapping groups and provide the first tractable
approaches for this and the latent k-group support norm (via extended formulations) under a bounded
treewidth condition. The projection algorithm for the best k-group sparse approximation problem
generalizes and improves on the algorithm by Baldassarre et al. [3]. Numerical results demonstrate
that the norms can provide significant improvement in support recovery and estimation.
A family of graphs with many applications and large treewidth is the set of grid graphs. Groups over
collections of adjacent pixels/voxels lead naturally to such group intersection graphs, and it remains
an open question whether polynomial time algorithms exist for this set of graphs. Another venue for
research is to derive and evaluate efficient approximations to these new norms.
It is tempting to apply recovery results on the latent group lasso here, since LG(k) can be cast as a
latent group lasso instance with groups {G0 : G0 is a union of up to k groups of G}. The consistency
results of [10] only applies under the strict condition that the target vector is supported exactly by a
unique set of k groups. The Gaussian width results of [16] do not give meaningful bounds even when
the groups are disjoint and k = 2. Developing theoretical guarantees on the performance of these
methods requires a much better understanding of the geometry of unions of overlapping groups.
We can easily extend the dynamic program to handle the case in which we want both k-group sparsity,
and overall sparsity of s. For tree-structured group intersection graphs, our dynamic program has
? 2 ks2 + mn) by [3]. This yields a variant of
time complexity O(mks + n log s) instead of the O(m
the above norms that again has a similar extended formulations. These variants could be employed as
an alternative to the sparse overlapping set LASSO by Rao et al. [14]. We leave this to future work.
Acknowledgements This work was supported by NSF award CMMI-1634597, ONR Award
N00014-13-1-0129, and AFOSR Award FA9550-13-1-0138.
8
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6,227 | 6,633 | A simple model of recognition and recall memory
Nisheeth Srivastava
Computer Science, IIT Kanpur
Kanpur, UP 208016
[email protected]
Edward Vul
Dept of Psychology, UCSD
9500 Gilman Drive La Jolla CA 92093
[email protected]
Abstract
We show that several striking differences in memory performance between recognition and recall tasks are explained by an ecological bias endemic in classic memory
experiments - that such experiments universally involve more stimuli than retrieval
cues. We show that while it is sensible to think of recall as simply retrieving
items when probed with a cue - typically the item list itself - it is better to think
of recognition as retrieving cues when probed with items. To test this theory, by
manipulating the number of items and cues in a memory experiment, we show
a crossover effect in memory performance within subjects such that recognition
performance is superior to recall performance when the number of items is greater
than the number of cues and recall performance is better than recognition when
the converse holds. We build a simple computational model around this theory,
using sampling to approximate an ideal Bayesian observer encoding and retrieving
situational co-occurrence frequencies of stimuli and retrieval cues. This model
robustly reproduces a number of dissociations in recognition and recall previously
used to argue for dual-process accounts of declarative memory.
1
Introduction
Over half a century, differences in memory performance in recognition and recall-based experiments
have been a prominent nexus of controversy and confusion. There is broad agreement among
memory researchers, following Mandler?s influential lead, that there are at least two different types
of memory activities - recollection, wherein we simply remember something we want to remember,
and familiarity, wherein we remember having seen something before, but nothing more beyond
it [8]. Recall-based experiments are obvious representatives of recollection. Mandler suggested that
recognition was a good example of familiarity activity.
Dual-process accounts of memory question Mandler?s premise that recognition is exclusively a
familiarity operation. They argue, phenomenologically, that recognition could also succeed successful
recollection, making the process a dual composition of recollection and familiarity [20]. Experimental
procedures and analysis methods have been designed to test for the relative presence of both processes
in recognition experiments, with variable success. These endeavors contrast with strength-based
single-process models of memory that treat recognition as the retrieval of a weak trace of item
memory, and recall as retrieval of a stronger trace of the same item [19].
The single/dual process dispute also spills over into the computational modeling of memory. Gillund
and Shiffrin?s influential SAM model is a single-process account of both recognition and recall [4]. In
SAM and other strength-based models of declarative memory, recognition is modeled as item-relevant
associative activation of memory breaching a threshold, while recall is modeled as sampling items
from memory using the relative magnitudes of these associative activations. In contrast, McClelland?s
equally influential CLS model is explicitly a dual-process model, where a fast learning hippocampal
component primarily responsible for recollection sits atop a slow learning neocortical component
responsible for familiarity [9]. Wixted?s signal detection model tries to bridge the gap between
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
these accounts by allowing dual process contributions to combine additively into a unidimensional
strength variable [19]. While such pragmatic syntheses are useful, the field is still looking for a more
satisfactory theoretical unification.
The depth of the difference between the postulated dual processes of recollection and familiarity
depends inevitably on the strength of the quantitative and qualitative dissociations that previous
research has documented in memory tasks, prominent among which are recognition and recall.
Mandler, for instance, postulated a one-to-one mapping between recognition and familarity on one
hand and recall and recollection on the other [8], although other authors hold more nuanced views [20].
Notwithstanding such differences of opinion, the road to discovering useful single-process accounts
of declarative memory has to go through explaining the multiple performance dissociations between
recognition and recall memory tasks. To the extent that single process accounts of both tasks can
explain such dissociations, differences between recollection and familarity will not seem nearly as
fundamental.
Improved strength-based models have competently modeled a large array of recognition-recall
dissociations [13], but fail, or have to make intricate assumptions, in the face of others [20]. More
importantly, the SAM model and its descendants are not purely single-process models. They model
recognition as a threshold event and recall as a sampling event, with the unification coming from
the fact that both events occur using the same information base of associative activation magnitude.
We present a much simpler single process model that capably reproduces many critical qualitative
recognition-recall dissociations. In the process, we rationalize the erstwhile abstract associative
activation of strength-based memory models as statistically efficient monitoring of environmental
co-occurrence frequencies. Finally, we show using simulations and a behavioral experiment, that the
large differences between recognition and recall in the literature can be explained by the responses of
an approximately Bayesian observer tracking these frequencies to two different questions.
2
Model
We use a very simple model, specified completely by heavily stylized encoding and retrieval processes.
The encoding component of our model simply learns the relative frequencies with which specific
conjunctions of objects are attended to in the world. We consider objects x of only two types: items
xi and lists xl . We model each timestep as as a Bernoulli trial between the propensity to attend to
any of the set of items or to the item-list itself, with a uniform prior probability of sampling any
of the objects. Observers update the probability of co-occurrence, defined in our case rigidly as
1-back occurrence, inductively as the items on the list are presented. We model this as the observer?s
sequential Bayesian updates of the probability p(x), stored at every time step as a discrete memory
engram m.
Thus, in this encoding model, information about the displayed list of items is available in distributed
form in memory as p(xi , xl |m), with each engram m storing one instance of co-occurrence. The true
joint distribution of observed items,to the extent that it is encoded within the set of all task-relevant
memory engrams M is then expressible as a simple probabilistic marginalization,
X
p(xi , xl ) =
p(xi , xl |m)p(m),
(1)
m?M
where we assume that p(m) is flat over M, i.e. we assume that within the set of memory engrams
relevant for the retrieval cue, memory access is random.
Our retrieval model is approximately Bayesian. It assumes that people sample a small subset of all
relevant engrams M0 ? M when making memory judgments. Thus, the joint distribution accessible
to the observer during retrieval becomes a function of the set of engrams actually retrieved,
X
pMk (xi , xl ) =
p(xi , xl |m)p(m),
(2)
m?Mk
where Mk denotes the set of first k engrams retrieved.
Following a common approach to sampling termination in strength-based sequential sampling memory
models, we use a novelty threshold that allows the memory retrieval process to self-terminate when
incoming engrams no longer convey significantly novel information [4, 13]. We treat the arrival of the
2
Recall
Recognition
Retrieval
Encoding
Retrieval
Encoding
A
A
B
B
C
C
Sample from p( |A)
p(x| )
p( |x)
Sample from p(x| )
A
A
B
X
C
Figure 1: Illustrating the ecological difference in retrieval during recognition and recall memory
experiments. We model recall retrieval as a probabilistic query about items conditioned on the item
list and recognition retrieval as a probabilistic query about the item list conditioned on the item
presented during retrieval. Since there are almost always more items than lists in classic memory
experiments, the second conditional distribution tends to be formed on a smaller discrete support set
than the former.
k th successive engram into working memory as a signal for the observer to probabilistically sample
from pMk . The stopping rule for memory retrieval in our model is for n consecutive identical samples
being drawn in succession during this internal sampling, n remaining a free parameter in the model.
This rule is designed to capture the fact that memory search is frugal and self-terminating [15]. The
sample drawn at the instant the novelty threshold is breached is overtly recalled. Since this sample is
drawn from a distribution constructed by approximately reconstructing the true encoded distribution
of situational co-occurrences, the retrieval model is approximately Bayesian. Finally, since our
encoding model ensures that the observer knows the joint distribution of event co-occurrences, which
contains all the information needed to compute marginals and conditionals also, we further assume
that these derivative distributions can also be sampled, using the same retrieval model, when required.
We show in this paper that this simple memory model yields both recognition and recall behavior.
The difference between recognition and recall is simply that these two retrieval modalities ask two
different questions of the same base of encoded memory - the joint distribution p(xi , xl ). We illustrate
this difference in Figure 1. During recall-based retrieval, experimenters ask participants to remember
all the items that were on a previously studied list. In this case, the probabilistic question being asked
is ?given xl , find xi ?, which our model would answer by sampling p(xi |xl ). In item-recognition
experiments, experimenters ask participants to determine whether each of several items was on a
previously shown list or not. We assert that in this case the probabilistic question being asked is
?given xi . find xl ?, which our model would answer by sampling p(xl |xi ).
Our operationalization of recognition as a question about the list rather than the item runs contrary to
previous formalizations, which have tended to model it as the associative activation engendered in
the brain by observing a previously seen stimulus - models of recognition memory assume that the
activation for previously seen stimuli is greater, for all sorts of reasons. In contrast, recall is modeled
in classical memory accounts much the same way as in ours - as a conditional activation of items
associated with retrieval cues, including both the item list and temporally contiguous items. Our
approach assumes that the same mechanism of conditional activation occurs in recognition as well the difference is that we condition on the item itself.
3
3
Basic prediction: fast recognition and slow recall
The sample-based threshold used to terminate memory retrieval in our model does not depend on
the size of the support of the probability distribution being sampled from. This immediately implies
that, for the same threshold sample value, the model will take longer to approach it when sampling
from a distribution with larger support than when sampling from distributions with smaller support.
In classical memory experiments, observers are typically asked to memorize multiple items associated
with one, or a few, lists. Thus, there is an ecological bias built into classic memory experiments such
that |items| |lists|. Making this assumption immediately rationalizes the apparent difference in
speed and effort between recognition and recall in our model. Because the recognition task samples
p(list|item), its sample complexity is lower than recall, which involves sampling p(item|list) from
memory.
To verify this numerically, starting from identical memory encodings in both cases, we ran 1000
simulations of recognition and recall respectively using our retrieval model, using a fixed n = 5 1 .
The results, measured in terms of the number of retrieval samples k drawn before termination in
each of the 1000 trials, are shown in the left panel of Figure 2. The sample complexity of recall is
evidently higher than for recognition2 . Thus, we suggest that the fundamental difference between
recognition and recall - that recognition is easier and recall is harder - is explicable simply by virtue
of the ecological bias of memory experiments that use fewer cues than stimuli.
The difference in speed between recollection and familiarity processes, as measured in recall and
recognition experiments, has been one of the fundamental motivations for proposing that two memory
processes are involved in declarative memory. Dual-process accounts have invoked priority arguments
instead, e.g. that information has to pass through semantic memory, which is responsible for
recognition, before accessing episodic memory which is responsible for recall [17].Single process
accounts following in the lineage of SAM [4] have explained the difference by arguing that recognition
involves a single comparison of activation values to a threshold, whereas recall involves competition
between multiple activations for sampling. Our model rationalizes this distinction made in SAM-style
sequential sampling models by arguing that recognition memory retrieval is identical to recall memory
retrieval; only the support of the distribution from which the memory trace is to be probabilistically
retrieved changes. Thus, instead of using a race to threshold for recognition and a sampling process
in recall, this model uses self-terminating sampling in both cases, explaining the main difference
between the two tasks - easy recognition and hard recall - as a function of typical ecological parameter
choices. This observation also explains the relative indifference of recognition tasks to divided
attention conditions, in contrast with recall which is heavily affected [2]. Because of the lower sample
complexity of recognition, fewer useful samples are needed to arrive at the correct conclusion.
4
An empirical test
The explanation our model offers is simple, but untested. To directly test it, we constructed a simple
behavioral experiment, where we would manipulate the number of items and cues keeping the total
number of presentations constant, and see how this affected memory performance in both recognition
and recall retrieval modalities. Our model predicts that memory performance difficulty scales up with
the size of the support of the conditional probability distribution relevant to the retrieval modality.
Thus recall, which samples from p(item|list), should become easier as the number of items to recall
per cue reduces. Similarly recognition, which samples from p(listlitem), should become harder as
the number of cues per item increases. Because classic memory experiments have tended to use more
items than cues (lists), our model predicts that such experiments would consistently find recognition
to be easier than recall. By inverting this pattern, having more cues than items, for instance, we would
expect to see the opposite pattern hold. We tested for this performance crossover using the following
experiment.
1
Our results are relatively independent of the choice of n, since for any value of n, recognition stays easier
than recall so long as the cue-item fan out remains large and vice versa.
2
Recall trials that timed out by not returning a sample beyond the maximum time limit (100 samples) are not
plotted. These corresponded to 55% of the trials, resulting in a recall hit rate of 45%. In contrast, the average
recognition hit rate was 82% for this simulation.
4
Recognition
400
Recall
2.5
Recognition
Recall
400
300
300
200
200
100
100
d'
Trials
2
1.5
1
0.5
0
0
50
100
0
0
50
100
0
{2,7}
Sample count
{3,5}
{4,4}
{5,3}
{7,2}
Condition
Figure 2: (Left) Simulation results show easier recognition and harder recall given typical ecological
choices for stimuli and cue set sizes. (Right) Results from experiment manipulating the stimuli and
cue set size ratio. By manipulating the number of stimuli and cues, we predicted that we would
be able to make recall harder than recognition for experiment participants. The results support our
prediction unambiguously.Error bars show s.e.m.
We used a 2?2 within subject factorial design for this experiment, testing for the effect of the retrieval
mode - recognition/recall and either a stimulus heavy, or cue heavy selection of task materials. In
addition, we ran two conditions between subjects, using different parameterization of the stimuli/cue
ratios. In the stimulus heavy condition, for instance, participants were exposed to 5 stimuli associated
with 3 cues, while for the cue heavy condition, they saw 3 stimuli associated with 5 cues. The semantic
identity of the stimuli and cue sets were varied across all four conditions randomly, and the order of
presentation of conditions to participants was counterbalanced. All participants worked on all four
of the memory tasks, with interference avoided with the use of semantically distinct category pairs
across the four conditions. Specifically, we used number-letter, vegetable-occupation, fruit-adjective
and animal-place category pairs for the four conditions. Within each category, stimuli/cues for a
particular presentation were sampled from a 16 item master list, such that a stimulus could not occur
twice in conjunction with the same cue, but could occur in conjunction with multiple cues.
120 undergraduates participated in the experiment for course credit. Voluntary consent was obtained
from all participants, and the experimental protocol was approved by an institutional IRB. We told
experiment participants that they would be participating in a memory experiment, and their goal was
to remember as many of the items we showed them as possible. We also told them that the experiment
would have four parts, and that once they started working on a part, there would be no opportunity to
take a break until it ended. 80 participants performed the experiment with 3/5 and 5/3 stimulus-to-cue
mappings, 40 did it with 2/7 and 7/2 stimulus-to-cue mappings. Note that in all cases, participants
saw approximately the same number of total stimulus-cue bindings (3x5 = 15 or 2x7 = 14), thus
undergoing equivalent cognitive load during encoding.
Stimuli and cues were presented onscreen, with each pair appearing on the screen for 3 seconds,
followed by an ITI of equal duration. To prevent mnemonic strategy use at the time of encoding, the
horizontal orientation of the stimulus-cue pair was randomly selected on each trial, and participants
were not told beforehand which item category would be the cue; they could only discover this at
the time of retrieval3 . Participants were permitted to begin retrieval at their own discretion once
the encoding segment of the trial had concluded within each condition. All participants chose to
commence retrieval without delay. Participants were also permitted to take breaks of between 2-5
minutes between working on the different conditions, with several choosing to do so.
Once participants had seen all item-pairs for one of the conditions, the experiment prompted them to,
when ready, click on a button to proceed to the testing phase. In the recall condition, they saw a text
box and a sentence asking them to recall all the items that occurred alongside item X, where X was
randomly chosen from the set of possible cues for that condition; they responded by typing in the
words they remembered. For recognition, participants saw a sentence asking them to identify if X had
occurred alongside Y, where Y was randomly chosen from the set of possible cues for that condition.
3
An active weblink to the actual experiment is available online at [anonymized weblink].
5
After each forced yes/no response, a new X was shown. Half the X?s shown in the recognition test
were ?lures? , they had not been originally displayed alongside Y.
Memory performance was measured using d?, which is simply the difference between the z-normed
hit rate and false alarm rate, as is conventional in recognition experiments. d? is generally not used
to measure recall performance, since the number of true negatives is undefined in classic recall
experiments, which leaves the false alarm rate undefined as well. In our setup, the number of true
negatives is obviously the number of stimuli the participant saw that were not on the specific list
being probed, which is what we used to calculate d-prime for recall as well.
The right panel in Figure 2 illustrates the results of our experiment. The predicted crossover is
unambiguously observed. Further, changes in memory performance across the stimulus-cue set size
manipulation is symmetric across recognition and recall. This is precisely what we?d expect if set
size dependence was symmetrically affecting memory performance across both tasks as occurs in
our model. While not wishing to read too much into the symmetry of the quantitative result, we note
that such symmetry under a simple manipulation of the retrieval conditions appears to suggest that
the manipulation does in fact affect memory performance very strongly. Overall, the data strongly
supports our thesis - that quantitative differences in memory performance in recognition and recall
tasks are driven by differences in the set size of the underlying memory distribution being sampled.
The set size of the distribution being sampled, in turn, is determined by task constraints - and ends up
being symmetric when comparing single-item recognition with cued recall.
5
Predicting more recognition-recall dissociations
The fact that recognition is usually easier than recall - more accurate and quicker for the same stimuli
sets - is simply the most prominent difference between the two paradigms. Experimentalists have
uncovered a number of interesting manipulations in memory experiments that affect performance
on these tasks differentially. These are called recognition-recall dissociations, and are prominent
challenges to single-process accounts of the two tasks. Why should a manipulation affect only one
task and not the other if they are both outcomes of the same underlying process? [20] Previous
single-process accounts have had success in explaining some such dissociations. We focus here
on some that have proved relatively hard to explain without making inelegant dissociation-specific
assumptions in earlier accounts [13].
5.1
List strength effects and part set cuing
Unidimensional strength-based models of memory like SAM and REM fail to predict the list strength
effect [12] where participants? memory performance in free recall is lower than a controlled baseline
for weaker items on mixed lists (lists containing both strongly and weakly encoded items). Such
behavior is predicted easily by strength-based models. What they find difficult to explain is that
performance does not deviate from baseline in recognition tasks. The classical explanation for this
discrepancy is the use of a differentiation assumption. It is assumed that stronger items are associated
more strongly to the encoding context, however differences between the item itself as shown, and
its encoded image are also stronger. In free recall, this second interaction does not have an effect,
since the item itself is not presented, so a positive list strength effect is seen. In recognition, it is
conjectured that the two influences cancel each other out, resulting in a null list strength effect [13].
A lot of intricate assumptions have to hold for the differentiation account to hold. Our model has
a much simpler explanation for the null list-strength effect in recognition. Recognition involves
sampling based on the strength of the associative activation of the list given a specific item and so
is independent of the encoding strength of other items. On the other hand, recall involves sampling
from p(item|list) across all items, in which case, having a distribution favoring other items will
reduce the probability that the unstrengthened items will be sampled. Thus, the difference in which
variable the retrieval operation conditions on explains the respective presence and absence of a list
strength effect in recall and recognition.
The left panel in Figure 3 presents simulation results from our model reproducing this effect, where we
implement mixed lists by presenting certain stimuli more frequently during encoding and retrieve in
the usual manner. Hit rates are calculated for less frequently presented stimuli. For either elicitation
modality, the actual outcome of the retrieval itself is sampled from the appropriate conditional
6
distribution as a specific item/cue. In this particular experiment, which manipulates how much
training observers have on some of the items on the list, the histories entering the simulation are
generated such that some items co-occur with the future retrieval cue more frequently than others, i.e.
two items occur with a probability of 0.4 and 0.3 respectively, and three items occur with a probability
of 0.1 each alongside the cue. The simulation shows a positive list strength effect for recall (weaker
hit rates for less studied items) and a null list strength effect for recognition, congruent with data.
Our model also reconciles the results of [1] who demonstrated that the list strength effect does not
occur if we examine only items that are the first in their category to be retrieved. For categoryinsensitive strength-based accounts, this is a serious problem. For our account, which is explicitly
concerned with how observers co-encode stimuli and retrieval cues, this result is no great mystery.
For multi-category memory tests, the presence of each semantic category instantiates a novel list
during encoding, such that the strength-dependent updates during retrieval apply to each individual
p(item|list) and do not apply across the other category lists.
More generally, the dynamic nature of the sampled distribution in our Bayesian theory accommodates
the theoretical views of both champions of strength-dependent activation and retrieval-dependent
suppression [1]. Strength-dependent activation is present in our model in the form of the Bayesian
posterior over cue-relevant targets at the time when cued recall commences; retrieval-dependent
suppression of competitors is present in the form of normalization of the distribution during further
sequential Bayesian updates as the retrieval process continues. Assigning credit differentially to
individual categories predicts an attenuation (though not removal) of the list strength effect, due
to the absence of learning-induced changes for the first-tested items, as well diminishing memory
performance with testing position seen in [1].
Mixed
0.8
0.6
0.4
Recognition FAR
0.8
Recall HR
Recall FAR
0.6
0.4
0.2
0.2
0
Recognition HR
Baseline
1.0
Fraction
Weak item hit rate
1
Recognition
Recall
0
0
0.2
0.4
0.6
0.8
1
Prepend ratio
Figure 3: Reproducing (left) list strength effects and (right) the word frequency mirror effect using
our model.
The part set cueing effect is the observation that showing participants a subset of the items to be
recalled during retrieval reduces their recall performance for non-shown items [11]. This effect does
not appear in recognition experiments, which is again problematic for unidimensional strength-based
memory models. Our model has a simple explanation. The presented items during retrieval are simply
treated as further encoding opportunities for the seen items, resulting in a list strength imbalance as
above. This affects recall, but not recognition for the same reasons the list strength effect does.
5.2
Mirror effect
Another interesting effect that strength-based memory models have found hard to explain is the
word-frequency mirror effect [5]. This is seen when participants see two different classes of items
in recognition experiments. It is found, for instance, that unique items are both recognized more
accurately as previously seen and unseen in such experiments than common items. Such a pattern of
memory performance is contrary to the predictions of nearly all accounts of memory that depend
on unidimensional measures of memory strength, who can only model adaptive changes in memory
performance via shifts in the response criterion [19] that do not permit both the hit rate and the false
alarm rate to improve simultaneously.
7
The essential insight of the mirror effect is that some types of stimuli are intrinsically more memorable
than others, a common-sense observation that has proved surprisingly difficult for strength-based
memory models to assimilate.This difficulty extends to our own model also, but our inductive framework allows us to express the assumptions about information that the stimuli base frequency adds
to the picture in a clean way. Specifically, in our model observers use p(list|item) for recognition, which is high for unique items and low for common items by Bayesian inversion because
p(item|list)/p(item) ? 1 for unique items, because they are unlikely to have been encountered
outside the experimental context, and 1 for common items. In contrast, observers sample from
p(item|list) during recall, removing the effect of the frequency base rate p(item), so that the pattern
of results is inverted: performance is equivalent or better than baseline for common stimuli than for
rare ones [6], since they are more likely to be retrieved in general.
The right panel in Figure 3 shows simulation results using our model wherein we used two possible
cues during encoding, one to test performance during retrieval and one to modify the non-retrieval
frequency of stimuli encounters. For this experiment, which manipulates where we have to influence
how often the relative frequency with which the observers have seen the items in task-irrelevant
contexts other than the retrieval task, we prepended the base case history (of size 50 time steps)
with differently sized prior history samples (between 10 and 50 time steps long, in steps of 5),
wherein items co-occurred with cues that were not used during retrieval. The simulation results show
that, in recognition, hit rates drop and false alarm rates rise with more exposure to items outside
the experimental list context (high frequency items). Since our model assumes unambiguous cue
conditioning, it predicts unchanged performance from baseline for recall. More intricate models that
permit cue-cue associations may reproduce the advantage for common items documented empirically.
6
Discussion
We have made a very simple proposal in this paper. We join multiple previous authors in arguing that
memory retrieval in cued recall tasks can be interpreted as a question about the likelihood of retrieving
an item given the retrieval cue, typically the list of items given at the time of encoding [17, 8, 4].
We depart from previous authors in arguing that memory retrieval in item recognition tasks asks the
precisely opposite question: what is the likelihood of a given item having been associated with the
list? We integrated this insight into a simple inference-based model of memory encoding, which
shares its formal motivations with recent inference-based models of conditioning [3, 14], and an
approximately Bayesian model of memory retrieval, which samples memory frugally along lines
motivated on information-theoretic [18] and ecological grounds [16] by recent work.
Our model is meant to be expository and ignores several large issues that other richer models typically
engage with. For instance, it is silent about the time decay of memory particles, the partitioning of
the world into items and cues, and how it would go about explaining other more intricate memory
tasks like plurality discrimination and remember-know judgments. These omissions are deliberate, in
the sense that we wanted to present a minimal model to deliver the core intuition behind our approach
- that differences in memory performance in recognition and recall are attributable to no deeper
issue than an ecological preference to test memory using more items than lists. This observation
can now subsequently guide and constrain the construction of more realistic models of declarative
memory [3]. To the extent that differences traditionally used to posit dual-process accounts of memory
can be accounted for using simpler models like ours, the need to proliferate neuroanatomical and
process-level distinctions for various memory operations can be concomitantly reduced.
The distinction between recall and recognition memory also has important implications for the
presumed architecture of machine learning systems. Modern ML systems increasingly rely on a
combination of distributed representation of sensory information (using deep nets) and state-centric
representation of utility information (using reinforcement learning) to achieve human-like learning
and transfer capabilities, for example in simple Atari games [10]. The elicitation of class or category
membership in neural networks is quintessentially a recognition task, while the elicitation of state
value functions, as well as other intermediate computations in RL are clearly recall tasks. Partly
in realization of the large differences in the sort of memory required to support these two classes
of learning models, researchers have taken to postulating dual-process artificial memories [7]. Our
demonstration of the fundamental unitarity of the two modes of memory performance can and should
constrain the design of deep RL models in simpler ways.
8
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6,228 | 6,634 | On Structured Prediction Theory with Calibrated
Convex Surrogate Losses
Anton Osokin
INRIA/ENS?, Paris, France
HSE?, Moscow, Russia
Francis Bach
INRIA/ENS?, Paris, France
Simon Lacoste-Julien
MILA and DIRO
Universit? de Montr?al, Canada
Abstract
We provide novel theoretical insights on structured prediction in the context of
efficient convex surrogate loss minimization with consistency guarantees. For any
task loss, we construct a convex surrogate that can be optimized via stochastic
gradient descent and we prove tight bounds on the so-called ?calibration function?
relating the excess surrogate risk to the actual risk. In contrast to prior related
work, we carefully monitor the effect of the exponential number of classes in the
learning guarantees as well as on the optimization complexity. As an interesting
consequence, we formalize the intuition that some task losses make learning harder
than others, and that the classical 0-1 loss is ill-suited for structured prediction.
1
Introduction
Structured prediction is a subfield of machine learning aiming at making multiple interrelated
predictions simultaneously. The desired outputs (labels) are typically organized in some structured
object such as a sequence, a graph, an image, etc. Tasks of this type appear in many practical domains
such as computer vision [34], natural language processing [42] and bioinformatics [19].
The structured prediction setup has at least two typical properties differentiating it from the classical
binary classification problems extensively studied in learning theory:
1. Exponential number of classes: this brings both additional computational and statistical challenges.
By exponential, we mean exponentially large in the size of the natural dimension of output, e.g., the
number of all possible sequences is exponential w.r.t. the sequence length.
2. Cost-sensitive learning: in typical applications, prediction mistakes are not all equally costly.
The prediction error is usually measured with a highly-structured task-specific loss function, e.g.,
Hamming distance between sequences of multi-label variables or mean average precision for ranking.
Despite many algorithmic advances to tackle structured prediction problems [4, 35], there have been
relatively few papers devoted to its theoretical understanding. Notable recent exceptions that made
significant progress include Cortes et al. [13] and London et al. [28] (see references therein) which
proposed data-dependent generalization error bounds in terms of popular empirical convex surrogate
losses such as the structured hinge loss [44, 45, 47]. A question not addressed by these works is
whether their algorithms are consistent: does minimizing their convex bounds with infinite data lead
to the minimization of the task loss as well? Alternatively, the structured probit and ramp losses are
consistent [31, 30], but non-convex and thus it is hard to obtain computational guarantees for them.
In this paper, we aim at getting the property of consistency for surrogate losses that can be efficiently
minimized with guarantees, and thus we consider convex surrogate losses.
The consistency of convex surrogates is well understood in the case of binary classification [50, 5, 43]
and there is significant progress in the case of multi-class 0-1 loss [49, 46] and general multi?
?
DI ?cole normale sup?rieure, CNRS, PSL Research University
National Research University Higher School of Economics
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
class loss functions [3, 39, 48]. A large body of work specifically focuses on the related tasks of
ranking [18, 9, 40] and ordinal regression [37].
Contributions. In this paper, we study consistent convex surrogate losses specifically in the context
of an exponential number of classes. We argue that even while being consistent, a convex surrogate
might not allow efficient learning. As a concrete example, Ciliberto et al. [10] recently proposed a
consistent approach to structured prediction, but the constant in their generalization error bound can
be exponentially large as we explain in Section 5. There are two possible sources of difficulties from
the optimization perspective: to reach adequate accuracy on the task loss, one might need to optimize
a surrogate loss to exponentially small accuracy; or to reach adequate accuracy on the surrogate loss,
one might need an exponential number of algorithm steps because of exponentially large constants
in the convergence rate. We propose a theoretical framework that jointly tackles these two aspects
and allows to judge the feasibility of efficient learning. In particular, we construct a calibration
function [43], i.e., a function setting the relationship between accuracy on the surrogate and task
losses, and normalize it by the means of convergence rate of an optimization algorithm.
Aiming for the simplest possible application of our framework, we propose a family of convex
surrogates that are consistent for any given task loss and can be optimized using stochastic gradient
descent. For a special case of our family (quadratic surrogate), we provide a complete analysis
including general lower and upper bounds on the calibration function for any task loss, with exact
values for the 0-1, block 0-1 and Hamming losses. We observe that to have a tractable learning
algorithm, one needs both a structured loss (not the 0-1 loss) and appropriate constraints on the
predictor, e.g., in the form of linear constraints for the score vector functions. Our framework also
indicates that in some cases it might be beneficial to use non-consistent surrogates. In particular, a
non-consistent surrogate might allow optimization only up to specific accuracy, but exponentially
faster than a consistent one.
We introduce the structured prediction setting suitable for studying consistency in Sections 2 and 3.
We analyze the calibration function for the quadratic surrogate loss in Section 4. We review the
related works in Section 5 and conclude in Section 6.
2
Structured prediction setup
In structured prediction, the goal is to predict a structured output y ? Y (such as a sequence, a graph,
an image) given an input x ? X . The quality of prediction is measured by a task-dependent loss
? y | x) ? 0 specifying the cost for predicting y? when the correct output is y. In this
function L(y,
paper, we consider the case when the number of possible predictions and the number of possible
labels are both finite. For simplicity,1 we also assume that the sets of possible predictions and correct
outputs always coincide and do not depend on x. We refer to this set as the set of labels Y, denote its
cardinality by k, and map its elements to 1, . . . , k. In this setting, assuming that the loss function
depends only on y? and y, but not on x directly, the loss is defined by a loss matrix L ? Rk?k . We
assume that all the elements of the matrix L are non-negative and will use Lmax to denote the maximal
element. Compared to multi-class classification, k is typically exponentially large in the size of the
natural dimension of y, e.g., contains all possible sequences of symbols from a finite alphabet.
Following standard practices in structured prediction [12, 44], we define the prediction model by
a score function f : X ? Rk specifying a score fy (x) for each possible output y ? Y. The final
prediction is done by selecting a label with the maximal value of the score
pred(f(x)) := argmax fy? (x),
(1)
?
y?Y
with some fixed strategy to resolve ties. To simplify the analysis, we assume that among the labels
with maximal scores, the predictor always picks the one with the smallest index.
The goal of prediction-based machine learning consists in finding a predictor that works well on
the unseen test set, i.e., data points coming from the same distribution D as the one generating the
training data. One way to formalize this is to minimize the generalization error, often referred to as
the actual (or population) risk based on the loss L,
RL (f) := IE(x,y)?D L pred(f(x)), y .
(2)
Minimizing the actual risk (2) is usually hard. The standard approach is to minimize a surrogate risk,
which is a different objective easier to optimize, e.g., convex. We define a surrogate loss as a function
1
Our analysis is generalizable to rectangular losses, e.g., ranking losses studied by Ramaswamy et al. [40].
2
? : Rk ? Y ? R depending on a score vector f = f(x) ? Rk and a target label y ? Y as input
arguments. We denote the y-th component of f with fy . The surrogate risk (the ?-risk) is defined as
R? (f) := IE(x,y)?D ?(f(x), y),
(3)
where the expectation is taken w.r.t. the data-generating distribution D. To make the minimization
of (3) well-defined, we always assume that the surrogate loss ? is bounded from below and continuous.
Examples of common surrogate
losses include the structured hinge-loss [44, 47] ?SSVM (f , y) :=
? y) ? fy , the logP
maxy?Y
fy? + L(y,
loss (maximum likelihood learning) used, e.g., in conditional
?
random fields [25], ?log (f , y) := log( y?Y
exp fy? ) ? fy , and their hybrids [38, 21, 22, 41].
?
? y) := [y? 6= y],2 and the
In terms of task losses, we consider the unstructured 0-1 loss L01 (y,
? y) :=
two following structured losses: block 0-1 loss with b equal blocks of labels L01,b (y,
[y? and y are not in the same block]; and (normalized) Hamming loss between tuples of T binary
P
? y) := T1 Tt=1 [?
yt 6= yt ]. To illustrate some aspects of our analysis, we also
variables yt : LHam,T (y,
look at the mixed loss L01,b,? : a convex combination of the 0-1 and block 0-1 losses, defined as
L01,b,? := ?L01 + (1 ? ?)L01,b for some ? ? [0, 1].
3
Consistency for structured prediction
3.1
Calibration function
We now formalize the connection between the actual risk RL and the surrogate ?-risk R? via the
so-called calibration function, see Definition 1 below [5, 49, 43, 18, 3]. As it is standard for this
kind of analysis, the setup is non-parametric, i.e. it does not take into account the dependency of
scores on input variables x. For now, we assume that a family of score functions FF consists of all
vector-valued Borel measurable functions f : X ? F where F ? Rk is a subspace of allowed score
vectors, which will play an important role in our analysis. This setting is equivalent to a pointwise
analysis, i.e, looking at the different input x independently. We bring the dependency on the input
back into the analysis in Section 3.3 where we assume a specific family of score functions.
Let DX represent the marginal distribution for D on x and IP(? | x) denote its conditional given x.
We can now rewrite the risk RL and ?-risk R? as
RL (f) = IEx?DX `(f(x), IP(? | x)), R? (f) = IEx?DX ?(f(x), IP(? | x)),
where the conditional risk ` and the conditional ?-risk ? depend on a vector of scores f and a
conditional distribution on the set of output labels q as
Xk
Xk
`(f , q) :=
qc L(pred(f ), c), ?(f , q) :=
qc ?(f , c).
c=1
c=1
The calibration function H?,L,F between the surrogate loss ? and the task loss L relates the excess
surrogate risk with the actual excess risk via the excess risk bound:
H?,L,F (?`(f , q)) ? ??(f , q), ?f ? F, ?q ? ?k ,
(4)
?
?
where ??(f , q) = ?(f , q) ? inf ? ?(f , q), ?`(f , q) = `(f , q) ? inf ? `(f , q) are the excess
f ?F
f ?F
risks and ?k denotes the probability simplex on k elements.
In other words, to find a vector f that yields an excess risk smaller than ?, we need to optimize the
?-risk up to H?,L,F (?) accuracy (in the worst case). We make this statement precise in Theorem 2
below, and now proceed to the formal definition of the calibration function.
Definition 1 (Calibration function). For a task loss L, a surrogate loss ?, a set of feasible scores F,
the calibration function H?,L,F (?) (defined for ? ? 0) equals the infimum excess of the conditional
surrogate risk when the excess of the conditional actual risk is at least ?:
H?,L,F (?) := inf ??(f , q)
(5)
f ?F , q??k
s.t. ?`(f , q) ? ?.
We set H?,L,F (?) to +? when the feasible set is empty.
(6)
By construction, H?,L,F is non-decreasing on [0, +?), H?,L,F (?) ? 0, the inequality (4) holds,
and H?,L,F (0) = 0. Note that H?,L,F can be non-convex and even non-continuous (see examples
in Figure 1). Also, note that large values of H?,L,F (?) are better.
2
Here we use the Iverson bracket notation, i.e., [A] := 1 if a logical expression A is true, and zero otherwise.
3
no constraints
tight constraints
H(")
H(")
no constraints
tight constraints
0
0.5
1
0
"
(a): Hamming loss LHam,T
0.2
0.4
"
(b): Mixed loss L01,b,0.4
Figure 1: Calibration functions for the quadratic surrogate ?quad (12) defined in Section 4 and two
different task losses. (a) ? the calibration functions for the Hamming loss LHam,T when used without
constraints on the scores, F = Rk (in red), and with the tight constraints implying consistency,
F = span(LHam,T ) (in blue). The red curve can grow exponentially slower than the blue one. (b) ?
the calibration functions for the mixed loss L01,b,? with ? = 0.4 (see Section 2 for the definition) when
used without constraints on the scores (red) and with tight constraints for the block 0-1 loss (blue).
The blue curve represents level-0.2 consistency. The calibration function equals zero for ? ? ?/2,
but grows exponentially faster than the red curve representing a consistent approach and thus could
be better for small ?. More details on the calibration functions in this figure are given in Section 4.
3.2
Notion of consistency
We use the calibration function H?,L,F to set a connection between optimizing the surrogate and
task losses by Theorem 2, which is similar to Theorem 3 of Zhang [49].
Theorem 2 (Calibration connection). Let H?,L,F be the calibration function between the surrogate
? ?,L,F be a convex non-decreasing
loss ? and the task loss L with feasible set of scores F ? Rk . Let H
lower bound of the calibration function. Assume that ? is continuous and bounded from below. Then,
? ?,L,F (?) and any f ? FF , we have
for any ? > 0 with finite H
? ?,L,F (?) ? RL (f) < R? + ?,
R? (f) < R? + H
(7)
?,F
L,F
where R??,F := inf f?FF R? (f) and R?L,F := inf f?FF RL (f).
Proof. We take the expectation of (4) w.r.t. x, where the second argument of ` is set to the conditional
? ?,L,F is convex) to get
distribution IP(? | x). Then, we apply Jensen?s inequality (since H
? ?,L,F (RL (f) ? R?L,F ) ? R? (f) ? R??,F < H
? ?,L,F (?),
H
(8)
? ?,L,F .
which implies (7) by monotonicity of H
? ?,L,F (?) required by Theorem 2 always exists, e.g.,
A suitable convex non-decreasing lower bound H
the zero constant. However, in this case Theorem 2 is not informative, because the l.h.s. of (7) is
? ?,L,F defined as the lower convex envelope of
never true. Zhang [49, Proposition 25] claims that H
? ?,L,F (?) > 0, ?? > 0, if H?,L,F (?) > 0, ?? > 0, and,
the calibration function H?,L,F satisfies H
? ?,L,F always exists and
e.g., the set of labels is finite. This statement implies that an informative H
allows to characterize consistency through properties of the calibration function H?,L,F .
We now define a notion of level-? consistency, which is more general than consistency.
Definition 3 (level-? consistency). A surrogate loss ? is consistent up to level ? ? 0 w.r.t. a task
loss L and a set of scores F if and only if the calibration function satisfies H?,L,F (?) > 0 for all
? > ? and there exists ?? > ? such that H?,L,F (?
?) is finite.
Looking solely at (standard level-0) consistency vs. inconsistency might be too coarse to capture
practical properties related to optimization accuracy (see, e.g., [29]). For example, if H?,L,F (?) = 0
only for very small values of ?, then the method can still optimize the actual risk up to a certain
level which might be good enough in practice, especially if it means that it can be optimized faster.
Examples of calibration functions for consistent and inconsistent surrogate losses are shown in
Figure 1.
Other notions of consistency. Definition 3 with ? = 0 and F = Rk results in the standard setting
often appearing in the literature. In particular, in this case Theorem 2 implies Fisher consistency as
4
formulated, e.g., by Pedregosa et al. [37] for general losses and Lin [27] for binary classification.
This setting is also closely related to many definitions of consistency used in the literature. For
example, for a bounded from below and continuous surrogate, it is equivalent to infinite-sample
consistency [49], classification calibration [46], edge-consistency [18], (L, Rk )-calibration [39],
prediction calibration [48]. See [49, Appendix A] for the detailed discussion.
Role of F. Let the approximation error for the restricted set of scores F be defined as R?L,F ?R?L :=
inf f?FF RL (f) ? inf f RL (f). For any conditional distribution q, the score vector f := ?Lq will
yield an optimal prediction. Thus the condition span(L) ? F is sufficient for F to have zero
approximation error for any distribution D, and for our 0-consistency condition to imply the standard
Fisher consistency with respect to L. In the following, we will see that a restricted F can both play a
role for computational efficiency as well as statistical efficiency (thus losses with smaller span(L)
might be easier to work with).
3.3
Connection to optimization accuracy and statistical efficiency
The scale of a calibration function is not intrinsically well-defined: we could multiply the surrogate
function by a scalar and it would multiply the calibration function by the same scalar, without
changing the optimization problem. Intuitively, we would like the surrogate loss to be of order 1. If
with this scale the calibration function is exponentially small (has a 1/k factor), then we have strong
evidence that the stochastic optimization will be difficult (and thus learning will be slow).
To formalize this intuition, we add to the picture the complexity of optimizing the surrogate loss with
a stochastic approximation algorithm. By using a scale-invariant convergence rate, we provide a
natural normalization of the calibration function. The following two observations are central to the
theoretical insights provided in our work:
1. Scale. For a properly scaled surrogate loss, the scale of the calibration function is a good indication
of whether a stochastic approximation algorithm will take a large number of iterations (in the worst
case) to obtain guarantees of small excess of the actual risk (and vice-versa, a large coefficient
indicates a small number of iterations). The actual verification requires computing the normalization
quantities given in Theorem 6 below.
2. Statistics. The bound on the number of iterations directly relates to the number of training
examples that would be needed to learn, if we see each iteration of the stochastic approximation
algorithm as using one training example to optimize the expected surrogate.
To analyze the statistical convergence of surrogate risk optimization, we have to specify the set of
score functions that we work with. We assume that the structure on input x ? X is defined by a
positive definite kernel K : X ? X ? R. We denote the corresponding reproducing kernel Hilbert
space (RKHS) by H and its explicit feature map by ?(x) ? H. By the reproducing property, we
have hf, ?(x)iH = f (x) for all x ? X , f ? H, where h?, ?iH is the inner product in the RKHS. We
define the subspace of allowed scores F ? Rk via the span of the columns of a matrix F ? Rk?r .
The matrix F explicitly defines the structure of the score function. With this notation, we will assume
that the score function is of the form f(x) = F W ?(x), where W : H ? Rr is a linear operator
to be learned (a matrix if H is of finite dimension) that represents a collection of r elements in H,
transforming ?(x) to a vector in Rr by applying the RKHS inner product r times.3 Note that for
structured losses, we usually have r k. The set of all score functions is thus obtained by varying W
in this definition and is denoted by FF,H . As a concrete example of a score family FF,H for structured
prediction, consider the standard sequence model with unary and pairwise potentials. In this case, the
dimension r equals T s + (T ? 1)s2 , where T is the sequence length and s is the number of labels
of each variable. The columns of the matrix F consist of 2T ? 1 groups (one for each unary and
pairwise potential). Each row of F has exactly one entry equal to one in each column group (with
zeros elsewhere).
In this setting, we use the online projected averaged stochastic subgradient descent ASGD4 (stochastic
w.r.t. data (x(n) , y (n) ) ? D) to minimize the surrogate risk directly [6]. The n-th update consists in
W (n) := PD W (n?1) ? ? (n) F T ???(x(n) )T ,
(9)
3
Note that if rank(F ) = r, our setup is equivalent to assuming a joint kernel [47] in the product form:
Kjoint ((x, c), (x0 , c0 )) := K(x, x0 )F (c, :)F (c0 , :)T , where F (c, :) is the row c for matrix F .
4
See, e.g., [36] for the formal setup of kernel ASGD.
5
where F T ???(x(n) )T : H ? Rr is the stochastic functional gradient, ? (n) is the step size
and PD is the projection on the ball of radius D w.r.t. the Hilbert?Schmidt norm5. The vector
?? ? Rk is a regular gradient of the sampled surrogate ?(f(x(n) ), y (n) ) w.r.t. the scores, ?? =
?f ?(f , y (n) )|f =f(x(n) ) . We wrote the above update using an explicit feature map ? for notational
simplicity, but kernel ASGD can also be implemented without it by using the kernel trick. The
convergence properties of ASGD in RKHS are analogous to the finite-dimensional ASGD because
they rely on dimension-free quantities. To use a simple convergence analysis, we follow Ciliberto
et al. [10] and make the following simplifying assumption:
Assumption 4 (Well-specified optimization w.r.t. the function class FF,H ). The distribution D is
such that R??,F := inf f?FF R? (f) has some global minimum f? that also belongs to FF,H .
Assumption 4 simply means that each row of W ? defining f? belongs to the RKHS H implying
a finite norm kW ? kHS . Assumption 4 can be relaxed if the kernel K is universal, but then the
convergence analysis becomes much more complicated [36].
Theorem 5 (Convergence rate). Under Assumption 4 and assuming that (i) the functions ?(f , y)
are bounded from below and convex w.r.t. f ? Rk for all y ? Y; (ii) the expected square of the norm
of the stochastic gradient is bounded, IE(x,y)?D kF T ???(x)T k2HS ? M 2 and (iii) kW ? kHS ? D,
?
then running the ASGD algorithm (9) with the constant step-size ? := M2D
for N steps admits the
N
(N )
?
following expected suboptimality for the averaged iterate f :
XN
?f(N ) := 1
?
IE[R? (?f(N ) )] ? R??,F ? 2DM
where
F W (n) ?(x(n) )T .
(10)
N
N
n=1
Theorem 5 is a straight-forward extension of classical results [33, 36].
By combining the convergence rate of Theorem 5 with Theorem 2 that connects the surrogate and
actual risks, we get Theorem 6 which explicitly gives the number of iterations required to achieve
? accuracy on the expected population risk (see App. A for the proof). Note that since ASGD is
applied in an online fashion, Theorem 6 also serves as the sample complexity bound, i.e., says how
many samples are needed to achieve ? target accuracy (compared to the best prediction rule if F has
zero approximation error).
Theorem 6 (Learning complexity). Under the assumptions of Theorem 5, for any ? > 0, the random
(w.r.t. the observed training set) output ?f(N ) ? FF,H of the ASGD algorithm after
N > N ? :=
4D 2 M 2
?2
H
?,L,F (?)
(11)
iterations has the expected excess risk bounded with ?, i.e., IE[RL (?f(N ) )] < R?L,F + ?.
4
Calibration function analysis for quadratic surrogate
A major challenge to applying Theorem 6 is the computation of the calibration function H?,L,F . In
App. C, we present a generalization to arbitrary multi-class losses of a surrogate loss class from Zhang
[49, Section 4.4.2] that is consistent for any task loss L. Here, we consider the simplest example of
this family, called the quadratic surrogate ?quad , which has the advantage that we can bound or even
compute exactly its calibration function. We define the quadratic surrogate as
?quad (f , y) :=
1
2k kf
+ L(:, y)k22 =
1
2k
k
X
(fc2 + 2fc L(c, y) + L(c, y)2 ).
(12)
c=1
One simple sufficient condition for the surrogate (12) to be consistent and also to have zero approximation error is that F fully contains span(L). To make the dependence on the score subspace explicit,
we parameterize it with a matrix F ? Rk?r with the number of columns r typically being much
smaller than the number of labels k. With this notation, we have F = span(F ) = {F ? | ? ? Rr },
and the dimensionality of F equals the rank of F , which is at most r.6
?
The Hilbert?Schmidt norm of a linear operator A is defined as kAkHS = trA? A where A? is the adjoint
operator. In the case of finite dimension, the Hilbert?Schmidt norm coincides with the Frobenius matrix norm.
6
Evaluating ?quad requires computing F T F and F T L(:, y) for which direct computation is intractable when
k is exponential, but which can be done in closed form for the structured losses we consider (the Hamming and
block 0-1 loss). More generally, these operations require suitable inference algorithms. See also App. F.
5
6
For the quadratic surrogate (12), the excess of the expected surrogate takes a simple form:
??quad (F ?, q) =
1
2k kF ?
+ Lqk22 .
(13)
Equation (13) holds under the assumption that the subspace F contains the column space of the
loss matrix span(L), which also means that the set F contains the optimal prediction for any q (see
Lemma 9 in App. B for the proof). Importantly, the function ??quad (F ?, q) is jointly convex in the
conditional probability q and parameters ?, which simplifies its analysis.
Lower bound on the calibration function. We now present our main technical result: a lower
bound on the calibration function for the surrogate loss ?quad (12). This lower bound characterizes
the easiness of learning with this surrogate given the scaling intuition mentioned in Section 3.3. The
proof of Theorem 7 is given in App. D.1.
Theorem 7 (Lower bound on H?quad ). For any task loss L, its quadratic surrogate ?quad , and a score
subspace F containing the column space of L, the calibration function can be lower bounded:
H?quad ,L,F (?) ?
?2
2k maxi6=j kPF ?ij k22
?
?2
4k ,
(14)
where PF is the orthogonal projection on the subspace F and ?ij = ei ? ej ? Rk with ec being
the c-th basis vector of the standard basis in Rk .
Lower bound for specific losses. We now discuss the meaning of the bound (14) for some specific
losses (the detailed derivations are given in App. D.3). For the 0-1, block 0-1 and Hamming losses
(L01 , L01,b and LHam,T , respectively) with the smallest possible score subspaces F, the bound (14)
?2 ?2
?2
gives 4k
, 4b and 8T
, respectively. All these bounds are tight (see App. E). However, if F = Rk
the bound (14) is not tight for the block 0-1 and mixed losses (see also App. E). In particular, the
bound (14) cannot detect level-? consistency for ? > 0 (see Def. 3) and does not change when the
loss changes, but the score subspace stays the same.
Upper bound on the calibration function. Theorem 8 below gives an upper bound on the calibration
function holding for unconstrained scores, i.e, F = Rk (see the proof in App. D.2). This result shows
that without some appropriate constraints on the scores, efficient learning is not guaranteed (in the
worst case) because of the 1/k scaling of the calibration function.
Theorem 8 (Upper bound on H?quad ). If a loss matrix L with Lmax > 0 defines a pseudometric7 on
labels and there are no constraints on the scores, i.e., F = Rk , then the calibration function for the
?2
quadratic surrogate ?quad can be upper bounded: H?quad ,L,Rk (?) ? 2k
, 0 ? ? ? Lmax .
From our lower bound in Theorem 7 (which guarantees consistency), the natural constraint on
the score is F = span(L), with the dimension of this space giving an indication of the intrinsic
?difficulty? of a loss. Computations for the lower bounds in some specific cases (see App. D.3 for
details) show that the 0-1 loss is ?hard? while the block 0-1 loss and the Hamming loss are ?easy?.
Note that in all these cases the lower bound (14) is tight, see the discussion below.
Exact calibration functions. Note that the bounds proven in Theorems 7 and 8 imply that, in the
case of no constraints on the scores F = Rk , for the 0-1, block 0-1 and Hamming losses, we have
?2
4k
? H?quad ,L,Rk (?) ?
?2
2k ,
(15)
where L is the matrix defining a loss. For completeness, in App. E, we compute the exact calibration
functions for the 0-1 and block 0-1 losses. Note that the calibration function for the 0-1 loss equals the
lower bound, illustrating the worst-case scenario. To get some intuition, an example of a conditional
distribution q that gives the (worst case) value to the calibration function (for several losses) is
qi = 21 + 2? , qj = 21 ? 2? and qc = 0 for c ?6= {i, j}. See the proof of Proposition 12 in App. E.1.
In what follows, we provide the calibration functions in the cases with constraints on the scores. For
the block 0-1 loss with b equal blocks and under constraints that the scores within blocks are equal,
the calibration function equals (see Proposition 14 of App. E.2)
H?quad ,L01,b ,F01,b (?) =
7
?2
4b ,
0 ? ? ? 1.
(16)
A pseudometric is a function d(a, b) satisfying the following axioms: d(x, y) ? 0, d(x, x) = 0 (but
possibly d(x, y) = 0 for some x 6= y), d(x, y) = d(y, x), d(x, z) ? d(x, y) + d(y, z).
7
For the Hamming loss defined over T binary variables and under constraints implying separable
scores, the calibration function equals (see Proposition 15 in App. E.3)
H?quad ,LHam,T ,FHam,T (?) =
?2
8T
, 0 ? ? ? 1.
(17)
The calibration functions (16) and (17) depend on the quantities representing the actual complexities
of the loss (the number of blocks b and the length of the sequence T ) and can be exponentially larger
than the upper bound for the unconstrained case.
In the case of mixed 0-1 and block 0-1 loss, if the scores f are constrained to be equal inside the
blocks, i.e., belong to the subspace F01,b = span(L01,b ) ( Rk , then the calibration function is equal
to 0 for ? ? ?2 , implying inconsistency (and also note that the approximation error can be as big as ?
for F01,b ). However, for ? > ?2 , the calibration function is of the order 1b (? ? ?2 )2 . See Figure 1b for
the illustration of this calibration function and Proposition 17 of App. E.4 for the exact formulation
and the proof. Note that while the calibration function for the constrained case is inconsistent, its
value can be exponentially larger than the one for the unconstrained case for ? big enough and when
the blocks are exponentially large (see Proposition 16 of App. E.4).
Computation of the SGD constants. Applying the learning complexity Theorem 6 requires to
compute the quantity DM where D bounds the norm of the optimal solution and M bounds the
expected square of the norm of the stochastic gradient. In App. F, we provide a way to bound this
quantity for our quadratic surrogate (12) under the simplifying assumption that each conditional qc (x)
(seen as function of x) belongs to the RKHS H (which implies Assumption 4). In particular, we get
?
DM = L2max ?(?(F ) rRQmax ), ?(z) = z 2 + z,
(18)
where ?(F ) is the condition number of the matrix F , R is an upper bound on the RKHS norm of
Pk
object feature maps k?(x)kH . We define Qmax as an upper bound on c=1 kqc kH (can be seen as
Pk
the generalization of the inequality c=1 qc ? 1 for probabilities). The constants R and Qmax depend
on the data, the constant Lmax depends on the loss, r and ?(F ) depend on the choice of matrix F .
We compute the constant DM for the specific losses that we considered in App. F.1. For the 0-1, block
0-1 and Hamming losses, we have DM = O(k), DM = O(b) and DM = O(log32 k), respectively.
These computations indicate that the quadratic surrogate allows efficient learning for structured block
0-1 and Hamming losses, but that the convergence could be slow in the worst case for the 0-1 loss.
5
Related works
Consistency for multi-class problems. Building on significant progress for the case of binary
classification, see, e.g. [5], there has been a lot of interest in the multi-class case. Zhang [49] and
Tewari & Bartlett [46] analyze the consistency of many existing surrogates for the 0-1 loss. Gao &
Zhou [20] focus on multi-label classification. Narasimhan et al. [32] provide a consistent algorithm
for arbitrary multi-class loss defined by a function of the confusion matrix. Recently, Ramaswamy &
Agarwal [39] introduce the notion of convex calibrated dimension, as the minimal dimensionality of
the score vector that is required for consistency. In particular, they showed that for the Hamming loss
on T binary variables, this dimension is at most T . In our analysis, we use scores of rank (T + 1),
see (35) in App. D.3, yielding a similar result.
The task of ranking has attracted a lot of attention and [18, 8, 9, 40] analyze different families of
surrogate and task losses proving their (in-)consistency. In this line of work, Ramaswamy et al.
[40] propose a quadratic surrogate for an arbitrary low rank loss which is related to our quadratic
surrogate (12). They also prove that several important ranking losses, i.e., precision@q, expected
rank utility, mean average precision and pairwise disagreement, are of low-rank. We conjecture that
our approach is compatible with these losses and leave precise connections as future work.
Structured SVM (SSVM) and friends. SSVM [44, 45, 47] is one of the most used convex surrogates
for tasks with structured outputs, thus, its consistency has been a question of great interest. It is
known that Crammer-Singer multi-class SVM [15], which SSVM is built on, is not consistent for
0-1 loss unless there is a majority class with probability at least 12 [49, 31]. However, it is consistent
for the ?abstain? and ordinal losses in the case of 3 classes [39]. Structured ramp loss and probit
surrogates are closely related to SSVM and are consistent [31, 16, 30, 23], but not convex.
8
Recently, Do?gan et al. [17] categorized different versions of multi-class SVM and analyzed them
from Fisher and universal consistency point of views. In particular, they highlight differences between
Fisher and universal consistency and give examples of surrogates that are Fisher consistent, but not
universally consistent and vice versa. They also highlight that the Crammer-Singer SVM is neither
Fisher, not universally consistent even with a careful choice of regularizer.
Quadratic surrogates
for structured prediction. Ciliberto et al. [10] and Brouard et al. [7] consider
Pn
minimizing i=1 kg(xi ) ? ?o (yi )k2H aiming to match the RKHS embedding of inputs g : X ? H
to the feature maps of outputs ?o : Y ? H. In their frameworks, the task loss is not considered at
the learning stage, but only at the prediction stage. Our quadratic surrogate (12) depends on the loss
directly. The empirical risk defined by both their and our objectives can be minimized analytically with
the help of the kernel trick and, moreover, the resulting predictors are identical. However, performing
such computation in the case of large dataset can be intractable and the generalization properties have
to be taken care of, e.g., by the means of regularization. In the large-scale scenario, it is more natural
to apply stochastic optimization (e.g., kernel ASGD) that directly minimizes the population risk and
has better dependency on the dataset size. When combined with stochastic optimization, the two
approaches lead to different behavior. In our framework, we need to estimate r = rank(L) scalar
functions, but the alternative needs to estimate k functions (if, e.g., ?o (y) = ey ? Rk ), which results
in significant differences for low-rank losses, such as block 0-1 and Hamming.
Calibration functions. Bartlett et al. [5] and Steinwart [43] provide calibration functions for most
existing surrogates for binary classification. All these functions differ in term of shape, but are
roughly similar in terms of constants. Pedregosa et al. [37] generalize these results to the case of
ordinal regression. However, their calibration functions have at best a 1/k factor if the surrogate is
normalized w.r.t. the number of classes. The task of ranking has been of significant interest. However,
most of the literature [e.g., 11, 14, 24, 1], only focuses on calibration functions (in the form of regret
bounds) for bipartite ranking, which is more akin to cost-sensitive binary classification.
?vila Pires et al. [3] generalize the theoretical framework developed by Steinwart [43] and present
results for the multi-class SVM of Lee et al. [26] (the score vectors are constrained
P to sum to zero)
that can
Pbe built for any task loss of interest. Their surrogate ? is of the form c?Y L(c, y)a(fc )
where c?Y fc = 0 and a(f ) is some convex function with all subgradients at zero being positive.
The recent work by ?vila Pires & Szepesv?ri [2] refines the results, but specifically for the case of
0-1 loss. In this line of work, the surrogate is typically not normalized by k, and if normalized the
calibration functions have the constant 1/k appearing.
Finally, Ciliberto et al. [10] provide the calibration function for their quadratic surrogate. Assuming
? y) = hV ?o (y),
? ?o (y)iHY , y,
? y ? Y (this assumption can
that the loss can be represented as L(y,
always be satisfied in the case of a finite number of labels, by taking V as the loss matrix L and
?o (y) := ey ? Rk where ey is the y-th vector of the standard basis in Rk ). In their Theorem 2, they
provide an excess risk bound leading to a lower bound on the corresponding calibration function
2
H?,L,Rk (?) ? c?2 where a constant c? = kV k2 maxy?Y k?o (y)k simply equals the spectral norm
?
of the loss matrix for the finite-dimensional construction provided above. However, the spectral
norm of the loss matrix is exponentially large even for highly structured losses such as the block 0-1
and Hamming losses, i.e., kL01,b k2 = k ? kb , kLHam,T k2 = k2 . This conclusion puts the objective
of Ciliberto et al. [10] in line with ours when no constraints are put on the scores.
6
Conclusion
In this paper, we studied the consistency of convex surrogate losses specifically in the context
of structured prediction. We analyzed calibration functions and proposed an optimization-based
normalization aiming to connect consistency with the existence of efficient learning algorithms.
Finally, we instantiated all components of our framework for several losses by computing the
calibration functions and the constants coming from the normalization. By carefully monitoring
exponential constants, we highlighted the difference between tractable and intractable task losses.
These were first steps in advancing our theoretical understanding of consistent structured prediction.
Further steps include analyzing more losses such as the low-rank ranking losses studied by Ramaswamy et al. [40] and, instead of considering constraints on the scores, one could instead put
constraints on the set of distributions to investigate the effect on the calibration function.
9
Acknowledgements
We would like to thank Pascal Germain for useful discussions. This work was partly supported
by the ERC grant Activia (no. 307574), the NSERC Discovery Grant RGPIN-2017-06936 and the
MSR-INRIA Joint Center.
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12
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6,229 | 6,635 | Best of Both Worlds: Transferring Knowledge from
Discriminative Learning to a Generative Visual
Dialog Model
Jiasen Lu1?, Anitha Kannan2?, Jianwei Yang1 , Devi Parikh3,1 , Dhruv Batra3,1
1
Georgia Institute of Technology, 2 Curai, 3 Facebook AI Research
{jiasenlu, jw2yang, parikh, dbatra}@gatech.edu
Abstract
We present a novel training framework for neural sequence models, particularly
for grounded dialog generation. The standard training paradigm for these models
is maximum likelihood estimation (MLE), or minimizing the cross-entropy of the
human responses. Across a variety of domains, a recurring problem with MLE
trained generative neural dialog models (G) is that they tend to produce ?safe?
and generic responses (?I don?t know?, ?I can?t tell?). In contrast, discriminative
dialog models (D) that are trained to rank a list of candidate human responses
outperform their generative counterparts; in terms of automatic metrics, diversity,
and informativeness of the responses. However, D is not useful in practice since it
can not be deployed to have real conversations with users.
Our work aims to achieve the best of both worlds ? the practical usefulness of
G and the strong performance of D ? via knowledge transfer from D to G. Our
primary contribution is an end-to-end trainable generative visual dialog model,
where G receives gradients from D as a perceptual (not adversarial) loss of the
sequence sampled from G. We leverage the recently proposed Gumbel-Softmax
(GS) approximation to the discrete distribution ? specifically, a RNN augmented
with a sequence of GS samplers, coupled with the straight-through gradient estimator to enable end-to-end differentiability. We also introduce a stronger encoder for
visual dialog, and employ a self-attention mechanism for answer encoding along
with a metric learning loss to aid D in better capturing semantic similarities in
answer responses. Overall, our proposed model outperforms state-of-the-art on the
VisDial dataset by a significant margin (2.67% on recall@10). The source code
can be downloaded from https://github.com/jiasenlu/visDial.pytorch
1
Introduction
One fundamental goal of artificial intelligence (AI) is the development of perceptually-grounded
dialog agents ? specifically, agents that can perceive or understand their environment (through vision,
audio, or other sensors), and communicate their understanding with humans or other agents in natural
language. Over the last few years, neural sequence models (e.g. [47, 44, 46]) have emerged as the
dominant paradigm across a variety of setting and datasets ? from text-only dialog [44, 40, 23, 3] to
more recently, visual dialog [7, 9, 8, 33, 45], where an agent must answer a sequence of questions
grounded in an image, requiring it to reason about both visual content and the dialog history.
The standard training paradigm for neural dialog models is maximum likelihood estimation (MLE)
or equivalently, minimizing the cross-entropy (under the model) of a ?ground-truth? human response.
Across a variety of domains, a recurring problem with MLE trained neural dialog models is that they
tend to produce ?safe? generic responses, such as ?Not sure? or ?I don?t know? in text-only dialog [23],
and ?I can?t see? or ?I can?t tell? in visual dialog [7, 8]. One reason for this emergent behavior is that
?
Work was done while at Facebook AI Research.
the space of possible next utterances in a dialog is highly multi-modal (there are many possible paths
a dialog may take in the future). In the face of such highly multi-modal output distributions, models
?game? MLE by latching on to the head of the distribution or the frequent responses, which by nature
tend to be generic and widely applicable. Such safe generic responses break the flow of a dialog and
tend to disengage the human conversing with the agent, ultimately rendering the agent useless. It is
clear that novel training paradigms are needed; that is the focus of this paper.
One promising alternative to MLE training proposed by recent work [36, 27] is sequence-level
training of neural sequence models, specifically, using reinforcement learning to optimize taskspecific sequence metrics such as BLEU [34], ROUGE [24], CIDEr [48]. Unfortunately, in the case
of dialog, all existing automatic metrics correlate poorly with human judgment [26], which renders
this alternative infeasible for dialog models.
In this paper, inspired by the success of adversarial training [16], we propose to train a generative
visual dialog model (G) to produce sequences that score highly under a discriminative visual dialog
model (D). A discriminative dialog model receives as input a candidate list of possible responses and
learns to sort this list from the training dataset. The generative dialog model (G) aims to produce a
sequence that D will rank the highest in the list, as shown in Fig. 1.
Note that while our proposed approach is inspired by adversarial training, there are a number of
subtle but crucial differences over generative adversarial networks (GANs). Unlike traditional GANs,
one novelty in our setup is that our discriminator receives a list of candidate responses and explicitly
learns to reason about similarities and differences across candidates. In this process, D learns a
task-dependent perceptual similarity [12, 19, 15] and learns to recognize multiple correct responses
in the feature space. For example, as shown in Fig. 1 right, given the image, dialog history, and
question ?Do you see any bird??, besides the ground-truth answer ?No, I do not?, D can also assign
high scores to other options that are valid responses to the question, including the one generated by
G: ?Not that I can see?. The interaction between responses is captured via the similarity between the
learned embeddings. This similarity gives an additional signal that G can leverage in addition to the
MLE loss. In that sense, our proposed approach may be viewed as an instance of ?knowledge transfer?
[17, 5] from D to G. We employ a metric-learning loss function and a self-attention answer encoding
mechanism for D that makes it particularly conducive to this knowledge transfer by encouraging
perceptually meaningful similarities to emerge. This is especially fruitful since prior work has
demonstrated that discriminative dialog models significantly outperform their generative counterparts,
but are not as useful since they necessarily need a list of candidate responses to rank, which is only
available in a dialog dataset, not in real conversations with a user. In that context, our work aims to
achieve the best of both worlds ? the practical usefulness of G and the strong performance of D ? via
this knowledge transfer.
Our primary technical contribution is an end-to-end trainable generative visual dialog model, where
the generator receives gradients from the discriminator loss of the sequence sampled from G. Note
that this is challenging because the output of G is a sequence of discrete symbols, which na?vely is not
amenable to gradient-based training. We propose to leverage the recently proposed Gumbel-Softmax
(GS) approximation to the discrete distribution [18, 30] ? specifically, a Recurrent Neural Network
(RNN) augmented with a sequence of GS samplers, which when coupled with the straight-through
gradient estimator [2, 18] enables end-to-end differentiability.
Our results show that our ?knowledge transfer? approach is indeed successful. Specifically, our
discriminator-trained G outperforms the MLE-trained G by 1.7% on recall@5 on the VisDial dataset,
essentially improving over state-of-the-art [7] by 2.43% recall@5 and 2.67% recall@10. Moreover,
our generative model produces more diverse and informative responses (see Table 3).
As a side contribution specific to this application, we introduce a novel encoder for neural visual
dialog models, which maintains two separate memory banks ? one for visual memory (where do
we look in the image?) and another for textual memory (what facts do we know from the dialog
history?), and outperforms the encoders used in prior work.
2
Related Work
GANs for sequence generation. Generative Adversarial Networks (GANs) [16] have shown to be
effective models for a wide range of applications involving continuous variables (e.g. images) c.f
[10, 35, 22, 55]. More recently, they have also been used for discrete output spaces such as language
generation ? e.g. image captioning [6, 41], dialog generation [23], or text generation [53] ? by either
viewing the generative model as a stochastic parametrized policy that is updated using REINFORCE
2
Generator
Question Qt
A gray tiger cat sitting underneath
a metal bench.
Is it in color? Yes it is.
Is it day time? Yes.
Is the tiger big? No, it?s a regular
cat.
?"
HCIAE
Encoder
I do not see any birds
No , I do not
Nope
?
yes
Mangoes
HCIAE
Encoder
White
Score
No
Answer
Encoder
?"
No ,
Nope
Not that I can see
yes
?
?
Mangoes
I see small shops
HQI
HQI
No , I do not
White
I see small shops
t rounds of history
HQI
No bird
No
Answer
Decoder
Option answers (D)
No bird
I do not see any
birds
Image I
Do you see any birds?
Discriminator
Not that I can see
Deep Metric Loss
Gumbel
Sampler
Figure 1: Model architecture of the proposed model. Given the image, history, and question, the discriminator
receives as additional input a candidate list of possible responses and learns to sort this list. The generator aims
to produce a sequence that discriminator will rank the highest in the list. The right most block is D?s score for
different candidate answers. Note that the multiple plausible responses all score high. Image from the COCO
dataset [25].
with the discriminator providing the reward [53, 6, 41, 23], or (closer to our approach) through
continuous relaxation of discrete variables through Gumbel-Softmax to enable backpropagating the
response from the discriminator [21, 41].
There are a few subtle but significant differences w.r.t. to our application, motivation, and approach.
In these prior works, both the discriminator and the generator are trained in tandem, and from scratch.
The goal of the discriminator in those settings has primarily been to discriminate ?fake? samples
(i.e. generator?s outputs) from ?real? samples (i.e. from training data). In contrast, we would like
to transfer knowledge from the discriminator to the generator. We start with pre-trained D and G
models suited for the task, and then transfer knowledge from D to G to further improve G, while
keeping D fixed. As we show in our experiments, this procedure results in G producing diverse
samples that are close in the embedding space to the ground truth, due to perceptual similarity learned
in D. One can also draw connections between our work and Energy Based GAN (EBGAN) [54] ?
without the adversarial training aspect. The ?energy? in our case is a deep metric-learning based
scoring mechanism, instantiated in the visual dialog application.
Modeling image and text attention. Models for tasks at the intersection of vision and language
? e.g., image captioning [11, 13, 20, 49], visual question answering [1, 14, 31, 37], visual dialog
[7, 9, 8, 45, 33] ? typically involve attention mechanisms. For image captioning, this may be attending
to relevant regions in the image [49, 51, 28]. For VQA, this may be attending to relevant image
regions alone [4, 50, 52] or co-attending to image regions and question words/phrases [29].
In the context of visual dialog, [7] uses attention to identify utterances in the dialog history that may
be useful for answering the current question. However, when modeling the image, the entire image
embedding is used to obtain the answer. In contrast, our proposed encoder HCIAE (Section 4.1)
localizes the region in the image that can help reliably answer the question. In particular, in addition
to the history and the question guiding the image attention, our visual dialog encoder also reasons
about the history when identifying relevant regions of the image. This allows the model to implicitly
resolve co-references in the text and ground them back in the image.
3
Preliminaries: Visual Dialog
We begin by formally describing the visual dialog task setup as introduced by Das et al. [7]. The
machine learning task is as follows. A visual dialog model is given as input an image I, caption c
describing the image, a dialog history till round t ? 1, H = ( c , (q1 , a1 ), . . . , (qt?1 , at?1 )), and
|{z} | {z }
|
{z
}
H0
H1
Ht?1
the followup question qt at round t. The visual dialog agent needs to return a valid response to the
question.
Given the problem setup, there are two broad classes of methods ? generative and discriminative
models. Generative models for visual dialog are trained by maximizing the log-likelihood of the
ground truth answer sequence agt
t ? At given the encoded representation of the input (I, H, qt ).
3
On the other hand, discriminative models receive both an encoding of the input (I, H, qt ) and as
(1)
(100)
additional input a list of 100 candidate answers At = {at , . . . , at
}. These models effectively
learn to sort the list. Thus, by design, they cannot be used at test time without a list of candidates
available.
4
Approach: Backprop Through Discriminative Losses for Generative
Training
In this section, we describe our approach to transfer knowledge from a discriminative visual dialog
model (D) to generative visual dialog model (G). Fig. 1 (a) shows the overview of our approach.
Given the input image I, dialog history H, and question qt , the encoder converts the inputs into a
joint representation et . The generator G takes et as input, and produces a distribution over answer
sequences via a recurrent neural network (specifically an LSTM). At each word in the answer
sequence, we use a Gumbel-Softmax sampler S to sample the answer token from that distribution.
The discriminator D in it?s standard form takes et , ground-truth answer agt
t and N ? 1 ?negative?
gt
N ?1
answers {a?
}
as
input,
and
learns
an
embedding
space
such
that
similarity(e
t , f (at )) >
t,i i=1
?
similarity(et , f (at,? )), where f (?) is the embedding function. When we enable the communication
between D and G, we feed the sampled answer a
?t into discriminator, and optimize the generator G
to produce samples that get higher scores in D?s metric space.
We now describe each component of our approach in detail.
4.1
History-Conditioned Image Attentive Encoder (HCIAE)
?
An important characteristic in dialogs is the use of co-reference to avoid repeating entities that can
be contextually resolved. In fact, in the VisDial dataset [7] nearly all (98%) dialogs involve at least
one pronoun. This means that for a model to correctly answer a question, it would require a reliable
mechanism for co-reference resolution.
A common approach is to use an encoder architecture with an attention mechanism that implicitly
performs co-reference resolution by identifying the portion of the dialog history that can help in
answering the current question [7, 38, 39, 32]. while using a holistic representation for the image.
Intuitively, one would also expect that the answer is also localized to regions in the image, and be
consistent with the attended history.
With this motivation, we propose a novel encoder architecture (called HCIAE) shown in Fig. 2. Our
encoder first uses the current question to attend to the exchanges in the history, and then use the
question and attended history to attend to the image, so as to obtain the final encoding.
Specifically, we use the spatial image features V ?
Rd?k from a convolution layer of a CNN. qt is
encoded with an LSTM to get a vector mqt ?
CNN
?
Rd . Simultaneously, each previous round of history
(H0 , . . . , Ht?1 ) is encoded separately with another
LSTM
?"
LSTM as Mth ? Rd?t . Conditioned on the question
embedding, the model attends to the history. The at?"
tended representation of the history and the question
LSTM
?$
embedding are concatenated, and used as input to
attend to the image:
LSTM
?"%&
?ht
= softmax(zth )
(2)
?
= waT tanh(Wh Mth + (Wq mqt )1T ) (1)
?
?
zth
Encoder
Figure 2: Structure of the proposed encoder.
where 1 ? R is a vector with all elements set to 1.
Wh , Wq ? Rt?d and wa ? Rk are parameters to be learned. ? ? Rk is the attention weight over
? ht is a convex combination of columns of Mt , weighted
history. The attended history feature m
h
? ht as the query vector and get
appropriately by the elements of ?t . We further concatenate mqt and m
the attended image feature v?t in the similar manner. Subsequently, all three components are used to
obtain the final embedding et :
t
? ht , v?t ])
et = tanh(We [mqt , m
where We ? Rd?3d is weight parameters and [?] is the concatenation operation.
4
(3)
4.2
Discriminator Loss
Discriminative visual dialog models produce a distribution over the candidate answer list At and
maximize the log-likelihood of the correct option agt
t . The loss function for D needs to be conducive
for knowledge transfer. In particular, it needs to encourage perceptually meaningful similarities.
Therefore, we use a metric-learning multi-class N-pair loss [43] defined as:
logistic loss
z
? N ?1
LD = Ln?pair {et , agt
t , {at,i }i=1 }, f = log 1 +
N
X
}|
gt
?
>
exp e>
t f (at,i ) ? et f (at )
|
{z
}
i=1
!{
(4)
score margin
where f is an attention based LSTM encoder for the answer. This attention can help the discriminator
better deal with paraphrases across answers. The attention weight is learnt through a 1-layer MLP
over LSTM output at each time step. The N-pair loss objective encourages learning a space in which
the ground truth answer is scored higher than other options, and at the same time, encourages options
similar to ground truth answers to score better than dissimilar ones. This means that, unlike the
multiclass logistic loss, the options that are correct but different from the correct option may not be
overly penalized, and thus can be useful in providing a reliable signal to the generator. See Fig. 1 for
an example. Follwing [43], we regularize the L2 norm of the embedding vectors to be small.
4.3
Discriminant Perceptual Loss and Knowledge Transfer from D to G
At a high-level, our approach for transferring knowledge from D to G is as follows: G repeatedly
queries D with answers a
?t that it generates for an input embedding et to get feedback and update
itself. In each such update, G?s goal is to update its parameters to try and have a
?t score higher than
the correct answer, agt
,
under
D?s
learned
embedding
and
scoring
function.
Formally,
the perceptual
t
loss that G aims to optimize is given by:
gt
>
>
LG = L1?pair {et , a
?t , agt
},
f
=
log
1
+
exp
e
f
(a
)
?
e
f
(?
a
)
(5)
t
t
t
t
t
where f is the embedding function learned by the discriminator as in (4). Intuitively, updating
generator parameters to minimize LG can be interpreted as learning to produce an answer sequence
a
?t that ?fools? the discriminator into believing that this answer should score higher than the human
response agt
t under the discriminator?s learned embedding f (?) and scoring function.
While it is straightforward to sample an answer a
?t from the generator and perform a forward pass
through the discriminator, na?vely, it is not possible to backpropagate the gradients to the generator
parameters since sampling discrete symbols results in zero gradients w.r.t. the generator parameters.
To overcome this, we leverage the recently introduced continuous relaxation of the categorical
distribution ? the Gumbel-softmax distribution or the Concrete distribution [18, 30].
At an intuitive level, the Gumbel-Softmax (GS) approximation uses the so called ?Gumbel-Max trick?
to reparametrize sampling from a categorical distribution and replaces argmax with softmax to obtain
a continuous relaxation of the discrete random variable. Formally, let x denote a K-ary categorical
K
random variable with parameters denoted by (p1 , . . . pK ), or x ? Cat(p). Let gi 1 denote K
?g
IID samples from the standard Gumbel distribution, gi ? F (g) = e?e . Now, a sample from the
Concrete distribution can be produced via the following transformation:
e(log pi +gi )/?
yi = PK (log p +g )
j
j /?
j=1 e
?i ? {1, . . . , K}
(6)
where ? is a temperature parameter that control how close samples y from this Concrete distribution
approximate the one-hot encoding of the categorical variable x.
As illustrated in Fig. 1, we augment the LSTM in G with a sequence of GS samplers. Specifically,
at each position in the answer sequence, we use a GS sampler to sample an answer token from
that conditional distribution. When coupled with the straight-through gradient estimator [2, 18]
this enables end-to-end differentiability. Specifically, during the forward pass we discretize the GS
samples into discrete samples, and in the backward pass use the continuous relaxation to compute
gradients. In our experiments, we held the temperature parameter fixed at 0.5.
5
5
Experiments
Dataset and Setup. We evaluate our proposed approach on the VisDial dataset [7], which was
collected by Das et al. by pairing two subjects on Amazon Mechanical Turk to chat about an image.
One person was assigned the role of a ?questioner? and the other of ?answerer?. One worker (the
questioner) sees only a single line of text describing an image (caption from COCO [25]); the image
remains hidden to the questioner. Their task is to ask questions about this hidden image to ?imagine
the scene better?. The second worker (the answerer) sees the image and caption and answers the
questions. The two workers take turns asking and answering questions for 10 rounds. We perform
experiments on VisDial v0.9 (the latest available release) containing 83k dialogs on COCO-train and
40k on COCO-val images, for a total of 1.2M dialog question-answer pairs. We split the 83k into 82k
for train, 1k for val, and use the 40k as test, in a manner consistent with [7]. The caption is
considered to be the first round in the dialog history.
Evaluation Protocol. Following the evaluation protocol established in [7], we use a retrieval setting
to evaluate the responses at each round in the dialog. Specifically, every question in VisDial is
coupled with a list of 100 candidate answer options, which the models are asked to sort for evaluation
purposes. D uses its score to rank these answer options, and G uses the log-likelihood of these
options for ranking. Models are evaluated on standard retrieval metrics ? (1) mean rank, (2) recall
@k, and (3) mean reciprocal rank (MRR) ? of the human response in the returned sorted list.
Pre-processing. We truncate captions/questions/answers longer than 24/16/8 words respectively. We
then build a vocabulary of words that occur at least 5 times in train, resulting in 8964 words.
Training Details In our experiments, all 3 LSTMs are single layer with 512d hidden state. We use
VGG-19 [42] to get the representation of image. We first rescale the images to be 224 ? 224 pixels,
and take the output of last pooling layer (512 ? 7 ? 7) as image feature. We use the Adam optimizer
with a base learning rate of 4e-4. We pre-train G using standard MLE for 20 epochs, and D with
supervised training based on Eq (4) for 30 epochs. Following [43], we regularize the L2 norm of the
embedding vectors to be small. Subsequently, we train G with LG + ?LM LE , which is a combination
of discriminative perceptual loss and MLE loss. We set ? to be 0.5. We found that including LM LE
(with teacher-forcing) is important for encouraging G to generate grammatically correct responses.
5.1
Results and Analysis
Baselines. We compare our proposed techniques to the current state-of-art generative and discriminative models developed in [7]. Specifically, [7] introduced 3 encoding architectures ? Late Fusion (LF),
Hierarchical Recurrent Encoder (HRE), Memory Network (MN) ? each trained with a generative
(-G) and discriminative (-D) decoder. We compare to all 6 models.
Our approaches. We present a few variants of our approach to systematically study the individual
contributions of our training procedure, novel encoder (HCIAE), self-attentive answer encoding
(ATT), and metric-loss (NP).
? HCIAE-G-MLE is a generative model with our proposed encoder trained under the MLE objective. Comparing this variant to the generative baselines from [7] establishes the improvement
due to our encoder (HCIAE).
? HCIAE-G-DIS is a generative model with our proposed encoder trained under the mixed MLE
and discriminator loss (knowledge transfer). This forms our best generative model. Comparing
this model to HCIAE-G-MLE establishes the improvement due to our discriminative training.
? HCIAE-D-MLE is a discriminative model with our proposed encoder, trained under the standard discriminative cross-entropy loss. The answer candidates are encoded using an LSTM
(no attention). Comparing this variant to the discriminative baselines from [7] establishes the
improvement due to our encoder (HCIAE) in the discriminative setting.
? HCIAE-D-NP is a discriminative model with our proposed encoder, trained under the n-pair
discriminative loss (as described in Section 4.2). The answer candidates are encoded using an
LSTM (no attention). Comparing this variant to HCIAE-D-MLE establishes the improvement
due to the n-pair loss.
? HCIAE-D-NP-ATT is a discriminative model with our proposed encoder, trained under the
n-pair discriminative loss (as described in Section 4.2), and using the self-attentive answer
encoding. Comparing this variant to HCIAE-D-NP establishes the improvement due to the
self-attention mechanism while encoding the answers.
6
Table 1: Results (generative) on VisDial dataset. ?MRR? Table 2: Results (discriminative) on VisDial dataset.
is mean reciprocal rank and ?Mean? is mean rank.
Model
MRR R@1 R@5 R@10 Mean
Model
MRR R@1 R@5 R@10 Mean
LF-D [7]
0.5807 43.82 74.68 84.07 5.78
0.5868 44.82 74.81 84.36 5.66
LF-G [7]
0.5199 41.83 61.78 67.59 17.07 HREA-D [7]
0.5965 45.55 76.22 85.37 5.46
HREA-G [7]
0.5242 42.28 62.33 68.17 16.79 MN-D [7]
MN-G [7]
0.5259 42.29 62.85 68.88 17.06
HCIAE-D-MLE
0.6140 47.73 77.50 86.35 5.15
0.6182 47.98 78.35 87.16 4.92
HCIAE-G-MLE 0.5386 44.06 63.55 69.24 16.01 HCIAE-D-NP
HCIAE-G-DIS 0.5467 44.35 65.28 71.55 14.23 HCIAE-D-NP-ATT 0.6222 48.48 78.75 87.59 4.81
Results. Tables 1, 2 present results for all our models and baselines in generative and discriminative
settings. The key observations are:
1. Main Results for HCIAE-G-DIS: Our final generative model with all ?bells and whistles?,
HCIAE-G-DIS, uniformly performs the best under all the metrics, outperforming the
previous state-of-art model MN-G by 2.43% on R@5. This shows the importance of the
knowledge transfer from the discriminator and the benefit from our encoder architecture.
2. Knowledge transfer vs. encoder for G: To understand the relative importance of the proposed history conditioned image attentive encoder (HCIAE) and the knowledge transfer, we
compared the performance of HCIAE-G-DIS with HCIAE-G-MLE, which uses our proposed encoder but without any feedback from the discriminator. This comparison highlights
two points: first, HCIAE-G-MLE improves R@5 by 0.7% over the current state-of-art
method (MN-D) confirming the benefits of our encoder. Secondly, and importantly, its
performance is lower than HCIAE-G-DIS by 1.7% on R@5, confirming that the modifications to encoder alone will not be sufficient to gain improvements in answer generation;
knowledge transfer from D greatly improves G.
3. Metric loss vs. self-attentive answer encoding: In the purely discriminative setting, our
final discriminative model (HCIAE-D-NP-ATT) also beats the performance of the corresponding state-of-art models [7] by 2.53% on R@5. The n-pair loss used in the discriminator
is not only helpful for knowledge transfer but it also improves the performance of the
discriminator by 0.85% on R@5 (compare HCIAE-D-NP to HCIAE-D-MLE). The improvements obtained by using the answer attention mechanism leads to an additional, albeit
small, gains of 0.4% on R@5 to the discriminator performance (compare HCIAE-D-NP to
HCIAE-D-NP-ATT).
5.2
Does updating discriminator help?
Recall that our model training happens as follows: we independently train the generative model
HCIAE-G-MLE and the discriminative model HCIAE-D-NP-ATT. With HCIAE-G-MLE as the
initialization, the generative model is updated based on the feedback from HCIAE-D-NP-ATT and
this results in our final HCIAE-G-DIS.
We performed two further experiments to answer the following questions:
? What happens if we continue training HCIAE-D-NP-ATT in an adversarial setting? In particular, we continue training by maximizing the score of the ground truth answer agt
t and
minimizing the score of the generated answer a
?t , effectively setting up an adversarial training regime LD = ?LG . The resulting discriminator HCIAE-GAN1 has significant drop in
performance, as can be seen in Table. 4 (32.97% R@5). This is perhaps expected because
HCIAE-GAN1 updates its parameters based on only two answers, the ground truth and the
generated sample (which is likely to be similar to ground truth). This wrecks the structure that
HCIAE-D-NP-ATT had previously learned by leveraging additional incorrect options.
? What happens if we continue structure-preserving training of HCIAE-D-NP-ATT? In addition
to providing HCIAE-D-NP-ATT samples from G as fake answers, we also include incorrect
options as negative answers so that the structure learned by the discriminator is preserved.
HCIAE-D-NP-ATT continues to train under loss LD . In this case (HCIAE-GAN2 in Table. 4),
we find that there is a small improvement in the performance of G. The additional computational
overhead to training the discriminator supersedes the performance improvement. Also note that
HCIAE-D-NP-ATT itself gets worse at the dialog task.
7
Table 3: Qualitative comparison. ?Ours? are samples from G-DIS model with different gumbel noise z. Images
from the COCO dataset [25].
Q: Is it a home or restaurant?
A: I think restaurant.
G-MLE: I can?t tell.
Ours (z1): Hard to say.
Ours (z2): It looks like a restaurant.
Ours (z3): I can?t tell because it
is too close.
Q: Can you see his face?
A: I am not sure.
G-MLE: Yes.
Ours (z1): I can only see the
back of his body.
Ours (z2): No.
Ours (z3): No , he?s too far
away.
Q: How old does the man seem to
be?
A: 20?s.
G-MLE: Late teens.
Ours (z1): On his 20s.
Ours (z2): In his twenties.
Ours (z3): Mid 20?s.
Q: Can you see broccoli?
A: Yes, 3 larger pieces and some
small.
G-MLE: No, just the broccoli.
Ours (z1): I can see broccoli,
slightly butter.
Ours (z2): Yes, there is broccoli.
Ours (z3): Yes, broccoli is
green.
Table 4: Adversarial training results on VisDial dataset.
Discriminative
Generative
Model
MRR
R@1
R@5
R@10
Mean
MRR
R@1
R@5
R@10
Mean
HCIAE-D-NP-ATT
HCIAE-G-DIS
0.6222
-
48.48
-
78.75
-
87.59
-
4.81
-
0.5467
44.35
65.28
71.55
14.23
HCIAE-GAN1
HCIAE-GAN2
0.2177
0.6050
8.82
46.20
32.97
77.92
52.14
87.20
18.53
4.97
0.5298
0.5459
43.12
44.33
62.74
65.05
68.58
71.40
16.25
14.34
One might wonder, why not train a GAN for visual dialog? Formulating the task in a GAN setting
would involve G and D training in tandem with D providing feedback as to whether a response that
G generates is real or fake. We found this to be a particularly unstable setting, for two main reasons:
First, consider the case when the ground truth answer and the generated answers are the same. This
happens for answers that are typically short or ?cryptic? (e.g. ?yes?). In this case, D can not train itself
or provide feedback, as the answer is labeled both positive and negative. Second, in cases where the
ground truth answer is descriptive but the generator provides a short answer, D can quickly become
powerful enough to discard generated samples as fake. In this case, D is not able to provide any
information to G to get better at the task. Our experience suggests that the discriminator, if one were
to consider a ?GANs for visual dialog? setting, can not merely be focused on differentiating fake
from real. It needs to be able to score similarity between the ground truth and other answers. Such a
scoring mechanism provides a more reliable feedback to G. In fact, as we show in the previous two
results, a pre-trained D that captures this structure is the key ingredient in sharing knowledge with G.
The adversarial training of D is not central.
5.3
Qualitative Comparison
In Table 3 we present a couple of qualitative examples that compares the responses generated by
G-MLE and G-DIS. G-MLE predominantly produces ?safe? and less informative answers, such as
?Yes? and or ?I can?t tell?. In contrast, our proposed model G-DIS does so less frequently, and often
generates more diverse yet informative responses.
6
Conclusion
Generative models for (visual) dialog are typically trained with an MLE objective. As a result, they
tend to latch on to safe and generic responses. Discriminative (or retrieval) models on the other hand
have been shown to significantly outperform their generative counterparts. However, discriminative
models can not be deployed as dialog agents with a real user where canned candidate responses
are not available. In this work, we propose transferring knowledge from a powerful discriminative
visual dialog model to a generative model. We leverage the Gumbel-Softmax (GS) approximation to
the discrete distribution ?specifically, a RNN augmented with a sequence of GS samplers, coupled
with a ST gradient estimator for end-to-end differentiability. We also propose a novel visual dialog
encoder that reasons about image-attention informed by the history of the dialog; and employ a
metric learning loss along with a self-attentive answer encoding to enable the discriminator to
learn meaningful structure in dialog responses. The result is a generative visual dialog model that
significantly outperforms state-of-the-art.
8
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11
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6,230 | 6,636 | MaskRNN: Instance Level Video Object
Segmentation
Yuan-Ting Hu
UIUC
[email protected]
Jia-Bin Huang
Virginia Tech
[email protected]
Alexander G. Schwing
UIUC
[email protected]
Abstract
Instance level video object segmentation is an important technique for video editing
and compression. To capture the temporal coherence, in this paper, we develop
MaskRNN, a recurrent neural net approach which fuses in each frame the output
of two deep nets for each object instance ? a binary segmentation net providing
a mask and a localization net providing a bounding box. Due to the recurrent
component and the localization component, our method is able to take advantage
of long-term temporal structures of the video data as well as rejecting outliers.
We validate the proposed algorithm on three challenging benchmark datasets,
the DAVIS-2016 dataset, the DAVIS-2017 dataset, and the Segtrack v2 dataset,
achieving state-of-the-art performance on all of them.
1
Introduction
Instance level video object segmentation of complex scenes is a challenging problem with applications
in areas such as object identification, video editing, and video compression. With the recent release
of the DAVIS dataset [39], the task of segmenting multiple object instances from videos has gained
considerable attention. However, just like for classical foreground-background segmentation, deforming shapes, fast movements, and multiple objects occluding each other pose significant challenges to
instance level video object segmentation.
Classical techniques [5, 10, 11, 17, 21, 41, 20, 44, 49] for video object segmentation often rely on
geometry and assume rigid scenes. Since these assumptions are often violated in practice, visually
apparent artifacts are commonly observed. To temporally and spatially smooth object mask estimates,
graphical model based techniques [22, 2, 14, 45, 47, 46] have been proposed in the past. While
graphical models enable an effective label propagation across the entire video sequences, they often
tend to be sensitive to parameters.
Recently, deep learning based approaches [7, 26, 23, 6, 25] have been applied to video object
segmentation. Early work in this direction predicts the segmentation mask frame by frame [7]. Later,
prediction of the current frame incorpoerates additional cues from the preceding frame using optical
flow [23, 26, 25], semantic segmentations [6], or mask propagation [26, 25]. Importantly, all these
methods only address the foreground-background segmentation of a single object and are not directly
applicable to instance level segmentation of multiple objects in videos.
In contrast to the aforementioned methods, in this paper, we develop MaskRNN, a framework that
deals with instance level segmentation of multiple objects in videos. We use a bottom-up approach
where we first track and segment individual objects before merging the results. To capture the
temporal structure, our approach employs a recurrent neural net while the segmentation of individual
objects is based on predictions of binary segmentation masks confined to a predicted bounding box.
We evaluate our approach on the DAVIS-2016 dataset [37], the DAVIS-2017 dataset [39], and the
Segtrack v2 dataset [30]. On all three we observe state-of-the-art performance.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Table 1: Comparisons with the state-of-the-art deep learning based video object segmentation
algorithms.
Method
Using flow
Temporal information
Location prior
Semantic prior
Post-processing
Finetuning on the 1st frame
2
OSVOS
[7]
MaskProp
[26]
FusionSeg
[23]
LucidTracker
[25]
SemanticProp
[6]
Ours
No
No
No
No
Boundary snapping
Yes
Yes
Short-term
Previous mask
No
denseCRF
Yes
Yes
Short-term
No
No
No
No
Yes
Short-term
Previous mask
No
denseCRF
Yes
No
No
No
Yes
No
Yes
Yes
Long-term (RNN)
Previous mask+Bounding box
No
No
Yes
Related Work
Video object segmentation has been studied extensively in recent years [45, 30, 34, 40, 29, 28, 36, 48,
16, 46, 37, 23, 6, 25]. In the following, we group the literature into two categories: (1) graph-based
approaches and (2) deep learning methods.
Video object segmentation via spatio-temporal graphs: Methods in this category construct a
three-dimensional spatio-temporal graph [45, 30, 16, 28] to model the inter- and the intra-frame
relationship of pixels or superpixels in a video. Evidence about a pixels assignment to the foreground
or background is then propagated along this spatio-temporal graph, to determine which pixels are
to be labeled as foreground and which pixel corresponds to the background of the observed scene.
Graph-based approaches are able to accept different degrees of human supervision. For example,
interactive video object segmentation approaches allow users to annotate the foreground segments
in several key frames to generate accurate results by propagating the user-specified masks to the
entire video [40, 13, 34, 31, 22]. Semi-supervised video object segmentation techniques [4, 16, 45,
22, 46, 33] require only one mask for the first frame of the video. Also, there are unsupervised
methods [9, 28, 50, 36, 35, 12, 48] that do not require manual annotation. Since constructing and
exploring the 3D spatio-temporal graphs is computationally expensive, the graph-based methods are
typically slow, and the running time of the graph-based video object segmentation is often far from
real time.
Video object segmentation via deep learning: With the success of deep nets on semantic segmentation [32, 42], deep learning based approaches for video object segmentation [7, 26, 23, 6, 25]
have been intensively studied recently and often yield state-of-the-art performance, outperforming
graph-based methods. Generally, the employed deep nets are pre-trained on object segmentation
datasets. In the semi-supervised setting where the ground truth mask of the first frame of a video
is given, the network parameters are then finetuned on the given ground truth of the first frame of
a particular video, to improve the results and the specificity of the network. Additionally, contour
cues [7] and semantic segmentation information [7] can be incorporated into the framework. Besides
those cues, optical flow between adjacent frames is another important key information for video
data. Several methods [26, 23, 25] utilize the magnitude of the optical flow between adjacent frames.
However, these methods do not explicitly model the location prior, which is important for object
tracking. In addition, these methods focus on separating foreground from background and do not
consider instance level segmentation of multiple objects in a video sequence.
In Tab. 1, we provide a feature-by-feature comparison of our video object segmentation technique
with representative state-of-the-art approaches. We note that the developed method is the only one
that takes long-term temporal information into account via back-propagation through time using a
recurrent neural net. In addition, the discussed method is the only one that estimates the bounding
boxes in addition to the segmentation masks, allowing us to incorporate a location prior of the tracked
object.
3
Instance Level Video Object Segmentation
Next, we present MaskRNN, a joint multi-object video segmentation technique, which performs
instance level object segmentation by combining binary segmentation with effective object tracking
via bounding boxes. To benefit from temporal dependencies, we employ a recurrent neural net
component to connect prediction over time in a unifying framework. In the following, we first provide
a general outline of the developed approach illustrated in Fig. 1 and detail the individual components
subsequently.
2
Figure 1: An illustration for the proposed algorithm. We show an example video with 2 objects (left).
Our method predicts the binary segmentation for each object using 2 deep nets (Section 3.3), one for
each object, which perform binary segmentation and object localization. The output instance-level
segmentation mask is obtained by combining the binary segmentation masks (Section 3.2).
3.1
Overview
We consider a video sequence I = {I1 , I2 , ..., IT } which consists of T frames It , t ? {1, . . . , T }.
Throughout, we assume the ground truth segmentation mask of the N object instances of interest
to be given for the first frame I1 . We refer to the ground truth segmentation mask of the first frame
via y?1 ? {0, 1, ..., N}H?W , where N is the number of object instances, and H and W are the height
and width of the video frames. In multi-instance video object segmentation, the goal is to predict
y2 , . . . , yT ? {0, . . . , N}H?W , which are the segmentation masks corresponding to frames I2 to IT .
The proposed method is outlined in Fig. 1. Motivated by the time-dependence of the frames in the
video sequence we formulate the task of instance level semantic video segmentation as a recurrent
neural net, where the prediction of the previous frame influences prediction of the current frame.
Beyond the prediction yt?1 for the previous frame t ? 1, our approach also takes into account both the
previous and the current frames, i.e., It?1 and It . We compute the optical flow from the two images.
We then use the predicted optical flow (i) as input feature to the neural nets and (ii) to warp the
previous prediction to roughly align with the current frame.
The warped prediction, the optical flow itself, and the appearance of the current frame are then
used as input for N deep nets, one for each of the N objects. Each of the deep nets consists of two
parts, a binary segmentation net which predicts a segmentation mask, and an object localization net
which performs bounding box regression. The latter is used to alleviate outliers. Both, bounding box
regression and segmentation map are merged into a binary segmentation mask bti ? [0, 1]H?W denoting
the foreground-background probability maps for each of the N object instances i ? {1, . . . , N}. The
binary semantic segmentations for all N objects are subsequently merged using an arg max operation.
The prediction for the current frame, i.e., yt , is computed via thresholding. Note that we combine the
binary predictions only at test time.
In the following, we first describe our fusion operation in detail, before discussing the deep net
performing binary segmentation and object localization.
3.2
Multiple instance level segmentation
Predicting the segmentation mask yt for the t-th frame, can be viewed as a multi-class prediction
problem, i.e., assigning to every pixel in the video a label, indicating whether the pixel p represents an
object instance (ytp = {1, ..., N}) or whether the pixel is considered background (ytp = 0). Following
a recent technique for instance level image segmentation [18], we cast this multi-class prediction
problem into multiple binary segmentations, one per object instance.
Assume availability of binary segmentation masks bti ? [0, 1]H?W which provide for each object
instance i ? {1, . . . , N} the probability that a pixel should be considered foreground or background.
To combine the binary segmentations bti into one final prediction yt such that every pixel is assigned
to only one object label, is achieved by assigning the class with the largest probability for every pixel.
3
Figure 2: An illustration of the binary object segmentation network and the object localization
network as described in Section 3.3. The binary segmentation network is a two-stream network
including appearance stream and flow stream. The inputs of the appearance stream are the current
i ). The inputs of the flow stream are the flow magnitude and the warped mask,
frame It and ?t (bt?1
i
?t (bt?1 ). The object localization net refines the bounding box proposal to estimate the location prior.
To compute the binary segmentation mask bti , the output of appearance stream and the flow stream
are linearly combined and the responses outside the refined bounding box are discarded.
To be more specific, we assign class label i ? {1, . . . , N} to the pixel if the probability for class i at
the pixel (indicated by bti ) is largest among the N probability maps for the N object instances. Note
that this operation is similar to a pooling operation, and permits back-propagation.
3.3
Binary Segmentation
To obtain the binary segmentations bti ? [0, 1]H?W employed in the fusion step, N deep nets are used,
one for each of the N considered object instances. One of the N deep nets is illustrated in Fig. 2. It
consists of two components, the binary segmentation net and the object localization net, which are
discussed in greater detail in the following.
Binary Segmentation Net: The objective for each of the binary segmentation nets is to predict the
foreground-background mask bti ? [0, 1]H?W for its corresponding object instance i ? {1, . . . , N}. To
achieve this task, the binary segmentation net is split into two streams, i.e., the appearance stream
and the flow stream. The input of the appearance stream is the concatenation of the current frame It
and the warped prediction of the previous frame yt?1 , denoted as ?t?1,t (yt?1 ). The warping function
?t?1,t (.) transforms the input based on the optical flow field from frame It?1 to frame It . The input of
the flow stream is the concatenation of the magnitude of the flow field from It to It?1 and It to It+1
and, again, the warped prediction of the previous frame ?t?1,t (yt?1 ). The architecture of both streams
is identical and follows the subsequent description.
The network architecture is inspired by [7] where the bottom of the network follows the structure of
the VGG-16 network [43]. The intermediate representations of the VGG-16 network, right before the
max-pooling layers and after the ReLU layers, are extracted, upsampled by bilinear interpolation and
linearly combined to form a single channel feature representation which has the same size as the input
image. By linearly combining the two representations, one from the appearance stream and the other
one from the flow stream, and by taking the sigmoid function on the combined single channel feature
response, we obtain a probability map which indicates the probability bti ? [0, 1]H?W of a pixel in the
t-th frame being foreground, i.e., corresponding to the i-th object. The network architecture of the
appearance stream is shown in Fig. 2 (right panel). During training, we use the weighted binary cross
entropy loss as suggested in [7].
Note that all the operations in our network are differentiable. Hence, we can train the developed
network end-to-end via back-propagation through time.
4
Table 2: Contribution of different components of our algorithm evaluated on DAVIS-2016 and
DAVIS-2017 dataset. The performance is in term of IoU (%).
Component
Enable (X) / Disable
AStream
X
X
X
X
X
X
Segmentation net FStream
X
X
X
X
X
Warp mask
X
X
X
X
Train
X
X
X
Localization net
Apply
X
X
RNN
X
DAVIS-2016 IoU(%), w/o Finetuning 54.17 55.87 56.88 52.29 53.90 56.32
DAVIS-2016 IoU(%), w/ Finetuning
76.63 79.77 79.92 78.43 80.10 80.38
DAVIS-2017 IoU(%), w/o Finetuning 41.29 43.33 44.52 38.95 41.57 45.53
DAVIS-2017 IoU(%), w/ Finetuning
58.66 59.46 59.71 56.12 60.41 60.51
Object Localization Net: Usage of an object localization net is inspired by tracking approaches
which regularize the prediction by assuming that the object is less likely to move drastically between
temporally adjacent frames. The object localization network computes the location for the i-th object
in the current frame via bounding box regression. First, we find the bounding box proposal on
i ). Similarly to the bounding box regression in Fast-RCNN [15], with the
the warped mask ?t (bt?1
bounding box proposal as the region of interest, we use the conv5_3 feature in the appearance stream
of the segmentation net to perform RoI-pooling, followed by two fully connected layers. Their output
is used to regress the bounding box position. We refer the reader to [15] for more details on bounding
box regression.
Given the bounding box, a pixel is classified as foreground if it is predicted as foreground by the
segmentation net and if it is inside a bounding box which is enlarged by a factor of 1.25 compared
to the predicted of the localization net. The estimated bounding box is then used to restrict the
segmentation to avoid outliers which are far away from the object.
3.4
Training and Finetuning
Our framework outlined in the preceding sections and illustrated in Fig. 1 can be trained end-to-end
via back-propagation through time given a training sequence. Note that back-propagation through
time is used because of the recurrence relation that connects multiple frames of the video sequence.
To further improve the predictive performance, we follow the protocol [39] for the semi-supervised
setting of video object segmentation and finetune our networks using the ground truth segmentation
mask provided for the first frame. Specifically, we further optimize the binary segmentation net
and localization net based on the given ground truth. Note that it is not possible to adjust the entire
architecture since only a single ground truth frame is provided in the supervised setting.
4
Implementation Details
In the following, we describe the implementation details of our approach, as well as the training data.
We also provide details about the offline training and online training in our experimental setup.
Training data: We use the training set of the DAVIS dataset to pre-train the appearance network
for general-purpose object segmentation. The DAVIS-2016 dataset [37] contains 30 training videos
and 20 testing videos and the DAVIS-2017 dataset [39] consists of 60 training videos and 30 testing
videos. Note that the annotation of the DAVIS-2016 dataset contains only one single object per video.
For a fair evaluation on the DAVIS-2016 and DAVIS-2017 datasets, the object segmentation net and
localization nets are trained on the training set of each dataset separately. During testing, the network
is further finetuned online on the given ground-truth of the first frame since we assume the ground
truth segmentation mask of the first frame, i.e., y?1 , to be available.
Offline training: During offline training, we first optimize the networks on static images. We
?
found it useful to randomly perturb the ground-truth segmentation mask yt?1
locally, to simulate
the imperfect prediction of the last frame. The random perturbation includes dilation, deformation,
resizing, rotation and translation. After having trained both the binary segmentation net and the
object localization net on single frames, we further optimize the segmentation net by taking long-term
5
Table 3: The quantitative evaluation on the validation set of DAVIS dataset [37]. The evaluation
matrics are the IoU measurement J , boundary precision F , and time stability T . Following [37],
we also report the recall and the decay of performance over time for J and F measurements.
Semi-supervised
OSVOS MSK VPN OFL BVS FCP JMP HVS SEA TSP OURS
[7]
[26] [24] [46] [33] [38] [13] [16] [1]
[8]
J
Mean M ?
Recall O ?
Decay D ?
79.8
93.6
14.9
79.7
93.1
8.9
70.2
82.3
12.4
68.0
75.6
26.4
60.0
66.9
28.9
58.4
71.5
-2.0
57.0
62.6
39.4
54.6
61.4
23.6
50.4 31.9
53.1 30.0
36.4 38.1
80.4
96.0
4.4
F
Mean M ?
Recall O ?
Decay D ?
80.6
92.6
15.0
75.4
87.1
9.0
65.5
69.0
14.4
63.4
70.4
27.2
58.8
67.9
21.3
49.2
49.5
-1.1
53.1
54.2
38.4
52.9
61.0
22.7
48.0 29.7
46.3 23.0
34.5 35.7
82.3
93.2
8.8
T
Mean M ?
37.8
21.8
32.4
22.2
34.7
30.6
15.9
36.0
15.4 41.7
19.0
information into account, i.e., training using the recurrence relation. We consider 7 frames at a time
due to the memory limitation imposed by the GPU.
During offline training all networks are optimized for 10 epochs using the Adam solver [27] and the
learning rate is gradually decayed during training, starting from 10?5 . Note that we use the pre-trained
flowNet2.0 [19] for optical flow computation. During training, we apply data augmentation with
randomly resizing, rotating, cropping, and left-right flipping the images and masks.
Online finetuning: In the semi-supervised setting of video object segmentation, the ground-truth
segmentation mask of the first frame is available. The object segmentation net and the localization
net are further finetuned on the first frame of the testing video sequence. We set the learning rate
to 10?5 . We train the network for 200 iterations, and the learning rate is gradually decayed over
time. To enrich the variation of the training data, for online finetuning the same data augmentation
techniques are applied as in offline training, namely randomly resizing, rotating, cropping and flipping
the images. Note that the RNN is not employed during online finetuning since only a single frame of
training data is available.
5
Experimental Results
Next, we first describe the evaluation metrics before we present an ablation study of our approach,
quantitative results, and qualitative results.
5.1
Evaluation Metrics
Intersection over union: We use the common mean intersection over union (IoU) metric which
calculates the average across all frames of the dataset. The IoU metric is particularly challenging for
small sized foreground objects.
Contour accuracy [37]: Besides an accurate object overlap measured by IoU, we are also interested
in an accurate delineation of the foreground objects. To assess the delineation quality of our approach,
we measure the precision, P, and the recall R of the two sets of points on the contours of the ground
truth segment and the output segment via a bipartite graph matching. The contour accuracy is
2PR
calculated as P+R
.
Temporal stability [37]: The temporal stability estimates the degree of deformation needed to
transform the segmentation masks from one frame to the next. The temporal stability is measured by
the dissimilarity of the shape context descriptors [3] which describe the points on the contours of the
segmentation between the two adjacent frames.
5.2
Ablation study
We validate the contributions of the components in our method by presenting an ablation study
summarized in Tab. 2 on two datasets, DAVIS-2016 and DAVIS-2017. We mark the enabled
components using the ?X? symbol. We analyze the contribution of the binary segmentation net
6
Table 4: The quantitative evaluation on DAVIS-2017 dataset [39] and SegTrack v2 dataset [30].
DAVIS-2017
SegTrack v2
OSVOS [7] OFL [46] OURS OSVOS [7] MSK [26] OFL [46] OURS
IoU(%)
52.1
54.9
60.5
61.9
67.4
67.5
72.1
including the appearance stream (?AStream?), the flow stream (?FStream?) and whether to warp
the input mask, yt?1 , based on the optical flow field (?Warp mask?). In addition, we analyze the
effects of the object localization net. Specifically, we assess the occurring performance changes of
two configurations: (i) by only adding the bounding box regression loss into the objective function
(?Train?), i.e., both the segmentation net and the object localization net are trained but only the
segmentation net is deployed; (ii) by training and applying the object localization net (?Apply?). The
contribution of the recurrent training (?RNN?) is also illustrated. The performances with and without
online finetuning as described in Section 4 are shown for each dataset as well.
In Tab. 2, we generally observe that online finetuning is important as the network is adjusted to the
specific object appearance in the current video.
For the segmentation net, the combination of the appearance stream and the flow stream performs
better than using only the appearance stream. This is due to the fact that the optical flow magnitude
provided in the flow stream provides complementary information by encoding motion boundaries,
which helps to discover moving objects in the cluttered background. The performance can be
further improved by using the optical flow to warp the mask so that the input to both streams of the
segmentation net also takes the motion into account.
For the localization net, we first show that adding the bounding box regression loss decreases
the performance of the segmentation net (adding ?Train? configuration). However, by applying the
bounding box to restrict the segmentation mask improves the results beyond the performance achieved
by only applying the segmentation net.
Training the network using the recurrence relationship further improves the results as the network
produces more consistent segmentation masks over time.
5.3
Quantitative evaluation
We compare the performance of our approach to several baselines on two tasks: foregroundbackground video object segmentation and multiple instance-level video object segmentation. More
specifically, we use DAVIS-2016 [37] for evaluating foreground-background segmentation, and
DAVIS-2017 [39] and Segtrack v2 [30] datasets for evaluating multiple instance-level segmentation.
The three datasets serve as a good testbed as they contain challenging variations, such as drastic
appearance changes, fast motion, and occlusion. We compare the performance of our approach to
several state-of-the-art benchmarks. We assess performance on the validation set when using the
DAVIS datasets and we use the whole dataset for Segtrack v2 as no split into train and validation
sets is available. The results on DAVIS-2016 are summarized in Tab. 3, where we report the IoU,
the contour accuracy, and the time stability metrics following [37]. The results on DAVIS-2017 and
SegTrack v2 are summarized in Tab. 4.
Foreground-background video object segmentation: We use the DAVIS-2016 dataset to evaluate
the performance of foreground-background video object segmentation. The DAVIS-2016 dataset
contains 30 training videos and 20 validation videos. The network is first trained on the 30 training
videos and finetuned on the first frame of the 20 validation videos, respectively during testing. The
performance evaluation is reported in Tab. 3. We outperform the other state-of-the-art semi-supervised
methods by 0.6%. Note that OSVOS [7], MSK [26], VPN [24] are also deep learning approach. In
contrast to our approach, these methods don?t employ the location prior.
Instance-level video object segmentation: We use the DAVIS-2017 and the Segtrack v2 datasets
to evaluate the performance of instance-level video object segmentation. The DAVIS-2017 dataset
contains 60 training videos and 30 validation videos. The Segtrack v2 dataset contains 14 videos.
There are 2.27 objects per video on average in the DAVIS-2017 dataset and 1.74 in the Segtrack v2
dataset. Again, as for DAVIS-2016, the network is trained on the training set and then finetuned using
the groundtruth of the given first frame. Since the Segtrack v2 dataset does not provide a training
set, we use the DAVIS-2017 training set to optimize and finetune the deep nets. The performance
evaluation is reported in Tab. 4. We outperform other state-of-the-art semi-supervised methods by
5.6% and 4.6% on DAVIS-2017 and Segtrack v2, respectively.
7
Figure 3: Visual results of our approach on DAVIS-2016 (1st and 2nd row), DAVIS-2017 (3rd and
4th row) and Segtrack v2 dataset (5th and 6th row).
Figure 4: Failure cases of our approach. The 1st and the 3rd column shows the results of the beginning
frames. Our method fails to track the object instances as shown in the 2nd and 4th column.
5.4
Qualitative evaluation
We visualize some of the qualitative results of our approach in Fig. 3 and Fig. 4. In Fig. 3, we show
some successful cases of our algorithm on the DAVIS and Segtrack datasets. We observe that the
proposed method accurately keeps track of the foreground objects even with complex motion and
cluttered background. We also observe accurate instance level segmentation of multiple objects which
occlude each other. In Fig. 4, we visualize two failure cases of our approach. Reasons for failures
are the similar appearance of instances of interest as can be observed for the leftmost two figures.
Another reason for failure is large variations in scale and viewpoint as shown for the two figures on
the right of Fig. 4.
6
Conclusion
We proposed MaskRNN, a recurrent neural net based approach for instance-level video object
segmentation. Due to the recurrent component and the combination of segmentation and localization
nets, our approach takes advantage of the long-term temporal information and the location prior to
improve the results.
Acknowledgments: This material is based upon work supported in part by the National Science
Foundation under Grant No. 1718221. We thank NVIDIA for providing the GPUs used in this
research.
8
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6,231 | 6,637 | Gated Recurrent Convolution Neural Network for
OCR
Jianfeng Wang?
Beijing University of Posts and Telecommunications
Beijing 100876, China
[email protected]
Xiaolin Hu
Tsinghua National Laboratory for Information Science and Technology (TNList)
Department of Computer Science and Technology
Center for Brain-Inspired Computing Research (CBICR)
Tsinghua University, Beijing 100084, China
[email protected]
Abstract
Optical Character Recognition (OCR) aims to recognize text in natural images.
Inspired by a recently proposed model for general image classification, Recurrent
Convolution Neural Network (RCNN), we propose a new architecture named
Gated RCNN (GRCNN) for solving this problem. Its critical component, Gated
Recurrent Convolution Layer (GRCL), is constructed by adding a gate to the
Recurrent Convolution Layer (RCL), the critical component of RCNN. The gate
controls the context modulation in RCL and balances the feed-forward information
and the recurrent information. In addition, an efficient Bidirectional Long ShortTerm Memory (BLSTM) is built for sequence modeling. The GRCNN is combined
with BLSTM to recognize text in natural images. The entire GRCNN-BLSTM
model can be trained end-to-end. Experiments show that the proposed model
outperforms existing methods on several benchmark datasets including the IIIT-5K,
Street View Text (SVT) and ICDAR.
1
Introduction
Reading text in scene images can be regarded as an image sequence recognition task. It is an important
problem which has drawn much attention in the past decades. There are two types of scene text
recognition tasks: constrained and unconstrained. In constrained text recognition, there is a fixed
lexicon or dictionary with known length during inference. In unconstrained text recognition, each
word is recognized without a dictionary. Most of the previous works are about the first task.
In recent years, deep neural networks have gained great success in many computer vision tasks
[39, 19, 34, 42, 9, 11]. The fast development of deep neural networks inspires researchers to use
them to solve the problem of scene text recognition. For example, an end-to-end architecture which
combines a convolutional network with a recurrent network is proposed [30]. In this framework, a
plain seven-layer-CNN is used as a feature extractor for the input image while a recurrent neural
network (RNN) is used for image sequence modeling. For another example, to let the recurrent
network focus on the most important segments of incoming features, an end-to-end system which
integrates attention mechanism, recurrent network and recursive CNN is developed [21].
?
This work was done when Jianfeng Wang was an intern at Tsinghua University.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Illustration of using RCL with T = 2 for OCR.
A Recurrent Convolution Neural Network (RCNN) is proposed for general image classification
[22], which simulates an anatomical fact that recurrent connections are ubiquitously existent in the
neocortex. It is believed that the recurrent synapses that exist in the neocortex play an important role
in context modulation during visual recognition. A feed-forward model can only capture the context
in higher layers where units have larger receptive fields, but this information cannot modulate the
units in lower layers which is responsible for recognizing smaller objects. Hence, using recurrent
connections within a convolutional layer (called Recurrent Convolution Layer or RCL) can bring
context information to all units in this layer. This implements a nonclassical receptive field [12, 18]
of a biological neuron as the effective receptive field is larger than that determined by feedforward
connections. However, with increasing iterations, the size of the effective receptive field will increase
unboundedly, which contradicts the biological fact. One needs a mechanism to constrain the growth
of the effective receptive field.
In addition, from the viewpoint of performance enhancing, one also needs to control the context
modulation of neurons in RCNN. For example, in Figure 1, it is seen that for recognizing a character,
not all of the context are useful. When the network recognizes the character "h", the recurrent
kernel which covers the other parts of the character is beneficial. However, when the recurrent
kernel is enlarged to the parts of other characters, such as "p", the context carried by the kernel is
unnecessary. Therefore, we need to weaken the signal that comes from unrelated context and combine
the feed-forward information with recurrent information in a flexible way.
To achieve the above goals, we introduce a gate to control the context modulation in each RCL layer,
which leads to the Gated Recurrent Convolution Neural Network (GRCNN).
In addition, the recurrent neural network is adopted in the recognition of words in natural images
since it is good at sequence modeling. In this work, we choose the Long Short Term Memory (LSTM)
as the top layer of the proposed model, which is trained in an end-to-end fashion.
2
Related Work
OCR is one of the most important challenges in computer vision and many methods have been
proposed for this task. The word locations and scores of detected characters are input into the
Pictorial Structure formulation to acquire an optimal configuration of a particular word [26]. A
complete OCR system that contains text detection as well as text recognition is designed, and it can
be well applied to both unconstrained and constrained text recognition task [3].
The OCR can be understood as a classification task by treating each word in the lexicon as an
object category [37]. Another classical method, the Conditional Random Fields, is proposed in text
recognition [17, 25, 31]. Besides those conventional methods, some deep network-based methods are
also proposed. The Convolutional Neural Network (CNN) is used to extract shared features, which
are fed into a character classifier [16]. To further improve the performance, more than one objective
functions are used in a CNN-based OCR system [15]. A CNN is combined with the Conditional
Random Field graphical model, which can be jointly optimized through back-propagation [14].
Moreover, some works introduced RNN, such as LSTM, to recognize constrained and unconstrained
words [30]. The attention mechanism is applied to a stacked RNN on the top of the recursive CNN
[21].
2
Figure 2: Illustration of GRCL with T = 2. The convolutional kernels in the same color use the same
weights.
Many new deep neural networks for general image classification have been proposed in these years.
The closely related model to the proposed model in this paper is RCNN [22], which is inspired by
the observation of abundant recurrent synapses in the brain. It adds recurrent connections within the
standard convolutional layers, and the recurrent connections improve the network capacity for object
recognition. When unfolded in time, it becomes a CNN with many shortcuts from the bottom layer
to upper layers. RCNN has been used to solve other problems such as scene labelling [23], action
recognition [35], and speech processing [43]. One related model to RCNN is the recursive neural
network [32], in which a recursive layer is unfolded into a stack of layers with tied weights. It can be
regarded as an RCNN without shortcuts. Another related model is the DenseNet [11], in which every
lower layer is connected to the upper layers. It can be regarded as an RCNN with more shortcuts and
without weight sharing. The idea of shortcut have also been explored in the residual network [9].
3
3.1
GRCNN-BLSTM Model
Recurrent Convolution Layer and RCNN
The RCL [22] is a module with recurrent connections in the convolutional layer of CNN. Consider a
generic RNN model with feed-forward input u(t). The internal state x(t) can be defined as:
x(t) = F(u(t), x(t ? 1), ?)
(1)
where the function F describes the nonlinearity of RNN (e.g. ReLU unit) and ? is the parameter. The
state of RCL evolves over discrete time steps:
x(t) = F((wf ? u(t) + wr ? x(t ? 1))
(2)
where "*" denotes convolution, u(t) and x(t ? 1) denote the feed-forward input and recurrent input
respectively, wf and wr denote the feed-forward weights and recurrent weights respectively.
Multiple RCLs together with other types of layers can be stacked into a deep model. A CNN that
contains RCL is called RCNN [22].
3
Figure 3: Overall pipeline of the architecture.
3.2
Gated Recurrent Convolution Layer and GRCNN
The Gated Recurrent Convolution Layer (GRCL) is the essential module in our framework. This
module is equipped with a gate to control the context modulation in RCL and it can weaken or even
cut off irrelevant context information. The gate of GRCL can be written as follows:
(
G(t) =
0
sigmoid(BN (wgf
? u(t)) +
t=0
BN (wgr
? x(t ? 1)))
(3)
t>0
Inspired by the Gated Recurrent Unit (GRU) [4], we let the controlling gate receive the signals from
the feed-forward input as well as the states at the last time step. We use two 1 ? 1 kernels, wgf
and wgr , to convolve with the feed-forward input and recurrent input separately. wgf denotes the
feed-forward weights for the gate and wgr denotes the recurrent weights for the gate. The recurrent
weights are shared over all time steps (Figure 2). Batch normalization (BN) [13] is used to improve
the performance and accelerate convergence. The GRCL can be described by:
(
x(t) =
ReLU (BN (wf ? u(t))
t=0
ReLU (BN (wf ? u(t)) + BN (BN (wr ? x(t ? 1)) G(t)))
t>0
(4)
In the equations, " " denotes element-wise multiplication. Batch normalization (BN) is applied after
each convolution and element-wise multiplication operation. The parameters and statistics in BN
are not shared over different time steps, though the recurrent weights are shared. It is assumed that
the input to GRCL is the same over time t, which is denoted by u(0). It is the output of the previous
layer. This assumption means that the feed-forward part contributes equally at each time step. It
is important to clarify that the time step in GRCL is not identical to the time associated with the
sequential data. The time steps denote the iterations in processing the input.
Figure 2 shows the diagram of the GRCL with T = 2. When t = 0, only the feed-forward
computation takes place. At t = 1, the gate?s output, which is determined by the feed-forward input
and the states at the previous time step (t = 0), acts on the recurrent component. It can modulate
the recurrent signals. When the output of the gate is 1, it becomes the standard RCL. When the
output of the gate is 0, the recurrent signal is dropped and it becomes the standard convolutional layer.
Therefore, the GRCL is a generalization of RCL and it can adjust context modulation dynamically.
The effective receptive field (RF) of each GRCL unit in the previous layer?s feature maps expands
while the iteration number increases. However, unlike the RCL, some regions that contain unrelated
information in large effective RF cannot provide strong signal to the center of RF. This mimics the
fact that human eyes only care about the context information that helps the recognition of the objects.
Multiple GRCLs together with other types of layers can be stacked into a deep model. Hereafter, a
CNN that contains GRCL is called a GRCNN.
3.3
Overall Architecture
The architecture consists of three parts: feature sequence extraction, sequence modelling, and
transcription. Figure 3 shows the overall pipeline which can be trained end-to-end.
Feature Sequence Extraction: We use the GRCNN in the first part and there are no fully-connected
layers. The input to the network is a whole image and the image is resized to fixed length and height.
4
Table 1: The GRCNN configuration
Conv MaxPool GRCL MaxPool GRCL MaxPool GRCL MaxPool Conv
3?3
2?2
3?3
2?2
3?3
2?2
3?3
2?2
2?2
num: 64
num: 64
num: 128
num: 256
num: 512
sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1
ph:1 pw:1ph:0 pw:0ph:1 pw:1ph:0 pw:0ph:1 pw:1ph:0 pw:1ph:1 pw:1ph:0 pw:1ph:0 pw:0
Specifically, the feature map in the last layer is sliced from left to right by column to form a feature
sequence. Therefore, the i-th feature vector is formed by concatenating the i-th columns of all of the
maps. We add some max pooling layers to the network in order to ensure the width of each column is
1. Each feature vector in the sequence represents a rectangle region of the input image, and it can be
regarded as the image descriptor for that region. Comparing the GRCL with the basic convolutional
layer, we find that each feature vector generated by the GRCL represents a larger region. This feature
vector contains more information than the feature vector generated by the basic convolutional layer,
and it is beneficial for the recognition of text.
Sequence Modeling: An LSTM [10] is used on the top of feature extraction module for sequence
modeling. The peephole LSTM is first proposed in [5], whose gates not only receive the inputs from
the previous layer, but also from the cell state. We add the peephole LSTM to the top of GRCNN and
investigate the effect of peephole connections to the whole network?s performance. The inputs to the
gates of LSTM can be written as:
i = ?(Wxi xt + Whi ht?1 + ?1 Wci ct?1 + bi ),
(5)
f = ?(Wxf xt + Whf ht?1 + ?2 Wcf ct?1 + bf ),
(6)
o = ?(Wxo xt + Who ht?1 + ?3 Wco ct + bo ),
(7)
?i ? {0, 1}.
(8)
?i is defined as an indication factor and its value is 0 or 1. When ?i is equal to 1, the gate receives the
modulation of the cell?s state.
However, LSTM only considers past events. In fact, the context information from both directions are
often complementary. Therefore, we use stacked bidirectional LSTM [29] in our architecture.
Transcription: The last part is transcription which converts per-frame predictions to real labels. The
Connectionist Temporal Classification (CTC) [8] method is used. Denote the dataset by S = {(I, z)},
where I is a training image and z is the corresponding ground truth label sequence. The objective
function to be minimized is defined as follows:X
O=?
logp(z|I).
(9)
(I,z)?S
Given an input image I, the prediction of RNN at each time step is denoted by ?t . The sequence
? may contain blanks and repeated labels (e.g. (a ? b ? ?b)=(ab ? ?bb)) and we need a concise
representation l (the two examples are both reduced to abb). Define a function ? which maps ? to l
by removing blanks and repeated labels. Then
X
p(l|I) =
logp(?|I),
(10)
?:?(?)=l
p(?|I) =
T
Y
y?t t ,
(11)
t=1
where y?t t denotes the probability of generating label ?t at time step t.
After training, for lexicon-free transcription, the predicted label sequence for a test image I is obtained
by [30]:
l? = ?(arg max p(?|I)).
(12)
?
The lexicon-based method needs a dictionary or lexicon. Each test image is associated with a fix
length lexicon D. The result is obtained by choosing the sequence in the lexicon that has highest
conditional probability [30]:
l? = arg max p(l|I).
(13)
l?D
5
Table 2: Model analysis over the IIIT5K and SVT (%). Mean and standard deviation of the results
are reported.
(a) GRCNN analysis
Model
IIIT5K
SVT
Plain CNN
77.21?0.54 77.69?0.59
RCNN(1 iter) 77.64?0.58 78.23?0.56
RCNN(2 iters) 78.17?0.56 79.11?0.63
RCNN(3 iters) 78.94?0.61 79.76?0.59
GRCNN(1 iter) 77.92?0.57 78.67?0.53
GRCNN(2 iters) 79.42?0.63 79.89?0.64
GRCNN(3 iters) 80.21?0.57 80.98?0.60
4
4.1
(b) LSTM?s variants analysis
LSTM variants
IIIT5K
SVT
LSTM{?1 =0,?2 =0,?3 =0} 77.92?0.57 78.67?0.53
LSTM-F{?1 =0,?2 =1,?3 =0} 77.26?0.61 78.23?0.53
LSTM-I{?1 =1,?2 =0,?3 =0} 76.84?0.58 76.89?0.63
LSTM-O{?1 =0,?2 =0,?3 =1} 76.91?0.64 78.65?0.56
LSTM-A{?1 =1,?2 =1,?3 =1} 76.52?0.66 77.88?0.59
Experiments
Datasets
ICDAR2003: ICDAR2003 [24] contains 251 scene images and there are 860 cropped images of the
words. We perform unconstrained text recognition and constrained text recognition on this dataset.
Each image is associated with a 50-word lexicon defined by wang et al. [36]. The full lexicon is
composed of all per-image lexicons.
IIIT5K: This dataset has 3000 cropped testing word images and 2000 cropped training images
collected from the Internet [31]. Each image has a lexicon of 50 words and a lexicon of 1000 words.
Street View Text (SVT): This dataset has 647 cropped word images from Google Street View [36].
We use the 50-word lexicon defined by Wang et al [36] in our experiment.
Synth90k: This dataset contains around 7 million training images, 800k validation images and 900k
test images [15]. All of the word images are generated by a synthetic text engine and are highly
realistic.
When evaluating the performance of our model on those benchmark dataset, we follow the evaluation
protocol in [36]. We perform recognition on the words that contain only alphanumeric characters
(A-Z and 0-9) and at least three characters. All recognition results are case-insensitive.
4.2
Implementation Details
The configuration of the network is listed in Table 1, where "sh" denotes the stride of the kernel along
the height; "sw" denotes the stride along the width; "ph" and "pw" denote the padding value of height
and width respectively; and "num" denotes the number of feature maps. The input is a gray-scale
image which is resized to 100?32. Before input to the network, the pixel values are rescaled to
the range (-1, 1). The final output of the feature extractor is a feature sequence of 26 frames. The
recurrent layer is a bidirectional LSTM with 512 units without dropout. The ADADELTA method
[41] is used for training with the parameter ?=0.9. The batch size is set to 192 and training is stopped
after 300k iterations.
All of the networks and related LSTM variants are trained on the training set of Synth90k. The
validation set of Synth90k is used for model selection. When a model is selected in this way, its
parameters are fixed and it is directly tested on other datasets (ICDAR2003, IIIT5K and SVT datasets)
without finetuning. The code and pre-trained model will be released at https://github.com/
Jianfeng1991/GRCNN-for-OCR.
4.3
Explorative Study
We empirically analyze the performance of the proposed model. The results are listed in Table 2. To
ensure robust comparison, for each configuration, after convergence during training, a different model
is saved at every 3000 iterations. We select ten models which perform the best on the Synth90k?s
validation set, and report the mean accuracy as well as the standard deviation on each tested dataset.
6
Table 3: The text recognition accuracies in natural images. "50","1k" and "Full" denote the lexicon
size used for lexicon-based recognition task. The dataset without lexicon size means the unconstrained
text recognition
Method
ABBYY [36]
wang et al. [36]
Mishra et al. [25]
Novikova et al. [27]
wang et al. [38]
Bissacco et al. [3]
Goel et al. [6]
Alsharif [2]
Almazan et al. [1]
Lee et al. [20]
Yao et al. [40]
Rodriguez et al. [28]
Jaderberg et al. [16]
Su and Lu et al. [33]
Gordo [7]
Jaderberg et al. [14]
Baoguang et al. [30]
Chen-Yu et al. [21]
ResNet-BLSTM
Ours
SVT-50
35.0%
57.0%
73.2%
72.9%
70.0%
90.4%
77.3%
74.3%
89.2%
80.0%
75.9%
70.0%
86.1%
83.0%
90.7%
93.2%
96.4%
96.3%
96.0%
96.3%
SVT IIIT5K-50 IIIT5K-1k IIIT5K IC03-50 IC03-Full IC03
24.3%
56.0%
55.0%
76.0%
62.0%
81.8%
67.8%
64.1%
57.5%
82.8%
90.0%
84.0%
78.0%
89.7%
93.1%
88.6%
91.2%
82.1%
88.0%
76.0%
80.2%
69.3%
88.5%
80.3%
76.1%
57.4%
96.2%
91.5%
92.0%
82.0%
93.3%
86.6%
71.1%
95.5%
89.6%
97.8%
97.0% 89.6%
80.8%
97.6%
94.4%
78.2% 98.7%
97.6% 89.4%
80.7%
96.8%
94.4%
78.4% 97.9%
97.0% 88.7%
80.2%
97.5%
94.9%
79.2% 98.1%
97.3% 89.9%
81.5% 98.0%
95.6% 80.8% 98.8%
97.8% 91.2%
First, a purely feed-forward CNN is constructed for comparison. To make this CNN have approximately the same number of parameters as GRCNN and RCNN, we use two convolutional layers
to replace each GRCL in Table 1, and each of them has the same number of feature maps as the
corresponding GRCL. Besides, this plain CNN has the same depth as the GRCNN with T = 1. The
results show that the plain CNN has lower accuracy than both RCNN and GRCNN.
Second, we compare GRCNN and RCNN to investigate the effect of adding gate to the RCL.
RCNN is constructed by replacing GRCL in Table 1 with RCL. Each RCL in RCNN has the same
number of feature maps as the corresponding GRCL. Batch normalization is also inserted after each
convolutional kernel in RCL. We fix T , and compare these two models. The results in Table 2(a)
show that each GRCNN model outperforms the corresponding RCNN on both IIIT5K and SVT.
Those results show the advantage of the introduced gate in the model.
Furthermore, we explore the effect of iterations in GRCL. From Table 2(a), we can conclude that
having more iterations is beneficial to GRCL. The increments of accuracy between each iteration
number are 1.50%, 0.79% on IIIT5K and 1.22%, 1.09% on SVT, respectively. This is reasonable
since GRCL with more iterations is able to receive more context information.
Finally, we compare various peephole LSTM units in the bidirectional LSTM for processing feature
sequences. Five types of LSTM variants are compared: full peephole LSTM (LSTM-A), input gate
peephole LSTM (LSTM-I), output gate peephole LSTM (LSTM-O), forget gate peephole LSTM
(LSTM-F) and none peephole LSTM (LSTM). In the feature extraction part, we use GRCNN with
T = 1 which is described in Table 1. Table 2(b) shows that the LSTM without peephole connections
(?1 = ?2 = ?3 = 0) gives the best result.
4.4
Comparison with the state-of-the-art
We use the GRCNN described in Table 1 as the feature extractor. Since having more iterations
is helpful to GRCL, we use GRCL with T = 5 in GRCNN. For sequence learning, we use the
bidirectional LSTM without peephole connections. The training details are described in Sec.4.2.
The best model on the validation set of Synth90k is selected for comparison. Note that this model
is trained on the training set of Synth90k and not finetuned with respect to any dataset. Table 3
shows the results. The proposed method outperforms most existing models for both constrained and
unconstrained text recognition.
7
Figure 4: Lexicon-free recognition results by the proposed GRCL-BLSTM framework on SVT,
ICDAR03 and IIIT5K
Moreover, we do an extra experiment by building a residual block in feature extraction part. We
use the ResNet-20 [9] in this experiment, since it has similar depth with GRCNN with T = 5. The
implementation is similar to what we have discussed in Sec.4.2. The results of ResNet-BLSTM are
listed in Table 3. This ResNet-based framework performs worse than the GRCNN-based framework.
The result indicates that GRCNN is more suitable for scene text recognition.
Some examples predicted by the proposed method under unconstrained scenario are shown in Figure
4. The correctly predicted examples are shown in the left. It is seen that our model can recognize
some long words that have missing part, such as "MEMENTO" in which "N" is not clear. Some other
bend words are also recognized perfectly, for instance, "COTTAGE" and "BADGER". However,
there are some words that cannot be distinguished precisely and some of them are showed in the right
side in Figure 4. The characters which are combined closely may lead to bad recognition results.
For the word "ARMADA", the network cannot accurately split "A" and "R", leading to a missing
character in the result. Moreover, some special symbols whose shapes are similar to the English
characters affect the results. For example, the symbol in "BLOOM" looks like the character "O", and
the word is incorrectly recognized as "OBLOOM". Finally, some words that have strange-shaped
character are also difficult to be recognized, such as "SBULT" (the ground truth is "SALVADOR").
5
Conclusion
we propose a new architecture named GRCNN which uses a gate to modulate recurrent connections
in a previous model RCNN. GRCNN is able to choose the context information dynamically and
combine the feed-forward part with recurrent part flexibly. The unrelated context information coming
from the recurrent part is inhibited by the gate. In addition, through experiments we find the LSTM
without peephole connection is suitable for scene text recognition. The experiments on scene text
recognition benchmarks demonstrate the superior performance of the proposed method.
Acknowledgements
This work was supported in part by the National Basic Research Program (973 Program) of China
under grant no. 2013CB329403, the National Natural Science Foundation of China under grant nos.
91420201, 61332007, 61621136008 and 61620106010, and in part by a grant from Sensetime.
8
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10
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6,232 | 6,638 | Towards Accurate Binary Convolutional Neural
Network
Xiaofan Lin
Cong Zhao
Wei Pan*
DJI Innovations Inc, Shenzhen, China
{xiaofan.lin, cong.zhao, wei.pan}@dji.com
Abstract
We introduce a novel scheme to train binary convolutional neural networks (CNNs)
? CNNs with weights and activations constrained to {-1,+1} at run-time. It has been
known that using binary weights and activations drastically reduce memory size
and accesses, and can replace arithmetic operations with more efficient bitwise operations, leading to much faster test-time inference and lower power consumption.
However, previous works on binarizing CNNs usually result in severe prediction
accuracy degradation. In this paper, we address this issue with two major innovations: (1) approximating full-precision weights with the linear combination of
multiple binary weight bases; (2) employing multiple binary activations to alleviate information loss. The implementation of the resulting binary CNN, denoted
as ABC-Net, is shown to achieve much closer performance to its full-precision
counterpart, and even reach the comparable prediction accuracy on ImageNet and
forest trail datasets, given adequate binary weight bases and activations.
1
Introduction
Convolutional neural networks (CNNs) have achieved state-of-the-art results on real-world applications such as image classification [He et al., 2016] and object detection [Ren et al., 2015], with the
best results obtained with large models and sufficient computation resources. Concurrent to these
progresses, the deployment of CNNs on mobile devices for consumer applications is gaining more
and more attention, due to the widespread commercial value and the exciting prospect.
On mobile applications, it is typically assumed that training is performed on the server and test
or inference is executed on the mobile devices [Courbariaux et al., 2016, Esser et al., 2016]. In
the training phase, GPUs enabled substantial breakthroughs because of their greater computational
speed. In the test phase, however, GPUs are usually too expensive to deploy. Thus improving the
test-time performance and reducing hardware costs are likely to be crucial for further progress,
as mobile applications usually require real-time, low power consumption and fully embeddable.
As a result, there is much interest in research and development of dedicated hardware for deep
neural networks (DNNs). Binary neural networks (BNNs) [Courbariaux et al., 2016, Rastegari et al.,
2016], i.e., neural networks with weights and perhaps activations constrained to only two possible
values (e.g., -1 or +1), would bring great benefits to specialized DNN hardware for three major
reasons: (1) the binary weights/activations reduce memory usage and model size 32 times compared
to single-precision version; (2) if weights are binary, then most multiply-accumulate operations can
be replaced by simple accumulations, which is beneficial because multipliers are the most space
and power-hungry components of the digital implementation of neural networks; (3) furthermore, if
both activations and weights are binary, the multiply-accumulations can be replaced by the bitwise
operations: xnor and bitcount Courbariaux et al. [2016]. This could have a big impact on dedicated
deep learning hardware. For instance, a 32-bit floating point multiplier costs about 200 Xilinx FPGA
slices [Govindu et al., 2004], whereas a 1-bit xnor gate only costs a single slice. Semiconductor
? indicates corresponding author.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
manufacturers like IBM [Esser et al., 2016] and Intel [Venkatesh et al., 2016] have been involved in
the research and development of related chips.
However, binarization usually cause severe prediction accuracy degradation, especially on complex
tasks such as classification on ImageNet dataset. To take a closer look, Rastegari et al. [2016] shows
that binarizing weights causes the accuracy of Resnet-18 drops from 69.3% to 60.8% on ImageNet
dataset. If further binarize activations, the accuracy drops to 51.2%. Similar phenomenon can also be
found in literatures such as [Hubara et al., 2016]. Clearly there is a considerable gap between the
accuracy of a full-precision model and a binary model.
This paper proposes a novel scheme for binarizing CNNs, which aims to alleviate, or even eliminate
the accuracy degradation, while still significantly reducing inference time, resource requirement and
power consumption. The paper makes the following major contributions.
? We approximate full-precision weights with the linear combination of multiple binary weight
bases. The weights values of CNNs are constrained to { 1, +1}, which means convolutions
can be implemented by only addition and subtraction (without multiplication), or bitwise
operation when activations are binary as well. We demonstrate that 3?5 binary weight bases
are adequate to well approximate the full-precision weights.
? We introduce multiple binary activations. Previous works have shown that the quantization
of activations, especially binarization, is more difficult than that of weights [Cai et al., 2017,
Courbariaux et al., 2016]. By employing five binary activations, we have been able to reduce
the Top-1 and Top-5 accuracy degradation caused by binarization to around 5% on ImageNet
compared to the full precision counterpart.
It is worth noting that the multiple binary weight bases/activations scheme is preferable to the fixedpoint quantization in previous works. In those fixed-point quantized networks one still needs to
employ arithmetic operations, such as multiplication and addition, on fixed-point values. Even though
faster than floating point, they still require relatively complex logic and can consume a lot of power.
Detailed discussions can be found in Section 5.2.
Ideally, combining more binary weight bases and activations always leads to better accuracy and will
eventually get very close to that of full-precision networks. We verify this on ImageNet using Resnet
network topology. This is the first time a binary neural network achieves prediction accuracy
comparable to its full-precision counterpart on ImageNet.
2
Related work
Quantized Neural Networks: High precision parameters are not very necessary to reach high
performance in deep neural networks. Recent research efforts (e.g., [Hubara et al., 2016]) have
considerably reduced a large amounts of memory requirement and computation complexity by using
low bitwidth weights and activations. Zhou et al. [2016] further generalized these schemes and
proposed to train CNNs with low bitwidth gradients. By performing the quantization after network
training or using the ?straight-through estimator (STE)" [Bengio et al., 2013], these works avoided the
issues of non-differentiable optimization. While some of these methods have produced good results
on datasets such as CIFAR-10 and SVHN, none has produced low precision networks competitive
with full-precision models on large-scale classification tasks, such as ImageNet. In fact, [Zhou et al.,
2016] and [Hubara et al., 2016] experiment with different combinations of bitwidth for weights and
activations, and show that the performance of their highly quantized networks deteriorates rapidly
when the weights and activations are quantized to less than 4-bit numbers. Cai et al. [2017] enhance
the performance of a low bitwidth model by addressing the gradient mismatch problem, nevertheless
there is still much room for improvement.
Binarized Neural Networks: The binary representation for deep models is not a new topic. At the
emergence of artificial neural networks, inspired biologically, the unit step function has been used as
the activation function [Toms, 1990]. It is known that binary activation can use spiking response for
event-based computation and communication (consuming energy only when necessary) and therefore
is energy-efficient [Esser et al., 2016]. Recently, Courbariaux et al. [2016] introduce BinarizedNeural-Networks (BNNs), neural networks with binary weights and activations at run-time. Different
from their work, Rastegari et al. [2016] introduce simple, efficient, and accurate approximations
to CNNs by binarizing the weights and even the intermediate representations in CNNs. All these
works drastically reduce memory consumption, and replace most arithmetic operations with bitwise
operations, which potentially lead to a substantial increase in power efficiency.
2
In all above mentioned works, binarization significantly reduces accuracy. Our experimental results
on ImageNet show that we are close to filling the gap between the accuracy of a binary model and its
full-precision counterpart. We relied on the idea of finding the best approximation of full-precision
convolution using multiple binary operations, and employing multiple binary activations to allow
more information passing through.
3
Binarization methods
In this section, we detail our binarization method, which is termed ABC-Net (Accurate-BinaryConvolutional) for convenience. Bear in mind that during training, the real-valued weights are
reserved and updated at every epoch, while in test-time only binary weights are used in convolution.
3.1
Weight approximation
Consider a L-layer CNN architecture. Without loss of generality, we assume the weights of each
convolutional layer are tensors of dimension (w, h, cin , cout ), which represents filter width, filter
height, input-channel and output-channel respectively. We propose two variations of binarization
method for weights at each layer: 1) approximate weights as a whole and 2) approximate weights
channel-wise.
3.1.1
Approximate weights as a whole
At each layer, in order to constrain a CNN to have binary weights, we estimate the real-value weight
filter W 2 Rw?h?cin ?cout using the linear combination of M binary filters B1 , B2 , ? ? ? , BM 2
{ 1, +1}w?h?cin ?cout such that W ? ?1 B1 +?2 B2 +? ? ?+?M BM . To find an optimal estimation,
a straightforward way is to solve the following optimization problem:
minJ(?, B) = ||w
?,B
B?||2 ,
s.t. Bij 2 { 1, +1},
(1)
where B = [vec(B1 ), vec(B2 ), ? ? ? , vec(BM )], w = vec(W ) and ? = [?1 , ?2 , ? ? ? , ?M ]T . Here
the notation vec(?) refers to vectorization.
Although a local minimum solution to (1) can be obtained by numerical methods, one could not
backpropagate through it to update the real-value weight filter W . To address this issue, assuming the
mean and standard deviation of W are mean(W ) and std(W ) respectively, we fix Bi ?s as follows:
? + ui std(W )), i = 1, 2, ? ? ? , M,
Bi = Fui (W ) := sign(W
(2)
? = W mean(W ), and ui is a shift parameter. For example, one can choose ui ?s to
where W
be ui = 1 + (i 1) M2 1 , i = 1, 2, ? ? ? , M, to shift evenly over the range [ std(W ), std(W )],
or leave it to be trained by the network. This is based on the observation that the full-precision
weights tend to have a symmetric, non-sparse distribution, which is close to Gaussian. To gain more
intuition and illustrate the approximation effectiveness, an example is visualized in Section S2 of the
supplementary material.
With Bi ?s chosen, (1) becomes a linear regression problem
min J(?) = ||w
?
B?||2 ,
(3)
in which Bi ?s serve as the bases in the design/dictionary matrix. We can then back-propagate
through Bi ?s using the ?straight-through estimator? (STE) [Bengio et al., 2013]. Assume c as the
cost function, A and O as the input and output tensor of a convolution respectively, the forward and
backward approach of an approximated convolution during training can be computed as follows:
Forward: B1 , B2 , ? ? ? , BM = Fu1 (W ), Fu2 (W ), ? ? ? , FuM (W ),
(4)
Solve (3) for ?,
(5)
O=
M
X
(6)
?m Conv(Bm , A).
m=1
@c
@c
Backward:
=
@W
@O
M
X
@O @Bm
?m
@B
m @W
m=1
!
3
@c
=
@O
STE
M
X
@O
?m
@B
m
m=1
!
=
M
X
m=1
?m
@c
.
@Bm
(7)
In test-time, only (6) is required. The block structure of this approximated convolution layer is shown
on the left side in Figure 1. With suitable hardwares and appropriate implementations, the convolution
can be efficiently computed. For example, since the weight values are binary, we can implement the
convolution with additions and subtractions (thus without multiplications). Furthermore, if the input
A is binary as well, we can implement the convolution with bitwise operations: xnor and bitcount
[Rastegari et al., 2016]. Note that the convolution with each binary filter can be computed in parallel.
Figure 1: An example of the block structure of the convolution in ABC-Net. M = N = 3. On the
left is the structure of the approximated convolution (ApproxConv). ApproxConv is expected to approximate the conventional full-precision convolution with linear combination of binary convolutions
(BinConv), i.e., convolution with binary and weights. On the right is the overall block structure of the
convolution in ABC-Net. The input is binarized using different functions Hv1 , Hv2 , Hv3 , passed into
the corresponding ApproxConv?s and then summed up after multiplying their corresponding n ?s.
With the input binarized, the BinConv?s can be implemented with highly efficient bitwise operations.
There are 9 BinConv?s in this example and they can work in parallel.
3.1.2
Approximate weights channel-wise
Alternatively, we can estimate the real-value weight filter Wi 2 Rw?h?cin of each output channel i 2 {1, 2, ? ? ? , cout } using the linear combination of M binary filters Bi1 , Bi2 , ? ? ? , BiM 2
{ 1, +1}w?h?cin such that Wi ? ?i1 Bi1 + ?i2 Bi2 + ? ? ? + ?iM BiM . Again, to find an optimal
estimation, we solve a linear regression problem analogy to (3) for each output channel. After
convolution, the results are concatenated together along the output-channel dimension. If M = 1,
this approach reduces to the Binary-Weights-Networks (BWN) proposed in [Rastegari et al., 2016].
Compared to weights approximation as a whole, the channel-wise approach approximates weights
more elaborately, however no extra cost is needed during inference. Since this approach requires
more computational resources during training, we leave it as a future work and focus on the former
approximation approach in this paper.
3.2
Multiple binary activations and bitwise convolution
As mentioned above, a convolution can be implemented without multiplications when weights are
binarized. However, to utilize the bitwise operation, the activations must be binarized as well, as they
are the inputs of convolutions.
Similar to the activation binarization procedure in [Zhou et al., 2016], we binarize activations after
passing it through a bounded activation function h, which ensures h(x) 2 [0, 1]. We choose the
bounded rectifier as h. Formally, it can be defined as:
hv (x) = clip(x + v, 0, 1),
(8)
where v is a shift parameter. If v = 0, then hv is the clip activation function in [Zhou et al., 2016].
We constrain the binary activations to either 1 or -1. In order to transform the real-valued activation
R into binary activation, we use the following binarization function:
Hv (R) := 2Ihv (R)
4
0.5
1,
(9)
where I is the indicator function. The conventional forward and backward approach of the activation
can be given as follows:
Forward: A = Hv (R).
@c
@c
Backward:
=
I0?R
@R
@A
v?1 .
(using STE)
(10)
Here denotes the Hadamard product. As can be expected, binaizing activations as above is kind of
crude and leads to non-trivial losses in accuracy, as shown in Rastegari et al. [2016], Hubara et al.
[2016]. While it is also possible to approximate activations with linear regression, as that of weights,
another critical challenge arises ? unlike weights, the activations always vary in test-time inference.
Luckily, this difficulty can be avoided by exploiting the statistical structure of the activations of deep
networks.
Our scheme can be described as follows. First of all, to keep the distribution of activations relatively
stable, we resort to batch normalization [Ioffe and Szegedy, 2015]. This is a widely used normalization
technique, which forces the responses of each network layer to have zero mean and unit variance. We
apply this normalization before activation. Secondly, we estimate the real-value activation R using
the linear combination of N binary activations A1 , A2 , ? ? ? , AN such that R ? 1 A1 + 2 A2 +
? ? ? + N AN , where
A1 , A2 , ? ? ? , AN = Hv1 (R), Hv2 (R), ? ? ? , HvN (R).
(11)
Different from that of weights, the parameters n ?s and vn ?s (n = 1, ? ? ? , N ) here are both trainable,
just like the scale and shift parameters in batch normalization. Without the explicit linear regression
approach, n ?s and vn ?s are tuned by the network itself during training and fixed in test-time. They
are expected to learn and utilize the statistical features of full-precision activations.
The resulting network architecture outputs multiple binary activations A1 , A2 , ? ? ? , AN and their
corresponding coefficients 1 , 2 , ? ? ? , N , which allows more information passing through compared
to the former one. Combining with the weight approximation, the whole convolution scheme is given
by:
!
M
N
M X
N
X
X
X
Conv(W , R) ? Conv
? m Bm ,
=
?m n Conv (Bm , An ) , (12)
n An
m=1
n=1
m=1 n=1
which suggests that it can be implemented by computing M ? N bitwise convolutions in parallel.
An example of the whole convolution scheme is shown in Figure 1.
3.3
Training algorithm
A typical block in CNN contains several different layers, which are usually in the following order:
(1) Convolution, (2) Batch Normalization, (3) Activation and (4) Pooling. The batch normalization
layer [Ioffe and Szegedy, 2015] normalizes the input batch by its mean and variance. The activation
is an element-wise non-linear function (e.g., Sigmoid, ReLU). The pooling layer applies any type of
pooling (e.g., max,min or average) on the input batch. In our experiment, we observe that applying
max-pooling on binary input returns a tensor that most of its elements are equal to +1, resulting in a
noticeable drop in accuracy. Similar phenomenon has been reported in Rastegari et al. [2016] as well.
Therefore, we put the max-pooling layer before the batch normalization and activation.
Since our binarization scheme approximates full-precision weights, using the full-precision pre-train
model serves as a perfect initialization. However, fine-tuning is always required for the weights
to adapt to the new network structure. The training procedure, i.e., ABC-Net, is summarized in
Section S1 of the supplementary material.
It is worth noting that as M increases, the shift parameters get closer and the bases of the linear
combination are more correlated, which sometimes lead to rank deficiency when solving (3). This
can be tackled with the `2 regularization.
4
Experiment results
In this section, the proposed ABC-Net was evaluated on the ILSVRC12 ImageNet classification
dataset [Deng et al., 2009], and visual perception of forest trails datasets for mobile robots [Giusti
et al., 2016] in Section S6 of supplementary material.
5
4.1
Experiment results on ImageNet dataset
The ImageNet dataset contains about 1.2 million high-resolution natural images for training that
spans 1000 categories of objects. The validation set contains 50k images. We use Resnet ([He et al.,
2016]) as network topology. The images are resized to 224x224 before fed into the network. We
report our classification performance using Top-1 and Top-5 accuracies.
4.1.1
Effect of weight approximation
We first evaluate the weight approximation technique by examining the accuracy improvement for a
binary model. To eliminate variables, we leave the activations being full-precision in this experiment.
Table 1 shows the prediction accuracy of ABC-Net on ImageNet with different choices of M . For
comparison, we add the results of Binary-Weights-Network (denoted ?BWN?) reported in Rastegari
et al. [2016] and the full-precision network (denoted ?FP?). The BWN binarizes weights and leaves
the activations being full-precision as we do. All results in this experiment use Resnet-18 as network
topology. It can be observed that as M increases, the accuracy of ABC-Net converges to its fullprecision counterpart. The Top-1 gap between them reduces to only 0.9 percentage point when
M = 5, which suggests that this approach nearly eliminates the accuracy degradation caused by
binarizing weights.
Table 1: Top-1 (left) and Top-5 (right) accuracy of ABC-Net on ImageNet, using full-precision
activation and different choices of the number of binary weight bases M .
Top-1
Top-5
BWN
60.8%
83.0%
M =1
62.8%
84.4%
M =2
63.7%
85.2%
M =3
66.2%
86.7%
M =5
68.3%
87.9%
FP
69.3%
89.2%
For interested readers, Figure S4 in section S5 of the supplementary material shows that the relationship between accuracy and M appears to be linear. Also, in Section S2 of the supplementary material,
a visualization of the approximated weights is provided.
4.1.2
Configuration space exploration
We explore the configuration space of combinations of number of weight bases and activations. Table
2 presents the results of ABC-Net with different configurations. The parameter settings for these
experiments are provided in Section S4 of the supplementary material.
Table 2: Prediction accuracy (Top-1/Top-5) for ImageNet with different choices of M and N in a
ABC-Net (approximate weights as a whole). ?res18?, ?res34? and ?res50? are short for Resnet-18,
Resnet-34 and Resnet-50 network topology respectively. M and N refer to the number of weight
bases and activations respectively.
Network
res18
res18
res18
res18
res18
res18
res18
res18
res34
res34
res34
res34
res50
res50
M -weight base N -activation base
1
1
3
1
3
3
3
5
5
1
5
3
5
5
Full Precision
1
1
3
3
5
5
Full Precision
5
5
Full Precision
Top-1
42.7%
49.1%
61.0%
63.1%
54.1%
62.5%
65.0%
69.3%
52.4%
66.7%
68.4%
73.3%
70.1%
76.1%
Top-5
67.6%
73.8%
83.2%
84.8%
78.1%
84.2%
85.9%
89.2%
76.5%
87.4%
88.2%
91.3%
89.7%
92.8%
Top-1 gap
26.6%
20.2%
8.3%
6.2%
15.2%
6.8%
4.3%
20.9%
6.6%
4.9%
6.0%
-
Top-5 gap
21.6%
15.4%
6.0%
4.4%
11.1%
5.0%
3.3%
14.8%
3.9%
3.1%
3.1%
-
As balancing between multiple factors like training time and inference time, model size and accuracy
is more a problem of practical trade-off, there will be no definite conclusion as which combination of
6
(M, N ) one should choose. In general, Table 2 shows that (1) the prediction accuracy of ABC-Net
improves greatly as the number of binary activations increases, which is analogous to the weight
approximation approach; (2) larger M or N gives better accuracy; (3) when M = N = 5, the Top-1
gap between the accuracy of a full-precision model and a binary one reduces to around 5%. To gain a
visual understanding and show the possibility of extensions to other tasks such object detection, we
print the a sample of feature maps in Section S3 of supplementary material.
4.1.3
Comparison with the state-of-the-art
Table 3: Classification test accuracy of CNNs trained on ImageNet with Resnet-18 network topology.
?W? and ?A? refer to the weight and activation bitwidth respectively.
Model
Full-Precision Resnet-18 [full-precision weights and activation]
BWN [full-precision activation] Rastegari et al. [2016]
DoReFa-Net [1-bit weight and 4-bit activation] Zhou et al. [2016]
XNOR-Net [binary weight and activation] Rastegari et al. [2016]
BNN [binary weight and activation] Courbariaux et al. [2016]
ABC-Net [5 binary weight bases, 5 binary activations]
ABC-Net [5 binary weight bases, full-precision activations]
W
32
1
1
1
1
1
1
A
32
32
4
1
1
1
32
Top-1
69.3%
60.8%
59.2%
51.2%
42.2%
65.0%
68.3%
Top-5
89.2%
83.0%
81.5%
73.2%
67.1%
85.9%
87.9%
Table 3 presents a comparison between ABC-Net and several other state-of-the-art models, i.e.,
full-precision Resnet-18, BWN and XNOR-Net in [Rastegari et al., 2016], DoReFa-Net in [Zhou
et al., 2016] and BNN in [Courbariaux et al., 2016] respectively. All comparative models use Resnet18 as network topology. The full-precision Resnet-18 achieves 69.3% Top-1 accuracy. Although
Rastegari et al. [2016]?s BWN model and DeReFa-Net perform well, it should be noted that they use
full-precision and 4-bit activation respectively. Models (XNOR-Net and BNN) that used both binary
weights and activations achieve much less satisfactory accuracy, and is significantly outperformed
by ABC-Net with multiple binary weight bases and activations. It can be seen that ABC-Net has
achieved state-of-the-art performance as a binary model.
One might argue that 5-bit width quantization scheme could reach similar accuracy as that of ABCNet with 5 weight bases and 5 binary activations. However, the former one is less efficient and
requires distinctly more hardware resource. More detailed discussions can be found in Section 5.2.
5
5.1
Discussion
Why adding a shift parameter works?
Intuitively, the multiple binarized weight bases/activations scheme works because it allows more
information passing through. Consider the case that a real value, say 1.5, is passed to a binarized
function f (x) = sign(x), where sign maps a positive x to 1 and otherwise -1. In that case, the
outputs of f (1.5) is 1, which suggests that the input value is positive. Now imagine that we have two
binarization function f1 (x) = sign(x) and f2 (x) = sign(x 2). In that case f1 outputs 1 and f2
outputs -1, which suggests that the input value is not only positive, but also must be smaller than 2.
Clearly we see that each function contributes differently to represent the input and more information
is gained from f2 compared to the former case.
From another point of view, both coefficients ( ?s) and shift parameters are expected to learn and
utilize the statistical features of full-precision tensors, just like the scale and shift parameters in batch
normalization. If we have more binarized weight bases/activations, the network has the capacity to
approximate the full-precision one more precisely. Therefore, it can be deduced that when M or N
is large enough, the network learns to tune itself so that the combination of M weight bases or N
binarized activations can act like the full-precision one.
5.2
Advantage over the fixed-point quantization scheme
It should be noted that there are key differences between the multiple binarization scheme (M
binarized weight bases or N binarized activations) proposed in this paper and the fixed-point quantization scheme in the previous works such as [Zhou et al., 2016, Hubara et al., 2016], though at first
Courbariaux et al. [2016] did not report their result on ImageNet. We implemented and presented the result.
7
thought K-bit width quantization seems to share the same memory requirement with K binarizations.
Specifically, our K binarized weight bases/activations is preferable to the fixed K-bit width scheme
for the following reasons:
(1) The K binarization scheme preserves binarization for bitwise operations. One or several bitwise
operations is known to be more efficient than a fixed-point multiplication, which is a major reason
that BNN/XNOR-Net was proposed.
(2) A K-bit width multiplier consumes more resources than K 1-bit multipliers in a digital chip: it
requires more than K bits to store and compute, otherwise it could easily overflow/underflow. For
example, if a real number is quantized to a 2-bit number, a possible choice is in range {0,1,2,4}.
In this 2-bit multiplication, when both numbers are 4, it outputs 4 ? 4 = 16, which is not within
the range. In [Zhou et al., 2016], the range of activations is constrained within [0,1], which seems
to avoid this situation. However, fractional numbers do not solve this problem, severe precision
deterioration will appear during the multiplication if there are no extra resources. The fact that the
complexity of a multiplier is proportional to THE SQUARE of bit-widths can be found in literatures
(e.g., sec 3.1.1. in [Grabbe et al., 2003]). In contrast, our K binarization scheme does not have this
issue ? it always outputs within the range {-1,1}. The saved hardware resources can be further used
for parallel computing.
(3) A binary activation can use spiking response for event-based computation and communication
(consuming energy only when necessary) and therefore is energy-efficient [Esser et al., 2016]. This
can be employed in our scheme, but not in the fixed K-bit width scheme. Also, we have mentioned the
fact that K-bit width multiplier consumes more resources than K 1-bit multipliers. It is noteworthy
that these resources include power.
To sum up, K-bit multipliers are the most space and power-hungry components of the digital
implementation of DNNs. Our scheme could bring great benefits to specialized DNN hardware.
5.3
Further computation reduction in run-time
On specialized hardware, the following operations in our scheme can be integrated with other
operations in run-time and further reduce the computation requirement.
(1) Shift operations. The existence of shift parameters seem to require extra additions/subtractions
(see (2) and (8)). However, the binarization operation with a shift parameter can be implemented
as a comparator where the shift parameter is the number for comparison, e.g., Hv (R) =
?
1, R 0.5 v;
1, R < 0.5 v. (0.5 v is a constant), so no extra additions/subtractions are involved.
(2) Batch normalization. In run-time, a batch normalization is simply an affine function, say,
BN(R) = aR + b, whose scale and shift parameters a, b are fixed and can be integrated with
vn ?s. More specifically, ?
a batch normalization can be integrated
into a binarization operation as
?
1, aR + b 0.5 v;
1, R (0.5 v b)/a;
follow: Hv (BN(R)) =
Therefore,
1, aR + b < 0.5 v. =
1, R < (0.5 v b)/a.
there will be no extra cost for the batch normalization.
6
Conclusion and future work
We have introduced a novel binarization scheme for weights and activations during forward and
backward propagations called ABC-Net. We have shown that it is possible to train a binary CNN with
ABC-Net on ImageNet and achieve accuracy close to its full-precision counterpart. The binarization
scheme proposed in this work is parallelizable and hardware friendly, and the impact of such a method
on specialized hardware implementations of CNNs could be major, by replacing most multiplications
in convolution with bitwise operations. The potential to speed-up the test-time inference might be
very useful for real-time embedding systems. Future work includes the extension of those results
to other tasks such as object detection and other models such as RNN. Also, it would be interesting
to investigate using FPGA/ASIC or other customized deep learning processor [Liu et al., 2016] to
implement ABC-Net at run-time.
7
Acknowledgement
We acknowledge Mr Jingyang Xu for helpful discussions.
8
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6,233 | 6,639 | Semi-Supervised Learning for Optical Flow
with Generative Adversarial Networks
Wei-Sheng Lai1
Jia-Bin Huang2
Ming-Hsuan Yang1,3
2
3
University of California, Merced
Virginia Tech
Nvidia Research
1
2
{wlai24|mhyang}@ucmerced.edu
[email protected]
1
Abstract
Convolutional neural networks (CNNs) have recently been applied to the optical
flow estimation problem. As training the CNNs requires sufficiently large amounts
of labeled data, existing approaches resort to synthetic, unrealistic datasets. On
the other hand, unsupervised methods are capable of leveraging real-world videos
for training where the ground truth flow fields are not available. These methods,
however, rely on the fundamental assumptions of brightness constancy and spatial
smoothness priors that do not hold near motion boundaries. In this paper, we
propose to exploit unlabeled videos for semi-supervised learning of optical flow
with a Generative Adversarial Network. Our key insight is that the adversarial
loss can capture the structural patterns of flow warp errors without making explicit
assumptions. Extensive experiments on benchmark datasets demonstrate that the
proposed semi-supervised algorithm performs favorably against purely supervised
and baseline semi-supervised learning schemes.
1
Introduction
Optical flow estimation is one of the fundamental problems in computer vision. The classical formulation builds upon the assumptions of brightness constancy and spatial smoothness [15, 25]. Recent
advancements in this field include using sparse descriptor matching as guidance [4], leveraging dense
correspondences from hierarchical features [2, 39], or adopting edge-preserving interpolation techniques [32]. Existing classical approaches, however, involve optimizing computationally expensive
non-convex objective functions.
With the rapid growth of deep convolutional neural networks (CNNs), several approaches have
been proposed to solve optical flow estimation in an end-to-end manner. Due to the lack of the
large-scale ground truth flow datasets of real-world scenes, existing approaches [8, 16, 30] rely on
training on synthetic datasets. These synthetic datasets, however, do not reflect the complexity of
realistic photometric effects, motion blur, illumination, occlusion, and natural image noise. Several
recent methods [1, 40] propose to leverage real-world videos for training CNNs in an unsupervised
setting (i.e., without using ground truth flow). The main idea is to use loss functions measuring
brightness constancy and spatial smoothness of flow fields as a proxy for losses using ground truth
flow. However, the assumptions of brightness constancy and spatial smoothness often do not hold
near motion boundaries. Despite the acceleration in computational speed, the performance of these
approaches still does not match up to the classical flow estimation algorithms.
With the limited quantity and unrealistic of ground truth flow and the large amounts of real-world
unlabeled data, it is thus of great interest to explore the semi-supervised learning framework. A
straightforward approach is to minimize the End Point Error (EPE) loss for data with ground truth
flow and the loss functions that measure classical brightness constancy and smoothness assumptions
for unlabeled training images (Figure 1 (a)). However, we show that such an approach is sensitive
to the choice of parameters and may sometimes decrease the accuracy of flow estimation. Prior
work [1, 40] minimizes a robust loss function (e.g., Charbonnier function) on the flow warp error
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
EPE loss
CNN
EPE loss
CNN
Ground truth flow
Predicted flow
Predicted flow
Labeled data
Flow
warp error
Unlabeled data
Smoothness loss
Predicted flow
Labeled data
Unlabeled data
Brightness
constancy loss
CNN
Ground truth flow
Adversarial
loss
CNN
Predicted flow
Flow
warp error
(a) Baseline semi-supervised learning
Flow
warp error
(b) The proposed semi-supervised learning
Figure 1: Semi-supervised learning for optical flow estimation. (a) A baseline semi-supervised
algorithm utilizes the assumptions of brightness constancy and spatial smoothness to train CNN from
unlabeled data (e.g., [1, 40]). (b) We train a generative adversarial network to capture the structure
patterns in flow warp error images without making any prior assumptions.
(i.e., the difference between the first input image and the warped second image) by modeling the
brightness constancy with a Laplacian distribution. As shown in Figure 2, although robust loss
functions can fit the likelihood of the per-pixel flow warp error well, the spatial structure in the warp
error images cannot be modeled by simple distributions. Such structural patterns often arise from
occlusion and dis-occlusion caused by large object motion, where the brightness constancy assumption
does not hold. A few approaches have been developed to cope with such brightness inconsistency
problem using the Fields-of-Experts (FoE) [37] or a Gaussian Mixture Model (GMM) [33]. However,
the inference of optical flow entails solving time-consuming optimization problems.
In this work, our goal is to leverage both the labeled and the unlabeled data without making explicit
assumptions on the brightness constancy and flow smoothness. Specifically, we propose to impose an
adversarial loss [12] on the flow warp error image to replace the commonly used brightness constancy
loss. We formulate the optical flow estimation as a conditional Generative Adversarial Network
(GAN) [12]. Our generator takes the input image pair and predicts the flow. We then compute the flow
warp error image using a bilinear sampling layer. We learn a discriminator to distinguish between the
flow warp error from predicted flow and ground truth optical flow fields. The adversarial training
scheme encourages the generator to produce the flow warp error images that are indistinguishable
from the ground truth. The adversarial loss serves as a regularizer for both labeled and unlabeled data
(Figure 1 (b)). With the adversarial training, our network learns to model the structural patterns of
flow warp error to refine the motion boundary. During the test phase, the generator can efficiently
predict optical flow in one feed-forward pass.
We make the following three contributions:
? We propose a generative adversarial training framework to learn to predict optical flow by leveraging both labeled and unlabeled data in a semi-supervised learning framework.
? We develop a network to capture the spatial structure of the flow warp error without making
primitive assumptions on brightness constancy or spatial smoothness.
? We demonstrate that the proposed semi-supervised flow estimation method outperforms the purely
supervised and baseline semi-supervised learning when using the same amount of ground truth
flow and network parameters.
2
Related Work
In the following, we discuss the learning-based optical flow algorithms, CNN-based semi-supervised
learning approaches, and generative adversarial networks within the context of this work.
Optical flow. Classical optical flow estimation approaches typically rely on the assumptions of
brightness constancy and spatial smoothness [15, 25]. Sun et al. [36] provide a unified review of
classical algorithms. Here we focus our discussion on recent learning-based methods in this field.
Learning-based methods aim to learn priors from natural image sequences without using hand-crafted
assumptions. Sun et al. [37] assume that the flow warp error at each pixel is independent and use a
set of linear filters to learn the brightness inconsistency. Rosenbaum and Weiss [33] use a GMM to
learn the flow warp error at the patch level. The work of Rosenbaum et al. [34] learns patch priors
2
Input image 2
Negative log likelihood
Input image 1
-0.15
Ground truth optical flow
Ground truth flow warp error
Flow warp error
Gaussian
Laplacian
Lorentzian
-0.1
-0.05
0
0.05
Flow warp error
0.1
0.15
Negative log likelihood
Figure 2: Modeling the distribution of flow warp error. The robust loss functions, e.g., Lorentzian
or Charbonnier functions, can model the distribution of per-pixel flow warp error well. However, the
spatial pattern resulting from large motion and occlusion cannot be captured by simple distributions.
to model the local flow statistics. These approaches incorporate the learned priors into the classical
formulation and thus require solving time-consuming alternative optimization to infer the optical flow.
Furthermore, the limited amount of training data (e.g., Middlebury [3] or Sintel [5]) may not fully
demonstrate the capability of learning-based optical flow algorithms. In contrast, we train a deep
CNN with large datasets (FlyingChairs [8] and KITTI [10]) in an end-to-end manner. Our model can
predict flow efficiently in a single feed-forward pass.
The FlowNet [8] presents a deep CNN approach for learning optical flow. Even though the network
is trained on a large dataset with ground truth flow, strong data augmentation and the variational
refinement are required. Ilg et al. [16] extend the FlowNet by stacking multiple networks and using
more training data with different motion types including complex 3D motion and small displacements.
To handle large motion, the SPyNet approach [30] estimates flow in a classical spatial pyramid
framework by warping one of the input images and predicting the residual flow at each pyramid level.
A few attempts have recently been made to learn optical flow from unlabeled videos in an unsupervised
manner. The USCNN method [1] approximates the brightness constancy with a Taylor series
expansion and trains a deep network using the UCF101 dataset [35]. Yu et al. [40] enables the backpropagation of the warping function using the bilinear sampling layer from the spatial transformer
network [18] and explicitly optimizes the brightness constancy and spatial smoothness assumptions.
While Yu et al. [40] demonstrate comparable performance with the FlowNet on the KITTI dataset,
the method requires significantly more sophisticated data augmentation techniques and different
parameter settings for each dataset. Our approach differs from these methods in that we use both
labeled and unlabeled data to learn optical flow in a semi-supervised framework.
Semi-supervised learning. Several methods combine the classification objective with unsupervised
reconstruction losses for image recognition [31, 41]. In low-level vision tasks, Kuznietsov et al. [21]
train a deep CNN using sparse ground truth data for single-image depth estimation. This method
optimizes a supervised loss for pixels with ground truth depth value as well as an unsupervised
image alignment cost and a regularization cost. The image alignment cost resembles the brightness
constancy, and the regularization cost enforces the spatial smoothness on the predicted depth maps.
We show that adopting a similar idea to combine the EPE loss with image reconstruction and
smoothness losses may not improve flow accuracy. Instead, we use the adversarial training scheme
for learning to model the structural flow warp error without making assumptions on images or flow.
Generative adversarial networks. The GAN framework [12] has been successfully applied to
numerous problems, including image generation [7, 38], image inpainting [28], face completion [23],
image super-resolution [22], semantic segmentation [24], and image-to-image translation [17, 42].
Within the scope of domain adaptation [9, 14], the discriminator learns to differentiate the features
from the two different domains, e.g., synthetic, and real images. Kozi?nski et al. [20] adopt the
adversarial training framework for semi-supervised learning on the image segmentation task where the
discriminator is trained to distinguish between the predictions produced from labeled and unlabeled
data. Different from Kozi?nski et al. [20], our discriminator learns to distinguish the flow warp errors
between using the ground truth flow and using the estimated flow. The generator thus learns to model
the spatial structure of flow warp error images and can improve flow estimation accuracy around
motion boundaries.
3
3
Semi-Supervised Optical Flow Estimation
In this section, we describe the semi-supervised learning approach for optical flow estimation, the
design methodology of the proposed generative adversarial network for learning the flow warp error,
and the use of the adversarial loss to leverage labeled and unlabeled data.
3.1
Semi-supervised learning
We address the problem of learning optical flow by using both labeled data (i.e., with the ground truth
dense optical flow) and unlabeled data (i.e., raw videos). Given a pair of input images {I1 , I2 }, we
train a deep network to generate the dense optical flow field f = [u, v]. For labeled data with the
ground truth optical flow (denoted by f? = [?
u, v?]), we optimize the EPE loss between the predicted
and ground truth flow:
q
2
2
LEPE (f, f?) = (u ? u
?) + (v ? v?) .
(1)
For unlabeled data, existing work [40] makes use of the classical brightness constancy and spatial
smoothness to define the image warping loss and flow smoothness loss:
Lwarp (I1 , I2 , f ) = ? (I1 ? W (I2 , f )) ,
Lsmooth (f ) = ?(?x u) + ?(?y u) + ?(?x v) + ?(?y v),
(2)
(3)
where ?x and ?y are horizontal and vertical gradient operators and ?(?) is the robust penalty function.
The warping function W (I2 , f ) uses the bilinear sampling [18] to warp I2 according to the flow
field f . The difference I1 ? W (I2 , f ) is the flow warp error as shown in Figure 2. Minimizing
Lwarp (I1 , I2 , f ) enforces the flow warp error to be close to zero at every pixel.
A baseline semi-supervised learning approach is to minimize LEPE for labeled data and minimize
Lwarp and Lsmooth for unlabeled data:
X
X
(j) (j)
LEPE f (i) , f?(i) +
?w Lwarp I1 , I2 , f (j) + ?s Lsmooth f (j) ,
(4)
i?Dl
j?Du
where Dl and Du represent labeled and unlabeled datasets, respectively. However, the commonly
used robust loss functions (e.g., Lorentzian and Charbonnier) assume that the error is independent
at each pixel and thus cannot model the structural patterns of flow warp error caused by occlusion.
Minimizing the combination of the supervised loss in (1) and unsupervised losses in (2) and (3)
may degrade the flow accuracy, especially when large motion present in the input image pair. As a
result, instead of using the unsupervised losses based on classical assumptions, we propose to impose
an adversarial loss on the flow warp images within a generative adversarial network. We use the
adversarial loss to regularize the flow estimation for both labeled and unlabeled data.
3.2
Adversarial training
Training a GAN involves optimizing the two networks: a generator G and a discriminator D. The
generator G takes a pair of input images to generate optical flow. The discriminator D performs
binary classification to distinguish whether a flow warp error image is produced by the estimated
flow from the generator G or by the ground truth flow. We denote the flow warp error image from the
ground truth flow and generated flow by y? = I1 ? W(I2 , f?) and y = I1 ? W(I2 , f ), respectively.
The objective function to train the GAN can be expressed as:
Ladv (y, y?) = Ey?[log D(?
y )] + Ey [log (1 ? D(y))].
(5)
We incorporate the adversarial loss with the supervised EPE loss and solve the following minmax
problem for optimizing G and D:
min max LEPE (G) + ?adv Ladv (G, D),
G
D
(6)
where ?adv controls the relative importance of the adversarial loss for optical flow estimation.
Following the standard procedure for GAN training, we alternate between the following two steps
to solve (6): (1) update the discriminator D while holding the generator G fixed and (2) update
generator G while holding the discriminator D fixed.
4
Frozen
Updated
Labeled data
Predicted flow
Flow warp error
Ground truth
flow
Ground truth
flow warp error
%
? "#$
(Eq. 7)
Generator G
Discriminator D
(a) Update discriminator D using labeled data
? "#" (Eq. 1)
Updated
Frozen
Predicted flow
Labeled data
Ground truth flow
Flow warp error
'
? $%&
(Eq. 9)
Generator G
Discriminator D
Predicted flow
Unlabeled data
Flow warp error
(b) Update generator G using both labeled and unlabeled data
Figure 3: Adversarial training procedure. Training a generative adversarial network involves the
alternative optimization of the discriminator D and generator G.
Updating discriminator D. We train the discriminator D to classify between the ground truth
flow warp error (real samples, labeled as 1) and the flow warp error from the predicted flow (fake
samples, labeled as 0). The maximization of (5) is equivalent to minimizing the binary cross-entropy
loss LBCE (p, t) = ?t log(p) ? (1 ? t) log(1 ? p) where p is the output from the discriminator and t
is the target label. The adversarial loss for updating D is defined as:
LD
?) = LBCE (D(?
y ), 1) + LBCE (D(y), 0)
adv (y, y
= ? log D(?
y ) ? log(1 ? D(y)).
(7)
As the ground truth flow is required to train the discriminator, only the labeled data Dl is involved in
this step. By fixing G in (6), we minimize the following loss function for updating D:
X
(i) (i)
LD
? ).
(8)
adv (y , y
i?Dl
Updating generator G. The goal of the generator is to ?fool? the discriminator by producing flow
to generate realistic flow warp error images. Optimizing (6) with respect to G becomes minimizing
log(1 ? D(y)). As suggested by Goodfellow et al. [12], one can instead minimize ? log(D(y)) to
speed up the convergence. The adversarial loss for updating G is then equivalent to the binary cross
entropy loss that assigns label 1 to the generated flow warp error y:
LG
(9)
adv (y) = LBCE (D(y), 1) = ? log(D(y)).
By combining the adversarial loss with the supervised EPE loss, we minimize the following function
for updating G:
X
X
LEPE f (i) , f?(i) + ?adv LG (y (i) ) +
?adv LG (y (j) ).
(10)
adv
i?Dl
adv
j?Du
We note that the adversarial loss is computed for both labeled and unlabeled data, and thus guides
the flow estimation for image pairs without the ground truth flow. Figure 3 illustrates the two main
steps to update the generator D and the discriminator G in the proposed semi-supervised learning
framework.
5
3.3
Network architecture and implementation details
Generator. We construct a 5-level SPyNet [30] as our generator. Instead of using simple stacks
of convolutional layers as sub-networks [30], we choose the encoder-decoder architecture with skip
connections to effectively increase the receptive fields. Each convolutional layer has a 3 ? 3 spatial
support and is followed by a ReLU activation. We present the details of our SPyNet architecture in
the supplementary material.
Discriminator. As we aim to learn the local structure of flow warp error at motion boundaries, it
is more effective to penalize the structure at the scale of local patches instead of the whole image.
Therefore, we use the PatchGAN [17] architecture as our discriminator. The PatchGAN is a fully
convolutional classifier that classifies whether each N ? N overlapping patch is real or fake. The
PatchGAN has a receptive field of 47 ? 47 pixels.
Implementation details. We implement the proposed method using the Torch framework [6]. We
use the Adam solver [19] to optimize both the generator and discriminator with ?1 = 0.9, ?2 = 0.999
and the weight decay of 1e ? 4. We set the initial learning rate as 1e ? 4 and then multiply by 0.5
every 100k iterations after the first 200k iterations. We train the network for a total of 600k iterations.
We use the FlyingChairs dataset [8] as the labeled dataset and the KITTI raw videos [10] as the
unlabeled dataset. In each mini-batch, we randomly sample 4 image pairs from each dataset. We
randomly augment the training data in the following ways: (1) Scaling between [1, 2], (2) Rotating
within [?17? , 17? ], (3) Adding Gaussian noise with a sigma of 0.1, (4) Using color jitter with respect
to brightness, contrast and saturation uniformly sampled from [0, 0.04]. We then crop images to
384 ? 384 patches and normalize by the mean and standard deviation computed from the ImageNet
dataset [13]. The source code is publicly available on http://vllab.ucmerced.edu/wlai24/
semiFlowGAN.
4
Experimental Results
We evaluate the performance of optical flow estimation on five benchmark datasets. We conduct
ablation studies to analyze the contributions of individual components and present comparisons with
the state-of-the-art algorithms including classical variational algorithms and CNN-based approaches.
4.1
Evaluated datasets and metrics
We evaluate the proposed optical flow estimation method on the benchmark datasets: MPI-Sintel [5],
KITTI 2012 [11], KITTI 2015 [27], Middlebury [3] and the test set of FlyingChairs [8]. The MPISintel and FlyingChairs are synthetic datasets with dense ground truth flow. The Sintel dataset
provides two rendered sets, Clean and Final, that contain both small displacements and large motion.
The training and test sets contain 1041 and 552 image pairs, respectively. The FlyingChairs test set
is composed of 640 image pairs with similar motion statistics to the training set. The Middlebury
dataset has only eight image pairs with small motion. The images from the KITTI 2012 and KITTI
2015 datasets are collected from driving real-world scenes with large forward motion. The ground
truth optical flow is obtained from a 3D laser scanner and thus only covers about 50% of image pixels.
There are 194 image pairs in the KITTI 2012 dataset, and 200 image pairs in the KITTI 2015 dataset.
We compute the average EPE (1) on pixels with the ground truth flow available for each dataset. On
the KITTI-2015 dataset, we also compute the Fl score [27], which is the ratio of pixels that have EPE
greater than 3 pixels and 5% of the ground truth value.
4.2
Ablation study
We conduct ablation studies to analyze the contributions of the adversarial loss and the proposed
semi-supervised learning with different training schemes.
Adversarial loss. We adjust the weight of the adversarial loss ?adv in (10) to validate the effect
of the adversarial training. When ?adv = 0, our method falls back to the fully supervised learning
setting. We show the quantitative evaluation in Table 1. Using larger values of ?adv may decrease the
performance and cause visual artifacts as shown in Figure 4. We therefore choose ?adv = 0.01.
6
Table 1: Analysis on adversarial loss. We train the proposed model using different weights for the
adversarial loss in (10).
?adv
Sintel-Clean
EPE
Sintel-Final
EPE
KITTI 2012
EPE
0
0.01
0.1
1
3.51
3.30
3.57
3.93
4.70
4.68
4.73
5.18
7.69
7.16
8.25
13.89
KITTI 2015
EPE
Fl-all
17.19
16.02
16.82
21.07
40.82%
38.77%
42.78%
63.43%
FlyingChairs
EPE
2.15
1.95
2.11
2.21
Table 2: Analysis on receptive field of discriminator. We vary the number of strided convolutional
layers in the discriminator to achieve different size of receptive fields.
# Strided
convolutions
Receptive field
Sintel-Clean
EPE
Sintel-Final
EPE
KITTI 2012
EPE
d=2
d=3
d=4
23 ? 23
47 ? 47
95 ? 95
3.66
3.30
3.70
4.90
4.68
5.00
7.38
7.16
7.54
KITTI 2015
EPE
Fl-all
16.28
16.02
16.38
40.19%
38.77%
41.52%
FlyingChairs
EPE
2.15
1.95
2.16
Receptive fields of discriminator. The receptive field of the discriminator is equivalent to the size
of patches used for classification. The size of the receptive field is determined by the number of
strided convolutional layers, denoted by d. We test three different values, d = 2, 3, 4, which are
corresponding to the receptive field of 23?23, 47?47, and 95?95, respectively. As shown in Table 2,
the network with d = 3 performs favorably against other choices on all benchmark datasets. Using
too large or too small patch sizes might not be able to capture the structure of flow warp error well.
Therefore, we design our discriminator to have a receptive field of 47 ? 47 pixels.
Training schemes. We train the same network (i.e., our generator G) with the following training
schemes: (a) Supervised: minimizing the EPE loss (1) on the FlyingChairs dataset. (b) Unsupervised:
minimizing the classical brightness constancy (2) and spatial smoothness (3) using the Charbonnier
loss function on the KITTI raw dataset. (c) Baseline semi-supervised: minimizing the combination
of supervised and unsupervised losses (4) on the FlyingChairs and KITTI raw datasets. For the
semi-supervised setting, we evaluate different combinations of ?w and ?s in Table 3. We note that it
is not easy to run grid search to find the best parameter combination for all evaluated datasets. We
choose ?w = 1 and ?s = 0.01 for the baseline semi-supervised and unsupervised settings.
We provide the quantitative evaluation of the above training schemes in Table 4 and visual comparisons
in Figure 5 and 6. As images in KITTI 2015 have large forward motion, there are large occluded/disoccluded regions, particularly on the image and moving object boundaries. The brightness constancy
does not hold in these regions. Consequently, minimizing the image warping loss (2) results in
inaccurate flow estimation. Compared to the fully supervised learning, our method further refines the
motion boundaries by modeling the flow warp error. By incorporating both labeled and unlabeled
data in training, our method effectively reduces EPEs on the KITTI 2012 and 2015 datasets.
Training on partially labeled data. We further analyze the effect of the proposed semi-supervised
method by reducing the amount of labeled training data. Specifically, we use 75%, 50% and 25%
Input images
?adv = 0
?adv = 0.01
Ground truth flow
?adv = 0.1
?adv = 1
Figure 4: Comparisons of adversarial loss ?adv . Using larger value of ?adv does not necessarily
improve the performance and may cause unwanted visual artifacts.
7
Table 3: Evaluation for baseline semi-supervised setting. We test different combinations of ?w
and ?s in (4). We note that it is difficult to find the best parameters for all evaluated datasets.
?w
?s
Sintel-Clean
EPE
Sintel-Final
EPE
KITTI 2012
EPE
1
1
1
0.1
0.01
0
0.1
0.01
0.01
0.01
3.77
3.75
3.69
3.64
3.57
5.02
5.05
4.86
4.81
4.82
10.90
11.82
10.38
10.15
8.63
KITTI 2015
EPE
Fl-all
18.52
19.98
18.07
18.94
18.87
FlyingChairs
EPE
39.94%
43.18%
39.33%
40.85 %
42.63 %
2.25
2.19
2.11
2.17
2.22
Table 4: Analysis on different training schemes. ?Chairs? represents the FlyingChairs dataset and
?KITTI? denotes the KITTI raw dataset. The baseline semi-supervised settings cannot improve the
flow accuracy as the brightness constancy assumption does not hold on occluded regions. In contrast,
our approach effectively utilizes the unlabeled data to improve the performance.
Method
Training Datasets
Sintel-Clean
EPE
Sintel-Final
EPE
KITTI 2012
EPE
Supervised
Unsupervised
Baseline semi-supervised
Proposed semi-supervised
Chairs
KITTI
Chairs + KITTI
Chairs + KITTI
3.51
8.01
3.69
3.30
4.70
8.97
4.86
4.68
7.69
16.54
10.38
7.16
KITTI 2015
EPE
Fl
17.19
25.53
18.07
16.02
40.82%
54.40%
39.33%
38.77%
FlyingChairs
EPE
2.15
6.66
2.11
1.95
of labeled data with ground truth flow from the FlyingChairs dataset and treat the remaining part as
unlabeled data to train the proposed semi-supervised method. We also train the purely supervised
method with the same amount of labeled data for comparisons. Table 5 shows that the proposed semisupervised method consistently outperforms the purely supervised method on the Sintel, KITTI2012
and KITTI2015 datasets. The performance gap becomes larger when using less labeled data, which
demonstrates the capability of the proposed method on utilizing the unlabeled data.
4.3 Comparisons with the state-of-the-arts
In Table 6, we compare the proposed algorithm with four variational methods: EpicFlow [32],
DeepFlow [39], LDOF [4] and FlowField [2], and four CNN-based algorithms: FlowNetS [8],
FlowNetC [8], SPyNet [30] and FlowNet 2.0 [16]. We further fine-tune our model on the Sintel
training set (denoted by ?+ft?) and compare with the fine-tuned results of FlowNetS, FlowNetC,
SPyNet, and FlowNet2. We note that the SPyNet+ft is also fine-tuned on the Driving dataset [26]
for evaluating on the KITTI2012 and KITTI2015 datasets, while other methods are fine-tuned on
the Sintel training data. The FlowNet 2.0 has significantly more network parameters and uses more
training datasets (e.g., FlyingThings3D [26]) to achieve the state-of-the-art performance. We show
that our model achieves competitive performance with the FlowNet and SPyNet when using the same
amount of ground truth flow (i.e., FlyingChairs and Sintel datasets). We present more qualitative
comparisons with the state-of-the-art methods in the supplementary material.
4.4 Limitations
As the images in the KITTI raw dataset are captured in driving scenes and have a strong prior of
forward camera motion, the gain of our semi-supervised learning over the supervised setting is mainly
on the KITTI 2012 and 2015 datasets. In contrast, the Sintel dataset typically has moving objects
with various types of motion. Exploring different types of video datasets, e.g., UCF101 [35] or
DAVIS [29], as the source of unlabeled data in our semi-supervised learning framework is a promising
future direction to improve the accuracy on general scenes.
Table 5: Training on partial labeled data. We use 75%, 50% and 25% of data with ground truth
flow from the FlyingChair dataset as labeled data and treat the remaining part as unlabeled data. The
proposed semi-supervised method consistently outperforms the purely supervised method.
Method
Amount of
labeled data
Sintel-Clean
EPE
Sintel-Final
EPE
KITTI 2012
EPE
Supervised
Proposed semi-supervised
75%
4.35
3.58
5.40
4.81
8.22
7.30
17.43
16.46
41.62%
41.00%
1.96
2.20
Supervised
Proposed semi-supervised
50%
4.48
3.67
5.46
4.92
9.34
7.39
18.71
16.64
42.14%
40.48%
2.04
2.28
Supervised
Proposed semi-supervised
25%
4.91
3.95
5.78
5.00
10.60
7.40
19.90
16.61
43.79%
40.68%
2.09
2.33
8
KITTI 2015
EPE
Fl-all
FlyingChairs
EPE
Input images
Unsupervised
Supervised
Ground truth flow
Baseline semi-supervised
Proposed semi-supervised
Figure 5: Comparisons of training schemes. The proposed method learns the flow warp error
using the adversarial training and improve the flow accuracy on motion boundary.
Baseline semi-supervised
Ground truth
Proposed semi-supervised
Figure 6: Comparisons of flow warp error. The baseline semi-supervised approach penalizes the
flow warp error on occluded regions and thus produce inaccurate flow.
Table 6: Comparisons with state-of-the-arts. We report the average EPE on six benchmark
datasets and the Fl score on the KITTI 2015 dataset.
Middlebury
Train
EPE
Method
5
Sintel-Clean
Train
Test
EPE
EPE
Sintel-Final
Train
Test
EPE
EPE
KITTI 2012
Train
Test
EPE
EPE
Train
EPE
KITTI 2015
Train
Test
Fl-all
Fl-all
Chairs
Test
EPE
EpicFlow
DeepFlow
LDOF
FlowField
[32]
[39]
[4]
[2]
0.31
0.25
0.44
0.27
2.27
2.66
4.64
1.86
4.12
5.38
7.56
3.75
3.57
4.40
5.96
3.06
6.29
7.21
9.12
5.81
3.47
4.58
10.94
3.33
3.8
5.8
12.4
3.5
9.27
10.63
18.19
8.33
27.18%
26.52%
38.11%
24.43%
27.10%
29.18%
-
2.94
3.53
3.47
-
FlowNetS
FlowNetC
SpyNet
FlowNet2
[8]
[8]
[30]
[16]
1.09
1.15
0.33
0.35
4.50
4.31
4.12
2.02
7.42
7.28
6.69
3.96
5.45
5.87
5.57
3.14
8.43
8.81
8.43
6.02
8.26
9.35
9.12
4.09
-
15.44
12.52
20.56
10.06
52.86%
47.93%
44.78%
30.37%
-
2.71
2.19
2.63
1.68
FlowNetS + ft [8]
FlowNetC + ft [8]
SpyNet + ft [30]
FlowNet2 + ft [16]
0.98
0.93
0.33
0.35
(3.66)
(3.78)
(3.17)
(1.45)
6.96
6.85
6.64
4.16
(4.44)
(5.28)
(4.32)
(2.01)
7.76
8.51
8.36
5.74
7.52
8.79
4.13
3.61
9.10
4.7
-
9.84
28.20%
-
3.04
2.27
3.07
-
Ours
Ours + ft
0.37
0.32
3.30
(2.41)
6.28
6.27
4.68
(3.16)
7.61
7.31
7.16
5.23
7.5
6.8
16.02
14.69
38.77%
30.30%
39.71%
31.01 %
1.95
2.41
Conclusions
In this work, we propose a generative adversarial network for learning optical flow in a semisupervised manner. We use a discriminative network and an adversarial loss to learn the structural
patterns of the flow warp error without making assumptions on brightness constancy and spatial
smoothness. The adversarial loss serves as guidance for estimating optical flow from both labeled and
unlabeled datasets. Extensive evaluations on benchmark datasets validate the effect of the adversarial
loss and demonstrate that the proposed method performs favorably against the purely supervised and
the straightforward semi-supervised learning approaches for learning optical flow.
Acknowledgement
This work is supported in part by the NSF CAREER Grant #1149783, gifts from Adobe and NVIDIA.
9
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11
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6,234 | 664 | Forecasting Demand for Electric Power
Jen-Lun Yuan and Terrence L. Fine
School of Electrical Engineering
Cornell University
Ithaca, NY 14853
Abstract
We are developing a forecaster for daily extremes of demand for
electric power encountered in the service area of a large midwestern utility and using this application as a testbed for approaches
to input dimension reduction and decomposition of network training. Projection pursuit regression representations and the ability
of algorithms like SIR to quickly find reasonable weighting vectors
enable us to confront the vexing architecture selection problem by
reducing high-dimensional gradient searchs to fitting single-input
single-output (SISO) subnets. We introduce dimension reduction
algorithms, to select features or relevant subsets of a set of many
variables, based on minimizing an index of level-set dispersions
(closely related to a projection index and to SIR), and combine
them with backfitting to implement a neural network version of
projection pursuit. The performance achieved by our approach,
when trained on 1989, 1990 data and tested on 1991 data, is comparable to that achieved in our earlier study of backpropagation
trained networks.
1
Introduction
Our work has the intertwined goals of:
(i) contributing to the improvement of the short-term electrical load (demand)
forecasts used by electric utilities to buy and sell power and ensure that they can
meet demand;
739
740
Yuan and Fine
(ii) reducing the computational burden entailed in gradient-based training of neural
networks and thereby enabling the exploration of architectures;
(iii) improving prospects for good statistical generalization by use of rational methods for reducing complexity through the identification of good small subsets of
variables drawn from a large set of candidate predictor variables (feature selection);
(iv) benchmarking backpropagation and neural networks as an approach to the
applied problem of load forecasting.
Our efforts proceed in the context of a problem suggested by the operational needs
of a particular electric utility to make daily forecasts of short-term load or demand.
Forecasts are made at midday (1 p.m.) on a weekday t ( Monday - Thursday), for
the next evening peak e(t) (occuring usually about 8 p.m. in the winter), the daily
minimum d(t + 1) (occuring about 4 a.m. the next morning) and the morning
peak m( t + 1) (about noon ). In addition, on Friday we are to forecast these
three variables for the weekend through the Monday morning peak. These daily
extremes of demand are illustrated in an excerpt from our hourly load data plotted
in Figure 1.
4600
4400
f~
4200
i
J
4000
3600
5am
3400
0
5
lam
I~am
10
15
Rpm
20
2S
6am
' lam
Ilam
30
35
9jlm
40
45
50
number of houri
Figure 1: Hourly demand for two consecutive days showing the intended forecasting
variables.
In this paper, we focus on forecasting these extremal demands up to three days
ahead (e.g. forecasting on Fridays). Neural network-based forecasters are developed which parallel the recently proposed method of slicing inverse regression (SIR)
(Li [1991]) and then use backfitting (Hastie and Tibshirani [1990]) to implement a
training algorithm for a projection pursuit model (Friedman [1987]' Huber [1985])
that can be implemented with a single hidden layer network. Our data consists of
hourly integrated system demand (MWH) and hourly temperatures measured at
three cities in the service area of a large midwestern utility during 1989-91. We use
1989 and 1990 for a training set and test over the whole of 1991, with the exception
of holidays that occur so infrequently that we have no training base.
Forecasting Demand for Electric Power
2
2.1
Baseline Performance
Previous Work on Load Forecasting
Since demand is a process which does not have a known physical or mathematical
model, we do not know the best achievable forecasting performance, and we are led
to making comparisons with methods and results reported elsewhere. There is a
substantial literature on short-term load forecasting, with Gross et al. [1987] and
Willis et al. [1984] providing good reviews of approaches based upon such statistical methods as linear least squares regression -1Ild Box-Jenkins and ARMAX time
series models. Many utilities rely upon the seemingly seat-of-the-pants estimates
produced by individuals who have been long employed at this task and who extrapolate from a large historical data base. In the past few years there have been several
efforts to employ neural networks trained through backpropagation. In two such
recent studies conducted at the Univ. of Washington an average peak error of 2.04%
was reported by Damborg et al. [1990] and an hourly load error of about 2.2% was
given by Connor et al. [1991]. However, the accuracies reported in the literature are
difficult to compare with since utilities are exposed to different operating conditions
(e.g., weather, residential/industrial balance). To provide a benchmark for the error
performance achieved by our method, we evaluated three basic forecasting models
on our data. These methods are based on a pair of features made plausible by the
scatter plots shown in Figure 2.
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Figure 2: Evening peaks ('IUe.-Fri.,1989-90) vs. morning peaks and temperatures.
2.2
Feature Selection and Homogeneous Data Types
Demand depends on predictable calendar factors such as the season, day-of-theweek and time-of-day considerations. We grouped separately Mondays, 'IUesdays
through Fridays, Saturdays, and Sundays, as well as holidays. In contrast to all
of the earlier work on this problem, we ignored seasonal considerations and let the
network and training algorithm adjust as needed. The advantage of this was the
ability to form larger training data sets. We thus constructed twelve networks, one
741
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Yuan and Fine
type
m(t+1)
e(t)
d(t+1)
Monday
Tue.-Fri.
Saturday
Sunday
m(t-3)
m(t-1)
m(t-1)
m(t-2}
m(t)
m(t)
m(t-1)
m(t-2)
d(t-3)
d(t-1)
d(t-1)
d(t- 2)
Table 1: Most recent peaks of a two- feature set
type
Monday
The.-Fri.
Saturday
Sunday
LLS
m(t+1}
LOESS
BP
LLS
e(t)
LOESS
BP
LLS
d(t+1)
LOESS
BP
3.78
3.01
3.37
4.83
2.45
2.44
2.60
3.28
2.42
1.98
2.36
3.79
1.73
1.89
4.54
4.89
2.43
3.04
3.76
2.74
1.59
1.65
3.10
3.81
4.40
3.29
3.48
4.26
3.30
3.81
3.25
2.44
2.69
2.49
2.06
3.03
Table 2: Forecasting accuracies (percentage absolute error) for three basic methods
for each pair consisting of one of these four types of days and one of the three
daily extremes to be forecast. Demand also depends heavily upon weather which is
the primary random factor affecting forecasts. This dependency can be seen in the
scatter plots of current demand vs. previous demand and temperature in Figure 2,
particularly in the projection onto the 'current demand-temperature' plane which
shows a pronounced "U"-shaped nonlinearity. A two-feature set consisting of the
most recent peaks and average temperatures over the three cities and the preceding
six hours is employed for testing all three models (Table 1).
2.3
Benchmark Results
The three basic forecasting models using the two-featured set are:
1) linear regression model fitted to the data in Figure 2;
2) demand vs. temperature models which roughly model the "U-shaped" nonlinear
relationship, (LOESS with .5 span was employed for scatter plot smoothing);
3) backpropagation trained neural networks using 5 logistic nodes in a single hidden
layer.
The test set errors are given in Table 2. Note that among these three models,
BP-trained neural networks gives superior test set performance on all but Sundays.
These models all give results comparable to those obtained in our earlier work on
forecasting demands for Thesday-Friday using autoregressive neural networks (Yuan
and Fine [1992]).
Forecasting Demand for Electric Power
3
Projection Pursuit Training
Satisfactory forecasting performance of the neural networks described above relies
on the appropriate choice of feature sets and network architectures. Unfortunately,
BP can only address the problem of appropriate architecture and relevant feature
sets through repeated time-consuming experiments. Modeling of high-dimensional
input features using gradient search requires extensive computation. We were thus
prompted to look at other network structures and at training algorithms that could
make it easier to explore architecture and training problems. Our initial attempt
combined the dimension reduction algorithm cf SIR (Li [1991]), currently replaced
by an algorithm of our devising sketched in Section 4, and backfitting (Hastie
et.al [1990]) to implement a neural network version of projection pursuit regression (PPR).
3.1
The Algorithm
A general nonlinear regression model for a forecast variable y in terms of a vector
x of input variables and model noise ?, independent of x, is gi ven by
y = f({3ix,{3~x, .. ,{3~X,?)
(*).
A least mean square predictor is the conditional expectation E(ylx). The projection
pursuit model/approximation of this conditional expectation is given in terms of a
family of SISO functions 2 1 ,22, ", 2k by
k
E(ylx)
= I: 2 i ({3:x) + {3o.
i=l
A single hidden layer neural network can approximate this representation by introducing subnets whose summed outputs approximate the individual 2 j .
We train such a 'projection pursuit network' with nodes partitioned into subnets,
representing the 2 i , by training the subnets individually in rotation. In this we
follow the statistical regression notion of backfitting. The subnet 2i is trained to
predict the residuals resulting from the difference between the weighted outputs of
the other k - 1 subnets and the true value of the demand variable. After a number
of training cycles one then proceeds to the next subnet and repeats the process. The
inputs to each subnet 2i are the low-dimensional projections {3:x of the regression
model. One termination criteria for determining the number of subnets k is to stop
adding subnets when the projection appears to be normally distributed; results
of Diaconis and Freedman point out that 'most' projections will be so distributed
and thus are 'uninteresting'. The directions {3i can be found by minimizing some
projection index which defines interesting projections that deviates from Gaussian
distributions (e.g., Friedman [1987]). Each {3i determines the weights connecting
the input to sub net 2 i .The whole projection pursuit regression process is simplified
by decoupling the direction {3 search from training the SISO subnets. Albeit, its
success depends upon an ability to rapidly discern the significant directions {3i.
3.2
Implementations
There are several variants in the implementation of projection pursuit training algorithms. General PPR procedure can be implemented in one stage by computa-
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Yuan and Fine
type
m(t+l)
e(t)
d(t+1)
Monday
Tue.-Fri.
Saturday
Sunday
2.35/3.45
2.37/2.83
2.67/3.16
3.15/5.38
1.25/1.60
1.65/1.66
2.78/3.96
2.63/3.67
2.76/3.49
2.15/2.66
2.57/3.04
2.29/3.61
Table 3: Forecasting performance (training/testing percentage error) of projection
pursuit trained networks
tionally intensive numerical methods, or in a two-stage heuristic (finding f3i, then
Bi ) as proposed here. It can be implemented with or without back fitting after the
PPR phase is done. Intrator [1992] has recently suggested incorporating the projection index into the objective function and then running an overall BPA. Other
variants in training each Bi net include using nonparametric smoothing techniques
such as LOESS or kernel methods. BP training can then be applied only in the
last stage to fit the smoothed curves so obtained. The complexity of each subnet
is then largely determined by the smoothing parameters, like window sizes, inherent in most nonparametric smoothing techniques. Another practical advantage of
this process is that one can incorporate easily fixed functions of a single variable
(e.g. linear nodes or quadratic nodes) when one's prior knowledge of the data
source suggests that such components may be present. Our current implementation
employs the two-stage algorithm with simple (either one or two nodes) logistic Bi
subnets. Each SISO Bi net runs a BP algorithm to fit the data. The directions f3i
are calculated based on minimizing a projection index (dispersion of level-sets, described in Section 4) which can be executed in a direct fashion. One can encourage
the convergence of backfitting by using a relaxation parameter (like a momentum
parameter in BPA ) to control the amount updated in the current direction. Training (fitting) of each (SISO) 3 i net can be carried out more efficiently than running
BP based on high-dimensional inputs, for example, it is less expensive to evaluate
the Hessian matrices in a Bi net than in a full BPA networks.
3.3
Forecasting Results
Experimental results were obtained using the two-component feature data sets
which gave the earlier baseline performance. To calibrate the performance we employed in all twelve projection pursuit trained networks an uniform architecture of
three subnets ( a (1,2, 2)-logistic network), matching the 5 nodes of the BP network of Section 2. The number of backfitting cycles was set to 20 with a relaxation
parameter w = 0.1. BPA was employed for fitting each Binet. The training/testing
percentage absolute errors are given in Table 3. The limited data sets in the cases
of individual days (Monday, Saturday, Sunday) led to failure in generalization that
could have been prevented by using one or two, rather than three, subnets.
Forecasting Demand for Electric Power
4
4.1
Dimension Reduction
Index of Level-Set Dispersion
A key step in the projection pursuit training algorithm is to find for each 3 i net
the projection direction f3i' an instance of the important problem of economically
choosing input features/variables in constructing a forecasting model. In general,
the fewer the number of input features, the more likely are the results to generalize
from training set performance to test set performance- reduction in variance at the
possible expense of increase in bias. Our controlled size subnet projection pursuit
training algorithm deals with part of the complexity problem, provided that the
input features are fixed. We turn now to our approach to finding input features
or search directions based on minimizing an index of dispersion of level-sets. Li
[1991] proposed taking an inverse ('slicing the y's') point of view to estimate the
directions f3i. The justification provided for this so-called slicing inverse regression
(SIR) method, however, requires that the input or feature vector x be elliptically
symmetrically distributed, and this is not likely to be the case in our electric load
forecasting problem. The basic idea behind minimizing dispersion of level-sets
is that from Eq. (*) we see that a fixed value of y, and small noise ?, implies
a highly constrained set of values for f3ix, ... ,f3~x, while leaving unconstrained the
components of x that lie in the subspace B~ orthogonal to that space B spanned by
the f3i.. Hence, if one has a good number of i.i.d. observations sharing a similar value
of the response y, then there should be more dispersion of input vectors projected
into Bl.. than along the projections into B. We implement this by quantizing the
observed y values into, say, H slices, with Lh denoting the h_th level-set containing
those inputs with y-value in the h_th slice, and X-h is their sample mean. The f3 are
then picked as the the eigenvector associated with the smallest eigenvalue of the
centered covariance matrix:
H
L L
(Xi - x"h)(Xi. - Xh)'.
h=l xiELh
4.2
Implementations
In practical implementations, one may discard both extremes of the family of H
level sets (trimming) to avoid large response values when it is believed that they
may correspond to large magnitudes of input components. One should also standardize initially the input data to a unit sample covariance matrix. Otherwise, our
results will reflect the distribution of x rather than the functional relationship of
Eq. (*). We have applied this projection index both in finding the f3i. during projection pursuit training and in reducing a high-dimensional feature set to a lowdimensional feature set. We have implemented such a feature selection scheme for
forecasting the Monday - Friday evening peaks. The initial feature set consists of
thirteen hourly loads from lam to 1pm, thirteen hourly temperatures from lam to
1pm and the temperature around the peak times. Three eigenvectors of the centered covariance matrix were chosen, thereby reducing a 27-dimensional feature set
to a 3-dimensional one. We then ran a standard BPA on this reduced featured set
and tested on the 1991 data. We obtained a percentage absolute error of 1.6% (rms
error about 100 MWH), which is as good as all of our previous efforts.
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Yuan and Fine
Acknow ledgements
Partial support for this research was provided by NSF Grant No. ECS-9017493.
We wish to thank Prof. J. Hwang, Mathematics Department, Cornell, for initial
discussions of SIR and are grateful to Dr. P.D. Yeshakul, American Electric Service
Corp., for providing the data set and instructing us patiently in the lore of shortterm load forecasting.
References
Connor, J., L. Atlas, R.D. Martin [1991], Recurrent networks and NARMA modeling, NIPS 91.
Damborg, M., M.EI-Sharkawi, R. Marks II [1990], Potential of artificial neural networks in power system operation, Proc. 1990 IEEE Inter.Symp. on Circuits and
Systems, 4, 2933-2937.
Friedman, J. [1987], Exploratory projection pursuit, J. Amer. Stat. Assn., 82,
249-266.
Gross,G., F. Galiana [1987], Short-term load forecasting, Proc. IEEE, 75, 15581573.
Hastie, T., R. Tibshirani [1990], Generalized Additive Models,Chapman and Hall.
Huber, P. [1985]' Projection pursuit, The Annals of Statistics, 13, 435-475.
Intrator, N. [1992] Combinining exploratory projection pursuit and projection pursuit regression with applicatons to neural networks, To appear in Neural Computation.
Li, K.-C. [1991] Slicing inverse regression for dimension reduction, Journal of American Statistical Assoc., 86.
Willis, H.1., J.F.D. Northcote-Green [1984], Comparison tests of fourteen load
forecasting methods, IEEE Trans. on Power Apparatus and Systems, PAS-103,
1190-1197.
Yuan, J-1., T.L.Fine [1992]' Forecasting demand for electric power using autoregressive neural networks, Proc. Con! on Info. Sci. and Systems, Princeton, NJ.
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6,235 | 6,640 | Learning a Multi-View Stereo Machine
Abhishek Kar
UC Berkeley
[email protected]
Christian H?ne
UC Berkeley
[email protected]
Jitendra Malik
UC Berkeley
[email protected]
Abstract
We present a learnt system for multi-view stereopsis. In contrast to recent learning
based methods for 3D reconstruction, we leverage the underlying 3D geometry of
the problem through feature projection and unprojection along viewing rays. By
formulating these operations in a differentiable manner, we are able to learn the
system end-to-end for the task of metric 3D reconstruction. End-to-end learning
allows us to jointly reason about shape priors while conforming to geometric
constraints, enabling reconstruction from much fewer images (even a single image)
than required by classical approaches as well as completion of unseen surfaces. We
thoroughly evaluate our approach on the ShapeNet dataset and demonstrate the
benefits over classical approaches and recent learning based methods.
1
Introduction
Multi-view stereopsis (MVS) is classically posed as the following problem - given a set of images
with known camera poses, it produces a geometric representation of the underlying 3D world. This
representation can be a set of disparity maps, a 3D volume in the form of voxel occupancies, signed
distance fields etc. An early example of such a system is the stereo machine from Kanade et al. [26]
that computes disparity maps from images streams from six video cameras. Modern approaches
focus on acquiring the full 3D geometry in the form of volumetric representations or polygonal
meshes [48]. The underlying principle behind MVS is simple - a 3D point looks locally similar when
projected to different viewpoints [29]. Thus, classical methods use the basic principle of finding
dense correspondences in images and triangulate to obtain a 3D reconstruction.
The question we try to address in this work is can we learn a multi-view stereo system? For the
binocular case, Becker and Hinton [1] demonstrated that a neural network can learn to predict a depth
map from random dot stereograms. A recent work [28] shows convincing results for binocular stereo
by using an end-to-end learning approach with binocular geometry constraints.
In this work, we present Learnt Stereo Machines (LSM) - a system which is able to reconstruct object
geometry as voxel occupancy grids or per-view depth maps from a small number of views, including
just a single image. We design our system inspired by classical approaches while learning each
component from data embedded in an end to end system. LSMs have built in projective geometry,
enabling reasoning in metric 3D space and effectively exploiting the geometric structure of the MVS
problem. Compared to classical approaches, which are designed to exploit a specific cue such as
silhouettes or photo-consistency, our system learns to exploit the cues that are relevant to the particular
instance while also using priors about shape to predict geometry for unseen regions.
Recent work from Choy et al. [5] (3D-R2N2) trains convolutional neural networks (CNNs) to
predict object geometry given only images. While this work relied primarily on semantic cues for
reconstruction, our formulation enables us to exploit strong geometric cues. In our experiments, we
demonstrate that a straightforward way of incorporating camera poses for volumetric occupancy
prediction does not lead to expected gains, while our geometrically grounded method is able to
effectively utilize the additional information.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Fused Feature
Grid
Recurrent Fusion
Unprojection
Image
Encoder
2D Feature
Maps
Final Grid
Voxel Occupancy Grid
3D Grid Reasoning
Projection
3D Feature
Grids
Depth Maps
Figure 1: Overview of a Learnt Stereo Machine (LSM). It takes as input one or more views and camera poses.
The images are processed through a feature encoder which are then unprojected into the 3D world frame using a
differentiable unprojection operation. These grids {Gfi }n
i=1 are then matched in a recurrent manner to produce a
fused grid G p which is then transformed by a 3D CNN into G o . LSMs can produce two kinds of outputs - voxel
occupancy grids (Voxel LSM) decoded from G o or per-view depth maps (Depth LSM) decoded after a projection
operation.
Classical multi-view stereopsis is traditionally able to handle both objects and scenes - we only
showcase our system for the case of objects with scenes left for future work. We thoroughly evaluate
our system on the synthetic ShapeNet [3] dataset. We compare to classical plane sweeping stereo,
visual hulls and several challenging learning-based baselines. Our experiments show that we are able
to reconstruct objects with fewer images than classical approaches. Compared to recent learning
based reconstruction approaches, our system is able to better use camera pose information leading to
significantly large improvements while adding more views. Finally, we show successful generalization
to unseen object categories demonstrating that our network goes beyond semantic cues and strongly
uses geometric information for unified single and multi-view 3D reconstruction.
2
Related Work
Extracting 3D information from images is one of the classical problems in computer vision. Early
works focused on the problem of extracting a disparity map from a binocular image pair [36]. We
refer the reader to [47] for an overview of classical binocular stereo matching algorithms. In the multiview setting, early work focused on using silhouette information via visual hulls [32], incorporating
photo-consistency to deal with concavities (photo hull) [29], and shape refinement using optimization
[55, 50, 7, 15]. [39, 35, 54] directly reason about viewing rays in a voxel grid, while [34] recovers
a quasi dense point cloud. In our work, we aim to learn a multi-view stereo machine grounded in
geometry, that learns to use these classical constraints while also being able to reason about semantic
shape cues from the data. Another approach to MVS involves representing the reconstruction as a
collection of depth maps [6, 57, 41, 13, 40]. This allows recovery of fine details for which a consistent
global estimate may be hard to obtain. These depth maps can then be fused using a variety of different
techniques [38, 8, 33, 59, 30]. Our learnt system is able to produce a set of per-view depth maps
along with a globally consistent volumetric representation which allows us to preserve fine details
while conforming to global structure.
Learning has been used for multi-view reconstruction in the form of shape priors for objects [2, 9, 58,
20, 27, 52], or semantic class specific surface priors for scenes [22, 17, 45]. These works use learnt
shape models and either directly fit them to input images or utilize them in a joint representation
that fuses semantic and geometric information. Most recently, CNN based learning methods have
been proposed for 3D reconstruction by learning image patch similarity functions [60, 18, 23] and
end-to-end disparity regression from stereo pairs [37, 28]. Approaches which predict shape from
a single image have been proposed in form of direct depth map regression [46, 31, 10], generating
multiple depth maps from novel viewpoints [51], producing voxel occupancies [5, 16], geometry
images [49] and point clouds [11]. [12] study a related problem of view interpolation, where a rough
depth estimate is obtained within the system.
A line of recent works, complementary to ours, has proposed to incorporate ideas from multi-view
geometry in a learning framework to train single view prediction systems [14, 56, 53, 42, 61] using
multiple views as supervisory signal. These works use the classical cues of photo-consistency and
2
De
2
pth
pla
2-D World Grid
ne
s
1
z=
z=
3
2-D Feature Grid
z=
z=3
z=2
z=1
1-D Projections
1-D
Feature Map
Sampling locations
1-D
Canvas
Camera
Camera
(a) Projection
(b) Unprojection
Figure 2: Illustrations of projection and unprojection operations between 1D maps and 2D grids. (a) The
projection operation samples values along the ray at equally spaced z-values into a 1D canvas/image. The
sampled features (shown by colors here) at the z planes are stacked into channels to form the projected feature
map. (b) The unprojection operation takes features from a feature map (here in 1-D) and places them along rays
at grid blocks where the respective rays intersect. Best viewed in color.
silhouette consistency only during training - their goal during inference is to only perform single
image shape prediction. In contrast, we also use geometric constraints during inference to produce
high quality outputs.
Closest to our work is the work of Kendall et al. [28] which demonstrates incorporating binocular
stereo geometry into deep networks by formulating a cost volume in terms of disparities and regressing
depth values using a differentiable arg-min operation. We generalize to multiple views by tracing
rays through a discretized grid and handle variable number of views via incremental matching using
recurrent units. We also propose a differentiable projection operation which aggregates features along
viewing rays and learns a nonlinear combination function instead of using the differentiable arg-min
which is susceptible to multiple modes. Moreover, we can also infer 3D from a single image during
inference.
3
Learnt Stereo Machines
Our goal in this paper is to design an end-to-end learnable system that produces a 3D reconstruction
given one or more input images and their corresponding camera poses. To this end, we draw inspiration from classical geometric approaches where the underlying guiding principle is the following - the
reconstructed 3D surface has to be photo-consistent with all the input images that depict this particular
surface. Such approaches typically operate by first computing dense features for correspondence
matching in image space. These features are then assembled into a large cost volume of geometrically
feasible matches based on the camera pose. Finally, the optimum of this matching volume (along
with certain priors) results in an estimate of the 3D volume/surface/disparity maps of the underlying
shape from which the images were produced.
Our proposed system, shown in Figure 1, largely follows the principles mentioned above. It uses
a discrete grid as internal representation of the 3D world and operates in metric 3D space. The
input images {Ii }ni=1 are first processed through a shared image encoder which produces dense
feature maps {Fi }ni=1 , one for each image. The features are then unprojected into 3D feature grids
{Gif }ni=1 by rasterizing the viewing rays with the known camera poses {Pi }ni=1 . This unprojection
operation aligns the features along epipolar lines, enabling efficient local matching. This matching
is modelled using a recurrent neural network which processes the unprojected grids sequentially to
produce a grid of local matching costs G p . This cost volume is typically noisy and is smoothed in
an energy optimization framework with a data term and smoothness term. We model this step by a
feed forward 3D convolution-deconvolution CNN that transforms G p into a 3D grid G o of smoothed
costs taking context into account. Based on the desired output, we propose to either let the final
grid be a volumetric occupancy map or a grid of features which is projected back into 2D feature
3
maps {Oi }ni=1 using the given camera poses. These 2D maps are then mapped to a view specific
representation of the shape such as a per view depth/disparity map. The key components of our
system are the differentiable projection and unprojection operations which allow us to learn the
system end-to-end while injecting the underlying 3D geometry in a metrically accurate manner. We
refer to our system as a Learnt Stereo Machine (LSM). We present two variants - one that produces
per voxel occupancy maps (Voxel LSM) and another that outputs a depth map per input image (Depth
LSM) and provide details about the components and the rationale behind them below.
2D Image Encoder. The first step in a stereo algorithm is to compute a good set of features to
match across images. Traditional stereo algorithms typically use raw patches as features. We model
this as a feed forward CNN with a convolution-deconvolution architecture with skip connections
(UNet) [44] to enable the features to have a large enough receptive field while at the same time having
access to lower level features (using skip connections) whenever needed. Given images {Ii }ni=1 , the
feature encoder produces dense feature maps {Fi }ni=1 in 2D image space, which are passed to the
unprojection module along with the camera parameters to be lifted into metric 3D space.
Differentiable Unprojection. The goal of the unprojection operation is to lift information from
2D image frame to the 3D world frame. Given a 2D point p, its feature representation F(p) and our
global 3D grid representation, we replicate F(p) along the viewing ray for p into locations along
the viewing ray in the metric 3D grid (a 2D illustration is presented in Figure 2). In the case of
perspective projection specified by an intrinsic camera matrix K and an extrinsic camera matrix [R|t],
the unprojection operation uses this camera pose to trace viewing rays in the world and copy the
image features into voxels in this 3D world grid. Instead of analytically tracing rays, given the centers
th
k
V
of blocks in our 3D grid {Xwk }N
block by projecting {Xw
} using
k=1 , we compute the feature for k
0
k
0
the camera projection equations pk = K[R|t]Xw into the image space. pk is a continuous quantity
whereas F is defined on at discrete 2D locations. Thus, we use the differentiable bilinear sampling
operation to sample from the discrete grid [25] to obtain the feature at Xwk .
Such an operation has the highly desirable property that features from pixels in multiple images
that may correspond to the same 3D world point unproject to the same location in the 3D grid trivially enforcing epipolar constraints. As a result, any further processing on these unprojected grids
has easy access to corresponding features to make matching decisions foregoing the need for long
range image connections for feature matching in image space. Also, by projecting discrete 3D points
into 2D and bilinearly sampling from the feature map rather than analytically tracing rays in 3D,
we implicitly handle the issue where the probability of a grid voxel being hit by a ray decreases
with distance from the camera due to their projective nature. In our formulation, every voxel gets
a ?soft" feature assigned based on where it projects back in the image, making the feature grids
G f smooth and providing stable gradients. This geometric procedure of lifting features from 2D
maps into 3D space is in contrast with recent learning based approaches [5, 51] which either reshape
flattened feature maps into 3D grids for subsequent processing or inject pose into the system using
fully connected layers. This procedure effectively saves the network from having to implicitly learn
projective geometry and directly bakes this given fact into the system. In LSMs, we use this operation
to unproject the feature maps {Fi }ni=1 in image space produced by the feature encoder into feature
grids {Gif }ni=1 that lie in metric 3D space.
For single image prediction, LSMs cannot match features from multiple images to reason about where
to place surfaces. Therefore, we append geometric features along the rays during the projection and
unprojection operation to facilitate single view prediction. Specifically, we add the depth value and
the ray direction at each sampling point.
Recurrent Grid Fusion. The 3D feature grids {Gif }ni=1 encode information about individual input
images and need to be fused to produce a single grid so that further stages may reason jointly over all
the images. For example, a simple strategy to fuse them would be to just use a point-wise function e.g. max or average. This approach poses an issue where the combination is too spatially local and
early fuses all the information from the individual grids. Another extreme is concatenating all the
feature grids before further processing. The complexity of this approach scales linearly with the
number of inputs and poses issues while processing a variable number of images. Instead, we choose
to processed the grids in a sequential manner using a recurrent neural network. Specifically, we use a
3D convolutional variant of the Gated Recurrent Unit (GRU) [24, 4, 5] which combines the grids
4
{Gif }ni=1 using 3D convolutions (and non-linearities) into a single grid G p . Using convolutions helps
us effectively exploit neighborhood information in 3D space for incrementally combining the grids
while keeping the number of parameters low. Intuitively, this step can be thought of as mimicking
incremental matching in MVS where the hidden state of the GRU stores a running belief about the
matching scores by matching features in the observations it has seen. One issue that arises is that we
now have to define an ordering on the input images, whereas the output should be independent of the
image ordering. We tackle this issue by randomly permuting the image sequences during training
while constraining the output to be the same. During inference, we empirically observe that the final
output has very little variance with respect to ordering of the input image sequence.
3D Grid Reasoning. Once the fused grid G p is constructed, a classical multi-view stereo approach
would directly evaluate the photo-consistency at the grid locations by comparing the appearance of
the individual views and extract the surface at voxels where the images agree. We model this step with
a 3D UNet that transforms the fused grid G p into G o . The purpose of this network is to use shape cues
present in G p such as feature matches and silhouettes as well as build in shape priors like smoothness
and symmetries and knowledge about object classes enabling it to produce complete shapes even
when only partial information is visible. The UNet architecture yet again allows the system to use
large enough receptive fields for doing multi-scale matching while also using lower level information
directly when needed to produce its final estimate G o . In the case of full 3D supervision (Voxel LSM),
this grid can be made to represent a per voxel occupancy map. G o can also be seen as a feature grid
containing the final representation of the 3D world our system produces from which views can be
rendered using the projection operation described below.
Differentiable Projection. Given a 3D feature grid G and a camera P, the projection operation
produces a 2D feature map O by gathering information along viewing rays. The direct method would
be to trace rays for every pixel and accumulate information from all the voxels on the ray?s path.
Such an implementation would require handling the fact that different rays can pass through different
number of voxels on their way. For example, one can define a reduction function along the rays to
aggregate information (e.g. max, mean) but this would fail to capture spatial relationships between
the ray features. Instead, we choose to adopt a plane sweeping approach where we sample from
z
locations on depth planes at equally spaced z-values {zk }N
k=1 along the ray.
Consider a 3D point Xw that lies along the ray corresponding to a 2D point p in the projected
feature grid at depth zw - i.e. p = K[R|t]Xw and z(Xw ) = zw . The corresponding feature O(p) is
computed by sampling from the grid G at the (continuous) location Xw . This sampling can be done
differentiably in 3D using trilinear interpolation. In practice, we use nearest neighbor interpolation
in 3D for computational efficiency. Samples along each ray are concatenated in ascending z-order
to produce the 2D map O where the features are stacked along the channel dimension. Rays in this
feature grid can be trivially traversed by just following columns along the channel dimension allowing
us to learn the function to pool along these rays by using 1x1 convolutions on these feature maps and
progressively reducing the number of feature channels.
Architecture Details. As mentioned above, we present two versions of LSMs - Voxel LSM (VLSM) and Depth LSM (D-LSM). Given one or more images and cameras, Voxel LSM (V-LSM)
produces a voxel occupancy grid whereas D-LSM produces a depth map per input view. Both
systems share the same set of CNN architectures (UNet) for the image encoder, grid reasoning and
the recurrent pooling steps. We use instance normalization for all our convolution operations and
layer normalization for the 3D convolutional GRU. In V-LSM, the final grid G o is transformed into
a probabilistic voxel occupancy map V ? Rvh ?vw ?vd by a 3D convolution followed by softmax
operation. We use simple binary cross entropy loss between ground truth occupancy maps and V. In
D-LSM, G o is first projected into 2D feature maps {Oi }ni=1 which are then transformed into metric
depth maps {di }ni=1 by 1x1 convolutions to learn the reduction function along rays followed by
deconvolution layers to upsample the feature map back to the size of the input image. We use absolute
L1 error in depth to train D-LSM. We also add skip connections between early layers of the image
encoder and the last deconvolution layers producing depth maps giving it access to high frequency
information in the images.
5
Figure 3: Voxel grids produced by V-LSM for example image sequences alongside a learning based baseline
which uses pose information in a fully connected manner. V-LSM produces geometrically meaningful reconstructions (e.g. the curved arm rests instead of perpendicular ones (in R2N2) in the chair on the top left and
the siren lights on top of the police car) instead of relying on purely semantic cues. More visualizations in
supplementary material.
4
Experiments
In this section, we demonstrate the ability of LSMs to learn 3D shape reconstruction in a geometrically
accurate manner. First, we present quantitative results for V-LSMs on the ShapeNet dataset [3] and
compare it to various baselines, both classical and learning based. We then show that LSMs generalize
to unseen object categories validating our hypothesis that LSMs go beyond object/class specific priors
and use photo-consistency cues to perform category-agnostic reconstruction. Finally, we present
qualitative and quantitative results from D-LSM and compare it to traditional multi-view stereo
approaches.
Dataset and Metrics. We use the synthetic ShapeNet dataset [3] to generate posed image-sets,
ground truth 3D occupancy maps and depth maps for all our experiments. More specifically, we
use a subset of 13 major categories (same as [5]) containing around 44k 3D models resized to lie
within the unit cube centered at the origin with a train/val/test split of [0.7, 0.1, 0.2]. We generated a
large set of realistic renderings for the models sampled from a viewing sphere with ?az ? [0, 360)
and ?el ? [?20, 30] degrees and random lighting variations. We also rendered the depth images
corresponding to each rendered image. For the volumetric ground truth, we voxelize each of the
models at a resolution of 32 ? 32 ? 32. In order to evaluate the outputs of V-LSM, we binarize the
probabilities at a fixed threshold (0.4 for all methods except visual hull (0.75)) and use the voxel
intersection over union (IoU) as the similarity measure. To aggregate the per model IoU, we compute
a per class average and take the mean as a per dataset measure. All our models are trained in a class
agnostic manner.
Implementation. We use 224 ? 224 images to train LSMs with a shape batch size of 4 and
4 views per shape. Our world grid is at a resolution of 323 . We implemented our networks in
Tensorflow and trained both the variants of LSMs for 100k iterations using Adam. The projection and
unprojection operations are trivially implemented on the GPU with batched matrix multiplications
and bilinear/nearest sampling enabling inference at around 30 models/sec on a GTX 1080Ti. We
unroll the GRU for upto 4 time steps while training and apply the trained models for arbitrary number
of views at test time.
Multi-view Reconstruction on ShapeNet. We evaluate V-LSMs on the ShapeNet test set and
compare it to the following baselines - a visual hull baseline which uses silhouettes to carve out
volumes, 3D-R2N2 [5], a previously proposed system which doesn?t use camera pose and performs
multi-view reconstruction, 3D-R2N2 w/pose which is an extension of 3D-R2N2 where camera pose
is injected using fully connected layers. For the experiments, we implemented the 3D-R2N2 system
6
2
3
4
3D-R2N2 [5]
55.6
59.6
61.3
62.0
Visual Hull
3D-R2N2 w/pose
V-LSM
18.0
55.1
61.5
36.9
59.4
72.1
47.0
61.2
76.2
52.4
62.1
78.2
V-LSM w/bg
60.5
69.8
73.7
75.6
3D-R2N2 w/pose
V-LSM
25
Gap in Performance
1
# Views
Table 1: Mean Voxel IoU on the ShapeNet test set. Note
that the original 3D-R2N2 system does not use camera
pose whereas the 3D-R2N2 w/pose system is trained
with pose information. V-LSM w/bg refers to voxel LSM
trained and tested with random images as backgrounds
instead of white backgrounds only.
20
15
10
5
1
2
3
4
5
6
Number of Views
7
8
Figure 4: Generalization performance for V-LSM
and 3D-R2N2 w/pose measured by gap in voxel
IoU when tested on unseen object categories.
Figure 5: Qualitative results for per-view depth map prediction on ShapeNet. We show the depth maps predicted
by Depth-LSM (visualized with shading from a shifted viewpoint) and the point cloud obtained by unprojecting
them into world coordinates.
(and the 3D-R2N2 w/pose) and trained it on our generated data (images and voxel grids). Due to the
difference in training data/splits and the implementation, the numbers are not directly comparable
to the ones reported in [5] but we observe similar performance trends. For the 3D-R2N2 w/pose
system, we use the camera pose quaternion as the pose representation and process it through 2 fully
connected layers before concatenating it with the feature passed into the LSTM. Table 1 reports the
mean voxel IoU (across 13 categories) for sequences of {1, 2, 3, 4} views. The accuracy increases
with number of views for all methods but it can be seen that the jump is much less for the R2N2
methods indicating that it already produces a good enough estimate at the beginning but fails to
effectively use multiple views to improve its reconstruction significantly. The R2N2 system with
naively integrated pose fails to improve over the base version, completely ignoring it in favor of
just image-based information. On the other hand, our system, designed specifically to exploit these
geometric multi-view cues improves significantly with more views. Figure 3 shows some example
reconstructions for V-LSM and 3D-R2N2 w/pose. Our system progressively improves based on
the viewpoint it receives while the R2N2 w/pose system makes very confident predictions early on
(sometimes ?retrieving" a completely different instance) and then stops improving as much. As we
use a geometric approach, we end up memorizing less and reconstruct when possible. More detailed
results can be found in the supplementary material.
Generalization. In order to test how well LSMs learn to generalize to unseen data, we split our
data into 2 parts with disjoint sets of classes - split 1 has data from 6 classes while split 2 has data
from the other 7. We train three V-LSMs - trained on split 1 (V-LSM-S1), on split 2 (V-LSM-S2) and
both splits combined (V-LSM-All). The quantity we are interested in is the change in performance
when we test the system on a category it hasn?t seen during training. We use the difference in test IoU
of a category C between V-LSM-All and V-LSM-S1 if C is not in split 1 and vice versa. Figure 4
shows the mean of this quantity across all classes as the number of views change. It can be seen that
for a single view, the difference in performance is fairly high and as we see more views, the difference
7
in performance decreases - indicating that our system has learned to exploit category agnostic shape
cues. On the other hand, the 3D-R2N2 w/pose system fails to generalize with more views. Note that
the V-LSMs have been trained with a time horizon of 4 but are evaluated till upto 8 steps here.
Mean Voxel IoU
Sensitivity to noisy camera pose and masks.
80
We conducted experiments to quantify the efAccurate Pose
Noisy Pose
Noise:0
Noise:0
fects of noisy camera pose and segmentations
Noise:10
Noise:10
75
Noise:20
Noise:20
on performance for V-LSMs. We evaluated
70
models trained with perfect poses on data with
perturbed camera extrinsics and observed that
65
performance degrades (as expected) yet still re60
mains better than the baseline (at 10? noise).
55
We also trained new models with synthetically
perturbed extrinsics and achieve significantly
50
higher robustness to noisy poses while maintain45
Views:1
Views:2
Views:3
Views:4
ing competitive performance (Figure 6). This is
illustrated in Figure 6. The perturbation is intro- Figure 6: Sensitivity to noise in camera pose estimates
duced by generating a random rotation matrix for V-LSM for systems trained with and without pose
which rotates the viewing axis by a max angular perturbation.
magnitude ? while still pointing at the object of
interest.
We also trained LSMs on images with random images backgrounds (V-LSM w/bg in Table 1) rather
than only white backgrounds and saw a very small drop in performance. This shows that our method
learns to match features rather than relying heavily on perfect segmentations.
Multi-view Depth Map Prediction. We show qualitative results from Depth LSM in Figure 5. We
manage to obtain thin structures in challenging examples (chairs/tables) while predicting consistent
geometry for all the views. We note that the skip connections from the image to last layers for D-LSM
do help in directly using low level image features while producing depth maps. The depth maps
are viewed with shading in order to point out that we produce metrically accurate geometry. The
unprojected point clouds also align well with each other showing the merits of jointly predicting the
depth maps from a global volume rather than processing them independently.
Comparision to Plane Sweeping. We qualitatively compare D-LSM to the popular plane sweeping
(PS) approach [6, 57] for stereo matching. Figure 7 shows the unprojected point clouds from per
view depths maps produced using PS and D-LSM using 5 and 10 images. We omit an evaluation with
less images as plane sweeping completely fails with fewer images. We use the publicly available
implementation for the PS algorithm [19] and use 5x5 zero mean normalized cross correlation as
matching windows with 300 depth planes. We can see that our approach is able to produce much
cleaner point clouds with less input images. It is robust to texture-less areas where traditional stereo
algorithms fail (e.g. the car windows) by using shape priors to reason about them. We also conducted
a quantitative comparison using PS and D-LSM with 10 views (D-LSM
was trained using only four
?
images). The evaluation region is limited to a depth range of ? 3/2 (maximally possible depth
range) around the origin as the original models lie in a unit cube centered at the origin. Furthermore,
pixels where PS is not able to provide a depth estimate are not taken into account. Note that all these
choices disadvantage our method. We compute the per depth map error as the median absolute depth
difference for the valid pixels, aggregate to a per category mean error and report the average of the
per category means for PS as 0.051 and D-LSM as 0.024. Please refer to the supplementary material
for detailed results.
5
Discussion
We have presented Learnt Stereo Machines (LSM) - an end-to-end learnt system that performs
multi-view stereopsis. The key insight of our system is to use ideas from projective geometry
to differentiably transfer features between 2D images and the 3D world and vice-versa. In our
experiments we showed the benefits of our formulation over direct methods - we are able to generalize
to new object categories and produce compelling reconstructions with fewer images than classical
8
(a) PS 5 Images
(b) LSM 5 Images
(c) PS 10 Images
(d) LSM 10 Images
(e) PS 20 Images
Figure 7: Comparison between Depth-LSM and plane sweeping stereo (PS) with varying numbers of images.
systems. However, our system also has some limitations. We discuss some below and describe how
they lead to future work.
A limiting factor in our current system is the coarse resolution (323 ) of the world grid. Classical
algorithms typically work on much higher resolutions frequently employing special data structures
such as octrees. We can borrow ideas from recent works [43, 21] which show that CNNs can predict
such high resolution volumes. We also plan to apply LSMs to more general geometry than objects,
eventually leading to a system which can reconstruct single/multiple objects and entire scenes. The
main challenge in this setup is to find the right global grid representation. In scenes for example, a
grid in terms of a per-view camera frustum might be more appropriate than a global aligned euclidean
grid.
In our experiments we evaluated classical multi-view 3D reconstruction where the goal is to produce
3D geometry from images with known poses. However, our system is more general and the projection
modules can be used wherever one needs to move between 2D image and 3D world frames. Instead
of predicting just depth maps from our final world representation, one can also predict other view
specific representations such as silhouettes or pixel wise part segmentation labels etc. We can also
project the final world representation into views that we haven?t observed as inputs (we would omit
the skip connections from the image encoder to make the projection unconditional). This can be used
to perform view synthesis grounded in 3D.
Acknowledgments
This work was supported in part by NSF Award IIS- 1212798 and ONR MURI-N00014-10-1-0933.
Christian H?ne is supported by an ?Early Postdoc.Mobility? fellowship No. 165245 from the Swiss
National Science Foundation. The authors would like to thank David Fouhey, Saurabh Gupta and
Shubham Tulsiani for valuable discussions and Fyusion Inc. for providing GPU hours for the work.
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6,236 | 6,641 | Phase Transitions in the Pooled Data Problem
Jonathan Scarlett and Volkan Cevher
Laboratory for Information and Inference Systems (LIONS)
?cole Polytechnique F?d?rale de Lausanne (EPFL)
{jonathan.scarlett,volkan.cevher}@epfl.ch
Abstract
In this paper, we study the pooled data problem of identifying the labels associated with a large collection of items, based on a sequence of pooled tests revealing
the counts of each label within the pool. In the noiseless setting, we identify an
exact asymptotic threshold on the required number of tests with optimal decoding, and prove a phase transition between complete success and complete failure.
In addition, we present a novel noisy variation of the problem, and provide an
information-theoretic framework for characterizing the required number of tests
for general random noise models. Our results reveal that noise can make the problem considerably more difficult, with strict increases in the scaling laws even at
low noise levels. Finally, we demonstrate similar behavior in an approximate recovery setting, where a given number of errors is allowed in the decoded labels.
1
Introduction
Consider the following setting: There exists a large population of items, each of which has an
associated label. The labels are initially unknown, and are to be estimated based on pooled tests.
Each pool consists of some subset of the population, and the test outcome reveals the total number
of items corresponding to each label that are present in the pool (but not the individual labels). This
problem, which we refer to as the pooled data problem, was recently introduced in [1,2], and further
studied in [3, 4]. It is of interest in applications such as medical testing, genetics, and learning with
privacy constraints, and has connections to the group testing problem [5] and its linear variants [6,7].
The best known bounds on the required number of tests under optimal decoding were given in [3];
however, the upper and lower bounds therein do not match, and can exhibit a large gap. In this
paper, we completely close these gaps by providing a new lower bound that exactly matches the
upper bound of [3]. These results collectively reveal a phase transition between success and failure,
with the probability of error vanishing when the number of tests exceeds a given threshold, but
tending to one below that threshold. In addition, we explore the novel aspect of random noise in the
measurements, and show that this can significantly increase the required number of tests. Before
summarizing these contributions in more detail, we formally introduce the problem.
1.1
Problem setup
We consider a large population of items [p] = {1, . . . , p}, each of which has an associated label
in [d] = {1, . . . , d}. We let ? = (?1 , . . . , ?d ) denote a vector containing the proportions of items
having each label, and we assume that the vector of labels itself, = ( 1 , . . . , p ), is uniformly
distributed over the sequences consistent with these proportions:
? Uniform(B(?)),
where B(?) is the set of length-p sequences whose empirical distribution is ?.
(1)
The goal is to recover based on a sequence of pooled tests. The i-th test is represented by a
(i)
(possibly random) vector X (i) 2 {0, 1}p , whose j-th entry Xj indicates whether the j-th item is
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Necessary for Pe 6! 1
(this paper)
p
p
1
p
?
max
f (r)
? f (1)
?
max
f (r)
log p r2{1,...,d 1}
log p 2
log p r2{1,...,d 1}
Table 1: Necessary and sufficient conditions on the number of tests n in the noiseless setting. The
function f (r) is defined in (5). Asymptotic multiplicative 1 + o(1) terms are omitted.
Sufficient for Pe ! 0 [3]
Necessary for Pe 6! 1 [3]
Noisy testing
Noisy testing
Noisy testing
Noiseless testing
(SNR = ?(1))
(SNR = (log p)?(1) )
(SNR = p?(1) )
? p ?
? p ?
?
?
p
?
?
?
? p log p
log p
log p
log log p
Table 2: Necessary and sufficient conditions on the number of tests n in the noisy setting. SNR
denotes the signal-to-noise ratio, and the noise model is given in Section 2.2.
included in the i-th test. We define a measurement matrix X 2 {0, 1}n?p whose i-th row is given
by X (i) for i = 1, . . . , n, where n denotes the total number of tests. We focus on the non-adaptive
testing scenario, where the entire matrix X must be specified prior to performing any tests.
(i)
(i)
In the noiseless setting, the i-th test outcome is a vector Y (i) = (Y1 , . . . , Yd ), with t-th entry
(i)
= Nt ( , X (i) ),
(2)
P
where for t = 1, . . . , d, we let Nt ( , X) = j2[p] 1{ j = t \ Xj = 1} denote the number of
items with label t that are included in the test described by X 2 {0, 1}p . More generally, in the
possible presence of noise, the i-th observation is randomly generated according to
Yt
Y (i) | X (i) ,
? PY |N1 ( ,X (i) )...Nd ( ,X (i) )
(3)
for some conditional probability mass function PY |N1 ,...,Nd (or density function in the case of continuous observations). We assume that the observations Y (i) (i = 1, . . . , n) are conditionally independent given X, but otherwise make no assumptions on PY |N1 ,...,Nd . Clearly, the noiseless model
(2) falls under this more general setup.
Similarly to X, we let Y denote an n ? d matrix of observations, with the i-th row being Y (i) . Given
X and Y, a decoder outputs an estimate ? of , and the error probability is given by
Pe = P[ ? 6= ],
(4)
where the probability is with respect to , X, and Y. We seek to find conditions on the number of
tests n under which Pe attains a certain target value in the limit as p ! 1, and our main results
provide necessary conditions (i.e., lower bounds on n) for this to occur. As in [3], we focus on the
case that d and ? are fixed and do not depend on p.1
1.2
Contributions and comparisons to existing bounds
Our focus in this paper is on information-theoretic bounds on the required number of tests that hold
regardless of practical considerations such as computation and storage. Among the existing works in
the literature, the one most relevant to this paper is [3], whose bounds strictly improve on the initial
bounds in [1]. The same authors also proved a phase transition for a practical algorithm based on
approximate message passing [4], but the required number of tests is in fact significantly larger than
the information-theoretic threshold (specifically, linear in p instead of sub-linear).
Table 1 gives a summary of the bounds from [3] and our contributions in the noiseless setting. To
define the function f (r) therein, we introduce the additional notation that for r = {1, . . . , d 1},
(r)
(r)
? (r) = (?1 , . . . , ?r ) is a vector whose first entry sums the largest d r + 1 entries of ?, and
whose remaining entries coincide with the remaining r 1 entries of ?. We have
2(H(?) H(? (r) ))
,
d r
r2{1,...,d 1}
meaning that the entries in Table 1 corresponding to the results of [3] are given as follows:
f (r) =
max
1
(5)
More precisely, ? should be rounded to the nearest empirical distribution (e.g., in `1 -norm) for sequences
2 [d]p of length p; we leave such rounding implicit throughout the paper.
2
1.4
Random :
Uniform :
Highly non-uniform :
1.2
f(r)
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
r
Figure 1: The function f (r) in (5), for several choices of ?, with d = 10.
The random ? are
drawn uniformly on the probability simplex, and the highly non-uniform choice of ? is given by ? =
(0.49, 0.49, 0.0025, . . . , 0.0025). When the maximum is achieved at r = 1, the bounds of [3] coincide up
to a factor of two, whereas if the maximum is achieved for r > 1 then the gap is larger.
? (Achievability) When the entries of X are i.i.d. on Bernoulli(q) for some q 2 (0, 1) (not
depending on p), there exists a decoder such that Pe ! 0 as p ! 1 with
?
?
p
2(H(?) H(? (r) ))
n?
max
(1 + ?)
(6)
log p r2{1,...,d 1}
d r
for arbitrarily small ? > 0.
? (Converse) In order to achieve Pe 6! 1 as p ! 1, it is necessary that
?
?
p
H(?)
n
(1 ?)
log p d 1
(7)
for arbitrarily small ? > 0.
Unfortunately, these bounds do not coincide. If the maximum in (6) is achieved by r = 1 (which
occurs, for example, when ? is uniform [3]), then the gap only amounts to a factor of two. However,
as we show in Figure 1, if we compute the bounds for some ?random? choices of ? then the gap is
typically larger (i.e., r = 1 does not achieve the maximum), and we can construct choices where the
gap is significantly larger. Closing these gaps was posed as a key open problem in [3].
We can now summarize our contributions as follows:
1. We give a lower bound that exactly matches (6), thus completely closing the above-mentioned
gaps in the existing bounds and solving the open problem raised in [3]. More specifically,
(r)
))
we show that Pe ! 1 whenever n ? logp p maxr2{1,...,d 1} 2(H(?)d H(?
(1 ?) for
r
some ? > 0, thus identifying an exact phase transition ? a threshold above which the error
probability vanishes, but below which the error probability tends to one.
2. We develop a framework for understanding variations of the problem consisting of random
noise, and give an example of a noise model where the scaling laws are strictly higher compared to the noiseless case. A summary is given in Table 2; the case SNR = (log p)?(1) reveals
a strict increase in the scaling laws even when the signal-to-noise ratio grows unbounded, and
the case SNR = ?(1) reveals that the required number of tests increases from sub-linear to
super-linear in the dimension when the signal-to-noise ratio is constant.
3. In the supplementary material, we discuss how our lower bounds extend readily to the approximate recovery criterion, where we only require to be identified up to a certain Hamming
distance. However, for clarity, we focus on exact recovery throughout the paper.
In a recent independent work [8], an adversarial noise setting was introduced. This turns out to
be fundamentally different to our noisy setting. In particular, the results of [8] state that exact recovery is impossible, and even with approximate recovery, a huge number of tests (i.e., higher than
1/2+o(1)
polynomial) is needed unless = O qmax
, where qmax is the maximum allowed reconstruction error measured by the Hamming distance, and is maximum adversarial noise amplitude. Of
course, both random and adversarial noise are of significant interest, depending on the application.
3
Notation. For a positive integer d, we write [d] = {1, . . . , d}. We use standard information-theoretic
notations for the (conditional) entropy and mutual information, e.g., H(X), H(Y |X), I(X; Y |Z)
[9]. All logarithms have base e, and accordingly, all of the preceding information measures are in
units of nats. The Gaussian distribution with mean ? and variance 2 is denoted by N(?, 2 ). We
use the standard asymptotic notations O(?), o(?), ?(?), !(?) and ?(?).
2
Main results
In this section, we present our main results for the noiseless and noisy settings. The proofs are given
in Section 3, as well as the supplementary material.
2.1
Phase transition in the noiseless setting
The following theorem proves that the upper bound given in (6) is tight. Recall that for r =
(r)
(r)
{1, . . . , d 1}, ? (r) = (?1 , . . . , ?r ) is a vector whose first entry sums the largest d r + 1
entries of ?, and whose remaining entries coincide with the remaining r 1 entries of ?.
Theorem 1. (Noiseless setting) Consider the pooled data problem described in Section 1.1 with a
given number of labels d and label proportion vector ? (not depending on the dimension p). For any
decoder, in order to achieve Pe 6! 1 as p ! 1, it is necessary that
?
?
p
2(H(?) H(? (r) ))
n
max
(1 ?)
(8)
log p r2{1,...,d 1}
d r
for arbitrarily small ? > 0.
Combined with (6), this result reveals an exact phase transition on the required number of measurer
))
ments: Denoting n? = logp p maxr2{1,...,d 1} 2(H(?)d H(?
, the error probability vanishes for
r
?
?
n n (1 + ?), tends to one for n ? n (1 ?), regardless of how small ? is chosen to be.
Remark 1. Our model assumes that is uniformly distributed over the sequences with empirical
distribution ?, whereas [3] assumes that is i.i.d. on ?. However, Theorem 1 readily extends to
the latter setting: Under the i.i.d. model, once we condition on a given empirical distribution, the
conditional distribution of is uniform. As a result, the converse bound for the i.i.d. model follows
directly from Theorem 1 by basic concentration and the continuity of the entropy function.
2.2
Information-theoretic framework for the noisy setting
We now turn to general noise models of the form (3), and provide necessary conditions for the noisy
pooled data problem in terms of the mutual information. General characterizations of this form were
provided previously for group testing [10, 11] and other sparse recovery problems [12, 13].
Our general result is stated in terms of a maximization over a vector parameter ` = (`1 , . . . , `d ) with
`t 2 {0, . . . , ?t p} for all t. We will see in the proof that `t represents the number of items of type t
that are unknown to the decoder after p?t `t are revealed by a genie. We define the following:
? Given ` and , we let S` be a random set of indices in [p] such that for each t 2 [d], the set
contains `t indices corresponding to entries where equals t. Specifically, we define S` to be
uniformly distributed over all such sets. Moreover, we define S`c = [p] \ S` .
? Given the above definitions, we define
?
j 2 S`c
j
c
(9)
S` =
? otherwise,
where ? can be thought of as representing an unknown value. Hence, knowing S`c amounts to
knowing the labels of all items in the set S`c .
? We define |B` (?)| to be the number of sequences 2 [d]p that coincide with a given S`c on
the entries not equaling ?, while also having empirical distribution ? overall. This number
does not depend on the specific choice of S`c . As an example, when `t = p?t for all t, we have
S` = [p], S`c = (?, . . . , ?), and |B` (?)| = |B(?)|, defined following (1)
? We let k`k0 denote the number of values in (`1 , . . . , `d ) that are positive.
4
With these definitions, we have the following result for general random noise models.
Theorem 2. (Noisy setting) Consider the pooled data problem described in Section 1.1 under a
general observation model of the form (3), with a given number of labels d and label proportion
vector ?. For any decoder, in order to achieve Pe ? for a given 2 (0, 1), it is necessary that
n
max
` : k`k0
log |B` (?)| (1
) log 2
Pn
.
(i)
| S`c , X (i) )
i=1 I( ; Y
(10)
1
2
n
In order to obtain more explicit bounds on n from (10), one needs to characterize the mutual information terms, ideally forming an upper bound that does not depend on the distribution of the
measurement matrix X. We do this for some specific models below; however, in general it can be a
difficult task. The following corollary reveals that if the entries of X are i.i.d. on Bernoulli(q) for
some q 2 (0, 1) (as was assumed in [3]), then we can simplify the bound.
Corollary 1. (Noisy setting with Bernoulli testing) Suppose that the entries of X are i.i.d. on
Bernoulli(q) for some q 2 (0, 1). Under the setup of Theorem 2, it is necessary that
n
max
` : k`k0 2
log |B` (?)| (1
) log 2
,
I(X0,` ; Y |X1,` )
(11)
where (X0,` , X1,` , Y ) are distributed as follows: (i) X0,` (respectively, X1,` ) is a concatenation
of the vectors X0,` (1), . . . , X0,` (d) (respectively, X1,` (1), . . . , X1,` (d)), the t-th of which contains
`t (respectively, ?t p `t ) entries independently drawn from Bernoulli(q); (ii) Letting each Nt
(t = 1, . . . , d) be the total number of ones in X0,` (t) and X1,` (t) combined, the random variable Y
is drawn from PY |N1 ,...,Nd according to (3).
As well as being simpler to evaluate, this corollary may be of interest in scenarios where one does
not have complete freedom in designing X, and one instead insists on using Bernoulli testing. For
instance, one may not know how to optimize X, and accordingly resort to generating it at random.
Example 1: Application to the noiseless setting. In the supplementary material, we show that in
the noiseless setting, Theorem 2 recovers a weakened version of Theorem 1 with 1 ? replaced by
1
o(1) in (8). Hence, while Theorem 2 does not establish a phase transition, it does recover
the exact threshold on the number of measurements required to obtain Pe ! 0.
An overview of the proof of this claim is as follows. We restrict the maximum in (10) to choices
of ` where each `t equals either its minimum value 0 or its maximum value p?t . Since we
are in the noiseless setting, each mutual information term reduces to the conditional entropy of
(i)
(i)
(i)
Y (i) = (Y1 , . . . , Yd ) given S`c and X (i) . For the values of t such that `t = 0, the value Yt is
(i)
deterministic (i.e., it has zero entropy), whereas for the values of t such that `t = p?t , the value Yt
follows a hypergeometric distribution, whose entropy behaves as 12 log p (1 + o(1)).
In the case that X is i.i.d. on Bernoulli(q), we can use Corollary 1 to obtain the following necessary
condition for Pe ? as as p ! 1, proved in the supplementary material:
?
?
p
2(H(?) H(? r ))
n
max
(1
o(1))
(12)
log(pq(1 q)) r2{1,...,d 1}
d r
for any q = q(p) such that both q and 1 q behave as ! p1 . Hence, while q = ?(1) recovers the
threshold in (8), the required number of tests strictly increases when q = o(1), albeit with a mild
logarithmic dependence.
Example 2: Group testing. To highlight the versatility of Theorem 2 and Corollary 1, we show
that the latter recovers the lower bounds given in the group testing framework of [11].
Set d = 2, and let label 1 represent ?defective? items, and label 2 represent ?non-defective? items.
Let PY |N1 N2 be of the form PY |N1 with Y 2 {0, 1}, meaning the observations are binary and
depend only on the number of defective items in the test. For brevity, let k = p?1 denote the total
number of defective items, so that p?2 = p k is the number of non-defective items.
Letting `2 = p
k in (11), and letting `1 remain arbitrary, we obtain the necessary condition
n
max
`1 2{1,...,k}
log
p k+`1
`1
(1
)
I(X0,`1 ; Y |X1,`1 )
5
log 2
,
(13)
where X0,`1 is a shorthand for X0,` with ` = (`1 , p k), and similarly for X1,`1 . This matches
the lower bound given in [11] for Bernoulli testing with general noise models, for which several
corollaries for specific models were also given.
Example 3: Gaussian noise. To give a concrete example of a noisy setting, consider the case that
we observe the values in (2), but with each such value corrupted by independent Gaussian noise:
(i)
Yt
(i)
= Nt ( , X (i) ) + Zt ,
(14)
(i)
Zt
where
? N(0, p 2 ) for some 2 > 0. Note that given X (i) , the values Nt themselves have
variance at most proportional to p (e.g., see Appendix C), so 2 = ?(1) can be thought of as the
constant signal-to-noise ratio (SNR) regime.
In the supplementary material, we prove the following bounds for this model:
? By letting each `t in (10) equal its minimum or maximum value analogously to the noiseless case
above, we obtain the following necessary condition for Pe ? as p ! 1:
?
?
pG H(?G )
P
n
max
(1
o(1)),
(15)
?t
1
G?[d] : |G| 2
t2G 2 log 1 + 4 2 )
P
where pG := t2G ?t p, and ?G has entries P 0 ?t ? 0 for t 2 G. Hence, we have the following:
t 2G
t
? In the case that 2 = p c for some c 2 (0, 1), each summand in the denominator simplifies
to 2c log p (1 + o(1)), and we deduce that compared to the noiseless case (cf., (8)), the
asymptotic number of tests increases by at least a constant factor of 1c .
? In the case that 2 = (log p) c for some c > 0, each summand in the denominator simplifies
to 2c log log p (1 + o(1)), and we deduce that compared to the noiseless case, the asymptotic
log p
number of tests increases by at least a factor of c log
log p . Hence, we observe a strict increase
in the scaling laws despite the fact that the SNR grows unbounded.
? While (15) also provides an ?(p) lower bound for the case 2 = ?(1), we can in fact do
better via a different choice of ` (see below).
? By letting `1 = p?1 , `2 = 1, and `t = 0 for t = 3, . . . , d, we obtain the necessary condition
n
4p 2 log p (1
o(1))
(16)
for Pe ? as p ! 1. Hence, if 2 = ?(1), we require n = ?(p log p); this is super-linear in
the dimension, in contrast with the sub-linear ? logp p behavior observed in the noiseless case.
Note that this choice of ` essentially captures the difficulty in identifying a single item, namely,
the one corresponding to `2 = 1.
These findings are summarized in Table 2; see also the supplementary material for extensions to the
approximate recovery setting.
Remark 2. While it may seem unusual to add continuous noise to discrete observations, this still
captures the essence of the noisy pooled data problem, and simplifies the evaluation of the mutual
information terms in (10). Moreover, this converse bound immediately implies the same bound for
the discrete model in which the noise consists of adding a Gaussian term, rounding, and clipping to
{0, . . . , p}, since the decoder could always choose to perform these operations as pre-processing.
3
Proofs
Here we provide the proof of Theorem 1, along with an overview of the proof of Theorem 2. The
remaining proofs are given in the supplementary material.
3.1
Proof of Theorem 1
Step 1: Counting typical outcomes. We claim that it suffices to consider the case that X is deterministic and ? is a deterministic function of Y; to see this, we note that when either of these are
random we have Pe = EX, ? [P [error]], and the average is lower bounded by the minimum.
(i)
The following lemma, proved in the supplementary material, shows
p that for any X , each entry of
(i)
the corresponding outcome Y lies in an interval of length O p log p with high probability.
6
Lemma 1. For any deterministic test vector X 2 {0, 1}p , and for uniformly distributed on B(?),
we have for each t 2 [d] that
h
i
p
2
P Nt ( , X) E[Nt ( , X)] > p log p ? 2 .
(17)
p
(i)
By Lemma 1 and the union bound, we have with probability at least 1 2nd
p2 that Nt ( , X )
p
(i)
E[Nt ( , X )] ? p log p for all i 2 [n] and t 2 [d]. Letting this event be denoted by A, we have
Pe
P[A \ no error]
P[A]
1
2nd
p2
P[A \ no error].
(18)
Next, letting Y( ) 2 [p]n?d denote Y explicitly as a function of and similarly for ?(Y) 2 [d]p ,
and letting YA denote the set of matrices Y under which the event A occurs, we have
X
1
P[A \ no error] =
1{Y(b) 2 YA \ ?(Y(b)) = b}
(19)
|B(?)|
b2B(?)
|YA |
?
,
|B(?)|
(20)
where (20) follows since each each Y 2 YA can only be counted once in the summation of (19),
due to the condition ?(Y(b)) = b.
Step 2: Bounding the set cardinalities. By a standard combinatorial argument (e.g., [14, Ch. 2])
and the fact that ? is fixed as p ! 1, we have
|B(?)| = ep(H(?)+o(1)) .
(21)
To bound |YA |, first note that the entries of each Y (i) 2 [p]d sum to a deterministic value, namely, the
number of ones in X (i) . Hence, each Y 2 YA is uniquely described by a sub-matrix of Y 2 [p]n?d
of size n ? (d 1). Moreover, since YA onlypincludes matrices under which A occurs, each value
in this sub-matrix only takes one of at most 2 p log p + 1 values. As a result, we have
p
n(d 1)
|YA | ? 2 p log p + 1
,
(22)
and combining (18)?(22) gives
p
n(d
2 p log p + 1
ep(H(?)+o(1))
Pe
1)
2nd
.
p2
(23)
pH(?)
p
Since d is constant, it immediately follows that Pe ! 1 whenever n ? (d 1) log(2
(1 ?)
p log p+1)
p
1
for some ? > 0. Applying log(2 p log p + 1) = 2 log p (1 + o(1)), we obtain the following
necessary condition for Pe 6! 1:
n
(d
2pH(?)
(1
1) log p
(24)
?).
This yields the term in (8) corresponding to r = 1.
Step 3: Genie argument. Let G be a subset of [d] of cardinality at least two, and define Gc = [d]\G.
Moreover, define Gc to be a length-p vector with
?
c
j
j 2G
( Gc ) j =
(25)
?
j 2 G,
where the symbol ? can be thought of as representing an unknown value. We consider a modified
setting in which a genie reveals Gc to the decoder, i.e., the decoder knows the labels of all items
for which the label lies in Gc , and is only left to estimate those in G. This additional knowledge
can only make the pooled data problem easier, and hence, any lower bound in this modified setting
remains valid in the original setting.
(i)
(i)
In the genie-aided setting, instead of receiving the full observation vector Y (i) = (Y1 , . . . , Yd ),
(i)
it is equivalent to only be given {Yj : j 2 G}, since the values in Gc are uniquely determined
7
from Gc and X (i) . This means that the genie-aided
Psetting can be cast in the original setting with
modified parameters: (i) p is replaced by pG =
t2G ?t p, the number of items with unknown
labels; (ii) d is replaced by |G|, the number of distinct remaining labels; (iii) ? is replaced by ?G ,
defined to be a |G|-dimensional probability vector with entries equaling P 0 ?t ? 0 (t 2 G).
t 2G
Due to this equivalence, the condition (24) yields the necessary condition n
and maximizing over all G with |G| 2 gives
2pG H(?G )
n
max
1 ? .
1) log p
G?[d] : |G| 2 (|G|
t
2pG H(?G )
(|G| 1) log p (1
?),
(26)
Step 4: Simplification. Define r = d |G| + 1. We restrict the maximum in (26) to sets G indexing
the highest |G| = d r + 1 values of ?, and consider the following process for sampling from ?:
? Draw a sample v from ? (r) (defined above Theorem 1);
? If v corresponds to the first entry of ? (r) , then draw a random sample from ?G and output it as
a label (i.e., the labels have conditional probability proportional to the top |G| entries of ?);
? Otherwise, if v corresponds to one of the other entries of ? (r) , then output v as a label.
By Shannon?s property ofP
entropy for sequentially-generated random variables
[15, p. 10], we find
P
that H(?) = H(? (r) ) +
?
H(?
).
Moreover,
since
p
=
p
?
?
,
this can be written
t
G
G
j
t2G
t2G
as pG H(?G ) = p H(?) H(? (r) ) . Substituting into (26), noting that |G| 1 = d r by the
definition of r, and maximizing over r = 1, . . . , d 1, we obtain the desired result (8).
3.2
Overview of proof of Theorem 2
We can interpret the pooled data problem as a communication problem in which a ?message?
(i)
(i)
is sent over a ?channel? PY |N1 ,...,Nd via ?codewords? of the form {(N1 , . . . , Nd )}ni=1 that are
constructed by summing various columns of X. As a result, it is natural to use Fano?s inequality [9,
Ch. 7] to lower bound the error probability in terms of information content (entropy) of and the
amount of information that Y reveals about (mutual information).
However, a naive application of Fano?s inequality only recovers the bound in (10) with ` = p?.
To handle the other possible choices of `, we again consider a genie-aided setting in which, for
each t 2 [d], the decoder is informed of p?t `t of the items whose label equals t. Hence, it only
remains to identify the remaining `t items of each type. This genie argument is a generalization of
that used in the proof of Theorem 1, in which each `t was either equal to its minimum value zero
or its maximum value p?t . In Example 3 of Section 2, we saw that this generalization can lead to a
strictly better lower bound in certain noisy scenarios.
The complete proof of Theorem 2 is given in the supplementary material.
4
Conclusion
We have provided novel information-theoretic lower bounds for the pooled data problem. In the
noiseless setting, we provided a matching lower bound to the upper bound of [3], establishing an
exact threshold indicating a phase transition between success and failure. In the noisy setting, we
provided a characterization of general noise models in terms of the mutual information. In the special
case of Gaussian noise, we proved an inherent added difficulty compared to the noiseless setting,
with strict increases in the scaling laws even when the signal-to-noise ratio grows unbounded.
An interesting direction for future research is to provide upper bounds for the noisy setting, potentially establishing the tightness of Theorem 2 for general noise models. This appears to be challenging using existing techniques; for instance, the pooled data problem bears similarity to group testing
with linear sparsity, whereas existing mutual information based upper bounds for group testing are
limited to the sub-linear regime [10, 11, 16]. In particular, the proofs of such bounds are based on
concentration inequalities which, when applied to the linear regime, lead to additional requirements
on the number of tests that prevent tight performance characterizations.
Acknowledgment: This work was supported in part by the European Commission under Grant
ERC Future Proof, SNF Sinergia project CRSII2-147633, SNF 200021-146750, and EPFL Fellows
Horizon2020 grant 665667.
8
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9
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6,237 | 6,642 | Universal Style Transfer via Feature Transforms
Yijun Li
UC Merced
[email protected]
Zhaowen Wang
Adobe Research
[email protected]
Chen Fang
Adobe Research
[email protected]
Xin Lu
Adobe Research
[email protected]
Jimei Yang
Adobe Research
[email protected]
Ming-Hsuan Yang
UC Merced, NVIDIA Research
[email protected]
Abstract
Universal style transfer aims to transfer arbitrary visual styles to content images.
Existing feed-forward based methods, while enjoying the inference efficiency, are
mainly limited by inability of generalizing to unseen styles or compromised visual
quality. In this paper, we present a simple yet effective method that tackles these
limitations without training on any pre-defined styles. The key ingredient of our
method is a pair of feature transforms, whitening and coloring, that are embedded
to an image reconstruction network. The whitening and coloring transforms reflect
a direct matching of feature covariance of the content image to a given style
image, which shares similar spirits with the optimization of Gram matrix based
cost in neural style transfer. We demonstrate the effectiveness of our algorithm by
generating high-quality stylized images with comparisons to a number of recent
methods. We also analyze our method by visualizing the whitened features and
synthesizing textures via simple feature coloring.
1
Introduction
Style transfer is an important image editing task which enables the creation of new artistic works.
Given a pair of examples, i.e., the content and style image, it aims to synthesize an image that
preserves some notion of the content but carries characteristics of the style. The key challenge is how
to extract effective representations of the style and then match it in the content image. The seminal
work by Gatys et al. [8, 9] show that the correlation between features, i.e., Gram matrix or covariance
matrix (shown to be as effective as Gram matrix in [20]), extracted by a trained deep neural network
has remarkable ability of capturing visual styles. Since then, significant efforts have been made to
synthesize stylized images by minimizing Gram/covariance matrices based loss functions, through
either iterative optimization [9] or trained feed-forward networks [27, 16, 20, 2, 6]. Despite the recent
rapid progress, these existing works often trade off between generalization, quality and efficiency,
which means that optimization-based methods can handle arbitrary styles with pleasing visual quality
but at the expense of high computational costs, while feed-forward approaches can be executed
efficiently but are limited to a fixed number of styles or compromised visual quality.
By far, the problem of universal style transfer remains a daunting task as it is challenging to develop
neural networks that achieve generalization, quality and efficiency at the same time. The main issue
is how to properly and effectively apply the extracted style characteristics (feature correlations) to
content images in a style-agnostic manner.
In this work, we propose a simple yet effective method for universal style transfer, which enjoys
the style-agnostic generalization ability with marginally compromised visual quality and execution
efficiency. The transfer task is formulated as image reconstruction processes, with the content features
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
C
S
C
VGG
Relu_5_1
S
VGG
Relu_X_1
VGG
Relu_X_1
Whitening & Coloring
Transform (WCT)
Recons
DecoderX
VGG
Relu_4_1
S
VGG
Relu_3_1
S
Recons
DecoderX
VGG
Relu_2_1
S
Output
S
(a) Reconstruction (b) Single-level stylization
VGG
Relu_1_1
Recons
Decoder5
WCT
WCT
WCT
WCT
WCT
I5
Recons
Decoder4
I4
Recons
Decoder3
Recons
Decoder2
Recons
Decoder1
I3
I2
Output I1
(c) Multi-level stylization
Figure 1: Universal style transfer pipeline. (a) We first pre-train five decoder networks DecoderX
(X=1,2,...,5) through image reconstruction to invert different levels of VGG features. (b) With both
VGG and DecoderX fixed, and given the content image C and style image S, our method performs
the style transfer through whitening and coloring transforms. (c) We extend single-level to multi-level
stylization in order to match the statistics of the style at all levels. The result obtained by matching
higher level statistics of the style is treated as the new content to continue to match lower-level
information of the style.
being transformed at intermediate layers with regard to the statistics of the style features, in the
midst of feed-forward passes. In each intermediate layer, our main goal is to transform the extracted
content features such that they exhibit the same statistical characteristics as the style features of the
same layer and we found that the classic signal whitening and coloring transforms (WCTs) on those
features are able to achieve this goal in an almost effortless manner.
In this work, we first employ the VGG-19 network [26] as the feature extractor (encoder), and train a
symmetric decoder to invert the VGG-19 features to the original image, which is essentially the image
reconstruction task (Figure 1(a)). Once trained, both the encoder and the decoder are fixed through all
the experiments. To perform style transfer, we apply WCT to one layer of content features such that
its covariance matrix matches that of style features, as shown in Figure 1(b). The transformed features
are then fed forward into the downstream decoder layers to obtain the stylized image. In addition
to this single-level stylization, we further develop a multi-level stylization pipeline, as depicted in
Figure 1(c), where we apply WCT sequentially to multiple feature layers. The multi-level algorithm
generates stylized images with greater visual quality, which are comparable or even better with much
less computational costs. We also introduce a control parameter that defines the degree of style
transfer so that the users can choose the balance between stylization and content preservation. The
entire procedure of our algorithm only requires learning the image reconstruction decoder with no
style images involved. So when given a new style, we simply need to extract its feature covariance
matrices and apply them to the content features via WCT. Note that this learning-free scheme is
fundamentally different from existing feed-forward networks that require learning with pre-defined
styles and fine-tuning for new styles. Therefore, our approach is able to achieve style transfer
universally.
The main contributions of this work are summarized as follows:
? We propose to use feature transforms, i.e., whitening and coloring, to directly match content
feature statistics to those of a style image in the deep feature space.
? We couple the feature transforms with a pre-trained general encoder-decoder network, such
that the transferring process can be implemented by simple feed-forward operations.
? We demonstrate the effectiveness of our method for universal style transfer with high-quality
visual results, and also show its application to universal texture synthesis.
2
Related Work
Existing style transfer methods are mostly example-based [13, 25, 24, 7, 21]. The image analogy
method [13] aims to determine the relationship between a pair of images and then apply it to stylize
2
other images. As it is based on finding dense correspondence, analogy-based approaches [25, 24, 7,
21] often require that a pair of image depicts the same type of scene. Therefore these methods do not
scale to the setting of arbitrary style images well.
Recently, Gatys et al. [8, 9] proposed an algorithm for arbitrary stylization based on matching the
correlations (Gram matrix) between deep features extracted by a trained network classifier within an
iterative optimization framework. Numerous methods have since been developed to address different
aspects including speed [27, 19, 16], quality [28, 18, 32, 31], user control [10], diversity [29, 20],
semantics understanding [7, 1] and photorealism [23]. It is worth mentioning that one of the
major drawbacks of [8, 9] is the inefficiency due to the optimization process. The improvement of
efficiency in [27, 19, 16] is realized by formulating the stylization as learning a feed-forward image
transformation network. However, these methods are limited by the requirement of training one
network per style due to the lack of generalization in network design.
Most recently, a number of methods have been proposed to empower a single network to transfer
multiple styles, including a model that conditioned on binary selection units [20], a network that
learns a set of new filters for every new style [2], and a novel conditional normalization layer that
learns normalization parameters for each style [6]. To achieve arbitrary style transfer, Chen et al. [3]
first propose to swap the content feature with the closest style feature locally. Meanwhile, inspired
by [6], two following work [30, 11] turn to learn a general mapping from the style image to style
parameters. One closest related work [15] directly adjusts the content feature to match the mean and
variance of the style feature. However, the generalization ability of the learned models on unseen
styles is still limited.
Different from the existing methods, our approach performs style transfer efficiently in a feed-forward
manner while achieving generalization and visual quality on arbitrary styles. Our approach is closely
related to [15], where content feature in a particular (higher) layer is adaptively instance normalized
by the mean and variance of style feature. This step can be viewed as a sub-optimal approximation
of the WCT operation, thereby leading to less effective results on both training and unseen styles.
Moreover, our encoder-decoder network is trained solely based on image reconstruction, while [15]
requires learning such a module particularly for stylization task. We evaluate the proposed algorithm
with existing approaches extensively on both style transfer and texture synthesis tasks and present
in-depth analysis.
3
Proposed Algorithm
We formulate style transfer as an image reconstruction process coupled with feature transformation,
i.e., whitening and coloring. The reconstruction part is responsible for inverting features back to the
RGB space and the feature transformation matches the statistics of a content image to a style image.
3.1
Reconstruction decoder
We construct an auto-encoder network for general image reconstruction. We employ the VGG-19 [26]
as the encoder, fix it and train a decoder network simply for inverting VGG features to the original
image, as shown in Figure 1(a). The decoder is designed as being symmetrical to that of VGG-19
network (up to Relu_X_1 layer), with the nearest neighbor upsampling layer used for enlarging
feature maps. To evaluate with features extracted at different layers, we select feature maps at five
layers of the VGG-19, i.e., Relu_X_1 (X=1,2,3,4,5), and train five decoders accordingly. The pixel
reconstruction loss [5] and feature loss [16, 5] are employed for reconstructing an input image,
L = kIo ? Ii k22 + ?k?(Io ) ? ?(Ii )k22 ,
(1)
where Ii , Io are the input image and reconstruction output, and ? is the VGG encoder that extracts
the Relu_X_1 features. In addition, ? is the weight to balance the two losses. After training, the
decoder is fixed (i.e., will not be fine-tuned) and used as a feature inverter.
3.2
Whitening and coloring transforms
Given a pair of content image Ic and style image Is , we first extract their vectorized VGG feature
maps fc ? <C?Hc Wc and fs ? <C?Hs Ws at a certain layer (e.g., Relu_4_1), where Hc , Wc (Hs ,
3
Figure 2: Inverting whitened features. We invert the whitened VGG Relu_4_1 feature as an example.
Left: original images, Right: inverted results (pixel intensities are rescaled for better visualization).
The whitened features still maintain global content structures.
(a) Style (b) Content
(c) HM
(d) WCT
(e) Style
(f) Content
(g) HM
(h) WCT
Figure 3: Comparisons between different feature transform strategies. Results are obtained by our
multi-level stylization framework in order to match all levels of information of the style.
Ws ) are the height and width of the content (style) feature, and C is the number of channels. The
decoder will reconstruct the original image Ic if fc is directly fed into it. We next propose to use a
whitening and coloring transform to adjust fc with respect to the statistics of fs . The goal of WCT
is to directly transform the fc to match the covariance matrix of fs . It consists of two steps, i.e.,
whitening and coloring transform.
Whitening transform. Before whitening, we first center fc by subtracting its mean vector mc .
Then we transform fc linearly as in (2) so that we obtain f?c such that the feature maps are uncorrelated
>
(f?c f?c = I),
1
?
f?c = Ec Dc 2 Ec> fc ,
(2)
where Dc is a diagonal matrix with the eigenvalues of the covariance matrix fc fc> ? <C?C , and Ec
is the corresponding orthogonal matrix of eigenvectors, satisfying fc fc> = Ec Dc Ec> .
To validate what is encoded in the whitened feature f?c , we invert it to the RGB space with our
previous decoder trained for reconstruction only. Figure 2 shows two visualization examples, which
indicate that the whitened features still maintain global structures of the image contents, but greatly
help remove other information related to styles. We note especially that, for the Starry_night example
on right, the detailed stroke patterns across the original image are gone. In other words, the whitening
step helps peel off the style from an input image while preserving the global content structure. The
outcome of this operation is ready to be transformed with the target style.
Coloring transform. We first center fs by subtracting its mean vector ms , and then carry out
the coloring transform [14], which is essentially the inverse of the whitening step to transform f?c
linearly as in (3) such that we obtain f?cs which has the desired correlations between its feature maps
>
(f?cs f?cs = fs f > ),
s
1
f?cs = Es Ds2 Es> f?c ,
(3)
fs fs>
C?C
where Ds is a diagonal matrix with the eigenvalues of the covariance matrix
?<
, and Es
is the corresponding orthogonal matrix of eigenvectors. Finally we re-center the f?cs with the mean
vector ms of the style, i.e., f?cs = f?cs + ms .
To demonstrate the effectiveness of WCT, we compare it with a commonly used feature adjustment
technique, i.e., histogram matching (HM), in Figure 3. The channel-wise histogram matching [12]
method determines a mapping function such that the mapped fc has the same cumulative histogram as
fs . In Figure 3, it is clear that the HM method helps transfer the global color of the style image well
4
(a) Style
(b) Relu_1_1
(c) Relu_2_1
(d) Relu_3_1
(e) Relu_4_1
(f) Relu_5_1
Figure 4: Single-level stylization using different VGG features. The content image is from Figure 2.
(a) I5
(b) I4
(c) I1
(d) Fine-to-coarse
Figure 5: (a)-(c) Intermediate results of our coarse-to-fine multi-level stylization framework in
Figure 1(c). The style and content images are from Figure 4. I1 is the final output of our multi-level
pipeline. (d) Reversed fine-to-coarse multi-level pipeline.
but fails to capture salient visual patterns, e.g., patterns are broken into pieces and local structures are
misrepresented. In contrast, our WCT captures patterns that reflect the style image better. This can
be explained by that the HM method does not consider the correlations between features channels,
which are exactly what the covariance matrix is designed for.
After the WCT, we may blend f?cs with the content feature fc as in (4) before feeding it to the decoder
in order to provide user controls on the strength of stylization effects:
f?cs = ? f?cs + (1 ? ?) fc ,
(4)
where ? serves as the style weight for users to control the transfer effect.
3.3
Multi-level coarse-to-fine stylization
Based on the single-level stylization framework shown in Figure 1(b), we use different layers of VGG
features Relu_X_1 (X=1,2,...,5) and show the corresponding stylized results in Figure 4. It clearly
shows that the higher layer features capture more complicated local structures, while lower layer
features carry more low-level information (e.g., colors). This can be explained by the increasing size
of receptive field and feature complexity in the network hierarchy. Therefore, it is advantageous to
use features at all five layers to fully capture the characteristics of a style from low to high levels.
Figure 1(c) shows our multi-level stylization pipeline. We start by applying the WCT on Relu_5_1
features to obtain a coarse stylized result and regard it as the new content image to further adjust
features in lower layers. An example of intermediate results are shown in Figure 5. We show the
intermediate results I5 , I4 , I1 with obvious differences, which indicates that the higher layer features
first capture salient patterns of the style and lower layer features further improve details. If we reverse
feature processing order (i.e., fine-to-coarse layers) by starting with Relu_1_1, low-level information
cannot be preserved after manipulating higher level features, as shown in Figure 5(d).
4
4.1
Experimental Results
Decoder training
For the multi-level stylization approach, we separately train five reconstruction decoders for features
at the VGG-19 Relu_X_1 (X=1,2,...,5) layer. It is trained on the Microsoft COCO dataset [22] and
the weight ? to balance the two losses in (1) is set as 1.
5
(a) Style
(b) [3]
(c) [15]
(d) [27]
(e) [9]
(f) Ours
Figure 6: Results from different style transfer methods. The content images are from Figure 2-3. We
evaluate various styles including paintings, abstract styles, and styles with obvious texton elements.
We adjust the style weight of each method to obtain the best stylized effect. For our results, we set
the style weight ? = 0.6.
Table 1: Differences between our approach and other methods.
Arbitrary
Efficient
Learning-free
4.2
Chen et al. [3]
?
?
Huang et al. [15]
?
?
TNet [27]
?
?
?
?
?
DeepArt [9]
?
?
?
Ours
?
?
?
Style transfer
To demonstrate the effectiveness of the proposed algorithm, we list the differences with existing
methods in Table 1 and present stylized results in Figure 6. We adjust the style weight of other
methods to obtain the best stylized effect. The optimization-based work of [9] handles arbitrary
styles but is likely to encounter unexpected local minima issues (e.g., 5th and 6th row of Figure 6(e)).
Although the method [27] greatly improves the stylization speed, it trades off quality and generality
for efficiency, which generates repetitive patterns that overlay with the image contents (Figure 6(d)).
6
Table 2: Quantitative comparisons between different stylization methods in terms of the covariance
matrix difference (Ls ), user preference and run-time, tested on images of size 256 ? 256 and a 12GB
TITAN X.
log(Ls )
Preference/%
Time/sec
Style
Chen et al. [3]
Huang et al. [15]
TNet [27]
Gatys et al. [9]
Ours
7.4
15.7
2.1
7.0
24.9
0.20
6.8
12.7
0.18
6.7
16.4
21.2
6.3
30.3
0.83
scale = 256
scale = 768
? = 0.4
? = 0.6
? = 1.0
Figure 7: Controlling the stylization on the scale and weight.
Closest to our work on generalization are the recent methods [3, 15], but the quality of the stylized
results are less appealing. The work of [3] replaces the content feature with the most similar style
feature based on patch similarity and hence has limited capability, i.e., the content is strictly preserved
while style is not well reflected with only low-level information (e.g., colors) transferred, as shown
in Figure 6(b). In [15], the content feature is simply adjusted to have the same mean and variance
with the style feature, which is not effective in capturing high-level representations of the style. Even
learned with a set of training styles, it does not generalize well on unseen styles. Results in Figure 6(c)
indicate that the method in [15] is not effective at capturing and synthesizing salient style patterns,
especially for complicated styles where there are rich local structures and non-smooth regions.
Figure 6(f) shows the stylized results of our approach. Without learning any style, our method is able
to capture visually salient patterns in style images (e.g., the brick wall on the 6th row). Moreover,
key components in the content images (e.g., bridge, eye, mouth) are also well stylized in our results,
while other methods only transfer patterns to relatively smooth regions (e.g., sky, face). The models
and code are available at https://github.com/Yijunmaverick/UniversalStyleTransfer.
In addition, we quantitatively evaluate different methods by computing the covariance matrix difference (Ls ) on all five levels of VGG features between stylized results and the given style image. We
randomly select 10 content images from [22] and 40 style images from [17], compute the averaged
difference over all styles, and show the results in Table 2 (1st row). Quantitative results show that we
generate stylized results with lower Ls , i.e., closer to the statistics of the style.
User study. Evaluating artistic style transfer has been an open question in the community. Since
the qualitative assessment is highly subjective, we conduct a user study to evaluate 5 methods shown
in Figure 6. We use 5 content images and 30 style images, and generate 150 results based on each
content/style pair for each method. We randomly select 15 style images for each subject to evaluate.
We display stylized images by 5 compared methods side-by-side on a webpage in random order. Each
subject is asked to vote his/her ONE favorite result for each style. We finally collect the feedback
from 80 subjects of totally 1,200 votes and show the percentage of the votes each method received in
Table 2 (2nd row). The study shows that our method receives the most votes for better stylized results.
It can be an interesting direction to develop evaluation metrics based on human visual perception for
general image synthesis problems.
Efficiency. In Table 2 (3rd row), we also compare our approach with other methods in terms of
efficiency. The method by Gatys et al. [9] is slow due to loops of optimization and usually requires at
least 500 iterations to generate good results. The methods [27] and [15] are efficient as the scheme is
based on one feed-forward pass with a trained network. The approach [3] is feed-forward based but
relatively slower as the feature swapping operation needs to be carried out for thousands of patches.
Our approach is also efficient but a little bit slower than [27, 15] because we have a eigenvalue
decomposition step in WCT. But note that the computational cost on this step will not increase along
with the image size because the the dimension of covariance matrix only depends on filter numbers (or
7
(a) Content
(b) Different masks and styles
(c) Our result
Figure 8: Spatial control in transferring, which enables users to edit the content with different styles.
Figure 9: Texture synthesis. In each panel, Left: original textures, Right: our synthesized results.
Texture images are mostly from the Describable Textures Dataset (DTD) [4].
channels), which is at most 512 (Relu_5_1). Currently the decomposition step is implemented based
on CPU. Our future work includes more efficient GPU implementations of the proposed algorithm.
User Controls. Given a content/style pair, our approach is not only as simple as a one-click
transferring, but also flexible enough to accommodate different requirements from users by providing
different controls on the stylization, including the scale, weight and spatial control. The style input
on different scales will lead to different extracted statistics due to the fixed receptive field of the
network. Therefore the scale control is easily achieved by adjusting the style image size. In the
middle of Figure 7, we show two examples where the brick can be transferred in either small or
large scale. The weight control refers to controlling the balance between stylization and content
preservation. As shown on right of Figure 7, our method enjoys this flexibility in simple feed-forward
passes by simply adjusting the style weight ? in (4). However in [9] and [27], to obtain visual results
of different weight settings, a new round of time-consuming optimization or model training is needed.
Moreover, our blending directly works on deep feature space before inversion/reconstruction, which
is fundamentally different from [9, 27] where the blending is formulated as the weighted sum of the
content and style losses that may not always lead to a good balance point.
The spatial control is also highly desired when users want to edit an image with different styles
transferred on different parts of the image. Figure 8 shows an example of spatially controlling the
stylization. A set of masks M (Figure 8(b)) is additionally required as input to indicate the spatial
correspondence between content regions and styles. By replacing the content feature fc in (3) with
M fc where is a simple mask-out operation, we are able to stylize the specified region only.
4.3
Texture synthesis
By setting the content image as a random noise image (e.g., Gaussian noise), our stylization framework
can be easily applied to texture synthesis. An alternative is to directly initialize the f?c in (3) to be white
noise. Both approaches achieve similar results. Figure 9 shows a few examples of the synthesized
textures. We empirically find that it is better to run the multi-level pipeline for a few times (e.g., 3) to
get more visually pleasing results.
Our method is also able to synthesize the interpolated result of two textures. Given two texture
?
examples s1 and s2 , we first perform the WCT on the input noise and get transformed features fcs
1
?
?
?
?
and fcs2 respectively. Then we blend these two features fcs = ? fcs1 + (1 ? ?)fcs2 and feed the
8
Texture s1
Texture s2
? = 0.75
? = 0.5
? = 0.25
? = 0.5
Figure 10: Interpolation between two texture examples. Left: original textures, Middle: our interpolation results, Right: interpolated results of [9]. ? controls the weight of interpolation.
Texture
TNet [27]
Ours
Figure 11: Comparisons of diverse synthesized results between TNet [27] and our model.
combined feature into the decoder to generate mixed effects. Note that our interpolation directly
works on deep feature space. By contrast, the method in [9] generates the interpolation by matching
the weighted sum of Gram matrices of two textures at the loss end. Figure 10 shows that the result
by [9] is simply overlaid by two textures while our method generates new textural effects, e.g., bricks
in the stripe shape.
One important aspect in texture synthesis is diversity. By sampling different noise images, our
method can generate diverse synthesized results for each texture. While [27] can generate different
results driven by the input noise, the learned networks are very likely to be trapped in local optima. In
other words, the noise is marginalized out and thus fails to drive the network to generate large visual
variations. In contrast, our approach explains each input noise better because the network is unlikely
to absorb the variations in input noise since it is never trained for learning textures. We compare the
diverse outputs of our model with [27] in Figure 11. Note that the common diagonal layout is shared
across different results of [27], which causes unsatisfying visual experiences. The comparison shows
that our method achieves diversity in a more natural and flexible manner.
5
Concluding Remarks
In this work, we propose a universal style transfer algorithm that does not require learning for each
individual style. By unfolding the image generation process via training an auto-encoder for image
reconstruction, we integrate the whitening and coloring transforms in the feed-forward passes to
match the statistical distributions and correlations between the intermediate features of content and
style. We also present a multi-level stylization pipeline, which takes all level of information of a
style into account, for improved results. In addition, the proposed approach is shown to be equally
effective for texture synthesis. Experimental results demonstrate that the proposed algorithm achieves
favorable performance against the state-of-the-art methods in generalizing to arbitrary styles.
Acknowledgments
This work is supported in part by the NSF CAREER Grant #1149783, gifts from Adobe and NVIDIA.
9
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11
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6,238 | 6,643 | On the Model Shrinkage Effect of
Gamma Process Edge Partition Models
Iku Ohama??
Issei Sato?
Takuya Kida?
Hiroki Arimura?
?
?
?
Panasonic Corp., Japan The Univ. of Tokyo, Japan Hokkaido Univ., Japan
[email protected] [email protected] {kida,arim}@ist.hokudai.ac.jp
Abstract
The edge partition model (EPM) is a fundamental Bayesian nonparametric model for extracting an overlapping structure from binary matrix. The
EPM adopts a gamma process (?P) prior to automatically shrink the number of active atoms. However, we empirically found that the model shrinkage of the EPM does not typically work appropriately and leads to an
overfitted solution. An analysis of the expectation of the EPM?s intensity
function suggested that the gamma priors for the EPM hyperparameters
disturb the model shrinkage effect of the internal ?P. In order to ensure that
the model shrinkage effect of the EPM works in an appropriate manner, we
proposed two novel generative constructions of the EPM: CEPM incorporating constrained gamma priors, and DEPM incorporating Dirichlet priors
instead of the gamma priors. Furthermore, all DEPM?s model parameters
including the infinite atoms of the ?P prior could be marginalized out, and
thus it was possible to derive a truly infinite DEPM (IDEPM) that can
be efficiently inferred using a collapsed Gibbs sampler. We experimentally
confirmed that the model shrinkage of the proposed models works well and
that the IDEPM indicated state-of-the-art performance in generalization
ability, link prediction accuracy, mixing efficiency, and convergence speed.
1
Introduction
Discovering low-dimensional structure from a binary matrix is an important problem in
relational data analysis. Bayesian nonparametric priors, such as Dirichlet process (DP) [1]
and hierarchical Dirichlet process (HDP) [2], have been widely applied to construct statistical
models with an automatic model shrinkage effect [3, 4]. Recently, more advanced stochastic
processes such as the Indian buffet process (IBP) [5] enabled the construction of statistical
models for discovering overlapping structures [6, 7], wherein each individual in a data matrix
can belong to multiple latent classes.
Among these models, the edge partition model (EPM) [8] is a fundamental Bayesian nonparametric model for extracting overlapping latent structure underlying a given binary matrix.
The EPM considers latent positive random counts for only non-zero entries in a given binary
matrix and factorizes the count matrix into two non-negative matrices and a non-negative
diagonal matrix. A link probability of the EPM for an entry is defined by transforming
the multiplication of the non-negative matrices into a probability, and thus the EPM can
capture overlapping structures with a noisy-OR manner [6]. By incorporating a gamma
process (?P) as a prior for the diagonal matrix, the number of active atoms of the EPM
shrinks automatically according to the given data. Furthermore, by truncating the infinite
atoms of the ?P with a finite number, all parameters and hyperparameters of the EPM
can be inferred using closed-form Gibbs sampler. Although, the EPM is well designed to
capture an overlapping structure and has an attractive affinity with a closed-form posterior
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
The proposed
IDEPM
successfully
found expected
5 overlapped
latent classes.
The EPM
extracted many
unexpected
latent classes.
(98 active
classes)
(a) Synthetic data
(b) EPM solution
(c) Proposed IDEPM solution
Figure 1: (Best viewed in color.) A synthetic example: (a) synthetic 90 ? 90 data (white
corresponds to one, and black to zero); (b) EPM solution; and (c) the proposed IDEPM solution. In (b) and (c), non-zero entries are colored to indicate their most probable assignment
to the latent classes.
inference, the EPM involves a critical drawback in its model shrinkage mechanism. As we
experimentally show in Sec. 5, we found that the model shrinkage effect of the EPM does
not typically work in an appropriate manner. Figure 1 shows a synthetic example. As shown
in Fig. 1a, there are five overlapping latent classes (white blocks). However, as shown in
Fig. 1b, the EPM overestimates the number of active atoms (classes) and overfits the data.
In this paper, we analyze the undesired property of the EPM?s model shrinkage mechanism
and propose novel generative constructions for the EPM to overcome the aforementioned
disadvantage. As shown in Fig. 1c, the IDEPM proposed in this paper successfully shrinks
unnecessary atoms. More specifically, we have three major contributions in this paper.
(1) We analyse the generative construction of the EPM and find a property that disturbs its
model shrinkage effect (Sec. 3). We derive the expectation of the EPM?s intensity function
(Theorem 1), which is the total sum of the infinite atoms for an entry. From the derived
expectation, we obtain a new finding that gamma priors for the EPM?s hyperparameters
disturb the model shrinkage effect of the internal ?P (Theorem 2). That is, the derived
expectation is expressed by a multiplication of the terms related to ?P and other gamma
priors. Thus, there is no guarantee that the expected number of active atoms is finite.
(2) Based on the analysis of the EPM?s intensity function, we propose two novel constructions of the EPM: the CEPM incorporating constrained gamma priors (Sec. 4.1) and the
DEPM incorporating Dirichlet priors instead of the gamma priors (Sec. 4.2). The model
shrinkage effect of the CEPM and DEPM works appropriately because the expectation of
their intensity functions depends only on the ?P prior (Sec. 4.1 and Theorem 3 in Sec. 4.2).
(3) Furthermore, for the DEPM, all model parameters, including the infinite atoms of the
?P prior, can be marginalized out (Theorem 4). Therefore, we can derive a truly infinite
DEPM (IDEPM), which has a closed-form marginal likelihood without truncating infinite
atoms, and can be efficiently inferred using collapsed Gibbs sampler [9] (Sec. 4.3).
2
The Edge Partition Model (EPM)
In this section, we review the EPM [8] as a baseline model. Let x be an I ? J binary matrix,
where an entry between i-th row and j-th column is represented by xi,j ? {0, 1}. In order to
extract an overlapping structure underlying x, the EPM [8] considers a non-negative matrix
factorization problem on latent Poisson counts as follows:
!
K
X
(1)
xi,j = I(mi,j,? ? 1), mi,j,? | U , V , ? ? Poisson
Ui,k Vj,k ?k ,
k=1
where U and V are I ? K and J ? K non-negative matrices, respectively, and ? is a K ? K
non-negative diagonal matrix. Note that I(?) is 1 if the predicate holds and is zero otherwise.
The latent counts m take positive values only for edges (non-zero entries) within a given
binary matrix and the generative model for each positive
count is equivalently expressed as
P
a sum of K Poisson random variables as mi,j,? = k mi,j,k , mi,j,k ? Poisson(Ui,k Vj,k ?k ).
This is the reason why the above model is called edge partition model. Marginalizing
m out from Eq. (1), the generative model of the EPM can be equivalently rewritten as
2
Q
xi,j | U , V , ? ? Bernoulli(1 ? k e?Ui,k Vj,k ?k ). As e?Ui,k Vj,k ?k ? [0, 1] denotes the probability that a Poisson random variable with mean Ui,k Vj,k ?k corresponds to zero, the EPM can
capture an overlapping structure with a noisy-OR manner [6].
In order to complete the Bayesian hierarchical model of the EPM, gamma priors are adopted
as Ui,k ? Gamma(a1 , b1 ) and Vj,k ? Gamma(a2 , b2 ), where a1 , a2 are shape parameters and
b1 , b2 are rate parameters for the gamma distribution, respectively. Furthermore, a gamma
process (?P) is incorporated as a Bayesian nonparametric prior for ? to make the EPM
automatically shrink its number of atoms K. Let Gamma(?0 /T, c0 ) denote a truncated ?P
with a concentration parameter ?0 and a rate parameter c0 , where T denotes a truncation
level that should be set large enough to ensure a good approximation to the true ?P. Then,
the diagonal elements of ? are drawn as ?k ? Gamma(?0 /T, c0 ) for k ? {1, . . . , T }.
The posterior inference for all parameters and hyperparameters of the EPM can be performed using Gibbs sampler (detailed in Appendix
conjugacy between
P A). Thanks to the P
gamma and Poisson distributions, given mi,?,k = j mi,j,k and m?,j,k = i mi,j,k , posterior
sampling for Ui,k and Vj,k is straightforward. As the ?P prior is approximated by a gamma
distribution, posterior sampling for ?k also can be performed straightforwardly. Given U ,
V , and ?, posterior sample for mi,j,? can be simulated using zero-truncated Poisson (ZTP)
distribution [10]. Finally, we can obtain sufficient statistics mi,j,k by partitioning mi,j,? into
T atoms using a multinomial distribution. Furthermore, all hyperparameters of the EPM
(i.e., ?0 , c0 , a1 , a2 , b1 , and b2 ) can also be sampled by assuming a gamma hyper prior
Gamma(e0 , f0 ). Thanks to the conjugacy between gamma distributions, posterior sampling
for c0 , b1 , and b2 is straightforward. For the remaining hyperparameters, we can construct
closed-form Gibbs samplers using data augmentation techniques [11, 12, 2].
3
Analysis for Model Shrinkage Mechanism
The EPM is well designed to capture an overlapping structure with a simple Gibbs inference.
However, the EPM involves a critical drawback in its model shrinkage mechanism.
For the EPM, a ?P prior is incorporated as a prior for the non-negative diagonal matrix as
?k ? Gamma(?0 /T, c0 ). From the form of the truncated ?P, thanks to the additive property
of independent gamma random P
variables, the total sum of ?k over countably infinite atoms
?
follows a gamma distribution as k=1P?k ? Gamma(?0 , c0 ), wherein the intensity function of
?
the ?P has a finite expectation as E[ k=1 ?k ] = ?c00 . Therefore, the ?P has a regularization
mechanism that automatically shrinks the number of atoms according to given observations.
However, as experimentally shown in Sec. 5, the model shrinkage mechanism of the EPM
does not work appropriately. More specifically, the EPM often overestimates the number of
active atoms and overfits the data. Thus, we analyse the intensity function of the EPM to
reveal the reason why the model shrinkage mechanism does not work appropriately.
P?
Theorem 1. The expectation of the EPM?s intensity function k=1 Ui,k Vj,k ?k for an entry
(i, j) is finite and can be expressed as follows:
#
"?
X
a2
?0
a1
?
? .
(2)
E
Ui,k Vj,k ?k =
b1
b2
c0
k=1
Proof. As U , V , and ? are independent of each other, the expected value operator is multiplicative for the EPM?s intensity function.PUsing the multiplicativity
and the low of total
P?
?
expectation, P
the proof is completed as E [ k=1 Ui,k Vj,k ?k ] = k=1 E[Ui,k ]E[Vj,k ]E[?k ] =
?
a1
a2
k=1 ?k ].
b1 ? b2 ? E[
As Eq. (2) in Theorem 1 shows, the expectation of the EPM?s intensity function is expressed
by multiplying individual expectations of a ?P and two gamma distributions. This causes
an undesirable property to the model shrinkage effect of the EPM. From Theorem 1, another
important theorem about the EPM?s model shrinkage effect is obtained as follows:
3
Theorem 2. Given an arbitrary non-negative constant
P? C, even if the expectation of the
EPM?s intensity function in Eq. (2) is fixed as E [ k=1 Ui,k Vj,k ?k ] = C, there exist cases
in which the model shrinkage effect of the ?P prior disappears.
P?
Proof. Substituting E [ k=1 Ui,k Vj,k ?k ] = C for Eq. (2), we obtain C = ab11 ? ab22 ? ?c00 . Since
a1 , a2 , b1 , and b2 are gamma random variables, even if the expectation of the EPM?s intensity
function, C, is fixed, ?c00 can take an arbitrary value so that equation C = ab11 ? ab22 ? ?c00
holds. Hence, ?0 can take an arbitrary large value such that ?0 = T ? ?
b0 . This implies that
the ?P prior for the EPM degrades to a gamma distribution without model shrinkage effect
as ?k ? Gamma(?0 /T, c0 ) = Gamma(b
?0 , c0 ).
Theorem 2 indicates that the EPM might overestimate the number of active atoms, and
lead to overfitted solutions.
4
Proposed Generative Constructions
We describe our novel generative constructions for the EPM with an appropriate model
shrinkage effect. According to the analysis described in Sec. 3, the model shrinkage mechanism of the EPM does not work because the expectation of the EPM?s intensity function
has an undesirable redundancy. This finding motivates the proposal of new generative constructions, in which the expectation of the intensity function depends only on the ?P prior.
First, we propose a naive extension of the original EPM using constrained gamma priors
(termed as CEPM). Next, we propose an another generative construction for the EPM by
incorporating Dirichlet priors instead of gamma priors (termed as DEPM). Furthermore,
for the DEPM, we derive truly infinite DEPM (termed as IDEPM) by marginalizing out all
model parameters including the infinite atoms of the ?P prior.
4.1
CEPM
In order to ensure that the EPM?s intensity function depends solely on the ?P prior, a naive
way is to introduce constraints for the hyperparameters of the gamma prior. In the CEPM,
the rate parameters of the gamma priors are constrained as b1 = C1 ? a1 and b2 = C2 ? a2 ,
respectively, where C1 > 0 and C2 > 0 are arbitrary constants. Based on the aforementioned
constraints and Theorem 1,Pthe expectation of the intensity function for the CEPM depends
?
only on the ?P prior as E[ k=1 Ui,k Vj,k ?k ] = C1?C02 c0 .
The posterior inference for the CEPM can be performed using Gibbs sampler in a manner
similar to that for the EPM. However, we can not derive closed-form samplers only for a1
and a2 because of the constraints. Thus, in this paper, posterior sampling for a1 and a2 are
performed using grid Gibbs sampling [13] (see Appendix B for details).
4.2
DEPM
We have another strategy to construct the EPM with efficient model shrinkage effect by
re-parametrizing the factorization problem. Let us denote transpose of a matrix A by A? .
According to the generative model of the EPM in Eq. (1), the original generative process
for counts m can be viewed as a matrix factorization as m ? U ?V ? . It is clear that the
optimal solution of the factorization problem is not unique. Let ?1 and ?2 be arbitrary
K ? K non-negative diagonal matrices. If a solution m ? U ?V ? is globally optimal, then
?1 ?
another solution m ? (U ?1 )(??1
is also optimal. In order to ensure that
1 ??2 )(V ?2 )
the EPM has only one optimal solution, we re-parametrize the original factorization problem
to an equivalent constrained factorization problem as follows:
m ? ??? ? ,
(3)
P
where ? denotes an I ? K non-negative matrix with l1 -constraints as Pi ?i,k = 1, ?k.
Similarly, ? denotes an J ? K non-negative matrix with l1 -constraints as j ?j,k = 1, ?k.
This parameterization ensures the uniqueness of the optimal solution for a given m because
each column of ? and ? is constrained such that it is defined on a simplex.
4
According to the factorization in Eq. (3), by incorporating Dirichlet priors instead of gamma
priors, the generative construction for m of the DEPM is as follows:
!
I
T
X
z }| {
mi,j,? | ?, ?, ? ? Poisson
?i,k ?j,k ?k , {?i,k }Ii=1 | ?1 ? Dirichlet(?1 , . . . , ?1 ),
k=1
J
z }| {
(4)
| ?2 ? Dirichlet(?2 , . . . , ?2 ), ?k | ?0 , c0 ? Gamma(?0 /T, c0 ).
P?
Theorem 3. The expectation of DEPM?s intensity
function
?
?
?
depends
sorely
k=1 i,k j,k k
P?
?0
.
on the ?P prior and can be expressed as E[ k=1 ?i,k ?j,k ?k ] = IJc
0
{?j,k }Jj=1
Proof. The expectations of Dirichlet random variables ?i,k and ?j,k are I1 and J1 , respectively. Similar to the proof for Theorem 1, using the multiplicativity of independent
random
P?
variables
and
the
low
of
total
expectation,
the
proof
is
completed
as
E
[
?
?
?k ] =
i,k
j,k
k=1
P?
P?
1
1
E[?
]E[?
]E[?
]
=
?
?
E[
?
].
i,k
j,k
k
k=1
k=1 k
I
J
Note that, if we set constants C1 = I and C2 = J for the CEPM in Sec. 4.1, then the
expectation of the intensity function for the CEPM is equivalent to that for the DEPM in
Theorem 3. Thus, in order to ensure the fairness of comparisons, we set C1 = I and C2 = J
for the CEPM in the experiments.
As the Gibbs sampler for ? and ? can be derived straightforwardly, the posterior inference
for all parameters and hyperparameters of the DEPM also can be performed via closedform Gibbs sampler (detailed in Appendix C). Differ from the CEPM, l1 -constraints in the
DEPM ensure the uniqueness of its optimal solution. Thus, the inference for the DEPM is
considered as more efficient than that for the CEPM.
4.3
Truly Infinite DEPM (IDEPM)
One remarkable property of the DEPM is that we can derive a fully marginalized likelihood
function. Similar to the beta-negative binomial topic model [13], we consider a joint distribumi,j,?
? {1, ? ? ? , T }mi,j,?
tion for mi,j,? Poisson customers and their assignments zi,j =Q
{zi,j,s }s=1
mi,j,?
to T tables as P (mi,j,? , zi,j | ?, ?, ?) = P (mi,j,? | ?, ?, ?) s=1 P (zi,j,s | mi,j,? , ?, ?, ?).
Thanks to the l1 -constraints we introduced in Eq. (3), the joint distribution P (m, z | ?, ?, ?)
has a fully factorized form (see Lemma 1 in Appendix D). Therefore, marginalizing ?, ?,
and ? out according to the prior construction in Eq. (4), we obtain an analytical marginal
likelihood P (m, z) for the truncated DEPM (see Appendix D for a detailed derivation).
Furthermore, by taking T ? ?, we can derive a closed-form marginal likelihood for the
truly infinite version of the DEPM (termed as IDEPM). In a manner similar to that in [14],
we consider the likelihood function for partition
P [z]
P instead of the assignments z. Assume
we have K+ of T atoms for which m?,?,k =
i
j mi,j,k > 0, and a partition of M (=
P P
m
)
customers
into
K
subsets.
Then,
joint
marginal likelihood of the IDEPM
i,j,?
+
i
j
for [z] and m is given by the following theorem, with the proof provided in Appendix D:
Theorem 4. The marginal likelihood function of the IDEPM is defined as P (m, [z])? =
T!
limT ?? P (m, [z]) = limT ?? (T ?K
P (m, z), and can be derived as follows:
+ )!
P (m, [z])? =
J
I Y
Y
i=1 j=1
1
mi,j,? !
?
K+
Y
k=1
I
Y
?(I?1 )
?(?1 + mi,?,k )
?(I?1 + m?,?,k ) i=1
?(?1 )
?0 Y
K+
c0
?(m?,?,k )
?(?2 + m?,j,k )
?(J?2 )
K+
? ?0
, (5)
?
?(J?2 + m?,?,k ) j=1
?(?2 )
c0 + 1
(c0 + 1)m?,?,k
k=1
k=1
P
P
P P
where mi,?,k =
j mi,j,k , m?,j,k =
i mi,j,k , and m?,?,k =
i
j mi,j,k . Note that ?(?)
denotes gamma function.
K+
Y
J
Y
From Eq. (5) in Theorem 4, we can derive collapsed Gibbs sampler [9] to perform posterior
inference for the IDEPM. Since ?, ?, and ? have been marginalized out, the only latent
variables we have to update are m and z.
5
Sampling z: Given m, similar to the Chinese restaurant process (CRP) [15], the posterior
probability that zi,j,s is assigned to k ? is given as follows:
?
\(ijs)
\(ijs)
? m\(ijs) ? ?1 +mi,?,k? ? ?2 +m?,j,k? if m\(ijs) > 0,
\(ijs)
\(ijs)
?,?,k?
k?
?
I?1 +m?,?,k?
I?2 +m?,?,k?
(6)
P (zi,j,s = k | z\(ijs) , m) ?
?
\(ijs)
1
1
?0 ? I ? J
if m?,?,k? = 0,
where the superscript \(ijs) denotes that the corresponding statistics are computed excluding the s-th customer of entry (i, j).
Sampling m: Given z, posteriors for the ? and ? are simulated as {?i,k }Ii=1 | ? ?
Dirichlet({?1 + mi,?,k }Ii=1 ) and {?j,k }Jj=1 | ? ? Dirichlet({?2 + m?,j,k }Jj=1 ) for k ?
{1, . . . , K+ }. Furthermore, the posterior sampling of the ?k for K+ active atoms can be
performed as ?k | ? ? Gamma(m?,?,k , c0 + 1). Therefore, similar to the sampler for the
EPM [8], we can update m as follows:
?(0)
if xi,j = 0,
PK +
mi,j,? | ?, ?, ? ?
(7)
ZTP( k=1 ?i,k ?j,k ?k ) if xi,j = 1,
?
(
)K+ ?
?
?
?
K+
i,k
j,k
k
?,
{mi,j,k }k=1 | mi,j,? , ?, ?, ? ? Multinomial ?mi,j,? ; PK+
(8)
? ?j,k ? ?k ?
?
?
i,k
k =1
k=1
where ?(0) denotes point mass at zero.
Sampling hyperparameters: We can construct closed-form Gibbs sampler for all hyperparameters of the IDEPM assuming a gamma prior (Gamma(e0 , f0 )). Using the additive
property
of the ?P, posterior sample for the sum of ?k over unused atoms is obtained as ??0 =
P?
k? =K+ +1 ?k? | ? ? Gamma(?0 , c0 + 1). Consequently, we obtain a closed-form posterior
PK+
?k ).
sampler for the rate parameter c0 of the ?P as c0 | ? ? Gamma(e0 + ?0 , f0 + ??0 + k=1
For all remaining hyperparameters (i.e., ?1 , ?2 , and ?0 ), we can derive posterior samplers
from Eq. (5) using data augmentation techniques [12, 8, 2, 11] (detailed in Appendix E).
5
Experimental Results
In previous sections, we theoretically analysed the reason why the model shrinkage of the
EPM does not work appropriately (Sec. 3) and proposed several novel constructions (i.e.,
CEPM, DEPM, and IDEPM) of the EPM with an efficient model shrinkage effect (Sec. 4).
The purpose of the experiments involves ascertaining the following hypotheses:
(H1) The original EPM overestimates the number of active atoms and overfits the data.
In contrast, the model shrinkage mechanisms of the CEPM and DEPM work appropriately. Consequently, the CEPM and DEPM outperform the EPM in generalization ability and link prediction accuracy.
(H2) Compared with the CEPM, the DEPM indicates better generalization ability and
link prediction accuracy because of the uniqueness of the DEPM?s optimal solution.
(H3) The IDEPM with collapsed Gibbs sampler is superior to the DEPM in generalization
ability, link prediction accuracy, mixing efficiency, and convergence speed.
Datasets: The first dataset was the Enron [16] dataset, which comprises e-mails sent
between 149 Enron employees. We extracted e-mail transactions from September 2001 and
constructed Enron09 dataset. For this dataset, xi,j = 1(0) was used to indicate whether
an e-mail was, or was not, sent by the i-th employee to the j-th employee. For larger
dataset, we used the MovieLens [17] dataset, which comprises five-point scale ratings of
movies submitted by users. For this dataset, we set xi,j = 1 when the rating was higher
than three and xi,j = 0 otherwise. We prepared two different sized MovieLens dataset:
MovieLens100K (943 users and 1,682 movies) and MovieLens1M (6,040 users and 3,706
movies). The densities of the Enron09, MovieLens100K and MovieLens1M datasets were
0.016, 0.035, and 0.026, respectively.
6
(a) Enron09
Estimated # of K
128
64
(b) MovieLens100K
128
IDEPM
DEPM-T
CEPM-T
EPM-T
64
64
32
32
16
16
16
8
8
8
4
4
4
2
2
32
2
4
8
16
32
Truncation level T
64
128
2
2
4
TDLL
TDAUC-PR
4
8
16
32
64
128
IDEPM
DEPM-T
CEPM-T
EPM-T
Truncation level T
(g) Enron09
0.350
0.300
0.250
0.200
0.150
0.100
0.050
0.000
8
16
32
Truncation level T
64
128
-0.084
-0.086 2
-0.088
-0.090
-0.092
-0.094
-0.096
-0.098
-0.100
8
16
32
Truncation level T
4
8
16
32
64
128
8
16 32 64
Truncation level T
2
4
64
128
8
16 32 64
Truncation level T
128
8
16
32
-0.072
-0.074
-0.076
-0.078
Truncation level T
Truncation level T
(i) MovieLens1M
0.440
0.420
0.400
0.380
0.360
0.340
0.320
0.300
0.390
0.370
128
128
-0.070
0.410
IDEPM
DEPM-T
CEPM-T
EPM-T
-0.068
(h) MovieLens100K
0.470
64
(f) MovieLens1M
0.430
4
4
-0.066
0.450
2
2
(e) MovieLens100K
(d) Enron09
-0.040
-0.050 2
-0.060
-0.070
-0.080
-0.090
-0.100
-0.110
-0.120
(c) MovieLens1M
128
2
4
8
16 32 64
Truncation level T
128
2
4
Figure 2: Calculated measurements as functions of the truncation level T for each dataset.
The horizontal line in each figure denotes the result obtained using the IDEPM.
Evaluating Measures: We adopted three measurements to evaluate the performance of
the models. The first is the estimated number of active atoms K for evaluating the model
shrinkage effect of each model. The second is the averaged Test Data Log Likelihood (TDLL)
for evaluating the generalization ability of each model. We calculated the averaged likelihood
that a test entry takes the actual value. For the third measurement, as many real-world
binary matrices are often sparse, we adopted the Test Data Area Under the Curve of the
Precision-Recall curve (TDAUC-PR) [18] to evaluate the link prediction ability. In order
to calculate the TDLL and TDAUC-PR, we set all the selected test entries as zero during
the inference period, because binary observations for unobserved entries are not observed
as missing values but are observed as zeros in many real-world situations.
Experimental Settings: Posterior inference for the truncated models (i.e., EPM, CEPM,
and DEPM) were performed using standard (non-collapsed) Gibbs sampler. Posterior inference for the IDEPM was performed using the collapsed Gibbs sampler derived in Sec. 4.3.
For all models, we also sampled all hyperparameters assuming the same gamma prior
(Gamma(e0 , f0 )). For the purpose of fair comparison, we set hyper-hyperparameters as
e0 = f0 = 0.01 throughout the experiments. We ran 600 Gibbs iterations for each model on
each dataset and used the final 100 iterations to calculate the measurements. Furthermore,
all reported measurements were averaged values obtained by 10-fold cross validation.
Results: Hereafter, the truncated models are denoted as EPM-T , CEPM-T , and DEPM-T
to specify the truncation level T . Figure 2 shows the calculated measurements.
(H1) As shown in Figs. 2a?c, the EPM overestimated the number of active atoms K for all
datasets especially for a large truncation level T . In contrast, the number of active atoms
K for the CEPM-T and DEPM-T monotonically converges to a specific value. This result
supports the analysis with respect to the relationship between the model shrinkage effect
and the expectation of the EPM?s intensity function, as discussed in Sec. 3. Consequently,
7
(a) Enron09
-0.050
TDLL
-0.055
0
-0.086
100 200 300 400 500 600
-0.088
0
(b) MovieLens100K
100 200 300 400 500 600 -0.066 0
-0.068
-0.060
-0.090
-0.070
-0.065
-0.092
-0.072
-0.070
-0.094
IDEPM
DEPM-128
-0.075
-0.080
-0.074
-0.076
-0.096
-0.078
-0.098
Gibbs sampling iteration
(c) MovieLens1M
100 200 300 400 500 600
-0.080
Gibbs sampling iteration
Gibbs sampling iteration
Figure 3: (Best viewed in color.) The TDLL as a function of the Gibbs iterations.
(a) Enron09
-0.050
-0.055
0
10
20
30
(b) MovieLens100K
-0.080
40
50
-0.085
0
200
400
600
800
(c) MovieLens1M
-0.065
1000
0
2000
4000
6000
8000 10000
TDLL
-0.070
-0.060
-0.090
-0.065
-0.095
-0.070
-0.075
-0.080
IDEPM
DEPM-128
DEPM-64
DEPM-32
DEPM-16
DEPM-8
DEPM-4
DEPM-2
Elapsed time (sec)
-0.075
-0.100
-0.080
-0.105
-0.110
Elapsed time (sec)
-0.085
Elapsed time (sec)
Figure 4: (Best viewed in color.) The TDLL as a function of the elapsed time (in seconds).
as shown by the TDLL (Figs. 2d?f) and TDAUC-PR (Figs. 2g?i), the CEPM and DEPM
outperformed the original EPM in both generalization ability and link prediction accuracy.
(H2) As shown in Figs. 2a?c, the model shrinkage effect of the DEPM is stronger than
that of the CEPM. As a result, the DEPM significantly outperformed the CEPM in both
generalization ability and link prediction accuracy (Figs. 2d?i). Although the CEPM slightly
outperformed the EPM, the CEPM with a larger T tends to overfit the data. In contrast,
the DEPM indicated its best performance with the largest truncation level (T = 128).
Therefore, we confirmed that the uniqueness of the optimal solution in the DEPM was
considerably important in achieving good generalization ability and link prediction accuracy.
(H3) As shown by the horizontal lines in Figs. 2d?i, the IDEPM indicated the state-of-theart scores for all datasets. Finally, the computational efficiency of the IDEPM was compared
with that of the truncated DEPM. Figure 3 shows the TDLL as a function of the number
of Gibbs iterations. In keeping with expectations, the IDEPM indicated significantly better
mixing property when compared with that of the DEPM for all datasets. Furthermore,
Fig. 4 shows a comparison of the convergence speed of the IDEPM and DEPM with several
truncation levels (T = {2, 4, 8, 16, 32, 64, 128}). As clearly shown in the figure, the convergence of the IDEPM was significantly faster than that of the DEPM with all truncation
levels. Therefore, we confirmed that the IDEPM indicated a state-of-the-art performance
in generalization ability, link prediction accuracy, mixing efficiency, and convergence speed.
6
Conclusions
In this paper, we analysed the model shrinkage effect of the EPM, which is a Bayesian
nonparametric model for extracting overlapping structure with an optimal dimension from
binary matrices. We derived the expectation of the intensity function of the EPM, and
showed that the redundancy of the EPM?s intensity function disturbs its model shrinkage
effect. According to this finding, we proposed two novel generative construction for the
EPM (i.e., CEPM and DEPM) to ensure that its model shrinkage effect works appropriately. Furthermore, we derived a truly infinite version of the DEPM (i.e, IDEPM), which
can be inferred using collapsed Gibbs sampler without any approximation for the ?P. We experimentally showed that the model shrinkage mechanism of the CEPM and DEPM worked
appropriately. Furthermore, we confirmed that the proposed IDEPM indicated a state-ofthe-art performance in generalization ability, link prediction accuracy, mixing efficiency, and
convergence speed. It is of interest to further investigate whether the truly infinite construction of the IDEPM can be applied to more complex and modern machine learning models,
including deep brief networks [19], and tensor factorization models [20].
8
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[8]
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[10]
[11]
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[14]
[15]
[16]
[17]
[18]
[19]
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9
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6,239 | 6,644 | Pose Guided Person Image Generation
Liqian Ma1
Xu Jia2? Qianru Sun3? Bernt Schiele3 Tinne Tuytelaars2 Luc Van Gool1,4
KU-Leuven/PSI, TRACE (Toyota Res in Europe) 2 KU-Leuven/PSI, IMEC
3
Max Planck Institute for Informatics, Saarland Informatics Campus
4
ETH Zurich
{liqian.ma, xu.jia, tinne.tuytelaars, luc.vangool}@esat.kuleuven.be
{qsun, schiele}@mpi-inf.mpg.de [email protected]
1
Abstract
This paper proposes the novel Pose Guided Person Generation Network (PG2 )
that allows to synthesize person images in arbitrary poses, based on an image of
that person and a novel pose. Our generation framework PG2 utilizes the pose
information explicitly and consists of two key stages: pose integration and image
refinement. In the first stage the condition image and the target pose are fed into a
U-Net-like network to generate an initial but coarse image of the person with the
target pose. The second stage then refines the initial and blurry result by training a
U-Net-like generator in an adversarial way. Extensive experimental results on both
128?64 re-identification images and 256?256 fashion photos show that our model
generates high-quality person images with convincing details.
1
Introduction
Generating realistic-looking images is of great value for many applications such as face editing,
movie making and image retrieval based on synthesized images. Consequently, a wide range of
methods have been proposed including Variational Autoencoders (VAE) [14], Generative Adversarial
Networks (GANs) [6] and Autoregressive models (e.g., PixelRNN [30]). Recently, GAN models have
been particularly popular due to their principle ability to generate sharp images through adversarial
training. For example in [21, 5, 1], GANs are leveraged to generate faces and natural scene images
and several methods are proposed to stabilize the training process and to improve the quality of
generation.
From an application perspective, users typically have a particular intention in mind such as changing
the background, an object?s category, its color or viewpoint. The key idea of our approach is to
guide the generation process explicitly by an appropriate representation of that intention to enable
direct control over the generation process. More specifically, we propose to generate an image by
conditioning it on both a reference image and a specified pose. With a reference image as condition,
the model has sufficient information about the appearance of the desired object in advance. The
guidance given by the intended pose is both explicit and flexible. So in principle this approach can
manipulate any object to an arbitrary pose. In this work, we focus on transferring a person from a
given pose to an intended pose. There are many interesting applications derived from this task. For
example, in movie making, we can directly manipulate a character?s human body to a desired pose or,
for human pose estimation, we can generate training data for rare but important poses.
Transferring a person from one pose to another is a challenging task. A few examples can be seen in
Figure 1. It is difficult for a complete end-to-end framework to do this because it has to generate both
correct poses and detailed appearance simultaneously. Therefore, we adopt a divide-and-conquer
strategy, dividing the problem into two stages which focus on learning global human body structure
?
Equal contribution.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Condition
image
Target
pose
Target
image (GT)
Coarse
result
Refined
result
Condition
image
(a) DeepFashion
Condition image
Target
pose
Target
image (GT)
Coarse
result
Refined
result
(b) Market-1501
Target pose sequence
Refined results
(c) Generating from a sequence of poses
Figure 1: Generated samples on DeepFashion dataset [16] (a)(c) and Market-1501 dataset [37] (b).
Our generation sequence (refined results)
and appearance details respectively similar to [35, 9, 3, 19]. At stage-I, we explore different ways to
model pose information. A variant of U-Net is employed to integrate the target pose with the person
image. It outputs a coarse generation result that captures the global structure of the human body in
the target image. A masked L1 loss is proposed to suppress the influence of background change
between condition image and target image. However, it would generate blurry result due to the use of
L1. At stage-II, a variant of Deep Convolutional GAN (DCGAN) model is used to further refine the
initial generation result. The model learns to fill in more appearance details via adversarial training
and generates sharper images. Different from the common use of GANs which directly learns to
generate an image from scratch, in this work we train a GAN to generate a difference map between
the initial generation result and the target person image. The training converges faster since it is an
easier task. Besides, we add a masked L1 loss to regularize the training of the generator such that it
will not generate an image with many artifacts. Experiments on two dataset, a low-resolution person
re-identification dataset and a high-resolution fashion photo dataset, demonstrate the effectiveness of
the proposed method.
Our contribution is three-fold. i) We propose a novel task of conditioning image generation on a
reference image and an intended pose, whose purpose is to manipulate a person in an image to an
arbitrary pose. ii) Several ways are explored to integrate pose information with a person image.
A novel mask loss is proposed to encourage the model to focus on transferring the human body
appearance instead of background information. iii) To address the challenging task of pose transfer,
we divide the problem into two stages, with the stage-I focusing on global structure of the human
body and the stage-II on filling in appearance details based on the first stage result.
2
Related works
Recently there have been a lot of works on generative image modeling with deep learning techniques.
These works fall into two categories. The first line of works follow an unsupervised setting. One
popular method under this setting is variational autoencoders proposed by Kingma and Welling [14]
and Rezende et al. [25], which apply a re-parameterization trick to maximize the lower bound of the
data likelihood. Another branch of methods are autogressive models [28, 30, 29] which compute the
product of conditional distributions of pixels in a pixel-by-pixel manner as the joint distribution of
pixels in an image. The most popular methods are generative adversarial networks (GAN) [6], which
simultaneously learn a generator to generate samples and a discriminator to discriminate generated
samples from real ones. Many works show that GANs can generate sharp images because of using
2
concat
coarse
result
skip connection
...
Condition
image
Target
pose
...
Target
image
Generator at Stage-I (G1)
ResUAE (G1) at Stage-I
Condition
image
Real
or
Fake
Real pair
skip connection
difference
map
Discriminator (D)
at Stage-II
concat
refined
result
Condition
image
Condition
image
Fake pair
Generator at Stage-II (G2)
Figure 2: The overall framework of our Pose Guided Person Generation Network (PG2 ). It contains
two stages. Stage-I focuses on pose integration and generates an initial result that captures the global
structure of the human. Stage-II focuses on refining the initial result via adversarial training and
generates sharper images.
the adversarial loss instead of L1 loss. In this work, we also use the adversarial loss in our framework
in order to generate high-frequency details in images.
The second group of works generate images conditioned on either category or attribute labels, texts or
images. Yan et al. [32] proposed a Conditional Variational Autoencoder (CVAE) to achieve attribute
conditioned image generation. Mirza and Osindero [18] proposed to condition both generator
and discriminator of GAN on side information to perform category conditioned image generation.
Lassner et al. [15] generated full-body people in clothing, by conditioning on the fine-grained body
part segments. Reed et al. proposed to generate bird image conditioned on text descriptions by adding
textual information to both generator and discriminator [24] and further explored the use of additional
location, keypoints or segmentation information to generate images [22, 23]. With only these visual
cues as condition and in contrast to our explicit condition on the intended pose, the control exerted
over the image generation process is still abstract. Several works further conditioned image generation
not only on labels and texts but also on images. Researchers [34, 33, 11, 8] addressed the task of face
image generation conditioned on a reference image and a specific face viewpoint. Chen et al. [4]
tackled the unseen view inference as a tensor completion problem, and use latent factors to impute
the pose in unseen views. Zhao et al. [36] explored generating multi-view cloth images from only a
single view input, which is most similar to our task. However, a wide range of poses is consistent
with any given viewpoint making the conditioning less expressive than in our work. In this work, we
make use of pose information in a more explicit and flexible way, that is, using poses in the format
of keypoints to model diverse human body appearance. It should be noted that instead of doing
expensive pose annotation, we use a state-of-the-art pose estimation approach to obtain the desired
human body keypoints.
3
Method
Our task is to simultaneously transfer the appearance of a person from a given pose to a desired pose
and keep important appearance details of the identity. As it is challenging to implement this as an
end-to-end model, we propose a two-stage approach to address this task, with each stage focusing on
one aspect. For the first stage we propose and analyze several model variants and for the second stage
we use a variant of a conditional DCGAN to fill in more appearance details. The overall framework
of the proposed Pose Guided Person Generation Network (PG2 ) is shown in Figure 2.
3.1
Stage-I: Pose integration
At stage-I, we integrate a conditioning person image IA with a target pose PB to generate a coarse
result I?B that captures the global structure of the human body in the target image IB .
3
FConv-ResUAE (G2) at Stage-II
Pose embedding. To avoid expensive annotation of poses, we apply a state-of-the-art pose estimator [2] to obtain approximate human body poses. The pose estimator generates the coordinates of
18 keypoints. Using those directly as input to our model would require the model to learn to map
each keypoint to a position on the human body. Therefore, we encode pose PB as 18 heatmaps. Each
heatmap is filled with 1 in a radius of 4 pixels around the corresponding keypoints and 0 elsewhere
(see Figure 3, target pose). We concatenate IA and PB as input to our model. In this way, we can
directly use convolutional layers to integrate the two kinds of information.
Generator G1. As generator at stage I, we adopt a U-Net-like architecture [20], i.e., convolutional
autoencoder with skip connections as is shown in Figure 2. Specifically, we first use several stacked
convolutional layers to integrate IA and PB from small local neighborhoods to larger ones so
that appearance information can be integrated and transferred to neighboring body parts. Then, a
fully connected layer is used such that information between distant body parts can also exchange
information. After that, the decoder is composed of a set of stacked convolutional layers which are
symmetric to the encoder to generate an image. The result of the first stage is denoted as I?B1 . In the
U-Net, skip connections between encoder and decoder help propagate image information directly
from input to output. In addition, we find that using residual blocks as basic component improves the
generation performance. In particular we propose to simplify the original residual block [7] and have
only two consecutive conv-relu inside a residual block.
Pose mask loss. To compare the generation I?B1
with the target image IB , we adopt L1 distance
as the generation loss of stage-I. However, since
we only have a condition image and a target
pose as input, it is difficult for the model to
generate what the background would look like if
the target image has a different background from
the condition image. Thus, in order to alleviate
the influence of background changes, we add
another term that adds a pose mask MB to the
L1 loss such that the human body is given more
weight than the background. The formulation of
pose mask loss is given in Eq. 1 with denoting
the pixels-wise multiplication:
Target
image
pose keypoints
estimation
Target
pose
Pose
skeleton
pose keypoints
connection
Pose
mask
morphological
operations
Figure 3: Process of computing the pose mask.
LG1 = k(G1(IA , PB ) ? IB ) (1 + MB )k1 .
(1)
The pose mask MB is set to 1 for foreground and 0 for background and is computed by connecting
human body parts and applying a set of morphological operations such that it is able to approximately
cover the whole human body in the target image, see the example in Figure 3.
The output of G1 is blurry because the L1 loss encourages the result to be an average of all possible
cases [10]. However, G1 does capture the global structural information specified by the target pose,
as shown in Figure 2, as well as other low-frequency information such as the color of clothes. Details
of body appearance, i.e. the high-frequency information, will be refined at the second stage through
adversarial training.
3.2
Stage-II: Image refinement
Since the model at the first stage has already synthesized an image which is coarse but close to
the target image in pose and basic color, at the second stage, we would like the model to focus on
generating more details by correcting what is wrong or missing in the initial result. We use a variant
of conditional DCGAN [21] as our base model and condition it on the stage-I generation result.
Generator G2. Considering that the initial result and the target image are already structurally similar,
we propose that the generator G2 at the second stage aims to generate an appearance difference map
that brings the initial result closer to the target image. The difference map is computed using a U-Net
similar to the first stage but with the initial result I?B1 and condition image IA as input instead. The
difference lies in that the fully-connected layer is removed from the U-Net. This helps to preserve
more details from the input because a fully-connected layer compresses a lot of information contained
in the input. The use of difference maps speeds up the convergence of model training since the model
focuses on learning the missing appearance details instead of synthesizing the target image from
4
scratch. In particular, the training already starts from a reasonable result. The overall architecture of
G2 can be seen in Figure 2.
Discriminator D. In traditional GANs, the discriminator distinguishes between real groundtruth
images and fake generated images (which is generated from random noise). However, in our
conditional network, G2 takes the condition image IA instead of a random noise as input. Therefore,
real images are the ones which not only are natural but also satisfy a specific requirement. Otherwise,
G2 will be mislead to directly output IA which is natural by itself instead of refining the coarse result
of the first stage I?B1 . To address this issue, we pair the G2 output with the condition image to make
the discriminator D to recognize the pairs? fakery, i.e., (I?B2 , IA ) vs (IB , IA ). This is diagrammed in
Figure 2. The pairwise input encourages D to learn the distinction between I?B2 and IB instead of
only the distinction between synthesized and natural images.
Another difference from traditional GANs is that noise is not necessary anymore since the generator is
conditioned on an image IA , which is similar to [17]. Therefore, we have the following loss function
for the discriminator D and the generator G2 respectively,
LD
= Lbce (D(IA , IB ), 1) + Lbce (D(IA , G2(IA , I?B1 )), 0),
(2)
adv
LG
= Lbce (D(IA , G2(IA , I?B1 )), 1),
(3)
adv
where Lbce denotes binary cross-entropy loss. Previous work [10, 17] shows that mixing the adversarial loss with a loss minimizing Lp distance can regularize the image generation process. Here
we use the same masked L1 loss as is used at the first stage such that it pays more attention to the
appearance of targeted human body than background,
LG2 = LG + ?k(G2(IA , I?B1 ) ? IB ) (1 + MB )k1 ,
(4)
adv
where ? is the weight of L1 loss. It controls how close the generation looks like the target image at
low frequencies. When ? is small, the adversarial loss dominates the training and it is more likely
to generate artifacts; when ? is big, the the generator with a basic L1 loss dominates the training,
making the whole model generate blurry results2 .
In the training process of our DCGAN, we alternatively optimize discriminator D and generator G2.
As shown in the left part of Figure 2, generator G2 takes the first stage result and the condition image
as input and aims to refine the image to confuse the discriminator. The discriminator learns to classify
the pair of condition image and the generated image as fake while classifying the pair including the
target image as real.
3.3
Network architecture
We summarize the network architecture of the proposed model PG2 . At stage-I, the encoder of G1
consists of N residual blocks and one fully-connected layer , where N depends on the size of input.
Each residual block consists of two convolution layers with stride=1 followed by one sub-sampling
convolution layer with stride=2 except the last block. At stage-II, the encoder of G2 has a fully
convolutional architecture including N -2 convolution blocks. Each block consists of two convolution
layers with stride=1 and one sub-sampling convolution layer with stride=2. Decoders in both G1
and G2 are symmetric to corresponding encoders. Besides, there are shortcut connections between
decoders and encoders, which can be seen in Figure 2. In G1 and G2, no batch normalization or
dropout are applied. All convolution layers consist of 3?3 filters and the number of filters are
increased linearly with each block. We apply rectified linear unit (ReLU) to each layer except the
fully connected layer and the output convolution layer. For the discriminator, we adopt the same
network architecture as DCGAN [21] except the size of the input convolution layer due to different
image resolutions.
4
Experiments
We evaluate the proposed PG2 network on two person datasets (Market-1501 [37] and DeepFashion [16]), which contain person images with diverse poses. We present quantitative and qualitative
results for three main aspects of PG2 : different pose embeddings; pose mask loss vs. standard L1
loss; and two-stage model vs. one-stage model. We also compare with the most related work [36].
2
The influence of ? on generation quality is analyzed in supplementary materials.
5
4.1
Datasets
The DeepFashion (In-shop Clothes Retrieval Benchmark) dataset [16] consists of 52,712 in-shop
clothes images, and 200,000 cross-pose/scale pairs. All images are in high-resolution of 256?256.
In the train set, we have 146,680 pairs each of which is composed of two images of the same person
but different poses. We randomly select 12,800 pairs from the test set for testing.
We also experiment on a more challenging re-identification dataset Market-1501 [37] containing
32,668 images of 1,501 persons captured from six disjoint surveillance cameras. Persons in this
dataset vary in pose, illumination, viewpoint and background, which makes the person generation
task more challenging. All images have size 128?64 and are split into train/test sets of 12,936/19,732
following [37]. In the train set, we have 439,420 pairs each of which is composed of two images of
the same person but different poses. We randomly select 12,800 pairs from the test set for testing.
Implementation details On both datasets, we use the Adam [13] optimizer with ?1 = 0.5 and
?2 = 0.999. The initial learning rate is set to 2e-5. On DeepFashion, we set the number of
convolution blocks N = 6. Models are trained with a minibatch of size 8 for 30k and 20k iterations
respectively at stage-I and stage-II. On Market-1501, we set the number of convolution blocks N = 5.
Models are trained with a minibatch of size 16 for 22k and 14k iterations respectively at stage-I and
stage-II. For data augmentation, we do left-right flip for both datasets3 .
4.2
Qualitative results
As mentioned above, we investigate three aspects of our proposed PG2 network. Different pose
embeddings and losses are compared within stage-I and then we demonstrate the advantage of our
two-stage model over a one-stage model.
Different pose embeddings. To evaluate our proposed pose embedding method, we implement two
alternative methods. For the first, coordinate embedding (CE), we pass the keypoint coordinates
through two fully connected layers and concatenate the embedded feature vector with the image
embedding vector at the bottleneck fully connected layer. For the second, called heatmap embedding
(HME), we feed the 18 keypoint heatmaps to an independent encoder and extract the fully connected
layer feature to concatenate with image embedding vector at the bottleneck fully connected layer.
Columns 4, 5 and 6 of Figure 4 show qualitative results of the different pose embedding methods
when used in stage-I, that is of G1 with CE (G1-CE-L1), with HME (G1-HME-L1) and our G1
(G1-L1). All three use standard L1 loss. We can see that G1-L1 is able to synthesize reasonable
looking images that capture the global structure of a person, such as pose and color. However, the
other two embedding methods G1-CE-L1 and G1-HME-L1 are quite blurry and the color is wrong.
Moreover, results of G1-CE-L1 all get wrong poses. This can be explained by the additional difficulty
to map the keypoint coordinates to appropriate image locations making training more challenging.
Our proposed pose embedding using 18 channels of pose heatmaps is able to guide the generation
process effectively, leading to correctly generated poses. Interestingly, G1-L1 can even generate
reasonable face details like eyes and mouth, as shown by the DeepFashion samples.
Pose mask loss vs. L1 loss. Comparing the results of G1 trained with L1 loss (G1-L1) and G1
trained with poseMaskLoss (G1-poseMaskLoss) for the Market-1501 dataset, we find that pose mask
loss indeed brings improvement to the performance (columns 6 and 7 in Figure 4). By focusing the
image generation on the human body, the synthesized image gets sharper and the color looks nicer.
We can see that for person ID 164, the person?s upper body generated by G1-L1 is more noisy in
color than the one generated by G1-poseMaskLoss. For person ID 23 and 346, the method with pose
mask loss generates more clear boundaries for shoulder and head. These comparisons validate that
our pose mask loss effectively alleviates the influence of noisy backgrounds and guides the generator
to focus on the pose transfer of the human body. The two losses generate similar results for the
DeepFashion samples because the background is much simpler.
Two-stage vs. one-stage. In addition, we demonstrate the advantage of our two-stage model over
a one-stage model. For this we use G1 as generator but train it in an adversarial way to directly
generate a new image given a condition image and a target pose as input. This one-stage model is
denoted as G1+D and our full model is denoted as G1+G2+D. From Figure 4, we can see that our full
model is able to generate photo-realistic results, which contain more details than the one-stage model.
3
More details about parameters of the network architecture are given in supplementary materials.
6
For example, for DeepFashion samples, more details in the face and the clothes are transferred to the
generated images. For person ID 245, the shorts on the result of G1+D have lighter color and more
blurry boundary than G1+G2+D. For person ID 346, the two-stage model is able to generate both
the right color and textures for the clothes, while the one-stage model is only able to generate the
right color. On Market-1501 samples, the quality of the images generated by both methods decreases
because of the more challenging setting. However, the two-stage model is still able to generate
better results than the one-stage method. We can see that for person ID 53, the stripes on the T-shirt
are retained by our full model while the one-stage model can only generate a blue blob as clothes.
Besides, we can also clearly see the stool in the woman?s hands (person ID 23).
1
2
3
Condition image Target pose Target image(GT)
4
5
6
G1-CE-L1
G1-HME-L1
G1-L1
7
G1-poseMaskLoss
(our coarse result)
8
9
G1+D
G1+G2+D
(our refined result)
ID. 245
ID. 346
ID. 116
ID. 53
ID. 164
ID. 23
Figure 4: Test results on DeepFashion (upper 3 rows, images are cut for the sake of display) and
Market-1501 dataset (lower 3 rows). We test G1 in two aspects: (1) three pose embedding methods,
i.e., coordinate embedding (CE), heatmap embedding (HME) and our pose heatmap concatenation
in G1-L1, and (2) two losses, i.e., the proposed poseMaskLoss and the standard L1 loss. Column
7, 8 and 9 show the differences among our stage-I (G1), one-stage adversarial model (G1+D) and
our two-stage adversarial model (G1+G2+D). Note that all three use poseMaskLoss. The IDs are
assigned randomly when splitting the datasets.
4.3
Quantitative results
We also give quantitative results on both datasets. Structural Similarity (SSIM) [31] and the Inception
Score (IS) [26] are adopted to measure the quality of synthesis. Note that in the Market-1501 dataset,
condition images and target images may have different background. Since there is no information
in the input about the background in the target image, our method is not able to imagine what the
7
Table 1: Quantitative evaluation. For all measures, higher is better.
DeepFashion
Market-1501
Model
SSIM
IS
SSIM
IS
mask-SSIM
mask-IS
G1-CE-L1
G1-HME-L1
G1-L1
G1-poseMaskLoss
G1+D
G1+G2+D
0.694
0.735
0.735
0.779
0.761
0.762
2.395
2.427
2.427
2.668
3.091
3.090
0.219
0.294
0.304
0.340
0.283
0.253
2.568
3.171
3.006
3.326
3.490
3.460
0.771
0.802
0.809
0.817
0.803
0.792
2.455
2.508
2.455
2.682
3.310
3.435
Table 2: User study results from AMT
DeepFashion
4
Market-1501
5
Model
R2G
G2R
R2G
G2R
G1+D
G1+G2+D
7.8%
9.2%
9.3%
14.9%
17.1%
11.2%
11.1%
5.5%
new background looks like. To reduce the influence of background in our evaluation, we propose
a variant of SSIM, called mask-SSIM. A pose mask is added to both the synthesis and the target
image before computing SSIM. In this way we only focus on measuring the synthesis quality of a
person?s appearance. Similarly, we employ mask-IS to eliminate the effect of background. However,
it should be noted that image quality does not always correspond to such image similarity metrics.
For example, in Figure 4, our full model generates sharper and more photo-realistic results than
G1-poseMaskLoss, but the latter one has a higher SSIM. This is also observed in super-resolution
papers [12, 27].
The advantages are also clearly shown in the numerical scores in Table 1. E.g. the proposed pose
embedding (G1-L1) consistently outperforms G1-CE-L1 across all measures and both datasets. G1HME-L1 obtains similar quantitative numbers probably due to the similarity of the two embeddings.
Changing the loss from L1 to the proposed poseMaskLoss (G1-poseMaskLoss) consistently improves
further across all measures and for both datasets. Adding the discriminator during training either
after the first stage (G1+D) or in our full model (G1+G2+D) leads to comparable numbers, even
though we have observed clear differences in the qualitative results as discussed above. This is
explained by the fact that blurry images often get good SSIM despite being less convincing and
photo-realistic [12, 27].
4.4
User study
We perform a user study on Amazon Mechanical Turk (AMT) for both datasets. For each one, we
show 55 real images and 55 generated images in a random order to 30 users. Following [10, 15],
each image is shown for 1 second. The first 10 images are used for practice thus are ignored when
computing scores. From the results reported in Table. 2, we can get some observations that (1)
On DeepFashion our generated images of G1+D and G1+G2+D manage to confuse users on 9.3%
and 14.9% trials respectively (see G2R), showing the advantage of G1+G2+D over G1+D; (2) On
Market-1501, the average score of G2R is lower, because the background is much more cluttered than
DeepFashion; (3) On Market-1501, G1+G2+D gets a lower score than G1+D, because G1+G2+D
transfers more backgrounds from the condition image, which can be figured out in Figure. 4, but
in the meantime it brings extra artifacts on backgrounds which lead users to rate ?Fake?; (4) With
respect to R2G, we notice that Market-1501 gets clearly high scores (>10%) because human users
sometimes get confused when facing low-quality surveillance images.
4
5
R2G means #Real images rated as generated / #Real images
G2R means #Generated images rated as Real / #Generated images
8
Condition
image
Target image
(GT)
Ours
(refined)
VariGAN [36]
(refined)
Condition
image
Target
pose
Target
Ours
image (GT) (coarse)
Ours
(refined)
ID. 215
Figure 5: Comparison examples with [36].
4.5
Figure 6: Our failure cases on DeepFashion.
Further analysis
ID. 2662
Since our task with pose condition is novel, there is no direct comparison work. We only compare
with the most related one6 [36], which did multi-view person image synthesis on the DeepFashion
dataset. It is noted that [36] used the condition image and an additional word vector of the target
view e.g. ?side? as network input. Comparison examples are shown in Figure 5. It is clear that
our refined results are much better than those of [36]. Taking the second row as an example, we
can generate high-quality whole body images conditioned on an upper body while the whole body
synthesis by [36] only has a rough body shape.
Additionally, we give two failure DeepFashion examples by our model in Figure 6. In the top row,
only the upper body is generated consistently. The ?pieces of legs? is caused by the rare training
data for such complicated poses. The bottom row shows inaccurate gender which is caused by the
imbalance of training data for male / female. Besides, the condition person wears a long-sleeve jacket
of similar color to his inner short-sleeve, making the generated cloth look like a mixture of both.
5
Conclusions
In this work, we propose the Pose Guided Person Generation Network (PG2 ) to address a novel task
of synthesizing person images by conditioning it on a reference image and a target pose. A divideand-conquer strategy is employed to divide the generation process into two stages. Stage-I aims to
capture the global structure of a person and generate an initial result. A pose mask loss is further
proposed to alleviate the influence of the background on person image synthesis. Stage-II fills in
more appearance details via adversarial training to generate sharper images. Extensive experimental
results on two person datasets demonstrate that our method is able to generate images that are both
photo-realistic and pose-wise correct. In the future work, we plan to generate more controllable and
diverse person images conditioning on both pose and attribute.
Acknowledgments
We gratefully acknowledge the support of Toyota Motors Europe, FWO Structure from Semantics
project, KU Leuven GOA project CAMETRON, and German Research Foundation (DFG CRC 1223).
We would like to thank Bo Zhao for his helpful discussions.
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11
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6,240 | 6,645 | Inference in Graphical Models
via Semidefinite Programming Hierarchies
Murat A. Erdogdu
Microsoft Research
[email protected]
Yash Deshpande
MIT and Microsoft Research
[email protected]
Andrea Montanari
Stanford University
[email protected]
Abstract
Maximum A posteriori Probability (MAP) inference in graphical models amounts
to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation
(GBP) are intimately related to linear programming (LP) relaxation within the
Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well
understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations
for a graph with n vertices require solving an SDP with n?(d) variables where
d is the degree in the hierarchy. In practice, for d
4, this approach does not
scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused
on computational efficiency. Firstly, in analogy to BP and its variants, we only
introduce decision variables corresponding to contiguous regions in the graphical
model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro
style method, and develop a sequential rounding procedure. We demonstrate that
the resulting algorithm can solve problems with tens of thousands of variables
within minutes, and outperforms BP and GBP on practical problems such as image
denoising and Ising spin glasses. Finally, for specific graph types, we establish a
sufficient condition for the tightness of the proposed partial SOS relaxation.
1
Introduction
Graphical models provide a powerful framework for analyzing systems comprised by a large number
of interacting variables. Inference in graphical models is crucial in scientific methodology with
countless applications in a variety of fields including causal inference, computer vision, statistical
physics, information theory, and genome research [WJ08, KF09, MM09].
In this paper, we propose a class of inference algorithms for pairwise undirected graphical models.
Such models are fully specified by assigning: (i) a finite domain X for the variables; (ii) a finite
graph G = (V, E) for V = [n] ? {1, . . . , n} capturing the interactions of the basic variables;
e
(iii) a collection of functions ? = ({?iv }i2V , {?ij
}(i,j)2E ) that quantify the vertex potentials and
interactions between the variables; whereby for each vertex i 2 V we have ?iv : X ! R and for each
e
edge (i, j) 2 E, we have ?ij
: X ? X ! R (an arbitrary ordering is fixed on the pair of vertices
{i, j}). These parameters can be used to form a probability distribution on X V for the random vector
x = (x1 , x2 , ..., xn ) 2 X V by letting,
X
X
1
e
p(x|?) =
eU (x;?) ,
U (x; ?) =
?ij
(xi , xj ) +
?iv (xi ) ,
(1.1)
Z(?)
(i,j)2E
i2V
where Z(?) is the normalization constant commonly referred to as the partition function. While such
models can encode a rich class of multivariate probability distributions, basic inference tasks are
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
intractable except for very special graph structures such as trees or small treewidth graphs [CD+ 06].
In this paper, we will focus on MAP estimation, which amounts to solving the combinatorial
optimization problem
? (?) ? arg max U (x; ?).
x
(1.2)
x2X V
Intractability plagues other classes of graphical models as well (e.g. Bayesian networks, factor
graphs), and has motivated the development of a wide array of heuristics. One of the simplest such
heuristics is the loopy belief propagation (BP) [WJ08, KF09, MM09]. In its max-product version
(that is well-suited for MAP estimation), BP is intimately related to the linear programming (LP)
relaxation of the combinatorial problem maxx2X V U (x; ?). Denoting the decision variables by
b = ({bi }i2V , {bij }(i,j)2E ), LP relaxation form of BP can be written as
X
X
X X
maximize
?ij (xi , xj )bij (xi , xj ) +
?i (xi )bi (xi ) ,
(1.3)
b
subject to
(i,j)2E xi ,xj 2X
X
i2V xi 2X
bij (xi , xj ) = bi (xi )
xj 2X
8(i, j) 2 E ,
(1.4)
bi 2 X 8i 2 V, bij 2 X ?X 8(i, j) 2 E ,
(1.5)
where S denotes the simplex of probability distributions over set S. The decision variables are
referred to as ?beliefs?, and their feasible set is a relaxation of the polytope of marginals of distributions.
The beliefs satisfy the constraints on marginals involving at most two variables connected by an edge.
Loopy belief propagation is successful on some applications, e.g. sparse locally tree-like graphs
that arise, for instance, decoding modern error correcting codes [RU08] or in random constraint
satisfaction problems [MM09]. However, in more structured instances ? arising for example in
computer vision ? BP can be substantially improved by accounting for local dependencies within
subsets of more than two variables. This is achieved by generalized belief propagation (GBP)
[YFW05] where the decision variables are beliefs bR that are defined on subsets of vertices (a
?region?) R ? [n], and that represent the marginal distributions of the variables
in that region. The
P
basic constraint on the beliefs is the linear marginalization constraint: xR\S bR (xR ) = bS (xS ),
holding whenever S ? R. Hence GBP itself is closely related to LP relaxation of the polytope
of marginals of probability distributions. The relaxation becomes tighter as larger regions are
incorporated. In a prototypical application, G is a two-dimensional grid, and regions are squares
induced by four contiguous vertices (plaquettes), see Figure 1, left frame. Alternatively in the right
frame of the same figure, the regions correspond to triangles.
The LP relaxations that correspond to GBP are closely related to the Sherali-Adams hierarchy
[SA90]. Similar to GBP, the variables within this hierarchy are beliefs
Pover subsets of variables
bR = (bR (xR ))xR 2X R which are consistent under marginalization:
xR\S bR (xR ) = bS (xS ).
However, these two approaches differ in an important point: Sherali-Adams hierarchy uses beliefs
over all subsets of |R| ? d variables, where d is the degree in the hierarchy; this leads to an LP of size
?(nd ). In contrast, GBP only retains regions that are contiguous in G. If G has maximum degree k,
this produces an LP of size O(nk d ), a reduction which is significant for large-scale problems.
Given the broad empirical success of GBP, it is natural to develop better methods for inference in
graphical models using tighter convex relaxations. Within combinatorial optimization, it is well
understood that the semidefinite programming (SDP) relaxations provide superior approximation
guarantees with respect to LP [GW95]. Nevertheless, SDP has found limited applications in inference
tasks for graphical models for at least two reasons. A structural reason: standard SDP relaxations (e.g.
[GW95]) do not account exactly for correlations between neighboring vertices in the graph which is
essential for structured graphical models. As a consequence, BP or GBP often outperforms basic
SDPs. A computational reason: basic SDP relaxations involve ?(n2 ) decision variables, and generic
interior point solvers do not scale well for the large-scale applications. An exception is [WJ04] which
employs the simplest SDP relaxation (degree 2 Sum-Of-Squares, see below) in conjunction with a
relaxation of the entropy and interior point methods ? higher order relaxations are briefly discussed
without implementation as the resulting program suffers from the aforementioned limitations.
In this paper, we revisit MAP inference in graphical models via SDPs, and propose an approach that
carries over the favorable performance guarantees of SDPs into inference tasks. For simplicity, we
focus on models with binary variables, but we believe that many of the ideas developed here can be
naturally extended to other finite domains. We present the following contributions:
2
Region 1
on
gi
Re
Region 2
1
on
gi
Re
2
on
gi
Re
3
on
gi
Re
4
Figure 1: A two dimensional grid, and two typical choices for regions for GBP and PSOS.
Left: Regions are plaquettes comprising four vertices. Right: Regions are triangles.
Partial Sum-Of-Squares relaxations. We use SDP hierarchies, specifically the Sum-Of-Squares
(SOS) hierarchy [Sho87, Las01, Par03] to formulate tighter SDP relaxations for binary MAP inference
that account exactly for the joint distributions of small subsets of variables xR , for R ? V . However,
SOS introduces decision variables for all subsets R ? V with |R| ? d/2 (d is a fixed even integer),
and hence scales poorly for large-scale inference problems. We propose a similar modification as in
GBP. Instead of accounting for all subsets R with |R| ? d/2, we only introduce decision variables to
represent a certain family of such subsets (regions) of vertices in G. The resulting SDP has (for d and
the maximum degree of G bounded) only O(n2 ) decision variables which is suitable for practical
implementations. We refer to these relaxations as Partial Sum-Of-Squares (PSOS), cf. Section 2.
Theoretical analysis. In Section 2.1, we prove that suitable PSOS relaxations are tight for certain
classes of graphs, including planar graphs, with ?v = 0. While this falls short of explaining the
empirical results (which uses simpler relaxations, and ?v 6= 0), it points in the right direction.
Optimization algorithm and rounding. Despite the simplification afforded by PSOS, interior-point
solvers still scale poorly to large instances. In order to overcome this problem, we adopt a non-convex
approach proposed by Burer and Monteiro [BM03]. We constrain the rank of the SDP matrix in
PSOS to be at most r, and solve the resulting non-convex problem using a trust-region coordinate
ascent method, cf. Section 3.1. Further, we develop a rounding procedure called Confidence Lift and
Project (CLAP) which iteratively uses PSOS relaxations to obtain an integer solution, cf. Section 3.2.
Numerical experiments. In Section 4, we present numerical experiments with PSOS by solving
problems of size up to 10, 000 within several minutes. While additional work is required to scale
this approach to massive sizes, we view this as an exciting proof-of-concept. To the best of our
knowledge, no earlier attempt was successful in scaling higher order SOS relaxations beyond tens
of dimensions. More specifically, we carry out experiments with two-dimensional grids ? an image
denoising problem, and Ising spin glasses. We demonstrate through extensive numerical studies that
PSOS significantly outperforms BP and GBP in the inference tasks we consider.
2
Partial Sum-Of-Squares Relaxations
For concreteness, throughout the paper we focus on pairwise models with binary variables. We do not
expect fundamental problems extending the same approach to other domains. For binary variables
x = (x1 , x2 , ..., xn ), MAP estimation amounts to solving the following optimization problem
X
X
e
maximize
?ij
xi xj +
?iv xi ,
(INT)
x
where ? =
e
e
(?ij
)1?i,j?n
i2V
(i,j)2E
subject to xi 2 {+1, 1} ,
and ? =
v
(?iv )1?i?n
8i 2 V ,
are the parameters of the graphical model.
For the reader?s convenience, we recall a few basic facts about SOS relaxations, referring to [BS16]
for further details. For an even integer d, SOS(d) is an SDP relaxation of INT with decision variable
[n]
[n]
X : ?d
! R where ?d
denotes the set of subsets S ? [n] of size |S| ? d; it is given as
X
X
e
maximize
?ij
X({i, j}) +
?iv X({i}) ,
(SOS)
X
i2V
(i,j)2E
subject to X(;) = 1, M(X) < 0 .
The moment matrix M(X) is indexed by sets S, T ? [n], |S|, |T | ? d/2, and has entries M(X)S,T =
X(S4T ) with 4 denoting the symmetric difference of two sets. Note that M(X)S,S = X(;) = 1.
3
Rank
20
10
5
3
2
1e+01
662
1e?01
Duality Gap
Objective Value
600
400
Rank
20
10
5
3
2
200
0
1e?03
1e?05
1e?07
0
50
100
150
Iterations
200
0
50
100
Iterations
150
200
Figure 2: Effect of the rank constraint r on n = 400 square lattice (20 ? 20): Left plot shows the
change in the value of objective at each iteration. Right plot shows the duality gap of the Lagrangian.
We can equivalently represent M(X) as a Gram matrix by letting M(X)S,T = h S , T i for a
[n]
[n]
collection of vectors S 2 Rr indexed by S 2 ?d/2
. The case r = ?d/2
can represent any
semidefinite matrix; however, in what follows it is convenient from a computational perspective to
consider smaller choices of r. The constraint M(X)S,S = 1 is equivalent to k S k = 1, and the
condition M (X)S,T = X(S4T ) can be equivalently written as
h
T1 i
S1 ,
=h
S2 ,
T2 i ,
8S1 4T1 = S2 4T2 .
(2.1)
In the case d = 2, SOS(2) recovers the classical Goemans-Williamson SDP relaxation [GW95].
In the following, we consider the simplest higher-order SDP, namely SOS(4) for which the general
constraints in Eq. (2.1) can be listed explicitly. Fixing a region R ? V , and defining the Gram vectors
; , ( i )i2V , ( ij ){i,j}?V , we list the constraints that involve vectors S for S ? R and |S| = 1, 2:
k ik = 1
h i, j i = h
h i , ij i = h
h
i,
h
ij ,
h
ij ,
jk i
jk i
kl i
=h
=h
;i
ij ,
=h
8i 2 S [ {;},
8i, j 2 S,
8i, j 2 S,
;i
j,
ij i
k,
ik ,
ik ,
8i, j, k 2 S,
;i
8i, j, k 2 S,
jl i
8i, j, k, l 2 S.
(Sphere i )
(Undirected i j)
(Directed i ! j)
(V-shaped ij V k )
i
(Triangle j 4k )
(Loop ik ?jl )
Given an assignment of the Gram vectors = ( ; , ( i )i2V , ( ij ){i,j}?V ), we denote by |R its
restriction to R, namely |R = ( ; , ( i )i2R , ( ij ){i,j}?R ). We denote by ?(R), the set of vectors
|R that satisfy the above constraints. With these notations, the SOS(4) SDP can be written as
X
X
e
maximize
?ij
h i, j i +
?iv h i , ; i ,
(SOS(4))
i2V
(i,j)2E
subject to
2 ?(V ) .
A specific Partial SOS (PSOS) relaxation is defined by a collection of regions R =
{R1 , R2 , . . . , Rm }, Ri ? V . We will require R to be a covering, i.e. [m
i=1 Ri = V and for
each (i, j) 2 E there exists ` 2 [m] such that {i, j} ? R` . Given such a covering, the PSOS(4)
relaxation is
X
X
e
maximize
?ij
h i, j i +
?iv h i , ; i ,
(PSOS(4))
i2V
(i,j)2E
subject to
|Ri 2 ?(Ri )
8i 2 {1, 2, . . . , m} .
Notice that variables ij only enter the above program ifP
{i, j} ? R` for some `. As a consequence,
m
the dimension of the above optimization problem is O(r `=1 |R` |2 ), which is O(nr) if the regions
have bounded size; this will be the case in our implementation. Of course, the specific choice of
regions R is crucial for the quality of this relaxation. A natural heuristic is to choose each region R`
to be a subset of contiguous vertices in G, which is generally the case for GBP algorithms.
4
Algorithm 1: Partial-SOS
Input :G = (V, E), ?e 2 Rn?n , ?v 2 Rn ,
Actives = V [ E \ Reliables, and = 1,
while > tol do
=0
for s 2 Actives do
if s 2 V then
P
e
v
cs = t2@s ?st
t + ?s ;
else
cs = ?se1 s2 ; + ?sv1 s2 + ?sv2
2 Rr?(1+|V |+|E|) , Reliables = ;
/* s 2 V is a vertex */
/* s = (s1 , s2 ) 2 E is an edge */
s1
Form matrix As , vector bs , and the corresponding Lagrange multipliers s (see text).
new
arg max hcs , i + ?2 kAs
bs + s k2
/* sub-problem */
s
s
s
k k=1
new
s
new
s
s + As s
+k
2
sk
+ kAs
s
bs k2
bs
/* update variables */
2.1 Tightness guarantees
Solving exactly INT is NP-hard even if G is a three-dimensional grid [Bar82]. Therefore, we do
not expect PSOS(4) to be tight for general graphs G. On the other hand, in our experiments (cf.
Section 4), PSOS(4) systematically achieves the exact maximum of INT for two-dimensional grids
e
with random edge and vertex parameters (?ij
)(i,j)2E , (?iv )i2V . This finding is quite surprising and
calls for a theoretical explanation. While full understanding remains an open problem, we present
here partial results in that direction.
Recall that a cycle in G is a sequence of distinct vertices (i1 , . . . , i` ) such that, for each j 2 [`] ?
{1, 2, . . . , `}, (ij , ij+1 ) 2 E (where ` + 1 is identified with 1). The cycle is chordless if there is no
j, k 2 [`], with j k 6= ?1 mod ` such that (ij , ik ) 2 E. We say that a collection of regions R
on graph G is circular if for each chordless cycle in G there exists a region in R 2 R such that all
vertices of the cycle belong to R. We also need the following straightforward notion of contractibility.
A contraction of G is a new graph obtained by identifying two vertices connected by an edge in G.
G is contractible to H if there exists a sequence of contractions transforming G into H.
The following theorem is a direct consequence of a result of Barahona and Mahjoub [BM86] (see
Supplement for a proof).
Theorem 1. Consider the problem INT with ?v = 0. If G is not contractible to K5 (the complete
graph over 5 vertices), then PSOS(4) with a circular covering R is tight.
The assumption that ?v = 0 can be made without loss of generality (see Supplement for the reduction
from the general case). Furthermore, INT can be solved in polynomial time if G is planar, and ?v = 0
[Bar82]. Note however, the reduction from ?v 6= 0 to ?v = 0 can transform a planar graph to a
non-planar graph. This theorem implies that (full) SOS(4) is also tight if G is not contractible to K5 .
Notice that planar graphs are not contractible to K5 , and we recover the fact that INT can be solved
in polynomial time if ?v = 0. This result falls short of explaining the empirical findings in Section 4,
for at least two reasons. Firstly the reduction to ?v = 0 induces K5 subhomomorphisms for grids.
Second, the collection of regions R described in the previous section does not include all chordless
cycles. Theoretically understanding the empirical performance of PSOS(4) as stated remains open.
However, similar cycle constraints have proved useful in analyzing LP relaxations [WRS16].
3
3.1
Optimization Algorithm and Rounding
Solving PSOS(4) via Trust-Region Coordinate Ascent
We will approximately solve PSOS(4) while keeping r = O(1). Earlier work implies
p that (under
suitable genericity condition on the SDP) there existspan optimal solution with rank 2 # constraints
[Pat98]. Recent work [BVB16] shows that for r > 2 # constraints, the non-convex optimization
problem has no non-global local maxima. For SOS(2), [MM+ 17] proves that setting r = O(1)
is sufficient for achieving O(1/r) relative error from the global maximum for specific choices of
potentials ?e , ?v . We find that there is little or no improvement beyond r = 10 (cf. Figure 2).
5
Algorithm 2: CLAP: Confidence Lift And Project
Input :G = (V, E), ?e 2 Rn?n , ?v 2 Rn , regions R = {R1 , ..., Rm }
Initialize variable matrix 2 Rr?(1+|V |+|E|) and set Reliables = ;.
while Reliables 6= V [ E do
Run Partial-SOS on inputs G = (V, E), ?e , ?v , , Reliables
/* lift procedure */
Promotions = ; and Confidence = 0.9
while Confidence > 0 and Promotions 6= ; do
for s 2 V [ E \ Reliables do
/* find promotions */
if |h ; , s i| > Confidence then
/* project procedure */
s = sign(h ; , s i) ? ;
Promotions
Promotions [ {sc }
if Promotions = ; then
/* decrease confidence level */
Confidence
Confidence 0.1
Reliables
Reliables [ Promotions
/* update Reliables */
Output :(h
i,
; i)i2V
2 { 1, +1}n
We will assume that R = (R1 , . . . , Rm ) is a covering of G (in the sense introduced in the previous
section), and ?without loss of generality? we will assume that the edge set is
E = (i, j) 2 V ? V :
9` 2 [m] such that {i, j} ? R` .
(3.1)
In other words, E is the maximal set of edges that is compatible with R being a covering. This can
e
always be achieved by adding new edges (i, j) to the original edge set with ?ij
= 0. Hence, the
decision variables s are indexed by s 2 S = {;} [ V [ E. Apart from the norm constraints, all other
consistency constraints take the form h s , r i = h t , p i for some 4-tuple of indices (s, r, t, p). We
denote the set of all such 4-tuples by C, and construct the augmented Lagrangian of PSOS(4) as
?2
X
X
? X ?
e
L( , ) =
?iv h i , ; i +
?ij
h i, j i +
h s , r i h t , p i + s,r,t,p .
2
i2V
(i,j)2E
(s,r,t,p)2C
At each step, our algorithm execute two operations: (i) maximize the cost function with respect to one
of the vectors s ; (ii) perform one step of gradient descent with respect to the corresponding subset
of Lagrangian parameters, to be denoted by s . More precisely, fixing s 2 S \ {;} (by rotational
invariance, it is not necessary to update ; ), we note that s appears in the constraints linearly (or
it does not appear). Hence, we can write these constraints in the form As s = bs where As , bs
depend on ( r )r6=s but not on s . We stack the corresponding Lagrangian parameters in a vector s ;
therefore the Lagrangian term involving s reads (?/2)kAs s bs + s k2 . On the other hand, the
graphical model contribution is that the first two terms in L( , ) are linear in s , and hence they
can be written as hcs , s i. Summarizing, we have
L( , ) =hcs ,
si
+ kAs
bs +
s
2
sk
+ Le (
r )r6=s ,
.
(3.2)
It is straightforward to compute As , bs , cs ; in particular, for (s, r, t, p) 2 C, the rows of As and bs
are indexed by r such that the vectors r form the rows of As , and h t , p i form
P the corresponding
e
v
entry of bs . Further, if s is a vertex and @s are its neighbors, we set cs = t2@s ?st
t + ?s ;
e
v
v
while if s = (s1 , s2 ) is an edge, we set cs = ?s1 s2 ; + ?s1 s2 + ?s2 s1 . Note that we are using the
equivalent representations h i , j i = h ij , ; i, h ij , j i = h i , ; i, and h ij , i i = h j , ; i.
Finally, we maximize Eq. (3.2) with respect to
s
by a Mor?-Sorenson style method [MS83].
3.2 Rounding via Confidence Lift and Project
After Algorithm 1 generates an approximate optimizer for PSOS(4), we reduce its rank to produce
a solution of the original combinatorial optimization problem INT. To this end, we interpret h i , ; i
as our belief about the value of xi in the optimal solution of INT, and h ij , ; i as our belief about
the value of xi xj . This intuition can be formalized using the notion of pseudo-probability [BS16].
We then recursively round the variables about which we have strong beliefs; we fix rounded variables
in the next iteration, and solve the induced PSOS(4) on the remaining ones.
More precisely, we set a confidence threshold Confidence. For any variable s such that |h s , ; i| >
Confidence, we let xs = sign(h s , ; i) and fix s = xs ; . These variables s are no longer
6
Noisy
BP-SP
BP-MP
GBP
PSOS(2)
PSOS(4)
Bernoulli p = 0.2
True
Blockwise p=0.006
U(x) : 25815 19237 26165 26134 26161 26015 26194
Time: - - 2826s 2150s
7894s
454s
5059s
U(x) : 27010 26808 27230 27012 27232 26942 27252
Time: - - 1674s 729s
8844s
248s
4457s
Figure 3: Denoising a binary image by maximizing the objective function Eq. (4.1). Top row: i.i.d.
Bernoulli error with flip probability p = 0.2 with ?0 = 1.26. Bottom row: blockwise noise where
each pixel is the center of a 3 ? 3 error block independently with probability p = 0.006 and ?0 = 1.
updated, and instead the reduced SDP is solved. If no variable satisfies the confidence condition, the
threshold is reduced until variables are found that satisfy it. After the first iteration, most variables
yield strong beliefs and are fixed; hence the consequent iterations have fewer variables and are faster.
4
Numerical Experiments
In this section, we validate the performance of the Partial SOS relaxation and the CLAP rounding
scheme on models defined on two-dimensional grids. Grid-like graphical models are common in a
variety of fields such as computer vision [SSZ02], and statistical physics [MM09]. In Section 4.1, we
study an image denoising example and in Section 4.2 we consider the Ising spin glass ? a model in
statistical mechanics that has been used as a benchmark for inference in graphical models.
Our main objective is to demonstrate that Partial SOS can be used successfully on large-scale
graphical models, and is competitive with the following popular inference methods:
? Belief Propagation - Sum Product (BP-SP): Pearl?s belief propagation computes exact marginal
distributions on trees [Pea86]. Given a graph structured objective function U (x), we apply BP-SP
to the Gibbs-Boltzmann distribution p(x) = exp{U (x)}/Z using the standard sum-product update
rules with an inertia of 0.5 to help convergence [YFW05], and threshold the marginals at 0.5.
? Belief Propagation - Max Product (BP-MP): By replacing the marginal probabilities in the sumproduct updates with max-marginals, we obtain BP-MP, which can be used for exact inference
on trees [MM09]. For general graphs, BP-MP is closely related to an LP relaxation of the
combinatorial problem INT [YFW05, WF01]. Similar to BP-SP, we use an inertia of 0.5. Note
that the Max-Product updates can be equivalently written as Min-Sum updates [MM09].
? Generalized Belief Propagation (GBP): The decision variables in GBP are beliefs (joint probability distributions) over larger subsets of variables in the graph G, and they are updated in a
message passing fashion [YFW00, YFW05]. We use plaquettes in the grid (contiguous groups of
four vertices) as the largest regions, and apply message passing with inertia 0.1 [WF01].
? Partial SOS - Degree 2 (PSOS(2)): By defining regions as single vertices and enforcing only
the sphere constraints, we recover the classical Goemans-Williamson SDP relaxation [GW95].
Non-convex Burer-Monteiro approach is extremely efficient in this case [BM03]. We round the
SDP solution by x
?i = sign(h i , ; i) which is closely related to the classical approach of [GW95].
? Partial SOS - Degree 4 (PSOS(4)): This is the algorithm developed p
in the present
paper. We
p
take
R` to be triangles, cf. Figure 1, right frame. In an n ? n grid, we have
p the regions
2( n 1)2 such regions resulting in O(n) constraints. In Figures 3 and 4, PSOS(4) refers to the
CLAP rounding scheme applied together with PSOS(4) in the lift procedure.
4.1 Image Denoising via Markov Random Fields
p
p
Given a n ? n binary image x0 2 {+1, 1}n , we generate a corrupted version of the same
image y 2 {+1, 1}n . We then try to denoise y by maximizing the following objective function:
X
X
U (x) =
xi xj + ?0
yi x i ,
(4.1)
i2V
(i,j)2E
7
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
PS
O
PS S(4
O )
GB S(2
P )
BP
-S
BP P
-M
P
Ratio to the best algorithm
Figure 4: Solving the MAP inference problem INT for Ising spin glasses on two-dimensional grids.
U and N represent uniform and normal distributions. Each bar contains 100 independent realizations.
We plot the ratio between the objective value achieved by that algorithm and the exact optimum for
n 2 {16, 25}, or the best value achieved by any of the 5 algorithms for n 2 {100, 400, 900}.
p
p
p
where the graph G is the n ? n grid, i.e., V = {i = (i1 , i2 ) : i1 , i2 2 {1, . . . , n}} and
E = {(i, j) : ki jk1 = 1}. In applying Algorithm 1, we add diagonals to the grid (see right plot
e
in Figure 1) in order to satisfy the condition (3.1) with corresponding weight ?ij
= 0.
In Figure 3, we report the output of various algorithms for a 100 ? 100 binary image. We are
not aware of any earlier implementation of SOS(4) beyond tens of variables, while PSOS(4) is
applied here to n = 10, 000 variables. Running times for CLAP rounding scheme (which requires
several runs of PSOS(4)) are of order an hour, and are reported in Figure 3. We consider two noise
models: i.i.d. Bernoulli noise and blockwise noise. The model parameter ?0 is chosen in each
case as to approximately optimize the performances under BP denoising. In these (as well as in 4
other experiments of the same type reported in the supplement), PSOS(4) gives consistently the best
reconstruction (often tied with GBP), in reasonable time. Also, it consistently achieves the largest
value of the objective function among all algorithms.
4.2
Ising Spin Glass
The Ising spin glass (also known as Edwards-Anderson model [EA75]) is one of the most studied
models in statistical physics. It is given by an objective function of the form INT with G a de
dimensional grid, and i.i.d. parameters {?ij
}(i,j)2E , {?iv }i2V . Following earlier work [YFW05],
we use Ising spin glasses as a testing ground for our algorithm. Denoting the uniform and normal
distributions by U and N respectively, we consider two-dimensional grids (i.e. d = 2), and the
e
e
following parameter distributions: (i) ?ij
? U ({+1, 1}) and ?iv ? U ({+1, 1}), (ii) ?ij
?
v
e
v
2
U ({+1, 1}) and ?i ? U ({+1/2, 1/2}), (iii) ?ij ? N(0, 1) and ?i ? N(0, ) with = 0.1
e
(this is the setting considered in [YFW05]), and (iv) ?ij
? N(0, 1) and ?iv ? N(0, 2 ) with = 1.
For each of these settings, we considered grids of size n 2 {16, 25, 100, 400, 900}.
In Figure 4, we report the results of 8 experiments as a box plot. We ran the five inference algorithms
described above on 100 realizations; a total of 800 experiments are reported in Figure 4. For each of
the realizations, we record the ratio of the achieved value of an algorithm to the exact maximum (for
n 2 {16, 25}), or to the best value achieved among these algorithms (for n 2 {100, 400, 900}). This
is because for lattices of size 16 and 25, we are able to run an exhaustive search to determine the true
maximizer of the integer program. Further details are reported in the supplement.
In every single instance of 800 experiments, PSOS(4) achieved the largest objective value, and
whenever this could be verified by exhaustive search (i.e. for n 2 {16, 25}) it achieved an exact
maximizer of the integer program.
8
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9
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6,241 | 6,646 | Variable Importance using Decision Trees
Jalil Kazemitabar
UCLA
[email protected]
Arash A. Amini
UCLA
[email protected]
Adam Bloniarz
UC Berkeley?
[email protected]
Ameet Talwalkar
CMU
[email protected]
Abstract
Decision trees and random forests are well established models that not only offer
good predictive performance, but also provide rich feature importance information.
While practitioners often employ variable importance methods that rely on this
impurity-based information, these methods remain poorly characterized from a
theoretical perspective. We provide novel insights into the performance of these
methods by deriving finite sample performance guarantees in a high-dimensional
setting under various modeling assumptions. We further demonstrate the effectiveness of these impurity-based methods via an extensive set of simulations.
1
Introduction
Known for their accuracy and robustness, decision trees and random forests have long been a
workhorse in machine learning [1]. In addition to their strong predictive accuracy, they are equipped
with measures of variable importance that are widely used in applications where model interpretability
is paramount. Importance scores are used for model selection: predictors with high-ranking scores
may be chosen for further investigation, or for building a more parsimonious model.
One common approach naturally couples the model training process with feature selection [2, 5].
This approach, which we call T REE W EIGHT, calculates the feature importance score for a variable
by summing the impurity reductions over all nodes in the tree where a split was made on that
variable, with impurity reductions weighted to account for the size of the node. For ensembles, these
quantities are averaged over constituent trees. T REE W EIGHT is particularly attractive because it can
be calculated without any additional computational expense above the standard training procedure.
However, as the training procedure in random forests combines several complex ingredients?bagging,
random selection of predictor subsets at nodes, line search for optimal impurity reduction, recursive
partitioning?theoretical investigation into T REE W EIGHT is extremely challenging. We propose a
new method called DS TUMP that is inspired by T REE W EIGHT but is more amenable to analysis.
DS TUMP assigns variable importance as the impurity reduction at the root node of a single tree.
In this work we characterize the finite sample performance of DS TUMP under an additive regression
model, which also yields novel results for variable selection under a linear model, both with correlated
and uncorrelated design. We corroborate our theoretical analyses with extensive simulations in which
we evaluate DS TUMP and T REE W EIGHT on the task of feature selection under various modeling
assumptions. We also compare the performance of these techniques against established methods
whose behaviors have been theoretically characterized, including Lasso, SIS, and SpAM [12, 3, 9].
?
Now at Google
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Our work provides the first finite-sample high-dimensional analyses of tree-based variable selection
techniques, which are commonly used in practice but lacking in theoretical grounding. Although
we focus on DS TUMP, which is a relatively simple tree-based variable selection approach, our
novel proof techniques are highly non-trivial and suggest a path forward for studying more general
multi-level tree-based techniques such as T REE W EIGHT. Moreover, our simulations demonstrate
that such algorithmic generalizations exhibit impressive performance relative to competing methods
under more realistic models, e.g., non-linear models with interaction terms and correlated design.
2
Related Work
Our analysis is distinct from existing work in analyzing variable importance measures of trees and
forests in several ways. To our knowledge, ours is the first analysis to consider the high-dimensional
setting, where the number of variables, p, and the size of the active set s, grow with the sample size
n, and potentially p n.
The closest related work is the analysis of [8], which considers a fixed set of variables, in the limit of
infinite data (n = ?). Unlike DS TUMP?s use of the root node only, [8] does consider importance
scores derived from the full set of splits in a tree as in T REE W EIGHT. However, they make crucial
simplifying (and unrealistic) assumptions that are distinct from those of our analysis: (1) each variable
is split on only once in any given path from the root to a leaf of the tree; (2) at each node a variable is
picked uniformly at random among those not yet used at the parent nodes, i.e., the splits themselves
are not driven by impurity reduction; and (3) all predictors are categorical, with splits being made on
all possible levels of a variable, i.e., the number of child nodes equals the cardinality of the variable
being split. Our analysis instead considers continuous-valued predictors, the split is based on actual
impurity reduction, and our results are nonasymptotic, i.e. they give high-probability bounds on
impurity measures for active and inactive variables that hold in finite samples.
A second line of related work is motivated by a permutation-based importance method [1] for feature
selection. In practice, this method is computationally expensive as it determines variable importance
by comparing the predictive accuracy of a forest before and after random permutation of a predictor.
Additionally, due to the algorithmic complexity of the procedure, it is not immediately amenable to
theoretical analysis, though the asymptotic properties of a simplified variant of the procedure have
been studied in [6].
While our work is the first investigation of finite-sample model selection performance of tree-based
regression methods, alternative methods performing both linear and nonparametric regression in high
dimensions have been studied in the literature. Considering model selection consistency results, most
of the attention has been focused on the linear setting, whereas the nonparametric (nonlinear) setup
has been mostly studied in terms of the prediction consistency. Under a high-dimensional linear
regression model, L ASSO has be extensively studied and is shown to be minimax optimal for variable
selection under appropriate regularity conditions, including the uncorrelated design with a moderate
?min condition. Remarkably, while not tailored to the linear setting, we show that DS TUMP is nearly
minimax optimal for variable selection in the same uncorrelated design setting (cf. Corollary 1). In
fact, DS TUMP can be considered a nonlinear version of SIS [4], itself a simplified form of the L ASSO
when one ignores correlation among features (cf. Section 3 for more details).
The Rodeo framework [7] performs automatic bandwidth selection and variable selection for local
linear smoothers, and is tailored to a more general nonparametric model with arbitrary interactions.
It was shown to possess model selection consistency in high dimensions; however, the results are
asymptotic and focus on achieving optimal prediction rate. In particular, there is no clear ?min
threshold as a function of n, s, and p. RODEO is also computationally burdensome for even modestsized problems (we thus omit it our experimental results in Section 4).
Among the nonlinear methods, S PAM is perhaps the most well-understood in terms of model
selection properties. Under a general high-dimensional sparse additive model, S PAM possesses the
sparsistency property (a term for model selection consistency); the analysis is reduced to a linear
setting by considering expansions in basis functions, and selection consistency is proved under an
irrepresentible condition on the coefficients in those bases. We show that DS TUMP is model selection
consistent in the sparse additive model with uncorrelated design. Compared to S PAM results, our
conditions are stated directly in terms of underlying functions and are not tied to a particular basis;
2
hence our proof technique is quite different. There is no implicit reduction to a linear setting via basis
expansions. Empirically, we show that DS TUMP indeed succeeds in the settings our theory predicts.
3
Selection consistency
The general model selection problem for non-parametric regression can be stated as follows: we
observe noisy samples yi = f (xi1 , . . . , xip ) + wi , i = 1, . . . , n where {wi } is an i.i.d. noise
sequence. Here, p is the total number of features (or covariates) and n is the total number of
observations (or the sample size). In general, f belongs to a class F of functions from Rp ? R.
One further assumes that the functions in F depend on at most s of the features, usually with s p.
That is, for every f ? F, there is some f0 : Rs ? R and a subset S ? [p] with |S| ? s such
that f (z1 , . . . , zp ) = f0 (zS ) where zS = (zi , i ? S). The subset S, i.e., the set of active features,
is unknown in advance and the goal of model selection is to recover it given {(yi , xi )}ni=1 . The
problem is especially challenging in the high-dimensional setting where p n. We will consider
various special cases of this general model when we analyze DS TUMP. For theoretical analysis it is
common to assume s to be known and we will make this assumption throughout. In practice, one
often considers s to be a tunable parameter that can be selected, e.g., via cross-validation or greedy
forward selection.
We characterize the model selection performance of DS TUMP by establishing its sample complexity:
that is, the scaling of n, p, and s that is sufficient to guarantee that DS TUMP identifies the active
set of features with probability converging to 1. Our general results, proved in the technical report,
allow for a correlated design matrix and additive nonlinearities in the true regression function. Our
results for the linear case, derived as a special case of the general theory, allow us to compare the
performance of DS TUMP to the information theoretic limits for sample complexity established in
[11], and to the performance of existing methods more tailored to this setting, such as the Lasso [12].
Given a generative model and the restriction of DS TUMP to using root-level impurity reduction, the
general thrust of our result is straightforward: impurity reduction due to active variables concentrates
at a significantly higher level than that of inactive variables. However, there are significant technical
challenges in establishing this result, mainly deriving from the fact that the splitting procedure
renders the data in the child nodes non-i.i.d., and hence standard concentration inequalities do not
immediately apply. We leverage the fact that the DS TUMP procedure considers splits at the median
of a predictor. Given this median point, the data in each child node is i.i.d., and hence we can
apply standard concentration inequalities in this conditional distribution. Removing this conditioning
presents an additional technical subtlety. For ease of exposition, we first present our results for the
linear setting in Section 3.1, and subsequently summarize our general results in Section 3.2. We
provide a proof of our result in the linear setting in Section 3.3, and defer the proof of our general
result to the supplementary material.
Algorithm 1 DS TUMP
k=p
input {xk ? Rn }k=1
, y ? Rn , # top features s
n
m= 2
for k = 1, . . . , p do
I(xk ) = SortFeatureValues(xk )
y k = SortLabelByFeature(y, I(xk ))
k
k
y[m]
, y[n]\[m]
= SplitAtMidpoint(y k )
k
k
ik = ComputeImpurity(y[m]
, y[n]\[m]
)
end for
S = FindTopImpurityReductions({ik }, s)
output top s features sorted by impurity reduction
The DS TUMP algorithm. In order to describe DS TUMP more precisely, let us introduce
some notation. We write [n] := {1, . . . , n}.
Throughout, y = (yi , i ? [n]) ? Rn will be the
response vector observed for a sample of size
n. For an ordered index set I = (i1 , i2 , . . . , ir ),
we set yI = (yi1 , yi2 . . . , yir ). A similar notation is used for unordered index sets. We write
xj = (x1j , x2j , . . . , xnj ) ? Rn for the vector
collecting values of the jth feature; xj forms the
jth column of the design matrix X ? Rn?p .
Let I(xj ) := (i1 , i2 . . . , in ) be an ordering of
[n] such that xi1 j ? xi2 j ? ? ? ? ? xin j and let
sor(y, xj ) := yI(xj ) ? Rn ; this is an operator
that sorts y relative
to xj . DS TUMP proceeds as
P
follows: Evaluate y k := sor(y, xk ) = sor
?
x
+
w,
x
,
for
k
= 1, . . . , p. Let m := n/2.
j
j
k
j?S
k
k
k
For each k, consider the midpoint split of y into y[m] and y[n]\[m] and evaluate the impurity of the
3
left-half, using empirical variance as impurity:
k
imp(y[m]
) :=
1
m
2
X
1?i<j?m
1 k
(y ? yjk )2 .
2 i
(1)
k
Let imp(y[m]
) be the score of feature k, and output the s features with the smallest scores (corresponding to maximal reduction in impurity). If the generative model is linear, the choice of the
midpoint is justified by our assumption of the uniform distribution for the features (Zi ), and we
further show that this simple choice is effective even under a nonlinear model. The choice of the
left-half in our analysis is for convenience; a similar analysis applies if we take the impurity to be that
of the sum of both halves (or their maximum). DS TUMP is summarized in Algorithm 1. Impurity
k
reduction imp(y[m] ) ? imp(y[m]
) can be considered a form of nonlinear correlation between y and
feature xk . The SIS algorithm is equivalent to replacing this nonlinear correlation with the (absolute)
linear correlation | n1 xTk y|. That is, both procedures assign a score to each feature by considering it
against the response separately, ignoring other features. In the uncorrelated (i.e. orthogonal design)
setting, this is more or less optimal, and as is the case with SIS, we show that DS TUMP also retains
some model selection performance even under correlated designs. In contrast to SIS, we show that
DS TUMP also enjoys performance guarantees in non-linear settings.
The models. We present our consistency results for models of various complexity. We start with the
well-known and extensively studied setting of a linear model with IID design. This basic setup serves
as the benchmark for comparison of model selection procedures. As will become clear in the course
of the proof, analyzing DS TUMP (or impurity-based feature selection in general) is challenging even
in this case, in contrast to linear model based approaches such as SIS or Lasso. Once we have a good
understanding of DS TUMP under the baseline model, we extend the analysis to correlated design and
nonlinear additive models. The structure of our proof is also most clearly seen in this simple case, as
outlined in Section 3.3. We now introduce our general models:
Model 1 (Sparse linear model with ICA-type design). A linear regression model y = X? + w with
e ? Rn?p
e where X
ICA-type (random) design X ? Rn?p has the following properties: (i) X = XM
e is an independent draw from a (column) vector Z = (Z1 , . . . , Zp ) with IID
and each row of X
entries drawn uniformly from [0, 1]. (ii) The noise vector w = (w1 , . . . , wn ) has IID sub-Gaussian
2
entries with variance with variance var(wi ) = vw
and sub-Gaussian norm kwi k?2 ? ?w , . (iii) The
p
? ? R is s-sparse, namely, ?j 6= 0 for j ? S = {1, . . . , s} and zero otherwise.
Model 1 serves both the correlated and uncorrelated design cases. Each row of the design matrix
X is a draw from the vector M T Z, which has covariance c M T M for some constant c. Thus, the
choice of M = I leads to an uncorrelated design. The choice of the interval [0, 1] for covariates
is for convenience; it can be replaced with any other compact interval, in the linear setting, since
variance impurity is invariant to a shift. Similarly the choice of the (active) support indices, S, is for
2
convenience. For simplicity, we often assume vw
= ? 2 and ?w ? C? (only ?w would affect the
results as examining of our proofs shows).
Model
2 (Sparse additive model with uncorrelated design). An additive regression model yi =
Pp
j=1 fj (xij ) + wi , is one with random design X = (xij ) and the noise (wi ) as in Model 1, with
M = I (uncorrelated design). We assume (fk ) to be s-sparse, namely, fj 6= 0 for j ? S = {1, . . . , s}
and zero otherwise.
3.1
Linear Setting
Uncorrelated design. Our baseline result is the following feature selection consistency guarantee
for DS TUMP, for the case M = I of Model 1. Throughout, we let p? := p?s, and C, C1 , . . . , c, c1 , . . .
are absolute positive constants which can be different in each occurrence. For any vector x, let
|x|min := mini |xi |, the minimum absolute value of its entries. The quantity |?S |2min = mink ? S ?k2
appearing in Theorem 1 is a well-known parameter controlling hardness of subset recovery. All our
results are stated in terms of constants ?, ? and ? that are related as:
? ? (0, 1/8),
? = log(1/(8?)),
4
? = 1 ? (1 ? ?)2 .
(2)
Theorem 1. Assume Model 1 with M = I, and (2). The DS TUMP algorithm, which selects
the ?s? least impure features at the root, succeeds in feature selection, with probability at least
1 ? p??c ? 2e??n/2 if log p?/n ? C1 and
r
C
log p?
2
2
2
|?S |min ?
(k?k2 + ? )
(3)
?
n
The result can be read by setting, e.g., ? = 1/16 leading to numerical constants for ? and ?. The
current form allows the flexibility to trade-off the constant (?) in the probability bound with the
constant (?) in the gap condition (3). Although Theorem 1 applies to a general ?, it is worthwhile to
see its consequence in a special regime of interest where |?S |2min 1/s, corresponding to k?k2 1.
We get the following immediate corollary:
Corollary 1. Assume |?S |2min 1/s, ? 2 1 and log p?/n = O(1). Then DS TUMP succeeds with
high probability if n & s2 log p?.
The minimax optimal threshold for support recovery in the regime of Corollary 1 is known to be
n s log p? [11], and achieved by LASSO [12]. Although this result is obtained for Gaussian design,
the same argument goes through for our uniform ensemble. Compared to the optimal threshold, using
DS TUMP we pay a small factor of s in the sample complexity. However, DS TUMP is not tied to the
linear model and as we discuss in Section 3.2, we can generalize the performance of DS TUMP to
nonlinear settings.
Correlated design. We take the following approach to generalize our result to the correlated case:
(1) We show a version of Theorem 1, which holds for an ?approximately sparse? parameter ?e with
uncorrelated design. (2) We derive conditions on M such that the correlated case can be turned into
the uncorrelated case with approximate sparsity. The following theorem details Step 1:
e a general vector in
Theorem 2. Assume Model 1(i)-(ii) with M = I, but instead of (iii) let ? = ?,
p
R . Let S be any subset of [p] of cardinality s. The DS TUMP algorithm, which selects the ?s? least
impure features at the root, recovers S, withp
probability at least 1 ? p??c ? 2e??n/2 if log p?/n ? C1
2
2
2
2
e + ? ) (log p?)/n.
and ?|?eS |min ? k?eS c k? > C(k?k
2
The theorem holds for any ?e and S, but the gap condition required is likely to be violated unless ?e
is approximately sparse w.r.t. S. Going back to Model 1, we see that setting ?e = M ? transforms
the model with correlated design X, and exact sparsity on ?, to the model with uncorrelated design
e The following corollary gives sufficient conditions on M , so
e and approximate sparsity on ?.
X,
that Theorem 2 is applicable. Recall theP
usual (vector) `? norm, kxk? = maxi |xi |, the matrix
`? ? `? operator norm |||A|||? = maxi j |Aij | , and the `2 ? `2 operator norm |||A|||2 .
Corollary 2. Consider a general ICA-type Model 1 with ? and M satisfying
k?S k? ? ?|?S |min ,
|||MSS ? I|||? ?
1??
,
?
|||MS c S |||? ?
?p
?(1 ? ?)
?
(4)
for some ?, ? ? (0, 1] and ? ? 1. Then, the conclusion of Theorem 1 holds, for DS TUMP applied to
e under the gap condition (3) with C/? replaced with C|||MSS |||2 /(? ? ?2 ).
input (y, X),
2
e is reasonable in cases where one can perform consistent ICA.
Access to decorrelated features, X,
This assumption is practically plausible, especially in the low-dimensional regimes, though it would
be desirable if this assumption can be removed theoretically. Moreover, we note that the response y
is based on the correlated features.
In this result, C|||MSS |||22 /(? ? ?2 ) plays the role of a new constant. There is a hard bound on how big
? can be, which via (4) controls how much correlation between off-support and on-support features
are
? tolerated. For example, taking ? = 1/9, we have ? = log(9/8) ? 0.1, ? = 17/81 ? 0.2 and
? ? 0.45 and this is about as big as it can get (the maximum we can allow is ? 0.48). ? can be
arbitrarily close to 0, relaxing the assumption (4), at the expense of increasing the constant in the
threshold. ? controls
deviation of |?j |, j ? S from uniform: in case of equal weights on the support,
?
i.e., |?j | = 1/ s for j ? S, we have ? = 1. Theorem 1 for the uncorrelated design is recovered, by
taking ? = ? = 1.
5
3.2
General Additive Model Setting
To prove results in this more general setting, we need some further regularity conditions on (fk ): Fix
some ? ? (0, 1), let U ? unif(0, 1) and assume the following about the underlying functions (fk ):
2
(F1) kfk (?U )k2?2 ? ?f,k
, ?? ? [0, 1]. (F2) var[fk (?U )] ? var[fk ((1 ? ?)U )], ?? ? 1 ? ?.
P
P
p
2
2
2
Next, we define ?f,?
:= k=1 ?f,k
= k?S ?f,k
along with the following key gap quantities:
gf,k (?) := var[fk (U ))] ? var[fk ((1 ? ?)U )].
1
Theorem 3. Assume additive Model 2 with (F1) and (F2). Let ? = log 8?
for ? ? (0, 1/8). The
DS TUMP algorithm, which selects the ?s? least impure features at the root, succeeds in model
selection, with probability at least 1 ? p??c ? 2e??n/2 if log p?/n ? C1 and
r
log p?
2
2
min gf,k (?) ? C(?f,? + ? )
(5)
k?S
n
In the supplementary material, we explore in detail the class of functions that satisfy conditions (F1)
and (F2), as well as the gap condition in (5). (F1) is relatively mild and satisfied if f is Lipschitz
or bounded. (F2) is more stringent and we show that it is satisfied for convex nondecreasing and
concave nonincreasing functions.2 The gap condition is less restrictive than (F2) and is related to the
slope of the function near the endpoint, i.e., x = 1. Notably, we study one such function that satisfies
all of these conditions, i.e., exp(?) on [?1, 1], in our simulations in Section 4.
3.3
Proof of Theorem 1
We provide the high-level proof of Theorem 1. For brevity, the proofs of the lemmas have been
omitted and can be found in the supplement, where we in fact prove them for the more general setup
of Theorem 3. The analysis boils down to understanding the behavior of y k = sor(y, xk ) as defined
k
earlier. Let yek be obtained from y k by random reshuffling of its left half y[m]
(i.e., rearranging the
entries according to a random permutation). This reshuffling has no effect on the impurity, that is,
k
k
imp(e
y[m]
) = imp(y[m]
), and the reason for it becomes clear when we analyze the case k ? S.
Understanding the distribution of y k . If k ?
/ S, the ordering according to which we sort y is
independent of y (since xk is independent of y), hence the sorted version, before and after reshuffling
has the same distribution as y. Thus, each entry of yek is an IID draw from the same distribution as
the pre-sort version:
X
iid
yeik ? W0 :=
?j Zj + w1 , i = 1, . . . , n.
(6)
j?S
On the other hand, if k ? S, then for i = 1, . . . , n
yik = ?k x(i)k + rik ,
iid
where rik ? Wk :=
X
?j Zj + w1 .
j?S\{k}
Here x(i)k is the ith order statistic of xk , that is, x(1)k ? x(2)k ? ? ? ? ? x(n)k . Note that the residual
terms are still IID since they gather the covariates (and the noise) that are independent of the kth one
and hence its ordering. Note also that rik is independent of the first term ?k x(i)k .
k
Recall that we split at the midpoint and focus on the left split, i.e., we look at y[n/2]
=
k k
k
k
k k
k
(y1 , y2 , . . . , yn/2 ), and its reshuffled version ye[n/2] = (e
y1 , ye2 , . . . , yen/2 ). Intuitively, we would
k
like to claim that the ?signal part? of the ye[n/2]
are approximately IID draws from ?k Unif(0, 1/2).
Unfortunately this is not true, in the sense that the distribution cannot be accurately approximated by
Unif(0, 1 ? ?) for any ? (Lemma 1). However, we show that the distribution can be approximated by
an infinite mixture of IID uniforms of reduced range (Lemma 2).
Let U(1) ? U(2) ? ? ? ? ? U(n) be the order statistics obtained by ordering an IID sample Ui ?
e := (U
e1 , U
e2 . . . , U
em ) be obtained from
Unif(0, 1), i = 1, . . . , n. Recall that m := n/2 and let U
2
We also observe that this condition holds for functions beyond these two categories.
6
e has an exchangeable distribution. We can write
(U(1) , . . . , U(m) ) by random permutation. Then, U
for k ? S,
yeik = ?k u
eki + reik ,
e,
u
ek ? U
and
iid
reik ? Wk , i ? [m]
where the m-vectors u
ek = (e
uki , i ? [m]) and rek = (e
rik , i ? [m]) are also independent.
e:
We have the following result regarding the distribution of U
e is a mixture of IID unif(0, ?) m-vectors with mixing variable
Lemma 1. The distribution of U
? ? Beta(m, m + 1).
Note that Beta(m, m + 1) has mean = m/(2m + 1) = (1 + o(1))/2 as m ? ?, and variance
e
= O(m?1 ). Thus, Lemma 1 makes our intuition precise in the sense that the distribution of U
is a ?range mixture? of IID uniform distributions, with the range concentrating around 1/2. We
now provide a reduced range, finite sample approximation in terms of the total variation distance
e, U
b ) between the distributions of random vectors U
e and U
b.
dTV (U
b be distributed according to a mixture of IID Unif(0, ?
b) m-vectors with ?
b distributed
Lemma 2. Let U
??
e
as a Beta(m, m + 1) truncated to (0, 1 ? ?) for ? = e /8 and ? > 0. With U as in Lemma 1, we
e, U
b ) ? 2 exp(??m).
have dTV (U
e by a truncated version, U
b , is an essential technique in
The approximation of the distribution of the U
our proof. As will become clear in the proof of Lemma 3, we will need to condition on the mixing
e , or its truncated approximation U
e , to allow for the use of concentration inequalities for
variable U
independent variables. The resulting bounds should be devoid of randomness so that by taking
expectation, we can get similar bounds for the exchangeable case. The truncation allows us to
maintain a positive gap in impurities (between on and off support features) throughout this process.
We expect the loss due to truncation to be minimal, only impacting the constants.
b described in Lemma 2,
For k ? S, let u
bk = (b
uki , i ? [m]) be drawn from the distribution of U
independently of anything else in the model, and let ?
bk be its corresponding mixing variable, which
has a Beta distribution truncated to (0, 1 ? ?). Let us define ybik = ?k u
bki + reik , i ? [m] where
k
k
k
k
re = (e
ri ) is as before. This construction provides a simple coupling between ye[m]
and yb[m]
giving
k
the same bound on the their total variation distance. Hence, we can safely work with yb[m] instead
k
of ye[m]
, and pay a price of at most 2 exp(??m) in probability bounds. To simplify discussion, let
k
ybi = yeik for k ?
/ S.
k
Concentration of empirical impurity. We will focus on yb[m]
due the discussion above. We would
k
k
k
like to control imp(b
y[m] ), the empirical variance impurity of yb[m] which is defined as in (1) with y[m]
k
k
replaced with yb[m]
. The idea is to analyze E[imp(b
y[m]
)], or proper bounds on it, and then show that
k
imp(b
y[m] ) concentrates around its mean. Let us consider the concentration first. (1) is a U-statistic of
order 2 with kernel h(u, v) = 21 (u ? v)2 . The classical Hoeffding inequality guarantees concentration
if h is uniformly bounded and the underlying variables are IID. Instead, we use a version of Hanson?
Wright concentration inequality derived in [10], which allows us to derive a concentration bound for
the empirical variance, for general sub-Gaussian vectors, avoiding the boundedness assumption:
Corollary 3. Let w = (w1 , . . . , wm ) ? Rm be a random vector with independent components wi
?1 P
2
which satisfy Ewi = ? and kwi ? ?k?2 ? K. Let imp(w) := m
1?i<j?m (wi ? wj ) be the
2
empirical variance of w. Then, for u ? 0,
P imp(w) ? E imp(w) > K 2 u ? 2 exp ?c (m ? 1) min(u, u2 ) .
(7)
We can immediately apply this result when k ?
/ S. However, for k ? S, a more careful application is
k
needed since we can only guarantee an exchangeable distribution for yb[m]
in this case. The following
lemma summarizes the conclusions:
7
k
Lemma 3. Let Ibm,k = imp(b
y[m]
) and recall that ? was introduced in the definition of ybik . Let
1
?21 := 12
be the variance of Unif(0, 1). Recall that p? := p ? s. Let L = k?k2 . There exist absolute
constants C1 , C2 , c such that if log p?/m ? C1 , then with probability at least 1 ? p??c ,
Ibm,k ? Ik1 + ?m , ?k ? S,
and,
Ibm,k ? I 0 ? ?m , ?k ?
/S
where, letting ? := 1 ? (1 ? ?)2 ,
Ik1 := ?21 (???k2 + L2 ) + ? 2 ,
I 0 := ?21 L2 + ? 2 , and ?m := C2 (L2 + ? 2 )
p
log p?/m.
The key outcome of Lemma 3 is that, on average, there is a positive gap I 0 ? Ik1 = ?21 ??k2 in
impurities between a feature on the support and those off of it, and that due to concentration, the
fluctuations in impurities will be less than this gap for large m. Combined with Lemma 2, we can
k
transfer the results to Iem,k := imp(e
y[m]
).
Corollary 4. The conclusion of Lemma 3 holds for Iem,k in place of Ibm,k , with probability at least
1
1 ? p??c ? 2e??m for ? = log 8?
.
Note that for ? < 1/8, the bound holds with high probability. Thus, as long as I 0 ? Ik1 > 2?m , the
selection algorithm correctly favors the kth feature in S, over the inactive ones (recall that lower
impurity is better). We have our main result after substituting n/2 for m.
4
Simulations
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1: Support recovery performance in a linear regression model augmented with possible
nonlinearities for n = 1024. (a) Linear case with uncorrelated design. (b) Linear case with correlated
design. (c) Nonlinear additive model with exponentials of covariates and uncorrelated design. (d)
Nonlinear model with interaction terms and uncorrelated design. (e) Nonlinear additive model with
exponentials of covariates, interaction terms, and uncorrelated design. (f) Nonlinear additive model
with exponentials of covariates, interaction terms, and correlated design.
In order to corroborate the theoretical analysis, we next present various simulation results. We
consider the following model: y = X? + f (XS ) + w, where f (XS ) is a potential nonlinearity, and
e ? Rn?p is a random
e where X
S is the true support of ?. We generate the training data as X = XM
p?p
matrix with IID Unif(?1, 1) entries, and M ? R
is an upper-triangular matrix that determines
whether the design is IID or correlated. In the IID case we set M = I. To achieve a correlated design
8
we randomly assign values from {0, ??, +?} to the upper triangular cells of M , with probabilities
(1 ? 2?, ?, ?). We observed qualitatively similar results for various values of ? and ? and here we
present results with ? = 0.04, and ?
? = 0.1. The noise is generated as w ? N (0, ? 2 In ). We fix
p = 200, ? = 0.1, and let ?i = ?1/ s over its support i ? S, where |S| = s. That is, only s of the
p = 200 variables are predictive of the response. The nonlinearity, f , optionally contains additive
terms in the form of exponentials of on-support covariates. It can also contain interaction terms across
on-support covariates, i.e., terms of the form ?2s xi xj for some randomly selected pairs of i, j ? S.
Notably, the choice of f is unknown to the variable selection methods. We vary s ? [5, 100] and note
that k?k2 = 1 remains fixed.
The plots in Figure 1 show the fraction of the true support recovered3 as a function of s, for various
methods under different modeling setups: f = 0 (linear), f = 2 exp(?) (additive), f = interaction
(interactions), and f = interaction + 2 exp(?) (interactions+additive) with IID or correlated designs.
Each data point is an average over 100 trials (see supplementary material for results with 95%
confidence intervals). In addition to DS TUMP, we evaluate T REE W EIGHT, S PAM, L ASSO, SIS and
random guessing for comparison. SIS refers to picking the indices of the top s largest values of X T y
in absolute value. When X is orthogonal and the generative model is linear, this approach is optimal,
and we use it as a surrogate for the optimal approach in our nearly orthogonal setup (i.e., the IID
linear case), due to its lack of any tuning parameters. Random guessing is used as a benchmark, and
as expected, on average recovers the fraction s/p = s/200 of the support.
The plots show that, in the linear setting, the performance of DS TUMP is comparable to, and only
slightly worse than, that of SIS or Lasso which are considered optimal in this case. Figure 1(b) shows
that under mildly correlated design the gap between DS TUMP and L ASSO widens. In this case, SIS
loses its optimality and performs at the same level as DS TUMP. This matches our intuition as both
SIS and DS TUMP are both greedy methods that consider covariates independently.
DS TUMP is more robust to nonlinearities, as characterized theoretically in Theorem 3 and evidenced
in Figure 1(c). In contrast, in the presence of exponential nonlinearities, SIS and Lasso are effective in
the very sparse regime of s p, but quickly approach random guessing as s grows. In the presence
of interaction terms, T REE W EIGHT and to a lesser extent S PAM outperform all other methods, as
shown in Figure 1(d), 1(e), and 1(f). We also note that the permutation-based importance method [1],
denoted by T REE W EIGHT P ERMUTATION in the plots in Figure 1, performs substantially worse than
T REE W EIGHT across the various modelling settings.
Overall, these simulations illustrate the promise of multi-level tree-based methods like T REE W EIGHT
under more challenging and realistic modeling settings. Future work involves generalizing our
theoretical analyses to extend to these more complex multi-level tree-based approaches.
5
Discussion
We presented a simple model selection algorithm for decision trees, which we called DS TUMP,
and analyzed its finite-sample performance in a variety of settings, including the high-dimensional,
nonlinear additive model setting. Our theoretical and experimental results show that even a simple
tree-based algorithm that selects at the root can achieve high dimensional selection consistency.
We hope these results pave the way for the finite-sample analysis of more refined tree-based model
selection procedures. Inspired by the empirical success of T REE W EIGHT in nonlinear settings, we are
actively looking at extensions of DS TUMP to a multi-stage algorithm capable of handling interactions
with high-dimensional guarantees.
Moreover, while we mainly focused on the regression problem, our proof technique based on
concentration of impurity reductions is quite general. We expect analogous results to hold, for example
for classification. However, aspects of the proof would be different, since impurity measures used for
classification are different than those of regression. One major hurdle involves deriving concentration
inequalities for the empirical versions of these measures, which are currently unavailable, and would
be of independent interest.
3
In the supplementary material we report analogous results using a more stringent performance metric,
namely the probability of exact support recovery. The results are qualitatively similar.
9
References
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10
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6,242 | 6,647 | Preventing Gradient Explosions
in Gated Recurrent Units
Sekitoshi Kanai, Yasuhiro Fujiwara, Sotetsu Iwamura
NTT Software Innovation Center
3-9-11, Midori-cho, Musashino-shi, Tokyo
{kanai.sekitoshi, fujiwara.yasuhiro, iwamura.sotetsu}@lab.ntt.co.jp
Abstract
A gated recurrent unit (GRU) is a successful recurrent neural network architecture
for time-series data. The GRU is typically trained using a gradient-based method,
which is subject to the exploding gradient problem in which the gradient increases
significantly. This problem is caused by an abrupt change in the dynamics of the
GRU due to a small variation in the parameters. In this paper, we find a condition
under which the dynamics of the GRU changes drastically and propose a learning
method to address the exploding gradient problem. Our method constrains the
dynamics of the GRU so that it does not drastically change. We evaluated our
method in experiments on language modeling and polyphonic music modeling. Our
experiments showed that our method can prevent the exploding gradient problem
and improve modeling accuracy.
1
Introduction
Recurrent neural networks (RNNs) can handle time-series data in many applications such as speech
recognition [14, 1], natural language processing [26, 30], and hand writing recognition [13]. Unlike
feed-forward neural networks, RNNs have recurrent connections and states to represent the data.
Back propagation through time (BPTT) is a standard approach to train RNNs. BPTT propagates the
gradient of the cost function with respect to the parameters, such as weight matrices, at each layer and
at each time step by unfolding the recurrent connections through time. The parameters are updated
using the gradient in a way that minimizes the cost function. The cost function is selected according
to the task, such as classification or regression.
Although RNNs are used in many applications, they have problems in that the gradient can be
extremely small or large; these problems are called the vanishing gradient and exploding gradient
problems [5, 28]. If the gradient is extremely small, RNNs can not learn data with long-term
dependencies [5]. On the other hand, if the gradient is extremely large, the gradient moves the RNNs
parameters far away and disrupts the learning process. To handle the vanishing gradient problem,
previous studies [18, 8] proposed sophisticated models of RNN architectures. One successful model
is a long short-term memory (LSTM). However, the LSTM has the complex structures and numerous
parameters with which to learn the long-term dependencies. As a way of reducing the number of
parameters while avoiding the vanishing gradient problem, a gated recurrent unit (GRU) was proposed
in [8]; the GRU has only two gate functions that hold or update the state which summarizes the past
information. In addition, Tang et al. [33] show that the GRU is more robust to noise than the LSTM
is, and it outperforms the LSTM in several tasks [9, 20, 33, 10].
Gradient clipping is a popular approach to address the exploding gradient problem [26, 28]. This
method rescales the gradient so that the norm of the gradient is always less than a threshold. Although
gradient clipping is a very simple method and can be used with GRUs, it is heuristic and does not
analytically derive the appropriate threshold. The threshold has to be manually tuned to the data
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
and tasks by trial and error. Therefore, a learning method is required to more effectively address the
exploding gradient problem in training of GRUs.
In this paper, we propose a learning method for GRUs that addresses the exploding gradient problem.
The method is based on an analysis of the dynamics of GRUs. GRUs suffer from gradient explosions
due to their nonlinear dynamics [11, 28, 17, 3] that enable GRUs to represent time-series data. The
dynamics can drastically change when the parameters cross certain values, called bifurcation points
[36], in the learning process. Therefore, the gradient of the state with respect to the parameters can
drastically increase at a bifurcation point. This paper presents an analysis of the dynamics of GRUs
and proposes a learning method to prevent the parameters from crossing the bifurcation point. It
describes evaluations of this method through language modeling and polyphonic music modeling
experiments. The experiments demonstrate that our method can train GRUs without gradient clipping
and that it can improve the accuracy of GRUs.
The rest of this paper is organized as follows: Section 2 briefly explains the GRU, dynamical systems
and the exploding gradient problem. It also outlines related work. The dynamics of GRUs is analyzed
and our training approach is presented in Section 3. The experiments that verified our method
are discussed in Section 4. The paper concludes in Section 5. Proofs of lemmas are given in the
supplementary material.
2
2.1
Preliminaries
Gated Recurrent Unit
Time-series data often have long and short-term dependencies. In order to model long and short-term
behavior, a GRU is designed to properly keep and forget past information. The GRU controls the past
information by having two gates: an update gate and reset gate. The update gate zt ? Rn?1 at a time
step t is expressed as
zt = sigm(Wxz xt + Whz ht?1 ),
(1)
where xt ? Rm?1 is the input vector, and ht ? Rn?1 is the state vector. Wxz ? Rn?m and
Whz ? Rn?n are weight matrices. sigm(?) represents the element-wise logistic sigmoid function.
The reset gate rt ? Rn?1 is expressed as
rt = sigm(Wxr xt + Whr ht?1 ),
(2)
where Wxr ? Rn?m and Whr ? Rn?n are weight matrices. The activation of the state ht is
expressed as
? t,
ht = zt ht?1 + (1 ? zt ) h
(3)
? t is a candidate for a new
where 1 is the vector of all ones, and means the element-wise product. h
state, expressed as
? t = tanh(Wxh xt + Whh (rt ht?1 )),
h
(4)
where tanh(?) is the element-wise hyperbolic tangent, and Wxh ? Rn?m and Whh ? Rn?n are
weight matrices. The initial value of ht is h0 = 0 where 0 represents the vector of all zeros; the
GRU completely forgets the past information when ht becomes 0.
The training of a GRU can be formulated as an optimization problem as follows:
PN
min? N1 j=1 C(x(j) , y (j) ; ?),
(j)
(j)
(j)
(5)
(j)
where ?, x , y , C(x , y ; ?), and N are all parameters of the model (e.g., elements of
Whh ), the j-th training input data, the j-th training output data, the loss function for the j-th data
(e.g., mean squared error or cross entropy), and the number of training data, respectively. This
optimization problem is usually solved through stochastic gradient descent (SGD). SGD iteratively
updates parameters according to the gradient of a mini-batch, which is randomly sampled data from
the training data. The parameter update at step ? is
P
(6)
? (? ) = ? (? ?1) ? ??? |D1? | (x(j) ,y(j) )?D? C(x(j) , y (j) ; ?),
where D? , |D? |, and ? represent the ? -th mini-batch, the size of the
P mini-batch, and the learning rate
of SGD, respectively. In gradient clipping, the norm of ?? |D1? | (x(j) ,y(j) )?D? C(x(j) , y (j) ; ?) is
clipped by the specified threshold. The size of the parameters ? is 3(n2 + mn) + ?, where ? is the
number of parameters except for the GRU, because the sizes of the six weight matrices of W? in eqs.
(1)-(4) are n?n or n?m. Therefore, the computational cost of gradient clipping is O(n2 + mn + ?).
2
2.2
Dynamical System and Gradient Explosion
An RNN is a nonlinear dynamical system that can be represented as follows:
ht = f (ht?1 , ?),
(7)
where ht is a state vector at time step t, ? is a parameter vector, and f is a nonlinear vector function.
The state evolves over time according to eq. (7). If the state ht? at some time step t? satisfies
ht? = f (ht? , ?), i.e., the new state equals the previous state, the state never changes until an external
input is applied to the system. Such a state point is called a fixed point h? . The state converges to or
goes away from the fixed point h? depending on f and ?. This property is important and is called
stability [36]. The fixed point h? is said to be locally stable if there exists a constant ? such that, for
ht whose initial value h0 satisfies |h0 ? h? | < ?, limt?? |ht ? h? | = 0 holds. In this case, a set of
points h0 such that |h0 ? h? | < ? is called a basin of attraction of the fixed point. Conversely, if
h? is not stable, the fixed point is said to be unstable. Stability and the behavior of ht near a fixed
point, e.g., converging or diverging, can be qualitatively changed by a smooth variation in ?. This
phenomenon is called a local bifurcation, and the value of the parameter of a bifurcation is called a
bifurcation point [36].
Doya [11], Pascanu et al. [28] and Baldi and Hornik [3] pointed out that gradient explosions are due
to bifurcations. The training of an RNN involves iteratively updating its parameters. This process
causes a bifurcation: a small change in parameters can result in a drastic change in the behavior of
the state. As a result, the gradient increases at a bifurcation point.
2.3
Related Work
Kuan et al. [23] established a learning method to avoid the exploding gradient problem. This method
restricts the dynamics of an RNN so that the state remains stable. Yu [37] proposed a learning rate for
stable training through Lyapunov functions. However, these methods mainly target Jordan and Elman
networks called simple RNNs which, unlike GRUs, are difficult to train long-term dependencies.
In addition, they suppose that the mean squared error is used as the loss function. By contrast, our
method targets the GRU, a more sophisticated model, and can be used regardless of the loss function.
Doya [11] showed that bifurcations cause gradient explosions and that real-time recurrent learning
(RTRL) can train an RNN without the gradient explosion. However, RTRL has a high computational
cost: O((n + u)4 ) for each update step where u is the number of output units [19]. More recently,
Arjovsky et al. [2] proposed unitary matrix constraints in order to prevent the gradient vanishing
and exploding. Vorontsov et al. [35], however, showed that it can be detrimental to maintain hard
constraints on matrix orthogonality.
Previous studies analyzed the dynamics of simple RNNs [12, 4, 31, 16, 27]. Barabanov and Prokhorov
[4] analyzed the absolute stability of multi-layer simple RNNs. Haschke and Steil [16] presented a
bifurcation analysis of a simple RNN in which inputs are regarded as the bifurcation parameter. Few
studies have analyzed the dynamics of the modern RNN models. Talathi and Vartak [32] analyzed
the nonlinear dynamics of an RNN with a Relu nonlinearity. Laurent and von Brecht [24] empirically
revealed that LSTMs and GRUs can exhibit chaotic behavior and proposed a novel model that has
stable dynamics. To the best of our knowledge, our study is the first to analyze the stability of GRUs.
3
Proposed Method
As mentioned in Section 2, a bifurcation makes the gradient explode. In this section, through an
analysis of the dynamics of GRUs, we devise a training method that avoids a bifurcation and prevents
the gradient from exploding.
3.1
Formulation of Proposed Training
In Section 3.1, we formulate our training approach. For the sake of clarity, we first explain the
formulation for a one-layer GRU; then, we apply the method to a multi-layer GRU.
3.1.1
One-Layer GRU
The training of a GRU is formulated as eq. (5). This training with SGD can be disrupted by a gradient
explosion. To prevent the gradient from exploding, we formulate the training of a one-layer GRU as
3
the following constrained optimization:
PN
min? N1 j=1 C(x(j) , y (j) ; ?), s.t. ?1 (Whh ) < 2,
(8)
where ?i (?) is the i-th largest singular value of a matrix, and ?1 (?) is called the spectral norm. This
constrained optimization problem keeps the one-layer GRU locally stable and prevents the gradient
from exploding due to a bifurcation of the fixed point on the basis of the following theorem:
Theorem 1. When ?1 (Whh ) < 2, a one-layer GRU is locally stable at a fixed point h? = 0.
We show the proof of this theorem later. This theorem indicates that our training approach of eq. (8)
maintains the stability of the fixed point h? = 0. Therefore, our approach prevents the gradient
explosion caused by the bifurcation of the fixed point h? . In order to prove this theorem, we need to
use the following three lemmas:
Lemma 1. A one-layer GRU has a fixed point at h? = 0.
Lemma 2. Let I be an n ? n identity matrix, ?i (?) be the eigenvalue that has the i-th largest absolute
value, and J = 14 Whh + 12 I. When the spectral radius 1 |?1 (J )| < 1, a one-layer GRU without
input can be approximated by the following linearized GRU near ht = 0:
ht = J ht?1 ,
(9)
and the fixed point h? = 0 of a one-layer GRU is locally stable.
Lemma 2 indicates that we can prevent a change in local stability by exploiting the constraint of
|?1 ( 14 Whh + 21 I)| < 1. This constraint can be represented as a bilinear matrix inequality (BMI)
constraint [7]. However, an optimization problem with a BMI constraint is NP-hard [34]. Therefore,
we relax the optimization problem to that of a singular value constraint as in eq. (8) by using the
following lemma:
Lemma 3. When ?1 (Whh ) < 2, we have |?1 ( 14 Whh + 12 I)| < 1.
By exploiting Lemmas 1, 2, and 3, we can prove Theorem 1 as follows:
Proof. From Lemma 1, there exists a fixed point h? = 0 in a one-layer GRU. This fixed point is
locally stable when |?1 ( 14 Whh + 21 I)| < 1 from Lemma 2. From Lemma 3, |?1 ( 14 Whh + 12 I)| < 1
holds when ?1 (Whh ) < 2. Therefore, when ?1 (Whh ) < 2, the one-layer GRU is locally stable at
the fixed point h? = 0
Lemma 1 indicates that a one-layer GRU has a fixed point. Lemma 2 shows the condition under
which this fixed point is kept stable. Lemma 3 shows that we can use a singular value constraint
instead of an eigenvalue constraint. These lemmas prove Theorem 1, and this theorem ensures that
our method prevents the gradient from exploding because of a local bifurcation.
In our method of eq. (8), h? = 0 is a fixed point. This fixed point is important since the initial
value of the state h0 is 0, and the GRU forgets all the past information when the state is reset to 0
as described in Section 2. If h? = 0 is stable, the state vector near 0 asymptotically converges to 0.
This means that the state vector ht can be reset to 0 after a sufficient time in the absence of an input;
i.e., the GRU can forget the past information entirely. On the other hand, when |?1 (J )| becomes
greater than one, the fixed point at 0 becomes unstable. This means that the state vector ht never
resets to 0; i.e., the GRU can not forget all the past information until we manually reset the state. In
this case, the forget gate and reset gate may not work effectively. In addition, Laurent and von Brecht
[24] show that an RNN model with state that asymptotically converges to zero achieves a level of
performance comparable to that of LSTMs and GRUs. Therefore, our constraint that the GRU is
locally stable at h? = 0 is effective for learning.
3.1.2
Multi-Layer GRU
Here, we extend our method in the multi-layer GRU. An L-layer GRU is represented as follows:
h1,t = f1 (h1,t?1 , xt ), h2,t = f2 (h2,t?1 , h1,t ), . . . , hL,t = fL (hL,t?1 , hL?1,t ),
where hl,t ? Rnl ?1 is a state vector with the length of nl at the l-th layer, and fl represents a
GRU that corresponds to eqs. (1)-(4) at the l-th layer. In the same way as the one-layer GRU,
T T
ht = [hT
1,t , . . . , hL,t ] = 0 is a fixed point, and we have the following lemma:
1
The spectral radius is the maximum value of the absolute value of the eigenvalues.
4
Lemma 4. When |?1 ( 14 Wl,hh + 12 I)| < 1 for l = 1, . . . , L, the fixed point h? = 0 of a multi-layer
GRU is locally stable.
From Lemma 3, we have |?1 (Wl,hh + 21 I)| < 1 when ?1 (Wl,hh ) < 2. Thus, we formulated our
training of a multi-layer GRU to prevent gradient explosions as
PN
min? N1 j=1 C(x(j) , y (j) ; ?), s.t. ?1 (Wl,hh ) < 2, ?1 (Wl,xh )? 2 for l = 1, . . . , L.
(10)
We added the constraint ?1 (Wl,xh ) ? 2 in order to prevent the input from pushing the state out of
the basin of attraction of the fixed point h? = 0. This constrained optimization problem keeps a
multi-layer GRU locally stable.
3.2
Algorithm
The optimization method for eq. (8) needs to find the optimal parameters in the feasible set, in which
the parameters satisfy the constraint: {Whh |Whh ? Rn?n , ?1 (Whh ) < 2}. Here, we modify SGD
in order to solve eq. (8). Our method updates the parameters as follows:
(? ?1)
(? )
(? )
(? ?1)
??Whh = ??Whh ? ??? CD? (?), Whh = P? (Whh
? ??Whh CD? (?)),
(11)
P
(?
)
where CD? (?) represents |D1? | (x(j) ,y(j) )?D? C(x(j) , y (j) ; ?), and ??Whh represents the parame(? )
ters except for Whh . In eq. (11), We compute P? (?) by using the following procedure:
(? )
(? ?1)
? := W
Step 1. Decompose W
hh
hh ???WhhCD?(?) by using singular value decomposition (SVD):
? (? ) = U ?V.
W
hh
Step 2. Replace the singular values that are greater than the threshold 2 ? ?:
? = diag(min(?1 , 2 ? ?), . . . min(?n , 2 ? ?)).
?
(12)
(13)
(? )
? in Steps 1 and 2:
Step 3. Reconstruct Whh by using U , V and ?
(? )
?
Whh ? U ?V.
(14)
By using this procedure, Whh is guaranteed to have a spectral norm of less than or equal to 2 ? ?.
When ? is 0 < ? < 2, our method constrains ?1 (Whh ) to be less than 2. P? (?) in our method brings
back the parameters into the feasible set when the parameters go out the feasible set after SGD. Our
procedure P? (?) is an optimal projection into the feasible set as shown by the following lemma:
(? )
Lemma 5. The weight matrix Whh obtained by P? (?) is a solution of the following optimization:
? (? ) ? W (? ) ||2 , s.t. ?1 (W (? ) ) ? 2??, where || ? ||2 represents the Frobenius norm.
minW (? ) ||W
F
hh
hh F
hh
hh
Lemma 5 indicates that our method can bring back the weight matrix into the feasible set with
minimal variations in the parameters. Therefore, our procedure P? (?) has minimal impact on the
minimization of the loss function. Note that our method does not depend on the learning rate schedule,
and an adaptive learning rate method (such as Adam [21]) can be used with it.
3.3
Computational Cost
Let n be the length of a state vector ht ; a naive implementation of SVD needs O(n3 ) time. Here, we
propose an efficient method to reduce the computational cost. First, let us reconsider the computation
of P? (?). Equations (12)-(14) can be represented as follows:
h
i
(? )
T
? (? ) ? Ps
? (? )
Whh = W
(15)
i=1 ?i (Whh ) ? (2 ? ?) ui vi ,
hh
where s is the number of the singular values greater than 2 ? ?, and ui and vi are the i-th left and
right singular vectors, respectively. Eq. (15) shows that our method only needs the singular values
? (? ) ) > 2 ? ?. In order to reduce the computational cost of our method,
and vectors such that ?i (W
hh
we use the truncated SVD [15] to efficiently compute the top s singular values in O(n2 log(s)) time,
where s is the specified number of singular values. Since the truncated SVD requires s to be set
beforehand, we need to efficiently estimate the number of singular values such that must meet the
? (? ) ) > 2 ? ?. Therefore, we compute upper bounds of the singular values that
condition of ?i (W
hh
meet the condition on the basis of the following lemma:
5
? (? ) ) ?
? (? ) are bounded with the following inequality: ?i (W
Lemma 6. The singular values of W
hh
hh
(? ?1)
?i (Whh ) + |?|||?Whh CD? (?)||F .
Using this upper bound, we can estimate s as the number of the singular values with upper bounds of
greater than 2??. This upper bound can be computed in O(n2 ) time since the size of ?Whh CD? (?) is
(? ?1)
n?n and ?i (Whh ) has already been obtained at step ? . If we did not compute the previous singular
? (? ) ) as ?i (W (? ?K?1) )+
values from ? ? K step to ? ? 1 step, we compute the upper bound of ?i (W
hh
hh
PK
(? )
|?|||?
C
(?)||
from
Lemma
6.
Since
our
training
originally
constrains
?
Whh D? ?k
F
1 (Whh ) <
k=0
? (? ) ) > 2,
2 as described in eq. (8), we can redefine s as the number of singular values such that ?i (W
hh
(?
)
?
instead of ?i (W
hh ) > 2 ? ?. This modification can further reduce the computational cost without
disrupting the training. In summary, our method can efficiently estimate the number of singular
values needed in O(n2 ) time, and we compute the truncated SVD in O(n2 log(s)) time only if we
need to compute singular values by using Lemma 6.
4
Experiments
4.1
Experimental Conditions
To evaluate the effectiveness of our method, we conducted experiments on language modeling and
polyphonic music modeling. We trained the GRU and examined the successful training rate, as well
as the average and standard deviation of the loss. We defined successful training as training in which
the validation loss at each epoch is never greater than the initial value. The experimental conditions
of each modeling are explained below.
4.1.1
Language Modeling
Penn Treebank (PTB) [25] is a widely used dataset to evaluate the performance of RNNs. PTB is
split into training, validation, and test sets, and the sets are composed of 930 k, 74 k, 80 k tokens.
This experiment used a 10 k word vocabulary, and all words outside the vocabulary were mapped
to a special token. The experimental conditions were based on the previous paper [38]. Our model
architecture was as follows: The first layer was a 650 ? 10, 000 linear layer without bias to convert the
one-hot vector input into a dense vector, and we multiplied the output of the first layer by 0.01 because
our method assumes small inputs. The second layer was a GRU layer with 650 units, and we used the
softmax function as the output layer. We applied 50 % dropout to the output of each layer except for
the recurrent connection [38]. We unfolded the GRU for 35 time steps in BPTT and set the mini-batch
size to 20. We trained the GRU with SGD for 75 epochs since the performance of the models trained
by Adam and RMSprop were worse than that trained by SGD in the preliminary experiments, and
Zaremba et al. [38] used SGD. The results and conditions of preliminary experiments are in the
supplementary material. We set the learning rate to one in the first 10 epochs, and then, divided
the learning rate by 1.1 after each epoch. In our method, ? was set to [0.2, 0.5, 0.8, 1.1, 1.4]. In
gradient clipping, a heuristic for setting the threshold is to look at the average norm of the gradient
[28]. We evaluated gradient clipping based on the gradient norm by following the study [28]. In
the supplementary material, we evaluated gradient elementwise clipping which is used practically.
Since the average norm of the gradient was about 10, we set the threshold to [5, 10, 15, 20]. We
initialized the weight matrices except for Whh with a normal distribution N (0, 1/650) , and Whh
as an orthogonal matrix composed of the left singular vectors of a random matrix [29, 8]. After
each epoch, we evaluated the validation loss. The model that achieved the least validation loss was
evaluated using the test set.
4.1.2
Polyphonic Music Modeling
In this modeling, we predicted MIDI note numbers at the next time step given the observed notes of
the previous time steps. We used the Nottingham dataset: a MIDI file containing 1200 folk tunes [6].
We represented the notes at each time step as a 93-dimensional binary vector. This dataset is split
into training, validation and test sets [6]. The experimental conditions were based on the previous
study [20]. Our model architecture was as follows: The first layer was a 200 ? 93 linear layer without
bias, and the output of the first layer was multiplied by 0.01. The second and third layers were GRU
6
Table 1: Language modeling results: success rate and perplexity.
Our method
Delta
0.2
0.5
0.8
1.1
1.4
Gradient clipping
Threshold
5
10
15
20
Success Rate
Validation Loss
Test Loss
100 %
102.0?0.3
97.6?0.4
100 %
102.8?0.3
98.4?0.3
100 %
103.7?0.2
99.0?0.4
100 %
105.2?0.2
100.3?0.2
100 %
107.0?0.4
102.1?0.2
Success Rate
Validation Loss
Test Loss
100 %
109.3?0.4
106.9?0.4
40 %
103.1?0.4
100.4?0.5
0%
N/A
N/A
0%
N/A
N/A
Table 2: Music modeling results: success rate and negative log-likelihood.
Our method
Delta
0.2
0.5
0.8
1.1
1.4
Gradient clipping
Threshold
15
30
45
60
Success Rate
Validation Loss
Test Loss
100 %
3.46?0.05
3.53?0.04
100 %
3.47?0.07
3.53?0.04
100 %
3.59?0.1
3.64?0.2
100 %
4.58?0.2
4.56?0.2
100 %
4.64?0.2
4.62?0.2
Success Rate
Validation Loss
Test Loss
100 %
3.57?0.01
3.64?0.04
100 %
3.61?0.2
3.64?0.2
100 %
3.88?0.2
3.89?0.2
100 %
5.26?3
5.36?3
layers with 200 units per layer, and we used the logistic function as the output layer. 50 % dropout
was applied to non-recurrent connections. We unfolded GRUs for 35 time steps and set the size
of the mini-batch to 20. We used SGD with a learning rate of 0.1 and divided the learning rate by
1.25 if we observed no improvement over 10 consecutive epochs. We repeated the same procedure
until the learning rate became smaller than 10?4 . In our method, ? was set to [0.2, 0.5, 0.8, 1.1,
1.4]. In gradient clipping, the threshold was set to [15, 30, 45, 60], since the average norm of the
gradient was about 30. We initialized the weight matrices except for Whh with a normal distribution
N (0, 10?4 /200) , and Whh as an orthogonal matrix. After each epoch, we evaluated the validation
loss, and the model that achieved the least validation loss was evaluated using the test set.
4.2
Success Rate and Accuracy
Tables 1 and 2 list the success rates of language modeling and music modeling, respectively. These
tables also list the averages and standard deviations of the loss in each modeling to show that our
method outperforms gradient clipping. In these tables, ?Threshold? means the threshold of gradient
clipping, and ?Delta? means ? in our method.
As shown in Table 1, in language modeling, gradient clipping failed to train even though its parameter
was set to 10, which is the average norm of the gradient as recommended by Pascanu et al. [28].
Although gradient clipping successfully trained the GRU when its threshold was five, it failed to
effectively learn the model with this setting; a threshold of 10 achieved lower perplexity than a
threshold of five. As shown in Table 2, in music modeling, gradient clipping successfully trained
the GRU. However, the standard deviation of the loss was high when the threshold was set to 60
(double the average norm). On the other hand, our method successfully trained the GRU in both
modelings. Tables 1 and 2 show that our approach achieved lower perplexity and negative loglikelihood compared with gradient clipping, while it constrained the GRU to be locally stable. This is
because our approach of constraining stability improves the performance of the GRU. The previous
study [22] showed that stabilizing the activation of the RNN can improve performance on several
tasks. In addition, Bengio et al. [5] showed that an RNN is robust to noise when the state remains
in the basin of attraction. Using our method, the state of the GRU tends to remain in the basin of
the attraction of h? = 0. Therefore, our method can improve robustness against noise, which is an
advantage of the GRU [33].
As shown in Table 2, when ? was set to 1.1 or 1.4, the performance of the GRU deteriorated. This is
because the convergence speed of the state depends on ?. As mentioned in Section 3.2, the spectral
norm of Whh is less than or equal to 2 ? ?. This spectral norm gives the upper bound of |?1 (J )|.
|?1 (J )| gives the rate of convergence of a linearized GRU (eq. (9)), which approximates GRU near
ht = 0 when |?1 (J )| < 1. Therefore, the state of the GRU near ht = 0 tends to converge quickly if
? is set to close to two. In this case, the GRU becomes robust to noise since the state affected by the
past noise converges to zero quickly, while the GRU loses effectiveness for long-term dependencies.
We can tune ? from the characteristics of the data: if data have the long-term dependencies, we should
set ? small, whereas we should set ? large for noisy data.
The threshold in gradient clipping is unbounded, and hence, it is difficult to tune. Although the
threshold can be heuristically set on the basis of the average norm, this may not be effective in
language modeling using the GRU, as shown in Table 1. In contrast, the hyper-parameter is bounded
in our method, i.e., 0 < ? < 2, and it is easy to understand its effect as mentioned above.
7
1.4
Norm of gradient
Spectral radius
1.2
1.2
Spectral radius
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0
Norm of gradient
100
200
300
400
No. of Iteration
0.0
0
500
(a) Gradient clipping (threshold of 5).
100
200
300
No. of Iteration
400
500
(b) Our method (delta of 0.2).
Figure 1: Gradient explosion in language modeling.
Table 3: Computation time in the language modeling (delta is 0.2, threshold is 5).
Naive SVD
4
5.02 ? 10
4.3
Computation time (s)
Truncated SVD
4.55 ? 10
4
Gradient clipping
4.96 ? 104
Relation between Gradient and Spectral Radius
Our method of constraining the GRU to be locally stable is based on the hypothesis that a change in
stability causes an exploding gradient problem. To confirm this hypothesis, we examined (i) the norm
of the gradient before clipping and (ii) the spectral radius of J (in Lemma 2), which determines local
stability, versus the number of iterations until the 500th iteration in Fig. 1. Fig. 1(a) and 1(b) show
the results of gradient clipping with a threshold of 5 and our method with ? of 0.2. Each norm of the
gradient was normalized so that its maximum value was one. The norm of the gradient significantly
increased when the spectral radius crossed one, such as at the 63rd, 79th, and 141st iteration (Fig. 1(a)).
In addition, the spectral radius decreased to less than one after the gradient explosion; i.e., when the
gradient explosion occurred, the gradient became in the direction of decreasing spectral radius. In
contrast, our method kept the spectral radius less than one by constraining the spectral norm of Whh
(Fig. 1(b)). Therefore, our method can prevent the gradient from exploding and effectively train the
GRU.
4.4
Computation Time
We evaluated computation time of the language modeling experiment. The detailed experimental
setup is described in the supplementary material. Table 3 lists the computation time of the whole
learning process using gradient clipping and our method with the naive SVD and with truncated
SVD. This table shows the computation time of our method is comparable to gradient clipping. As
mentioned in Section 2.1, the computational cost of gradient clipping is proportional to the number
of parameters including weight matrices of input and output layers. In language modeling, the sizes
of input and output layers tend to be large due to the large vocabulary size. On the other hand, the
computational cost of our method only depends on the length of the state vector, and our method
can be efficiently computed if the number of singular values greater than 2 is small as described
in Section 3.3. As a result, our method could reduce the computation time comparing to gradient
clipping.
5
Conclusion
We analyzed the dynamics of GRUs and devised a learning method that prevents the exploding
gradient problem. Our analysis of stability provides new insight into the behavior of GRUs. Our
method constrains GRUs so that the states near 0 asymptotically converge to 0. Through language
and music modeling experiments, we confirmed that our method can successfully train GRUs and
found that our method can improve their performance.
8
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6,243 | 6,648 | On the Power of Truncated SVD for General
High-rank Matrix Estimation Problems
Simon S. Du
Carnegie Mellon University
[email protected]
Yining Wang
Carnegie Mellon University
[email protected]
Aarti Singh
Carnegie Mellon University
[email protected]
Abstract
? that is close to a general high-rank positive semiWe show that given an estimate A
?
de?nite (PSD) matrix A in spectral norm (i.e., ?A?A?
2 ? ?), the simple truncated
? produces a multiplicative approximation of A
Singular Value Decomposition of A
in Frobenius norm. This observation leads to many interesting results on general
high-rank matrix estimation problems:
1. High-rank matrix completion: we show that it is possible to recover a general
high-rank matrix A up to (1 + ?) relative error in Frobenius norm from partial
observations, with sample complexity independent of the spectral gap of A.
2. High-rank matrix denoising: we design an algorithm that recovers a matrix A
with error in Frobenius norm from its noise-perturbed observations, without
assuming A is exactly low-rank.
3. Low-dimensional approximation of high-dimensional covariance: given N
i.i.d. samples of dimension n from Nn (0, A), we show that it is possible to
approximate the covariance matrix A with relative error in Frobenius norm with
N ? n, improving over classical covariance estimation results which requires
N ? n2 .
1
Introduction
Let A be an unknown general high-rank n ? n PSD data matrix that one wishes to estimate. In many
machine learning applications, though A is unknown, it is relatively easy to obtain a crude estimate
? ? A?2 ? ?). For example, in matrix completion
? that is close to A in spectral norm (i.e., ?A
A
a simple procedure that ?lls all unobserved entries with 0 and re-scales observed entries produces
an estimate that is consistent in spectral norm (assuming the matrix satis?es a spikeness condition,
standard assumption in matrix completion literature). In matrix de-noising, an observation that is
corrupted by Gaussian noise is close to the underlying signal, because Gaussian noise is isotropic and
has small spectral norm. In covariance estimation, the sample covariance in low-dimensional settings
is close to the population covariance in spectral norm under mild conditions [Bunea and Xiao, 2015].
However, in most such applications it is not suf?cient to settle for a spectral norm approximation. For
example, in recommendation systems (an application of matrix completion) the zero-?lled re-scaled
rating matrix is close to the ground truth in spectral norm, but it is an absurd estimator because
most of the estimated ratings are zero. It is hence mandatory to require a more stringent measure of
? ? A?F ,
performance. One commonly used measure is the Frobenius norm of the estimation error ?A
which ensures that (on average) the estimate is close to the ground truth in an element-wise sense. A
? is in general not a good estimate under Frobenius norm, because in
spectral norm approximation A
?
?
? ? A?2 .
high-rank scenarios ?A ? A?F can be n times larger than ?A
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this paper, we show that in many cases a powerful multiplicative low-rank approximation in
Frobenius norm can be obtained by applying a simple truncated SVD procedure on a crude, easyto-?nd spectral norm approximate. In particular, given the spectral norm approximation condition
? ? A?2 ? ?, the top-k SVD of A
? k of A
? multiplicatively approximates A in Frobenius norm; that
?A
?
is, ?Ak ? A?F ? C(k, ?, ?k+1 (A))?A ? Ak ?F , where Ak is the best rank-k approximation of A
in Frobenius and spectral norm. To our knowledge, the best existing result under the assumption
? ? A?2 ? ? is due to Achlioptas and McSherry [2007], who showed that ?A
? k ? A?F ?
?A
?
?
1/4
??Ak ?F , which depends on ?Ak ?F and is not multiplicative in
?A ? Ak ?F + k? + 2k
?A ? Ak ?F .
Below we summarize applications in several matrix estimation problems.
High-rank matrix completion Matrix completion is the problem of (approximately) recovering a
data matrix from very few observed entries. It has wide applications in machine learning, especially
in online recommendation systems. Most existing work on matrix completion assumes the data matrix
is exactly low-rank [Candes and Recht, 2012, Sun and Luo, 2016, Jain et al., 2013]. Candes and Plan
[2010], Keshavan et al. [2010] studied the problem of recovering a low-rank matrix corrupted by
stochastic noise; Chen et al. [2016] considered sparse column corruption. All of the aforementioned
work assumes that the ground-truth data matrix is exactly low-rank, which is rarely true in practice.
Negahban and Wainwright [2012] derived minimax rates of estimation error when the spectrum
of the data matrix lies in an ?q ball. Zhang et al. [2015], Koltchinskii et al. [2011] derived oracle
?
inequalities for general matrix completion; however their error bound has an additional O( n)
multiplicative factor. These results also require solving computationally expensive nuclear-norm
penalized optimization problems whereas our method only requires solving a single truncated
singular value decomposition. Chatterjee et al. [2015] also used the truncated SVD estimator for
matrix completion.
However, his bound depends on the nuclear norm of the underlying matrix
?
which may be n times larger than our result. Hardt and Wootters [2014] used a ?soft-de?ation?
technique to remove condition number dependency in the sample complexity; however, their error
bound for general high-rank matrix completion is additive and depends on the ?consecutive? spectral
gap (?k (A) ? ?k+1 (A)), which can be small in practical settings [Balcan et al., 2016, Anderson
et al., 2015]. Eriksson et al. [2012] considered high-rank matrix completion with additional union-ofsubspace structures.
In this paper, we show that if the n ? n data matrix A satis?es ?0 -spikeness condition, 1 then for
? k ? A?F ? (1 + O(?))?A ?
? k satis?es ?A
any ? ? (0, 1), the truncated SVD of zero-?lled matrix A
n max{??4 ,k2 }?20 ?A?2F log n
) ,which can be
Ak ?F if the sample complexity is lower bounded by ?(
?k+1 (A)2
further simpli?ed to ?(?20 max{??4 , k 2 }?k (A)2 ? nrs (A) log n), where ?k (A) = ?1 (A)/?k+1 (A)
is the kth-order condition number and rs (A) = ?A?2F /?A?22 ? rank(A) is the stable rank of
A. Compared to existing work, our error bound is multiplicative, gap-free, and the estimator is
computationally ef?cient. 2
? = A + E be a noisy observation of A, where E is a PSD
High-rank matrix de-noising Let A
Gaussian noise matrix with zero mean and ? 2 /n variance on each entry. By simple concentration
? ? A?2 = ? with high probability; however, A
? is in general not a good estimator
results we have ?A
?
?
of A in Frobenius norm when A is high-rank. Speci?cally, ?A ? A?F can be as large as n?.
?
Applying our main result, we show that
? if ? < ?k+1 (A) for some k ??n, then the top-k SVD Ak of
?
?
A satis?es ?Ak ? A?F ? (1 + O( ?/?k+1 (A)))?A ? Ak ?F + k?. This suggests a form of
bias-variance decomposition as larger rank threshold k induces smaller bias ?A ? Ak ?F but larger
variance k? 2 . Our results generalize existing work on matrix de-noising [Donoho and Gavish, 2014,
Donoho et al., 2013, Gavish and Donoho, 2014], which focus primarily on exact low-rank A.
1
n?A?max ? ?0 ?A?F ; see also De?nition 2.1.
We remark that our relative-error analysis does not, however, apply to exact rank-k matrix where ?k+1 = 0.
This is because for exact rank-k matrix a bound of the form (1 + O(?))?A ? Ak ?F requires exact recovery of
A, which truncated SVD cannot achieve. On the other hand, in the case of ?k+1 = 0 a weaker additive-error
bound is always applicable, as we show in Theorem 2.3.
2
2
Low-rank estimation of high-dimensional covariance The (Gaussian) covariance estimation
problem asks to estimate an n ? n PSD covariance matrix A, either in spectral or Frobenius
norm, from N i.i.d. samples X1 , ? ? ? , XN ? N (0, A). The high-dimensional regime of covariance
estimation, in which N ? n or even N ? n, has attracted enormous interest in the mathematical
statistics literature [Cai et al., 2010, Cai and Zhou, 2012, Cai et al., 2013, 2016]. While most existing
work focus on sparse or banded covariance matrices, the setting where A has certain low-rank
structure has seen rising interest recently [Bunea and Xiao, 2015, Kneip and Sarda, 2011]. In
particular, Bunea and Xiao [2015] shows that if n = O(N ? ) for some ? ? 0 then the sample
? = 1 ?N Xi X ? satis?es
covariance estimator A
i
i=1
N
?
?
?
log
N
? ? A?F = OP ?A?2 re (A)
,
(1)
?A
N
where re (A) = tr(A)/?A?2 ? rank(A) is the effective rank of A. For high-rank matrices where
re (A) ? n, Eq. (1) requires N = ?(n2 log n) to approximate A consistently in Frobenius norm.
? k and show that, if
consider a reduced-rank estimator A
? k ? A?F admits
? c for some small universal constant c > 0, then ?A
a relative Frobenius-norm error bound (1+O(?))?A?Ak ?F with high probability. Our result allows
reasonable approximation of A in Frobenius norm under the regime of N = ?(npoly(k) log n)
if ?k = O (poly (k)), which is signi?cantly more ?exible than N = ?(n2 log n), though the
dependency of ? is worse than [Bunea and Xiao, 2015]. The error bound is also agnostic in nature,
making no assumption on the actual or effective rank of A.
In
this
paper
we
re (A) max{??4 ,k2 }?k (A)2 log N
N
Notations For an n?n PSD matrix A, denote A = U?U? as its eigenvalue decomposition, where
U is an orthogonal matrix and ? = diag(?1 , ? ? ? , ?n ) is a diagonal matrix, with eigenvalues sorted in
descending order ?1 ? ?2 ??
? ? ? ? ?n ? 0. The spectral norm and Frobenius norm of A are de?ned
as ?A?2 = ?1 and ?A?F = ?12 + ? ? ? + ?n2 , respectively. Suppose u1 , ? ? ? , un are eigenvectors as?k
?n
sociated with ?1 , ? ? ? , ?n . De?ne Ak = i=1 ?i ui u?
= Uk ? k U ?
, An?k = i=k+1 ?i ui u?
i
i =
k
?m 2
?
?
and
A
=
?
u
u
=
U
?
U
.
For
a
tall
matrix
Un?k ?n?k U?
m1 :m2
m1 :m2 m1 :m2 m1 :m2
n?k
i=m1 +1 i i i
U ? Rn?k , we use U = Range(U) to denote the linear subspace spanned by the columns of U. For
two linear subspaces U and V, we write W = U ?V if U ?V = {0} and W = {u+v : u ? U , v ? V}.
For a sequence of random variables {Xn }?
n=1 and real-valued function f : N ? R, we say
Xn = OP (f (n)) if for any ? > 0, there exists N ? N and C > 0 such that Pr[|Xn | ? C ? |f (n)|] ? ?
for all n ? N .
2
Multiplicative Frobenius-norm Approximation and Applications
We ?rst state our main result, which shows that truncated SVD on a weak estimator with small
approximation error in spectral norm leads to a strong estimator with multiplicative Frobenius-norm
error bound. We remark that truncated SVD in general has time complexity
?
?
? ?
??
? + npoly (k) ,
O min n2 k, nnz A
? is the number of non-zero entries in A,
? and the time complexity is at most linear in
where nnz(A)
matrix sizes when k is small. We refer readers to [Allen-Zhu and Li, 2016] for details.
Theorem 2.1. Suppose A is an n ? n PSD matrix with eigenvalues ?1 (A) ? ? ? ? ? ?n (A) ? 0,
? ? A?2 ? ? = ?2 ?k+1 (A) for some ? ? (0, 1/4]. Let Ak and
? satis?es ?A
and a symmetric matrix A
? Then
?
Ak be the best rank-k approximations of A and A.
?
? k ? A?F ? (1 + 32?)?A ? Ak ?F + 102 2k?2 ?A ? Ak ?2 .
?A
(2)
?
Remark 2.1. Note when ? = O(1/ k) we obtain an (1 + O (?)) error bound.
Remark 2.2. This theorem only studies PSD matrices. Using similar arguments in the proof, we
believe similar results for general asymmetric matrices can be obtained as well.
3
? k ? A?F assuming ?A
? ? A?2 ? ? is due to
To our knowledge, the best existing bound for ?A
Achlioptas and McSherry [2007], who showed that
?
? k ? A?F ? ?A ? Ak ?F + ?(A
? ? A)k ?F + 2 ?(A
? ? A)k ?F ?Ak ?F
?A
? ?
?
? ?A ? Ak ?F + k??A ? Ak ?2 + 2k 1/4 ? ?Ak ?F .
(3)
?
Compared to Theorem 2.1, Eq. (3) is not relative because the third term 2k 1/4 ?Ak ?F depends on
the k largest eigenvalues of A, which could be much larger than the remainder term ?A ? Ak ?F . In
? k ? A?F could be upper bounded
contrast, Theorem 2.1, together with Remark 2.1, shows that ?A
by a small factor multiplied with the remainder term ?A ? Ak ?F .
We also provide a gap-dependent version.
Theorem 2.2. Suppose A is an n ? n PSD matrix with eigenvalues ?1 (A) ? ? ? ? ? ?n (A) ? 0,
? ? A?2 ? ? = ? (?k (A) ? ?k+1 (A)) for some ? ? (0, 1/4].
? satis?es ?A
and a symmetric matrix A
?
? Then
Let Ak and Ak be the best rank-k approximations of A and A.
?
? k ? A?F ? ?A ? Ak ?F + 102 2k? (?k (A) ? ?k+1 (A)) .
?A
(4)
?
If A is an exact rank-k matrix, Theorem 2.2 implies that truncated SVD gives an ? 2k?k error
approximation in Frobenius norm, which has been established by many previous works [Yi et al.,
2016, Tu et al., 2015, Wang et al., 2016].
Before we proceed to the applications and proof of Theorem 2.1, we ?rst list several examples of A
with classical distribution of eigenvalues and discuss how Theorem 2.1 could be applied to obatin
good Frobenius-norm approximations of A. We begin with the case where eigenvalues of A have a
polynomial decay rate (i.e., power law). Such matrices are ubiquitous in practice [Liu et al., 2015].
? ? A?2 ? ? for some ? ? (0, 1/2] and
Corollary 2.1 (Power-law spectral decay). Suppose ?A
??
?1/?
?j (A) = j for some ? > 1/2. Set k = ?min{C1 ?
, n} ? 1?. If k ? 1 then
? 2??1
?
2??1
? k ? A?F ? C ? ? max ? 2? , n? 2? ,
?A
1
where C1 , C1? > 0 are constants that only depend on ?.
We remark that the
?j (A) = j ?? implies that the eigenvalues lie in an ?q ball for
?nassumption
q
q = 1/?; that is, j=1 ?j (A) = O(1). The error bound in Corollary 2.1 matches the minimax
rate (derived by Negahban and Wainwright ?
[2012]) for matrix completion when the spectrum is
constrained in an ?q ball, by replacing ? with n/N where N is the number of observed entries.
Next, we consider the case where eigenvalues satisfy a faster decay rate.
? ? A?2 ? ? for some ? ? (0, e?16 ) and
Corollary 2.2 (Exponential spectral decay). Suppose ?A
?j (A) = exp{?cj} for some c > 0. Set k = ?min{c?1 log(1/?) ? c?1 log log(1/?), n} ? 1?. If
k ? 1 then
? ?
?
? k ? A?F ? C ? ? max ? log(1/?)3 , n1/2 exp(?cn) ,
?A
2
where C2? > 0 is a constant that only depends on c.
Both corollaries are proved in the appendix. The error bounds in both Corollaries 2.1 and 2.2 are
? ? A?F ? n1/2 ?. We also remark
? which satis?es ?A
signi?cantly better than the trivial estimate A,
that the bound in Corollary 2.1 cannot be obtained by a direct application of the weaker bound Eq. (3),
?
which yields a ? 2??1 bound.
We next state results that are consequences of Theorem 2.1 in several matrix estimation problems.
2.1
High-rank Matrix Completion
Suppose A is a high-rank n ? n PSD matrix that satis?es ?0 -spikeness condition de?ned as follows:
4
De?nition 2.1 (Spikeness condition). An n ? n PSD matrix A satis?es ?0 -spikeness condition if
n?A?max ? ?0 ?A?F , where ?A?max = max1?i,j?n |Aij | is the max-norm of A.
Spikeness condition makes uniform sampling of matrix entries powerful in matrix completion
problems. If A is exactly low rank, the spikeness condition is implied by an upper bound on
max1?i?n ?e?
i Uk ?2 , which is the standard incoherence assumption on the top-k space of A [Candes
and Recht, 2012]. For general high-rank A, the spikeness condition is implied by a more restrictive
incoherence condition that imposes an upper bound on max1?i?n ?e?
i Un?k ?2 and ?An?k ?max ,
which are assumptions adopted in [Hardt and Wootters, 2014].
? is a symmetric re-scaled zero-?lled matrix of observed entries. That is,
Suppose A
?
Aij /p, with probability p;
? ij =
?1 ? i ? j ? n.
[A]
0,
with probability 1 ? p;
(5)
Here p ? (0, 1) is a parameter that controls the probability of observing a particular entry in A,
? and A are symmetric so we only
corresponding to a sample complexity of O(n2 p). Note that both A
?
specify the upper triangle of A. By a simple application of matrix Bernstein inequality [Mackey
? is close to A in spectral norm when A satis?es ?0 -spikeness. Here we
et al., 2014], one can show A
cite a lemma from [Hardt, 2014] to formally establish this observation:
Lemma 2.1 (Corollary of [Hardt, 2014], Lemma A.3). Under the model of Eq. (5) and ?0 -spikeness
condition of A, for t ? (0, 1) it holds with probability at least 1 ? t that
??
??
?
? ?2 ?A?2 log(n/t) ? ?A? log(n/t) ?
0
F
0
F
? ? A?2 ? O ?max
?.
?A
,
?
?
np
np
? k be the best rank-k approximation of A
? in Frobenius/spectral norm. Applying Theorem 2.1
Let A
and 2.2 we obatin the following result:
Theorem 2.3. Fix t ? (0, 1). Then with probability 1 ? t we have
? 2
?
?
?0 ?A?2F log(n/t)
? k ? A?F ? O( k) ? ?A ? Ak ?F
?A
if p = ?
.
n?k+1 (A)2
Furthermore, for ?xed ? ? (0, 1/4], with probability 1 ? t we have
?
? 2
?4 2
2
? k ? A?F ? (1 + O(?)) ?A ? Ak ?F if p = ? ?0 max{? , k }?A?F log(n/t)
?A
n?k+1 (A)2
? k ? A?F ? ?A ? Ak ?F + ? (?k (A) ? ?k+1 (A)) if p = ?
?A
?
?20 k?A?2F log(n/t)
n?2 (?k (A) ? ?k+1 (A))
2
?
.
?
As a remark, because ?0 ? 1 and ?A?F /?k+1 (A) ? k always hold, the sample complexity is
lower bounded by ?(nk log n), the typical sample complexity in noiseless matrix completion. In
the case of high rank A, the results in Theorem 2.3 are the strongest when A has small stable rank
rs (A) = ?A?2F /?A?22 and the top-k condition number ?k (A) = ?1 (A)/?k+1 (A) is not too large.
?
? k ? A?F has an O( k) multiplicative error
For example, if A has stable rank rs (A) = r then ?A
bound with sample complexity ?(?20 ?k (A)2 ? nr log n); or an (1 + O(?)) relative error bound with
sample complexity ?(?20 max{??4 , k 2 }?k (A)2 ? nr log n). Finally, when ?k+1 (A) is very small
and the ?gap? ?k (A) ? ?k+1 (A) is large, a weaker additive-error bound is applicable with sample
complexity independent of ?k+1 (A)?1 .
Comparing with previous works, if? the gap (1 ? ?k+1 /?k ) is of order ?, then sample complexity
of[Hardt, 2014] Theorem 1.2 and [Hardt and Wootters, 2014] Theorem 1 scale with 1/?7 . Our result
improves their results to the scaling of 1/?4 with a much simpler algorithm (truncated SVD).
5
2.2
High-rank matrix de-noising
Let A be an n ? n PSD signal matrix and E a symmetric random Gaussian matrix with zero mean and
i.i.d.
? = A + E.
? 2 /n variance. That is, Eij ? N (0, ? 2 /n) for 1 ? i ? j ? n and Eij = Eji . De?ne A
?
The matrix de-noising problem is then to recover the signal matrix A from noisy observations A.
We refer the readers to [Gavish and Donoho, 2014] for a list of references that shows the ubiquitous
application of matrix de-noising in scienti?c ?elds.
? ? A?2 = ?E?2 =
It is well-known by concentration results of Gaussian random matrices, that ?A
? k be the best rank-k approximation of A
? in Frobenius/spectral norm. Applying
OP (?). Let A
Theorem 2.1, we immediately have the following result:
Theorem 2.4. There exists an absolute constant c > 0 such that, if ? < c ? ?k+1 (A) for some
1 ? k < n, then with probability at least 0.8 we have that
??
?
??
?
?
? k ? A?F ? 1 + O
?A ? Ak ?F + O( k?).
(6)
?A
?k+1 (A)
Eq.?(6) can be understood from a classical bias-variance tradeoff perspective: the ?rst (1 +
cut-off rank k,
O( ?/?k+1 (A)))?A ? Ak ?F acts as a bias term, which decreases as we increase
?
corresponding to a more complicated model; on the other hand, the second O( k?) term acts as the
(square root of) variance, which does not depend on the signal A and increases with k.
2.3
Low-rank estimation of high-dimensional covariance
Suppose A is an n ? n PSD matrix and X1 , ? ? ? , XN are i.i.d. samples drawn from the multivariate
Gaussian distribution Nn (0, A). The question is to estimate A from samples X1 , ? ? ? , XN . A
? = 1 ?N Xi X ? . While in low-dimensional
common estimator is the sample covariance A
i
i=1
N
? is obvious (cf. [Van der Vaart,
regimes (i.e., n ?xed and N ? ?) the asymptotic ef?ciency of A
2000]), its statistical power in high-dimensional regimes where n and N are comparable are highly
? ? A?? , ? = 2/F
non-trivial. Below we cite results by Bunea and Xiao [2015] for estimation error ?A
when n is not too large compared to N :
Lemma 2.2 (Bunea and Xiao [2015]). Suppose n = O(N ? ) for some ? ? 0 and let re (A) =
? =
tr(A)/?A?2 denote the effective rank of the covariance A. Then the sample covariance A
?N
1
?
i=1 Xi Xi satis?es
N
?
?
?
log
N
? ? A?F = OP ?A?2 re (A)
(7)
?A
N
and
? ? A?2 = OP
?A
?
?A?2 max
??
re (A) log(N n) re (A) log(N n)
,
N
N
??
.
(8)
? k be the best rank-k approximation of A
? in Frobenius/spectral norm. Applying Theorem 2.1
Let A
and 2.2 together with Eq. (8), we immediately arrive at the following theorem.
Theorem 2.5. Fix ? ? (0, 1/4] and 1 ? k < n. Recall that re (A) = tr(A)/?A?2 and ?k (A) =
?1 (A)/?k+1 (A). There exists a universal constant c > 0 such that, if
re (A) max{??4 , k 2 }?k (A)2 log(N )
?c
N
then with probability at least 0.8,
and if
? k ? A?F ? (1 + O(?)) ?A ? Ak ?F
?A
re (A)k?A?22 log(N )
N ?2 (?k (A) ? ?k+1 (A))
6
2
?c
then with probability at least 0.8,
? k ? A?F ? ?A ? Ak ?F + ? (?k (A) ? ?k+1 (A)) .
?A
?
Theorem 2.5 shows that it is possible to obtain a reasonable Frobenius-norm approximation of A
by truncated SVD in the asymptotic regime of N = ?(re (A)poly(k) log N ), which is much more
?exible than Eq. (7) that requires N = ?(re (A)2 log N ).
3
Proof Sketch of Theorem 2.1
In this section we give a proof sketch of Theorem 2.1. The proof of Theorem 2.2 is similar and less
challenging so we defer it to appendix. We defer proofs of technical lemmas to Section A.
?
? k ? Ak ?F is upper bounded by an O( k) factor of
? k and Ak are low-rank, ?A
Because both A
? k ? Ak ?2 . From the condition that ?A
? ? A?2 ? ?, a straightforward approach to upper
?A
? k ? Ak ?2 is to consider the decomposition ?A
? k ? A k ? 2 ? ?A
? ? A?2 + 2?Uk U? ?
bound ?A
k
? kU
? k ?2 , where Uk U? and U
? ? ?2 ?A
? kU
? ? are projection operators onto the top-k eigenspaces
U
k
k
k
? respectively. Such a naive approach, however, has two major disadvantages. First,
of A and A,
? k ?2 , which is additive and may be much larger than ?A
? ? A?2 .
the upper bound depends on ?A
? kU
? ? ?2 depends on the ?consecutive? sepctral
Perhaps more importantly, the quantity ?Uk U?
?
U
k
k
gap (?k (A) ? ?k+1 (A)), which could be very small for large matrices.
The key idea in the proof of Theorem 2.1 is to ?nd an ?envelope? m1 ? k ? m2 in the spectrum of
A surrounding k, such that the eigenvalues within the envelope are relatively close. De?ne
m1
=
m2
=
argmax0?j?k {?j (A) ? (1 + 2?)?k+1 (A)};
argmaxk?j?n {?j (A) ? ?k (A) ? 2??k+1 (A)},
where we let ?0 (A) = ? for convenience. Let Um , U?m be basis of the top m-dimensional linear
? respectively. Also denote Un?m and U?n?m as basis of the orthogonal
subspaces of A and A,
complement of Um and U?m . By asymmetric Davis-Kahan inequality (Lemma C.1) and Wely?s
inequality we can obtain the following result.
? ? A?2 ? ?2 ?k+1 (A) for ? ? (0, 1) then ?U
? ? Um ?2 , ?U
? ? Un?m ?2 ? ?.
Lemma 3.1. If ?A
n?k
1
k
2
Let Um1 :m2 be the linear subspace of A associated with eigenvalues ?m1 +1 (A), ? ? ? , ?m2 (A).
Intuitively, we choose a (k ? m1 )-dimensional linear subspace in Um1 :m2 that is ?most aligned? with
? Formally, de?ne
the top-k subspace U?k of A.
?
?
?k .
W = argmaxdim(W)=k?m1 ,W?Um1 :m2 ?k?m1 W? U
W is then a d ? (k ? m1 ) matrix with orthonormal columns that corresponds to a basis of W. W is
carefully constructed so that it is closely aligned with U?k , yet still lies in Uk . In particular, Lemma
? ? W?2 is upper bounded by ?.
3.2 shows that sin ?(W, U?k ) = ?U
n?k
? ? W?2 ? ?.
? ? A?2 ? ?2 ?k+1 (A) for ? ? (0, 1) then ?U
Lemma 3.2. If ?A
n?k
Now de?ne
? = Am + WW? AWW? .
A
1
? as the ?reference matrix" because we can decompose ?A
? k ? A?F as
We use A
?
? k ? A?F ? ?A ? A?
? F + ?A
? k ? A?
? F ? ?A ? A?
? F + 2k?A
? k ? A?
? 2
?A
(9)
and bound each term on the right hand side separately. Here the last inequality holds because both
? have rank at most k. The following lemma bounds the ?rst term.
? k and A
A
? ? A?2 ? ?2 ?k+1 (A)2 for ? ? (0, 1/4] then ?A ? A?
? F ? (1 + 32?)?A ? Ak ?F .
Lemma 3.3. If ?A
7
The proof of this lemma relies Pythagorean theorem and Poincar? separation theorem. Let Um1 :m2
be the (m2 ? m1 )-dimensional linear subspace such that Um2 = Um1 ? Um1 :m2 . De?ne Am1 :m2 =
Um1 :m2 ?m1 :m2 U?
m1 :m2 , where ?m1 :m2 = diag(?m1 +1 (A), ? ? ? , ?m2 (A)) and Um1 :m2 is an orthonormal basis associated with Um1 :m2 . Applying Pythagorean theorem (Lemma C.2), we can
decompose
? 2 = ?A ? Am ?2 + ?Am :m ?2 ? ?WW? Am :m WW? ?2 .
?A ? A?
F
F
2 F
1
2 F
1
2
Applying Poincar? separation theorem (Lemma C.3) where X = ?m1 :m2 and P = U?
m1 :m2 W, we
?m2 ?m1
?m 2
2
2
have ?W? Am1 :m2 W?2F ? j=m
?
(A
)
=
?
(A)
. With some
j
m1 :m2
j=m1 +m2 ?k+1 j
2 ?k+1
routine algebra we can prove Lemma 3.3.
To bound the second term of Eq. (9) we use the following lemma.
? ? A?2 ? ?2 ?k+1 (A) for ? ? (0, 1/4] then ?A
? k ? A?
? 2 ? 102?2 ?A ? Ak ?2 .
Lemma 3.4. If ?A
? k and A.
? Recall the de?nition that
The proof of Lemma 3.4 relies on the low-rankness of A
? and U?? = Null(A).
? Consider ?v?2 = 1 such that v ? (A
? k ? A)v
? = ?A
?
? 2.
U? = Range(A)
? k ? A??
? ?
?
?
?
Because v maximizes v (Ak ?A)v over all unit-length vectors, it must lie in the range of Ak ? A
because otherwise the component outside the range will not contribute. Therefore, we can choose v
? k ) = U?k and v 2 ? Range(A)
? = U?. Subsequently, we have
that v = v 1 + v 2 where v 1 ? Range(A
that
?U
? ?U
? ?v + U
? n?k U
?? v
? kU
(10)
v = U
k
=
n?k
? kU
?U
? ?v + U
? ?U
? ?U
? ? v.
U
k
?
(11)
Consider the following decomposition:
? ?
? ?
? ?
?
?
? ? ?
? ?? ? ??v ? (A
? ? A)v ?? + ??v ? (A
? k ? A)v
? ?? + ??v ? (A ? A)v
? ?? .
?v (Ak ? A)v
? ? A)v| is trivially upper bounded by ?A
? ? A?2 ? ?2 ?k+1 (A). The second
The ?rst term |v ? (A
?
and the third term can be bounded by Wely?s inequality (Lemma C.4) and basic properties of A
(Lemma A.3). See Section A for details.
4
Discussion
We mention two potential directions to further extend results of this paper.
4.1
Model selection for general high-rank matrices
? ? A?2 ? ?2 ?k+1 (A), which could be
The validity of Theorem 2.1 depends on the condition ?A
hard to verify if ?k+1 (A) is unknown and dif?cult to estimate. Furthermore, for general high-rank
matrices, the model selection problem of determining an appropriate (or even optimal) cut-off rank k
requires knowledge of the distribution of the entire spectrum of an unknown data matrix, which is
even more challenging to obtain.
One potential approach is to impose a parametric pattern of decay of the eigenvalues (e.g., polynomial
and exponential decay), and to estimate a small set of parameters (e.g., degree of polynomial) from
? Afterwards, the optimal cut-off rank k could be determined by a theoretical
the noisy observations A.
analysis, similar to the examples in Corollaries 2.1 and 2.2. Another possibility is to use repeated
sampling techniques such as boostrap in a stochastic problem (e.g.,
? matrix de-noising) to estimate the
?bias? term ?A ? Ak ?F for different k, as the variance term k? is known or easy to estimate.
4.2
Minimax rates for polynomial spectral decay
Consider the class of PSD matrices whose eigenvalues follow a polynomial (power-law) decay:
?(?, n) = {A ? Rn?n : A ? 0, ?j (A) = j ?? }. We are interested in the following minimax rates
for completing or de-noising matrices in ?(?, n):
8
Question 1 (Completion of ?(?, n)). Fix n ? N, p ? (0, 1) and de?ne N = pn2 . For M ? ?(?, n),
? ij = Mij with probability p and A
? ij = 0 with probability 1 ? p. Also let ?(?0 , n) = {M ?
let A
Rn?n : n?M?max ? ?0 ?M?F } be the class of all non-spiky matrices. Determine
R1 (?0 , ?, n, N ) := inf
sup
? ?M
? M??(?,n)??(?0 ,n)
A?
? ? M?2 .
E?M
F
? = M + ?/?nZ, where Z is a
Question 2 (De-noising of ?(?, n)). Fix n ? N, ? > 0 and let A
symmetric matrices with i.i.d. standard Normal random variables on its upper triangle. Determine
R2 (?, ?, n) := inf
sup
? ?M
? M??(?,n)
A?
? ? M?2 .
E?M
F
Compared to existing settings on matrix completion and de-noising, we believe ?(?, n) is a more
natural matrix class which allows for general high-rank matrices, but also imposes suf?cient spectral
decay conditions so that spectrum truncation algorithms result in signi?cant bene?ts. Based on
Corollary 2.1 and its matching lower bounds for a larger ?p class [Negahban and Wainwright, 2012],
we make the following conjecture:
Conjecture 4.1. For ? > 1/2 and ? not too small, we conjecture that
R1 (?0 , ?, n, N ) ? C(?0 ) ?
? n ? 2??1
2?
N
and
where C(?0 ) > 0 is a constant that depends only on ?0 .
5
? ? 2??1
R2 (?, ?, n) ? ? 2 2? ,
Acknowledgements
S.S.D. was supported by ARPA-E Terra program. Y.W. and A.S. were supported by the NSF CAREER
grant IIS-1252412.
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6,244 | 6,649 | f -GANs in an Information Geometric Nutshell
Richard Nock?,?,?
Zac Cranko?,?
Aditya Krishna Menon?,?
?,?
Lizhen Qu
Robert C. Williamson?,?
?
?
Data61, the Australian National University and ? the University of Sydney
{firstname.lastname, aditya.menon, bob.williamson}@data61.csiro.au
Abstract
Nowozin et al showed last year how to extend the GAN principle to all f divergences. The approach is elegant but falls short of a full description of the
supervised game, and says little about the key player, the generator: for example,
what does the generator actually converge to if solving the GAN game means
convergence in some space of parameters? How does that provide hints on the generator?s design and compare to the flourishing but almost exclusively experimental
literature on the subject? In this paper, we unveil a broad class of distributions for
which such convergence happens ? namely, deformed exponential families, a wide
superset of exponential families ?. We show that current deep architectures are
able to factorize a very large number of such densities using an especially compact
design, hence displaying the power of deep architectures and their concinnity in the
f -GAN game. This result holds given a sufficient condition on activation functions
? which turns out to be satisfied by popular choices. The key to our results is a
variational generalization of an old theorem that relates the KL divergence between
regular exponential families and divergences between their natural parameters. We
complete this picture with additional results and experimental insights on how
these results may be used to ground further improvements of GAN architectures,
via (i) a principled design of the activation functions in the generator and (ii) an
explicit integration of proper composite losses? link function in the discriminator.
1
Introduction
In a recent paper, Nowozin et al. [30] showed that the GAN principle [15] can be extended to the
variational formulation of all f -divergences. In the GAN game, there is an unknown distribution P
which we want to approximate using a parameterised distribution Q. Q is learned by a generator
by finding a saddle point of a function which we summarize for now as f -GAN(P, Q), where f is
a convex function (see eq. (7) below for its formal expression). A part of the generator?s training
involves as a subroutine a supervised adversary ? hence, the saddle point formulation ? called
discriminator, which tries to guess whether randomly generated observations come from P or Q.
Ideally, at the end of this supervised game, we want Q to be close to P, and a good measure of this is
the f -divergence If (PkQ), also known as Ali-Silvey distance [1, 12]. Initially, one choice of f was
considered [15]. Nowozin et al. significantly grounded the game and expanded its scope by showing
that for any f convex and suitably defined, then [30, Eq. 4]:
f -GAN(P, Q) ? If (PkQ) .
(1)
The inequality is an equality if the discriminator is powerful enough. So, solving the f -GAN game
can give guarantees on how P and Q are distant to each other in terms of f -divergence. This elegant
characterization of the supervised game unfortunately falls short of justifying or elucidating all
parameters of the supervised game [30, Section 2.4], and the paper is also silent regarding a key
part of the game: the link between distributions in the variational formulation and the generator, the
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
main player which learns a parametric model of a density. In doing so, the f -GAN approach and
its members remain within an information theoretic framework that relies on divergences between
distributions only [30]. In the GAN world at large, this position contrasts with other prominent
approaches that explicitly optimize geometric distortions between the parameters or support of
distributions [6, 14, 16, 21, 22], and raises the problem of connecting the f -GAN approach to any
sort of information geometric optimization. One such information-theoretic/information-geometric
identity is well known: The Kullback-Leibler (KL) divergence between two distributions of the
same (regular) exponential family equals a Bregman divergence D between their natural parameters
[2, 4, 7, 9, 35], which we can summarize as:
IfKL (PkQ) = D(?k?) .
(2)
Here, ? and ? are respectively the natural parameters of P and Q. Hence, distributions are points on
a manifold on the right-hand side, a powerful geometric statement [4]; however, being restricted to
KL divergence or "just" exponential families, it certainly falls short of the power to explain the GAN
game. To our knowledge, the only generalizations known fall short of the f -divergence formulation
and are not amenable to the variational GAN formulation [5, Theorem 9], [13, Theorem 3].
Our first contribution is such an identity that connects the general If -divergence formulation in eq.
(1) to the general D (Bregman) divergence formulation in eq. (2). We now briefly state it, postponing
the details to Section 3:
f -GAN(P, escort(Q)) = D(?k?) + Penalty(Q) ,
(3)
for P and Q (with respective parameters ? and ?) which happen to lie in a superset of exponential
families called deformed exponential families, that have received extensive treatment in statistical
physics and differential information geometry over the last decade [3, 25]. The right-hand side of
eq. (3) is the information geometric part [4], in which D is a Bregman divergence. Therefore, the
f -GAN problem can be equivalent to a geometric optimization problem [4], like for the Wasserstein
GAN and its variants [6]. Notice also that Q appears in the game in the form of an escort [5]. The
difference vanish only for exponential families (escort(Q) = Q, Penalty(Q) = 0 and f = KL).
Our second contribution drills down into the information-theoretic and information-geometric
parts of (3). In particular, from the former standpoint, we completely specify the parameters of the
supervised game, unveiling a key parameter left arbitrary in [30] (explicitly incorporating the link
function of proper composite losses [32]). From the latter standpoint, we show that the standard deep
generator architecture is powerful at modelling complex escorts of any deformed exponential family,
factorising a number of escorts in order of the total inner layers? dimensions, and this factorization
happens for an especially compact design. This hints on a simple sufficient condition on the activation
function to guarantee the escort modelling, and it turns out that this condition is satisfied, exactly or
in a limit sense, by most popular activation functions (ELU, ReLU, Softplus, ...). We also provide
experiments1 that display the uplift that can be obtained through a principled design of the activation
function (generator), or tuning of the link function (discriminator).
Due to the lack of space, a supplement (SM) provides the proof of the results in the main file and
additional experiments. A longer version with a more exhaustive treatment of related results is
available [27]. The rest of this paper is as follows. Section ? 2 presents definition, ? 3 formally
presents eq. (3), ? 4 derives consequences for deep learning, ? 5 completes the supervised game
picture of [30], Section ? 6 presents experiments and a last Section concludes.
2
Definitions
Throughout this paper, the domain X of observations is a measurable set. We begin with two
important classes of distortion measures, f -divergences and Bregman divergences.
Definition 1 For any two distributions P and Q having respective densities P and Q absolutely
continuous with respect to a base measure ?, the f -divergence between P and Q, where f : R+ ? R
is convex with f (1) = 0, is
Z
P (X)
P (x)
.
If (PkQ) = EX?Q f
=
Q(x) ? f
d?(x) .
(4)
Q(X)
Q(x)
X
1
The code used for our experiments is available through https://github.com/qulizhen/fgan_info_geometric
2
For any convex differentiable ? : Rd ? R, the (?-)Bregman divergence between ? and % is:
.
D? (?k%) = ?(?) ? ?(%) ? (? ? %)> ??(%) ,
where ? is called the generator of the Bregman divergence.
(5)
f -divergences are the key distortion measure of information theory, while Bregman divergences are
the key distortion measure of information geometry. A distribution P from a (regular) exponential
.
family with cumulant C : ? ? R and sufficient statistics ? : X ? Rd has density PC (x|?, ?) =
>
exp(?(x) ? ? C(?)), where ? is a convex open set, C is convex and ensures normalization on the
simplex (we leave implicit the associated dominating measure [3]). A fundamental Theorem ties
Bregman divergences and f -divergences: when P and Q belong to the same exponential family, and
denoting their respective densities PC (x|?, ?) and QC (x|?, ?), it holds that IKL (PkQ) = DC (?k?).
.
Here, IKL is Kullback-Leibler (KL) f -divergence (f = x 7? x log x). Remark that the arguments in
the Bregman divergence are permuted with respect to those in eq. (2) in the introduction. This also
holds if we consider fKL in eq. (2) to be the Csisz?r dual of f [8], namely fKL : x 7? ? log x, since
in this case IfKL (PkQ) = IKL (QkP) = DC (?k?). We made this choice in the introduction for the
sake of readability in presenting eqs. (1 ? 3). We now define generalizations of exponential families,
following [5, 13]. Let
? : R+ ? R+ be non-decreasing [25, Chapter 10].
R z We define the ?-logarithm,
. Rz 1
.
log? , as log? (z) = 1 ?(t)
dt. The ?-exponential is exp? (z) = 1 + 0 ?(t)dt, where ? is defined
.
by ?(log? (z)) = ?(z). In the case where the integrals are improper, we consider the corresponding
limit in the argument / integrand.
Definition 2 [5] A distribution P from a ?-exponential family (or deformed exponential family, ?
being implicit) with convex cumulant C : ? ? R and sufficient statistics ? : X ? Rd has density
.
given by P?,C (x|?, ?) = exp? (?(x)> ? ? C(?)), with respect to a dominating measure ?. Here, ?
is a convex open set and ? is called the coordinate of P. The escort density (or ?-escort) of P?,C is
Z
1
.
.
? ?(P?,C ) , Z =
?(P?,C (x|?, ?))d?(x) .
(6)
P??,C =
Z
X
Z is the escort?s normalization constant.
? the escort distribution of P whose density
We leaving implicit the dominating measure and denote P
is given by eq. (6). We shall name ? the signature of the deformed (or ?-)exponential family, and
sometimes drop indexes to save readability without ambiguity, noting e.g. P? for P??,C . Notice that
normalization in the escort is ensured by a simple integration [5, Eq. 7]. For the escort to exist, we
require that Z < ? and therefore ?(P ) is finite almost everywhere. Such a requirement would be
satisfied in the GAN game. There is another generalization of regular exponential families, known as
generalized exponential families [13, 27]. The starting point of our result is the following Theorem,
in which the information-theoretic part is not amenable to the variational GAN formulation.
Theorem 3 [5][36] for any two ?-exponential distributions P and Q with respective densities
P?,C , Q?,C and coordinates ?, ?, DC (?k?) = EX?Q? [log? (Q?,C (X)) ? log? (P?,C (X))].
We now briefly frame the now popular (f -)GAN adversarial learning [15, 30]. We have a true
unknown distribution P over a set of objects, e.g. 3D pictures, which we want to learn. In the GAN
setting, this is the objective of a generator, who learns a distribution Q? parameterized by vector
?. Q? works by passing (the support of) a simple, uninformed distribution, e.g. standard Gaussian,
through a possibly complex function, e.g. a deep net whose parameters are ? and maps to the support
of the objects of interest. Fitting Q. involves an adversary (the discriminator) as subroutine, which
fits classifiers, e.g. deep nets, parameterized by ?. The generator?s objective is to come up with
arg min? Lf (?) with Lf (?) the discriminator?s objective:
.
Lf (?) = sup{EX?P [T? (X)] ? EX?Q? [f ? (T? (X))]} ,
(7)
?
where ? is Legendre conjugate [10] and T? : X ? R integrates the classifier of the discriminator
and is therefore parameterized by ?. Lf is a variational approximation to a f -divergence [30]; the
discriminator?s objective is to segregate true (P) from fake (Q. ) data. The original GAN choice, [15]
.
fGAN (z) = z log z ? (z + 1) log(z + 1) + 2 log 2
(8)
(the constant ensures f (1) = 0) can be replaced by any convex f meeting mild assumptions.
3
3
A variational information geometric identity for the f -GAN game
We deliver a series of results that will bring us to formalize eq. (3). First, we define a new set of
distortion measures, that we call KL? divergences.
Definition 4 For any ?-logarithm and distributions P, Q having respective densities P and Q
absolutely continuous with respect
to base measure ?, the KL? divergence between P and Q is
.
defined as KL? (PkQ) = EX?P ? log? (Q(X)/P (X)) .
Since ? is non-decreasing, ? log? is convex and so any KL? divergence is an f -divergence.
.
When ?(z) = z, KL? is the KL divergence. In what follows, base measure ? and absolute continuity are implicit, as well as that P (resp. Q) is the density of P (resp. Q). We
now relate KL? divergences vs f -divergences. Let ?f be the subdifferential of convex f and
.
IP,Q = [inf x P (x)/Q(x), supx P (x)/Q(x)) ? R+ denote the range of density ratios of P over Q.
Our first result states that if there is a selection of the subdifferential which is upperbounded on IP,Q ,
the f -divergence If (PkQ) is equal to a KL? divergence.
Theorem 5 Suppose that P, Q are such that there exists a selection ? ? ?f with sup ?(IP,Q ) < ?.
Then ?? : R+ ? R+ non decreasing such that If (PkQ) = KL? (QkP).
Theorem 5 essentially covers most if not all relevant GAN cases, as the assumption has to be satisfied
in the GAN game for its solution not to be vacuous up to a large extent (eq. (7)). We provide a
more complete treatment in the extended version [27]. The proof of Theorem 5 (in SM, Section
I) is constructive: it shows how to pick ? which satisfies all requirements. It brings the following
interesting corollary: under mild assumptions on f , there exists a ? that fits for all densities P and Q.
A prominent example of f that fits is the original GAN choice for which we can pick
.
?GAN (z) =
1
.
log 1 + z1
(9)
We now define a slight generalization of KL? -divergences and allow for ? to depend on the choice
of the expectation?s X, granted that for any of these choices, it will meet the constraints to be
R+ ? R+ and also increasing,
h and therefore define a ivalid signature. For any f : X ? R+ , we
.
.
denote KL?f (PkQ) = EX?P ? log?f (X) (Q(X)/P (X)) , where for any p ? R+ , ?p (t) = p1 ? ?(tp).
Whenever f = 1, we just write KL? as we already did in Definition 4. We note that for any x ? X,
?f (x) is increasing and non negative because of the properties of ? and f , so ?f (x) (t) defines a
?-logarithm. We are ready to state a Theorem that connects KL? -divergences and Theorem 3.
.
.
Theorem 6 Letting P = P?,C and Q = Q?,C for short in Theorem 3, we have EX?Q? [log? (Q(X)) ?
.
?
?
log? (P (X))] = KL?Q? (QkP)
? J(Q), with J(Q) = KL?Q? (QkQ).
(Proof in SM, Section II) To summarize, we know that under mild assumptions relatively to the GAN
game, f -divergences coincide with KL? divergences (Theorems 5). We also know from Theorem
6 that KL?. divergences quantify the geometric proximity between the coordinates of generalized
exponential families (Theorem 3). Hence, finding a geometric (parameter-based) interpretation of
the variational f -GAN game as described in eq. (7) can be done via a variational formulation of the
KL? divergences appearing in Theorem 6. Since penalty J(Q) does not belong to the GAN game (it
?
does not depend on P), it reduces our focus on KL?Q? (QkP).
?
Theorem 7 KL?Q? (QkP)
admits the variational formulation
n
o
?
KL?Q? (QkP)
=
sup
EX?P [T (X)] ? EX?Q? [(? log?Q? )? (T (X))] ,
T ?R++
(10)
X
.
with R++ = R\R++ . Furthermore, letting Z denoting the normalization constant of the ?-escort of
Q, the optimum T ? : X ? R++ to eq. (10) is T ? (x) = ?(1/Z) ? (?(Q(x))/?(P (x))).
4
(Proof in SM, Section III) Hence, the variational f -GAN formulation can be captured in an
information-geometric framework by the following identity using Theorems 3, 5, 7.
Corollary 8 (the variational information-geometric f -GAN identity) Using notations from Theorems
6, 7 and letting ? (resp. ?) denote the coordinate of P (resp. Q), we have:
sup
T ?R++
X
n
o
EX?P [T (X)] ? EX?Q? [(? log?Q? )? (T (X))] = DC (?k?) + J(Q) .
(11)
We shall also name for short vig-f -GAN the identity in eq. (11). We note that we can drill down
further the identity, expressing in particular the Legendre conjugate (? log?Q? )? with an equivalent
"dual" (negative) ?-logarithm in the variational problem [27]. The left hand-side of Eq. (11) has the
exact same overall shape as the variational objective of [30, Eqs 2, 6]. However, it tells the formal
story of GANs in significantly greater details, in particular for what concerns the generator. For
example, eq. (11) yields a new characterization of the generators? convergence: because DC is a
Bregman divergence, it satisfies the identity of the indiscernibles. So, solving the f -GAN game [30]
can guarantees convergence in the parameter space (? vs ?). In the realm of GAN applications, it
makes sense to consider that P (the true distribution) can be extremely complex. Therefore, even when
deformed exponential families are significantly more expressive than regular exponential families
[25], extra care should be put before arguing that complex applications comply with such a geometric
convergence in the parameter space. One way to circumvent this problem is to build distributions in
Q that factorize many deformed exponential families. This is one strong point of deep architectures
that we shall prove next.
4
Deep architectures in the vig-f -GAN game
In the GAN game, distribution Q in eq. (11) is built by the generator (call it Qg ), by passing the
support of a simple distribution (e.g. uniform, standard Gaussian), Qin , through a series of non-linear
transformations. Letting Qin denote the corresponding density, we now compute Qg . Our generator
g : X ? Rd consists of two parts: a deep part and a last layer. The deep part is, given some L ? N,
the computation of a non-linear transformation ?L : X ? RdL as
.
Rdl 3 ?l (x) =
.
?0 (x) =
v(Wl ?l?1 (x) + bl ) , ?l ? {1, 2, ..., L} ,
x?X .
(12)
(13)
v is a function computed coordinate-wise, such as (leaky) ReLUs, ELUs [11, 17, 23, 24], Wl ?
.
Rdl ?dl?1 , bl ? Rdl . The last layer computes the generator?s output from ?L : g(x) = vOUT (??L (x)+
d?dL
d
?), with ? ? R
, ? ? R ; in general, vOUT 6= v and vOUT fits the output to the domain at hand,
ranging from linear [6, 20] to non-linear functions like tanh [30]. Our generator captures the
high-level features of some state of the art generative approaches [31, 37].
To carry our analysis, we make the assumption that the network is reversible, which is going to reguire
that vOUT , ?, Wl (l ? {1, 2, ..., L}) are invertible. We note that popular examples can be invertible (e.g.
DCGAN, if we use ?-ReLU, dimensions match and fractional-strided convolutions are invertible).
At this reasonable price, we get in closed form the generator?s density and it shows the following:
for any continuous signature ?net , there exists an activation function v such that the deep part in the
network factors as escorts for the ?net -exponential family. Let 1i denote the ith canonical basis vector.
Theorem 9 ?vOUT , ?, Wl invertible (l ? {1, 2, ..., L}), for any continuous signature ?net , there exists
.
activation v and bl ? Rd (?l ? {1, 2, ..., L}) such that for any output z, letting x = g ?1 (z),
? deep (x)) ? 1/(Hout (x) ? Znet ), with Znet > 0 a constant,
Qg (z) factorizes as Qg (z) = (Qin (x)/Q
. Qd
.
.
0
>
Hout (x) = i=1 |vOUT (?i ?L (x) + ?i )|, ?i = ?> 1i , and (letting wl,i = W>
l 1i ):
.
? deep (x) =
Q
L Y
d
Y
P??net ,bl,i (x|wl,i , ?l?1 ) .
l=1 i=1
5
(14)
Name
ReLU(?)
Leaky-ReLU(?)
(?, ?)-ELU(?)
prop-? (?)
Softplus(?)
?-ReLU(?)
LSU
v(z)
max{0,
z}
z if z > 0
z if z ? 0
?z
if z > 0
?(exp(z) ? 1) if z ? 0
1
1
?
z
? 0?1 ?(? ? )?1 (? ? (0)z)
? ? (0)
1
?
1 ? 2?z
log 2
?
k + ?? ? (z)
(0)
k + log2 (1 + exp(z))
?
z+ (1??)2 +z 2
k+
2
?
0
if z < ?1
?
k+
(1 + z)2 if z ? [?1, 1]
?
4z
if z > 1
?(z)
1z>0
if z > ??
if z ? ??
if z > ?
if z ? ?
4z 2
(1??)2 +4z 2
?
2 z
4
if
if
z<4
z>4
Table 1: Some strongly/weakly admissible couples (v, ?). (?) : 1. is the indicator function; (?) :
? ? 0, 0 < ? 1 and dom(v) = [?/, +?). (?) : ? ? ? > 0; (?) : ? is Legendre conjugate; (?) :
? ? [0, 1). Shaded: prop-? activations; k is a constant (e.g. such that v(0) = 0) (see text).
(Proof in SM, Section IV) The relationship between the inner layers of a deep net and deformed
exponential families (Definition 2) follows from the theorem: rows in Wl s define coordinates, ?l
define "deep" sufficient statistics, bl are cumulants and the crucial part, the ?-family, is given by the
activation function v. Notice also that the bl s are learned, and so the deformed exponential families?
? deep factors escorts, and in number.
normalization is in fact learned and not specified. We see that Q
What is notable is the compactness achieved by the deep representation: the total dimension of all
? deep (eq. (14)) is L ? d. To handle this, a shallow net with a single inner
deep sufficient statistics in Q
layer would require a matrix W of space ?(L2 ? d2 ). The deep net g requires only O(L ? d2 ) space to
store all Wl s. The proof of Theorem 9 is constructive: it builds v as a function of ?. In fact, the proof
? deep factors ?-escorts.
also shows how to build ? from the activation function v in such a way that Q
The following Lemma essentially says that this is possible for all strongly admissible activations v.
Definition 10 Activation function v is strongly admissible iff dom(v) ? R+ 6= ? and v is C 1 ,
lowerbounded, strictly increasing and convex. v is weakly admissible
iff for any > 0, there exists
. R
v strongly admissible such that ||v ? v ||L1 < , where ||f ||L1 = |f (t)|dt.
Lemma 11 The following holds: (i) for any strongly admissible v, there exists signature ? such that
Theorem 9 holds; (ii) (?,?)-ELU (for any ? > 0), Softplus are strongly admissible. ReLU is weakly
admissible.
(proof in SM, Section V) The proof uses a trick for ReLU which can easily be repeated for (?, ?)ELU, and for leaky-ReLU, with the constraint that the domain has to be lowerbounded. Table 1
provides some examples of strongly / weakly admissible activations. It includes a wide class of
so-called "prop-? activations", where ? is negative a concave entropy, defined on [0, 1] and symmetric
around 1/2 [29]. This concludes our treatment of the information geometric part of the vig-f -GAN
identity. We now complete it with a treatment of its information-theoretic part.
5
A complete proper loss picture of the supervised GAN game
In their generalization of the GAN objective, Nowozin et al. [30] leave untold a key part of the
supervised game: they split in eq. (7) the discriminator?s contribution in two, T? = gf ? V? , where
V? : X ? R is the actual discriminator, and gf is essentially a technical constraint to ensure that
V? (.) is in the domain of f ? . They leave the choice of gf "somewhat arbitrary" [30, Section 2.4]. We
now show that if one wants the supervised loss to have the desirable property to be proper composite
[32]2 , then gf is not arbitrary. We proceed in three steps, first unveiling a broad class of proper
f -GANs that deal with this property. The initial motivation of eq. (7) was that the inner maximisation
may be seen as the f -divergence between P and Q? [26], Lf (?) = If (PkQ? ). In fact, this variational
2
informally, Bayes rule realizes the optimum and the loss accommodates for any real valued predictor.
6
representation of an f -divergence holds more generally: by [33, Theorem 9], we know that for any
convex f , and invertible link function ? : (0, 1) ? R, we have:
inf
E
T : X?R (X,Y)?D
1
[`? (Y, T (X))] = ? ? If (P k Q)
2
(15)
where D is the distribution over (observations ? {fake, real}) and the loss function `? is defined by:
??1 (z)
??1 (z)
.
.
?
0
;
`
(?1,
z)
=
f
f
, (16)
`? (+1, z) = ?f 0
?
1 ? ??1 (z)
1 ? ??1 (z)
.
assuming f differentiable. Note now that picking ?(z) = f 0 (z/(1 ? z)) with z = T (x) and
simplifying eq. (15) with P[Y = fake] = P[Y = real] = 1/2 in the GAN game yields eq. (7). For
other link functions, however, we get an equally valid class of losses whose optimisation will yield
a meaningful estimate of the f -divergence. The losses of eq. (16) belong to the class of proper
composite losses with link function ? [32]. Thus (omitting parameters ?, ?), we rephrase eq. (7) and
refer to the proper f -GAN formulation as inf Q L? (Q) with (` is as per eq. (16)):
.
L? (Q) = sup
E [?`? (+1, T (X))] + E [?`? (?1, T (X))] .
(17)
T : X?R
X?P
X?Q
Note also that it is trivial to start from a suitable proper composite loss, and derive the corresponding
generator f for the f -divergence as per eq. (15). Finally, our proper composite loss view of the
f -GAN game allows us to elicitate gf in [30]: it is the composition of f 0 and ? in eq. (16).
The use of proper composite losses as part of the supervised GAN formulation sheds further light
on another aspect the game: the connection between the value of the optimal discriminator, and
the density ratio between the generator and discriminator distributions. Instead of the optimal
T ? (x) = f 0 (P (x)/Q(x)) for eq. (7) [30, Eq. 5], we now have with the more general eq. (17) the
result T ? (x) = ?((1 + Q(x)/P (x))?1 ). We now show that proper f -GANs can easily be adapted
.
to eq. (11). We let ?? (t) = 1/??1 (1/t).
.
Theorem 12 For any ?, define `x (?1, z) = ? log(?? )
.
1
?
Q(x)
(?z), and let `(+1, z) = ?z. Then
L? (Q) in eq. (17) equals eq. (11). Its link in eq. (17) is ?x (z) = ?1/?Q(x)
(z/(1 ? z)).
?
(Proof in SM, Section VI) Hence, in the proper composite view of the vig-f -GAN identity, the
generator rules over the supervised game: it tempers with both the link function and the loss
? but only for fake examples. Notice also that when z = ?1, the fake examples loss satisfies
`x (?1, ?1) = 0 regardless of x by definition of the ?-logarithm.
6
Experiments
Two of our theoretical contributions are (A) the fact that on the generator?s side, there exists numerous
activation functions v that comply with the design of its density as factoring escorts (Lemma 11),
and (B) the fact that on the discriminator?s side, the so-called output activation function gf of [30]
aggregates in fact two components of proper composite losses, one of which, the link function ?,
should be a fine knob to operate (Theorem 12). We have tested these two possibilities with the idea
that an experimental validation should provide substantial ground to be competitive with mainstream
approaches, leaving space for a finer tuning in specific applications. Also, in order not to mix their
effects, we have treated (A) and (B) separately.
Architectures and datasets ? We provide in SM (Section VI) the detail of all experiments. To summarize, we consider two architectures in our experiments: DCGAN [31] and the multilayer feedforward
network (MLP) used in [30]. Our datasets are MNIST [19] and LSUN tower category [38].
Comparison of varying activations in the generator (A) ? We have compared ?-ReLUs with varying
? in [0, 0.1, ..., 1] (hence, we include ReLU as a baseline for ? = 1), the Softplus and the Least
Square Unit (LSU, Table 1) activation (Figure 1). For each choice of the activation function, all
inner layers of the generator use the same activation function. We evaluate the activation functions
by using both DCGAN and the MLP used in [30] as the architectures. As training divergence, we
adopt both GAN [15] and Wasserstein GAN (WGAN, [6]). Results are shown in Figure 1 (left).
7
Softplus
LSU
ReLU
?
?-ReLU
Softplus / LS / ReLU
Discriminator: varying link
Figure 1: Summary of our results on MNIST, on experiment A (left+center) and B (right). Left:
comparison of different values of ? for the ?-ReLU activation in the generator (ReLU = 1-ReLU, see
text). Thicker horizontal dashed lines present the ReLU average baseline: for each color, points above
the baselines represent values of ? for which ReLU is beaten on average. Center: comparison of
different activations in the generator, for the same architectures as in the left plot. Right: comparison
of different link function in the discriminator (see text, best viewed in color).
Three behaviours emerge when varying ?: either it is globally equivalent to ReLU (GAN DCGAN)
but with local variations that can be better (? = 0.7) or worse (? = 0), or it is almost consistently
better than ReLU (WGAN MLP) or worse (GAN MLP). The best results were obtained for GAN
DCGAN, and we note that the ReLU baseline was essentially beaten for values of ? yielding smaller
variance, and hence yielding smaller uncertainty in the results. The comparison between different
activation functions (Figure 1, center) reveals that (?-)ReLU performs overall the best, yet with some
variations among architectures. We note in particular that, in the same way as for the comparisons
intra ?-ReLU (Figure 1, left), ReLU performs relatively worse than the other criteria for WGAN
MLP, indicating that there may be different best fit activations for different architectures, which is
good news. Visual results on LSUN (SM, Table A5) also display the quality of results when changing
the ?-ReLU activation.
Comparison of varying link functions in the discriminator (B) ? We have compared the replacement
of the sigmoid function by a link which corresponds to the entropy which is theoretically optimal
in
?
.
boosting algorithms, Matsushita entropy [18, 28], for which ?MAT (z) = (1/2) ? (1 + z/ 1 + z 2 ).
Figure 1 (right) displays the comparison Matsushita vs "standard" (more specifically, we use sigmoid
in the case of GAN [30], and none in the case of WGAN to follow current implementations [6]). We
evaluate with both DCGAN and MLP on MNIST (same hyperparameters as for generators, ReLU
activation for all hidden layer activation of generators). Experiments tend to display that tuning the
link may indeed bring additional uplift: for GANs, Matsushita is indeed better than the sigmoid link
for both DCGAN and MLP, while it remains very competitive with the no-link (or equivalently an
identity link) of WGAN, at least for DCGAN.
7
Conclusion
It is hard to exaggerate the success of GAN approaches in modelling complex domains, and with
their success comes an increasing need for a rigorous theoretical understanding [34]. In this paper,
we complete the supervised understanding of the generalization of GANs introduced in [30], and
provide a theoretical background to understand its unsupervised part, showing in particular how deep
architectures can be powerful at tackling the generative part of the game. Experiments display that
the tools we develop may help to improve further the state of the art.
8
Acknowledgments
The authors thank the reviewers, Shun-ichi Amari, Giorgio Patrini and Frank Nielsen for numerous
comments.
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6,245 | 665 | Generalization Abilities of
Cascade Network Architectures
E. Littmann*
H. Ritter
Department of Information Science
Bielefeld University
D-4800 Bielefeld, FRG
[email protected]
Department of Information Science
Bielefeld University
D-4800 Bielefeld, FRG
[email protected] bielefeld.de
Abstract
In [5], a new incremental cascade network architecture has been
presented. This paper discusses the properties of such cascade
networks and investigates their generalization abilities under the
particular constraint of small data sets. The evaluation is done for
cascade networks consisting of local linear maps using the MackeyGlass time series prediction task as a benchmark. Our results indicate that to bring the potential of large networks to bear on the
problem of extracting information from small data sets without running the risk of overjitting, deeply cascaded network architectures
are more favorable than shallow broad architectures that contain
the same number of nodes.
1
Introduction
For many real-world applications, a major constraint for the successful learning
from examples is the limited number of examples available. Thus, methods are
required, that can learn from small data sets. This constraint makes the problem
of generalization particularly hard. If the number of adjustable parameters in a
* to whom correspondence should be sent
188
Generalization Abilities of Cascade Network Architectures
network approaches the number of training examples, the problem of overfitting occurs and generalization becomes very poor. This severely limits the size of networks
applicable to a learning task with a small data set. To achieve good generalization
also in these cases, particular attention must be paid to a proper architecture chosen
for the network. The better the architecture matches the structure of the problem
at hand, the better is the chance to achieve good results even with small data sets
and small numbers of units.
In the present paper, we address this issue for the class of so called Cascade Network
Architectures [5, 6] on the basis of an empirical approach, where we use the MackeyGlass time series prediction as a benchmark problem. In our experiments we want
to exploit the potential of large networks to bear on the problem of extracting
information from small data sets without running the risk of overfitting. Our results
indicate that it is more favorable to use deeply cascaded network architectures than
shallow broad architectures, provided the same number of nodes is used in both
cases. The width of each individual layer is essentially determined by the size of
the training data set. The cascade depth is then matched to the total number of
nodes available.
2
Cascade Architecture
So far, mainly architectures with few layers containing many units have been considered, while there has been very little research on narrow, but deeply cascaded
networks. One of the few exceptions is the work of Fahlman [1], who proposed
networks trained by the cascade-correlation algorithm. In his original approach,
training is strictly feed-forward and the nonlinearity is achieved by incrementally
adding percept ron units trained to maximize the covariance with the residual error.
2.1
Construction Algorithm
In [5] we presented a new incremental cascade network architecture based 011 error
minimization instead of covariance maximization. This leads to all architecture
that differs significantly from Fahlman's proposal and allows an inversion of the
construction process of the network. Thus, at each stage of the construction of the
network all cascaded modules provide an approximation of the target function t(e),
albeit corresponding to different states of convergence (Fig. 1).
The algorithm starts with the training of a neural module with output yeo) to
approximate a target function t(e), yielding
(1)
the superscript (0) indicating the cascade level. After an arbitrary number of training epochs, the weight vector w(O) becomes "frozen". Now we add the output y(O)
of this module as a virtual input unit and train another neural module as new output
189
190
Littmann and Ritter
Neural
Module
Output
Cascade layer2
Neural
Module
(Output)
Cascade layer 1
(Output)
Neural
Module
Xn
Input
{
X2
X,
Bias
1
Figure 1: Cascade Network Architecture
unit
y(1)
with
(2)
=
where x(1)(~)
{x(O)(~),y(O)(~)} denotes the extended input. This procedure can
be iterated arbitrarily and generates a network sttucture as shown in Fig. 1.
2.2
Cascade Modules
The details and advantages of this approach are discussed in [5, 6]. In particular,
this architecture can be applied to any arbitrary nonlinear module. It does not rely
on the availability of a procedure for error backpropagation. Therefore, it is also
applicable to (and has been extensively tested with) pure feed-forward approaches
like simple perceptrons [5] and vector quantization or "Local linear maps" ("LLM
networks") [6, 7].
2.3
Local Linear Maps
LLM networks have been introduced earlier ((Fig. 2); for details, d. [11, 12]) and
are related to the GRBF -approach [10] and the self-organizing maps [2, 3, 11].
They consist of N units r = 1, ... ,N, with an input weight vector w~in) E }RL, an
output weight vector w~out) E }RM and a MxL-matrix Ar for each unit r.
Generalization Abilities of Cascade Network Architectures
Output space
Input space
Figure 2: LLM Network Architecture
The output
y(net)
of a single LLM-network for an input feature vector x E }RL is
(3)
the "winner" node s determined by the minimality condition
(4)
This leads to the learning steps for a training sample (x(o), yeo?):
f
1
(x(o) _
W(in?)
8'
(5)
(6)
(7)
applied for T samples (x(o),y(o?),a = 1,2, ... T, and 0 < fi ? 1, i = 1,2,3
denote learning step sizes. The additional term in (6), not given in [11, 121, leads
to a better decoupling of the effects of (5) and (6,7).
3
Experiments
In order to evaluate the generalization performance of this architecture, we consider the problem of time series prediction based on the Mackey-Glass differential
equation, for which results of other networks already have been reported in the
literature.
191
192
Littmann and Ritter
1.25
]
:=
t
1
0.75
0.50
o
100
200
300
400
Time (t)
Figure 3: Mackey-Glass function
3.1
Time Series Prediction
Lapedes and Farber [4] introduced the prediction of chaotic time series as a benchmark problem. The data is based on the Mackey-Glass differential equation [8]:
x(t)
= -bx(t) + (ax(t -
r))J(l
+ xlO(t -
r)).
(8)
With the parameters a = 0.2, b = 0.1, and r = 17, this equation produces a chaotic
time series with a strange attractor of fractal dimension d ~ 2.1 (Fig. 3). The
input data is a vector x(t) = {x(t),x(t - ~),x(t - 2~),x(t - 3~)}T. The learning
task is defined to predict the value x(t + P). To facilitate comparison, we adopt
the standard choice ~ = 6 and P = 85. Results with these parameters have been
reported in [4, 9, 13].
The data was generated by integration with 30 steps per time unit. We performed
different numbers of training epochs with samples randomly chosen from training
sets consisting of 500 (5000 resp.) samples. The performance was measured on an
independent test set of 5000 samples. All results are averages over ten runs. The
error measure is the normalized root mean squ.are error (NRMSE), i.e. predicting
the average value yields an error value of 1.
4
Results and Discussion
The training of the single LLM networks was performed without extensive parameter tuning. If fine tuning for each cascade unit would be necessary, the training
would be unattractively expensive.
The first results were achieved with cascade networks consisting of LLM units after
30 training epochs per layer on a learning set of 500 samples. Figs. 4 and 5 represent
the performance of such LLM cascade networks on the independent test set for
different numbers of cascaded layers as a function of the number of nodes per layer
Generalization Abilities of Cascade Network Architectures
NRMSE
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0.4
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. . ..L...?~~-~~j
0.2
0.1 .....................................
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:
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O~------------------------------~?
10 20 30 40 50 60 70 80 90 100
# nodes per layer
Figure 4: Iso-Layer-Dependence
lO),8I"S
Nodes
90
ID
100
Figure 5: Error Landscape
("iso-layer-curves"). The graphs indicate that there is an optimal number N~!~
of nodes for which the performance of the single layer network has a best value
p!~:. Within the single layer architecture, additional nodes lead to a decrease of
performance due to overfitting. This can only be avoided if the training set is
enlarged, since N~!~ grows with the number of available training examples.
However, Figs. 4 and 5 show that adding more units in the form of an additional,
cascaded layer allows to increase performance significantly beyond p!~:. Similarly,
the optimal performance of the resulting two-layer network cannot be improved
beyond an optimal value p!;~ by arbitrarily increasing the number of nodes in the
two-layer system. However, adding a third cascaded layer again allows to make use
of more nodes to improve performance further, although this time the relative gain
is smaller than for the first cascade step. The same situation repeats for larger
numbers of cascaded layers. This suggests that the cascade architecture is very
suitable to exploit the computational capabilities of large numbers of nodes for the
task of building networks that generalize well from small data sets without running
into the problem of overfitting when many nodes are used .
A second way of comparing the benefits of shallow and broad versus narrow and
deep architectures is to compare the performance achieveable by distributing a fixed
number N of nodes over different numbers L of cascaded layers. Fig. 6 shows the
result for the same benchmark problem as in Fig. 4, each graph belonging to one of
the values N = 40,60,120,240 nodes and representing the NRMSE for distributing
the N nodes among L layers of N / L nodes each t, L ranging from 1 to 10 layers
("iso-nodes-curves" ).
1 rounding
to the nearest integral, whenever N / L is nonintegral.
193
194
Littmann and Ritter
----t-------
E3 240 E3
120 B 60 B 40 Nodes
0.5 ........................\.............. ................. ..... ... ...................... ;.. ....:
NRMSE
-1:------_
G??
G.3
0.4
0.2
0.3
0.2
D.5
,l,- :.'
:....,'..: .... '
?
'
.... ___
~......
.
~.~_~
.
0.1 .......... t.~.~.~.~~.~ ..~
????
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! ???? .,
.:
:
:
~~~:r.'=
. . ??????????1????????????:
???????????,
:
j
o~----~--------------------~
4
6
7
8
9 10
1
2
3
5
I
2
J
,?
l"'....
L = # cascaded layers
Figure 6: Iso-Nodes-Dependence
Figure 7: Nodes-Layer-Dependence
The results show that
(i) the optimal number of layers increases monotonously with - and is roughly
proportional to - the number of nodes to be used.
(ii) if for each number of nodes the optimal number of layers is used, performance increases monotonously with the number of available nodes, and
thus, as a consequence of (i), with the number of cascaded layers.
These results are not restricted to small data sets only. The application of the
cascade algorithm is also useful if larger training sets are available. Fig. 7 represents
the performance of LLM cascade networks on the test set after 300 training epochs
overall on a learning set consisting of 5000 samples. As could be expected, there is
still no sign of overfitting, even using LLM networks with 100 nodes per layer. But
regardless of the size of the single LLM unit, network performance is improved by
the cascade process at least in a zone involving a total of some 300 nodes in the
whole cascade.
5
Conclusions
Summarizing, we find that Cascade Network Architectures allow to use the benefits
of large numbers of nodes even for small training data sets, and still bypass the
problem of overfitting. To achieve this, the "width" of each layer must be matched
to the size of the training set. The "depth" of the cascade then is determined by
the total number of nodes available.
Generalization Abilities of Cascade Network Architectures
Acknowledgements
This work was supported by the German Ministry of Research and Technology
(BMFT), Grant No. ITN9104AO. Any responsibility for the contents of this publication is with the authors.
References
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195
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6,246 | 6,650 | Toward Multimodal Image-to-Image Translation
Jun-Yan Zhu
UC Berkeley
Trevor Darrell
UC Berkeley
Richard Zhang
UC Berkeley
Alexei A. Efros
UC Berkeley
Oliver Wang
Adobe Research
Deepak Pathak
UC Berkeley
Eli Shechtman
Adobe Research
Abstract
Many image-to-image translation problems are ambiguous, as a single input image
may correspond to multiple possible outputs. In this work, we aim to model
a distribution of possible outputs in a conditional generative modeling setting.
The ambiguity of the mapping is distilled in a low-dimensional latent vector,
which can be randomly sampled at test time. A generator learns to map the given
input, combined with this latent code, to the output. We explicitly encourage the
connection between output and the latent code to be invertible. This helps prevent
a many-to-one mapping from the latent code to the output during training, also
known as the problem of mode collapse, and produces more diverse results. We
explore several variants of this approach by employing different training objectives,
network architectures, and methods of injecting the latent code. Our proposed
method encourages bijective consistency between the latent encoding and output
modes. We present a systematic comparison of our method and other variants on
both perceptual realism and diversity.
1
Introduction
Deep learning techniques have recently made steep progress in conditional image generation. For
example, networks have been used to inpaint missing image regions [15, 28, 41], create sentence
conditioned generations [43], add color to grayscale images [14, 15, 21, 44], and sketch-to-photo [15,
34]. However, most techniques in this space have focused on generating a single result conditioned
on the input. In this work, our focus is to model a distribution of potential results, as many of these
problems may be multimodal or ambiguous in nature. For example, in a night-to-day translation task
(see Figure 1), an image captured at night may correspond to many possible day images with different
types of lighting, sky and clouds. There are two main goals of the conditional generation problem:
producing results which are (1) perceptually realistic and (2) diverse, while remaining faithful to the
input. This multimodal mapping from a high-dimensional input to a distribution of high-dimensional
outputs makes the conditional generative modeling task challenging. In existing approaches, this
leads to the common problem of mode collapse [10], where the generator learns to generate only
a small number of unique outputs. We systematically study a family of solutions to this problem,
which learn a low-dimensional latent code for aspects of possible outputs which are not contained in
the input image. The generator then produces an output conditioned on both the given input and this
learned latent code.
We start with the pix2pix framework [15] which has previously been shown to produce good-quality
results for a variety of image-to-image translation tasks. The pix2pix method works by training a
generator network, conditioned on the input image, with two losses (1) a regression loss to produce
a similar output to the known paired ground truth image and (2) a learned discriminator loss to
encourage realism. The authors note that trivially appending a randomly drawn latent code did not
help produce diverse results, and using random dropout at test time only helped marginally. Instead,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
?
(a) Input night image
(b) Diverse day images sampled by our model
Figure 1: Multimodal image-to-image translation using our proposed method: given an input image from one
domain (night image of a scene), we aim to model a distribution of potential outputs (corresponding day images)
in the target domain, producing both realistic and diverse results.
we propose encouraging a bijection between the output and latent space. Not only do we perform
the direct task of mapping from the input and latent code to the output, we also jointly learn an
encoder from the output back to the latent space. This discourages two different latent codes from
generating the same output (non-injective mapping). During training, the learned encoder is trained
to find a latent code vector that corresponds to the ground truth output image, while passing enough
information to the generator to resolve any ambiguities about the mode of output. For example, when
generating a day image from a night one (Figure 1), the latent vector may encode information about
the sky color, lighting effects on the ground and the density and shape of clouds. In both cases,
applying the encoder and generator, in either order, should be self-consistent, and result in either the
same latent code or the same image.
In this work, we instantiate this idea by exploring several objective functions inspired by the literature
in unconditional generative modeling:
? cVAE-GAN (Conditional Variational Autoencoder GAN): One popular approach to model multimodal output distribution is by learning the distribution of latent encoding given the output as
popularized by VAEs [18]. In the conditional setup (similar to unconditional analogue [20]), we
enforce that the latent distribution encoded by the desired output image maps back to the output via
conditional generator. The latent distribution is regularized using KL-divergence to be close to a
standard normal so as to sample random codes during inference. This variational objective is then
optimized jointly with the discriminator loss.
? cLR-GAN (Conditional Latent Regressor GAN): Another approach to enforce mode-capture in latent
encoding is to explicitly model the inverse mapping. Starting from a randomly sampled latent
encoding, the conditional generator should result into an output which when given itself as input to
the encoder should result back into the same latent code, enforcing self-consistency. This method
could be seen as a conditional formulation of the ?latent regressor" model [6, 8] and also related to
InfoGAN [2].
? BiCycleGAN: Finally, we combine both these approaches to enforce the connection between latent
encoding and output in both directions jointly and achieve improved performance. We show that
our method can produce both diverse and visually appealing results across a wide range of imageto-image translation problems, significantly more diverse than other baselines, including naively
adding noise in the pix2pix framework. In addition to the loss function, we study the performance
with respect to several encoder networks, as well as different ways of injecting the latent code into
the generator network.
We perform a systematic evaluation of these variants by using real humans to judge photo-realism
and an automated distance metric to assess output diversity.
2
Related Work
Generative modeling Parametric modeling of the natural image distribution is a challenging
problem. Classically, this problem has been tackled using autoencoders [13, 38] or Restricted
Boltzmann machines [36]. Variational autoencoders [18] provide an effective approach to model
2
Input Image
?
Ground truth output
Network output
?
?"
?
?(?)
Loss
?"
?
?(?)
?* + ?
?
Deep network
Target latent distribution
Sample from distribution
(a) Testing Usage for all models
(b) Training pix2pix+noise
?*
?* + ?
?
?
?
?
?
?"
?
?(?|?)
z
?"
?
??
?(?)
??
?
?(?)
(c) Training cVAE-GAN
?
(d) Training cLR-GAN
(e) Training BiCycleGAN
Figure 2: Overview: (a) Test time usage of all the methods. To produce a sample output, a latent code z is
first randomly sampled from a known distribution (a standard normal distribution). A generator G maps input
? (yellow). (b) pix2pix+noise [15] baseline,
image A (blue) and latent sample z to produce output sample B
with additional input B (blue) that corresponds to A. (c) cVAE-GAN (and cAE-GAN) start from ground truth
target image B and encode it into the latent space. The generator then attempts to map the input image A
along with a sampled z back into original image B. (d) cLR-GAN randomly samples a latent code from a known
? and then tries to reconstruct the latent code from it. (e) Our
distribution, uses it to map A into the output B,
hybrid BiCycleGAN method combines constraints in both directions.
stochasticity within the network by re-parametrization of a latent distribution. Other approaches
to modeling stochasticity include auto-regressive models [26, 27] which can model multimodality
via sequential conditional prediction. These approaches are trained with a pixel-wise independent
loss on samples of natural images using maximum likelihood and stochastic back-propagation. This
is a disadvantage because two images, which are close regarding a pixel-wise independent metric,
may be far apart on the manifold of natural images. Generative adversarial networks [11] overcome
this issue by learning the loss function using a discriminator network, and have recently been very
successful [1, 2, 4, 6, 8, 29, 30, 43, 46, 47]. Our method builds on the conditional version of VAE [18]
and InfoGAN [2] or latent regressor [6] model via an alternating joint optimization to learn diverse
and realistic samples. We revisit this connection in Section 3.4.
Conditional image generation Potentially, all of the methods defined above could be easily
conditioned. While conditional VAEs [37] and conditional auto-regressive models [26, 27] have
shown promise [39, 40], image-to-image conditional GANs have lead to a substantial boost in
the quality of the results. However, the quality has been attained at the expense of multimodality,
as the generator learns to largely ignore the random noise vector when conditioned on a relevant
context [15, 28, 34, 41, 48]. In fact, it has even been shown that ignoring the noise leads to more
stable training [15, 23, 28].
Explicitly-encoded multimodality One way to express multiple modes in the output is to encode
them, conditioned on some mode-related context in addition to the input image. For example, color
and shape scribbles and other interfaces were used as conditioning in iGAN [47], pix2pix [15],
Scribbler [34] and interactive colorization [45]. Another option is to explicitly force diversity amongst
a mixture of models [9]. While there has been some degree of success for generating multimodal
outputs in unconditional setups, and conditional text generation [5, 11, 20, 25, 30], conditional
image-to-image generation is still far from achieving the same results, unless explicitly encoded as
discussed above. In this work, we learn conditional generation models for modeling multiple modes
of output by enforcing tight connections in both image and latent space.
3
3
Multimodal Image-to-Image Translation
Our goal is to learn a multi-modal mapping between two image domains, for example, edges and
photographs, or day and night images, etc. Consider the input domain A ? RH?W?3 which is to be
mapped to an output domain
B ? RH?W?3
. During training, we are given a dataset of paired instances
from these domains, (A ? A, B ? B) , which is representative of a joint distribution p(A, B). It is
important to note that there could be multiple plausible paired instances B that would correspond to
an input instance A, but the training dataset usually contains only one such pair. However, given a
b
new instance A during test time, our model should be able to generate a diverse set of output B?s,
corresponding to different modes in the distribution p(B|A).
While conditional GANs have achieved success in image-to-image translation tasks [15, 48], they
b given the input image A. On the other
are primarily limited to generating deterministic output B
b from true conditional
hand, we would like to learn the mapping that could sample the output B
distribution given A, and produce results which are both diverse and realistic. To do so, we learn a
low-dimensional latent space z ? RZ , which encapsulates the ambiguous aspects of the output mode
which are not present in the input image. For example, a sketch of a shoe could map to a variety of
colors and textures, which could get compressed in this latent code. We then learn a deterministic
mapping G : (A, z) ? B to the output. To enable stochastic sampling, we desire the latent code
vector z to be drawn from some prior distribution p(z); we use a standard Gaussian distribution
N (0, I) in this work.
We first discuss a simple extension of existing methods and discuss its strengths and weakness,
motivating the development of our proposed approach in the subsequent subsections.
3.1
b
Baseline: pix2pix+noise (z ? B)
The recently proposed pix2pix model [15] has shown high quality results in image-to-image
translation setting. It uses conditional adversarial networks [11, 24] to help produce perceptually
realistic results. GANs train a generator G and discriminator D by formulating their objective as an
adversarial game. The discriminator attempts to differentiate between real images from the dataset
and fake samples produced by the generator. Randomly drawn noise z is added to attempt to induce
stochasticity. We illustrate the formulation in Figure 2(b) and describe it below.
LGAN (G, D) = EA,B?p(A,B) [log(D(A, B))] + EA?p(A),z?p(z) [log(1 ? D(A, G(A, z)))] (1)
To encourage the output of the generator to match the input as well as stabilize the GANs training,
we use an `1 loss between the output and the ground truth image.
Limage
(G) = EA,B?p(A,B),z?p(z) ||B ? G(A, z)||1
1
(2)
The final loss function is comprised of both the GANs term and the `1 term, where ? balances the
two terms.
G? = arg min max LGAN (G, D) + ?Limage
(G)
(3)
1
G
D
In this scenario, there is no incentive for the generator to make use of the noise vector which encodes
random information. It was also noted in the preliminary experiments in [15] that the generator simply
ignored the added noise and hence the noise was removed in their final experiments. This observation
is consistent with Mathieu et al. [23], Pathak et al. [28] and the mode collapse phenomenon observed
in unconditional cases [10, 33]. In this paper, we explore different ways to explicitly enforce the
generator to use the latent encoding by making it capture relevant information than being random.
3.2
b
Conditional Variational Autoencoder GAN: cVAE-GAN (B ? z ? B)
One way to force the latent code z to be ?useful" is to directly map the ground truth B to it
using an encoding function E. The generator G then uses both the latent code and the input
b The overall model can be easily understood as the
image A to synthesize the desired output B.
reconstruction of B, with latent encoding b
z concatenated with the paired A in the middle ? similar to
an autoencoder [13]. This interpretation is better shown in Figure 2(c).
4
This approach has been successfully investigated in Variational Auto-Encoders [18] in the unconditional scenario without the adversarial objective. Extending it to conditional scenario, the distribution
Q(z|B) of latent code z using the encoder E with a Gaussian assumption, Q(z|B) , E(B). To
reflect this, Equation 1 is modified to sampling z ? E(B) using the re-parameterization trick,
allowing direct back-propagation [18].
LVAE
GAN = EA,B?p(A,B) [log(D(A, B))] + EA,B?p(A,B),z?E(B) [log(1 ? D(A, G(A, z)))]
(4)
We make the corresponding change in the `1 loss term in Equation 2 as well to obtain LVAE
1 (G) =
EA,B?p(A,B),z?E(B) ||B ? G(A, z)||1 . Further, the latent distribution encoded by E(B) is encouraged to be close to random gaussian so as to sample z at inference (i.e., B is not known).
LKL (E) = EB?p(B) [DKL (E(B)|| N (0, I))],
R
where DKL (p||q) = ? p(z) log
of the VAE-GAN [20] as
p(z)
q(z) dz.
G? = arg min max
G,E
This forms our cVAE-GAN objective, a conditional version
VAE
LVAE
GAN (G, D, E) + ?L1 (G, E) + ?KL LKL (E).
D
(5)
(6)
As a baseline, we also consider the deterministic version of this approach, i.e., dropping KLdivergence and encoding z = E(B). We call it cAE-GAN and show comparison in the experiments.
However, there is no guarantee in cAE-GAN on the distribution of the latent space z, which makes
test-time sampling of z difficult.
3.3
b ?b
Conditional Latent Regressor GAN: cLR-GAN (z ? B
z)
We explore another method of enforcing the generator network to utilize the latent code embedding z,
while staying close to the actual test time distribution p(z), but from the latent code?s perspective.
We start from the latent code z, as shown in Figure 2(d), and then enforce that E(G(A, z)) map back
to the randomly drawn latent code with an `1 loss. Note that the encoder E here is producing a point
estimate for b
z, whereas the encoder in the previous section was predicting a Gaussian distribution.
Llatent
(G, E) = EA?p(A),z?p(z) ||z ? E(G(A, z))||1
1
(7)
We also include the discriminator loss LGAN (G, D) (Equation 1) on to encourage the network to
generate realistic results, and the full loss can be written as:
G? = arg min max
G,E
D
LGAN (G, D) + ?latent Llatent
(G, E)
1
(8)
The `1 loss for the ground truth image B is not used. In this case, since the noise vector is randomly
b to be the ground truth, but rather a realistic result. The above
drawn, we do not want the predicted B
b is
objective bears similarity to the ?latent regressor" model [2, 6, 8], where the generated sample B
encoded to generate a latent vector.
3.4
Our Hybrid Model: BiCycleGAN
To take the advantages of both directions, we present a hybrid model by combining cVAE-GAN and
b and z ? B
b ?b
cLR-GAN objectives, as well as combining the two GANs cycles: B ? z ? B
z
(hence the name BiCycleGAN). For cVAE-GAN, the encoding is learned from real data, but a random
latent code may not yield realistic images at test time due to two reasons. First, the KL loss may not
be well optimized. Secondly, our adversarial classifier D does not have a chance to see randomly
sampled results during training. In cLR-GAN, the latent space is easily sampled from a simple
distribution, but the generator is trained without the benefit of seeing ground truth input-output pairs.
We propose to train with constraints in both directions.
G? = arg min max
G,,E
D
VAE
LVAE
GAN (G, D, E) + ?L1 (G, E)
+LGAN (G, D) + ?latent Llatent
(G, E) + ?KL LKL (E),
1
where the hyper-parameters ?, ?latent , and ?KL control the importance of each term.
5
(9)
+
+
z
z
Figure 3: Alternatives for injecting z into generator. Latent code z is injected by spatial replication and
concatenation into the generator network. We tried two alternatives, injecting into (left) the input layer and
(right) every intermediate layer in the encoder.
In the unconditional GAN setting, it has been observed that using samples both from random z and
encoded vector E(B) help further improves the results [20]. Hence, we also report one variant which
is the full objective shown above (Equation 9), but without the reconstruction loss on the latent space
Llatent
. We call it cVAE-GAN++, as it is based on cVAE-GAN with additional loss LGAN (G, D), that
1
encourages the discriminator to also see the randomly generated samples.
4
Implementation Details
The code and additional results are publicly available on our website. Please refer to supplement for
more details about the datasets, architectures, and training procedures.
Network architecture For generator G, we use the U-Net [31], which contains an encoder-decoder
architecture, with symmetric skip connections. The architecture has been shown to produce strong
results in the unimodal image prediction setting, when there is spatial correspondence between input
and output pairs. For discriminator D, we use two PatchGAN discriminators [15] at different scales,
which aims to predict real vs. fake for 70 ? 70 and 140 ? 140 overlapping image patches respectively.
For the encoder E, we experiment with two networks: (1) E_CNN: CNN with a few convolutional and
downsampling layers and (2) E_ResNet: a classifier with several residual blocks [12].
Training details We build our model on the Least Squares GANs (LSGANs) variant [22], which
uses a least-squares objective instead of a cross entropy loss. LSGANs produce high quality results
with stable training. We also find that not conditioning the discriminator D leads to better results
(also discussed in [28]), and hence choose to do the same for all methods. We set the parameters
?image = 10, ?latent = 0.5 and ?KL = 0.01 in all our experiments. In practice, we tie the weights for
the generators in the cVAE-GAN and cLR-GAN. We observe that using two separate discriminators
yields slightly better visual results compared to discriminators with shared weights. We train our
networks from scratch using Adam [17] with a batch size of 1 and with learning rate of 0.0002. We
choose latent dimension to be 8 across all the datasets.
Injecting the latent code z to generator G. How to propagate the information encoded by latent
code z to the image generation process is critical to our applications. We explore two choices as
shown in Figure 3: (1) add_to_input: We simply extend a Z-dimensional latent code z to an
H ? W ? Z spatial tensor and concatenate it with the H ? W ? 3 input image. (2) add_to_all:
Alternatively, we add z to each intermediate layer of the network G.
5
Experiments
Datasets We test our method on several image-to-image translation problems from prior work,
including edges ? photos [42, 47], Google maps ? satellite [15], labels ? images [3], and outdoor
night ? day images [19]. These problems are all one-to-many mappings. We train all the models on
256 ? 256 images.
Methods We train the following models described in Section 3: pix2pix+noise, cAE-GAN,
cVAE-GAN, cVAE-GAN++, cLR-GAN and the hybrid model BiCycleGAN.
5.1
Qualitative Evaluation
We show qualitative comparison results on Figure 5. We observe that pix2pix+noise typically
produces a single realistic output, but does not produce any meaningful variation. cAE-GAN adds
6
Input
Ground truth
Generated samples
Figure 4: Example Results We show example results of our hybrid model BiCycleGAN. The left column show
cAE-GAN
BiCycleGAN
Ground truth
cLR-GAN
Input
cVAE-GAN++
Pix2pix
+noise
cVAE-GAN
shows the input. The second shows the ground truth output. The final four columns show randomly generated
samples. We show results of our method on night?day, edges?shoes, edges?handbags, and maps?satellites.
Figure 5: Qualitative method comparison We show an example on the label ? facades dataset
across different methods. The BiCycleGAN method produces results which are both realistic and
diverse.
7
Diversity
VGG-16
Distance
3.520?.021
0.338?.002
2.304?.012
1.350?.013
1.425?.014
a
1.374?.022
1.469?.014
a
We found that cLR-GAN resulted in severe mode collapse, resulting in 15% of the images producing the same
result. Those images were omitted from this calculation.
40
Realism (AMT Fooling Rate [%])
Method
Random real images
pix2pix+noise [15]
cAE-GAN
cVAE-GAN
cVAE-GAN++
cLR-GAN
BiCycleGAN
Realism
AMT Fooling
Rate [%]
50.0%
27.93?2.40 %
13.64?1.80 %
24.93?2.27 %
29.19?2.43 %
29.23?2.48 %
34.33?2.69 %
35
30
25
20
15
10
5
0
0.0
pix2pix+noise
cAE-GAN
cVAE-GAN
cVAE-GAN++
cLR-GAN
BiCycleGAN
0.5
1.0
1.5
2.0
Diversity (Average VGG Distance)
2.5
Figure 6: Realism vs Diversity. We measure diversity using average feature distance in the VGG-16 space
using cosine distance summed across five layers, and realism using a real vs. fake Amazon Mechanical Turk
test on the Google maps ? satellites task. The pix2pix+noise baseline produces realistic results, but little
diversity. Using only cAE-GAN method produces diverse results, but large artifacts during sampling. The hybrid
BiCycleGAN method, which combines cVAE-GAN++ and cLR-GAN, produces results which have higher realism
while maintaining diversity.
variation to the output, but typically reduces quality of results, as shown for an example on facades
on Figure 4. We typically observe more variation in the cVAE-GAN, as the latent space is encouraged
to encode information about ground truth outputs. However, the space is not densely populated,
so drawing random samples may cause artifacts in the output. The cLR-GAN typically shows less
variation in the output, and sometimes suffers from mode collapse. When combining these methods,
however, in the hybrid method BiCycleGAN, we observe results which are both diverse and realistic.
We show example results in Figure 4. Please see our website for a full set of results.
5.2
Quantitative Evaluation
We perform a quantitative analysis on the diversity, realism, and latent space distribution on our six
variants and baselines. We quantitatively test the Google maps ? satellites dataset.
Diversity We randomly draw samples from our model and compute average distance in a deep
feature space. In the context of style transfer, image super-resolution [16], and feature inversion [7],
pre-trained networks have been used as a ?perceptual loss" and explicitly optimized over. In the
context of generative modeling, they have been used as a held-out ?validation" score, for example to
assess how semantic samples from a generative model [33] or the semantic accuracy of a grayscale
colorization [44].
In Figure 5, we show the diversity-score using the cosine distance, averaged across spatial dimensions,
and summed across the five conv layers preceding the pool layers on the VGG-16 network [35],
pre-trained for Imagenet classification [32]. The maximum score is 5.0, as all the feature responses
are non-negative. For each method, we compute the average distance between 1900 pairs of randomly
b images (sampled from 100 input A images). Random pairs of ground truth real
generated output B
images in the B ? B domain produce an average variation of 3.520 using cosine distance. As we are
b which correspond to a specific input, a system which stays faithful to the input
measuring samples B
should definitely not exceed this score.
The pix2pix system [15] produces a single point estimate. Adding noise to the system
pix2pix+noise produces a diversity score of 0.338, confirming the finding in [15] that adding
noise does not produce large variation. Using an cAE-GAN model to encode ground truth image
B into latent code z does increase the variation. The cVAE-GAN, cVAE-GAN++, and BiCycleGAN
models all place explicit constraints on the latent space, and the cLR-GAN model places an implicit
constraint through sampling. These four methods all produce similar diversity scores. We note
that high diversity scores may also indicate that non-realistic images are being generated, causing
meaningless variation. Next, we investigate the visual realism of our samples.
Perceptual Realism To judge the visual realism of our results, we use human judgments, as proposed
in [44] and later used in [15, 48]. The test sequentially presents a real and generated image to a human
for 1 second each, in a random order, asks them to identify the fake, and measures the ?fooling"
8
Encoder
E_ResNet
E_ResNet
E_CNN
E_CNN
Injecting z
add_to_all add_to_input add_to_all add_to_input
label?photo
0.292 ? 0.058 0.292 ? 0.054 0.326 ? 0.066 0.339 ? 0.069
map ? satellite 0.268 ? 0.070 0.266 ? 0.068 0.287 ? 0.067 0.272 ? 0.069
Table 1: The encoding performance with respect to the different encoder architectures and methods
of injecting z. Here we report the reconstruction loss ||B ? G(A, E(B))||1
.
Input label
|z| = 2
|z| = 8
|z| = 256
Figure 7: Different label ? facades results trained with varying length of the latent code |z| ? {2, 8, 256}.
rate. Figure 5(left) shows the realism across methods. The pix2pix+noise model achieves high
realism score, but without large diversity, as discussed in the previous section. The cAE-GAN helps
produce diversity, but this comes at a large cost to the visual realism. Because the distribution of
the learned latent space is unclear, random samples may be from unpopulated regions of the space.
Adding the KL-divergence loss in the latent space, used in the model recovers the visual realism.
Furthermore, as expected, checking randomly drawn z vectors in the cVAE-GAN++ model slightly
increases realism. The cLR-GAN, which draws z vectors from the predefined distribution randomly,
produces similar realism and diversity scores. However, the cLR-GAN model resulted in large mode
collapse - approximately 15% of the outputs produced the same result, independent of the input
image. The full hybrid BiCycleGAN gets the best of both worlds, as it does not suffer from mode
collapse and also has the highest realism score by a significant margin.
Encoder architecture The pix2pix framework [15] have conducted extensive ablation studies
on discriminators and generators. Here we focus on the performance of two encoders E_CNN and
E_ResNet for our applications on maps and facades datasets, and we find that E_ResNet can better
encode the output image, regarding the image reconstruction loss ||B ? G(A, E(B))||1 on validation
datasets as shown in Table 1.
Methods of injecting latent code We evaluate two ways of injecting latent code z: add_to_input
and add_to_all (Section 4), regarding the same reconstruction loss ||B ? G(A, E(B))||1 . Table 1
shows that two methods give similar performance. This indicates that the U_Net [31] can already
propagate the information well to the output without the additional skip connections from z.
Latent code length We study the BiCycleGAN model results with respect to the varying number
of dimensions of latent codes {2, 8, 256} in Figure 7. A low-dimensional latent code may limit the
amount of diversity that can be expressed by the model. On the contrary, a high-dimensional latent
code can potentially encode more information about an output image at the cost of making sampling
quite difficult. The optimal length of z largely depends on individual datasets and applications, and
how much ambiguity there is in the output.
6
Conclusions
In conclusion, we have presented a novel method for obtaining and exploring diverse results in a
conditional image generation application. Our approach produces optimal results in terms of diversity
and realism, while being simple to train and requiring no additional supervision. We see many
interesting avenues of future work, including directly enforcing a distribution in the latent space that
encodes semantically meaningful attributes to allow for image-to-image transformations with user
controllable parameters.
Acknowledgments We thank Phillip Isola and Tinghui Zhou for helpful discussions. This work was
supported in part by Adobe Inc., DARPA, AFRL, DoD MURI award N000141110688, NSF awards
IIS-1633310, IIS-1427425, IIS-1212798, the Berkeley Artificial Intelligence Research (BAIR) Lab,
and hardware donations from NVIDIA. JYZ is supported by Facebook Graduate Fellowship, RZ by
Adobe Research Fellowship, and DP by NVIDIA Graduate Fellowship.
9
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6,247 | 6,651 | Mixture-Rank Matrix Approximation
for Collaborative Filtering
Dongsheng Li1
Chao Chen1
Wei Liu2? Tun Lu3,4
Ning Gu3,4
Stephen M. Chu1
1
IBM Research - China
2
Tencent AI Lab, China
3
School of Computer Science, Fudan University, China
4
Shanghai Key Laboratory of Data Science, Fudan University, China
{ldsli, cshchen, schu}@cn.ibm.com, [email protected], {lutun, ninggu}@fudan.edu.cn
Abstract
Low-rank matrix approximation (LRMA) methods have achieved excellent accuracy among today?s collaborative filtering (CF) methods. In existing LRMA
methods, the rank of user/item feature matrices is typically fixed, i.e., the same rank
is adopted to describe all users/items. However, our studies show that submatrices
with different ranks could coexist in the same user-item rating matrix, so that
approximations with fixed ranks cannot perfectly describe the internal structures
of the rating matrix, therefore leading to inferior recommendation accuracy. In
this paper, a mixture-rank matrix approximation (MRMA) method is proposed, in
which user-item ratings can be characterized by a mixture of LRMA models with
different ranks. Meanwhile, a learning algorithm capitalizing on iterated condition
modes is proposed to tackle the non-convex optimization problem pertaining to
MRMA. Experimental studies on MovieLens and Netflix datasets demonstrate that
MRMA can outperform six state-of-the-art LRMA-based CF methods in terms of
recommendation accuracy.
1
Introduction
Low-rank matrix approximation (LRMA) is one of the most popular methods in today?s collaborative
filtering (CF) methods due to high accuracy [11, 12, 13, 17]. Given a targeted user-item rating matrix
R ? Rm?n , the general goal of LRMA is to find two rank-k matrices U ? Rm?k and V ? Rn?k
? = U V T . After obtaining the user and item feature matrices, the recommendation
such that R ? R
score of the i-th user on the j-th item can be obtained by the dot product between their corresponding
feature vectors, i.e., Ui Vj T .
In existing LRMA methods [12, 13, 17], the rank k is considered fixed, i.e., the same rank is adopted
to describe all users and items. However, in many real-world user-item rating matrices, e.g., Movielens
and Netflix, users/items have a significantly varying number of ratings, so that submatrices with
different ranks could coexist. For instance, a submatrix containing users and items with few ratings
should be of a low rank, e.g., 10 or 20, and a submatrix containing users and items with many ratings
may be of a relatively higher rank, e.g., 50 or 100. Adopting a fixed rank for all users and items
cannot perfectly model the internal structures of the rating matrix, which will lead to imperfect
approximations as well as degraded recommendation accuracy.
In this paper, we propose a mixture-rank matrix approximation (MRMA) method, in which user-item
ratings are represented by a mixture of LRMA models with different ranks. For each user/item, a
probability distribution with a Laplacian prior is exploited to describe its relationship with different
?
This work was conducted while the author was with IBM.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
LRMA models, while a joint distribution of user-item pairs is employed to describe the relationship
between the user-item ratings and different LRMA models. To cope with the non-convex optimization
problem associated with MRMA, a learning algorithm capitalizing on iterated condition modes
(ICM) [1] is proposed, which can obtain a local maximum of the joint probability by iteratively
maximizing the probability of each variable conditioned on the rest. Finally, we evaluate the proposed
MRMA method on Movielens and Netflix datasets. The experimental results show that MRMA can
achieve better accuracy compared against state-of-the-art LRMA-based CF methods, further boosting
the performance for recommender systems leveraging matrix approximation.
2
Related Work
Low-rank matrix approximation methods have been leveraged by much recent work to achieve
accurate collaborative filtering, e.g., PMF [17], BPMF [16], APG [19], GSMF [20], SMA [13],
etc. These methods train one user feature matrix and one item feature matrix first and use these
feature matrices for all users and items without any adaptation. However, all these methods adopt
fixed rank values for the targeted user-item rating matrices. Therefore, as analyzed in this paper,
submatrices with different ranks could coexist in the rating matrices and only adopting a fixed rank
cannot achieve optimal matrix approximation. Besides stand-alone matrix approximation methods,
ensemble methods, e.g., DFC [15], LLORMA [12], WEMAREC [5], etc., and mixture models, e.g.,
MPMA [4], etc., have been proposed to improve the recommendation accuracy and/or scalability
by weighing different base models across different users/items. However, the above methods do not
consider using different ranks to derive different base models. In addition, it is desirable to borrow
the idea of mixture-rank matrix approximation (MRMA) to generate more accurate base models in
the above methods and further enhance their accuracy.
In many matrix approximation-based collaborative filtering methods, auxiliary information, e.g.,
implicit feedback [9], social information [14], contextual information [10], etc., is introduced to
improve the recommendation quality of pure matrix approximation methods. The idea of MRMA is
orthogonal to these methods, and can thus be employed by these methods to further improve their
recommendation accuracy. In general low-rank matrix approximation methods, it is non-trivial to
directly determine the maximum rank of a targeted matrix [2, 3]. Cand?s et al. [3] proved that a
non-convex rank minimization problem can be equivalently transformed into a convex nuclear norm
minimization problem. Based on this finding, we can easily determine the range of ranks for MRMA
and choose different K values (the maximum rank in MRMA) for different datasets.
3
Problem Formulation
In this paper, upper case letters such as R, U, V denote matrices, and k denotes the rank for matrix
approximation. For a targeted user-item rating matrix R ? Rm?n , m denotes the number of users, n
? denotes
denotes the number of items, and Ri,j denotes the rating of the i-th user on the j-th item. R
the low-rank approximation of R. The general goal of k-rank matrix approximation is to determine
? = U V T . The rank k is
user and item feature matrices, i.e., U ? Rm?k , V ? Rn?k , such that R ? R
considered low, because k min{m, n} can achieve good performance in many CF applications.
In real-world rating matrices, e.g., Movielens and Netflix, users/items have a varying number of
ratings, so that a lower rank which best describes users/items with less ratings will easily underfit the
users/items with more ratings, and similarly a higher rank will easily overfit the users/items with less
ratings. A case study is conducted on the Movielens (1M) dataset (with 1M ratings from 6,000 users
on 4,000 movies), which confirms that internal submatrices with different ranks indeed coexist in the
rating matrix. Here, we run the probabilistic matrix factorization (PMF) method [17] using k = 5
and k = 50, and then compare the root mean square errors (RMSEs) for the users/items with less
than 10 ratings and more than 50 ratings.
As shown in Table 1, when the rank is 5, the users/items with less than 10 ratings achieve lower
RMSEs than the cases when the rank is 50. This indicates that the PMF model overfits the users/items
with less than 10 ratings when k = 50. Similarly, we can conclude that the PMF model underfits
the users/items with more than 50 ratings when k = 5. Moreover, PMF with k = 50 achieves lower
RMSE (higher accuracy) than PMF with k = 5, but the improvement comes with sacrificed accuracy
for the users and items with a small number of ratings, e.g., less than 10. This study shows that PMF
2
Table 1: The root mean square errors (RMSEs) of PMF [17] for users/items with different numbers
of ratings when rank k = 5 and k = 50.
rank = 5
rank = 50
#user ratings < 10
#user ratings > 50
#item ratings < 10
#item ratings > 50
0.9058
0.8416
0.9338
0.8520
0.9165
0.8352
0.9598
0.8418
All
0.8614
0.8583
with fixed rank values cannot perfectly model the internal mixture-rank structure of the rating matrix.
To this end, it is desirable to model users and items with different ranks.
4
Mixture-Rank Matrix Approximation (MRMA)
2
?U
??
b?
b?
??
Uik
?ik
?jk
Vjk
?V2
k = {1, ..., K}
Ri,j
?2
j = {1, ..., n}
i = {1, ..., m}
Figure 1: The graphical model for the proposed mixture-rank matrix approximation (MRMA) method.
Following the idea of PMF, we exploit a probabilistic model with Gaussian noise to model the
ratings [17]. As shown in Figure 1, the conditional distribution over the observed ratings for the
mixture-rank model can be defined as follows:
m Y
n X
K
Y
T
Pr(R|U, V, ?, ?, ? 2 ) =
[
?ik ?jk N (Ri,j |Uik Vjk , ? 2 )]1i,j ,
(1)
i=1 j=1 k=1
where N (x|?, ? 2 ) denotes the probability density function of a Gaussian distribution with mean ?
and variance ? 2 . K is the maximum rank among all internal structures of the user-item rating matrix.
?k and ? k are the weight vectors of the rank-k matrix approximation model for all users and items,
respectively. Thus, ?ik and ?jk denote the weights of the rank-k model for the i-th user and j-th item,
respectively. U k and V k are the feature matrices of the rank-k matrix approximation model for all
users and items, respectively. Likewise, Uik and Vjk denote the feature vectors of the rank-k model
for the i-th user and j-th item, respectively. 1i,j is an indication function, which will be 1 if Ri,j is
observed and 0 otherwise.
By placing a zero mean isotropic Gaussian prior [6, 17] on the user and item feature vectors, we have
m
n
Y
Y
2
2
Pr(U k |?U
)=
N (Uik |0, ?U
I), Pr(V k |?V2 ) =
N (Vjk |0, ?V2 I).
(2)
i=1
j=1
For ?k and ? k , we choose a Laplacian prior here, because the models with most suitable ranks
for each user/item should be with large weights, i.e., ?k and ? k should be sparse. By placing the
Laplacian prior on the user and item weight vectors, we have
m
n
Y
Y
Pr(?k |?? , b? ) =
L(?ik |?? , b? ), Pr(? k |?? , b? ) =
L(?jk |?? , b? ),
(3)
i=1
j=1
3
where ?? and b? are the location parameter and scale parameter of the Laplacian distribution for ?,
respectively, and accordingly ?? and b? are the location parameter and scale parameter for ?.
The log of the posterior distribution over the user and item features and weights can be given as
follows:
2
l = ln Pr(U, V, ?, ?|R, ? 2 , ?U
, ?V2 , ?? , b? , ?? , b? )
2
? ln Pr(R|U, V, ?, ?, ? 2 ) Pr(U |?U
) Pr(V |?V2 ) Pr(?|?? , b? ) Pr(?|?? , b? )
=
m X
n
X
1i,j ln
i=1 j=1
K
X
?ik ?jk N (Ri,j |Uik (Vjk )T , ? 2 I)
k=1
K m
K
n
1 XX k 2
1
1 XX k 2 1
2
? 2
(Ui ) ? 2
(Vi ) ? Km ln ?U
? Kn ln ?V2
2?U
2?
2
2
V
i=1
j=1
k=1
?
(4)
k=1
K
K
K m
K
n
1 XX k
1X
1 XX k
1X
m ln b2? ?
n ln b2? + C,
|?i ? ?? | ?
|?j ? ?? | ?
b?
b
2
2
?
i=1
j=1
k=1
k=1
k=1
k=1
where C is a constant that does not depend on any parameters. Since the above optimization problem
is difficult to solve directly, we obtain its lower bound using Jensen?s inequality and then optimize
the following lower bound:
l0
=
?
K
m n
m X
n
1X
X
1 XX
k k
k
k T 2
?
?
(R
?
U
(V
)
)
?
1
1i,j ln ?2
i,j
i,j
i j
i
j
2? 2 i=1 j=1
2 i=1 j=1
k=1
?
?
1
2
2?U
K X
m
X
(Uik )2 ?
k=1 i=1
K
n
1 XX k 2 1
1
2
(Vi ) ? Km ln ?U
? Kn ln ?V2
2
2?V
2
2
j=1
(5)
k=1
K m
K
n
1 XX k
1 XX k
1
1
|?i ? ?? | ?
|?j ? ?? | ? Km ln b2? ? Kn ln b2? + C.
b?
b?
2
2
i=1
j=1
k=1
k=1
If we keep the hyperparameters of the prior distributions fixed, then maximizing l0 is similar to the
popular least square error minimization with `2 regularization on U and V and `1 regularization on ?
and ?. However, keeping the hyperparameters fixed may easily lead to overfitting because MRMA
models have many parameters.
5
Learning MRMA Models
The optimization problem defined in Equation 5 is very likely to overfit if we cannot precisely
estimate the hyperparameters, which automatically control the generalization capacity of the MRMA
model. For instance, ?U and ?V will control the regularization of U and V . Therefore, it is more
desirable to estimate the parameters and hyperparameters simultaneously during model training. One
possible way is to estimate each variable by its maximum a priori (MAP) value while conditioned
on the rest variables and then iterate until convergence, which is also known as iterated conditional
modes (ICM) [1].
The ICM procedure for maximizing Equation 5 is presented as follows.
Initialization: Choose initial values for all variables and parameters.
ICM Step: The values of U , V , ? and ? can be updated by solving the following minimization
problems when conditioned on other variables or hyperparameters.
?k ? {1, ..., K}, ?i ? {1, ..., m} :
Uik ? arg min
0
U
?ik ? arg min
0
?
n
K
K
1 X
X
1 X k 2
k k
k
k T 2
1
?
?
(R
?
U
(V
)
)
+
(Ui ) ,
i,j
i,j
i j
i
j
2
2? 2 j=1
2?U
k=1
k=1
1
2? 2
n
X
j=1
1i,j
K
X
k=1
1
?ik ?jk (Ri,j ? Uik (Vjk )T )2 +
b?
4
K
X
k=1
|?ik ? ?? | .
?k ? {1, ..., K}, ?j ? {1, ..., n} :
Vjk ? arg min
0
V
?jk ? arg min
0
?
m
K
K
X
1 X
1 X k 2
k k
k
k T 2
1
(Vj ) ,
?
+
?
(R
?
U
(V
)
)
i,j
i,j
i j
i
j
2? 2 i=1
2?V2
k=1
k=1
1
2? 2
m
X
i=1
1i,j
K
X
k=1
1
?ik ?jk (Ri,j ? Uik (Vjk )T )2 +
b?
K
X
|?jk ? ?? | .
k=1
The hyperparameters can be learned as their maximum likelihood estimates by setting their partial
derivatives on l0 to 0.
?2 ?
m X
n
X
1i,j
i=1 j=1
2
?U
?
?V2 ?
K
X
m X
n
X
?ik ?jk (Ri,j ? Uik (Vjk )T )2 /
1i,j ,
i=1 j=1
k=1
K X
m
X
K X
m
X
k=1 i=1
k=1 i=1
K X
n
X
K X
n
X
k=1 j=1
k=1 j=1
(Uik )2 /Km, ?? ?
(Vjk )2 /Kn, ?? ?
?ik /Km, b? =
?jk /Kn, b? =
K X
m
X
|?ik ? ?? |/Km,
k=1 i=1
K X
n
X
|?jk ? ?? |/Kn.
k=1 j=1
Repeat: until convergence or the maximum number of iterations reached.
Note that ICM is sensitive to initial values. Our empirical studies show that setting the initial values
of U k and V k by solving the classic PMF method ?
can achieve good performance. Regarding ?
and ?, one of the proper initial values should be 1/ K (K denotes the number of sub-models in
the mixture model). To improve generalization performance and enable online learning [7], we can
update U, V, ?, ? using stochastic gradient descent. Meanwhile, the `1 norms in learning ? and ?
can be approximated by the smoothed `1 method [18]. To deal with massive datasets, we can use the
alternating least squares (ALS) method to learn the parameters of the proposed MRMA model, which
is amenable to parallelization.
6
Experiments
This section presents the experimental results of the proposed MRMA method on three well-known
datasets: 1) MovieLens 1M dataset (?1 million ratings from 6,040 users on 3,706 movies); 2)
MovieLens 10M dataset (?10 million ratings from 69,878 users on 10,677 movies); 3) Netflix Prize
dataset (?100 million ratings from 480,189 users on 17,770 movies). For all accuracy comparisons,
we randomly split each dataset into a training set and a test set by the ratio of 9:1. All results are
reported by averaging over 5 different splits. The root mean square error (RMSE) is adopted to
measure the
rating prediction accuracy of different algorithms, which can be computed as follows:
qP
P
?
? 2 P P 1i,j (1i,j indicates that entry (i, j) appears in the
D(R) =
i
j 1i,j (Ri,j ? Ri,j ) /
i
j
test set). The normalized discounted cumulative gain (NDCG) is adopted to measure the item
ranking accuracy of different algorithms, which can be computed as follows: N DCG@N =
PN
DCG@N/IDCG@N (DCG@N = i=1 (2reli ? 1)/ log2 (i + 1), and IDCG is the DCG value
with perfect ranking).
In ICM-based learning, we adopt = 0.00001 as the convergence threshold and T = 300 as the
maximum number of iterations. Considering efficiency, we only choose a subset of ranks, e.g.,
{10, 20, 30, ..., 300} rather than {1, 2, 3, ..., 300}, in MRMA. The parameters of all the compared
algorithms are adopted from their original papers because all of them are evaluated on the same
datasets.
We compare the recommendation accuracy of MRMA with six matrix approximation-based collaborative filtering algorithms as follows: 1) BPMF [16], which extends the PMF method from a Baysian
view and estimates model parameters using a Markov chain Monte Carlo scheme; 2) GSMF [20],
which learns user/item features with group sparsity regularization in matrix approximation; 3) LLORMA [12], which ensembles the approximations from different submatrices using kernel smoothing;
4) WEMAREC [5], which ensembles different biased matrix approximation models to achieve higher
5
PMF
MRMA
RMSE
computation time
0.86
1000
RMSE
RMSE
0.86
0.84
0.82
0.82
0.80
0.84
k=
10
k=
20
k=
50
k=
0.80
k
k
k
k
M
10 =15 =20 =25 =30 RM
0
0
0
0
0
A
se
t1
se
t2
se
t3
se
t4
se
t
computation time (s)
0.88
0
5
rank setting
Model
Figure 2: Root mean square error comparison Figure 3: The accuracy and efficiency tradeoff of
between MRMA and PMF with different ranks. MRMA.
accuracy; 5) MPMA [4], which combines local and global matrix approximations using a mixture
model; 6) SMA [13], which yields a stable matrix approximation that can achieve good generalization
performance.
6.1
Mixture-Rank Matrix Approximation vs. Fixed-Rank Matrix Approximation
Given a fixed rank k, the corresponding rank-k model in MRMA is identical to probabilistic matrix
factorization (PMF) [17]. In this experiment, we compare the recommendation accuracy of MRMA
with ranks in {10, 20, 50, 100, 150, 200, 250, 300} against those of PMF with fixed ranks on the
MovieLens 1M dataset. For PMF, we choose 0.01 as the learning rate, 0.01 as the user feature
regularization coefficient, and 0.001 as the item feature regularization coefficient, respectively. The
convergence condition is the same as MRMA.
As shown in Figure 2, when the rank increases from 10 to 300, PMF can achieve RMSEs between
0.86 and 0.88. However, the RMSE of MRMA is about 0.84 when mixing all these ranks from 10 to
300. Meanwhile, the accuracy of PMF is not stable when k ? 100. For instance, PMF with k = 10
achieves better accuracy than k = 20 but worse accuracy than k = 50. This is because fixed rank
matrix approximation cannot be perfect for all users and items, so that many users and items either
underfit or overfit at a fixed rank less than 100. Yet when k > 100, only overfitting occurs and PMF
achieves consistently better accuracy when k increases, which is because regularization terms can
help improve generalization capacity. Nevertheless, PMF with all ranks achieves lower accuracy
than MRMA, because individual users/items can give the sub-models with the optimal ranks higher
weights in MRMA and thus alleviate underfitting or overfitting.
6.2
Sensitivity of Rank in MRMA
In MRMA, the set of ranks decide the performance of the final model. However, it is neither efficient
nor necessary to choose all the ranks in [1, 2, ..., K]. For instance, a rank-k approximation will
be very similar to rank-(k ? 1) and rank-(k + 1) approximations, i.e., they may have overlapping
structures. Therefore, a subset of ranks will be sufficient. Figure 3 shows 5 different settings of
rank combinations, in which set 1 = {10, 20, 30, ..., 300}, set 2 = {20, 40, ..., 300}, set 3 =
{30, 60, ..., 300}, set 4 = {50, 100, ..., 300}, and set 5 = {100, 200, 300}. As shown in this figure,
RMSE decreases when more ranks are adopted in MRMA, which is intuitive because more ranks will
help users/items better choose the most appropriate components. However, the computation time
also increases when more ranks are adopted in MRMA. If a tradeoff between accuracy and efficiency
is required, then set 2 or set 3 will be desirable because they achieve slightly worse accuracies but
significantly less computation overheads.
MRMA only contains three sub-models with different ranks in set 5 = {100, 200, 300}, but it still
significantly outperforms PMF with ranks ranging from 10 to 300 in recommendation accuracy (as
shown in Figure 2). This further confirms that MRMA can indeed discover the internal mixture-rank
structure of the user-item rating matrix and thus achieve better recommendation accuracy due to
better approximation.
6
Table 2: RMSE comparison between MRMA and six state-of-the-art matrix approximation-based
collaborative filtering algorithms on MovieLens (10M) and Netflix datasets. Note that MRMA
statistically significantly outperforms the other algorithms with 95% confidence level.
MovieLens (10M)
Netflix
0.8197 ? 0.0004
0.8012 ? 0.0011
0.7855 ? 0.0002
0.7775 ? 0.0007
0.7712 ? 0.0002
0.7682 ? 0.0003
0.7634 ? 0.0009
0.8421 ? 0.0002
0.8420 ? 0.0006
0.8275 ? 0.0004
0.8143 ? 0.0001
0.8139 ? 0.0003
0.8036 ? 0.0004
0.7973 ? 0.0002
BPMF [16]
GSMF [20]
LLORMA [12]
WEMAREC [5]
MPMA [4]
SMA [13]
MRMA
Table 3: NDCG comparison between MRMA and six state-of-the-art matrix approximation-based
collaborative filtering algorithms on Movielens (1M) and Movielens (10M) datasets. Note that
MRMA statistically significantly outperforms the other algorithms with 95% confidence level.
Metric
N=1
N=5
N=10
N=20
Movielens 1M
BPMF
GSMF
LLORMA
WEMAREC
MPMA
SMA
MRMA
0.6870 ? 0.0024
0.6909 ? 0.0048
0.7025 ? 0.0027
0.7048 ? 0.0015
0.7020 ? 0.0005
0.7042 ? 0.0033
0.7153 ? 0.0027
0.6981 ? 0.0029
0.7031 ? 0.0023
0.7101 ? 0.0005
0.7089 ? 0.0016
0.7114 ? 0.0018
0.7109 ? 0.0011
0.7182 ? 0.0005
0.7525 ? 0.0009
0.7555 ? 0.0017
0.7626 ? 0.0023
0.7617 ? 0.0041
0.7606 ? 0.0006
0.7607 ? 0.0008
0.7672 ? 0.0013
0.8754 ? 0.0008
0.8769 ? 0.0011
0.8811 ? 0.0010
0.8796 ? 0.0005
0.8805 ? 0.0007
0.8801 ? 0.0004
0.8837 ? 0.0004
Movielens 10M
Data | Method
NDCG@N
BPMF
GSMF
LLORMA
WEMAREC
MPMA
SMA
MRMA
0.6563 ? 0.0005
0.6708 ? 0.0012
0.6829 ? 0.0014
0.7013 ? 0.0003
0.6908 ? 0.0006
0.7002 ? 0.0006
0.7048 ? 0.0006
0.6845 ? 0.0003
0.6995 ? 0.0008
0.7066 ? 0.0005
0.7176 ? 0.0006
0.7133 ? 0.0002
0.7134 ? 0.0004
0.7219 ? 0.0001
0.7467 ? 0.0007
0.7566 ? 0.0017
0.7632 ? 0.0004
0.7703 ? 0.0002
0.7680 ? 0.0001
0.7679 ? 0.0003
0.7743 ? 0.0001
0.8691 ? 0.0002
0.8748 ? 0.0004
0.8782 ? 0.0012
0.8824 ? 0.0006
0.8808 ? 0.0004
0.8809 ? 0.0002
0.8846 ? 0.0001
6.3
6.3.1
Accuracy Comparison
Rating Prediction Comparison
Table 2 compares the rating prediction accuracy between MRMA and six matrix approximationbased collaborative filtering algorithms on MovieLens (10M) and Netflix datasets. Note that among
the compared algorithms, BPMF, GSMF, MPMA and SMA are stand-alone algorithms, while
LLORMA and WEMAREC are ensemble algorithms. In this experiment, we adopt the set of ranks
as {10, 20, 50, 100, 150, 200, 250, 300} due to efficiency reason, which means that the accuracy of
MRMA should not be optimal. However, as shown in Table 2, MRMA statistically significantly
outperforms all the other algorithms with 95% confidence level. The reason is that MRMA can
choose different rank values for different users/items, which can achieve not only globally better
approximation but also better approximation in terms of individual users or items. This further
confirms that mixture-rank structure indeed exists in user-item rating matrices in recommender
systems. Thus, it is desirable to adopt mixture-rank matrix approximations rather than fixed-rank
matrix approximations for recommendation tasks.
6.3.2
Item Ranking Comparison
Table 3 compares the NDCGs of MRMA with the other six state-of-the-art matrix approximationbased collaborative filtering algorithms on Movielens (1M) and Movielens (10M) datasets. Note that
for each dataset, we keep 20 ratings in the test set for each user and remove users with less than 5
7
ratings in the training set. As shown in the results, MRMA can also achieve higher item ranking
accuracy than the other compared algorithms thanks to the capability of better capturing the internal
mixture-rank structures of the user-item rating matrices. This experiment demonstrates that MRMA
can not only provide accurate rating prediction but also achieve accurate item ranking for each user.
6.4
Interpretation of MRMA
Table 4: Top 10 movies with largest ? values for sub-models with rank k = 20 and k = 200 in MRMA.
Here, #ratings stands for the average number of ratings in the training set for the corresponding
movies.
rank=20
rank=200
movie name
?
Smashing Time
Gate of Heavenly Peace
Man of the Century
Mamma Roma
Dry Cleaning
Dear Jesse
Skipped Parts
The Hour of the Pig
Inheritors
Dangerous Game
0.6114
0.6101
0.6079
0.6071
0.6071
0.6063
0.6057
0.6055
0.6042
0.6034
#ratings
movie name
?
#ratings
2.4
American Beauty
Groundhog Day
Fargo
Face/Off
2001: A Space Odyssey
Shakespeare in Love
Saving Private Ryan
The Fugitive
Braveheart
Fight Club
0.9219
0.9146
0.8779
0.8693
0.8608
0.8553
0.8480
0.8404
0.8247
0.8153
1781.4
To better understand how users/items weigh different sub-models in the mixture model of MRMA,
we present the top 10 movies which have largest ? values for sub-models with rank=20 and rank=200,
show their ? values, and compare their average numbers of ratings in the training set in Table 4.
Intuitively, the movies with more ratings (e.g., over 1000 ratings) should weigh higher towards more
complex models, and the movies with less ratings (e.g., under 10 ratings) should weigh higher towards
simpler models in MRMA.
As shown in Table 4, the top 10 movies with largest ? values for the sub-model with rank 20 have
only 2.4 ratings on average in the training set. On the contrary, the top 10 movies with largest ? values
for the sub-model with rank 200 have 1781.4 ratings on average in the training set, and meanwhile
these movies are very popular and most of them are Oscar winners. This confirms our previous claim
that MRMA can indeed weigh more complex models (e.g., rank=200) higher for movies with more
ratings to prevent underfitting, and weigh less complex models (e.g., rank=20) higher for the movies
with less ratings to prevent overfitting. A similar phenomenon has also been observed from users
with different ? values, and we omit the results due to space limit.
7
Conclusion and Future Work
This paper proposes a mixture-rank matrix approximation (MRMA) method, which describes useritem ratings using a mixture of low-rank matrix approximation models with different ranks to achieve
better approximation and thus better recommendation accuracy. An ICM-based learning algorithm is
proposed to handle the non-convex optimization problem pertaining to MRMA. The experimental
results on MovieLens and Netflix datasets demonstrate that MRMA can achieve better accuracy than
six state-of-the-art matrix approximation-based collaborative filtering methods, further pushing the
frontier of recommender systems. One of the possible extensions of this work is to incorporate other
inference methods into learning the MRMA model, e.g., variational inference [8], because ICM may
be trapped in local maxima and therefore cannot achieve global maxima without properly chosen
initial values.
Acknowledgement
This work was supported in part by the National Natural Science Foundation of China under Grant
No. 61332008 and NSAF under Grant No. U1630115.
8
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9
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6,248 | 6,652 | Non-monotone Continuous DR-submodular
Maximization: Structure and Algorithms
An Bian
ETH Zurich
[email protected]
Kfir Y. Levy
ETH Zurich
[email protected]
Andreas Krause
ETH Zurich
[email protected]
Joachim M. Buhmann
ETH Zurich
[email protected]
Abstract
DR-submodular continuous functions are important objectives with wide real-world
applications spanning MAP inference in determinantal point processes (DPPs),
and mean-field inference for probabilistic submodular models, amongst others.
DR-submodularity captures a subclass of non-convex functions that enables both
exact minimization and approximate maximization in polynomial time.
In this work we study the problem of maximizing non-monotone DR-submodular
continuous functions under general down-closed convex constraints. We start
by investigating geometric properties that underlie such objectives, e.g., a strong
relation between (approximately) stationary points and global optimum is proved.
These properties are then used to devise two optimization algorithms with provable
guarantees. Concretely, we first devise a ?two-phase? algorithm with 1/4 approximation guarantee. This algorithm allows the use of existing methods for finding
(approximately) stationary points as a subroutine, thus, harnessing recent progress
in non-convex optimization. Then we present a non-monotone F RANK -W OLFE
variant with 1/e approximation guarantee and sublinear convergence rate. Finally,
we extend our approach to a broader class of generalized DR-submodular continuous functions, which captures a wider spectrum of applications. Our theoretical
findings are validated on synthetic and real-world problem instances.
1
Introduction
Submodularity is classically most well known for set function optimization, where it enables efficient
minimization [23] and approximate maximization [31; 25] in polynomial time. Submodularity has
recently been studied on the integer lattice [34; 33] and on continuous domains [3; 4; 36; 21], with
significant theoretical results and practical applications. For set functions, it is well known that
submodularity is equivalent to the diminishing returns (DR) property. However, this does not hold
for integer-lattice functions or continuous functions, where the DR property defines a subclass of
submodular functions, called DR-submodular functions.
In continuous domains, applying convex optimization techniques enables efficient minimization of
submodular continuous functions [3; 36] (despite the non-convex nature of such objectives). In [4]
it is further shown that continuous submodularity enables constant-factor approximation schemes
for constrained monotone DR-submodular maximization and ?box? constrained non-monotone
submodular maximization problems.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Many real-world non-convex problems, such as maximizing the softmax extension of DPPs, require
maximizing a non-monotone DR-submodular function over a general down-closed convex constraint.
Yet, current theory [3; 4; 36] does not apply to this general problem setting, which motivates us to
develop guaranteed and efficient algorithms for such problems.
Exploring the structure that underlies DR-submodularity is crucial to deriving guaranteed algorithms.
Combined with a notion of non-stationarity for constrained optimization problems and a new notion
of ?strong DR-submodularity?, we find a rich structure in the problem of continuous DR-submodular
maximization. This in turn gives rise to two approximation algorithms with provable guarantees.
Specifically, we make the following contributions:
- We bound the difference between objective values of stationary points and the global optimum.
Our analysis shows that the bound is even tighter if the objective is strongly DR-submodular
(see Definition 3).
- Based on the geometric properties, we present two algorithms: (i) A two-phase
p F RANK W OLFE-style algorithm with 1/4 approximation guarantee converges with a 1/ k rate; (ii) a
non-monotone F RANK -W OLFE variant exhibits a 1/e approximation guarantee and converges
sublinearly. Even though the worst-case guarantee of the first one is worse than the second, it
yields several practical advantages, which we discuss in Section 4.2.
- We investigate a generalized class of submodular functions on ?conic? lattices. This allows
us to model a larger class of non-trivial applications. These include logistic regression with
a non-convex separable regularizer, non-negative PCA, etc. To optimize them, we provide a
reduction that enables to invoke algorithms for continuous submodular optimization problems.
- We experimentally demonstrate the applicability of our methods on both synthetic and realworld problem instances.
1.1
Problem Statement
Notation. We use boldface letters, e.g., x to represent a vector, boldface capital letters, e.g., A
to denote a matrix. xi is the ith entry of x, Aij is the (ij)th entry of A. We use ei to denote the
standard ith basis vector. f (?) is used to denote a continuous function, and F (?) to represent a set
function. [n] := {1, ..., n} for an integer n 1. k ? k means the Euclidean norm by default. Given
two vectors x, y, x ? y means xi ? yi , 8i. x _ y and x ^ y denote coordinate-wise maximum and
coordinate-wise minimum, respectively.
The general setup of constrained non-monotone DR-submodular (see Definition 1 below) maximization is,
max f (x),
(P)
x2P
Qn
where f : X ! R is continuous DR-submodular, X = i=1 Xi , each Xi is an interval [3; 4]. Wlog1 ,
? The set P ? [0, u]
? is assumed to be a
we assume that the lower bound u of X is 0, i.e., X = [0, u].
down-closed convex set, where down-closedness means: x 2 P and 0 ? y ? x implies that y 2 P.
? We use x? to denote the
The diameter of P is D := maxx,y2P kx yk, and it holds that D ? kuk.
global maximum of (P). One can assume f is non-negative over X , since otherwise one just needs to
? (and box-constrained submodular
find a lower bound for the minimum function value of f over [0, u]
minimization can be solved to arbitrary precision in polynomial time [3]). Over continuous domains,
a DR-submodular function [4] is a submodular function with the diminishing returns (DR) property,
Definition 1 (DR-submodular & DR property). A function f : X 7! R is DR-submodular (has the
DR property) if 8a ? b 2 X , 8i 2 [n], 8k 2 R+ s.t. (kei + a) and (kei + b) are still in X , it holds,
f (kei + a)
f (a)
f (kei + b)
f (b).
(1)
If f is differentiable, one can show that Definition 1 is equivalent to rf being an antitone mapping
from Rn to Rn . Furthermore, if f is twice-differentiable, the DR property is equivalent to all of
the entries of its Hessian being non-positive, i.e., r2ij f (x) ? 0, 8x 2 X , i, j 2 [n]. A function
f : X 7! R is DR-supermodular iff f is DR-submodular. We also assume that f has Lipschitz
gradients,
1
Since otherwise one can work on a new function g(x) := f (x + u) that has 0 as the lower bound of its
domain, and all properties of the function are still preserved.
2
Definition 2. A function f has L-Lipschitz gradients if for all x, y 2 X it holds that,
krf (x)
rf (y)k ? Lkx
yk.
(2)
A brief summary of related work appears in Section 6.
2
Motivating Real-world Examples
Many continuous objectives in practice turn out to be DR-submodular. Here we list several of them.
More can be found in Appendix B.
Softmax extension. Determinantal point processes (DPPs) are probabilistic models of repulsion,
that have been used to model diversity in machine learning [26]. The constrained MAP (maximum a
posteriori) inference problem of a DPP is an NP-hard combinatorial problem in general. Currently,
the methods with the best approximation guarantees are based on either maximizing the multilinear
extension [6] or the softmax extension [20], both of which are DR-submodular functions (details in
Appendix F.1). The multilinear extension is given as an expectation over the original set function
values, thus evaluating the objective of this extension requires expensive sampling. In constast, the
softmax extension has a closed form expression, which is much more appealing from a computational
perspective. Let L be the positive semidefinite kernel matrix of a DPP, its softmax extension is:
I) + I) , x 2 [0, 1]n ,
f (x) = log det (diag(x)(L
(3)
where I is the identity matrix, diag(x) is the diagonal matrix with diagonal elements set as x.
The problem of MAP inference in DPPs corresponds to the problem maxx2P f (x), where P is a
down-closed convex constraint, e.g., a matroid polytope or a matching polytope.
Mean-field inference for log-submodular models. Log-submodular models [9] are a class of
probabilistic models over subsets of a ground set V = [n], where the log-densities
P are submodular
set functions F (S): p(S) = Z1 exp(F (S)). The partition function Z =
S?V exp(F (S)) is
typically hard to evaluate.
One
can
use
mean-field
inference
to
approximate
p(S)
by some factorized
Q
Q
distribution qx (S) := i2S xi j 2S
xj ), x 2 [0, 1]n , by minimizing the distance measured
/ (1
P
x (S)
w.r.t. the Kullback-Leibler divergence between qx and p, i.e., S?V qx (S) log qp(S)
. It is,
XY Y
Xn
KL(x) =
xi
(1 xj )F (S) +
[xi log xi + (1 xi ) log(1 xi )] + log Z.
S?V i2S
i=1
j 2S
/
KL(x) is DR-supermodular w.r.t. x (details in Appendix F.1). Minimizing the Kullback-Leibler
divergence KL(x) amounts to maximizing a DR-submodular function.
2.1
Motivating Example Captured by Generalized Submodularity on Conic Lattices
Submodular continuous functions can already model many scenarios. Yet, there are several interesting cases which are in general not (DR-)Submodular, but can still be captured by a generalized
notion. This generalization enables to develop polynomial algorithms with guarantees by using ideas
from continuous submodular optimization. We present one representative objective here (more in
Appendix B). In Appendix A we show the technical details on how they are covered by a class of
submodular continuous functions over conic lattices.
Consider the logistic regression model with a non-convex separable regularizer. This flexibility may
result in better statistical performance (e.g., in recovering discontinuities, [2]) compared to classical
models with convex regularizers. Let z 1 , ..., z m in Rn be m training points with corresponding
binary labels y 2 {?1}m . Assume that the following mild assumption is satisfied: For any fixed
dimension i, all the data points have the same sign, i.e., sign(zij ) is the same for all j 2 [m] (which
can be achieved by easily scaling if not). The task is to solve the following non-convex optimization
problem,
Xm
minn f (x) := m 1
fj (x) + r(x),
(4)
j=1
x2R
where fj (x) = log(1 + exp( yj x> z j )) is the logistic loss; > 0 is the regularization parameter,
and r(x) is some non-convex separable regularizer. Such separable regularizers are popular in
3
statistics, and two notable choices are r(x) =
Pn
x2i
i=1 1+ x2i ,
sign(zij ), i 2
and r(x) =
Pn
i=1 min{
Pm
1
:= m
j=1
x2i , 1} (see
[2]). Let us define a vector ? 2 {?1}n as ?i =
[n] and l(x)
fj (x). One
can show that l(x) is not DR-submodular or DR-supermodular. Yet, in Appendix A we will show
that l(x) is K? -DR-supermodular, where the latter generalizes DR-supermodularity. Usually, one
? Then the problem is an instance of
can assume the optimal solution x? lies in some box [u, u].
constrained non-monotone K? -DR-submodular maximization.
3
Underlying Properties of Constrained DR-submodular Maximization
In this section we present several properties arising in DR-submodular function maximization. First
we show properties related to concavity of the objective along certain directions, then we establish the
relation between locally stationary points and the global optimum (thus called ?local-global relation?).
These properties will be used to derive guarantees for the algorithms in Section 4. All omitted proofs
are in Appendix D.
3.1
Properties Along Non-negative/Non-positive Directions
A DR-submodular function f is concave along any non-negative/non-positive direction [4]. Notice
that DR-submodularity is a stronger condition than concavity along directions v 2 ?Rn+ : for instance,
a concave function is concave along any direction, but it may not be a DR-submodular function.
For a DR-submodular function with L-Lipschitz gradients, one can get the following quadratic lower
bound using standard techniques by combing the concavity and Lipschitz gradients in (2).
Quadratic lower bound. If f is DR-submodular with a L-Lipschitz gradient, then for all x 2 X
and v 2 ?Rn+ , it holds,
L
kvk2 .
(5)
2
It will be used in Section 4.2 for analyzing the non-monotone F RANK -W OLFE variant (Algorithm 2).
f (x + v)
f (x) + hrf (x), vi
Strong DR-submodularity. DR-submodular objectives may be strongly concave along directions
v 2 ?Rn+ , e.g., for DR-submodular quadratic functions. We will show that such additional structure
may be exploited to obtain stronger guarantees for the local-global relation.
Definition 3 (Strongly DR-submodular). A function f is ?-strongly DR-submodular (? 0) if for
all x 2 X and v 2 ?Rn+ , it holds that,
f (x + v) ? f (x) + hrf (x), vi
3.2
?
kvk2 .
2
(6)
Relation Between Approximately Stationary Points and Global Optimum
First of all, we present the following Lemma, which will motivate us to consider a non-stationarity
measure for general constrained optimization problems.
Lemma 1. If f is ?-strongly DR-submodular, then for any two points x, y in X , it holds:
?
(y x)> rf (x) f (x _ y) + f (x ^ y) 2f (x) + kx yk2 .
(7)
2
Lemma 1 implies that if x is stationary (i.e., rf (x) = 0), then 2f (x) f (x _ y) + f (x ^ y) +
?
yk2 , which gives an implicit relation between x and y. While in practice finding an exact
2 kx
stationary point is not easy, usually non-convex solvers will arrive at an approximately stationary
point, thus requiring a proper measure of non-stationarity for the constrained optimization problem.
Non-stationarity measure. Looking at the LHS of (7), it naturally suggests to use maxy2P (y
x)> rf (x) as the non-stationarity measure, which happens to coincide with the measure proposed
by recent work of [27], and it can be calculated for free for Frank-Wolfe-style algorithms (e.g.,
Algorithm 3). In order to adapt it to the local-global relation, we give a slightly more general
definition here: For any constraint set Q ? X , the non-stationarity of a point x 2 Q is,
gQ (x) := maxhv
v2Q
x, rf (x)i
4
(non-stationarity).
(8)
It always holds that gQ (x) 0, and x is a stationary point in Q iff gQ (x) = 0, so (8) is a natural
generalization of the non-stationarity measure krf (x)k for unconstrained optimization. As the next
statement shows, gQ (x) plays an important role in characterizing the local-global relation.
Proposition 1 (Local-Global Relation). Let x be a point in P with non-stationarity gP (x), and
? x}. Let z be a point in Q with non-stationarity gQ (z). It holds that,
Q := {y 2 P | y ? u
max{f (x), f (z)}
where z ? := x _ x?
1
[f (x? )
4
gP (x)
gQ (z)] +
?
kx
8
x? k2 + kz
z ? k2 ,
(9)
x.
Proof sketch of Proposition 1: The proof uses Lemma 1, the non-stationarity in (8) and a key
observation in the following Claim. The detailed proof is in Appendix D.2.
Claim 1. It holds that f (x _ x? ) + f (x ^ x? ) + f (z _ z ? ) + f (z ^ z ? )
f (x? ).
Note that [7; 20] propose a similar relation for the special cases of multilinear/softmax extensions by
mainly proving the same conclusion as in Claim 1. Their relation does not incorporate the properties
of non-stationarity or strong DR-submodularity. They both use the proof idea of constructing a
complicated auxiliary set function tailored to specific DR-submodular functions. We present a
different proof method by directly utilizing the DR property on carefully constructed auxiliary points
(e.g., (x + z) _ x? in the proof of Claim 1).
4
Algorithms for Constrained DR-submodular Maximization
Based on the properties, we present two algorithms for solving (P). The first is based on the localglobal relation, and the second is a F RANK -W OLFE variant adapted for the non-monotone setting.
All the omitted proofs are deferred to Appendix E.
4.1
An Algorithm Based on the Local-Global Relation
Algorithm 1: TWO - PHASE F RANK -W OLFE for non-monotone DR-submodular maximization
Input: maxx2P f (x), stopping tolerance ?1 , ?2 , #iterations K1 , K2
1 x
N ON - CONVEX F RANK -W OLFE(f, P, K1 , ?1 , x(0) ) ;
// x(0) 2 P
n
? x};
2 Q
P \ {y 2 R+ | y ? u
3 z
N ON - CONVEX F RANK -W OLFE(f, Q, K2 , ?2 , z (0) ) ;
// z (0) 2 Q
Output: arg max{f (x), f (z)} ;
We summarize the TWO - PHASE algorithm in Algorithm 1. It is generalized from the ?two-phase?
method in [7; 20]. It invokes some non-convex solver (we use the NON - CONVEX F RANK -W OLFE
by [27]; pseudocode is included in Algorithm 3 of Appendix C) to find approximately stationary
points in P and Q, respectively, then returns the solution with the larger function value. Though
we use N ON - CONVEX F RANK -W OLFE as the subroutine here, it is worth noting that any algorithm
that is guaranteed to find an approximately stationary point can be plugged into Algorithm 1 as
the subroutine. We give an improved approximation bound by considering more properties of
DR-submodular functions. Borrowing the results from [27] for the NON - CONVEX F RANK -W OLFE
subroutine, we get the following,
Theorem 1. The output of Algorithm 1 satisfies,
?
max{f (x), f (z)}
kx x? k2 + kz z ? k2
8
?
?
1
max{2h1 , Cf (P)}
?
p
+
f (x ) min
, ?1
4
K1 + 1
(10)
min
?
max{2h2 , Cf (Q)}
p
, ?2
K2 + 1
,
where h1 := maxx2P f (x) f (x(0) ), h2 := maxz2Q f (z) f (z (0) ) are the initial suboptimalities,
Cf (P) := supx,v2P, 2[0,1],y=x+ (v x) 22 (f (y) f (x) (y x)> rf (x)) is the curvature of f
w.r.t. P, and z ? = x _ x? x.
5
p
Theorem 1 indicates that Algorithm 1 has a 1/4 approximation guarantee and 1/ k rate. However,
it has good empirical performance as demonstrated by the experiments in Section 5. Informally, this
can be partially explained by the term ?8 kx x? k2 + kz z ? k2 in (10): if x is away from x? ,
this term will augment the bound; if x is close to x? , by the smoothness of f , it should be close to
optimal.
4.2
The Non-monotone F RANK -W OLFE Variant
Algorithm 2: Non-monotone F RANK -W OLFE variant for DR-submodular maximization
Input: maxx2P f (x), prespecified step size 2 (0, 1]
(0)
1 x
0, t(0)
0, k
0;
// k : iteration index, t(k) : cumulative step size
(k)
2 while t
< 1 do
3
v (k)
arg maxv2P,v?u? x(k) hv, rf (x(k) )i;
// shrunken LMO
(k)
4
use uniform step size k = ; set k
min{ k , 1 t };
5
x(k+1)
x(k) + k v (k) , t(k+1)
t(k) + k , k
k + 1;
Output: x(K) ;
// assuming there are K iterations in total
Algorithm 2 summarizes the non-monotone F RANK -W OLFE variant, which is inspired by the unified
continuous greedy algorithm in [13] for maximizing the multilinear extension of a submodular set
function. It initializes the solution x(0) to be 0, and maintains t(k) as the cumulative step size. At
iteration k, it maximizes the linearization of f over a ?shrunken? constraint set: {v|v 2 P, v ?
? x(k) }, which is different from the classical LMO of Frank-Wolfe-style algorithms (hence we
u
refer to it as the ?shrunken LMO?). Then it employs an update step in the direction v (k) chosen by
the LMO with a uniform step size k = . The cumulative step size t(k) is used to ensure that the
overall step sizes sum to one, thus the output solution x(K) is a convex combination of the LMO
outputs, hence also lies in P.
The shrunken LMO (Step 3) is the key difference compared to the monotone F RANK -W OLFE variant
? x(k) is added to prevent too aggressive growth of the solution,
in [4]. The extra constraint v ? u
since in the non-monotone setting such aggressive growth may hurt the overall performance. The
next theorem states the guarantees of Algorithm 2.
Theorem 2. Consider Algorithm 2 with uniform step size . For k = 1, ..., K it holds that,
f (x(k) )
t(k) e
t(k)
LD2
k
2
f (x? )
2
O(
2
)f (x? ).
(11)
By observing that t(K) = 1 and applying Theorem 2, we get the following Corollary:
Corollary 1. The output of Algorithm 2 satisfies f (x(K) )
e
1
f (x? )
LD 2
2K
O
1
K2
f (x? ).
Corollary 1 shows that Algorithm 2 enjoys a sublinear convergence rate towards some point x(K)
inside P, with a 1/e approximation guarantee.
Proof sketch of Theorem 2: The proof is by induction. To prepare the building blocks, we first of
all show that the growth of x(k) is indeed bounded,
Lemma 2. Assume x(0) = 0. For k = 0, ..., K
(k)
1, it holds xi
?u
?i [1
)t
(1
(k)
/
], 8i 2 [n].
Then the following Lemma provides a lower bound, which gets the global optimum involved,
? let 0 = mini2[n] u??ii . Then for all
Lemma 3 (Generalized from Lemma 7 in [8]). Given ? 2 (0, u],
1
?
?
x 2 [0, ?], it holds f (x _ x ) (1
0 )f (x ).
Then the key ingredient for induction is the relation between f (x(k+1) ) and f (x(k) ) indicated by:
Claim 2. For k = 0, ..., K
1 it holds f (x(k+1) )
(1
)f (x(k) )+ (1
)t
(k)
/
f (x? )
LD 2
2
which is derived by a combination of the quadratic lower bound in (5), Lemma 2 and Lemma 3.
6
2
,
Remarks on the two algorithms. Notice that though the TWO - PHASE algorithm has a worse
guarantee than the non-monotone F RANK -W OLFE variant, it is still of interest: i) It allows flexibility
in using a wide range of existing solvers for finding an (approximately) stationary point. ii) The
guarantees that we present rely on a worst-case analysis. The empirical performance of the TWO PHASE algorithm is often comparable or better than that of the F RANK -W OLFE variant. This suggests
to explore more properties in concrete problems that may favor the TWO - PHASE algorithm, which we
leave for future work.
5
Experimental Results
We test the performance of the analyzed algorithms, while considering the following baselines: 1)
QUADPROG IP [39], which is a global solver for non-convex quadratic programming; 2) Projected
1
gradient ascent (P ROJ G RAD) with diminishing step sizes ( k+1
, k starts from 0). We run all the
algorithms for 100 iterations. For the subroutine (Algorithm 3) of TWO - PHASE F RANK -W OLFE,
we set ?1 = ?2 = 10 6 , K1 = K2 = 100. All the synthetic results are the average of 20 repeated
experiments. All experiments were implemented using MATLAB. Source code can be found at:
https://github.com/bianan.
5.1
DR-submodular Quadratic Programming
As a state-of-the-art global solver, QUADPROG IP2 [39] can find the global optimum (possibly in
exponential time), which were used to calculate the approximation ratios. Our problem instances are
synthetic DR-submodular quadratic objectives with down-closed polytope constraints, i.e., f (x) =
m?n
1 >
>
n
m
?
2 x Hx + h x + c and P = {x 2 R+ | Ax ? b, x ? u, A 2 R++ , b 2 R+ }. Both objective
and constraints were randomly generated, in the following two manners:
1) Uniform distribution. H 2 Rn?n is a symmetric matrix with uniformly distributed entries in
[ 1, 0]; A 2 Rm?n has uniformly distributed entries in [?, ? + 1], where ? = 0.01 is a small positive
constant in order to make entries of A strictly positive.
0.95
0.9
1
1
0.95
0.95
Approx. ratio
Approx. ratio
Approx. ratio
1
0.9
0.9
0.85
0.85
8
10
12
14
Dimensionality
(a) m = b0.5nc
16
8
10
12
Dimensionality
(b) m = n
14
16
8
10
12
14
16
Dimensionality
(c) m = b1.5nc
Figure 1: Results on DR-submodular quadratic instances with uniform distribution.
2) Exponential distribution. The entries of H and A were sampled from exponential distributions
Exp( ) (For a random variable y 0, its probability density function is e y , and for y < 0, its
density is 0). Specifically, each entry of H was sampled from Exp(1), then the matrix H was
made to be symmetric. Each entry of A was sampled from Exp(0.25) + ?, where ? = 0.01 is a small
positive constant.
? to be the tightest upper bound of P by
In both the above two cases, we set b = 1m , and u
i
? To
u
?j = mini2[m] Abij
, 8j 2 [n]. In order to make f non-monotone, we set h = 0.2 ? H> u.
1 >
>
make sure that f is non-negative, we first of all solve the problem minx2P 2 x Hx + h x using
? then set c = f (x)
? + 0.1 ? |f (x)|.
?
QUADPROG IP, let the solution to be x,
The approximation ratios w.r.t. dimensionalities (n) are plotted in Figures 1 and 2, for the two
manners of data generation. We set the number of constraints to be m = b0.5nc, m = n and
m = b1.5nc in Figures 1a to 1c (and Figures 2a to 2c), respectively.
2
We used the open source code provided by [39], and the IBM CPLEX optimization studio https://www.
ibm.com/jm-en/marketplace/ibm-ilog-cplex as the subroutine.
7
1
0.95
0.9
0.85
0.8
Approx. ratio
1
0.95
Approx. ratio
Approx. ratio
1
0.95
0.9
0.85
0.9
0.85
0.8
0.8
0.75
0.75
8
10
12
14
16
0.75
8
Dimensionality
8
10
12
14
(a) m = b0.5nc
10
12
14
16
Dimensionality
16
Dimensionality
(b) m = n
(c) m = b1.5nc
Figure 2: Results on quadratic instances with exponential distribution.
One can see that TWO - PHASE F RANK -W OLFE usually performs the best, P ROJ G RAD follows, and
non-monotone F RANK -W OLFE variant is the last. The good performance of TWO - PHASE F RANK W OLFE can be partially explained by the strong DR-submodularity of quadratic functions according
to Theorem 1. Performance of the two analyzed algorithms is consistent with the theoretical bounds:
the approximation ratios of F RANK -W OLFE variant are always much higher than 1/e.
5.2
Maximizing Softmax Extensions
0.2
0.2
0.15
0.15
0.1
0.05
0
-0.05
8
10
12
14
Dimensionality
(a) m = b0.5nc
16
Function value
0.2
0.15
Function value
Function value
With some derivation, one can see the derivative of the softmax extension in (3) is: ri f (x) =
tr((diag(x)(L I) + I) 1 (L I)i ), 8i 2 [n], where (L I)i denotes the matrix obtained by
zeroing all entries except the ith row of (L I). Let C := (diag(x)(L I) + I) 1 , D := (L I),
one can see that ri f (x) = D>
i? C?i , which gives an efficient way to calculate the gradient rf (x).
0.1
0.05
0
-0.05
0.1
0.05
0
-0.05
8
10
12
Dimensionality
(b) m = n
14
16
8
10
12
14
16
Dimensionality
(c) m = b1.5nc
Figure 3: Results on softmax instances with polytope constraints generated from uniform distribution.
Results on synthetic data. We generate the softmax objectives (see (3)) in the following way: first
generate the n eigenvalues d 2 Rn+ , each randomly distributed in [0, 1.5], and set D = diag(d).
After generating a random unitary matrix U, we set L = UDU> . One can verify that L is positive
semidefinite and has eigenvalues as the entries of d.
We generate the down-closed polytope constraints in the same form and same way as that for DRsubmodular quadratic functions, except for setting b = 2 ? 1m . Function values returned by different
solvers w.r.t. n are shown in Figure 3, for which the random polytope constraints were generated
with uniform distribution (results for which the random polytope constraints were generated with
exponential distribution are deferred to Appendix G). The number of constraints was set to be
m = b0.5nc, m = n and m = b1.5nc in Figures 3a to 3c, respectively. One can observe that
TWO - PHASE F RANK -W OLFE still has the best performance, the non-monotone F RANK -W OLFE
variant follows, and P ROJ G RAD has the worst performance.
Real-world results on matched summarization. The task of ?matched summarization? is to select
a set of document pairs out of a corpus of documents, such that the two documents within a pair
are similar, and the overall set of pairs is as diverse as possible. The motivation for this task is very
practical: it could be, for example, to compare the opinions of various politicians on a range of
representative topics.
In our experiments, we used a similar setting to the one in [20]. We experimented on the 2012
US Republican debates data, which consists of 8 candidates: Bachman, Gingrich, Huntsman, Paul,
Perry, Romney and Santorum. Each task involves one pair of candidates, so in total there are
8
6
Function value
Function value
28 = 7 ? 8/2 tasks. Figure 4a plots the averaged function values returned by the three solvers over 28
tasks, w.r.t. different values of a hyperparameter reflecting the matching quality (details see [20]).
Figure 4b traces the objectives
w.r.t.
iterations for a spe2.5
cific
candidate
pair (Bachman,
10
2
Romney). For TWO - PHASE
8
F RANK -W OLFE, the objectives
1.5
6
of
the selected phase were plot1
4
ted.
One can see that TWO 0.5
2
PHASE F RANK -W OLFE also
0
0.2
0.4
0.6
0.8
1
0
20
40
60
80
100
achieves the best performance,
Match quality controller
Iteration
while
the performance of non(a) Average on 28 tasks
(b) Objectives w.r.t. iterations
monotone F RANK -W OLFE variant and P ROJ G RAD is comparaFigure 4: Results on 2012 US Republican debates data.
ble.
Related Work
Submodular optimization and, more broadly, non-convex optimization are extensively studied in the
literature, which renders it very difficult comprehensively surveying all previous work. Here we only
briefly summarize some of the most related papers.
Submodular optimization over integer-lattice and continuous domains. Many results from submodular set function optimization have been generalized to the integer-lattice case [34; 33; 12; 24].
Of particular interest is the reduction [12] from an integer-lattice DR-submodular maximization problem to a submodular set function maximization problem. Submodular optimization over continuous
domains has attracted considerable attention recently [3; 4; 36]. Two classes of functions that are
covered by continuous submodularity are the Lovasz extensions [28] and multilinear extensions
[6] of submodular set functions. Particularly, multilinear extensions of submodular set functions
are also continuous DR-submodular [3], but with the special property that they are coordinate-wise
linear. Combined with the rounding technique of contention resolution [7], maximizing multilinear
extensions [38; 19; 13; 8; 11] has become the state-of-the-art method for submodular set function
maximization. Some of the techniques in maximizing multilinear extensions [13; 7; 8] have inspired
this work. However, we are the first to explore the rich properties and devise algorithms for the
general constrained DR-submodular maximization problem over continuous domains.
Non-convex optimization. Non-convex optimization receives a surge of attention in the past years.
One active research topic is to reach a stationary point for unconstrained optimization [35; 32; 1] or
constrained optimization [18; 27]. However, without proper assumptions, a stationary point may not
lead to any global approximation guarantee. The local-global relation (in Proposition 1) provides a
strong relation between (approximately) stationary points and global optimum, thus making it flexible
to incorporate progress in this area.
7
Conclusion
We have studied the problem of constrained non-monotone DR-submodular continuous maximization.
We explored the structural properties of such problems, and established a local-global relation. Based
on these properties, we presented a TWO - PHASE algorithm with a 1/4 approximation guarantee, and a
non-monotone F RANK -W OLFE variant with a 1/e approximation guarantee. We further generalized
submodular continuous function over conic lattices, which enabled us to model a larger class of
applications. Lastly, our theoretical findings were verified by synthetic and real-world experiments.
Acknowledgement. This research was partially supported by ERC StG 307036, by the Max Planck
ETH Center for Learning Systems, and by the ETH Z?rich Postdoctoral Fellowship program.
9
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11
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6,249 | 6,653 | Learning with Average Top-k Loss
Yanbo Fan3,4,1 , Siwei Lyu1?, Yiming Ying2 , Bao-Gang Hu3,4
1
Department of Computer Science, University at Albany, SUNY
2
Department of Mathematics and Statistics, University at Albany, SUNY
3
National Laboratory of Pattern Recognition, CASIA
4
University of Chinese Academy of Sciences (UCAS)
{yanbo.fan,hubg}@nlpr.ia.ac.cn, [email protected], [email protected]
Abstract
In this work, we introduce the average top-k (ATk ) loss as a new aggregate loss
for supervised learning, which is the average over the k largest individual losses
over a training dataset. We show that the ATk loss is a natural generalization of the
two widely used aggregate losses, namely the average loss and the maximum loss,
but can combine their advantages and mitigate their drawbacks to better adapt to
different data distributions. Furthermore, it remains a convex function over all
individual losses, which can lead to convex optimization problems that can be
solved effectively with conventional gradient-based methods. We provide an intuitive interpretation of the ATk loss based on its equivalent effect on the continuous
individual loss functions, suggesting that it can reduce the penalty on correctly
classified data. We further give a learning theory analysis of MATk learning on
the classification calibration of the ATk loss and the error bounds of ATk -SVM.
We demonstrate the applicability of minimum average top-k learning for binary
classification and regression using synthetic and real datasets.
1
Introduction
Supervised learning concerns the inference of a function f : X 7? Y that predicts a target y ? Y
from data/features x ? X using a set of labeled training examples {(xi , yi )}ni=1 . This is typically
achieved by seeking a function f that minimizes an aggregate loss formed from individual losses
evaluated over all training samples.
To be more specific, the individual loss for a sample (x, y) is given by `(f (x), y), in which ` is
a nonnegative bivariate function that evaluates the quality of the prediction made by function f .
For example, for binary classification (i.e., yi ? {?1}), commonly used forms for individual loss
include the 0-1 loss, Iyf (x)?0 , which is 1 when y and f (x) have different sign and 0 otherwise, the
hinge loss, max(0, 1 ? yf (x)), and the logistic loss, log2 (1 + exp(?yf (x))), all of which can be
further simplified as the so-called margin loss, i.e., `(y, f (x)) = `(yf (x)). For regression, squared
difference (y ? f (x))2 and absolute difference |y ? f (x)| are two most popular forms for individual
loss, which can be simplified as `(y, f (x)) = `(|y ? f (x)|). Usually the individual loss is chosen
to be a convex function of its input, but recent works also propose various types of non-convex
individual losses (e.g., [10, 15, 27, 28]).
The supervised learning problem is then formulated as minf {L(Lz (f )) + ?(f )}, where L(Lz (f ))
is the aggregate loss accumulates all individual losses over training samples, i.e., Lz (f ) =
{`i (f )}ni=1 , with `i (f ) being the shorthand notation for `(f (xi ), yi ), and ?(f ) is the regularizer
on f . However, in contrast to the plethora of the types of individual losses, there are only a few
choices when we consider the aggregate loss:
?
Corresponding author.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Classification Boundary
3
Misclassification Rate
0.1
3
Misclassification Rate
0.03
0.025
0.08
2
Classification Boundary
2
0.02
0.06
1
0.015
1
0.04
0.01
0
0
0.02
0
-1
-1
0
1
2
3
1
10
Classification Boundary
3
100 200
k
Misclassification Rate
0.025
3
0.02
2
-1
-1
0.005
0
0
1
2
1
3
Classification Boundary
10
k
100 200
Misclassification Rate
0.025
0.02
2
0.015
0.015
1
1
0.01
0.01
0
0
0.005
0
-1
-1
0
1
2
3
1
10
k
100 200
-1
-1
0.005
0
0
1
2
3
1
10
k
100 200
Figure 1: Comparison of different aggregate losses on 2D synthetic datasets with n = 200 samples for binary
classification on a balanced but multi-modal dataset and with outliers (top) and an imbalanced dataset with
outliers (bottom) with logistic loss (left) and hinge loss (right). Outliers in data are shown as an enlarged ?
and the optimal Bayes classifications are shown as shaded areas. The figures in the second and fourth columns
show the misclassification rate of ATk vs. k for each case.
Pn
? the average loss: Lavg (Lz (f )) = n1 i=1 `i (f ), i.e., the mean of all individual losses;
? the maximum loss: Lmax (Lz (f )) = max1?k?n `i (f ), i.e., the largest individual loss;
? the top-k loss [20]: Ltop-k (Lz (f )) = `[k] (f )2 for 1 ? k ? n, i.e., the k-th largest (top-k)
individual loss.
The average loss is unarguably the most widely used aggregate loss, as it is a unbiased approximation
to the expected risk and leads to the empirical risk minimization in learning theory [1, 7, 22, 25, 26].
Further, minimizing the average loss affords simple and efficient stochastic gradient descent algorithms [3, 21]. On the other hand, the work in [20] shows that constructing learning objective based
on the maximum loss may lead to improved performance for data with separate typical and rare subpopulations. The top-k loss [20] generalizes the maximum loss, as Lmax (Lz (f )) = Ltop-1 (Lz (f )),
and can alleviate the sensitivity to outliers of the latter. However, unlike the average loss or the
maximum loss, the top-k loss in general does not lead to a convex learning objective, as it is not
convex of all the individual losses Lz (f ).
In this work, we propose a new type of aggregate loss that we term as the average top-k (ATk ) loss,
which is the average of the largest k individual losses, that is defined as:
Pk
Lavt-k (Lz (f )) = k1 i=1 `[i] (f ).
(1)
We refer to learning objectives based on minimizing the ATk loss as MATk learning.
The ATk loss generalizes the average loss (k = n) and the maximum loss (k = 1), yet it is less
susceptible to their corresponding drawbacks, i.e., it is less sensitive to outliers than the maximum
loss and can adapt to imbalanced and/or multi-modal data distributions better than the average loss.
This is illustrated with two toy examples of synthesized 2D data for binary classification in Fig.1
(see supplementary materials for a complete illustration). As these plots show, the linear classifier
obtained with the maximum loss is not optimal due to the existence of outliers while the linear classifier corresponding to the average loss has to accommodate the requirement to minimize individual
losses across all training data, and sacrifices smaller sub-clusters of data (e.g., the rare population of
+ class in the top row and the smaller dataset of ? class in the bottom row). In contrast, using ATk
loss with k = 10 can better protect such smaller sub-clusters and leads to linear classifiers closer
to the optimal Bayesian linear classifier. This is also corroborated by the plots of corresponding
misclassification rate of ATk vs. k value in Fig.1, which show that minimum misclassification rates
occur at k value other than 1 (maximum loss) or n (average loss).
The ATk loss is a tight upper-bound of the top-k loss, as Lavt-k (Lz (f )) ? Ltop-k (Lz (f )) with
equality holds when k = 1 or `i (f ) = constant, and it is a convex function of the individual
losses (see Section 2). Indeed, we can express `[k] (f ) as the difference of two convex functions
kLavt-k (Lz (f ))?(k ?1)Lavt-(k?1) (Lz (f )), which shows that in general Ltop-k (Lz (f )) is not convex
with regards to the individual losses.
2
We define the top-k element of a set S = {s1 , ? ? ? , sn } as s[k] , such that s[1] ? s[2] ? ? ? ? ? s[n] .
2
In sequel, we will provide a detailed analysis of the ATk loss and MATk learning. First, we establish
a reformulation of the ATk loss as the minimum of the average of the individual losses over all training examples transformed by a hinge function. This reformulation leads to a simple and effective
stochastic gradient-based algorithm for MATk learning, and interprets the effect of the ATk loss as
shifting down and truncating at zero the individual loss to reduce the undesirable penalty on correctly classified data. When combined with the hinge function as individual loss, the ATk aggregate loss
leads to a new variant of SVM algorithm that we term as ATk SVM, which generalizes the C-SVM
and the ?-SVM algorithms [19]. We further study learning theory of MATk learning, focusing on
the classification calibration of the ATk loss function and error bounds of the ATk SVM algorithm.
This provides a theoretical lower-bound for k for reliable classification performance. We demonstrate the applicability of minimum average top-k learning for binary classification and regression
using synthetic and real datasets.
The main contributions of this work can be summarized as follows.
? We introduce the ATk loss for supervised learning, which can balance the pros and cons
of the average and maximum losses, and allows the learning algorithm to better adapt to
imbalanced and multi-modal data distributions.
? We provide algorithm and interpretation of the ATk loss, suggesting that most existing
learning algorithms can take advantage of it without significant increase in computation.
? We further study the theoretical aspects of ATk loss on classification calibration and error
bounds of minimum average top-k learning for ATk -SVM.
? We perform extensive experiments to validate the effectiveness of the MATk learning.
2
Formulation and Interpretation
The original ATk loss, though intuitive, is not convenient to work with because of the sorting procedure involved. This also obscures its connection with the statistical view of supervised learning as
minimizing the expectation of individual loss with regards to the underlying data distribution. Yet,
it affords an equivalent form, which is based on the following result.
Pk
Lemma 1 (Lemma 1, [16]).
of (x1 , ? ? ? , xn ). Furthermore, for
i=1 x[i] is a convex function
Pk
Pn
xi ? 0 and i = 1, ? ? ? , n, we have i=1 x[i] = min??0 k? + i=1 [xi ? ?]+ , where [a]+ =
max{0, a} is the hinge function.
For completeness, we include a proof of Lemma 1 in supplementary materials. Using Lemma 1, we
can reformulate the ATk loss (1) as
( n
)
k
1X
k
1X
`[i] (f ) ? min
[`i (f ) ? ?]+ + ? .
(2)
Lavt-k (Lz (f )) =
??0
k i=1
n i=1
n
In other words, the ATk loss is equivalent to minimum of the average of individual losses that are
shifted and truncated by the hinge function controlled by ?. This sheds more lights on the ATk loss,
which is particularly easy to illustrate in the context of binary classification using the margin losses,
`(f (x), y) = `(yf (x)).
In binary classification, the ?gold standard? of individual loss is the 0-1 loss Iyf (x)?0 , which exerts
a constant penalty 1 to examples that are misclassified by f and no penalty to correctly classified
examples. However, the 0-1 loss is difficult to work as it is neither continuous nor convex. In
practice, it is usually replaced by a surrogate convex loss. Such convex surrogates afford efficient
algorithms, but as continuous and convex upper-bounds of the 0-1 loss, they typically also penalize
correctly classified examples, i.e., for y and x that satisfy yf (x) > 0, `(yf (x)) > 0, whereas
Iyf (x)?0 = 0 (Fig.2). This implies that when the average of individual losses across all training
examples is minimized, correctly classified examples by f that are ?too close? to the classification
boundary may be sacrificed to accommodate reducing the average loss, as is shown in Fig.1.
In contrast, after the individual loss is combined with the hinge function, i.e., [`(yf (x)) ? ?]+ with
? > 0, it has the effect of ?shifting down? the original individual loss function and truncating it at
zero, see Fig.2. The transformation of the individual loss reduces penalties of all examples, and in
particular benefits correctly classified data. In particular, if such examples are ?far enough? from the
decision boundary, like in the 0-1 loss, their penalty becomes zero. This alleviates the likelihood of
misclassification on those rare sub-populations of data that are close to the decision boundary.
3
Loss
3
Algorithm: The reformulation of the ATk loss in Eq.(2) also facilitates development of optimization algorithms for the
2.5
minimum ATk learning. As practical supervised learning prob2
lems usually use a parametric form of f , as f (x; w), where w
is the parameter, the corresponding minimum ATk objective
1.5
becomes
1
( n
)
1X
k
0.5
min
[`(f (xi ; w), yi ) ? ?]+ + ? + ?(w) ,
w,??0
n i=1
n
0
-1.5
-1
-0.5
0
0.5
1
1.5
(3)
It is not hard to see that if `(f (x; w), y) is convex with respect Figure 2: The ATk loss interpreted at
to w, the objective function of in Eq.(3) is a convex function the individual loss level. Shaded area
for w and ? jointly. This leads to an immediate stochastic (pro- corresponds to data/target with correct
jected) gradient descent [3, 21] for solving (3). For instance, classification.
1
kwk2 , where C > 0 is a regularization facwith ?(w) = 2C
tor, at the t-th iteration, the corresponding MATk objective can be minimized by first randomly
sampling (xit , yit ) from the training set and then updating the parameters as
(t)
w(t+1) ? w(t) ? ?t ?w `(f (xit ; w(t) ), yit ) ? I[`(f (xit ;w(t) ),yit )>?(t) ] + wC
i
h
(4)
?(t+1) ? ?(t) ? ?t nk ? I[`(f (xit ;w(t) ,yit )>?(t) ]
+
where ?w `(f (x; w), y) denotes the sub-gradient with respect to w, and ?t ?
1
?
t
is the step size.
ATk -SVM: As a general aggregate loss, the ATk loss can be combined with any functional form
for individual losses. In the case of binary classification, the ATk loss combined with the individual
hinge loss for a prediction function f from a reproducing kernel Hilbert space (RKHS) [18] leads to
the ATk -SVM model. Specifically, we consider function f as a member of RKHS HK with norm
k ? kK , which is induced from a reproducing kernel K : X ? X ? R. Using the individual hinge
loss, [1 ? yi f (xi )]+ , the corresponding MATk learning objective in RKHS becomes
n
1 X
k
1
min
[1 ? yi f (xi )]+ ? ? + + ? +
kf k2K ,
(5)
f ?HK ,??0 n
n
2C
i=1
where C > 0 is the regularization factor. Furthermore, the outer hinge function in (5) can be
removed due to the following result.
Lemma 2. For a ? 0, b ? 0, there holds [a ? `]+ ? b + = [a ? b ? `]+ .
Proof of Lemma 2 can be found in the supplementary materials. In addition, note that for any
minimizer (fz , ?z ) of (5), setting f (x) = 0, ? = 1 in the objective function of (5), we have nk ?z ?
Pn
k
k
1
1
2
i=1 [1 ? yi fz (xi )]+ ? ?z + + n ?z + 2C kfz kK ? n , so we have 0 ? ?z ? 1 which means
n
that the minimization can be restricted to 0 ? ? ? 1. Using these results and introducing ? = 1 ? ?,
Eq.(5) can be rewritten as
n
1X
k
1
min
[? ? yi f (xi )]+ ? ? +
kf k2K .
(6)
f ?HK ,0???1 n
n
2C
i=1
The ATk -SVM objective generalizes many several existing SVM models. For example, when k = n,
it equals to the standard C-SVM [5]. When C = 1 and with conditions K(xi , xi ) ? 1 for any i,
ATk -SVM reduces to ?-SVM [19] with ? = nk . Furthermore, similar to the conventional SVM
model, writing in the dual form of (6) can lead to a convex quadratic programming problem that can
be solved efficiently. See supplementary materials for more detailed explanations.
Choosing k. The number of top individual losses in the ATk loss is a critical parameter that affects
the learning performance. In concept, using ATk loss will not be worse than using average or
maximum losses as they correspond to specific choices of k. In practice, k can be chosen during
training from a validation dataset as the experiments in Section 4. As k is an integer, a simple grid
search usually suffices to find a satisfactory value. Besides, Theorem 1 in Section 3 establishes a
theoretical lower bound for k to guarantee reliable classification based on the Bayes error. If we
have information about the proportion of outliers, we can also narrow searching space of k based on
the fact that ATk loss is the convex upper bound of the top-k loss, which is similar to [20].
4
3
Statistical Analysis
In this section, we address the statistical properties of the ATk objective in the context of binary
classification. Specifically, we investigate the property of classification calibration [1] of the ATk
general objective, and derive bounds for the misclassification error of the ATk -SVM model in the
framework of statistical learning theory (e.g. [1, 7, 23, 26]).
3.1
Classification Calibration under ATk Loss
We assume the training data z = {(xi , yi )}ni=1 are i.i.d. samples from an unknown distribution p on
X ?{?1}. Let pX be the marginal distribution of p on the input space X . Then, the misclassification
error of a classifier f : X ? {?1} is denoted by R(f ) = Pr(y 6= f (x)) = E[Iyf (x)?0 ]. The Bayes
error is given by R? = inf f R(f ), where the infimum is over all measurable functions. No function
can achieve less risk than the Bayes rule fc (x) = sign(?(x) ? 21 ), where ?(x) = Pr(y = 1|x) [8].
In practice, one uses a surrogate loss ` : R ? [0, ?) which is convex and upper-bound the 0-1
loss. The population `-risk (generalization error) is given by E` (f ) = E[`(yf (x))]. Denote the
optimal `-risk by E`? = inf f E` (f ). A very basic requirement for using such a surrogate loss ` is
the so-called classification calibration (point-wise form of Fisher consistency) [1, 14]. Specifically,
a loss ` is classification calibrated with respect to distribution p if, for any x, the minimizer f`? =
inf f E` (f ) should have the same sign as the Bayes rule fc (x), i.e., sign(f`? (x)) = sign(fc (x))
whenever fc (x) 6= 0.
An appealing result concerning the classification calibration of a loss function ` was obtained in [1],
which states that ` is classification calibrated if ` is convex, differentiable at 0 and `0 (0) < 0. In the
same spirit, we investigate the classification calibration property of the ATk loss. Specifically, we
first obtain the population form of the ATk objective using the infinite limit of (2)
n
1X
k nk ??
[`(yi f (xi )) ? ?]+ + ? ??
??? E [[`(yf (x)) ? ?]+ ] + ??.
n i=1
n n??
We then consider the optimization problem
(f ? , ?? ) = arg inf E [[`(yf (x)) ? ?]+ ] + ??,
f,??0
(7)
where the infimum is taken over all measurable function f : X ? R. We say the ATk (aggregate)
loss is classification calibrated with respect to p if f ? has the same sign as the Bayes rule fc . The
following theorem establishes such conditions.
Theorem 1. Suppose the individual loss ` : R ? R+ is convex, differentiable at 0 and `0 (0) < 0.
Without loss of generality, assume that `(0) = 1. Let (f ? , ?? ) be defined in (7),
(i) If ? > E`? then the ATk loss is classification calibrated.
(ii) If, moreover, ` is monotonically
decreasing and the ATk aggregate loss is classification
R
calibrated then ? ? ?(x)6= 1 min(?(x), 1 ? ?(x))dpX (x).
2
The proof of Theorem 1 can be found in the supplementary materials. Part (i) and (ii) of the above
theorem address respectively the sufficient and necessary conditions on ? such that the ATk loss
becomes classification calibrated. Since ` is an upper bound surrogate of the 0-1 loss, the optimal
`-risk E`? is larger than the Bayes error R? , i.e., E`? ? R? . In particular, if the individual loss `
is the hinge loss then E`? = 2R? . Part (ii) of the above theorem indicates that the ATk aggregate
loss is classification calibrated if ? = limn?? k/n is larger than the optimal generalization error E`?
associated with the individual loss. The choice of k > nE`? thus guarantees classification calibration,
which gives a lower bound of k. This result also provides a theoretical underpinning of the sensitivity
to outliers of the
R maximum loss (ATk loss with k =R 1). If the probability of the set {x : ?(x) = 1/2}
is zero, R? = X min(?(x), 1 ? ?(x))dpX (x) = ?(x)6=1/2 min(?(x), 1 ? ?(x))dpX (x). Theorem
1 indicates that in this case, if the maximum loss is calibrated, one must have n1 ? ? ? R? . In
other words, as the number of training data increases, the Bayes error has to be arbitrarily small,
which is consistent with the empirical observation that the maximum loss works well under the
well-separable data setting but are sensitive to outliers and non-separable data.
5
3.2
Error bounds of ATk -SVM
We next study the excess misclassification error of the ATk -SVM model i.e., R(sign(fz )) ? R? .
Let (fz , ?z ) be the minimizer of the ATk -SVM objective (6) in the RKHS setting. Let fH be the
minimizer of the generalization error over the RKHS space HK , i.e., fH = argminf ?HK Eh (f ),
where we use the notation Eh (f ) = E [[1 ? yf (x)]+ ] to denote the `-risk of the hinge loss. In the
finite-dimension case, the existence of fH follows from the direct method in the variational calculus,
as Eh (?) is lower bounded by zero, coercive, and weakly sequentially lower semi-continuous by its
convexity. For an infinite dimensional HK , we assume the existence of fH . We also assume that
Eh (fH ) < 1 since even a na??ve zero classifier can achieve Eh (0) = 1. Denote the approximation
p
error by A(HK ) = inf f ?HK Eh (f ) ? Eh (fc ) = Eh (fH ) ? Eh (fc ), and let ? = supx?X K(x, x).
The main theorem can be stated as follows.
Theorem 2. Consider the ATk -SVM in RKHS (6). For any ? ? (0, 1] and ? ? (0, 1 ? Eh (fH )),
choosing k = dn(Eh (fH ) + ?)e. Then, it holds
1 + C?,H
n?2 ?2
Pr R(sign(fz )) ? R? ? ? + A(H) + ? + ?
,
? 2 exp ?
(1 + C?,H )2
n?
?
where C?,H = ?(2 2C + 4kfH kK ).
The complete proof of Theorem 2 is given in the supplementary materials. The main idea is to
show that ?z is bounded from below by a positive constant with high probability, and then bound the
excess misclassification error R(sign(fz? )) ? R? by Eh (fz /?z ) ? Eh (fc ). If K is a universal kernel
then A(HK ) = 0 [23]. In this case, let ? = ? ? (0, 1 ? Eh (fH )), then from Theorem 2 we have
1 + C?,H
n?4
Pr R(sign(fz )) ? R? ? 2? + ?
? 2 exp ?
,
(1 + C?,H )2
n?
Consequently, choosing C such that limn?? C/n = 0, which is equivalent to
limn?? (1 + C?,H )2 /n = 0, then R(sign(fz )) can be arbitrarily close to the Bayes error R? ,
with high probability, as long as n is sufficiently large.
4
Experiments
We have demonstrated that ATk loss provides a continuum between the average loss and the maximum loss, which can potentially alleviates their drawbacks. A natural question is whether such
an advantage actually benefits practical learning problems. In this section, we demonstrate the
behaviors of MATk learning coupled with different individual losses for binary classification and
regression on synthetic and real datasets, with minimizing the average loss and the maximum loss
treated as special cases for k = n and k = 1, respectively. For simplicity, in all experiments, we use
homogenized linear prediction functions f (x) = wT x with parameters w and the Tikhonov regu1
larizer ?(w) = 2C
||w||2 , and optimize the MATk learning objective with the stochastic gradient
descent method given in (4).
Binary Classification: We conduct experiments on binary classification using eight benchmark
datasets from the UCI3 and KEEL4 data repositories to illustrate the potential effects of using ATk
loss in practical learning to adapt to different underlying data distributions. A detailed description
of the datasets is given in supplementary materials. The standard individual logistic loss and hinge
loss are combined with different aggregate losses. Note that average loss combined with individual
logistic loss corresponds to the logistic regression model and average loss combined with individual
hinge loss leads to the C-SVM algorithm [5].
For each dataset, we randomly sample 50%, 25%, 25% examples as training, validation and testing
sets, respectively. During training, we select parameters C (regularization factor) and k (number of
top losses) on the validation set. Parameter C is searched on grids of log10 scale in the range of
[10?5 , 105 ] (extended when optimal value is on the boundary), and k is searched on grids of log10
scale in the range of [1, n]. We use k ? to denote the optimal k selected from the validation set.
3
4
https://archive.ics.uci.edu/ml/datasets.html
http://sci2s.ugr.es/keel/datasets.php
6
Logistic Loss
Hinge Loss
Maximum
Average
ATk?
Maximum
Average
Monk
22.41(2.95) 20.46(2.02) 16.76(2.29)
22.04(3.08) 18.61(3.16)
Australian 19.88(6.64) 14.27(3.22) 11.70(2.82)
19.82(6.56) 14.74(3.10)
Madelon
47.85(2.51) 40.68(1.43) 39.65(1.72)
48.55(1.97) 40.58(1.86)
Splice
23.57(1.93) 17.25(0.93) 16.12(0.97)
23.40(2.10) 16.25(1.12)
Spambase
21.30(3.05)
8.36(0.97)
8.36(0.97)
21.03(3.26)
7.40(0.72)
German
28.24(1.69) 25.36(1.27) 23.28(1.16)
27.88(1.61) 24.16(0.89)
Titanic
26.50(3.35) 22.77(0.82) 22.44(0.84)
25.45(2.52) 22.82(0.74)
Phoneme
28.67(0.58) 25.50(0.88) 24.17(0.89)
28.81(0.62) 22.88(1.01)
Table 1: Average misclassification rate (%) of different learning objectives over 8 datasets.
are shown in bold with results that are not significant different to the best results underlined.
0.24
0.22
0.2
0.18
Splice
0.28
Misclassification Rate
Misclassification Rate
Misclassification Rate
Australian
0.35
0.26
0.3
0.25
0.2
0.15
Phoneme
0.3
0.26
Misclassification Rate
Monk
0.28
0.24
0.22
0.2
0.18
ATk?
17.04(2.77)
12.51(4.03)
40.18(1.64)
16.23(0.97)
7.40(0.72)
23.80(1.05)
22.02(0.77)
22.88(1.01)
The best results
0.28
0.26
0.24
0.16
0.16
1
50
100
150
k
216
0.1
0.22
1
100
k
200
300 346
1
500
k
1000
1588
1
500
1000
1500
k
2000
2702
Figure 3: Plots of misclassification rate on testing set vs. k on four datasets.
We report the average performance over 10 random splitting of training/validation/testing for each
dataset with MATk learning objectives formed from individual logistic loss and hinge loss. Table 1
gives their experimental results in terms of misclassification rate (results in terms of other classification quality metrics are given in supplementary materials). Note that on these datasets, the average
loss consistently outperforms the maximum loss, but the performance can be further improved with
the ATk loss, which is more adaptable to different data distributions. This advantage of the ATk loss
is particularly conspicuous for datasets Monk and Australian.
To further understand the behavior of MATk learning on individual datasets, we show plots of misclassification rate on testing set vs. k for four representative datasets in Fig.3 (in which C is fixed to
102 and k ? [1, n]). As these plots show, on all four datasets, there is a clear range of k value with
better classification performance than the two extreme cases k = 1 and k = n, corresponding to the
maximum and average loss, respectively. To be more specific, when k = 1, the potential noises and
outliers will have the highest negative effects on the learned classifier and the related classification
performance is very poor. As k increases, the negative effects of noises and outliers will reduce and
the classification performance becomes better, this is more significant on dataset Monk, Australian
and Splice. However, if k keeps increase, the classification performance may decrease (e.g., when
k = n). This may because that as k increases, more and more well classified samples will be included and the non-zero loss on these samples will have negative effects on the learned classifier (see
our analysis in Section 2), especifically for dataset Monk, Australian and Phoneme.
Regression. Next, we report experimental results of linear regression on one synthetic dataset
(Sinc) and three real datasets from [4], with a detailed description of these datasets given in supplementary materials. The standard square loss and absolute loss are adopted as individual losses.
Note that average loss coupled with individual square loss is standard ridge regression model and
average loss coupled with individual absolute loss reduces to ?-SVR [19]. We normalize the target
output to [0, 1] and report their root mean square error (RMSE) in Table 2, with optimal C and k ?
obtained by a grid search as in the case of classification (performance in terms of mean absolute
square error (MAE) is given in supplementary materials). Similar to the classification cases, using
the ATk loss usually improves performance in comparison to the average loss or maximum loss.
5
Related Works
Most work on learning objectives focus on designing individual losses, and only a few are dedicated
to new forms of aggregate losses. Recently, aggregate loss considering the order of training data have
been proposed in curriculum learning [2] and self-paced learning [11, 9], which suggest to organize
the training process in several passes and samples are included from easy to hard gradually. It is
interesting to note that each pass of self-paced learning [11] is equivalent to minimum the average of
7
Square Loss
Absolute Loss
Maximum
Average
ATk?
Maximum
Average
ATk?
Sinc 0.2790(0.0449) 0.1147(0.0060) 0.1139(0.0057) 0.1916(0.0771) 0.1188(0.0067) 0.1161(0.0060)
Housing 0.1531(0.0226) 0.1065(0.0132) 0.1050(0.0132) 0.1498(0.0125) 0.1097(0.0180) 0.1082(0.0189)
Abalone 0.1544(0.1012) 0.0800(0.0026) 0.0797(0.0026) 0.1243(0.0283) 0.0814(0.0029) 0.0811(0.0027)
Cpusmall 0.2895(0.0722) 0.1001(0.0035) 0.0998(0.0037) 0.2041(0.0933) 0.1170(0.0061) 0.1164(0.0062)
Table 2: Average RMSE on four datasets. The best results are shown in bold with results that are not significant
different to the best results underlined.
Pn
the k smallest individual losses, i.e., k1 i=n?k+1 `[i] (f ), which we term it as the average bottom-k
loss in contrast to the average top-k losses in our case. In [20], the pros and cons of the maximum
loss and the average loss are compared, and the top-k loss, i.e., `[k] (f ), is advocated as a remedy to
the problem of both. However, unlike the ATk loss, in general, neither the average bottom-k loss nor
the top-k loss are convex functions with regards to the individual losses.
Minimizing top-k errors has also been used in individual losses. For ranking problems, the work of
[17, 24] describes a form of individual loss that gives more weights to the top examples in a ranked
list. In multi-class classification, the top-1 loss is commonly used which causes penalties when the
top-1 predicted class is not the same as the target class label [6]. This has been further extended
in [12, 13] to the top-k multi-class loss, in which for a class label that can take m different values,
the classifier is only penalized when the correct value does not show up in the top k most confident
predicted values. As an individual loss, these works are complementary to the ATk loss and they
can be combined to improve learning performance.
6
Discussion
In this work, we introduce the average top-k (ATk ) loss as a new aggregate loss for supervised
learning, which is the average over the k largest individual losses over a training dataset. We show
that the ATk loss is a natural generalization of the two widely used aggregate losses, namely the
average loss and the maximum loss, but can combine their advantages and mitigate their drawbacks
to better adapt to different data distributions. We demonstrate that the ATk loss can better protect
small subsets of hard samples from being swamped by a large number of easy ones, especially for
imbalanced problems. Furthermore, it remains a convex function over all individual losses, which
can lead to convex optimization problems that can be solved effectively with conventional gradientbased methods. We provide an intuitive interpretation of the ATk loss based on its equivalent effect
on the continuous individual loss functions, suggesting that it can reduce the penalty on correctly
classified data. We further study the theoretical aspects of ATk loss on classification calibration and
error bounds of minimum average top-k learning for ATk -SVM. We demonstrate the applicability
of minimum average top-k learning for binary classification and regression using synthetic and real
datasets.
There are many interesting questions left unanswered regarding using the ATk loss as learning objectives. Currently, we use conventional gradient-based algorithms for its optimization, but we are
investigating special instantiations of MATk learning for which more efficient optimization methods
can be developed. Furthermore, the ATk loss can also be used for unsupervised learning problems
(e.g., clustering), which is a focus of our subsequent study. It is also of practical importance to
combine ATk loss with other successful learning paradigms such as deep learning, and to apply it to
large scale real life dataset. Lastly, it would be very interesting to derive error bounds of MATk with
general individual loss functions.
7
Acknowledgments
We thank the anonymous reviewers for their constructive comments. This work was completed when
the first author was a visiting student at SUNY Albany, supported by a scholarship from University of
Chinese Academy of Sciences (UCAS). Siwei Lyu is supported by the National Science Foundation
(NSF, Grant IIS-1537257) and Yiming Ying is supported by the Simons Foundation (#422504) and
the 2016-2017 Presidential Innovation Fund for Research and Scholarship (PIFRS) program from
SUNY Albany. This work is also partially supported by the National Science Foundation of China
(NSFC, Grant 61620106003) for Bao-Gang Hu and Yanbo Fan.
8
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6,250 | 6,654 | Learning multiple visual domains with residual
adapters
Sylvestre-Alvise Rebuffi1
Hakan Bilen1,2
1
Visual Geometry Group
University of Oxford
{srebuffi,hbilen,vedaldi}@robots.ox.ac.uk
Andrea Vedaldi1
2
School of Informatics
University of Edinburgh
Abstract
There is a growing interest in learning data representations that work well for many
different types of problems and data. In this paper, we look in particular at the
task of learning a single visual representation that can be successfully utilized in
the analysis of very different types of images, from dog breeds to stop signs and
digits. Inspired by recent work on learning networks that predict the parameters of
another, we develop a tunable deep network architecture that, by means of adapter
residual modules, can be steered on the fly to diverse visual domains. Our method
achieves a high degree of parameter sharing while maintaining or even improving
the accuracy of domain-specific representations. We also introduce the Visual
Decathlon Challenge, a benchmark that evaluates the ability of representations to
capture simultaneously ten very different visual domains and measures their ability
to perform well uniformly.
1
Introduction
While research in machine learning is often directed at improving the performance of algorithms on
specific tasks, there is a growing interest in developing methods that can tackle a large variety of
different problems within a single model. In the case of perception, there are two distinct aspects of
this challenge. The first one is to extract from a given image diverse information, such as image-level
labels, semantic segments, object bounding boxes, object contours, occluding boundaries, vanishing
points, etc. The second aspect is to model simultaneously many different visual domains, such as
Internet images, characters, glyph, animal breeds, sketches, galaxies, planktons, etc (fig. 1).
In this work we explore the second challenge and look at how deep learning techniques can be used
to learn universal representations [5], i.e. feature extractors that can work well in several different
image domains. We refer to this problem as multiple-domain learning to distinguish it from the more
generic multiple-task learning.
Multiple-domain learning contains in turn two sub-challenges. The first one is to develop algorithms
that can learn well from many domains. If domains are learned sequentially, but this is not a
requirement, this is reminiscent of domain adaptation. However, there are two important differences.
First, in standard domain adaptation (e.g. [9]) the content of the images (e.g. ?telephone?) remains
the same, and it is only the style of the images that changes (e.g. real life vs gallery image). Instead
in our case a domain shift changes both style and content. Secondly, the difficulty is not just to adapt
the model from one domain to another, but to do so while making sure that it still performs well on
the original domain, i.e. to learn without forgetting [21].
The second challenge of multiple-domain learning, and our main concern in this paper, is to construct
models that can represent compactly all the domains. Intuitively, even though images in different
domains may look quite different (e.g. glyph vs. cats), low and mid-level visual primitives may still
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure 1: Visual Decathlon. We explore deep architectures that can learn simultaneously different
tasks from very different visual domains. We experiment with ten representative ones: (a) Aircraft, (b)
CIFAR-100, (c) Daimler Pedestrians, (d) Describable Textures, (e) German Traffic Signs, (f) ILSVRC
(ImageNet) 2012, (g) VGG-Flowers, (h) OmniGlot, (i) SVHN, (j) UCF101 Dynamic Images.
be largely shareable. Sharing knowledge between domains should allow to learn compact multivalent
representations. Provided that sufficient synergies between domains exist, multivalent representations
may even work better than models trained individually on each domain (for a given amount of training
data).
The primary contribution of this paper (section 3) is to introduce a design for multivalent neural
network architectures for multiple-domain learning (section 3 fig. 2). The key idea is reconfigure
a deep neural network on the fly to work on different domains as needed. Our construction is based on
recent learning-to-learn methods that showed how the parameters of a deep network can be predicted
from another [2, 16]. We show that these formulations are equivalent to packing the adaptation
parameters in convolutional layers added to the network (section 3). The layers in the resulting
parametric network are either domain-agnostic, hence shared between domains, or domain-specific,
hence parametric. The domain-specific layers are changed based on the ground-truth domain of the
input image, or based on an estimate of the latter obtained from an auxiliary network. In the latter
configuration, our architecture is analogous to the learnet of [2].
Based on such general observations, we introduce in particular a residual adapter module and
use it to parameterize the standard residual network architecture of [13]. The adapters contain a
small fraction of the model parameters (less than 10%) enabling a high-degree of parameter sharing
between domains. A similar architecture was concurrently proposed in [31], which also results in
the possibility of learning new domains sequentially without forgetting. However, we also show a
specific advantage of the residual adapter modules: the ability to modulate adaptation based on the
size of the target dataset.
Our proposed architectures are thoroughly evaluated empirically (section 5). To this end, our second
contribution is to introduce the visual decathlon challenge (fig. 1 and section 4), a new benchmark
for multiple-domain learning in image recognition. The challenge consists in performing well
simultaneously on ten very different visual classification problems, from ImageNet and SVHN to
action classification and describable texture recognition. The evaluation metric, also inspired by the
decathlon discipline, rewards models that perform better than strong baselines on all the domains
simultaneously. A summary of our finding is contained in section 6.
2
Related Work
Our work touches on multi-task learning, learning without forgetting, domain adaptation, and other
areas. However, our multiple-domain setup differs in ways that make most of the existing approaches
not directly applicable to our problem.
Multi-task learning (MTL) looks at developing models that can address different tasks, such as
detecting objects and segmenting images, while sharing information and computation among them.
Earlier examples of this paradigm have focused on kernel methods [10, 1] and deep neural network
(DNN) models [6]. In DNNs, a standard approach [6] is to share earlier layers of the network, training
the tasks jointly by means of back-propagation. Caruana [6] shows that sharing network parameters
between tasks is beneficial also as a form of regularization, putting additional constraints on the
learned representation and thus improving it.
MTL in DNNs has been applied to various problems ranging from natural language processing [8, 22],
speech recognition [14] to computer vision [41, 42, 4]. Collobert et al. [8] show that semi-supervised
learning and multi-task learning can be combined in a DNN model to solve several language
processing prediction tasks such as part-of-speech tags, chunks, named entity tags and semantic
2
roles. Huang et al. [14] propose a shared multilingual DNN which shares hidden layers across
many languages. Liu et al. [22] combine multiple-domain classification and information retrieval for
ranking web search with a DNN. Multi-task DNN models are also reported to achieve performance
gains in computer vision problems such as object tracking [41], facial-landmark detection [42], object
and part detection [4], a collection of low-level and high-level vision tasks [18]. The main focus of
these works is learning a diverse set of tasks in the same visual domain. In contrast, our paper focuses
on learning a representation from a diverse set of domains.
Our investigation is related to the recent paper of [5], which studied the ?size? of the union of different
visual domains measured in terms of the capacity of the model required to learn it. The authors
propose to absorb different domain in a single neural network by tuning certain parameters in batch
and instance normalization layers throughout the architecture; we show that our residual adapter
modules, which include the latter as a special case, lead to far superior results.
Life-long learning. A particularly important aspect of MTL is the ability of learning multiple tasks
sequentially, as in Never Ending Learning [25] and Life-long Learning [38]. Sequential learning
typically suffers in fact from forgetting the older tasks, a phenomenon aptly referred to as ?catastrophic
forgetting? in [11]. Recent work in life-long learning try to address forgetting in two ways. The first
one [37, 33] is to freeze the network parameters for the old tasks and learn a new task by adding
extra parameters. The second one aims at preserving knowledge of the old tasks by retaining the
response of the original network on the new task [21, 30], or by keeping the network parameters of
the new task close to the original ones [17]. Our method can be considered as a hybrid of these two
approaches, as it can be used to retain the knowledge of previous tasks exactly, while adding a small
number of extra parameters for the new tasks.
Transfer learning. Sometimes one is interested in maximizing the performance of a model on a
target domain. In this case, sequential learning can be used as a form of initialization[29]. This is
very common in visual recognition, where most DNN are initialize on the ImageNet dataset and then
fine-tuned on a target domain and task. Note, however, that this typically results in forgetting the
original domain, a fact that we confirm in the experiments.
Domain adaptation. When domains are learned sequentially, our work can be related to domain
adaptation. There is a vast literature in domain adaptation, including recent contributions in deep
learning such as [12, 39] based on the idea of minimizing domain discrepancy. Long et al. [23]
propose a deep network architecture for domain adaptation that can jointly learn adaptive classifiers
and transferable features from labeled data in the source domain and unlabeled data in the target
domain. There are two important differences with our work: First, in these cases different domains
contain the same objects and is only the visual style that changes (e.g. webcam vs. DSLR), whereas
in our case the object themselves change. Secondly, domain adaptation is a form of transfer learning,
and, as the latter, is concerned with maximizing the performance on the target domain reagardless of
potential forgetting.
3
Method
Our primary goal is to develop neural network architectures that can work well in a multiple-domain
setting. Modern neural networks such as residual networks (ResNet [13]) are known to have very high
capacity, and are therefore good candidates to learn from diverse data sources. Furthermore, even
when domains look fairly different, they may still share a significant amount of low and mid-level
visual patterns. Nevertheless, we show in the experiments (section 5) that learning a ResNet (or a
similar model) directly from multiple domains may still not perform well.
In order to address this problem, we consider a compact parametric family of neural networks
?? : X ? V indexed by parameters ?. Concretely, X ? RH?W ?3 can be a space of RGB images
and V = RHv ?Wv ?Cv a space of feature tensors. ?? can then be obtained by taking all but the last
classification layer of a standard ResNet model. The parametric feature extractors ?? is then used to
construct predictors for each domain d as ?d = ?d ? ??d , where ?d are domain-specific parameters
and ?d (v) = softmax(Wd v) is a domain-specific linear classifier V ? Yd mapping features to
image labels.
If ? comprises all the parameters of the feature extractor ?? , this approach reduces to learning
independent models for each domain. On the contrary, our goal is to maximize parameter sharing,
which we do below by introducing certain network parametrizations.
3
w1
BN
?
0
(?1s , ?1b ) ?1w
0
w2
(?1s , ?1b )
?
BN
+
BN
?
[?]+
0
(?2s , ?2n ) ?2w
0
(?2s , ?2b )
?
+
BN
+
[?]+
Figure 2: Residual adapter modules. The figure shows a standard residual module with the inclusion
of adapter modules (in blue). The filter coefficients (w1 , w2 ) are domain-agnostic and contains the
vast majority of the model parameters; (?1 , ?2 ) contain instead a small number of domain-specific
parameters.
3.1
Learning to learn and filter prediction
The problem of adapting a neural network dynamically to variations of the input data is similar to the
one found in recent approaches to learning to learn. A few authors [34, 16, 2], in particular, have
proposed to learn neural networks that predict, in a data-dependent manner, the parameters of another.
Formally, we can write ?d = Aedx where edx is the indicator vector of the domain dx of image x
and A is a matrix whose columns are the parameter vectors ?d . As shown later, it is often easy to
construct an auxiliary network that can predict d from x, so that the parameter ? = ?(x) can also be
expressed as the output of a neural network. If d is known, then ?(x, d) = ?d as before, and if not ?
can be constructed as suggested above or from scratch as done in [2].
The result of this construction is a network ??(x) (x) whose parameters are predicted by a second
network ?(x). As noted in [2], while this construction is conceptually simple, its implementation is
more subtle. Recall that the parameters w of a deep convolutional neural network consist primarily of
the coefficients of the linear filters in the convolutional layers. If w = ?, then ? = ?(x) would need
to predict millions of parameters (or to learn independent models when d is observed). The solution
of [2] is to use a low-rank decomposition of the filters, where w = ?(w0 , ?) is a function of a filter
basis w0 and ? is a small set of tunable parameters.
Here we build on the same idea, with some important extensions. First, we note that linearly
parametrizing a filter bank is the same as introducing a new, intermediate convolutional layer in the
network. Specifically, let Fk ? RHf ?Wf ?Cf be a basis of K filters of size Hf ? Wf operating on
Cf input feature channels. Given parameters [?tk ] ? RT ?K , we can express a bank of T filters as
PK
linear combinations Gt = k=1 ?tk Fk . Applying the bank to a tensor x and using associativity and
PK
linearity of convolution results in G ? x = k=1 ?:k (Fk ? x) = ? ? F ? x where we interpreted ? as
a 1 ? 1 ? T ? K filter bank. While [2] used a slightly different low-rank filter decomposition, their
parametrization can also be seen as introducing additional filtering layers in the network.
An advantage of this parametrization is that it results in a useful decomposition, where part of the
convolutional layers contain the domain-agnostic parameters F and the others contain the domainspecific ones ?d . As discussed in section 5, this is particularly useful to address the forgetting
problem. In the next section we refine these ideas to obtain an effective parametrization of residual
networks.
3.2
Residual adapter modules
As an example of parametric network, we propose to modify a standard residual network. Recall that
a ResNet is a chain gm ? ? ? ? ? g1 of residual modules gt . In the simplest variant of the model, each
residual module g takes as input a tensor RH?W ?C and produces as output a tensor of the same size
using g(x; w) = x + ((w2 ? ?) ? [?]+ ? (w1 ? ?))(x). Here w1 and w2 are the coefficients of banks of
small linear filters, [z]+ = max{0, z} is the ReLU operator, w ? z is the convolution of z by the filter
bank w, and ? denotes function composition. Note that, for the addition to make sense, filters must
be configured such that the dimensions of the output of the last bank are the same as x.
Our goal is to parametrize the ResNet module. As suggested in the previous section, rather than
changing the filter coefficients directly, we introduce additional parametric convolutional layers. In
fact, we go one step beyond and make them small residual modules in their own right and call them
4
residual adapter modules (blue blocks in fig. 2). These modules have the form:
g(x; ?) = x + ? ? x.
In order to limit the number of domain-specific parameters, ? is selected to be a bank of 1 ? 1 filters.
A major advantage of adopting a residual architecture for the adapter modules is that the adapters
reduce to the identity function when their coefficients are zero. When learning the adapters on small
domains, this provides a simple way of controlling over-fitting, resulting in substantially improved
performance in some cases.
Batch normalization and scaling. Batch Normalization (BN) [15] is an important part of very deep
neural networks. This module is usually inserted after convolutional layers in order to normalize
their outputs and facilitate learning (fig. 2). The normalization operation is followed by rescaling and
shift operations s x + b, where (s, b) are learnable parameters. In our architecture, we incorporate
the BN layers into the adapter modules (fig. 2). Furthermore, we add a BN module right before the
adapter convolution layer.1 Note that the BN scale and bias parameters are also dataset-dependent ?
as noted in the experiments, this alone provides a certain degree of model adaptation.
Domain-agnostic vs domain-specific parameters. If the residual module of fig. 2 is configured to
process an input tensor with C feature channels, and if the domain-agnostic filters w1 , w2 are of size
h ? h ? C, then the model has 2(h2 C 2 + hC) domain-agnostic parameters (including biases in the
convolutional layers) and 2(C 2 + 5C) domain-specific parameters.2 Hence, there are approximately
h2 more domain-agnostic parameters than domain specific ones (usually h2 = 9).
3.3
Sequential learning and avoiding forgetting
While in this paper we are not concerned with sequential learning, we have found it to be a good
strategy to bootstrap a model when a large number of domains have to be learned. However, the most
popular approach to sequential learning, fine-tuning (section 2), is often a poor choice for learning
shared representations as it tends to quickly forget the original tasks.
The challenge in learning without forgetting is to maintain information about older tasks as new
ones are learned (section 2). With respect to forgetting, our adapter modules are similar to the
tower model [33] as they preserve the original model exactly: one can pre-train the domain-agnostic
parameters w on a large domain such as ImageNet, and then fine-tune only the domain-specific
parameters ?d for each new domain. Like the tower method, this preserves the original task exactly,
but it is far less expensive as it does not require to introduce new feature channels for each new
domain (a quadratic cost). Furthermore, the residual modules naturally reduce to the identity function
when sufficient shrinking regularization is applied to the adapter weights ?w . This allows the adapter
to be tuned depending on the availability of data for a target domain, sometimes significantly reducing
overfitting.
4
Visual decathlon
In this section we introduce a new benchmark, called visual decathlon, to evaluate the performance
of algorithms in multiple-domain learning. The goal of the benchmark is to assess whether a method
can successfully learn to perform well in several different domains at the same time. We do so by
choosing ten representative visual domains, from Internet images to characters, as well as by selecting
an evaluation metric that rewards performing well on all tasks.
Datasets. The decathlon challenge combines ten well-known datasets from multiple visual domains:
FGVC-Aircraft Benchmark [24] contains 10,000 images of aircraft, with 100 images for each of
100 different aircraft model variants such as Boeing 737-400, Airbus A310. CIFAR100 [19] contains
60,000 32 ? 32 colour images for 100 object categories. Daimler Mono Pedestrian Classification
Benchmark (DPed) [26] consists of 50,000 grayscale pedestrian and non-pedestrian images, cropped
and resized to 18 ? 36 pixels. Describable Texture Dataset (DTD) [7] is a texture database, consisting of 5640 images, organized according to a list of 47 terms (categories) such as bubbly, cracked,
1
While the bias and scale parameters of the latter can be incorporated in the following filter bank, we found
it easier to leave them separated from the latter
2
Including all bias and scaling vectors; 2(C 2 + 3C) if these are absorbed in the filter banks when possible.
5
marbled. The German Traffic Sign Recognition (GTSR) Benchmark [36] contains cropped images for 43 common traffic sign categories in different image resolutions. Flowers102 [28] is a
fine-grained classification task which contains 102 flower categories from the UK, each consisting of
between 40 and 258 images. ILSVRC12 (ImNet) [32] is the largest dataset in our benchmark contains 1000 categories and 1.2 million images. Omniglot [20] consists of 1623 different handwritten
characters from 50 different alphabets. Although the dataset is designed for one-shot learning, we use
the dataset for standard multi-class classification task and include all the character categories in train
and test splits. The Street View House Numbers (SVHN) [27] is a real-world digit recognition
dataset with around 70,000 32 ? 32 images. UCF101 [35] is an action recognition dataset of realistic
human action videos, collected from YouTube. It contains 13,320 videos for 101 action categories.
In order to make this dataset compatible with our benchmark, we convert the videos into images by
using the Dynamic Image encoding of [3] which summarizes each video into an image based on a
ranking principle.
Challenge and evaluation. Each dataset Dd , d = 1, . . . , 10 is formed of pairs (x, y) ? Dd where x
is an image and y ? {1, . . . , Cd } = Yd is a label. For each dataset, we specify a training, validation
and test subsets. The goal is to train the best possible model to address all ten classification tasks
using only the provided training and validation data (no external data is allowed). A model ? is
evaluated on the test data, where, given an image x and its ground-truth domain dx label, it has to
predict the corresponding label y = ?(x, dx ) ? Yd .
Performance is measured in terms of a single scalar score S determined as in the decathlon discipline.
Performing well at this metric requires algorithms to perform well in all tasks, compared to a
minimum level of baseline performance for each. In detail, S is computed as follows:
S=
10
X
?d max{0, Edmax ? Ed }?d ,
Ed =
d=1
1
|Ddtest |
X
1{y6=?(x,d)} .
(1)
(x,y)?Ddtest
where Ed is the average test error for each domain. Edmax the baseline error (section 5), above which
no points are scored. The exponent ?d ? 1 rewards more reductions of the classification error
as this becomes close to zero and is set to ?d = 2 for all domains. The coefficient ?d is set to
1, 000 (Edmax )??d so that a perfect result receives a score of 1,000 (10,000 in total).
Data preprocessing. Different domains contain a different set of image classes as well as a different
number of images. In order to reduce the computational burden, all images have been resized
isotropically to have a shorter side of 72 pixels. For some datasets such as ImageNet, this is a
substantial reduction in resolution which makes training models much faster (but still sufficient to
obtain excellent classification results with baseline models). For the datasets for which there exists
training, validation, and test subsets, we keep the original splits. For the rest, we use 60%, 20% and
20% of the data for training, validation, and test respectively. For the ILSVRC12, since the test labels
are not available, we use the original validation subset as the test subset and randomly sample a new
validation set from their training split. We are planning to make the data and an evaluation server
public soon.
5
Experiments
In this section we evaluate our method quantitatively against several baselines (section 5.1), investigate
the ability of the proposed techniques to learn models for ten very diverse visual domains.
Implementation details. In all experiments we choose to use the powerful ResNets [13] as base
architectures due to their remarkable performance. In particular, as a compromise of accuracy and
speed, we chose the ResNet28 model [40] which consists of three blocks of four residual units. Each
residual unit contains 3 ? 3 convolutional, BN and ReLU modules (fig. 2). The network accepts
64 ? 64 images as input, downscales the spatial dimensions by two at each block and ends with a
global average pooling and a classifier layer followed by a softmax. We set the number of filters to
64, 128, 256 for these blocks respectively. Each network is optimized to minimize its cross-entropy
loss with stochastic gradient descent. The network is run for 80 epochs and the initial learning rate of
0.1 is lowered to 0.01 and then 0.001 gradually.
6
Model
#par. ImNet Airc. C100 DPed DTD GTSR Flwr
1.3m
# images
7k
50k
30k
4k
40k
OGlt SVHN UCF
2k
26k
70k
mean
S
9k
Scratch
Scratch+
10? 59.87 57.10 75.73 91.20 37.77 96.55 56.30 88.74 96.63 43.27 70.32 1625
11? 59.67 59.59 76.08 92.45 39.63 96.90 56.66 88.74 96.78 44.17 71.07 1826
Feature extractor
Finetune
LwF [21]
1? 59.67 23.31 63.11 80.33 45.37 68.16 73.69 58.79 43.54 26.80 54.28 544
10? 59.87 60.34 82.12 92.82 55.53 97.53 81.41 87.69 96.55 51.20 76.51 2500
10? 59.87 61.15 82.23 92.34 58.83 97.57 83.05 88.08 96.10 50.04 76.93 2515
BN adapt. [5]
? 1? 59.87 43.05
2? 59.67 56.68
Res. adapt.
2? 59.67 61.87
Res. adapt. decay
Res. adapt. finetune all
2? 59.23 63.73
Res. adapt. dom-pred
Res. adapt. (large)
78.62
81.20
81.20
81.31
92.07
93.88
93.88
93.30
51.60
50.85
57.13
57.02
95.82
97.05
97.57
97.47
74.14
66.24
81.67
83.43
84.83
89.62
89.62
89.82
94.10
96.13
96.13
96.17
43.51
47.45
50.12
50.28
71.76
73.88
76.89
77.17
1363
2118
2621
2643
2.5? 59.18 63.52 81.12 93.29 54.93 97.20 82.29 89.82 95.99 50.10 76.74 2503
? 12? 67.00 67.69 84.69 94.28 59.41 97.43 84.86 89.92 96.59 52.39 79.43 3131
Table 1: Multiple-domain networks. The figure reports the (top-1) classification accuracy (%) of
different models on the decathlon tasks and final decathlon score (S). ImageNet is used to prime the
network in every case, except for the networks trained from scratch. The model size is the number of
parameters w.r.t. the baseline ResNet. The fully-finetuned model, written blue, is used as a baseline
to compute the decathlon score.
Model
Airc.
C100
DPed
1.1 60.3 3.6 63.1 0.6
Finetune
4.1 61.1 21.0 82.2 23.8
LwF [21] high lr
38.0 50.6 33.0 80.7 53.3
LwF [21] low lr
Res. adapt. finetune all 59.2 63.7 59.2 81.3 59.2
DTD
GTSR
80.3 0.7 45.3 1.4
92.3 36.7 58.8 11.5
92.2 47.0 57.2 23.7
93.3 59.2 57.0 59.2
68.1
97.6
96.6
97.5
Flwr
27.2
34.2
45.7
59.2
OGlt
SVHN
UCF
73.6 13.4 87.7 0.2 96.6 5.4 51.2
83.1 3.0 88.1 0.2 96.1 18.6 50.0
75.7 21.0 86.0 13.3 94.8 29.0 44.6
83.4 59.2 89.8 59.2 96.1 59.2 50.3
Table 2: Pairwise forgetting. Each pair of numbers report the top-1 accuracy (%) on the old task
(ImageNet) and a new target task after the network is fully finetuned on the latter. We also show
the performance of LwF when it is finetuned on the new task with a high and low learning rate,
trading-off forgetting ImageNet and improving the results on the target domain. By comparison, we
show the performance of tuning only the residual adapters, which by construction does not result in
any performance loss in ImageNet while still achieving very good performance on each target task.
5.1
Results
There are two possible extremes. The first one is to learn ten independent models, one for each
dataset, and the second one is to learn a single model where all feature extractor parameters are
shared between the ten domains. We evaluate next different approaches to learn such models.
Pairwise learning. In the first experiment (table 1), we start by learning a ResNet model on ImageNet,
and then use different techniques to extend it to the remaining nine tasks, one at a time. Depending
on the method, this may produce an overall model comprising ten ResNet architectures, or just one
ResNet with a few domain-specific parameters; thus we also report the total number of parameters
used, where 1? is the size of a single ResNet (excluding the last classification layer, which can never
be shared).
As baselines, we evaluate four cases: i) learning an individual ResNet model from scratch for each
task, ii) freezing all the parameters of the pre-trained network, using the network as feature extractor
and only learn a linear classifier, iii) standard finetuning and iv) applying a reimplementation of the
LwF technique of [21] that encourages the fine-tuned network to retain the responses of the original
ImageNet model while learning the new task.
In terms of accuracy, learning from scratch performs poorly on small target datasets and, by learning
10 independent models, requires 10? parameters in total. Freezing the ImageNet feature extraction is
very efficient in terms of parameter sharing (1? parameters in total), preserves the original domain
exactly, but generally performs very poorly on the target domain. Full fine-tuning leads to accurate
results both for large and small datasets; however, it also forgets the ImageNet domain substantially
(table 2), so it still requires learning 10 complete ResNet models for good overall performance.
When LwF is run as intended by the original authors [21], is still leads to a noticeable performance
drop on the original task, even when learning just two domains (table 2), particularly if the target
domain is very different from ImageNet (e.g. Omniglot and SVHN). Still, if one chooses a different
trade-off point and allows the method to forget ImageNet more, it can function as a good regularizer
that slightly outperforms vanilla fine-tuning overall (but still resulting in a 10? model).
7
Next, we evaluate the effect of sharing the majority of parameters between tasks, whereas still
allowing a small number of domain-specific parameters to change. First, we consider specializing
only the BN layer scaling and bias parameters, which is equivalent to the approach of [5]. In this
case, less than the 0.1% of the model parameters are domain-specific (for the ten domains, this results
in a model with 1.01? parameters overall). Hence the model is very similar to the one with the
frozen feature extractor; nevertheless, the performances increase very substantially in most cases (e.g.
23.31% ? 43.05% accuracy on Aircraft).
As the next step, we introduce the residual adapter modules, which increase by 11% the number
of parameters per domain, resulting in a 2? model. In the pre-training phase, we first pretrain on
ImageNet the network with the added modules. Then, we freeze the task agnostic parameters and
train the task specific parameters on the different datasets. Differently from vanilla fine-tuning, there
is no forgetting in this setting. While most of the parameters are shared, our method is either close or
better than full fine-tuning. As a further control, we also train 10 models from scratch with the added
parameters (denoted as Scratch+), but do not observe any noticeable performance gain in average,
demonstrating that parameters sharing is highly beneficial. We also contrast learning the adapter
modules with two values of weight decay (0.002 and 0.005) higher than the default 0.0005. These
parameters are obtained after a coarse grid search using cross-validation for each dataset. Using
higher decay significantly improves the performance on smaller datasets such as Flowers, whereas
the smaller decay is best for larger datasets. This shows both the importance and utility of controlling
overfitting in the adaptation process. In practice, there is an almost direct correspondence between
the size of the data and which one of these values to use. The optimal decay can be selected via
validation, but a rough choice can be performed by simply looking at the dataset size.
We also compare to another baseline where we only finetune the last two convolutional layers and
freeze the others, which may be thought to be generic. This amounts to having a network with twice
the number of total parameters in a vanilla ResNet which is equal to our proposed architecture. This
model obtains 64.7% mean accuracy over ten datasets, which is significantly lower than our 73.9%,
likely due to overfitting (controlling overfitting is one of the advantages of our technique).
Furthermore, we also assess the quality of our adapter without residual connections, which corresponds to the low rank filter parametrization of section 3.1; this approach achieves an accuracy of
70.3%, which is worse than our 73.9%. We also observe that this configuration requires notably more
iterations to converge. Hence, the residual architecture for the adapters results in better performances,
better control of overfitting, and a faster convergence.
End-to-end learning. So far, we have shown that our method, by learning only the adapter modules
for each new domain, does not suffer from forgetting. However, for us sequential learning is just a
scalable learning strategy. Here, we also show (table 1) that we can further improve the results by
fine-tuning all the parameters of the network end-to-end on the ten tasks. We do so by sampling a
batch from each dataset in a round robin fashion, allowing each domain to contribute to the shared
parameters. A final pass is done on the adapter modules to take into account the change in the shared
parameters.
Domain prediction. Up to now we assume that the domain of each image is given during test
time for all the methods. If this is unavailable, it can be predicted on the fly by means of a small
neural-network predictor. We train a light ResNet, which is composed three stacks of two residual
networks, half deep as the original net, obtaining 99.8% accuracy in domain prediction, resulting in a
barely noticeable drop in the overall multiple-domain challenge (see Res. adapt dom-pred in table 1).
Note that similar performance drop would be observed for the other baselines.
Decathlon evaluation: overall performance. While so far we have looked at results on individual
domain, the Decathlon score eq. (1) can be used to compare performance overall. As baseline error
rates in eq. (1), we double the error rates of the fully finetuned networks on each task. In this manner,
this 10? model achieves a score of 2,500 points (over 10,000 possible ones, see eq. (1)). The last
column of table 1 reports the scores achieved by the other architectures. As intended, the decathlon
score favors the methods that perform well overall, emphasizes their consistency rather than just their
average accuracy. For instance, although the Res. adapt. model (trained with single decay coefficient
for all domains) performs well in terms of average accuracy (73.88%), its decathlon score (2118) is
relatively low because the model performs poorly in DTD and Flowers. This also shows that, once
the weight decays are configured properly, our model achieves superior performance (2643 points) to
all the baselines using only 2? the capacity of a single ResNet.
8
Finally we show that using a higher capacity ResNet28 (12?, ResNet adapt. (large) in table 1), which
is comparable to 10 independent networks, significantly improves our results and outperforms the
finetuning baseline by 600 point in decathlon score. As a matter of fact, this model outperforms the
state-of-the-art [40] (81.2%) by 3.5 points in CIFAR100. In other cases, our performances are in
general in line to current state-of-the-art methods. When this is not the case, this is due to reduced
image resolution (ImageNet, Flower) or due to the choice of a specific video representation in UCF
(dynamic image).
6
Conclusions
As machine learning applications become more advanced and pervasive, building data representations
that work well for multiple problems will become increasingly important. In this paper, we have
introduced a simple architectural element, the residual adapter module, that allows compressing many
visual domains in relatively small residual networks, with substantial parameter sharing between
them. We have also shown that they allow addressing the forgetting problem, as well as adapting to
target domain for which different amounts of training data are available. Finally, we have introduced
a new multi-domain learning challenge, the Visual Decathlon, to allow a systematic comparison of
algorithms for multiple-domain learning.
Acknowledgments: This work acknowledges the support of Mathworks/DTA DFR02620 and ERC 677195IDIU.
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6,251 | 6,655 | Dykstra?s Algorithm, ADMM, and Coordinate
Descent: Connections, Insights, and Extensions
Ryan J. Tibshirani
Department of Statistics and Machine Learning Department
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
We study connections between Dykstra?s algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADMM,
and block coordinate descent. We prove that coordinate descent for a regularized
regression problem, in which the penalty is a separable sum of support functions,
is exactly equivalent to Dykstra?s algorithm applied to the dual problem. ADMM
on the dual problem is also seen to be equivalent, in the special case of two sets,
with one being a linear subspace. These connections, aside from being interesting
in their own right, suggest new ways of analyzing and extending coordinate descent. For example, from existing convergence theory on Dykstra?s algorithm over
polyhedra, we discern that coordinate descent for the lasso problem converges at
an (asymptotically) linear rate. We also develop two parallel versions of coordinate
descent, based on the Dykstra and ADMM connections.
1
Introduction
In this paper, we study two seemingly unrelated but closely connected convex optimization problems,
and associated algorithms. The first is the best approximation problem: given closed, convex sets
C1 , . . . , Cd ? Rn and y ? Rn , we seek the point in C1 ? ? ? ? ? Cd (assumed nonempty) closest to y,
and solve
minn ky ? uk22 subject to u ? C1 ? ? ? ? ? Cd .
(1)
u?R
The second problem is the regularized regression problem: given a response y ? Rn and predictors
X ? Rn?p , and a block decomposition Xi ? Rn?pi , i = 1, . . . , d of the columns of X (i.e., these
could be columns, or groups of columns), we build a working linear model by applying blockwise
regularization over the coefficients, and solve
minp
w?R
d
X
1
ky ? Xwk22 +
hi (wi ),
2
i=1
(2)
where hi : Rpi ? R, i = 1, . . . , d are convex functions, and we write wi ? Rpi , P
i = 1, . . . , d for the
d
appropriate block decomposition of a coefficient vector w ? Rp (so that Xw = i=1 Xi wi ).
Two well-studied algorithms for problems (1), (2) are Dykstra?s algorithm (Dykstra, 1983; Boyle and
Dykstra, 1986) and (block) coordinate descent (Warga, 1963; Bertsekas and Tsitsiklis, 1989; Tseng,
1990), respectively. The jumping-off point for our work in this paper is the following fact: these two
algorithms are equivalent for solving (1) and (2). That is, for a particular relationship between the
sets C1 , . . . , Cd and penalty functions h1 , . . . , hd , the problems (1) and (2) are duals of each other,
and Dykstra?s algorithm on the primal problem (1) is exactly the same as coordinate descent on the
dual problem (2). We provide details in Section 2.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
This equivalence between Dykstra?s algorithm and coordinate descent can be essentially found in
the optimization literature, dating back to the late 1980s, and possibly earlier. (We say ?essentially?
here because, to our knowledge, this equivalence has not been stated for a general regression matrix
X, and only in the special case X = I; but, in truth, the extension to a general matrix X is fairly
straightforward.) Though this equivalence has been cited and discussed in various ways over the
years, we feel that it is not as well-known as it should be, especially in light of the recent resurgence
of interest in coordinate descent methods. We revisit the connection between Dykstra?s algorithm
and coordinate descent, and draw further connections to a third method?the augmented Lagrangian
method of multipliers or ADMM (Glowinski and Marroco, 1975; Gabay and Mercier, 1976)?that
has also received a great deal of attention recently. While these basic connections are interesting in
their own right, they also have important implications for analyzing and extending coordinate descent.
Below we give a summary of our contributions.
1. We prove in Section 2 (under a particular relationship between C1 , . . . , Cd and h1 , . . . , hd )
that Dykstra?s algorithm for (1) is equivalent to block coordinate descent for (2). (This is a
mild generalization of the previously known connection when X = I.)
2. We also show in Section 2 that ADMM is closely connected to Dykstra?s algorithm, in that
ADMM for (1), when d = 2 and C1 is a linear subspace, matches Dykstra?s algorithm.
3. Leveraging existing results on the convergence of Dykstra?s algorithm for an intersection of
halfspaces, we establish in Section 3 that coordinate descent for the lasso problem has an
(asymptotically) linear rate of convergence, regardless of the dimensions of X (i.e., without
assumptions about strong convexity of the problem). We derive two different explicit forms
for the error constant, which shed light onto how correlations among the predictor variables
affect the speed of convergence.
4. Appealing to parallel versions of Dykstra?s algorithm and ADMM, we present in Section 4
two parallel versions of coordinate descent (each guaranteed to converge in full generality).
5. We extend in Section 5 the equivalence between coordinate descent and Dykstra?s algorithm
to the case of nonquadratic loss in (2), i.e., non-Euclidean projection in (1). This leads to a
Dykstra-based parallel version of coordinate descent for (separably regularized) problems
with nonquadratic loss, and we also derive an alternative ADMM-based parallel version of
coordinate descent for the same class of problems.
2
Preliminaries and connections
Dykstra?s algorithm. Dykstra?s algorithm was first proposed by Dykstra (1983), and was extended
to Hilbert spaces by Boyle and Dykstra (1986). Since these seminal papers, a number of works have
analyzed and extended Dykstra?s algorithm in various interesting ways. We will reference many of
these works in the coming sections, when we discuss connections between Dykstra?s algorithm and
other methods; for other developments, see the comprehensive books Deutsch (2001); Bauschke and
Combettes (2011) and review article Bauschke and Koch (2013).
Dykstra?s algorithm for the best approximation problem (1) can be described as follows. We initialize
u(0) = y, z (?d+1) = ? ? ? = z (0) = 0, and then repeat, for k = 1, 2, 3, . . .:
u(k) = PC[k] (u(k?1) + z (k?d) ),
z (k) = u(k?1) + z (k?d) ? u(k) ,
(3)
where PC (x) = argminc?C kx ? ck22 denotes the (Euclidean) projection of x onto a closed, convex
set C, and [?] denotes the modulo operator taking values in {1, . . . , d}. What differentiates Dykstra?s
algorithm from the classical alternating projections method of von Neumann (1950); Halperin (1962)
is the sequence of (what we may call) dual variables z (k) , k = 1, 2, 3, . . .. These track, in a cyclic
fashion, the residuals from projecting onto C1 , . . . , Cd . The simpler alternating projections method
will always converge to a feasible point in C1 ? ? ? ? ? Cd , but will not necessarily converge to the
solution in (1) unless C1 , . . . , Cd are subspaces (in which case alternating projections and Dykstra?s
algorithm coincide). Meanwhile, Dykstra?s algorithm converges in general (for any closed, convex
sets C1 , . . . , Cd with nonempty intersection, see, e.g., Boyle and Dykstra (1986); Han (1988); Gaffke
and Mathar (1989)). We note that Dykstra?s algorithm (3) can be rewritten in a different form, which
2
(0)
(0)
(0)
will be helpful for future comparisons. First, we initialize ud = y, z1 = ? ? ? = zd = 0, and then
repeat, for k = 1, 2, 3, . . .:
(k)
(k?1)
u0 = ud
(k)
ui
(k)
zi
,
(k)
(k?1)
= PCi (ui?1 + zi
=
(k)
ui?1
+
(k?1)
zi
?
)
),
(k)
ui ,
for i = 1, . . . , d.
(4)
Coordinate descent. Coordinate descent methods have a long history in optimization, and have
been studied and discussed in early papers and books such as Warga (1963); Ortega and Rheinboldt
(1970); Luenberger (1973); Auslender (1976); Bertsekas and Tsitsiklis (1989), though coordinate
descent was still likely in use much earlier. (Of course, for solving linear systems, coordinate descent
reduces to Gauss-Seidel iterations, which dates back to the 1800s.) Some key papers analyzing the
convergence of coordinate descent methods are Tseng and Bertsekas (1987); Tseng (1990); Luo and
Tseng (1992, 1993); Tseng (2001). In the last 10 or 15 years, a considerable interest in coordinate
descent has developed across the optimization community. With the flurry of recent work, it would be
difficult to give a thorough account of the recent progress on the topic. To give just a few examples,
recent developments include finite-time (nonasymptotic) convergence rates for coordinate descent,
and exciting extensions such as accelerated, parallel, and distributed versions of coordinate descent.
We refer to Wright (2015), an excellent survey that describes this recent progress.
In (block) coordinate descent1 for (2), we initialize say w(0) = 0, and repeat, for k = 1, 2, 3, . . .:
2
X
X
1
(k)
(k)
(k?1)
wi = argmin
y
?
X
w
?
X
w
?
X
w
i = 1, . . . , d. (5)
j j
j j
i i
+ hi (wi ),
wi ?Rpi 2
2
j<i
j>i
We assume here and throughout that Xi ? Rn?pi , i = 1, . . . , d each have full column rank so that
the updates in (5) are uniquely defined (this is used for convenience, and is not a strong assumption;
note that there is no restriction on the dimensionality of the full problem in (2), i.e., we could still
have X ? Rn?p with p n). The precise form of these updates, of course, depends on the penalty
functions. Suppose that each hi is the support function of a closed, convex set Di ? Rpi , i.e.,
hi (v) = max hd, vi,
d?Di
for i = 1, . . . , d.
Suppose also that Ci = (XiT )?1 (Di ) = {v ? Rn : XiT v ? Di }, the inverse image of Di under the
linear map XiT , for i = 1, . . . , d. Then, perhaps surprisingly, it turns out that the coordinate descent
iterations (5) are exactly the same as the Dykstra iterations (4), via a duality argument. We extract
the key relationship as a lemma below, for future reference, and then state the formal equivalence.
Proofs of these results, as with all results in this paper, are given in the supplement.
Lemma 1. Assume that Xi ? Rn?pi has full column rank and hi (v) = maxd?Di hd, vi for a closed,
convex set Di ? Rpi . Then for Ci = (XiT )?1 (Di ) ? Rn and any b ? Rn ,
w
?i = argmin
wi ?Rpi
1
kb ? Xi wi k22 + hi (wi ) ?? Xi w
?i = (Id ? PCi )(b).
2
where Id(?) denotes the identity mapping.
Theorem 1. Assume the setup in Lemma 1, for each i = 1, . . . , d. Then problems (1), (2) are dual to
each other, and their solutions, denoted u
?, w,
? respectively, satisfy u
? = y ? X w.
? Further, Dykstra?s
algorithm (4) and coordinate descent (5) are equivalent, and satisfy at all iterations k = 1, 2, 3, . . .:
X
X
(k)
(k)
(k)
(k)
(k?1)
zi = Xi wi
and ui = y ?
Xj wj ?
Xj wj
, for i = 1, . . . , d.
j>i
j?i
The equivalence between coordinate descent and Dykstra?s algorithm dates back to (at least) Han
(1988); Gaffke and Mathar (1989), under the special case X = I. In fact, Han (1988), presumably
unaware of Dykstra?s algorithm, seems to have reinvented the method and established convergence
1
To be precise, this is cyclic coordinate descent, where exact minimization is performed along each block of
coordinates. Randomized versions of this algorithm have recently become popular, as have inexact or proximal
versions. While these variants are interesting, they are not the focus of our paper.
3
through its relationship to coordinate descent. This work then inspired Tseng (1993) (who must have
also been unaware of Dykstra?s algorithm) to improve the existing analyses of coordinate descent,
which at the time all assumed smoothness of the objective function. (Tseng continued on to become
arguably the single most important contributor to the theory of coordinate descent of the 1990s and
2000s, and his seminal work Tseng (2001) is still one of the most comprehensive analyses to date.)
References to this equivalence can be found speckled throughout the literature on Dykstra?s method,
but given the importance of the regularized problem form (2) for modern statistical and machine
learning estimation tasks, we feel that the connection between Dykstra?s algorithm and coordinate
descent and is not well-known enough and should be better explored. In what follows, we show that
some old work on Dykstra?s algorithm, fed through this equivalence, yields new convergence results
for coordinate descent for the lasso and a new parallel version of coordinate descent.
ADMM. The augmented Lagrangian method of multipliers or ADMM was invented by Glowinski
and Marroco (1975); Gabay and Mercier (1976). ADMM is a member of a class of methods generally
called operator splitting techniques, and is equivalent (via a duality argument) to Douglas-Rachford
splitting (Douglas and Rachford, 1956; Lions and Mercier, 1979). Recently, there has been a strong
revival of interest in ADMM (and operator splitting techniques in general), arguably due (at least in
part) to the popular monograph of Boyd et al. (2011), where it is argued that the ADMM framework
offers an appealing flexibility in algorithm design, which permits parallelization in many nontrivial
situations. As with coordinate descent, it would be difficult thoroughly describe recent developments
on ADMM, given the magnitude and pace of the literature on this topic. To give just a few examples,
recent progress includes finite-time linear convergence rates for ADMM (see Nishihara et al. 2015;
Hong and Luo 2017 and references therein), and accelerated extensions of ADMM (see Goldstein
et al. 2014; Kadkhodaie et al. 2015 and references therein).
To derive an ADMM algorithm for (1), we introduce auxiliary variables and equality constraints to
put the problem in a suitable ADMM form. While different formulations for the auxiliary variables
and constraints give rise to different algorithms, loosely speaking, these algorithms generally take on
similar forms to Dykstra?s algorithm for (1). The same is also true of ADMM for the set intersection
problem, a simpler task than the best approximation problem (1), in which we only seek a point in
the intersection C1 ? ? ? ? ? Cd , and solve
d
X
minn
ICi (ui ),
(6)
u?R
i=1
where IC (?) denotes the indicator function of a set C (equal to 0 on C, and ? otherwise). Consider
the case of d = 2 sets, in which case the translation of (6) into ADMM form is unambiguous. ADMM
for (6), properly initialized, appears highly similar to Dykstra?s algorithm for (1); so similar, in fact,
that Boyd et al. (2011) mistook the two algorithms for being equivalent, which is not generally true,
and was shortly thereafter corrected by Bauschke and Koch (2013).
Below we show that when d = 2, C1 is a linear subspace, and y ? C1 , an ADMM algorithm for (1)
(and not the simpler set intersection problem (6)) is indeed equivalent to Dykstra?s algorithm for (1).
Introducing auxiliary variables, the problem (1) becomes
min n ky ? u1 k22 + IC1 (u1 ) + IC2 (u2 ) subject to u1 = u2 .
u1 ,u2 ?R
The augmented Lagrangian is L(u1 , u2 , z) = ky ? u1 k22 + IC1 (u1 ) + IC2 (u2 ) + ?ku1 ? u2 + zk22
??kzk22 , where ? > 0 is an augmented Lagrangian parameter. ADMM repeats, for k = 1, 2, 3, . . .:
(k?1)
? z (k?1) )
y
?(u2
(k)
u1 = PC1
+
,
1+?
1+?
(7)
(k)
(k)
u2 = PC2 (u1 + z (k?1) ),
(k)
(k)
z (k) = z (k?1) + u1 ? u2 .
Suppose we initialize
= y, z (0) = 0, and set ? = 1. Using linearity of PC1 , the fact that y ? C1 ,
and a simple inductive argument, the above iterations can be rewritten as
(0)
u2
(k)
(k?1)
u1 = PC1 (u2
(k)
u2
=
(k)
PC2 (u1
),
+ z (k?1) ),
(k)
(k)
z (k) = z (k?1) + u1 ? u2 ,
4
(8)
which is precisely the same as Dykstra?s iterations (4), once we realize that, due again to linearity of
(k)
PC1 , the sequence z1 , k = 1, 2, 3, . . . in Dykstra?s iterations plays no role and can be ignored.
Though d = 2 sets in (1) may seem like a rather special case, the strategy for parallelization in both
Dykstra?s algorithm and ADMM stems from rewriting a general d-set problem as a 2-set problem, so
the above connection between Dykstra?s algorithm and ADMM can be relevant even for problems
with d > 2, and will reappear in our later discussion of parallel coordinate descent. As a matter of
conceptual interest only, we note that for general d (and no constraints on the sets being subspaces),
Dykstra?s iterations (4) can be viewed as a limiting version of the ADMM iterations either for (1) or
for (6), as we send the augmented Lagrangian parameters to ? or to 0 at particular scalings. See the
supplement for details.
3
Coordinate descent for the lasso
The lasso problem (Tibshirani, 1996; Chen et al., 1998), defined for a tuning parameter ? ? 0 as
1
min ky ? Xwk22 + ?kwk1 ,
(9)
w?Rp 2
is a special case of (2) where the coordinate blocks are of each size 1, so that Xi ? Rn , i = 1, . . . , p
are just the columns of X, and wi ? R, i = 1, . . . , p are the components of w. This problem fits into
the framework of (2) with hi (wi ) = ?|wi | = maxd?Di dwi for Di = [??, ?], for each i = 1, . . . , d.
Coordinate descent is widely-used for the lasso (9), both because of the simplicity of the coordinatewise updates, which reduce to soft-thresholding, and because careful implementations can achieve
state-of-the-art performance, at the right problem sizes. The use of coordinate descent for the lasso
was popularized by Friedman et al. (2007, 2010), but was studied earlier or concurrently by several
others, e.g., Fu (1998); Sardy et al. (2000); Wu and Lange (2008).
As we know from Theorem 1, the dual of problem (9) is the best approximation problem (1), where
Ci = (XiT )?1 (Di ) = {v ? Rn : |XiT v| ? ?} is an intersection of two halfspaces, for i = 1, . . . , p.
This makes C1 ? ? ? ? ? Cd an intersection of 2p halfspaces, i.e., a (centrally symmetric) polyhedron.
For projecting onto a polyhedron, it is well-known that Dykstra?s algorithm reduces to Hildreth?s
algorithm (Hildreth, 1957), an older method for quadratic programming that itself has an interesting
history in optimization. Theorem 1 hence shows coordinate descent for the lasso (9) is equivalent not
only to Dykstra?s algorithm, but also to Hildreth?s algorithm, for (1).
This equivalence suggests a number of interesting directions to consider. For example, key practical
speedups have been developed for coordinate descent for the lasso that enable this method to attain
state-of-the-art performance at the right problem sizes, such as clever updating rules and screening
rules (e.g., Friedman et al. 2010; El Ghaoui et al. 2012; Tibshirani et al. 2012; Wang et al. 2015).
These implementation tricks can now be used with Dykstra?s (Hildreth?s) algorithm. On the flip side,
as we show next, older results from Iusem and De Pierro (1990); Deutsch and Hundal (1994) on
Dykstra?s algorithm for polyhedra, lead to interesting new results on coordinate descent for the lasso.
Theorem 2 (Adaptation of Iusem and De Pierro 1990). Assume the columns of X ? Rn?p are in
general position, and ? > 0. Then coordinate descent for the lasso (9) has an asymptotically linear
convergence rate, in that for large enough k,
1/2
a2
kw(k+1) ? wk
? ?
?
,
(10)
T X )/ max
2
kw(k) ? wk
? ?
a2 + ?min (XA
A
i?A kXi k2
where w
? is the lasso solution in (9), ? = X T X, and kzk2? = z T ?z for z ? Rp , A = supp(w)
? is
the active set of w,
? a = |A| is its size, XA ? Rn?a denotes the columns of X indexed by A, and
T
T
?min (XA
XA ) denotes the smallest eigenvalue of XA
XA .
Theorem 3 (Adaptation of Deutsch and Hundal 1994). Assume the same conditions and notation
as in Theorem 2. Then for large enough k,
!1/2
?
a?1
Y kP{i
Xij k22
kw(k+1) ? wk
? ?
j+1 ,...,ia }
? 1?
,
(11)
kXij k22
kw(k) ? wk
? ?
j=1
?
where we enumerate A = {i1 , . . . , ia }, i1 < . . . < ia , and we denote by P{i
the projection
j+1 ,...,ia }
onto the orthocomplement of the column span of X{ij+1 ,...,ia } .
5
The results in Theorems 2, 3 both rely on the assumption of general position for the columns of X.
This is only used for convenience and can be removed at the expense of more complicated notation.
Loosely put, the general position condition simply rules out trivial linear dependencies between small
numbers of columns of X, but places no restriction on the dimensions of X (i.e., it still allows for
p n). It implies that the lasso solution w
? is unique, and that XA (where A = supp(w))
? has full
column rank. See Tibshirani (2013) for a precise definition of general position and proofs of these
facts. We note that when XA has full column rank, the bounds in (10), (11) are strictly less than 1.
Remark 1 (Comparing (10) and (11)). Clearly, both the bounds in (10), (11) are adversely affected
by correlations among Xi , i ? A (i.e., stronger correlations will bring each closer to 1). It seems to
us that (11) is usually the smaller of the two bounds, based on simple mathematical and numerical
comparisons. More detailed comparisons would be interesting, but is beyond the scope of this paper.
Remark 2 (Linear convergence without strong convexity). One striking feature of the results in
Theorems 2, 3 is that they guarantee (asymptotically) linear convergence of the coordinate descent
iterates for the lasso, with no assumption about strong convexity of the objective. More precisely,
there are no restrictions on the dimensionality of X, so we enjoy linear convergence even without an
assumption on the smooth part of the objective. This is in line with classical results on coordinate
descent for smooth functions, see, e.g., Luo and Tseng (1992). The modern finite-time convergence
analyses of coordinate descent do not, as far as we understand, replicate this remarkable property.
For example, Beck and Tetruashvili (2013); Li et al. (2016) establish finite-time linear convergence
rates for coordinate descent, but require strong convexity of the entire objective.
Remark 3 (Active set identification). The asymptotics developed in Iusem and De Pierro (1990);
Deutsch and Hundal (1994) are based on a notion of (in)active set identification: the critical value of
k after which (10), (11) hold is based on the (provably finite) iteration number at which Dykstra?s
algorithm identifies the inactive halfspaces, i.e., at which coordinate descent identifies the inactive
set of variables, Ac = supp(w)
? c . This might help explain why in practice coordinate descent for the
lasso performs exceptionally well with warm starts, over a decreasing sequence of tuning parameter
values ? (e.g., Friedman et al. 2007, 2010): here, each coordinate descent run is likely to identify the
(in)active set?and hence enter the linear convergence phase?at an early iteration number.
4
Parallel coordinate descent
Parallel-Dykstra-CD. An important consequence of the connection between Dykstra?s algorithm
and coordinate descent is a new parallel version of the latter, stemming from an old parallel version
of the former. A parallel version of Dykstra?s algorithm is usually credited to Iusem and Pierro (1987)
for polyhedra and Gaffke and Mathar (1989) for general sets, but really the idea dates back to the
product space formalization of Pierra (1984). We rewrite problem (1) as
d
X
min
u=(u1 ,...,ud )?Rnd
?i ky ? ui k22
subject to
u ? C0 ? (C1 ? ? ? ? ? Cd ),
(12)
i=1
where C0 = {(u1 , . . . , ud ) ? Rnd : u1 = ? ? ? = ud }, and ?1 , . . . , ?d > 0 are weights that sum to 1.
After rescaling appropriately to turn (12) into an unweighted best approximation problem, we can
(0)
(0)
(0)
(0)
apply Dykstra?s algorithm, which sets u1 = ? ? ? = ud = y, z1 = ? ? ? = zd = 0, and repeats:
(k)
u0 =
d
X
(k?1)
?i ui
,
i=1
(k)
= PCi (u0 + zi
(k)
= u0 + zi
ui
zi
(k)
(k)
(k?1)
(k?1)
),
(k)
(13)
)
for i = 1, . . . , d,
? ui ,
for k = 1, 2, 3, . . .. The steps enclosed in curly brace above can all be performed in parallel, so that
(13) is a parallel version of Dykstra?s algorithm (4) for (1). Applying Lemma 1, and a straightforward
inductive argument, the above algorithm can be rewritten as follows. We set w(0) = 0, and repeat:
2
1
(k)
(k?1)
wi = argmin
y ? Xw(k?1) + Xi wi
/?i ? Xi wi /?i
+ hi (wi /?i ), i = 1, . . . , d, (14)
2
wi ?Rpi 2
for k = 1, 2, 3, . . ., which we call parallel-Dykstra-CD (with CD being short for coordinate descent).
Again, note that the each of the d coordinate updates in (14) can be performed in parallel, so that
6
(14) is a parallel version of coordinate descent (5) for (2). Also, as (14) is just a reparametrization of
Dykstra?s algorithm (13) for the 2-set problem (12), it is guaranteed to converge in full generality, as
per the standard results on Dykstra?s algorithm (Han, 1988; Gaffke and Mathar, 1989).
Theorem 4. Assume that Xi ? Rn?pi has full column rank and hi (v) = maxd?Di hd, vi for a closed,
convex set Di ? Rpi , for i = 1, . . . , d. If (2) has a unique solution, then the iterates in (14) converge
to this solution. More generally, if the interior of ?di=1 (XiT )?1 (Di ) is nonempty, then the sequence
w(k) , k = 1, 2, 3, . . . from (14) has at least one accumulation point, and any such point solves (2).
Further, Xw(k) , k = 1, 2, 3, . . . converges to X w,
? the optimal fitted value in (2).
There have been many recent exciting contributions to the parallel coordinate descent literature; two
standouts are Jaggi et al. (2014); Richtarik and Takac (2016), and numerous others are described in
Wright (2015). What sets parallel-Dykstra-CD apart, perhaps, is its simplicity: convergence of the
iterations (14), given in Theorem 4, just stems from the connection between coordinate descent and
Dykstra?s algorithm, and the fact that the parallel Dykstra iterations (13) are nothing more than the
usual Dykstra iterations after a product space reformulation. Moreover, parallel-Dykstra-CD for the
lasso enjoys an (asymptotic) linear convergence rate under essentially no assumptions, thanks once
again to an old result on the parallel Dykstra (Hildreth) algorithm from Iusem and De Pierro (1990).
The details can be found in the supplement.
Parallel-ADMM-CD. As an alternative to the parallel method derived using Dykstra?s algorithm,
ADMM can also offer a version of parallel coordinate descent. Since (12) is a best approximation
problem with d = 2 sets, we can refer back to our earlier ADMM algorithm in (7) for this problem.
By passing these ADMM iterations through the connection developed in Lemma 1, we arrive at what
(0)
we call parallel-ADMM-CD, which initializes u0 = y, w(?1) = w(0) = 0, and repeats:
Pd
(k?1)
( i=1 ?i )u0
y ? Xw(k?1)
X(w(k?2) ? w(k?1) )
(k)
u0 =
+
+
,
Pd
Pd
Pd
1 + i=1 ?i
1 + i=1 ?i
1 + i=1 ?i
(15)
2
1
(k)
(k?1)
(k)
/?i ? Xi wi /?i
+ hi (wi /?i ), i = 1, . . . , d,
wi = argmin
u0 + Xi wi
2
wi ?Rpi 2
for k = 1, 2, 3, . . ., where ?1 , . . . , ?d > 0 are augmented Lagrangian parameters. In each iteration,
(k)
the updates to wi , i = 1, . . . , d above can be done in parallel. Just based on their form, it seems
that (15) can be seen as a parallel version of coordinate descent (5) for problem (2). The next result
confirms this, leveraging standard theory for ADMM (Gabay, 1983; Eckstein and Bertsekas, 1992).
Theorem 5. Assume that Xi ? Rn?pi has full column rank and hi (v) = maxd?Di hd, vi for a closed,
convex set Di ? Rpi , for i = 1, . . . , d. Then the sequence w(k) , k = 1, 2, 3, . . . in (15) converges to
a solution in (2).
The parallel-ADMM-CD iterations in (15) and parallel-Dykstra-CD iterations in (14) differ in that,
(k)
where the latter uses a residual y ? Xw(k?1) , the former uses an iterate u0 that seems to have a
(k?1)
more complicated form, being a convex combination of u0
and y ? Xw(k?1) , plus a quantity
that acts like a momentum term. It turns out that when ?1 , . . . , ?d sum to 1, the two methods (14),
(15) are exactly the same. While this may seem like a surprising coincidence, it is in fact nothing
more than a reincarnation of the previously established equivalence between Dykstra?s algorithm (4)
and ADMM (8) for a 2-set best approximation problem, as here C0 is a linear subspace.
Of course, with ADMM we need not choose probability weights for ?1 , . . . , ?d , and the convergence
in Theorem 5 is guaranteed for any fixed values of these parameters. Thus, even though they were
derived from different perspectives, parallel-ADMM-CD subsumes parallel-Dykstra-CD, and it is a
strictly more general approach. It is important to note that larger values of ?1 , . . . , ?d can often lead
to faster convergence in practice, as we show in Figure 1. More detailed study and comparisons to
related parallel methods are worthwhile, but are beyond the scope of this work.
5
Discussion and extensions
We studied connections between Dykstra?s algorithm, ADMM, and coordinate descent. Leveraging
these connections, we established an (asymptotically) linear convergence rate for coordinate descent
for the lasso, as well as two parallel versions of coordinate descent (one based on Dykstra?s algorithm
and the other on ADMM). Some extensions and possibilities for future work are described below.
7
No parallelization
1e+01
1e+04
Coordinate descent
Par?Dykstra?CD
Par?ADMM?CD, rho=10
Par?ADMM?CD, rho=50
Par?ADMM?CD, rho=200
1e?02
1e?08
1e?08
1e?05
1e?02
Suboptimality
1e+01
1e+04
Coordinate descent
Par?Dykstra?CD
Par?ADMM?CD, rho=10
Par?ADMM?CD, rho=50
Par?ADMM?CD, rho=200
1e?05
Suboptimality
10% parallelization
0
500
1000
1500
2000
0
50
Actual iteration number
100
150
Effective iteration number
Figure 1: Suboptimality curves for serial coordinate descent,
Ppparallel-Dykstra-CD, and three tunings
of parallel-ADMM-CD (i.e., three different values of ? = i=1 ?i ), each run over the same 30 lasso
problems with n = 100 and p = 500. For details of the experimental setup, see the supplement.
Nonquadratic loss: Dykstra?s algorithm and coordinate descent. Given a convex function f , a
generalization of (2) is the regularized estimation problem
minp f (Xw) +
w?R
d
X
hi (wi ).
(16)
i=1
Regularized regression (2) is given byP
f (z) = 12 ky ? zk22 , and e.g., regularized classification (under
n
T
the logistic loss) by f (z) = ?y z + i=1 log(1 + ezi ). In (block) coordinate descent for (16), we
(0)
initialize say w = 0, and repeat, for k = 1, 2, 3, . . .:
X
X
(k)
(k)
(k?1)
wi = argmin f
Xj wj +
Xj wj
+ Xi wi + hi (wi ), i = 1, . . . , d.
(17)
wi ?Rpi
j<i
j>i
On the other hand, given a differentiable and strictly convex function g, we can generalize (1) to the
following best Bregman-approximation problem,
min Dg (u, b)
u?Rn
subject to
u ? C1 ? ? ? ? ? Cd .
(18)
where Dg (u, b) = g(u) ? g(b) ? h?g(b), u ? bi is the Bregman divergence between u and b with
respect to g. When g(v) = 12 kvk22 (and b = y), this recovers the best approximation problem (1). As
shown in Censor and Reich (1998); Bauschke and Lewis (2000), Dykstra?s algorithm can be extended
(0)
(0)
(0)
to apply to (18). We initialize ud = b, z1 = ? ? ? = zd = 0, and repeat for k = 1, 2, 3, . . .:
(k)
(k?1)
u0 = ud
(k)
ui
(k)
zi
,
?
(k)
(k?1)
? ?g ? ) ?g(ui?1 ) + zi
, ?
=
(PCg i
=
(k)
?g(ui?1 )
+
(k?1)
zi
?
(k)
?g(ui ),
PCg (x)
for i = 1, . . . , d,
(19)
?
where
= argminc?C Dg (c, x) denotes the Bregman (rather than Euclidean) projection of x
onto a set C, and g ? is the conjugate function of g. Though it may not be immediately obvious, when
g(v) = 12 kvk22 the above iterations (19) reduce to the standard (Euclidean) Dykstra iterations in (4).
Furthermore, Dykstra?s algorithm and coordinate descent are equivalent in the more general setting.
Theorem 6. Let f be a strictly convex, differentiable function that has full domain. Assume that
Xi ? Rn?pi has full column rank and hi (v) = maxd?Di hd, vi for a closed, convex set Di ? Rpi , for
i = 1, . . . , d. Also, let g(v) = f ? (?v), b = ??f (0), and Ci = (XiT )?1 (Di ) ? Rn , i = 1, . . . , d.
8
Then (16), (18) are dual to each other, and their solutions w,
? u
? satisfy u
? = ??f (X w).
? Moreover,
Dykstra?s algorithm (19) and coordinate descent (17) are equivalent, i.e., for k = 1, 2, 3, . . .:
X
X
(k)
(k)
(k)
(k)
(k?1)
zi = Xi wi
and ui = ??f
Xj wj +
Xj wj
, for i = 1, . . . , d.
j>i
j?i
Nonquadratic loss: parallel coordinate descent methods. For a general regularized estimation
problem (16), parallel coordinate descent methods can be derived by applying Dykstra?s algorithm
and ADMM to a product space reformulation of the dual. Interestingly, the subsequent coordinate
descent algorithms are no longer equivalent (for a unity augmented Lagrangian parameter), and they
feature quite different computational structures. Parallel-Dykstra-CD for (16) initializes w(0) = 0,
and repeats:
(k)
(k)
wi = argmin f Xw(k) ? Xi wi /?i + Xi wi /?i + hi (wi /?i ), i = 1, . . . , d,
(20)
wi ?Rpi
for k = 1, 2, 3, . . ., and weights ?1 , . . . , ?d > 0 that sum to 1. In comparison, parallel-ADMM-CD
(0)
for (16) begins with u0 = 0, w(?1) = w(0) = 0, and repeats:
!
X
d
(k)
(k)
(k)
(k?1)
(k?2)
(k?1)
Find u0 such that: u0 = ??f
?i (u0 ? u0
) ? X(w
? 2w
) ,
i=1
(21)
2
1
(k)
(k)
(k?1)
wi = argmin
u0 + Xi wi
/?i ? Xi wi /?i
+ hi (wi /?i ), i = 1, . . . , d,
2
wi ?Rpi 2
for k = 1, 2, 3, . . ., and parameters ?1 , . . . , ?d > 0. Derivation details are given in the supplement.
Notice the stark contrast between the parallel-Dykstra-CD iterations (20) and the parallel-ADMMCD iterations (21). In (20), we perform (in parallel) coordinatewise hi -regularized minimizations
involving f , for i = 1, . . . , d. In (21), we perform a single quadratically-regularized minimization
involving f for the u0 -update, and then for the w-update, we perform (in parallel) coordinatewise
hi -regularized minimizations involving a quadratic loss, for i = 1, . . . , d (these are typically much
cheaper than the analogous minimizations for typical nonquadratic losses f of interest).
We note that the u0 -update in the parallel-ADMM-CD iterations (21) simplifies
Pn for many losses f
of interest; in particular, for separable loss functions of the form f (v) = i=1 fi (vi ), for convex,
univariate functions fi , i = 1, . . . , n, the u0 -update separates into n univariate minimizations. As an
example, consider the logistic lasso problem,
n
X
T
T
minp ?y Xw +
log(1 + exi w ) + ?kwk1 ,
(22)
w?R
i=1
Pp
where xi ? Rp , i = 1, . . . , n denote the rows of X. Abbreviating ? = i=1 ?i , and denoting by
?x
?(x) = 1/(1 + e ) the sigmoid function, and by St (x) = sign(x)(|x| ? t)+ the soft-thresholding
function at a level t > 0, the parallel-ADMM-CD iterations (21) for (22) reduce to:
(k)
(k)
(k?1)
Find u0i such that: (? ? 1)u0i = ?u0i
+ xTi (w(k?2) ? 2w(k?1) ) +
(k)
(k?1)
? ?u0i ? ?u0i
? xTi (w(k?2) ? 2w(k?1) ) , i = 1, . . . , n,
(23)
(k)
(k?1)
T
?i Xi (u0 + Xi wi
/?i )
(k)
, i = 1, . . . , p,
wi = S??i /kXi k22
kXi k22
for k = 1, 2, 3, . . .. Now we see that both the u0 -update and w-update in (23) can be parallelized,
and each coordinate update in the former can be done with, say, a simple bisection search.
Asynchronous parallel algorithms, and coordinate descent in Hilbert spaces. We finish with
some directions for possible future work. Asynchronous variants of parallel coordinate descent are
currently of great interest, e.g., see the review in Wright (2015). Given the link between ADMM and
coordinate descent developed in this paper, it would be interesting to investigate the implications of
the recent exciting progress on asynchronous ADMM, e.g., see Chang et al. (2016a,b) and references
therein, for coordinate descent. In a separate direction, much of the literature on Dykstra?s algorithm
emphasizes that this method works seamlessly in Hilbert spaces. It would be interesting to consider
the connections to (parallel) coordinate descent in infinite-dimensional function spaces, which we
would encounter, e.g., in alternating conditional expectation algorithms or backfitting algorithms in
additive models.
9
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12
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6,252 | 6,656 | Learning Spherical Convolution
for Fast Features from 360? Imagery
Yu-Chuan Su
Kristen Grauman
The University of Texas at Austin
Abstract
While 360? cameras offer tremendous new possibilities in vision, graphics, and
augmented reality, the spherical images they produce make core feature extraction non-trivial. Convolutional neural networks (CNNs) trained on images from
perspective cameras yield ?flat" filters, yet 360? images cannot be projected to a
single plane without significant distortion. A naive solution that repeatedly projects
the viewing sphere to all tangent planes is accurate, but much too computationally
intensive for real problems. We propose to learn a spherical convolutional network
that translates a planar CNN to process 360? imagery directly in its equirectangular projection. Our approach learns to reproduce the flat filter outputs on 360?
data, sensitive to the varying distortion effects across the viewing sphere. The key
benefits are 1) efficient feature extraction for 360? images and video, and 2) the
ability to leverage powerful pre-trained networks researchers have carefully honed
(together with massive labeled image training sets) for perspective images. We
validate our approach compared to several alternative methods in terms of both raw
CNN output accuracy as well as applying a state-of-the-art ?flat" object detector
to 360? data. Our method yields the most accurate results while saving orders of
magnitude in computation versus the existing exact reprojection solution.
1
Introduction
Unlike a traditional perspective camera, which samples a limited field of view of the 3D scene
projected onto a 2D plane, a 360? camera captures the entire viewing sphere surrounding its optical
center, providing a complete picture of the visual world?an omnidirectional field of view. As such,
viewing 360? imagery provides a more immersive experience of the visual content compared to
traditional media.
360? cameras are gaining popularity as part of the rising trend of virtual reality (VR) and augmented
reality (AR) technologies, and will also be increasingly influential for wearable cameras, autonomous
mobile robots, and video-based security applications. Consumer level 360? cameras are now common
on the market, and media sharing sites such as Facebook and YouTube have enabled support for
360? content. For consumers and artists, 360? cameras free the photographer from making real-time
composition decisions. For VR/AR, 360? data is essential to content creation. As a result of this great
potential, computer vision problems targeting 360? content are capturing the attention of both the
research community and application developer.
Immediately, this raises the question: how to compute features from 360? images and videos?
Arguably the most powerful tools in computer vision today are convolutional neural networks (CNN).
CNNs are responsible for state-of-the-art results across a wide range of vision problems, including
image recognition [14, 39], object detection [11, 27], image and video segmentation [13, 18, 25], and
action detection [10, 29]. Furthermore, significant research effort over the last five years (and really
decades [24]) has led to well-honed CNN architectures that, when trained with massive labeled image
datasets [8], produce ?pre-trained" networks broadly useful as feature extractors for new problems.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
?
Strategy I
Equirectangular
Projection
?
Input: 360? image
?
Np
???
Fully Convolution
?
Sample n
???
?
???
Perspective
Projection
Output
Np
???
Np
???
Np
Strategy II
Figure 1: Two existing strategies for applying CNNs to 360? images. Top: The first strategy unwraps the
360? input into a single planar image using a global projection (most commonly equirectangular projection),
then applies the CNN on the distorted planar image. Bottom: The second strategy samples multiple tangent
planar projections to obtain multiple perspective images, to which the CNN is applied independently to obtain
local results for the original 360? image. Strategy I is fast but inaccurate; Strategy II is accurate but slow. The
proposed approach learns to replicate flat filters on spherical imagery, offering both speed and accuracy.
Indeed such networks are widely adopted as off-the-shelf feature extractors for other algorithms and
applications (c.f., VGG [30], ResNet [14], and AlexNet [22] for images; C3D [33] for video).
However, thus far, powerful CNN features are awkward if not off limits in practice for 360? imagery.
The problem is that the underlying projection models of current CNNs and 360? data are different.
Both the existing CNN filters and the expensive training data that produced them are ?flat", i.e., the
product of perspective projection to a plane. In contrast, a 360? image is projected onto the unit
sphere surrounding the camera?s optical center.
To address this discrepancy, there are two common, though flawed, approaches. In the first, the
spherical image is projected to a planar one,1 then the CNN is applied to the resulting 2D image [16,23]
(see Fig. 1, top). However, any sphere-to-plane projection introduces distortion, making the resulting
convolutions inaccurate. In the second existing strategy, the 360? image is repeatedly projected
to tangent planes around the sphere, each of which is then fed to the CNN [31, 32, 35, 38] (Fig. 1,
bottom). In the extreme of sampling every tangent plane, this solution is exact and therefore accurate.
However, it suffers from very high computational cost. Not only does it incur the cost of rendering
each planar view, but also it prevents amortization of convolutions: the intermediate representation
cannot be shared across perspective images because they are projected to different planes.
We propose a learning-based solution that, unlike the existing strategies, sacrifices neither accuracy
nor efficiency. The main idea is to learn a CNN that processes a 360? image in its equirectangular
projection (fast) but mimics the ?flat" filter responses that an existing network would produce on
all tangent plane projections for the original spherical image (accurate). Because convolutions are
indexed by spherical coordinates, we refer to our method as spherical convolution (S PH C ONV). We
develop a systematic procedure to adjust the network structure in order to account for distortions.
Furthermore, we propose a kernel-wise pre-training procedure which significantly accelerates the
training process.
In addition to providing fast general feature extraction for 360? imagery, our approach provides a
bridge from 360? content to existing heavily supervised datasets dedicated to perspective images.
In particular, training requires no new annotations?only the target CNN model (e.g., VGG [30]
pre-trained on millions of labeled images) and an arbitrary collection of unlabeled 360? images.
We evaluate S PH C ONV on the Pano2Vid [32] and PASCAL VOC [9] datasets, both for raw convolution accuracy as well as impact on an object detection task. We show that it produces more precise
outputs than baseline methods requiring similar computational cost, and similarly precise outputs as
the exact solution while using orders of magnitude less computation. Furthermore, we demonstrate
that S PH C ONV can successfully replicate the widely used Faster-RCNN [27] detector on 360? data
when training with only 1,000 unlabeled 360? images containing unrelated objects. For a similar cost
as the baselines, S PH C ONV generates better object proposals and recognition rates.
1
e.g., with equirectangular projection, where latitudes are mapped to horizontal lines of uniform spacing
2
2
Related Work
360? vision Vision for 360? data is quickly gaining interest in recent years. The SUN360 project
samples multiple perspective images to perform scene viewpoint recognition [35]. PanoContext [38]
parses 360? images using 3D bounding boxes, applying algorithms like line detection on perspective
images then backprojecting results to the sphere. Motivated by the limitations of existing interfaces
for viewing 360? video, several methods study how to automate field-of-view (FOV) control for
display [16, 23, 31, 32], adopting one of the two existing strategies for convolutions (Fig. 1). In these
methods, a noted bottleneck is feature extraction cost, which is hampered by repeated sampling of
perspective images/frames, e.g., to represent the space-time ?glimpses" of [31, 32]. This is exactly
where our work can have positive impact.
Knowledge distillation Our approach relates to knowledge distillation [3, 5, 12, 15, 26, 28, 34],
though we explore it in an entirely novel setting. Distillation aims to learn a new model given existing
model(s). Rather than optimize an objective function on annotated data, it learns the new model
that can reproduce the behavior of the existing model, by minimizing the difference between their
outputs. Most prior work explores distillation for model compression [3, 5, 15, 28]. For example,
a deep network can be distilled into a shallower [3] or thinner [28] one, or an ensemble can be
compressed to a single model [15]. Rather than compress a model in the same domain, our goal is to
learn across domains, namely to link networks on images with different projection models. Limited
work considers distillation for transfer [12, 26]. In particular, unlabeled target-source paired data can
help learn a CNN for a domain lacking labeled instances (e.g., RGB vs. depth images) [12], and
multi-task policies can be learned to simulate action value distributions of expert policies [26]. Our
problem can also be seen as a form of transfer, though for a novel task motivated strongly by image
processing complexity as well as supervision costs. Different from any of the above, we show how to
adapt the network structure to account for geometric transformations caused by different projections.
Also, whereas most prior work uses only the final output for supervision, we use the intermediate
representation of the target network as both input and target output to enable kernel-wise pre-training.
Spherical image projection Projecting a spherical image into a planar image is a long studied
problem. There exists a large number of projection approaches (e.g., equirectangular, Mercator,
etc.) [4]. None is perfect; every projection must introduce some form of distortion. The properties of
different projections are analyzed in the context of displaying panoramic images [37]. In this work,
we unwrap the spherical images using equirectangular projection because 1) this is a very common
format used by camera vendors and researchers [1, 32, 35], and 2) it is equidistant along each row and
column so the convolution kernel does not depend on the azimuthal angle. Our method in principle
could be applied to other projections; their effect on the convolution operation remains to be studied.
CNNs with geometric transformations There is an increasing interest in generalizing convolution in CNNs to handle geometric transformations or deformations. Spatial transformer networks
(STNs) [17] represent a geometric transformation as a sampling layer and predict the transformation
parameters based on input data. STNs assume the transformation is invertible such that the subsequent
convolution can be performed on data without the transformation. This is not possible in spherical images because it requires a projection that introduces no distortion. Active convolution [19]
learns the kernel shape together with the weights for a more general receptive field, and deformable
convolution [7] goes one step further by predicting the receptive field location. These methods are
too restrictive for spherical convolution, because they require a fixed kernel size and weight. In
contrast, our method adapts the kernel size and weight based on the transformation to achieve better
accuracy. Furthermore, our method exploits problem-specific geometric information for efficient
training and testing. Some recent work studies convolution on a sphere [6, 21] using spectral analysis,
but those methods require manually annotated spherical images as training data, whereas our method
can exploit existing models trained on perspective images as supervision. Also, it is unclear whether
CNNs in the spectral domain can reach the same accuracy and efficiency as CNNs on a regular grid.
3
Approach
We describe how to learn spherical convolutions in equirectangular projection given a target network
trained on perspective images. We define the objective in Sec. 3.1. Next, we introduce how to adapt
the structure from the target network in Sec. 3.2. Finally, Sec. 3.3 presents our training process.
3
? = 180?
? = 108?
? = 36?
Figure 2: Inverse perspective projections P ?1 to equirectangular projections at different polar angles ?. The
same square image will distort to different sizes and shapes depending on ?. Because equirectangular projection
unwraps the 180? longitude, a line will be split into two if it passes through the 180? longitude, which causes
the double curve in ? = 36?.
3.1
Problem Definition
Let Is be the input spherical image defined on spherical coordinates (?, ?), and let Ie ? IWe ?He ?3
be the corresponding flat RGB image in equirectangular projection. Ie is defined by pixels on the
image coordinates (x, y) ? De , where each (x, y) is linearly mapped to a unique (?, ?). We define
the perspective projection operator P which projects an ?-degree field of view (FOV) from Is to
W pixels on the the tangent plane n
? = (?, ?). That is, P(Is , n
? ) = Ip ? IW ?W ?3 . The projection
operator is characterized by the pixel size ?p ? = ?/W in Ip , and Ip denotes the resulting perspective
image. Note that we assume ?? = ?? following common digital imagery.
Given a target network2 Np trained on perspective images Ip with receptive field (Rf) R ? R, we
define the output on spherical image Is at n
? = (?, ?) as
Np (Is )[?, ?] = Np (P(Is , (?, ?))),
(1)
where w.l.o.g. we assume W = R for simplicity. Our goal is to learn a spherical convolution network
Ne that takes an equirectangular map Ie as input and, for every image position (x, y), produces as
output the results of applying the perspective projection network to the corresponding tangent plane
for spherical image Is :
Ne (Ie )[x, y] ? Np (Is )[?, ?],
?(x, y) ? De , (?, ?) = (
180? ? y 360? ? x
,
).
He
We
(2)
This can be seen as a domain adaptation problem where we want to transfer the model from the
domain of Ip to that of Ie . However, unlike typical domain adaptation problems, the difference
between Ip and Ie is characterized by a geometric projection transformation rather than a shift
in data distribution. Note that the training data to learn Ne requires no manual annotations: it
consists of arbitrary 360? images coupled with the ?true" Np outputs computed by exhaustive planar
reprojections, i.e., evaluating the rhs of Eq. 1 for every (?, ?). Furthermore, at test time, only a single
equirectangular projection of the entire 360? input will be computed using Ne to obtain the dense
(inferred) Np outputs, which would otherwise require multiple projections and evaluations of Np .
3.2
Network Structure
The main challenge for transferring Np to Ne is the distortion introduced by equirectangular projection. The distortion is location dependent?a k ? k square in perspective projection will not be a
square in the equirectangular projection, and its shape and size will depend on the polar angle ?. See
Fig. 2. The convolution kernel should transform accordingly. Our approach 1) adjusts the shape
of the convolution kernel to account for the distortion, in particular the content expansion, and 2)
reduces the number of max-pooling layers to match the pixel sizes in Ne and Np , as we detail next.
We adapt the architecture of Ne from Np using the following heuristic. The goal is to ensure each
kernel receives enough information from the input in order to compute the target output. First, we
untie the weight of convolution kernels at different ? by learning one kernel Key for each output row
y. Next, we adjust the shape of Key such that it covers the Rf of the original kernel. We consider
Key ? Ne to cover Kp ? Np if more than 95% of pixels in the Rf of Kp are also in the Rf of Ke
in Ie . The Rf of Kp in Ie is obtained by backprojecting the R ? R grid to n
? = (?, 0) using P ?1 ,
where the center of the grid aligns on n
? . Ke should be large enough to cover Kp , but it should also
be as small as possible to avoid overfitting. Therefore, we optimize the shape of Kel,y for layer l as
follows. The shape of Kel,y is initialized as 3 ? 3. We first adjust the height kh and increase kh by 2
2
e.g., Np could be AlexNet [22] or VGG [30] pre-trained for a large-scale recognition task.
4
Kel+1
l+1
Ke
Kel+1
..
....
Kel
Kel
l
Ke
..
....
?
?
Figure 3: Spherical convolution. The kernel weight in spherical convolution is tied only along each row of the
equirectangular image (i.e., ?), and each kernel convolves along the row to generate 1D output. Note that the
kernel size differs at different rows and layers, and it expands near the top and bottom of the image.
until the height of the Rf is larger than that of Kp in Ie . We then adjust the width kw similar to kh .
Furthermore, we restrict the kernel size kh ? kw to be smaller than an upper bound Uk . See Fig. 4.
Because the Rf of Kel depends on Kel?1 , we search for the kernel size starting from the bottom layer.
It is important to relax the kernel from being square to being rectangular, because equirectangular
projection will expand content horizontally near the poles of the sphere (see Fig. 2). If we restrict the
kernel to be square, the Rf of Ke can easily be taller but narrower than that of Kp which leads to
overfitting. It is also important to restrict the kernel size, otherwise the kernel can grow wide rapidly
near the poles and eventually cover the entire row. Although cutting off the kernel size may lead to
information loss, the loss is not significant in practice because pixels in equirectangular projection do
not distribute on the unit sphere uniformly; they are denser near the pole, and the pixels are by nature
redundant in the region where the kernel size expands dramatically.
Besides adjusting the kernel sizes, we also adjust the number of pooling layers to match the pixel
size ?? in Ne and Np . We define ??e = 180?/He and restrict We = 2He to ensure ??e = ??e .
Because max-pooling introduces shift invariance up to kw pixels in the image, which corresponds to
kw ? ?? degrees on the unit sphere, the physical meaning of max-pooling depends on the pixel size.
Since the pixel size is usually larger in Ie and max-pooling increases the pixel size by a factor of kw ,
we remove the pooling layer in Ne if ??e ? ??p .
Fig. 3 illustrates how spherical convolution differs from ordinary CNN. Note that we approximate
one layer in Np by one layer in Ne , so the number of layers and output channels in each layer is
exactly the same as the target network. However, this does not have to be the case. For example,
we could use two or more layers to approximate each layer in Np . Although doing so may improve
accuracy, it would also introduce significant overhead, so we stick with the one-to-one mapping.
3.3
Training Process
Given the goal in Eq. 2 and the architecture described in Sec. 3.2, we would like to learn the network
Ne by minimizing the L2 loss E[(Ne (Ie ) ? Np (Is ))2 ]. However, the network converges slowly,
possibly due to the large number of parameters. Instead, we propose a kernel-wise pre-training
process that disassembles the network and initially learns each kernel independently.
To perform kernel-wise pre-training, we further require Ne to generate the same intermediate representation as Np in all layers l:
Nel (Ie )[x, y] ? Npl (Is )[?, ?] ?l ? Ne .
(3)
Given Eq. 3, every layer l ? Ne is independent of each other. In fact, every kernel is independent and
can be learned separately. We learn each kernel by taking the ?ground truth? value of the previous
layer Npl?1 (Is ) as input and minimizing the L2 loss E[(Nel (Ie ) ? Npl (Is ))2 ], except for the first layer.
Note that Npl refers to the convolution output of layer l before applying any non-linear operation,
e.g. ReLU, max-pooling, etc. It is important to learn the target value before applying ReLU because
it provides more information. We combine the non-linear operation with Kel+1 during kernel-wise
pre-training, and we use dilated convolution [36] to increase the Rf size instead of performing
max-pooling on the input feature map.
For the first convolution layer, we derive the analytic solution directly.
P The projection operator P is
linear in the pixels in equirectangular projection: P(Is , n
? )[x, y] = ij cij Ie [i, j], for coefficients cij
5
Target Kernel Kp
Target Network Np
Receptive Field
Perspective Projection
(Inverse)
Kernel Ke
Yes
>
Network Ne
Receptive Field
No
kw
No
Increase
kh
Output kh
Yes
kw? kh > Uk
kh
Figure 4: Method to select the kernel height kh . We project the receptive field of the target kernel to equirectangular projection Ie and increase kh until it is taller than the target kernel in Ie . The kernel width kw is determined
using the same procedure after kh is set. We restrict the kernel size kw ? kh by an upper bound Uk .
from,
P e.g., bilinear interpolation. Because convolution is a weighted sum of input pixels Kp ? Ip =
xy wxy Ip [x, y], we can combine the weight wxy and interpolation coefficient cij as a single
convolution operator:
X
X
XX
Kp1 ? Is [?, ?] =
wxy
cij Ie [i, j] =
wxy cij Ie [i, j] = Ke1 ? Ie .
(4)
xy
ij
ij
xy
Ne1
The output value of
will be exact and requires no learning. Of course, the same is not possible for
l > 1 because of the non-linear operations between layers.
After kernel-wise pre-training, we can further fine-tune the network jointly across layers and kernels
by minimizing the L2 loss of the final output. Because the pre-trained kernels cannot fully recover
the intermediate representation, fine-tuning can help to adjust the weights to account for residual
errors. We ignore the constraint introduced in Eq. 3 when performing fine-tuning. Although Eq. 3
is necessary for kernel-wise pre-training, it restricts the expressive power of Ne and degrades the
performance if we only care about the final output. Nevertheless, the weights learned by kernel-wise
pre-training are a very good initialization in practice, and we typically only need to fine-tune the
network for a few epochs.
One limitation of S PH C ONV is that it cannot handle very close objects that span a large FOV. Because
the goal of S PH C ONV is to reproduce the behavior of models trained on perspective images, the
capability and performance of the model is bounded by the target model Np . However, perspective
cameras can only capture a small portion of a very close object in the FOV, and very close objects are
usually not available in the training data of the target model Np . Therefore, even though 360? images
offer a much wider FOV, S PH C ONV inherits the limitations of Np , and may not recognize very close
large objects. Another limitation of S PH C ONV is the resulting model size. Because it unties the
kernel weights along ?, the model size grows linearly with the equirectangular image height. The
model size can easily grow to tens of gigabytes as the image resolution increases.
4
Experiments
To evaluate our approach, we consider both the accuracy of its convolutions as well as its applicability
for object detections in 360? data. We use the VGG architecture3 and the Faster-RCNN [27] model as
our target network Np . We learn a network Ne to produce the topmost (conv5_3) convolution output.
Datasets We use two datasets: Pano2Vid for training, and Pano2Vid and PASCAL for testing.
Pano2Vid: We sample frames from the 360? videos in the Pano2Vid dataset [32] for both training
and testing. The dataset consists of 86 videos crawled from YouTube using four keywords: ?Hiking,?
?Mountain Climbing,? ?Parade,? and ?Soccer?. We sample frames at 0.05fps to obtain 1,056 frames
for training and 168 frames for testing. We use ?Mountain Climbing? for testing and others for
training, so the training and testing frames are from disjoint videos. See Supp. for sampling process.
Because the supervision is on a per pixel basis, this corresponds to N ? We ? He ? 250M (non
i.i.d.) samples. Note that most object categories targeted by the Faster-RCNN detector do not appear
in Pano2Vid, meaning that our experiments test the content-independence of our approach.
PASCAL VOC: Because the target model was originally trained and evaluated on PASCAL VOC 2007,
we ?360-ify? it to evaluate the object detector application. We test with the 4,952 PASCAL images,
which contain 12,032 bounding boxes. We transform them to equirectangular images as if they
3
https://github.com/rbgirshick/py-faster-rcnn
6
originated from a 360? camera. In particular, each object bounding box is backprojected to 3 different
scales {0.5R, 1.0R, 1.5R} and 5 different polar angles ??{36?, 72?, 108?, 144?, 180?} on the 360?
image sphere using the inverse perspective projection, where R is the resolution of the target network?s
Rf. Regions outside the bounding box are zero-padded. See Supp. for details. Backprojection allows
us to evaluate the performance at different levels of distortion in the equirectangular projection.
Metrics We generate the output widely used in the literature (conv5_3) and evaluate it with the
following metrics.
Network output error measures the difference between Ne (Ie ) and Np (Is ). In particular, we report
the root-mean-square error (RMSE) over all pixels and channels. For PASCAL, we measure the error
over the Rf of the detector network.
Detector network performance measures the performance of the detector network in Faster-RCNN
using multi-class classification accuracy. We replace the ROI-pooling in Faster-RCNN by pooling
over the bounding box in Ie . Note that the bounding box is backprojected to equirectangular projection
and is no longer a square region.
Proposal network performance evaluates the proposal network in Faster-RCNN using average
Intersection-over-Union (IoU). For each bounding box centered at n
? , we project the conv5_3 output
to the tangent plane n
? using P and apply the proposal network at the center of the bounding box on
the tangent plane. Given the predicted proposals, we compute the IoUs between foreground proposals
and the bounding box and take the maximum. The IoU is set to 0 if there is no foreground proposal.
Finally, we average the IoU over bounding boxes.
We stress that our goal is not to build a new object detector; rather, we aim to reproduce the behavior
of existing 2D models on 360? data with lower computational cost. Thus, the metrics capture how
accurately and how quickly we can replicate the exact solution.
Baselines We compare our method with the following baselines.
? E XACT ? Compute the true target value Np (Is )[?, ?] for every pixel. This serves as an upper
bound in performance and does not consider the computational cost.
? D IRECT ? Apply Np on Ie directly. We replace max-pooling with dilated convolution to produce
a full resolution output. This is Strategy I in Fig. 1 and is used in 360? video analysis [16, 23].
? I NTERP ? Compute Np (Is )[?, ?] every S-pixels and interpolate the values for the others. We set
S such that the computational cost is roughly the same as our S PH C ONV. This is a more efficient
variant of Strategy II in Fig. 1.
? P ERSPECT ? Project Is onto a cube map [2] and then apply Np on each face of the cube, which is
a perspective image with 90? FOV. The result is backprojected to Ie to obtain the feature on Ie .
We use W =960 for the cube map resolution so ?? is roughly the same as Ip . This is a second
variant of Strategy II in Fig. 1 used in PanoContext [38].
S PH C ONV variants We evaluate three variants of our approach:
? O PT S PH C ONV ? To compute the output for each layer l, O PT S PH C ONV computes the exact
output for layer l?1 using Np (Is ) then applies spherical convolution for layer l. O PT S PH C ONV
serves as an upper bound for our approach, where it avoids accumulating any error across layers.
? S PH C ONV-P RE ? Uses the weights from kernel-wise pre-training directly without fine-tuning.
? S PH C ONV ? The full spherical convolution with joint fine-tuning of all layers.
Implementation details We set the resolution of Ie to 640?320. For the projection operator P, we
map ?=65.5? to W =640 pixels following SUN360 [35]. The pixel size is therefore ??e =360?/640
for Ie and ??p =65.5?/640 for Ip . Accordingly, we remove the first three max-pooling layers so
Ne has only one max-pooling layer following conv4_3. The kernel size upper bound Uk =7 ? 7
following the max kernel size in VGG. We insert batch normalization for conv4_1 to conv5_3. See
Supp. for details.
4.1
Network output accuracy and computational cost
Fig. 5a shows the output error of layers conv3_3 and conv5_3 on the Pano2Vid [32] dataset (see
Supp. for similar results on other layers.). The error is normalized by that of the mean predictor. We
evaluate the error at 5 polar angles ? uniformly sampled from the northern hemisphere, since error is
roughly symmetric with the equator.
7
conv3 3 RMSE
conv5 3 RMSE
2
2
1
1
0
18? 36? 54? 72? 90?
?
0
18? 36? 54? 72? 90?
Direct
Interp
Perspective
Exact
OptSphConv
SphConv-Pre
SphConv
0.8
Accuracy
1
0.6
0.5
0.5
0.4
conv5 3 RMSE
1.5
0.7
101
102
103
0
0
2
4
6
Tera-MACs
(a) Network output errors vs. polar angle
(b) Cost vs. accuracy
Figure 5: (a) Network output error on Pano2Vid; lower is better. Note the error of E XACT is 0 by definition.
Our method?s convolutions are much closer to the exact solution than the baselines?. (b) Computational cost
vs. accuracy on PASCAL. Our approach yields accuracy closest to the exact solution while requiring orders of
magnitude less computation time (left plot). Our cost is similar to the other approximations tested (right plot).
Plot titles indicate the y-labels, and error is measured by root-mean-square-error (RMSE).
Figure 6: Three AlexNet conv1 kernels (left squares) and their corresponding four S PH C ONV-P RE kernels at
? ? {9?, 18?, 36?, 72?} (left to right).
First we discuss the three variants of our method. O PT S PH C ONV performs the best in all layers
and ?, validating our main idea of spherical convolution. It performs particularly well in the lower
layers, because the Rf is larger in higher layers and the distortion becomes more significant. Overall,
S PH C ONV-P RE performs the second best, but as to be expected, the gap with O PT C ONV becomes
larger in higher layers because of error propagation. S PH C ONV outperforms S PH C ONV-P RE in
conv5_3 at the cost of larger error in lower layers (as seen here for conv3_3). It also has larger error
at ?=18? for two possible reasons. First, the learning curve indicates that the network learns more
slowly near the pole, possibly because the Rf is larger and the pixels degenerate. Second, we optimize
the joint L2 loss, which may trade the error near the pole with that at the center.
Comparing to the baselines, we see that ours achieves lowest errors. D IRECT performs the worst
among all methods, underscoring that convolutions on the flattened sphere?though fast?are inadequate. I NTERP performs better than D IRECT, and the error decreases in higher layers. This is because
the Rf is larger in the higher layers, so the S-pixel shift in Ie causes relatively smaller changes in
the Rf and therefore the network output. P ERSPECTIVE performs similarly in different layers and
outperforms I NTERP in lower layers. The error of P ERSPECTIVE is particularly large at ?=54?, which
is close to the boundary of the perspective image and has larger perspective distortion.
Fig. 5b shows the accuracy vs. cost tradeoff. We measure computational cost by the number of
Multiply-Accumulate (MAC) operations. The leftmost plot shows cost on a log scale. Here we
see that E XACT?whose outputs we wish to replicate?is about 400 times slower than S PH C ONV,
and S PH C ONV approaches E XACT?s detector accuracy much better than all baselines. The second
plot shows that S PH C ONV is about 34% faster than I NTERP (while performing better in all metrics).
P ERSPECTIVE is the fastest among all methods and is 60% faster than S PH C ONV, followed by
D IRECT which is 23% faster than S PH C ONV. However, both baselines are noticeably inferior in
accuracy compared to S PH C ONV.
To visualize what our approach has learned, we learn the first layer of the AlexNet [22] model
provided by the Caffe package [20] and examine the resulting kernels. Fig. 6 shows the original
kernel Kp and the corresponding kernels Ke at different polar angles ?. Ke is usually the re-scaled
version of Kp , but the weights are often amplified because multiple pixels in Kp fall to the same
pixel in Ke like the second example. We also observe situations where the high frequency signal in
the kernel is reduced, like the third example, possibly because the kernel is smaller. Note that we
learn the first convolution layer for visualization purposes only, since l = 1 (only) has an analytic
solution (cf. Sec 3.3). See Supp. for the complete set of kernels.
4.2
Object detection and proposal accuracy
Having established our approach provides accurate and efficient Ne convolutions, we now examine
how important that accuracy is to object detection on 360? inputs. Fig. 7a shows the result of the
Faster-RCNN detector network on PASCAL in 360? format. O PT S PH C ONV performs almost as well
as E XACT. The performance degrades in S PH C ONV-P RE because of error accumulation, but it still
8
2
Output RMSE
1.5
0.6
1
0.4
0.5
0.2
18? 36? 54? 72? 90?
0
18? 36? 54? 72? 90?
Direct
Interp
Perspective
Exact
OptSphConv
SphConv-Pre
SphConv
Scale = 0.5R
0.3
IoU
Accuracy
0.8
Scale = 1.0R
0.3
0.2
0.2
0.1
0.1
0
18? 36? 54? 72? 90?
0
18? 36? 54? 72? 90?
(a) Detector network performance.
(b) Proposal network accuracy (IoU).
Figure 7: Faster-RCNN object detection accuracy on a 360? version of PASCAL across polar angles ?, for both
the (a) detector network and (b) proposal network. R refers to the Rf of Np . Best viewed in color.
Figure 8: Object detection examples on 360? PASCAL test images. Images show the top 40% of equirectangular
projection; black regions are undefined pixels. Text gives predicted label, multi-class probability, and IoU, resp.
Our method successfully detects objects undergoing severe distortion, some of which are barely recognizable
even for a human viewer.
significantly outperforms D IRECT and is better than I NTERP and P ERSPECTIVE in most regions.
Although joint training (S PH C ONV) improves the output error near the equator, the error is larger near
the pole which degrades the detector performance. Note that the Rf of the detector network spans
multiple rows, so the error is the weighted sum of the error at different rows. The result, together
with Fig. 5a, suggest that S PH C ONV reduces the conv5_3 error in parts of the Rf but increases it at
the other parts. The detector network needs accurate conv5_3 features throughout the Rf in order to
generate good predictions.
D IRECT again performs the worst. In particular, the performance drops significantly at ?=18?,
showing that it is sensitive to the distortion. In contrast, I NTERP performs better near the pole
because the samples are denser on the unit sphere. In fact, I NTERP should converge to E XACT at the
pole. P ERSPECTIVE outperforms I NTERP near the equator but is worse in other regions. Note that
??{18?, 36?} falls on the top face, and ?=54? is near the border of the face. The result suggests that
P ERSPECTIVE is still sensitive to the polar angle, and it performs the best when the object is near the
center of the faces where the perspective distortion is small.
Fig. 7b shows the performance of the object proposal network for two scales (see Supp. for more).
Interestingly, the result is different from the detector network. O PT S PH C ONV still performs almost
the same as E XACT, and S PH C ONV-P RE performs better than baselines. However, D IRECT now
outperforms other baselines, suggesting that the proposal network is not as sensitive as the detector
network to the distortion introduced by equirectangular projection. The performance of the methods
is similar when the object is larger (right plot), even though the output error is significantly different.
The only exception is P ERSPECTIVE, which performs poorly for ??{54?, 72?, 90?} regardless of the
object scale. It again suggests that objectness is sensitive to the perspective image being sampled.
Fig. 8 shows examples of objects successfully detected by our approach in spite of severe distortions.
See Supp. for more examples.
5
Conclusion
We propose to learn spherical convolutions for 360? images. Our solution entails a new form of
distillation across camera projection models. Compared to current practices for feature extraction on
360? images/video, spherical convolution benefits efficiency by avoiding performing multiple perspective projections, and it benefits accuracy by adapting kernels to the distortions in equirectangular
projection. Results on two datasets demonstrate how it successfully transfers state-of-the-art vision
models from the realm of limited FOV 2D imagery into the realm of omnidirectional data.
Future work will explore S PH C ONV in the context of other dense prediction problems like segmentation, as well as the impact of different projection models within our basic framework.
9
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6,253 | 6,657 | MarrNet: 3D Shape Reconstruction via 2.5D Sketches
Jiajun Wu*
MIT CSAIL
Yifan Wang*
ShanghaiTech University
Tianfan Xue
MIT CSAIL
William T. Freeman
MIT CSAIL, Google Research
Xingyuan Sun
Shanghai Jiao Tong University
Joshua B. Tenenbaum
MIT CSAIL
Abstract
3D object reconstruction from a single image is a highly under-determined problem, requiring strong prior knowledge of plausible 3D shapes. This introduces
challenges for learning-based approaches, as 3D object annotations are scarce in
real images. Previous work chose to train on synthetic data with ground truth 3D
information, but suffered from domain adaptation when tested on real data.
In this work, we propose MarrNet, an end-to-end trainable model that sequentially
estimates 2.5D sketches and 3D object shape. Our disentangled, two-step formulation has three advantages. First, compared to full 3D shape, 2.5D sketches are much
easier to be recovered from a 2D image; models that recover 2.5D sketches are also
more likely to transfer from synthetic to real data. Second, for 3D reconstruction
from 2.5D sketches, systems can learn purely from synthetic data. This is because
we can easily render realistic 2.5D sketches without modeling object appearance
variations in real images, including lighting, texture, etc. This further relieves the
domain adaptation problem. Third, we derive differentiable projective functions
from 3D shape to 2.5D sketches; the framework is therefore end-to-end trainable on
real images, requiring no human annotations. Our model achieves state-of-the-art
performance on 3D shape reconstruction.
1
Introduction
Humans quickly recognize 3D shapes from a single image. Figure 1a shows a number of images
of chairs; despite their drastic difference in object texture, material, environment lighting, and
background, humans easily recognize they have very similar 3D shapes. What is the most essential
information that makes this happen?
Researchers in human perception argued that our 3D perception could rely on recovering 2.5D
sketches [Marr, 1982], which include intrinsic images [Barrow and Tenenbaum, 1978, Tappen et al.,
2003] like depth and surface normal maps (Figure 1b). Intrinsic images disentangle object appearance
variations in texture, albedo, lighting, etc., with its shape, which retains all information from the
observed image for 3D reconstruction. Humans further combine 2.5D sketches and a shape prior
learned from past experience to reconstruct a full 3D shape (Figure 1c). In the field of computer
vision, there have also been abundant works exploiting the idea for reconstruction 3D shapes of
faces [Kemelmacher-Shlizerman and Basri, 2011], objects [Tappen et al., 2003], and scenes [Hoiem
et al., 2005, Saxena et al., 2009].
Recently, researchers attempted to tackle the problem of single-image 3D reconstruction with deep
learning. These approaches usually regress a 3D shape from a single RGB image directly [Tulsiani
et al., 2017, Choy et al., 2016, Wu et al., 2016b]. In contrast, we propose a two-step while end-to-end
trainable pipeline, sequentially recovering 2.5D sketches (depth and normal maps) and a 3D shape.
? indicates equal contributions.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(b) 2.5D Sketches
(c) 3D Shape
(a) Images
Figure 1: Objects in real images (a) are subject to appearance variations regarding color, texture,
lighting, material, background, etc. Despite this, their 2.5D sketches like surface normal and depth
maps remain constant (b). The 2.5D sketches can be seen as an abstraction of the image, retaining all
information about the 3D shape of the object inside. We combine the sketches with learned shape
priors to reconstruct the full 3D shape (c).
We use an encoder-decoder structure for each component of the framework, and also enforce the
reprojection consistency between the estimated sketch and the 3D shape. We name it MarrNet, for its
close resemblance to David Marr?s theory of perception [Marr, 1982].
Our approach offers several unique advantages. First, the use of 2.5D sketches releases the burden on
domain transfer. As single-image 3D reconstruction is a highly under-constrained problem, strong
prior knowledge of object shapes is needed. This poses challenges to learning-based methods, as
accurate 3D object annotations in real images are rare. Most previous methods turned to training
purely on synthetic data [Tulsiani et al., 2017, Choy et al., 2016, Girdhar et al., 2016]. However, these
approaches often suffer from the domain adaption issue due to imperfect rendering. Learning 2.5D
sketches from images, in comparison, is much easier and more robust to transfer from synthetic to
real images, as shown in Section 4.
Further, as our second step recovers 3D shape from 2.5D sketches ? an abstraction of the raw input
image, it can be trained purely relying on synthetic data. Though rendering diverse realistic images is
challenging, it is straightforward to obtain almost perfect object surface normals and depths from a
graphics engine. This further relieves the domain adaptation issue.
We also enforce differentiable constraints between 2.5D sketches and 3D shape, making our system
end-to-end trainable, even on real images without any annotations. Given a set of unlabeled images,
our algorithm, pre-trained on synthetic data, can infer the 2.5D sketches of objects in the image,
and use it to refine its estimation of objects? 3D shape. This self-supervised feature enhances its
performance on images from different domains.
We evaluate our framework on both synthetic images of objects from ShapeNet [Chang et al., 2015],
and real images from the PASCAL 3D+ dataset [Xiang et al., 2014]. We demonstrate that our
framework performs well on 3D shape reconstruction, both qualitatively and quantitatively.
Our contributions are three-fold: inspired by visual cognition theory, we propose a two-step, disentangled formulation for single-image 3D reconstruction via 2.5D sketches; we develop a novel,
end-to-end trainable model with a differentiable projection layer that ensures consistency between
3D shape and mid-level representations; we demonstrate its effectiveness on 2.5D sketch transfer and
3D shape reconstruction on both synthetic and real data.
2
Related Work
2.5D Sketch Recovery Estimating 2.5D sketches has been a long-standing problem in computer
vision. In the past, researchers have explored recovering 2.5D shape from shading, texture, or color
images [Horn and Brooks, 1989, Zhang et al., 1999, Tappen et al., 2003, Barron and Malik, 2015,
Weiss, 2001, Bell et al., 2014]. With the development of depth sensors [Izadi et al., 2011] and larger
scale RGB-D datasets [Silberman et al., 2012, Song et al., 2017, McCormac et al., 2017], there
have also been papers on estimating depth [Chen et al., 2016, Eigen and Fergus, 2015], surface
normals [Bansal and Russell, 2016, Wang et al., 2015], and other intrinsic images [Shi et al., 2017,
2
2.5D Sketches
(c) Reprojection Consistency
normal
??
depth
3D Shape
2D Image
(a) 2.5D Sketch Estimation
silhouette
(b) 3D Shape Estimation
Normal Ball
Figure 2: Our model (MarrNet) has three major components: (a) 2.5D sketch estimation, (b) 3D
shape estimation, and (c) a loss function for reprojection consistency. MarrNet first recovers object
normal, depth, and silhouette images from an RGB image. It then regresses the 3D shape from the
2.5D sketches. In both steps, it uses an encoding-decoding network. It finally employs a reprojection
consistency loss to ensure the estimated 3D shape aligns with the 2.5D sketches. The entire framework
can be trained end-to-end.
Janner et al., 2017] with deep networks. Our method employs 2.5D estimation as a component, but
targets reconstructing full 3D shape of an object.
Single-Image 3D Reconstruction The problem of recovering object shape from a single image
is challenging, as it requires both powerful recognition systems and prior shape knowledge. With
the development of large-scale shape repository like ShapeNet [Chang et al., 2015], researchers
developed models encoding shape prior for this task [Girdhar et al., 2016, Choy et al., 2016, Tulsiani
et al., 2017, Wu et al., 2016b, Kar et al., 2015, Kanazawa et al., 2016, Soltani et al., 2017], with
extension to scenes [Song et al., 2017]. These methods typically regress a voxelized 3D shape
directly from an input image, and rely on synthetic data or 2D masks for training. In comparison, our
formulation tackles domain difference better, as it can be end-to-end fine-tuned on images without
any annotations.
2D-3D Consistency It is intuitive and practically helpful to constrain the reconstructed 3D shape
to be consistent with 2D observations. Researchers have explored this idea for decades [Lowe, 1987].
This idea is also widely used in 3D shape completion from depths or silhouettes [Firman et al.,
2016, Rock et al., 2015, Dai et al., 2017]. Recently, a few papers discussed enforcing differentiable
2D-3D constraints between shape and silhouettes, enabling joint training of deep networks for 3D
reconstruction [Wu et al., 2016a, Yan et al., 2016, Rezende et al., 2016, Tulsiani et al., 2017]. In our
paper, we exploit this idea to develop differentiable constraints on the consistency between various
2.5D sketches and 3D shape.
3
Approach
To recover the 3D structure from a single view RGB image, our MarrNet contains three parts: first, a
2.5D sketch estimator, which predicts the depth, surface normal, and silhouette images of the object
(Figure 2a); second, a 3D shape estimator, which infers 3D object shape using a voxel representation
(Figure 2b); third, a reprojection consistency function, enforcing the alignment between the estimated
3D structure and inferred 2.5D sketches (Figure 2c).
3.1
2.5D Sketch Estimation
The first component of our network (Figure 2a) takes a 2D RGB image as input, and predicts its 2.5D
sketch: surface normal, depth, and silhouette. The goal of the 2.5D sketch estimation step is to distill
intrinsic object properties from input images, while discarding properties that are non-essential for
the task of 3D reconstruction, such as object texture and lighting.
We use an encoder-decoder network architecture for 2.5D sketch estimation. Our encoder is a
ResNet-18 [He et al., 2015], encoding a 256?256 RGB image into 512 feature maps of size 8?8.
3
?
?=4
?
?
? =?
Voxels that should be 1 for
reprojected depth consistency
Voxels that should be 0 for
reprojected depth consistency
?=
Voxels that should be 1 for reprojected
surface normal consistency
Figure 3: Reprojection consistency between 2.5D sketches and 3D shape. Left and middle: the
criteria for depths and silhouettes; right: the criterion for surface normals. See Section 3.3 for details.
The decoder contains four sets of 5?5 fully convolutional and ReLU layers, followed by four sets
of 1?1 convolutional and ReLU layers. It outputs the corresponding depth, surface normal, and
silhouette images, also at the resolution of 256?256.
3.2
3D Shape Estimation
The second part of our framework (Figure 2b) infers 3D object shape from estimated 2.5D sketches.
Here, the network focuses on learning the shape prior that explains input well. As it takes only surface
normal and depth images as input, it can be trained on synthetic data, without suffering from the
domain adaption problem: it is straightforward to render nearly perfect 2.5D sketches, but much
harder to render realistic images.
The network architecture is inspired by the TL network [Girdhar et al., 2016], and the 3D-VAEGAN [Wu et al., 2016b], again with an encoding-decoding style. It takes a normal image and a depth
image as input (both masked by the estimated silhouette), maps them to a 200-dim vector via five sets
of convolutional, ReLU, and pooling layers, followed by two fully connected layers. The detailed
encoder structure can be found in Girdhar et al. [2016]. The vector then goes through a decoder,
which consists of five fully convolutional and ReLU layers to output a 128?128?128 voxel-based
reconstruction of the input. The detailed encoder structure can be found in Wu et al. [2016b].
3.3
Reprojection Consistency
There have been works attempting to enforce the consistency between estimated 3D shape and 2D
representations in a neural network [Yan et al., 2016, Rezende et al., 2016, Wu et al., 2016a, Tulsiani
et al., 2017]. Here, we explore novel ways to include a reprojection consistency loss between the
predicted 3D shape and the estimated 2.5D sketch, consisting of a depth reprojection loss and a
surface normal reprojection loss.
We use vx,y,z to represent the value at position (x, y, z) in a 3D voxel grid, assuming that vx,y,z ?
[0, 1], ?x, y, z. We use dx,y to denote the estimated depth at position (x, y), and nx,y = (na , nb , nc )
to denote the estimated surface normal. We assume orthographic projection in this work.
Depths The projected depth loss tries to guarantee that the voxel with depth vx,y,dx,y should be 1,
and all voxels in front of it should be 0. This ensures the estimated 3D shape matches the estimated
depth values.
As illustrated in Figure 3a, we define projected depth loss as follows:
? 2
z < dx,y
?vx,y,z ,
Ldepth (x, y, z) = (1 ? vx,y,z )2 , z = dx,y .
?
0,
z > dx,y
(1)
The gradients are
?
2v
,
?Ldepth (x, y, z) ? x,y,z
= 2(vx,y,z ? 1),
?
?vx,y,z
0,
4
z < dx,y
z = dx,y .
z > dx,y
(2)
When dx,y = ?, our depth criterion reduces to a special case ? the silhouette criterion. As shown
in Figure 3b, for a line that has no intersection with the shape, all voxels in it should be 0.
Surface Normals As vectors nx = (0, ?nc , nb ) and ny = (?nc , 0, na ) are orthogonal to the normal vector nx,y = (na , nb , nc ), we can normalize them to obtain two vectors, n0x = (0, ?1, nb /nc )
and n0y = (?1, 0, na /nc ), both on the estimated surface plane at (x, y, z). The projected surface
normal loss tries to guarantee that the voxels at (x, y, z) ? n0x and (x, y, z) ? n+y should be 1 to
match the estimated surface normals. These constraints only apply when the target voxels are inside
the estimated silhouette.
As shown in Figure 3c, let z = dx,y , the projected surface normal loss is defined as
Lnormal (x, y, z) =
2
+ 1 ? vx,y+1,z? nb +
nc
nc
2
2
1 ? vx?1,y,z+ nna + 1 ? vx+1,y,z? nna .
1 ? vx,y?1,z+ nb
c
2
c
(3)
Then the gradients along the x direction are
?Lnormal (x, y, z)
= 2 vx?1,y,z+ nna ? 1
c
?vx?1,y,z+ nna
and
c
?Lnormal (x, y, z)
= 2 vx+1,y,z? nna ? 1 .
c
?vx+1,y,z? nna
c
(4)
The gradients along the y direction are similar.
3.4
Training Paradigm
We employ a two-step training paradigm. We first train the 2.5D sketch estimation and the 3D shape
estimation components separately on synthetic images; we then fine-tune the network on real images.
For pre-training, we use synthetic images of ShapeNet objects. The 2.5D sketch estimator is trained
using the ground truth surface normal, depth, and silhouette images with a L2 loss. The 3D interpreter
is trained using ground truth voxels and a cross-entropy loss. Please see Section 4.1 for details on
data preparation.
The reprojection consistency loss is used to fine-tune the 3D estimation component of our model on
real images, using the predicted normal, depth, and silhouette. We observe that a straightforward
implementation leads to shapes that explain 2.5D sketches well, but with unrealistic appearance. This
is because the 3D estimation module overfits the images without preserving the learned 3D shape
prior. See Figure 5 for examples, and Section 4.2 for more details.
We therefore choose to fix the decoder of the 3D estimator and only fine-tune the encoder. During
testing, our method can be self-supervised, i.e., we can fine-tune even on a single image without any
annotations. In practice, we fine-tune our model separately on each image for 40 iterations. For each
test image, fine-tuning takes up to 10 seconds on a modern GPU; without fine-tuning, testing time
is around 100 milliseconds. We use SGD for optimization with a batch size of 4, a learning rate of
0.001, and a momentum of 0.9. We implemented our framework in Torch7 [Collobert et al., 2011].
4
Evaluation
In this section, we present both qualitative and quantitative results on single-image 3D reconstruction
using variants of our framework. We evaluate our entire framework on both synthetic and real-life
images on three datasets.
4.1
3D Reconstruction on ShapeNet
Data We start with experiments on synthesized images of ShapeNet chairs [Chang et al., 2015]. We
put objects in front of random backgrounds from the SUN database [Xiao et al., 2010], and render the
corresponding RGB, depth, surface normal, and silhouette images. We use a physics-based renderer,
Mitsuba [Jakob, 2010], to obtain more realistic images. For each of the 6,778 ShapeNet chairs, we
render 20 images of random viewpoints.
5
Images
Estimated
normals
Estimated
depths
Direct predictions
MarrNet
Ground truth
Figure 4: Results on rendered images of ShapeNet objects [Chang et al., 2015]. From left to right:
input, estimated normal map, estimated depth map, our prediction, a baseline algorithm that predicts
3D shape directly from RGB input without modeling 2.5D sketch, and ground truth. Both normal and
depth maps are masked by predicted silhouettes. Our method is able to recover shapes with smoother
surfaces and finer details.
Methods We follow the training paradigm described in Section 3.4, but without the final fine-tuning
stage, as ground truth 3D shapes are available on this synthetic dataset. Specifically, the 2.5D sketch
estimator is trained using ground truth depth, normal and silhouette images and a L2 reconstruction
loss. The 3D shape estimation module takes in the masked ground truth depth and normal images as
input, and predicts 3D voxels of size 128?128?128 with a binary cross entropy loss.
We compare MarrNet with a baseline that predicts 3D shape directly from an RGB image, without
modeling 2.5D sketches. The baseline employs the same architecture as our 3D shape estimator
(Section 3.2). We show qualitative results in Figure 4. Our estimated surface normal and depth images
abstract out non-essential information like textures and lighting in the RGB image, while preserving
intrinsic information about object shape. Compared with the direct prediction baseline, our model
outputs objects with more details and smoother surfaces. For quantitative evaluation, previous works
usually compute the Intersection-over-Union (IoU) [Tulsiani et al., 2017, Choy et al., 2016]. Our full
model achieves a higher IoU (0.57) than the direct prediction baseline (0.52).
4.2
3D Reconstruction on Pascal 3D+
Data PASCAL 3D+ dataset [Xiang et al., 2014] provides (rough) 3D models for objects in real-life
images. Here, we use the same test set of PASCAL 3D+ with earlier works [Tulsiani et al., 2017].
Methods We follow the paradigm described in Section 3.4: we first train each module separately
on the ShapeNet dataset, and then fine-tune them on the PASCAL 3D+ dataset. Unlike previous
works [Tulsiani et al., 2017], our model requires no silhouettes as input during fine-tuning; it instead
estimates silhouette jointly.
As an ablation study, we compare three variants of our model: first, the model trained using ShapeNet
data only, without fine-tuning; second, the fine-tuned model whose decoder is not fixed during
6
Images
Estimated
normals and
silhouettes
Estimated
depths and
silhouettes
Estimated 3D Two views of refined 3D shape Two views of refined 3D shape
not fine-tuned fine-tuned, not fixing decoder
fine-tuned, fixing decoder
Figure 5: We present an ablation study, where we compare variants of our models. From left to
right: input, estimated normal, estimated depth, 3D prediction before fine-tuning, two views of the
3D prediction after fine-tuning without fixing decoder, and two views of the 3D prediction after
fine-tuning with the decoder fixed. When the decoder is not fixed, the model explains the 2.5D sketch
well, but fails to preserve the learned shape prior. Fine-tuning with a fixed decoder resolves the issue.
fine-tuning; and third, the full model whose decoder is fixed during fine-tuning. We also compare
with the state-of-the-art method (DRC) [Tulsiani et al., 2017], and the provided ground truth shapes.
Results The results of our ablation study are shown in Figure 5. The model trained on synthetic
data provides a reasonable shape estimate. If we fine-tune the model on Pascal 3D+ without fixing
the decoder, the output voxels explain the 2.5D sketch data well, but fail to preserve the learned shape
prior, leading to impossible shapes from certain views. Our final model, fine-tuned with the decoder
fixed, keeps the shape prior and provides more details of the shape.
We show more results in Figure 6, where we compare with the state-of-the-art (DRC) [Tulsiani et al.,
2017], and the provided ground truth shapes. Quantitatively, our algorithm achieves a higher IoU
over these methods (MarrNet 0.39 vs. DRC 0.34). However, we find the IoU metric sub-optimal
for three reasons. First, measuring 3D shape similarity is a challenging yet unsolved problem, and
IoU prefers models that predict mean shapes consistently, with no emphasize on details. Second, as
object shape can only be reconstructed up to scale from a single image, it requires searching over all
possible scales during the computation of IoU, making it less efficient. Third, as discussed in Tulsiani
et al. [2017], PASCAL 3D+ has only rough 3D annotations (10 CAD chair models for all images).
Computing IoU with these shapes would thus not be the most informative evaluation metric.
We instead conduct human studies, where we show users the input image and two reconstructions,
and ask them to choose the one that looks closer to the shape in the image. We show each test image
to 10 human subjects. As shown in Table 1, our reconstruction is preferred 74% of the time to DRC,
and 42% of the time to ground truth, showing a clear advantage.
We present some failure cases in Figure 7. Our algorithm does not perform well on recovering
complex, thin structure, and sometimes fails when the estimated mask is very inaccurate. Also, while
DRC may benefit from multi-view supervision, we have only evaluated MarrNet given a single view
of the shape, though adapting our formulation to multi-view data should be straightforward.
7
Images Ground truth
DRC
MarrNet
Images
Ground truth DRC
MarrNet
Figure 6: 3D reconstructions of chairs on the Pascal 3D+ [Xiang et al., 2014] dataset. From left to
right: input, the ground truth shape from the dataset, 3D estimation by DRC [Tulsiani et al., 2017],
and two views of MarrNet predictions. Our model recovers more accurate 3D shapes.
DRC
MarrNet
Ground truth
DRC
MarrNet
GT
50
74
83
26
50
58
17
42
50
Table 1: Human preferences on chairs in PASCAL 3D+ [Xiang et al., 2014]. We compare
MarrNet with the state-of-the-art (DRC) [Tulsiani et al., 2017], and the ground truth provided
by the dataset. Each number shows the percentage of humans prefer the left method to the top
one. MarrNet is preferred 74% of the time to
DRC, and 42% of the time to ground truth.
4.3
Images
Estimated Estimated
normals
depths
MarrNet
Figure 7: Failure cases on Pascal 3D+. Our algorithm does not perform well on recovering complex, thin structure, and sometimes fails when
the estimated mask is very inaccurate.
3D Reconstruction on IKEA
Data The IKEA dataset [Lim et al., 2013] contains images of IKEA furniture, along with accurate
3D shape and pose annotations. These images are challenging, as objects are often heavily occluded
or cropped. We also evaluate our model on the IKEA dataset.
Results We show qualitative results in Figure 8, where we compare with estimations by 3D-VAEGAN [Wu et al., 2016b] and the ground truth. As shown in the figure, our model can deal with mild
occlusions in real life scenarios. We also conduct human studies on the IKEA dataset. Results show
that 61% of the subjects prefer our reconstructions to those of 3D-VAE-GAN.
4.4
Extensions
We also apply our framework on cars and airplanes. We use the same setup as that for chairs. As
shown in Figure 9, shape details like the horizontal stabilizer and rear-view mirrors are recovered
8
Images
Ground truth
3DVAE-GAN
MarrNet
Images
Ground truth
3DVAE-GAN
MarrNet
Figure 8: 3D reconstruction of chairs on the IKEA [Lim et al., 2013] dataset. From left to right:
input, ground truth, 3D estimation by 3D-VAE-GAN [Wu et al., 2016b], and two views of MarrNet
predictions. Our model recovers more details compared to 3D-VAE-GAN.
Images
Ground truth
DRC
MarrNet
Images
Ground truth
DRC
MarrNet
Figure 9: 3D reconstructions of airplanes and cars from PASCAL 3D+. From left to right: input, the
ground truth shape from the dataset, 3D estimation by DRC [Tulsiani et al., 2017], and two views of
MarrNet predictions.
Images
MarrNet
Ground truth
Images
MarrNet
Ground truth
Figure 10: 3D reconstruction of objects from multiple categories on the PASCAL 3D+ [Xiang et al.,
2014] dataset. MarrNet also recovers 3D shape well when it is trained on multiple categories.
by our model. We further train MarrNet jointly on all three object categories, and show results in
Figure 10. Our model successfully recovers shapes of different categories.
5
Conclusion
We proposed MarrNet, a novel model that explicitly models 2.5D sketches for single-image 3D
shape reconstruction. The use of 2.5D sketches enhanced the model?s performance, and made it
easily adaptive to images across domains or even categories. We also developed differentiable loss
functions for the consistency between 3D shape and 2.5D sketches, so that MarrNet can be end-to-end
fine-tuned on real images without annotations. Experiments demonstrated that our model performs
well, and is preferred by human annotators over competitors.
9
Acknowledgements
We thank Shubham Tulsiani for sharing the DRC results, and Chengkai Zhang for the help on shape
visualization. This work is supported by NSF #1212849 and #1447476, ONR MURI N00014-161-2007, the Center for Brain, Minds and Machines (NSF #1231216), Toyota Research Institute,
Samsung, and Shell.
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6,254 | 6,658 | Multimodal Learning and Reasoning for Visual
Question Answering
Ilija Ilievski
Integrative Sciences and Engineering
National University of Singapore
[email protected]
Jiashi Feng
Electrical and Computer Engineering
National University of Singapore
[email protected]
Abstract
Reasoning about entities and their relationships from multimodal data is a key goal
of Artificial General Intelligence. The visual question answering (VQA) problem
is an excellent way to test such reasoning capabilities of an AI model and its
multimodal representation learning. However, the current VQA models are oversimplified deep neural networks, comprised of a long short-term memory (LSTM)
unit for question comprehension and a convolutional neural network (CNN) for
learning single image representation. We argue that the single visual representation
contains a limited and general information about the image contents and thus limits
the model reasoning capabilities. In this work we introduce a modular neural network model that learns a multimodal and multifaceted representation of the image
and the question. The proposed model learns to use the multimodal representation
to reason about the image entities and achieves a new state-of-the-art performance
on both VQA benchmark datasets, VQA v1.0 and v2.0, by a wide margin.
1
Introduction
One of the hallmarks of human intelligence is the ability to reason about entities and their relationships
using data from different modalities [40]. Humans can easily reason about the entities in a complex
scene by building multifaceted mental representations of the scene contents. Thus, any plausible
attempt for Artificial General Intelligence must also be able to reason about entities and their
relationships.
Deep learning models have demonstrated excellent performance on many computer vision tasks such
as image classification, object recognition, and scene classification [37, 23]. However, the models
are limited to a single task over single data modality and thus are still far from complete scene
understanding and reasoning.
The visual question answering (VQA), a task to provide answers to natural language questions about
the contents of an image, has been proposed to fill this gap [3, 22]. Solving the VQA task requires
understanding of the image contents, the question words, and their relationships. The human-posed
questions are arbitrary and thus besides being a challenging machine comprehension problem, their
answering also involves many computer vision tasks such as scene classification, object detection
and classification, and face analysis. Thus, VQA represents a problem comprised of multiple subproblems over multimodal data and as such can serve as a proxy test for the general reasoning
capabilities of an AI model [30].
However, the current state-of-the-art VQA approaches employ oversimplified deep neural network
models, comprised of a long short-term memory (LSTM) unit [9] for question comprehension and
a convolutional neural network (CNN) [24] for learning a single representation of the image. The
single visual representation encodes a limited and general information about the image contents and
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
thus hampers the model?s reasoning about the image entities and their relationships. The analysis of
the model?s behavior is also made difficult because of the single visual representation, as one does
not have evidence of the visual information used by the model to produce the answer. The problem is
partially alleviated by incorporating a visual attention module, as most VQA models do [6, 18, 45],
but still, the VQA models remain black box in most cases. There have also been approaches that
employ modular networks for VQA [2], however they achieved limited success. Most VQA models
join the image and text representations via element-wise multiplication [18, 45], exception is the
model proposed by Fukui et al. [6], which learns the joint representation via compact bilinear pooling.
Goyal et al. [7] recently showed that the current state-of-the-art VQA methods actually learn dataset
biases instead of reasoning about the image objects or learning multifaceted image representations.
In this work we introduce ReasonNet, a model that learns to reason about entities in an image by
learning a multifaceted representation of the image contents. ReasonNet employs a novel multimodal
representation learning and fusion module that enables the model to develop a complete image and
question understanding. ReasonNet then learns to utilize the multiple image representations, each
encoding different visual aspects of the image, by explicitly incorporating a neural reasoning module.
In contrast to current VQA models, the ReasonNet model is fully interpretable and provides evidence
of its reasoning capabilities. The proposed model outperforms the state-of-the-art by a significant
margin on the two largest benchmark VQA datasets, VQA v1.0 [3] and VQA v2.0 [7]. In summary,
the contributions of our work are as follows:
? We develop a novel multimodal representation learning and fusion method, crucial for
obtaining the complete image understanding necessary for multimodal reasoning.
? We introduce a new VQA reasoning model that learns multifaceted image representations to
reason about the image entities.
? We perform an extensive evaluation and achieve new state-of-the-art performance on the
two VQA benchmark datasets.
2
Related work
Multimodal reasoning models Recently, several works introduced modular neural networks for
reasoning evaluated on natural language question answering and visual question answering [2, 1, 11].
The modular networks use conventional natural language parser [21] to obtain a network layout for
composing the network architecture [11]. Later work also incorporated a dynamic network layout
prediction by learning to rank the parser proposed modules [1]. The neural modules are then jointly
trained to execute the task on which they are applied.
In contrast to our model, the existing modular networks learn to compose a network architecture using
a hand-designed or parser proposed layout structure. These models were shown [2] to be unable to
capture the complex nature of the natural language and perform poorly on the complex VQA v1.0 [3].
Concurrently to this work, Hu et al. [10] and Johnson et al. [16] proposed a similar modular network
model that learns the network layout structure and thus do not require a parser. However, the models
have been applied with success only to the synthetic dataset CLEVR [15] because for training the
layout prediction module they require the ground-truth programs that generated the questions.
Visual question answering The visual question answering (VQA) task has received great interest [5, 45, 50, 38] since the release of the first large-scale VQA v1.0 dataset by Antol et al. [3].
Typically, a VQA model is comprised of two modules for learning the question and the image representations, and a third module for fusing the representations into a single multimodal representation.
The multimodal representation is then fed to multiple fully-connected layers and a softmax layer at
the end outputs the probabilities of each answer (e.g. see [12, 6, 44]).
The question representation is learned by mapping each question word to a vector via a lookup
table matrix, which is often initialized with word2vec [31] or skip-thought [20] vectors. The word
vectors are then sequentially fed to a Long Short-Term Memory (LSTM) unit [9], and the final hidden
LSTM state is considered as the question representation. The image representation is obtained from a
pretrained convolutional neural network (CNN) [24], such as ResNet [8] or VGG [39], and the output
from the penultimate layer is regarded as the image representation. Some, increase the information
available in the image representation by using feature maps from a convolutional layer [45, 29, 6].
2
3
ReasonNet
We develop a novel modular deep neural network model that learns to reason about entities and their
relationships in a complex scene using multifaceted scene representations. The model, named as ReasonNet, takes as inputs an image I and a natural language text L. Then, ReasonNet passes the image
and the text through its different modules that encode multiple image aspects into multiple vector
representations. At the same time, ReasonNet uses a language encoder module to encode the text into
a vector representation. Finally, ReasonNet?s reasoning unit fuses the different representations into a
single reasoning vector. In the following we give details of the network modules and its reasoning
unit, while in Section 4 we ensemble and describe a ReasonNet applied to the VQA task (Figure 1).
3.1
Multimodal representation learning
ReasonNet incorporates two types of representation learning modules: visual classification modules
and visual attention modules. A visual classification module outputs a vector that contains the class
probabilities of the specific image aspect assigned to that module. While a visual attention module
outputs a vector that contains visual features focused on the module?s specific image aspect.
We denote the classification modules as ? and the visual attention modules as ?. We use bold lower
case letters to represent vectors and bold upper case letters to represent tensors. Subscript indices
denote countable variables while superscript indices label the variables. For example, P ?k denotes the
k-th matrix P of the module ?. In the following we describe each module type.
Visual classification modules Each visual classification module maps a different aspect of the
image contents to a module specific class representation space. Consequently, ReasonNet needs to
transfer the output of each module to a common representation space. For example, if one module
classifies the image scene as ?train station?, while another module classifies a detected object as
?train?, ReasonNet needs to be aware that the outputs are semantically related. For this reason,
ReasonNet transfers each classification module?s output to a common representation space using a
lookup table matrix.
Formally, a classification module ? outputs a matrix P ? of n one-hot vectors with lookup table
indices of the n highest probability class labels. The matrix P ? is then mapped to a vector c? in a
common representation space with:
c? = vec(P ? W LT ),
where W
LT
(1)
is the lookup table matrix with learned parameters.
Visual attention modules ReasonNet passes the image I through a residual neural network [8] to
obtain a visual feature tensor V ? RF ?W ?H representing the global image contents. Then, each
visual attention module ? focuses the visual feature tensor to the specific image aspect assigned to
the said module using an attention probability distribution ?? ? RW ?H .
The module-specific visual representation vector v ? ? RF is then obtained with:
?
v =
W X
H
X
??
i,j V i,j .
(2)
i=1 j=1
Encoder units A problem with this approach is that the common representation vectors c? from
the classification modules, being distributed representations of the modules? class labels, are highdimensional vectors. On the other hand, the visual feature vectors v ? of the attention modules are
also high-dimensional vectors, but sparse and with different dimensionality and scale.
As a solution, ReasonNet appends to each classification and to each attention modules an encoder
unit Enc that encodes a module?s output vector x (equal to c? if classification module and to v ? if
attention module) to a condensed vector r in a common low-dimensional representation space. The
encoder units are implemented as two fully-connected layers followed by a non-linear activation
function, and max pooling over the magnitude while preserving the sign:
r = Enc(x)
Enc(x) := sgn(f (x)) ? max(|f (x)|),
(3)
E
E
E
E
f (x) := ? W 2 ?(W 1 x + b1 ) + b2 ,
3
E
where W E
k and bk are the parameters and biases of the k-th layer in an encoder E, and ?(?) :=
tanh(?). The max pooling is performed over the row dimension as f (x) outputs a matrix of rowstacked encoded vectors from one mini-batch.
Text encoding ReasonNet treats the words in L as class labels and correspondingly uses the lookup
table from Eq. (1) and an encoder unit to map the text to a vector r l in the same low-dimensional
representation space of the other modules.
3.2
Multimodal reasoning
The reasoning capabilities of the ReasonNet model come from its ability to learn the interaction
between each module?s representation r k and the question representation r l . The ReasonNet model
parameterizes the r k ? r l interaction with a bilinear model [42]. Bilinear models have excellent representational expressiveness by allowing the vector representations to adapt each other multiplicatively.
The bilinear model is defined as:
s
s l
sk = r >
(4)
k W k r + bk ,
where k = 1, . . . , K and K is the number of representation learning modules, provides a rich vector
representation sk of the k-th module?s output ? language interaction.
Note that while other multimodal representation learning works, e.g. [6], have criticized the use of
bilinear models for representation fusion because of their high dimensionality as the tensor W sk is
cubic in the dimensions of r k , r l , and sk . However, ReasonNet mitigates this issue by employing
encoder units to reduce the dimension of the representation vectors and thus also reduce the dimension
of the bilinear model. We evaluate different fusion methods with ablation studies in Section 5.3.
ReasonNet builds a complete image and language representation by concatenating each interaction
fK
f
vector sk into a vector g = k=1 sk , where denotes concatenation of vectors. The vector concatenation is crucial for disentangling the contributions of each module in the model?s task. By partially
ignoring some of the inputs of the vector g, ReasonNet can learn to ?softly? utilize the different
modules only when their outputs help in predicting the correct answer for the given question. In
contrast to the recent works on module networks, e.g. [10, 16], ReasonNet can choose to partially use
a module if it is helpful for the task instead of completely removing a module as other module network
models do. For example, for the visual question answering (VQA) task, the soft module usage is
particularly useful when answering a question which implicitly requires a module, e.g. answering the
question ?Is it raining?? implicitly requires a scene classification module.
The concatenation of the modules? representations also enables the interpretability of the reasoning
behavior of our model. Specifically, by observing which elements of the vector g are most active we
can infer which modules ReasonNet used in the reasoning process and thus explain the reasons for its
behavior. We visualize the reasoning process on the VQA task in Section 5.4.
Finally, the multimodal representation vector g can be used as an input to an answer classification
network, when applied to the VQA task, or as an input to an LSTM unit when applied to the
image captioning task. In the next section, we use the challenging VQA task as a proxy test for
ReasonNet reasoning capabilities.
4
ReasonNet for VQA
The visual question answering problem is an excellent way to test the reasoning capabilities of
ReasonNet and its use of multifaceted image representations. For example, answering the question
?Does the woman look happy?? requires a face detection, gender and emotion classification, while
answering the question ?How many mice are on the desk?? requires object detection and classification.
Thus, for the VQA task, ReasonNet incorporates the following set of modules: 1. question-specific
visual attention module, 2. object-specific visual attention module, 3. face-specific visual attention
module, 4. object classification module, 5. scene classification module, 6. face analysis classification
module. In the following we give a formal definition of the VQA problem and details of the model?s
modules when applied on the VQA task. The network architecture is visualized in Figure 1.
Namely, the VQA problem can be solved by modeling the likelihood probability distribution pvqa
which for each answer a in the answer set ? outputs the probability of being the correct answer, given
a question Q about an image I:
a
? = arg max pvqa (a|Q, I; ?),
(5)
a??
4
Figure 1: Network architecture diagram of the ReasonNet model applied on the VQA task. Round
rectangles represent attention modules, squared rectangles represent classification modules, small
trapezoids represent encoder units (Eq. (3)), thin rectangles represent the learned multimodal representation vectors, N represents the bilinear interaction model (Eq. (4)), and the big trapezoid is a
multi-layer perceptron network that classifies the reasoning vector g to an answer a (Eq. (7))
where ? are the model parameters, a
? is the predicted answer, and ? is the set of possible answers.
4.1
ReasonNet VQA modules
First, ReasonNet obtains the question representation vector r l and a global visual feature tensor V as
described in Section 3.1. ReasonNet then learns a question-specific image representation by using
the question representation vector r l to learn an attention probability distribution ?v ? RW ?H over
the global visual feature tensor V ? RF ?W ?H :
h
i
?v = softmax W ? ? (W v r l + bv ) ? 1 ? ?(W V V + bV ) + b? ,
where ?(?) := tanh(?), W ? , W v , W V and the corresponding biases are learned parameters, 1 ?
1W ?H is used to tile the question vector representation to match the V tensor dimensionality, and ?
denotes element-wise matrix multiplication. The question-specific visual feature vector v v is then
obtained with Eq. (2) and the representation vector r v with Eq. (3).
Naturally, many of the VQA questions are about image objects so ReasonNet incorporates a fully
convolutional network (FCN) [28] for object detection. Given an image, the FCN will output a
set of object bounding boxes and their corresponding confidence scores. Each bounding box is
represented as a four-element vector d> = [x, y, w, h], where (x, y) is the coordinate of the top-left
box corner and w, h is the size of the box. Using a confidence score threshold ReasonNet obtains
a set B containing high confidence bounding boxes. ReasonNet then uses the set B to compute an
attention probability distribution ?o that focuses the visual feature tensor V on the image objects.
To ground the image pixels to visual feature maps, all images are resized to the same pre-fixed
dimension before feeding them to the object detection module. Thus each feature vector v ? RF from
a corresponding element in the tensor slice V slice ? RW ?H from the feature tensor V ? RF ?W ?H
maps to a fixed sized image region.
Formally, for each dk ? B, ReasonNet calculates a single object attention ? k . The |B| object-specific
attentions are then used to calculate the overall object attention distribution ?o :
?o = softmax(?
? o ),
k
?i,j
o
k
?
? i,j
= max (?i,j
),
k=1,...,|B|
?
dkx ? i ? (dkx + dkw ),
?1
dky ? j ? (dky + dkh ),
=
?
0.1 otherwise.
5
(6)
As before, the object-specific visual feature vector v o is then obtained with Eq. (2) and the representation vector r o with Eq. (3).
ReasonNet further uses the object bounding boxes to obtain a vector of object class labels. Namely,
for each bounding box in B, ReasonNet crops out the image part corresponding to the box coordinates
and then uses a residual network to classify the cropped-out image part and obtain a class label. The
n class labels of the boxes with highest class probability are represented as n one-hot vectors of
lookup table indices. The matrix P c , obtained by stacking the n vectors, is then mapped to a dense
low-dimensional vector r c with Eq. (1) and Eq. (3).
Next, ReasonNet uses a scene classification network as many of the questions explicitly or implicitly
necessitate the knowledge of the image setting. The scene classification network is implemented as a
residual network trained on the scene classification task. As before, the top n predicted class labels
are represented as a matrix of n one-hot vectors P s from which the module?s representation vector
r s is obtained (Eq. (1) and Eq. (3)).
Since the VQA datasets [7, 3] contain human-posed questions, many of the questions are about
people. Thus, ReasonNet also incorporates a face detector module, and a face analysis classification
module.
The face detector module is a fully convolutional network that outputs a set of face bounding boxes
and confidence scores. As with the object detector, ReasonNet uses a threshold to filter out bounding
boxes with low confidence scores and obtain a set of face detections F. Then, from F, using Eq. (6),
ReasonNet obtains an attention probability distribution ? f that focuses the visual feature tensor V
on people?s faces. The face-specific visual feature vector v f is then obtained with Eq. (2) and the
representation vector r f with Eq. (3).
The face bounding boxes from F are also used to crop out the image regions that contain a face and
using a convolutional neural network to obtain three class labels for each detected face representing
the age group, the gender, and the emotion. As with the other classification modules, ReasonNet represents the three class labels as a matrix of one-hot vectors P a and uses Equations (1) and (3) to
obtain the face analysis representation vector r a .
ReasonNet obtains a complete multimodal understanding by learning the interaction of the learned
representations H = {r v , r o , r c , r s , r f , r a } with the question representation r q :
n
s q
s
g=
(r >
h W h r + bh ),
r h ?H
f
where W sh is a learned bilinear tensor for a representation r h , and denotes concatenation of vectors.
Finally, ReasonNet forwards the vector g, containing the question representation and multifaceted
image representations, to a two-layer perceptron network which outputs the probabilities pvqa (Eq. (5)):
h
i
pvqa (a|Q, I; ?) = softmax ? W g2 ?(W g1 g + bg1 ) + bg2 ,
(7)
where ? represents all the model parameters.
5
5.1
Experiments
Datasets
We evaluate our model on the two benchmark VQA datasets, VQA v1.0 [3] and VQA v2.0 [7]. The
VQA v1.0 dataset is the first large-scale VQA dataset. The dataset contains three human-posed
questions and answers for each one of the 204,721 images found in the MS-COCO [27] dataset.
We also evaluate our model on the second version of this dataset, the VQA v2.0. The new version
includes about twice as many question-answer pairs and addresses the dataset bias issues [7] of the
VQA v1.0 dataset. We report results according to the evaluation metric provided by the authors of
the VQA datasets, where an answer is counted as correct if at least three annotators gave that answer:
P10
j=1 1(a = aj )
Acc(a) = min(
, 1).
3
For fair evaluation, we use the publicly available VQA evaluation servers to compute the overall and
per question type results.
6
Table 1: Results of the ablation study on the VQA v2.0 validation.
Method
All
Y/N
Num Other Q-type changed
VQA
55.13 69.07 34.29 48.01
VQA+Sc
56.80 70.62 35.14 49.99 +2.74% Which
VQA+Sc+oDec
58.46 71.05 36.16 52.86 +5.73% What color is the
VQA+Sc+oDec+oCls
59.82 72.88 37.38 54.47 +3.68% How
VQA+Sc+oDec+oCls+fDec
60.35 74.21 37.46 53.79 +12.63% Is the man
VQA+Sc+oDec+oCls+fDec+fAna 60.60 73.78 36.98 54.81 +0.88% Is he
ReasonNet-HadamardProduct
58.37 71.05 35.99 52.72
ReasonNet-MCB [6]
58.78 71.04 36.96 53.35
ReasonNet
60.60 73.78 36.98 54.81
5.2
Implementation details
Given a question Q about an image I, our model works as follows. Following [18, 6] the images
are scaled and center-cropped to a dimensionality of 3 ? 448 ? 448, then are fed through a ResNet152 [8] pretrained on ImageNet [36]. We utilize the output of the last convolutional layer as the image
representation V ? R2048?14?14 .
The question words are converted to lowercase and all punctuation characters are removed. We
further remove some uninformative words such as ?a, ?an?, ?the?, etc. We then trim the questions to
contain at most ten question words by removing the words after the first ten. The lookup table matrix
uses 300-dimensional vectors, initialized with word2vec [31] vectors.
The module parameters used to produce the module?s outputs are pretrained on each specific task and
are kept fixed when applying ReasonNet to the VQA problem. In the following we give details of
each module.
The object detection module is implemented and pretrained as in [32, 33]. The object classification
and scene classification modules are implemented as ResNet-152, the only difference is the object
classification module is pretrained on MS-COCO while the scene classification module is pretrained
on Places365 [48, 49]. The object classification module outputs 80 different class labels [27], while
the scene classification module outputs 365 class labels [48].
We implement and pretrain the face detection module following Zhang et al. [46, 47], while the age
and gender classification is performed as [34, 35] and the emotion recognition following [25, 26]. The
output of the age classification network is an integer from zero to hundred, so we group the integers
into four named groups, 0 ? 12 as ?child?, 13 ? 30 as ?young?, 31 ? 65 as ?adult? and +65 as ?old?.
This enables as to map the integer outputs to a class labels. Similarly, the output from the gender
classification module is 0 for ?woman? and 1 for ?man?. Finally, the emotion recognition module
classifies a detected face to the following seven emotions ?Angry?, ?Disgust?, ?Fear?, ?Happy?,
?Neutral?, ?Sad?, and ?Surprise?.
The encoder units encodes the module outputs to 500-dimensional vectors, with a hidden layer of
1,500 dimensions. Each bilinear interaction model outputs a 500-dimensional interaction vector, i.e.
500 ? 500 ? 500. The classification network classifies the reasoning vector g using one hidden
layer of 2,500 dimensions to one of 4,096 most common answers in the training set.
We jointly optimize the parameters of the encoder units, the bilinear models, and the answer classification network using Adam [19] with a learning rate of 0.0007, without learning rate decay. We
apply a gradient clipping threshold of 5 and use dropout[41] (with p(keep) = 0.5) layers before and
batch normalization[13] after each fully-conected layer as a regularization.
5.3
Ablation study
To assess the contribution of each ReasonNet module we perform an ablation study where we train a
model that only uses one module and then subsequently add the rest of the proposed VQA modules.
A VQA model with only question-specific attention module is denoted as ?VQA?, the addition of the
scene classification module is denoted as ?Sc?, the object detection module as ?oDec?, the object
classification module as ?oCls?, the face detection module as ?fDec?, and the face analysis as ?fAna?.
7
To evaluate the bilinear model as representation fusion mechanism, we compare ReasonNet to models
where we only swap the bilinear interaction learning (Eq. (4)) with (1) Hadamard product (denoted
as ReasonNet-HadamardProduct) and with (2) multimodal compact bilinear pooling [6] (denoted
as ReasonNet-MCB). The bilinear interaction model maps the two vectors to an interaction vector,
by learning a projection matrix W that projects the vectorized outer product of the two vectors
to an interaction vector. When using Hadamard product the interaction vector is just an elementwise multiplication of the two vectors. On the other hand, the MCB uses Count Sketch projection
function [4] to project the two vectors to a lower dimension and then applies convolution of the two
count sketch vectors to produce an interaction vector. As opposed to a bilinear model, the MCB does
not learn a projection matrix. We train the models on the VQA v2.0 train set and evaluate them on
the validation set. The results are shown in Table 1.
From Table 1 we can observe that each module addition improves the overall score. The results
show that the object detection module is responsible for the highest increase in accuracy, specifically
for question of type ?Other?. The addition of the object classification module further improves the
accuracy on the ?Other? question types, but the addition of the face detection module reduces the
accuracy of ?Other? question types and increases the accuracy on the ?Yes/No? questions. Possible
reasons for this is that the two attention modules (object detection and face detection) bring too much
noise in the image representation. The increase in accuracy for the ?Yes/No? questions is likely
because most ?Yes/No? questions are about people. Finally, the addition of the face analysis module
brings the highest accuracy by returning the accuracy of the ?Other? question types, possibly due to
the face class labels help in understanding the face attention.
The results in Table 1 clearly show the representational expressiveness of the bilinear models as
representation fusion. The bilinear model improves the accuracy for all question types, while there is
a small difference between the Hadamard product and compact bilinear pooling, as discussed in [18].
Figure 2: Qualitative results: we visualize the concatenation vector g from Eq. (4.1) to investigate
the module utilization given an image and two questions about the same image. The question-image
pairs are from the VQA v2.0 test set.
5.4
Qualitative analysis and failure cases
To investigate the contribution of each of the network modules, we visualize the concatenation vector
g from Eq. (4.1) in Figure 2. We show two images from the VQA v2.0 test set and the corresponding
g vector values for two questions. Higher values are displayed with lighter shade of gray.
From Figure 2 we can observe that for the question ?Does the man look happy?? the network strongly
use the representation from the face analysis module and partially use the question-only representation
and the question-specific attention representation. We can observe the same trend in the next two
questions. It is interesting to observe that for complex questions such as ?What color is the building
the bikes are in front of?? most of the network modules are used which means the network does
actually need multifaceted image representation to answer the questions.
The first example in Figure 2 also serves as a failure case. Namely, for the question ?Does the man
look happy?? the network correctly learns to use the face analysis module when the word ?happy? is
8
Table 2: Results on the VQA v1.0 and v2.0 test-standard datasets for single models and without
data augmentation. NMN is the only other modular neural. Results are taken from the official VQA
evaluation servers.
VQA v1.0 test
VQA v2.0 test
Method
All Y/N Num Other
All
Y/N
Num Other
VQA-LangOnly 48.9 78.1 34.9
27.0 44.26 67.01 31.55 27.37
D-LSTM-nI [14] 58.2 80.6 36.5
43.7 54.22 73.46 35.18 41.83
NMN [2]
58.7
DMN+ [43]
60.4 80.4 36.8
48.3
MRN [17]
61.8 82.4 38.2
49.4
HieCoAtt [29]
62.1 79.9 38.2
51.9
MCB [6] 1
64.7 82.5 37.6
55.6 62.27 78.82 38.28 53.36
MLB [18]
65.1 84.0 38.0
54.8
ReasonNet2
67.9 84.0 38.7
60.4 64.61 78.86 41.98 57.39
present in the question. However, the face analysis module incorrectly classifies the face as ?Angry?
misleading the network to give a wrong answer. Such error propagation from individual network
modules is the main limitation of the proposed model. Future work can possibly overcome this
limitation by backpropagating the error through the network modules. On the other hand, there is a
constant improvement by the research community for each individual computer vision sub-task, that
the limitation might be alleviated by simply incorporating the latest state-of-the-art network module.
5.5
Comparison with the state-of-the-art
Compared with the previous state-of-the-art on the VQA v1.0 dataset, the ReasonNet model achieves
2.8% higher accuracy (Table 2). The improvement in accuracy predominately comes from the
ability of the ReasonNet model to answer complex questions as evident from the 5.6% increase in
accuracy (denoted as ?Other? in Table 2). This validates the ability of the ReasonNet to learn complex
question-image relationships and to perform reasoning over the learned multimodal representations.
We observe the same improvement in accuracy of 2.34% on the more challenging VQA v2.0 dataset.
As on the VQA v1.0, the main improvement comes in answering the questions of type ?Other? where
there is a 4.03% difference. The improvement in the ?Other? questions likely comes from learning
complex interactions of all modules outputs. There is also an improvement of 3.7% in the counting
questions denoted as ?Num?, which serves as evidence of the contribution of the object detection and
object classification modules.
The new state-of-the-art on these datasets indicates the superiority of ReasonNet and the need
for reasoning when answering complicated questions whose answering requires reasoning and
understanding the relationship of multiple image objects.
6
Conclusion
We have presented a novel model for multimodal representation learning and reasoning. Our proposed
reasoning model learns to reason over a learned multifaceted image representation conditioned on a
text data. We validated the proposed reasoning neural network on the challenging VQA task and the
model achieved a new state-of-the-art performance by a wide margin. The proposed reasoning model
is general and probably applicable to other tasks involving multimodal representations, such as image
captioning. We leave this promising direction for future work.
Acknowledgments
The work of Jiashi Feng was partially supported by National University of Singapore startup grant
R-263-000-C08-133, Ministry of Education of Singapore AcRF Tier One grant R-263-000-C21-112
and NUS IDS grant R-263-000-C67-646.
1
Fukui et al. in [6] only report the test-dev results for VQA v1.0. The VQA v2.0 results are obtained from an
implementation of their model.
2
Due to a bug in the answer generating script the reviewed draft reported slightly lower VQA v2.0 results.
9
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12
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6,255 | 6,659 | Adversarial Surrogate Losses for Ordinal Regression
Rizal Fathony
Mohammad Bashiri
Brian D. Ziebart
Department of Computer Science
University of Illinois at Chicago
Chicago, IL 60607
{rfatho2, mbashi4, bziebart}@uic.edu
Abstract
Ordinal regression seeks class label predictions when the penalty incurred for
mistakes increases according to an ordering over the labels. The absolute error
is a canonical example. Many existing methods for this task reduce to binary
classification problems and employ surrogate losses, such as the hinge loss. We
instead derive uniquely defined surrogate ordinal regression loss functions by
seeking the predictor that is robust to the worst-case approximations of training
data labels, subject to matching certain provided training data statistics. We
demonstrate the advantages of our approach over other surrogate losses based on
hinge loss approximations using UCI ordinal prediction tasks.
1
Introduction
For many classification tasks, the discrete class labels being predicted have an inherent order (e.g.,
poor, fair, good, very good, and excellent labels). Confusing two classes that are distant from one
another (e.g., poor instead of excellent) is more detrimental than confusing two classes that are nearby.
The absolute error, |?
y ? y| between label prediction (?
y ? Y) and actual label (y ? Y) is a canonical
ordinal regression loss function. The ordinal regression task seeks class label predictions for new
datapoints that minimize losses of this kind.
Many prevalent methods reduce the ordinal regression task to subtasks solved using existing supervised learning techniques. Some view the task from the regression perspective and learn both a linear
regression function and a set of thresholds that define class boundaries [1?5]. Other methods take a
classification perspective and use tools from cost-sensitive classification [6?8]. However, since the
absolute error of a predictor on training data is typically a non-convex (and non-continuous) function
of the predictor?s parameters for each of these formulations, surrogate losses that approximate the
absolute error must be optimized instead. Under both perspectives, surrogate losses for ordinal regression are constructed by transforming the surrogate losses for binary zero-one loss problems?such as
the hinge loss, the logistic loss, and the exponential loss?to take into account the different penalties
of the ordinal regression problem. Empirical evaluations have compared the appropriateness of
different surrogate losses, but these still leave the possibility of undiscovered surrogates that align
better with the ordinal regression loss.
To address these limitations, we seek the most robust [9] ordinal regression predictions by focusing
on the following adversarial formulation of the ordinal regression task: what predictor best minimizes
absolute error in the worst case given partial knowledge of the conditional label distribution? We
answer this question by considering the Nash equilibrium for a game defined by combining the loss
function with Lagrangian potential functions [10]. We derive a surrogate loss function for empirical
risk minimization that realizes this same adversarial predictor. We show that different types of
available knowledge about the conditional label distribution lead to thresholded regression-based
predictions or classification-based predictions. In both cases, the surrogate loss is novel compared to
existing surrogate losses. We also show that our surrogate losses enjoy Fisher consistency, a desirable
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
theoretical property guaranteeing that minimizing the surrogate loss produces Bayes optimal decisions
for the original loss in the limit. We develop two different approaches for optimizing the loss: a
stochastic optimization of the primal objective and a quadratic program formulation of the dual
objective. The second approach enables us to efficiently employ the kernel trick to provide a richer
feature representation without an overly burdensome time complexity. We demonstrate the benefits
of our adversarial formulation over previous ordinal regression methods based on hinge loss for a
range of prediction tasks using UCI datasets.
2
Background and Related Work
2.1
Ordinal Regression Problems
Ordinal regression is a discrete label prediction problem characterized by Table 1: Ordinal rean ordered penalty for making mistakes: loss(?
y1 , y) < loss(?
y2 , y) if y < gression loss matrix.
?
?
y?1 < y?2 or y > y?1 > y?2 . Though many loss functions possess this property,
0 1 2 3
the absolute error |?
y ? y| is the most widely studied. We similarly restrict
?1 0 1 2?
our consideration to this loss function in this paper. The full loss matrix
?2 1 0 1?
L for absolute error with four labels is shown in Table 1. The expected
3 2 1 0
loss incurred using a probabilistic predictor P? (?
y |x) evaluated on true data
P
? y |x)Ly?,y . The supervised
distribution P (x, y) is: EX,Y ?P ;Y? |X?P? [LY? ,Y ] =
x,y,?
y P (x, y)P (?
y |x) in a way
learning objective for this problem setting is to construct a probabilistic predictor P? (?
that minimizes this expected loss using training samples distributed according to the empirical
distribution P? (x, y), which are drawn from the unknown true data generating distribution, P (x, y).
A na?ve ordinal regression approach relaxes the task to a continuous prediction problem, minimizes
the least absolute deviation [11], and then rounds predictions to nearest integral label [12]. More
sophisticated methods range from using a cumulative link model [13] that assumes the cumulative
conditional probability P (Y ? j|x) follows a link function, to Bayesian non-parametric approaches
[14] and many others [15?22]. We narrow our focus over this broad range of methods found in the
related work to those that can be viewed as empirical risk minimization methods with piece-wise
convex surrogates, which are more closely related to our approach.
2.2
Threshold Methods for Ordinal Regression
Threshold methods are one popular family of techniques that treat the ordinal response variable,
f? , w ? x, as a continuous real-valued variable and introduce |Y| ? 1 thresholds ?1 , ?2 , ..., ?|Y|?1
that partition the real line into |Y| segments: ?0 = ?? < ?1 < ?2 < ... < ?|Y|?1 < ?|Y| = ?
[4]. Each segment corresponds to a label with y?i assigned label j if ?j?1 < f? ? ?j . There are two
different approaches for constructing surrogate losses based on the threshold methods to optimize the
choice of w and ?1 , . . . , ?|Y|?1 : one is based on penalizing all thresholds involved when a mistake is
made and one is based on only penalizing the most immediate thresholds.
All thresholds methods penalize every erroneous threshold using a surrogate loss, ?, for sets of binary
Py?1
P|Y|
classification problems: lossAT (f?, y) = k=1 ?(?(?k ? f?)) + k=y ?(?k ? f?). Shashua and Levin
[1] studied the hinge loss under the name of support vector machines with a sum-of margin strategy,
while Chu and Keerthi [2] proposed a similar approach under the name of support vector ordinal
regression with implicit constraints (SVORIM). Lin and Li [3] proposed ordinal regression boosting,
an all thresholds method using the exponential loss as a surrogate. Finally, Rennie and Srebro [4]
proposed a unifying approach for all threshold methods under a variety of surrogate losses.
Rather than penalizing all erroneous thresholds when an error is made, immediate thresholds methods
only penalize the threshold of the true label and the threshold immediately beneath the true label:
lossIT (f?, y) = ?(?(?y?1 ? f?)) + ?(?y ? f?).1 Similar to the all thresholds methods, immediate threshold methods have also been studied in the literature under different names. For hinge loss surrogates,
Shashua and Levin [1] called the model support vector with fixed-margin strategy while Chu and
Keerthi [2] use the term support vector ordinal regression with explicit constraints (SVOREX). For
1
For the boundary labels, the method defines ?(?(?0 ? f?)) = ?(?y+1 ? f?) = 0.
2
the exponential loss, Lin and Li [3] introduced ordinal regression boosting with left-right margins.
Rennie and Srebro [4] also proposed a unifying framework for immediate threshold methods.
2.3
Reduction Framework from Ordinal Regression to Binary Classification
Li and Lin [5] proposed a reduction framework to convert ordinal regression problems to binary
classification problems by extending training examples. For each training sample (x, y), the reduction
framework creates |Y| ? 1 extended samples (x(j) , y (j) ) and assigns weight wy,j to each extended
sample. The binary label associated with the extended sample is equivalent to the answer of the
question: ?is the rank of x greater than j?? The reduction framework allows a choice for how extended
samples x(j) are constructed from original samples x and how to perform binary classification. If
the threshold method is used to construct the extended sample and SVM is used as the binary
classification algorithm, the classifier can be obtained by solving a family of quadratic optimization
problems that includes SVORIM and SVOREX as special instances.
2.4
Cost-sensitive Classification Methods for Ordinal Regression
Rather than using thresholding or the reduction framework, ordinal regression can also be cast as a
special case of cost-sensitive multiclass classification. Two of the most popular classification-based
ordinal regression techniques are extensions of one-versus-one (OVO) and one-versus-all (OVA) costsensitive classification [6, 7]. Both algorithms leverage a transformation that converts a cost-sensitive
classification problem to a set of weighted binary classification problems. Rather than reducing
to binary classification, Tu and Lin [8] reduce cost-sensitive classification to one-sided regression
(OSR), which can be viewed as an extension of the one-versus-all (OVA) technique.
2.5
Adversarial Prediction
Foundational results establish a duality between adversarial logarithmic loss minimization and
constrained maximization of the entropy [23]. This takes the form of a zero-sum game between
a predictor seeking to minimize expected logarithmic loss and an adversary seeking to maximize
this same loss. Additionally, the adversary is constrained to choose a distribution that matches
certain sample statistics. Ultimately, through the duality to maximum entropy, this is equivalent
to maximum likelihood estimation of probability distributions that are members of the exponential
family [23]. Gr?nwald and Dawid [9] emphasize this formulation as a justification for the principle of
maximum entropy [24] and generalize the adversarial formulation to other loss functions. Extensions
to multivariate performance measures [25] and non-IID settings [26] have demonstrated the versatility
of this perspective.
Recent analysis [27, 28] has shown that for the special case of zero-one loss classification, this
adversarial formulation is equivalent to empirical risk minimization with a surrogate loss function:
X
AL0-1
max
(?j,yi (xi ) + |S| ? 1)/|S|,
(1)
f (xi , yi ) =
S?{1,...,|Y|},S6=?
j?S
where ?j,yi (xi ) is the potential difference ?j,yi (xi ) = fj (xi ) ? fyi (xi ). This surrogate loss function
provides a key theoretical advantage compared to the Crammer-Singer hinge loss surrogate for
multiclass classification [29]: it guarantees Fisher consistency [27] while Crammer-Singer?despite
its popularity in many applications, such as Structured SVM [30, 31]?does not [32, 33]. We extend
this type of analysis to the ordinal regression setting with the absolute error as the loss function in
this paper, producing novel surrogate loss functions that provide better predictions than other convex,
piece-wise linear surrogates.
3
3.1
Adversarial Ordinal Regression
Formulation as a zero-sum game
We seek the ordinal regression predictor that is the most robust to uncertainty given partial knowledge
of the evaluating distribution?s characteristics. This takes the form of a zero-sum game between a
predictor player choosing a predicted label distribution P? (?
y |x) that minimizes loss and an adversarial
3
player choosing an evaluation distribution P? (?
y |x) that maximizes loss while closely matching the
feature-based statistics of the training data:
min
i
h
?
max EX?P ;Y? |X?P? ;Y? |X?P? Y? ? Y? such that: EX?P ;Y? |X?P? [?(X, Y? )] = ?.
y |x)
P? (?
y |x) P? (?
(2)
The vector of feature moments, ?? = EX,Y ?P? [?(X, Y )], is measured from sample training data
distributed according to the empirical distribution P? (x, y).
An ordinal regression problem can be viewed as a cost-sensitive loss with the entries of the cost
matrix defined by the absolute loss between the row and column labels (an example of the cost
matrix for the case of a problem with four labels is shown in Table 1). Following the construction of
adversarial prediction games for cost-sensitive classification [10], the optimization of Eq. (2) reduces
to minimizing the equilibrium game values of a new set of zero-sum games characterized by matrix
L0xi ,w :
f1 ? fyi
}|
{
f1 ? fyi + 1
Xz
?
? Txi L0xi ,w p
? xi ; L0xi ,w=?
min
max min p
..
?
w
? xi p
p
? xi
.
i
{z
}
|
f1 ? fyi + |Y| ? 1
zero-sum game
?
convex optimization of w
???
???
..
.
???
?
f|Y| ? fyi + |Y| ? 1
f|Y| ? fyi + |Y| ? 2?
? , (3)
..
?
.
f|Y| ? fyi
where: w represents a vector of Lagrangian model parameters; fj = w ? ?(xi , j) is a Lagrangian
? xi is a vector representation of the conditional label distribution, P? (Y? = j|xi ), i.e.,
potential; p
?
? xi = [P (Y? = 1|xi ) P? (Y? = 2|xi ) . . .]T ; and p
? xi is similarly defined. The matrix L0xi ,w =
p
(|?
y ? y?| + fy? ? fyi ) is a zero-sum game matrix for each example. This optimization problem (Eq. (3))
is convex in w and the inner zero-sum game can be solved using a linear program [10]. To address
finite sample estimation errors, the difference between expected and sample feature can be bounded
? ? , leading to Lagrangian parameter regularization in
in Eq. (2), ||EX?P ;Y? |X?P? [?(X, Y? )] ? ?||
Eq. (3) [34].
3.2
Feature representations
We consider two feature representations corresponding to different training data summaries:
?
?
?
?th (x, y) = ?
?
?
yx
I(y ? 1)
I(y ? 2)
..
.
I(y ? |Y| ? 1)
?
?
I(y = 1)x
? I(y = 2)x ?
? I(y = 3)x ?
?.
?mc (x, y) = ?
?
?
..
?
?
.
I(y = |Y|)x
?
?
?
? ; and
?
?
(4)
The first, which we call the thresholded regression representation,
has size m + |Y| ? 1, where m is the dimension of our input space. It
induces a single shared vector of feature weights and a set of thresholds. If we denote the weight vector associated with the yx term as
w and the terms associated with each sum of class indicator functions as ?1 , ?2 , . . ., ?|Y|?1 , then thresholds for switching between
class j and j + 1 (ignoring other classes) occur when w ? xi = ?j .
The second feature representation, ?mc , which we call the multiclass representation, has size m|Y| and can be equivalently interpreted as inducing a set of class-specific feature weights, fj = wj ?xi .
This feature representation is useful when ordered labels cannot be
thresholded according to any single direction in the input space, as Figure 1: Example where mulshown in the example dataset of Figure 1.
tiple weight vectors are useful.
4
3.3
Adversarial Loss from the Nash Equilibrium
We now present the main technical contribution of our paper: a surrogate loss function that, when
minimized, produces a solution to the adversarial ordinal regression problem of Eq. (3).2
Theorem 1. An adversarial ordinal regression predictor is obtained by choosing parameters w that
minimize the empirical risk of the surrogate loss function:
fj + fl + j ? l
fj + j
fl ? l
? fyi = max
+ max
? fyi ,
j
l
2
2
2
j,l?{1,...,|Y|}
ALord
w (xi , yi ) =
max
(5)
where fj = w ? ?(xi , j) for all j ? {1, . . . , |Y|}.
f +f +j?l
Proof sketch. Let j ? , l? be the solution of argmaxj,l?{1,...,|Y|} j 2l
, we show that the Nash
equilibrium value of a game matrix that contains only row j ? and l? and column j ? and l? from
f ? ,+fl? +j ? ?l?
matrix L0xi ,w is exactly j
. We then show that adding other rows and columns in L0xi ,w
2
to the game matrix does not change the game value. Given the resulting closed form solution of the
game (instead of a minimax), we can recast the adversarial framework for ordinal regression as an
empirical risk minimization with the proposed loss.
We note that the ALord
w surrogate is the maximization over pairs of different potential functions
associated with each class (including pairs of identical class labels) added to the distance between the
pair. For both of our feature representations, we make use of the fact that maximization over each
element of the pair can be independently realized, as shown on the right-hand side of Eq. (5).
Thresholded regression surrogate loss
In the thresholded regression feature representation, the parameter contains a single shared vector of
feature weights w and |Y| ? 1 terms ?k associated with thresholds. Following Eq. (5), the adversarial
ordinal regression surrogate loss for this feature representation can be written as:
ord-th
AL
(xi , yi ) = max
j(w ? xi + 1) +
j
P
k?j
?k
2
+ max
l
l(w ? xi ? 1) +
2
P
k?l
?k
? yi w ? x i ?
X
?k .
k?yi
(6)
This loss has a straight-forward interpretation in terms of the thresholded regression
perspective, as shown in Figure 2: it is
based on averaging the thresholded label
predictions for potentials w ? xi + 1 and
w ? xi ? 1. This penalization of the pair of
thresholds differs from the thresholded surrogate losses of related work, which either
penalize all violated thresholds or penalize
only the thresholds adjacent to the actual
Figure 2: Surrogate loss calculation for datapoint xi
class label.
(projected to w ? xi ) with a label prediction of 4 for preUsing a binary search procedure over
dictive purposes, the surrogate loss is instead obtained
?1 , . . . , ?|Y|?1 , the largest lower bounding
using potentials for the classes based on w ?xi +1 (label
threshold for each of these potentials can
5) and w ? xi ? 1 (label 2) averaged together.
be obtained in O(log |Y|) time.
Multiclass ordinal surrogate loss
In the multiclass feature representation, we have a set of specific feature weights wj for each label
and the adversarial multiclass ordinal surrogate loss can be written as:
ALord-mc (xi , yi ) =
max
j,l?{1,...,|Y|}
wj ? xi + wl ? xi + j ? l
? wyi ? xi .
2
2
(7)
The detailed proof of this theorem and others are contained in the supplementary materials. Proof sketches
are presented in the main paper.
5
(a)
(b)
(c)
Figure 3: Loss function contour plots of ALord over the space of potential differences ?j , fj ? fyi
for the prediction task with three classes when the true label is yi = 1 (a), yi = 2 (b), and yi = 3 (c).
We can also view this as the maximization over |Y|(|Y| + 1)/2 linear hyperplanes. For an ordinal
regression problem with three classes, the loss has six facets with different shapes for each true label
value, as shown in Figure 3. In contrast with ALord-th , the class label potentials for ALord-mc may
differ from one another in more-or-less arbitrary ways. Thus, searching for the maximal j and l class
labels requires O(|Y|) time.
3.4
Consistency Properties
The behavior of a prediction method in ideal learning settings?i.e., trained on the true evaluation
distribution and given an arbitrarily rich feature representation, or, equivalently, considering the space
of all measurable functions?provides a useful theoretical validation. Fisher consistency requires that
the prediction model yields the Bayes optimal decision boundary [32, 33, 35] in this setting. Given
the true label conditional probability Pj (x) , P (Y = j|x), a surrogate loss function ? is said to
be Fisher consistent with respect to the loss ` if the minimizer f ? of the surrogate loss achieves the
Bayes optimal risk, i.e.,:
f ? = argmin EY |X?P [?f (X, Y )|X = x]
(8)
f
? EY |X?P [`f ? (X, Y )|X = x] = min EY |X?P [`f (X, Y )|X = x] .
f
Ramaswamy and Agarwal [36] provide a necessary and sufficient condition for a surrogate loss to be
Fisher consistent with respect to general multiclass losses, which includes ordinal regression losses.
A recent analysis by Pedregosa et al. [35] shows that the all thresholds and the immediate thresholds
methods are Fisher consistent provided that the base binary surrogates losses they use are convex
with a negative derivative at zero.
For our proposed approach, the condition for Fisher consistency above is equivalent to:
f ? = argmin
f
X
y
X
fj + fl + j ? l
Py max
? fy ? argmax fj? (x) ? argmin
Py |j ? y| .
j,l
2
j
j
y
(9)
Since adding a constant to all fj does not change the value of both ALord
f and argmaxj fj (x), we
employ the constraint maxj fj (x) = 0, to remove redundant solutions for the consistency analysis.
We establish an important property of the minimizer for ALord
f in the following theorem.
ord
?
Theorem 2. The minimizer vector f of EY |X?P ALf (X, Y )|X = x satisfies the loss reflective
property, i.e., it complements the absolute error by starting with a negative integer value, then
increasing by one until reaching zero, and then incrementally decreases again.
Proof sketch. We show that for any f 0 that does not satisfy the loss reflective property, we can
construct f 1 using several steps that satisfy the loss reflective property and has the expected loss value
less than the expected loss of f 0 .
6
Example vectors f ? that satisfy Theorem 2 are [0, ?1, ?2]T , [?1, 0, ?1]T and [?2, ?1, 0]T for
three-class problems, and [?3, ?2, ?1, 0, ?1] for five-class problems. Using the key property of the
minimizer above, we establish the consistency of our loss functions in the following theorem.
Theorem 3. The adversarial ordinal regression surrogate loss ALord from Eq. (5) is Fisher consistent.
Proof sketch. We only consider |Y| possible values of f that satisfy the loss reflective property. For
the f that corresponds to class j, the value of the expected loss is equal to the Bayes loss if we predict
j as the label. Therefore minimizing over f that satisfy the loss reflective property is equivalent to
finding the Bayes optimal response.
3.5
Optimization
3.5.1
Primal Optimization
To optimize the regularized adversarial ordinal regression loss from the primal, we employ stochastic
average gradient (SAG) methods [37, 38], which have been shown to converge faster than standard
stochastic gradient optimization. The idea of SAG is to use the gradient of each example from the last
iteration where it was selected to take a step. However, the na?ve implementation of SAG requires
storing the gradient of each sample, which may be expensive in terms of the memory requirements.
Fortunately, for our loss ALord
w , we can drastically reduce this memory requirement by just storing a
f +f +j?l
pair of number, (j ? , l? ) = argmaxj,l?{1,...,|Y|} j 2l
, rather than storing the gradient for each
sample. Appendix C explains the details of this technique.
3.5.2
Dual Optimization
Dual optimization is often preferred when optimizing piecewise linear losses, such as the hinge loss,
since it enables one to easily perform the kernel trick and obtain a non-linear decision boundary
without heavily sacrificing computational efficiency. Optimizing the regularized adversarial ordinal
regression loss in the dual can be performed by solving the following quadratic optimization:
max
?,?
X
1 X
(?i,j + ?i,j ) (?k,l + ?k,l ) (?(xi , j) ? ?(xi , yi )) ? (?(xk , l) ? ?(xl , yk ))
2
i,j,k,l
X
X
? 0;
?i,j = C2 ;
?i,j = C2 ; i, k ? {1, . . . , n}; j, l ? {1, . . . , |Y|}.
(10)
j(?i,j ? ?i,j ) ?
i,j
subject to: ?i,j ? 0; ?i,j
j
j
Note that our dual formulation only depends on the dot product of the features. Therefore, we can
also easily apply the kernel trick to our algorithm. Appendix D describes the derivation from the
primal optimization to the dual optimization above.
4
4.1
Experiments
Table 2: Dataset properties.
Experiment Setup
Dataset
We conduct our experiments on a benchmark
dataset for ordinal regression [14], evaluate the
performance using mean absolute error (MAE),
and perform statistical tests on the results of different hinge loss surrogate methods. The benchmark contains datasets taken from the UCI Machine Learning repository [39] ranging from relatively small to relatively large in size. The characteristics of the datasets, including the number
of classes, the training set size, the testing set
size, and the number of features, are described
in Table 2.
#class
diabetes
pyrimidines
triazines
wisconsin
machinecpu
autompg
boston
stocks
abalone
bank
computer
calhousing
5
5
5
5
10
10
5
5
10
10
10
10
#train
#test #features
30
51
130
135
146
274
354
665
2923
5734
5734
14447
13
23
56
59
63
118
152
285
1254
2458
2458
6193
2
27
60
32
6
7
13
9
10
8
21
8
In the experiment, we consider different methods using the original feature space and a kernelized
feature space using the Gaussian radial basis function kernel. The methods that we compare include
two variations of our approach, the threshold based (ALord-th ), and the multiclass-based (ALord-mc ).
7
Table 3: The average of the mean absolute error (MAE) for each model. Bold numbers in each case
indicate that the result is the best or not significantly worse than the best (paired t-test with ? = 0.05).
Threshold-based models
Dataset
ord-th
AL
th
RED
Multiclass-based models
AT
IT
ord-mc
REDmc
CSOSR
CSOVO
AL
CSOVA
diabetes
pyrimidines
triazines
wisconsin
machinecpu
autompg
boston
stocks
abalone
bank
computer
calhousing
0.696
0.654
0.607
1.077
0.449
0.551
0.316
0.324
0.551
0.461
0.640
1.190
0.715
0.678
0.683
1.067
0.456
0.550
0.304
0.317
0.547
0.460
0.635
1.183
0.731
0.615
0.649
1.097
0.458
0.550
0.306
0.315
0.546
0.461
0.633
1.182
0.827
0.626
0.654
1.175
0.467
0.617
0.298
0.324
0.571
0.461
0.683
1.225
0.629
0.509
0.670
1.136
0.518
0.599
0.311
0.168
0.521
0.445
0.625
1.164
0.700
0.565
0.673
1.141
0.515
0.602
0.311
0.175
0.520
0.446
0.624
1.144
0.715
0.520
0.677
1.208
0.646
0.741
0.353
0.204
0.545
0.732
0.889
1.237
0.738
0.576
0.738
1.275
0.602
0.598
0.294
0.147
0.558
0.448
0.649
1.202
0.762
0.526
0.732
1.338
0.702
0.731
0.363
0.213
0.556
0.989
1.055
1.601
average
# bold
0.626
5
0.633
5
0.629
4
0.661
2
0.613
5
0.618
5
0.706
2
0.652
2
0.797
1
The baselines we use for the threshold-based models include a SVM-based reduction framework
algorithm (REDth ) [5], an all threshold method with hinge loss (AT) [1, 2], and an immediate threshold
method with hinge loss (IT) [1, 2]. For the multiclass-based models, we compare our method with an
SVM-based reduction algorithm using multiclass features (REDmc ) [5], with cost-sensitive one-sided
support vector regression (CSOSR) [8], with cost-sensitive one-versus-one SVM (CSOVO) [7], and
with cost-sensitive one-versus-all SVM (CSOVA) [6]. For our Gaussian kernel experiment, we
compare our threshold based model (ALord-th ) with SVORIM and SVOREX [2].
In our experiments, we first make 20 random splits of each dataset into training and testing sets. We
performed two stages of five-fold cross validation on the first split training set for tuning each model?s
regularization constant ?. In the first stage, the possible values for ? are 2?i , i = {1, 3, 5, 7, 9, 11, 13}.
i
Using the best ? in the first stage, we set the possible values for ? in the second stage as 2 2 ?0 , i =
{?3, ?2, ?1, 0, 1, 2, 3}, where ?0 is the best parameter obtained in the first stage. Using the selected
parameter from the second stage, we train each model on the 20 training sets and evaluate the MAE
performance on the corresponding testing set. We then perform a statistical test to find whether the
performance of a model is different with statistical significance from other models. We perform the
Gaussian kernel experiment similarly with model parameter C equals to 2i , i = {0, 3, 6, 9, 12} and
kernel parameter ? equals to 2i , i = {?12, ?9, ?6, ?3, 0} in the first stage. In the second stage, we
set C equals to 2i C0 , i = {?2, ?1, 0, 1, 2} and ? equals to 2i ?0 , i = {?2, ?1, 0, 1, 2}, where C0
and ?0 are the best parameters obtained in the first stage.
4.2
Results
We report the mean absolute error (MAE) averaged over the dataset splits as shown in Table 3 and
Table 4. We highlight the result that is either the best or not worse than the best with statistical
significance (under paired t-test with ? = 0.05) in boldface font. We also provide the summary for
each model in terms of the averaged MAE over all datasets and the number of datasets for which
each model marked with boldface font in the bottom of the table.
As we can see from Table 3, in the experiment with the original feature space, threshold-based
models perform well on relatively small datasets, whereas multiclass-based models perform well on
relatively large datasets. A possible explanation for this result is that multiclass-based models have
more flexibility in creating decision boundaries, hence they perform better if the training data size is
sufficient. However, since multiclass-based models have many more parameters than threshold-based
models (m|Y| parameters rather than m + |Y| ? 1 parameters), multiclass methods may need more
data, and hence, may not perform well on relatively small datasets.
In the threshold-based models comparison, ALord-th , REDth , and AT perform competitively on
relatively small datasets like triazines, wisconsin, machinecpu, and autompg. ALord-th has a
8
slight advantage over REDth on the overall accuracy, and a slight advantage over AT on the number
of ?indistinguishably best? performance on all datasets. We can also see that AT is superior to IT in
the experiments under the original feature space.
Among the multiclass-based models, ALord-mc Table 4: The average of MAE for models with
and REDmc perform competitively on datasets Gaussian kernel.
like abalone, bank, and computer, with a
Dataset
ALord-th SVORIM SVOREX
slight advantage of ALord-mc model on the overall accuracy. In general, the cost-sensitive moddiabetes
0.696
0.665
0688
pyrimidines
0.478
0.539
0.550
els perform poorly compared with ALord-mc and
triazines
0.609
0.612
0.604
REDmc . A notable exception is the CSOVO
wisconsin
1.090
1.113
1.049
model which perform very well on the stocks
machinecpu
0.452
0.652
0.628
and the boston datasets.
autompg
0.529
0.589
0.593
In the Gaussian kernel experiment, we can see
boston
0.278
0.324
0.316
stocks
0.103
0.099
0.100
from Table 4 that the kernelized version of
ALord-th performs significantly better than the
average
0.531
0.574
0.566
threshold-based models SVORIM and SVOREX
# bold
7
3
4
in terms of both the overall accuracy and the
number of ?indistinguishably best? performance
on all datasets. We also note that immediate-threshold-based model (SVOREX) performs better than
all-threshold-based model (SVORIM) in our experiment using Gaussian kernel. We can conclude
that our proposed adversarial losses for ordinal regression perform competitively compared to the
state-of-the-art ordinal regression models using both original feature spaces and kernel feature spaces
with a significant performance improvement in the Gaussian kernel experiments.
5
Conclusion and Future Work
In this paper, we have proposed a novel surrogate loss for ordinal regression, a classification problem
where the discrete class labels have an inherent order and penalty for making mistakes based on that
order. We focused on the absolute loss, which is the most widely used ordinal regression loss. In
contrast with existing methods, which typically reduce ordinal regression to binary classification
problems and then employ surrogates for the binary zero-one loss, we derive a unique surrogate
ordinal regression loss by seeking the predictor that is robust to a worst case constrained approximation of the training data. We derived two versions of the loss based on two different feature
representation approaches: thresholded regression and multiclass representations. We demonstrated
the benefit of our approach on a benchmark of datasets for ordinal regression tasks. Our approach
performs competitively compared to the state-of-the-art surrogate losses based on hinge loss. We
also demonstrated cases when the multiclass feature representations works better than thresholded
regression representation, and vice-versa, in our experiments.
Our future work will investigate less prevalent ordinal regression losses, such as the discrete quadratic
loss and arbitrary losses that have v-shaped penalties. Furthermore, we plan to investigate the
characteristics required of discrete ordinal losses for their optimization to have a compact analytical
solution. In terms of applications, one possible direction of future work is to combine our approach
with deep neural network models to perform end-to-end representation learning for ordinal regression
applications like age estimation and rating prediction. In that setting, our proposed loss can be used
in the last layer of a deep neural network to serve as the gradient source for the backpropagation
algorithm.
Acknowledgments
This research was supported as part of the Future of Life Institute (futureoflife.org) FLI-RFP-AI1
program, grant#2016-158710 and by NSF grant RI-#1526379.
9
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6,256 | 666 | Transient Signal Detection with Neural Networks:
The Search for the Desired Signal
Jose C. Principe and Abir Zahalka
Computational NeuroEngineering Laboratory
Department of Electrical Engineering
University of Florida, CSE 447
Gainesville, FL 32611
[email protected]
Abstract
Matched filtering has been one of the most powerful techniques
employed for transient detection. Here we will show that a dynamic
neural network outperforms the conventional approach. When the
artificial neural network (ANN) is trained with supervised learning
schemes there is a need to supply the desired signal for all time,
although we are only interested in detecting the transient. In this
paper we also show the effects on the detection agreement of
different strategies to construct the desired signal. The extension of
the Bayes decision rule (011 desired signal), optimal in static
classification, performs worse than desired signals constructed by
random noise or prediction during the background.
1
INTRODUCTION
Detection of poorly defined waveshapes in a nonstationary high noise background is
an important and difficult problem in signal processing. The matched filter is the
optimal linear filter assuming stationary noise [Thomas, 1969]. The application area
that we are going to discuss is epileptic spike detection in the electrocorticogram
(ECoG), where the matched filtering produce poor results [Barlow and Dubinsky,
1976], [Pola and Romagnoli, 1979] due to the variability of the spike shape and the
ever changing background (the brain electric activity). Recently artificial neural
networks have been applied to spike detection [Eberhart et ai, 1989], [Gabor and
688
Transient Signal Detection with Neural Networks: The Search for the Desired Signal
Seydal, 1992], but static neural network architectures were chosen. Here a static
multilayer perceptron (MLP) will be augmented with a short term memory
mechanism to detect spikes. In the way we utilized the dynamic neural net, the ANN
can be thought of as an extension of the matched filter to nonlinear models, which
we will refer to as a neural template matcher. In our implementation, the ANN looks
directly at the input signal with a time window larger than the longest spike for the
sampling frequency utilized. The input layer of the dynamic network is a delay line,
and the data is clocked one sample at a time through the network. The desired signal
is "I" following the occurrence of a spike. With this strategy we teach the ANN to
produce an output of" 1" when a waveform similar to the spike is present within the
time window. A spike will be recognized when the ANN output is above a given
threshold.
Unlike the matched filter, the ANN does not require a single, explicit waveform for
the template (due to the spike shape variability some form of averaging is needed to
create the "average" spike shape which is normally a poor compromise). Rather, the
ANN will learn the important features of the transient class by training on many
sample spikes, refining its approximation with each presentation. Moreover, the
ANN using the sigmoid nonlinearity will have to necessarily represent the
background activity, since the discriminant function in pattern space is established
from information pertaining to all input classes. Therefore, the nonstationary nature
of the background can be accommodated during network training and we can expect
that the performance of the neural template matcher will be improved with respect to
the matched filter.
We will not address here the normalization of the ECoG, nor the issues associated
with the learning criterion [Zahalka, 1992]. The purpose of this paper is to delve on
the design of the desired signal, and quantify the effect on performance. What should
be the shape of the desired sinal for transient detection, when on-line supervised
learning is employed? In our approach we decided to construct a desired signal that
exists for all time. We shall point out that the existence of a desired signal for every
sample will simplify supervised learning, since in principle the conventional
backpropagation algorithm [Rumelhart et aI, 1986] can be utilized instead of the
more time consuming backpropagation through time [Werbos, 1990] or real-time
recurrent learning [Williams and Zipzer, 1989]. The simplified learning algorithm
may very well be one of the factors which will make ANNs learn on-line as adaptive
linear filters do. The main decision regarding the desired signal for spike detection is
to decide the value of the desired signal during the background (which for spikes
represent 99% of the time), since we decided already that it should be "1" following
the spike. Similar problems have been found in speech recognition, when patterns
that exist in time need to be learned [Unnikrishnan et aI, 1991] [Watrous et aI, 1990].
Here we will experimentally compare three desired signals (Figure 1):
Desired signal 1- Extrapolating the Bayes rule for static patterns [Makhoul], we will
create a target of zero during the background, and a value of 1 following the spike ,
with a duration equal to the amount of time the spike is in the input window.
Desired signal 11- During the background, a random, uniformly distributed (between
-0.5 and 0.5), zero mean target signal. Same target as above following the spike .
689
690
Principe and Zahalka
Desired signal 111- During the background the network will be trained as a one step
predictor. Same targe,t as above following the spike.
+~--+--------
o--~I-foI----'desired
output
-lr----------------
(I). Hardlimited 01+ 1 signal
desired output .
\,
tnput
+~---------+~.-~~----
+
+0.
o
-l~-------------------
-1~--------~-------
(II). Random noise signal
(III). Prediction signal
Figure 1. Desired signals considered.
2
NEURAL NETWORK ARCmTECTURE AND TRAINING
One ECoG channel was sampled at 500 Hz (12 bit AID converter) and was preprocessed for normalization between -1 and 0.4, and DC removal before being
presented to the ANN [Zahalka, 1992]. An epileptic spike has a duration between 20
and 70 msec.
The dynamic network used for this application is a time delay neural network
(TDNN) consisting of an input layer with 45 taps, 12 hidden processing elements
(PEs) and one linear output PE. The input window corresponds to 90 msec, so even
the longest spike is fully present in the window during at least 10 samples.
The training set consists of 60 hand picked 2 sec. segments containing one spike
each, embedded in 6,860 points of background activity. In principle the data could be
streamed on-line to the ANN, provided that we could create also on-line the desired
signal. But at this point of the research we preferred to control the segment length
and choose well defined spikes to study training issues. The test data set consists of
another set (belonging to the same individual) of 49 spikes embedded in 6,970
samples. A spike was defined when the ANN output was above 0.9.
The ANN was trained with the backpropagation algorithm [Rumelhart et al 1986].
The weights were updated after every sample (real-time mode). A momentum term
(a.=0.9) was used, and 0.1 was added to the sigmoid derivative to speedup learning
[Fahlman, 1988]. The training was stopped when the test set performance decreased.
The network typically learned in less than 50 presentations of the training set. All the
Transient Signal Detection with Neural Networks: The Search for the Desired Signal
results presented next use the same trainingltest sets, the same learning and stop
criterion and the same network topology.
3
RESULTS
Desired Signal I.
We begin with the most commonly used desired signal for static classification, the
hardlimited 01+ 1 signal of Figure 1(I), but now extended in time (0 during the
background, 1 after the occurrence of the spike). This desired signal has been shown
to be optimal in the sense that it is a least square approximation of the Bayes decision
rule [Makhoul, 1991 J. However it performs poorly for transient detection (73% of
correct detections and 8% of false positives). We suspect that the problem lies in the
different waveshapes present in the background. When an explicit 0 value is given as
the desired signal for the background, the network has difficulties extracting common
features to all the waves and biases the decision rule. Samples of the network input
and the corresponding output are shown in Figure 2(1). Notice that the ANN output
gets close to "1" during high amplitude background, and fails to be above 0.9 during
small amplitude spikes. In Table 1, the test set performance is better than the training
set due to the fact that the spikes chosen for training happen to include more difficult
cases.
Table 1. Performance Results.
Desired
Training Set
Training Set
Test Set
Signal
detections
false positives
detections
Test Set
false positives
0<--> +1
38/60 = 63%
2/38 = 5%
36/49 = 73%
3/36 = 8%
noise
54/60 = 90%
0154 = 0%
46/49 = 94%
1/47 = 2%
prediction
50/60 = 83%
1150 = 2%
45/49 = 92%
2/45 = 4%
Detections mean number of events in agreement between the human expert
and the ANN (normalized by the number of spikes in the set).
False positives mean the number of events picked by the ANN, but not considered
spikes by the human expert (nonnalized by the number of ANN detections)
The "random noise" desired signal is shown in Figure 1(II). This signal consists
simply of uniformly distributed random values bounded between -0.5 and +0.5 for
the nonspike regions and again a value of "+ 1" after the spikes. This approach is
based on the fact that the random number generator will have a zero mean
distribution. Therefore, during training over the nonspike regions, the errors
backpropagated through the network will normally average out to zero, yielding in
practice a "don't care" target for the background. This effectively should give the
least bias to the decision rule . The net result is that consistent training is only
performed over the spike patterns, allowing the network to better learn these patterns
and provide improved performance. The results of this configuration are very
691
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Principe and Zahalka
promising, as can be seen in Table 1 and in Figure 2(11). The "noise" signal performs
better than all of the other desired signal configurations examined (94% correct
detections and 2% false detections).
Ok
~~~~V-\f"1t
II. 1\
D."
1\
.'
. Input to the network
~jPL~ if;. ZL:~, (
--- 1 0 0 - 0 2
-0.2
a.t~ ~ ~o
Output of network
(I) 0/+1 signal
D?"b
II
~}\fJrrwrifO
A
0 .7 5
0 ."
1\
""./\
_ r
11 "l!1
~~~;~y ~VdWV'V 7~00
o ? "..
A
. mput to the l~~twork
~~.
~
Input for the network.
~.
~~k~jUl~
?
100
200-0
300
.00
:~:~f' h II
o
500
Uutput ot' network
(II) noise desired signal
""
~~~~~ vV~v1M\[0V\J wrtJ' \.ft
Output for the network
(III) prediction signal
Figure 2. Input and output of neural network
The last signal configuration is the prediction paradigm. The network is performing
one-step prediction during the nonspike portions of the signal and a saturation value
of "+ 1" is added to the desired signal after the spike, as in Figure 1(111). The rationale
behind such a configuration is that the desired signal is readily available and may
require fewer network resources than a target of "0", decreasing the bias in the
decision rule for the spike class. The results are given in Table 1, where we see a
marked improvement over the hardlimited desired signal (92% correct detection and
4% false positives).
4
COMPARISON WITH MATCHED FILTER
The template for the matched filter was formed by averaging five spikes from the
training set. While averaging reduces morphological "crispness," it enhances the
robustness of spike shape representation, since the spike shape is variable. The 45
point template is correlated continuously with the time signal from the test set, hence
Transient Signal Detection with Neural Networks: The Search for the Desired Signal
no correction to the nonwhite nature of the ECoG is being done. Performance in both
the matched filter and the ANN approach will be dependent upon the threshold levels
chosen for detection, which determines the receiver operating characteristic (ROC)
for the detector, but requires large data sets to be meaningful. Table 2 shows a partial
result, wht;~;,! bvth detectors are compared in terms of false positives for 90%
detection rate and detection rate for 2 false positives. The results presented for the
ANN are for the random noise desired signal. In both cases, we see the superiority of
the ANN approach.
Table 2. Comparison ANNlMatched filter
90% detection rate
NN
ofalse +'s
5
MF
21 false +'s
2 false positives
NN
MF
92%
13%
CONCLUSIONS
The ultimate conclusion of this experimental work is that neural networks can be
used to implement transient detectors, outperforming the conventional matched filter
for the detection of spike transients embedded in the ECoG. However, the difficulty
in the ANN approach comes in how to setup the training, and the desired signal.
Although the training was not discussed here, we would like to point several issues
that are important. When a cost function that is sensitive to the a priori probability
of the classes is used (as the error signal in backpropagation) the proper balance of
background versus spike data sizes is relevant for adequate training. Our results
show that at a low spike concentration of 4% (ratio of spike samples over background
samples in the training set), the network never learns the waveform and always
outputs the value chosen to represent the background. For our application, these
problems disappear at an 18% concentration level.
Another important issue is the selection of the class exemplars for training. Transient
detection is a one class classification problem (A versus the universe), rather than a
two class classification problem. It is not possible to cover appropriately the
unconstrained background with examples, and even when a large number of
background waves is utilized (in our case 80% of the training samples belonged to
the background) the training produced bad results. We found out that only the waves
similar in shape to the spikes are important to bound the spike class in feature space.
As a solution, we included in the training set the false positives, i.e. waves that the
system detects as spikes but the human expert classifies as background, to improve
the detection agreement with the human [Zahalka, 1992].
The issues regarding the choice of the desired signal are also important. We found
experimentally that the best desired signal for our case is the one that uses random
noise during the background. We can not, at this point, explain this result
693
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Principe and Zahalka
theoretically. This desired signal is also the one that uses fewer network resources (i.
e. number of hidden units) without degrading the performance [Zahalka, 1992]. This
result came as a surprise since the 0/1 signal has been shown to be optimal for static
patterns in the sense that makes the classifier approximate the Bayes decision rule.
The explanation may be simply a matter of local minima in the performance surface
of the network trained with the 0/1 desired signal, but it may also reflect deeper
causes. With the OIl desired signal, the network weights are updated equally with the
information of the background waveforms and with the information regarding the
spikes. The training paradigm should emphasize the spikes, since the spikes are the
waves we are interested in. Moreover, since the background class is unconstrained,
many more degrees of freedom are necessary to represent well the background. When
not enough hidden units are available, the network biases the discriminant function,
and the performance is poor. In the prediction paradigm the background is selected
as the desired signal and the required number of hidden nodes to predict the next
sample is much smaller (actually only three hidden nodes are sufficient to keep the
reported performance [Zahalka, 1992]). The network resources naturally self
organize as two template matching nodes and one prediction node. However, we have
found out that the signal has to be properly normalized in the sense that the target for
the spike ("I") must be outside the range of the input signal voltages (that is the
reason we normalize the input signal between -1 and 0.4). From a classification point
of view we "do not care" what the net output is provided it is far from the target of
"1". The random noise target with zero mean achieves this goal very easily, because
the error gradient during the background averages out to zero. Therefore the weights
reflect primarily the information containing in the shape of the spike. We found out
that only two hidden nodes are sufficient to keep the reported performance. [Zahalka,
1992]. It seems that the theory of time varying signal classification with neural
networks is not a straight forward extension of the static classification case, and
requires further theoretical analysis.
An implication of this work regards the role of supervised learning in biological
neural networks. This work shows that during non-interesting events, there is no need
to provide a target to the neural assembly (noise, which is so readily available in
biological systems, suffices). Re-enforcement stimulus are only needed during or
after relevant events, which is compatible with the information processing models of
the olfactory bulb [Freeman and DiPrisco, 1989]. Therefore, looking at learning and
adaptation in biological systems as a signal detection instead of a classification
problem seems promising.
Acknowledgments
This work has been partially supported by NSF grants ECS-9208789 and DDM8914084.
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6,257 | 6,660 | Hypothesis Transfer Learning via
Transformation Functions
Simon S. Du
Carnegie Mellon University
[email protected]
Jayanth Koushik
Carnegie Mellon University
[email protected]
Barnab?s P?czos
Carnegie Mellon University
[email protected]
Aarti Singh
Carnegie Mellon University
[email protected]
Abstract
We consider the Hypothesis Transfer Learning (HTL) problem where one incorporates a hypothesis trained on the source domain into the learning procedure
of the target domain. Existing theoretical analysis either only studies speci?c
algorithms or only presents upper bounds on the generalization error but not on the
excess risk. In this paper, we propose a uni?ed algorithm-dependent framework
for HTL through a novel notion of transformation function, which characterizes
the relation between the source and the target domains. We conduct a general risk
analysis of this framework and in particular, we show for the ?rst time, if two
domains are related, HTL enjoys faster convergence rates of excess risks for Kernel
Smoothing and Kernel Ridge Regression than those of the classical non-transfer
learning settings. Experiments on real world data demonstrate the effectiveness of
our framework.
1
Introduction
In a classical transfer learning setting, we have a large amount of data from a source domain and
a relatively small amount of data from a target domain. These two domains are related but not
necessarily identical, and the usual assumption is that the hypothesis learned from the source domain
is useful in the learning task of the target domain.
In this paper, we focus on the regression problem where the functions we want to estimate of the source
and the target domains are different but related. Figure 1a shows a 1D toy example of this setting,
where the source function is f so (x) = sin(4?x) and the target function is f ta (x) = sin(4?x) + 4?x.
Many real world problems can be formulated as transfer learning problems. For example, in the task
of predicting the reaction time of an individual from his/her fMRI images, we have about 30 subjects
but each subject has only about 100 data points. To learn the mapping from neural images to the
reaction time of a speci?c subject, we can treat all but this subject as the source domain, and this
subject as the target domain. In Section 6, we show how our proposed method helps us learn this
mapping more accurately.
This paradigm, hypothesis transfer learning (HTL) has been explored empirically with success in
many applications [Fei-Fei et al., 2006, Yang et al., 2007, Orabona et al., 2009, Tommasi et al.,
2010, Kuzborskij et al., 2013, Wang and Schneider, 2014]. Kuzborskij and Orabona [2013, 2016]
pioneered the theoretical analysis of HTL for linear regression and recently Wang and Schneider
[2015] analyzed Kernel Ridge Regression. However, most existing works only provide generalization
bounds, i.e. the difference between the true risk and the training error or the leave-one-out error.
These analyses are not complete because minimizing the generalization error does not necessarily
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
12
10
6
1
0.5
2
Offset KS
Only Target KS
fta
Scale KS
Only Target KS
fta
1
0
Y
Y
8
3
fso
Source data
ta
f
Target Data
Y
14
4
?1
0
2
?2
0
?2
0
0.2
0.4
0.6
0.8
1
?0.5
0
0.1
0.2
X
X
(a) A toy example of transfer
learning. We have many more
samples from the source domain
than the target domain.
0.3
0.4
(b) Transfer learning with Offset
Transformation.
?3
0
0.1
0.2
X
0.3
0.4
(c) Transfer learning with Scale
Transformation.
Figure 1: Experimental results on synthetic data.
reduce the true risk. Further, these works often rely on a particular form of transformation from the
source domain to the target domain. For example, Wang and Schneider [2015] studied the offset
transformation that instead of estimating the target domain function directly, they learn the residual
between the target domain function and the source domain function. It is natural to ask what if we
use other transfer functions and how it affects the risk on the target domain.
In this paper, we propose a general framework of HTL. Instead of analyzing a speci?c form of
transfer, we treat it as an input of our learning algorithm. We call this input transformation function
since intuitively, it captures the relevance between these two domains.1 This framework uni?es many
previous works Wang and Schneider [2014], Kuzborskij and Orabona [2013], Wang et al. [2016] and
naturally induces a class of new learning procedures.
Theoretically, we develop excess risk analysis for this framework. The performance depends on the
stability [Bousquet and Elisseeff, 2002] of the algorithm used as a subroutine that if the algorithm
is stable then the estimation error in the source domain will not affect the estimation in the target
domain much. To our knowledge, this connection was ?rst established by Kuzborskij et al. [2013] in
the linear regression setting but here we generalize it to a broader context. In particular, we provide
explicit risk bounds for two widely used nonlinear estimators, Kernel Smoothing (KS) estimators
and Kernel Ridge Regression (KRR) as subroutines. To the best of our knowledge, these are the ?rst
results showing when two domains are related, transfer learning techniques have faster statistical
convergence rate of excess risk than that of non-transfer learning of kernel based methods. Further,
we accompany this framework with a theoretical analysis showing a small amount of data for crossvalidation enables us (1) avoid using HTL when it is not useful and (2) choose the best transformation
function as input from a large pool.
The rest of the paper is organized as follows. In Section 2 we introduce HTL and provide necessary
backgrounds for KS and KRR. We formalize our transformation function based framework in
Section 3. Our main theoretical results are in Section 4 and speci?cally in Section 4.1 and Section 4.2
we provide explicit risk bounds for KS and KRR, respectively. In Section 5 we analyze crossvalidation in HTL setting and in Section 6 we conduct experiments on real world data data. We
conclude with a brief discussion of avenues for future work.
2
Preliminaries
2.1
Problem Setup
In this paper, we assume both X ? Rd and Y ? R lie in compact subsets: ||X||2 ? ?X , |Y | ? ?Y
n
for some ?X , ?Y ? R+ . Throughout the paper, we use T = {(Xi , Yi )}i=1 to denote a set of
so
so
ta
ta
samples. Let (X , Y ) be the sample from the source domain, and (X , Y ) the sample from the
target domain. In our setting, there are nso samples drawn i.i.d from the source distribution: T so =
nta
nso
{(Xiso , Yiso )}i=1
, and nta samples drawn i.i.d from the target distribution: T ta = {(Xita , Yita )}i=1
.
In addition, we also use nval samples drawn i.i.d from the target domain for cross-validation. We
model the joint relation between X and Y by: Y so = f so (X so ) + ?so and Y ta = f ta (X ta ) + ?ta
where f so and f ta are regression functions and we assume the noise E [?so ] = E [?ta ] = 0, i.i.d,
1
We formally de?ne the transformation functions in Section 3.
2
and bounded. We use A : T ? f? to denote an algorithm that takes a set of samples
?? and produce
?2 ?
an estimator. Given an estimator f?, we de?ne the integrated L2 risk as R(f?) = E f?(X) ? Y
where the expectation is taken over the distribution of (X, Y ). Similarly, the empirical L2 risk on
?2
?
? f?) = 1 ?n
a set of sample T is de?ned as R(
Yi ? f? (Xi ) . In HTL setting, we use f?so an
n
i=1
estimator from the source domain to facilitate the learning procedure for f ta .
2.2
Kernel Smoothing
We say a function f is in the (?, ?) H?lder class [Wasserman, 2006], if for any x, x? ? Rd , f satis?es
?
|f (x) ? f (x? )| ? ? ||x ? x? ||2 , for some ? ? (0, 1). The kernel smoothing
? method uses a positive
kernel K on [0, 1], highest at 0, decreasing on [0, 1], 0 outside [0, 1], and Rd u2 K(u) < ?. Using
?n
n
T = {(Xi , Yi )}i=1 , the kernel smoothing estimator is de?ned as follows: f?(x) = i=1 wi (x)Yi ,
i ||/h)
where wi (x) = ?nK(||x?X
K(||x?Xj ||/h) ? [0, 1].
j=1
2.3
Kernel Ridge Regression
Another popular non-linear estimator is the kernel ridge regression (KRR) which uses the theory
of reproducing kernel Hilbert space (RKHS) for regression [Vovk, 2013]. Any symmetric positive
semide?nite kernel function K : Rd ? Rd ? R de?nes a RKHS H. For each x ? Rd , the function
z ? K(z, x) is contained in the Hilbert space H; moreover, the Hilbert space is endowed with
an inner product ??, ??H such that K(?, x) acts as the kernel of the evaluation functional, meaning
?f, K(x, ?)?H = f (x) for f ? H. In this paper we assume K is bounded: supx?Rd K (x, ?
x) = k <
?. Given the inner product, the H norm of a function g ? H is de?ned as ||g||H ? ?g, g?H
??
?1/2
and similarly the L2 norm, ||g||2 ? Rd g(x)2 dPX
for a given
? PX . Also, the kernel induces
an integral operator TK : L2 (PX ) ? L2 (PX ): TK [f ] (x) = Rd K (x? , x) f (x? ) dPx (x? ) with
countably many non-zero eigenvalues: {?i }i?1 . For a given function f , the approximation error is
?
?
2
2
de?ned as: Af (?) ? inf h?H ||h ? f ||L2 (PX ) + ? ||h||H for ? ? 0. Finally the estimated function
?1
evaluated at point x can be written as f? (x) = K(X, x) (K(X, X) + n?I) Y where X ? Rn?d
n?1
are the inputs of training samples and Y ? R
are the training labels Vovk [2013].
2.4
Related work
Before we present our framework, it is helpful to give a brief overview of existing literature on
theoretical analysis of transfer learning. Many previous works focused on the settings when only
unlabeled data from the target domain are available [Huang et al., 2006, Sugiyama et al., 2008, Yu
and Szepesv?ri, 2012]. In particular, a line of research has been established based on distribution
discrepancy, a loss induced metric for the source and target distributions [Mansour et al., 2009,
Ben-David et al., 2007, Blitzer et al., 2008, Cortes and Mohri, 2011, Mohri and Medina, 2012]. For
example, recently Cortes and Mohri [2014] gave generalization bounds for kernel based methods
under convex loss in terms of discrepancy.
In many real world applications such as yield prediction from pictures [Nuske et al., 2014], or
prediction of response time from fMRI [Verstynen, 2014], some labeled data from the target domain
is also available. Cortes et al. [2015] used these data to improve their discrepancy minimization
algorithm. Zhang et al. [2013] focused on modeling target shift (P (Y ) changes), conditional shift
(P (X|Y ) changes), and a combination of both. Recently, Wang and Schneider [2014] proposed a
kernel mean embedding method to match the conditional probability in the kernel space and later
derived generalization bound for this problem Wang and Schneider [2015]. Kuzborskij and Orabona
[2013, 2016], Kuzborskij et al. [2016] gave excess risk bounds for target domain estimator in the form
of a linear combination of estimators from multiple source domains and an additional linear function.
Ben-David and Urner [2013] showed a similar bound of the same setting with different quantities
capturing the relatedness. Wang et al. [2016] showed that if the features of source and target domain
are [0, 1]d , using orthonormal basis function estimator, transfer learning achieves better excess risk
3
guarantee if f ta ? f so can be approximated by the basis functions easier than f ta . Their work can be
viewed as a special case of our framework using the transformation function G(a, b) = a + b.
3
Transformation Functions
In this section, we ?rst de?ne our class of models and give a meta-algorithm to learn the target
regression function. Our models are based on the idea that transfer learning is helpful when one
transforms the target domain regression problem into a simpler regression problem using source
domain knowledge. Consider the following example.
?
?
?
2.1?
and f ta (x) = f so (x) + x. f so
Example: Offset Transfer. Let f so (x) = x (1 ? x) sin x+0.05
is the so called Doppler function. It requires a large number of samples to estimate well because of its
lack of smoothness Wasserman [2006]. For the same reason, f ta is also dif?cult to estimate directly.
However, if we have enough data from the source domain, we can have a fairly good estimate of f so .
Further, notice that the offset function w(x) = f ta (x) ? f so (x) = x, is just a linear function. Thus,
instead of directly using T ta to estimate f ta , we can use the target domain samples to ?nd an estimate
of w(x), denoted by w(x),
?
and our estimator for the target domain is just: f?ta (x) = f?so (x) + w(x).
?
Figure 1b shows this technique gives improved ?tting for f ta .
The previous example exploits the fact that function w(x) = f ta (x) ? f so (x) is a simpler function
than f ta . Now we generalize this idea further. Formally, we de?ne the transformation function as
G(a, b) : R2 ? R where we assume that given a ? R, G(a, ?) is invertible. Here a will be the
regression function of the source domain evaluated at some point and the output of G will be the
regression function of?the target domain
evaluated at the same point. Let G?1
a (?) denote the inverse
?
of G(a, ?) such that G a, G?1
(c)
=
c.
For
example
if
G(a,
b)
=
a
+
b
and
G?1
a
a (c) = c ? a. For a
?1
so
ta
given G and a pair (f , f ), they together induce a function wG (x) = Gf so (x) (f ta (x)). In the offset
transfer example, wG (x) = x. By this de?nition, for any x, we have G (f so (x) , wG (x)) = f ta (x) .
We call wG the auxiliary function of the transformation function G. In the HTL setting, G is a userde?ned transformation that represents users? prior knowledge on the relation between the source and
target domains. Now we list some other examples:
Example: Scale-Transfer. Consider G(a, b) = ab. This transformation function is useful when
f so and f ta satisfy a smooth scale transfer. For example, if f ta = cf so , for some constant c, then
wG (x) = c because f ta (x) = G (f so (x) , wG (x)) = f so (x) wG (x) = f so (x) c. See Figure 1c.
Example: Non-Transfer. Consider G(a, b) = b. Notice that f ta (x) = wG (x) and so f so is
irrelevant. Thus this model is equivalent to traditional regression on the target domain since data from
the source domain does not help.
3.1
A Meta Algorithm
Given the transformation G and data, we provide a general procedure to estimate f ta . The spirit of
the algorithm is turning learning a complex function f ta into an easier function wG . First we use an
algorithm Aso that takes T so to obtain f?so . Since we have suf?cient data from the source domain,
. Second, we? construct a new???
data set using
f?so should be close to the true regression function f so??
nta
wG
ta
so
ta
ta
?
the nta data points from the target domain: T
=
X , HG f (X ) , Y
where
i
HG : R2 ? R and satis?es
?
?
?
?
??
E HG f so Xita , Yita = G?1
f so
(Xita )
?
i
i
i=1
?
??
?
?
f ta Xita = wG Xita
where and the expectation is taken over ?ta . ?Thus, ?we can use these newly constructed data to
?G = AWG T WG . Finally, we plug trained f?so and w
?G into
learn wG with algorithm AWG : w
?G (X)). Pseudocode is
transformation G to obtain an estimation for f ta : f?ta (X) = G(f?so (X) , w
shown in Algorithm 1.
Unbiased Estimator HG (f so (X ta ) , Y ta ): In Algorithm 1, we require an unbiased estimator for
so
wG (X ta ). Note that if G (a, b) is linear b or ?ta = 0, we can simply set
? HG (f (X) , Y? ) =
G?1
f so (X) (Y ).
For other scenarios, G?1
f so
(Y ta ) is biased: E G?1
(Yita )
f so (Xita )
(Xita ) i
ta
G?1
(x)) and we need to design estimator using the structure of G.
f so (x) (f
4
?=
Algorithm 1 Transformation Function based Transfer Learning
so
Inputs: Source domain data: T so = {(Xiso , Yiso )}ni=1
, target domain data: T ta =
ta
ta nta
{(Xi , Yi )}i=1 , transformation function: G, algorithm to train f so : Aso , algorithm to train
wG : AwG and HG an unbiased estimator for estimating wG .
Outputs: Regression function for the target domain: f?ta .
1: Train the source domain regression function f?so = Aso (T so ).
nta
2: Construct new data using f?so and T ta : T wG = {(Xita , Wi )}i=1
, where Wi =
?
?
so
ta
ta
?
HG f (X ) , Y
.
i
i
3: Train the auxiliary function: w
?G = AWG (T wG ).
?
?
4: Output the estimated regression for the target domain: f?ta (X) = G f?so (X), w
?G (X) .
Remark 1: Many transformation functions are equivalent to a transformation function G? (a, b)
2
where G? (a, b) is linear in b. For example, for G (a, b) = ab2 , i.e., f ta (x) = f so (x) wG
(x),
?
?
2
ta
so
?
consider G (a, b) = ab where b in G stands for b in G, i.e., f (x) = f (x) wG (x). Therefore
?
2
?
wG
= wG
and we only need to estimate wG
well instead of estimating wG . More generally, if
G (a, b) can be factorized as G (a, b) = g1 (a) g2 (b), i.e., f ta (x) = g1 (f so (x)) g2 (wG (x)), we
only need to estimate g2 (wG (x)) and the convergence rate depends on the structure of g2 (wG (x)).
Remark 2: When G is not linear in b and ?ta ?= 0, observe that in Algorithm 1, we treat Yita s as
noisy covariates to estimate wG (Xi )s. This problem is called error-in-variable or measurement error
and has been widely studied in statistics literature. For details, we refer the reader to the seminal
work by Carroll et al. [2006]. There is no universal estimator for the measurement error problem. In
Section B, we provide a common technique, regression calibration to deal with measurement error
problem.
4
Excess Risk Analyses
In this section, we present theoretical analyses for the proposed class of models and estimators. First,
we need to impose some conditions on G. The ?rst assures that if the estimations of f so and wG are
close to the source regression and auxiliary function, then our estimator for f ta is close to the true
target regression function. The second assures that we are estimating a regular function.
Assumption 1 G (a, b) is L-Lipschitz: |G(a, b) ? G(a? , b? )| ? L ||(a, b) ? (a? , b? )||2 and is invertible with respect to b given a, i.e. if G (x, y) = z then G?1
x (z) = y.
Assumption 2 Given G, the induced auxiliary function wG is bounded: for x : ||x||2 ? ?X ,
wG (x) ? B for some B > 0.
Offset Transfer and Non-Transfer satisfy these conditions with L = 1 and B = ?Y . Scale Transfer
satis?es these assumptions when f so is lower bounded from away 0. Lastly, we assume our unbiased
estimator is also regular.
Assumption 3 For x : ||x||2 ? ?X and y : |y| ? ?Y , HG (x, y) ? B for some B > 0 and HG is
Lipschitz continuous in the ?rst argument:|HG (x, y) ? HG (x? , y)| ? L |x ? x? | for some L > 0.
We begin with a general result which only requires the stability of AWG :
of samples that have same features but different labels: T =
Theorem 1 Suppose for any
??two sets ??
nta
nta
ta
ta ?
?
{(X , Wi )}
and T =
X , Wi
, the algorithm Aw for training wG satis?es:
i
i=1
i
G
i=1
??
? ???
??
??
??AwG (T ) ? AwG T? ??
?
?
5
nta
?
i=1
?
?
? ??
?i ?? ,
ci Xita ?Wi ? W
(1)
where ci only depends on Xita . Then for any x,
??
?2
?2
?
?
?
?
? ?ta
2
ta
?G (x) ? wG (x)| +
?f (x) ? f (x)? =O ?f?so (x) ? f so (x)? + |w
? ?
?n
ta
? 2
?
? ta ? ?? so ? ta ?
?
?
?
ci Xi ?f? Xi ? f so Xita ? ?
i=1
?
nta ?
where w
?G = AwG {(Xita , HG (f so (Xita ) , Yita ))}i=1
, the estimated auxiliary function trained
based on true source domain regression function.
Theorem 1 shows how the estimation error in the source domain function propagates to our estimation
of ?the target domain function.
Notice that if we happen to know f so , then the error is bounded by
?
2
O |w
?G (x) ? wG (x)| , the estimation error of wG . However, since we are using estimated f so to
construct training samples for wG , the error might accumulate as nta increases. Though the third
term in Theorem 1 might increase with nta , it also depends on the estimation error of f so which is
relatively small because of the large amount of source domain data.
The stability condition (1) we used is related to the uniform stability introduced by Bousquet and
Elisseeff Bousquet and Elisseeff [2002] where they consider how much will the output change if one
of the training instance is removed or replaced by another whereas ours depends on two different
training data sets. The connection between transfer learning and stability has been discovered
by Kuzborskij and Orabona [2013], Liu et al. [2016] and Zhang [2015] in different settings, but they
only showed bounds for generalization, not for excess risk.
4.1
Kernel Smoothing
We ?rst analyze kernel smoothing method.
Theorem 2 Suppose the support of X ta is a subset of the support of X so and the probability density
of PX so and PX ta are uniformly bounded away from below on their supports. Further assume f so is
(?so , ?so ) H?lder and wG is (?wG , ?wG ) H?lder . If we use kernel smoothing estimation for f so and
?1/(2?wG +d)
?1/(2?so +d)
wG with bandwidth hso ? nso
and hwG ? nta
, with probability at least 1 ? ?
the risk satis?es:
?
?
? ?
?2?w
? ? ??
G
?2?so
? ta ?
1
2?w +d
2?
+d
ta
G
so
log
+ nta
E R f?
?R f
= O nso
?
where the expectation is taken over T so and T ta .
Theorem 2 suggests that the risk depends on two sources, one from estimation of f so and one from
estimation of wG . For the ?rst term, since in the typical transfer learning scenarios nso >> nta , it is
relatively small in the setting we focus on. The second terms shows the power of transfer learning on
transforming a possibly complex target regression function into a simpler auxiliary
function. It ?
is
?
?2?
/
2?
+d
ta
ta
(
)
f
f
well known that learning f ta only using target domain has risk of the order ? nta
.
Thus, if the auxiliary function is smoother than the target regression function, i.e. ?wG > ?f ta , we
obtain better statistical rate.
4.2
Kernel Ridge Regression
Next, we give an upper bound for the excess risk using KRR:
Theorem 3 Suppose PX so = PX ta and the eigenvalues of the integral operator TK satisfy ?i ?
ai?1/p for
i ? 1 a ? 16?4Y , p ? (0, 1) and there exists a constant C ? 1 such that for f ? H,
so
p
1?p
||f ||? ? C ||f ||H ? ||f ||L2 (PX ) . Furthur assume that Af (?) ? c??so and AwG (?) ? c??wG .
?1/(?so +p)
If we use KRR for estimating f so and wG with regularization parameters ?so ? nso
6
and
?1/(?wG +p)
, then with probability at least 1 ? ? the excess risk satis?es:
?
??
? ??
??w
2
? ? ??
G
??so
? ta ?
1
?w +p
?w +p
?
+p
ta
G
G
so
log (nta ) ? nso
+ nta
log
?R f
E R f?
=O
nta
?
?wG ? nta
where the expectation is taken over T so and T ta .
Similar to Theorem 2, Theorem 3 suggests that the estimation error comes from two sources. For
estimating the auxiliary function wG , the statistical rate depends on properties of the kernel induced
RKHS, and how far the auxiliary function is from this space. For the ease of presentation, we assume
so
ta
PX so = PX ta , so the approximation errors
Af and A? f are de?ned on the same domain. The
?
error of estimating f so is ampli?ed by O ??2
wG log (nta ) , which is worse than that of nonparametric
kernel smoothing. We believe this ??2
is
nearly
tight because
and Elisseeff have shown
wG
? Bousquet
?
the uniform algorithmic stability parameter for KRR is O ??2
Bousquet
and Elisseeff [2002].
wG
Steinwart et al. Steinwart?et al. [2009]
showed
that
for
non-transfer
learning,
the
optimal statistical
?
??ta
?ta +p
rate for excess risk is ? nta
, so if ?wg ? ?ta and nso is suf?ciently large then we achieve
improved convergence rate through transfer learning.
Remark: Theorem 2 and 3 are not directly comparable because our assumptions on the function
spaces of these two theorems are different. In general, H?lder space is only a Banach space but not a
Hilbert space. We refer readers to Theorem 1 in Zhou [2008] for details.
5
Finding the Best Transformation Function
In the previous section we showed for a speci?c transformation function G, if auxiliary function
is smoother than the target regression function then we have smaller excess risk. In practice,
we would like to try out a class of transformation functions G , which is possibly uncountable.
We can construct a subset of G ? G, which is ?nite and satis?es that each G in G there is a
G in G that is close to G. Here we give an example. Consider the transformation functions
that have the form: G = {G(a, b) = ?a + b where |?| ? L? , |a| ? La } . We can quantize this set
of transformation functions by considering a subset of G: G = {G(a, b) = k?a + b} where ? =
L?
2K , k = ?K, ? ? ? , 0, ? ? ? , K and |a| ? La . Here ? is the quantization unit.
The next theorem shows we only need to search the transformation function G in G whose correta
sponding estimator f?G
has the lowest empirical risk on the validation dataset.
Theorem 4 Let G be a class of transformation functions and G be its ||?||? norm ?-cover.
Suppose wG satis?es the same assumption in Theorem 1 and for any two G1 , G2 ?? G,
?
ta
||wG1 ? wG2 ||? ? L ||G1 ? G2 ||? for some constant L. Denote G? = argminG?G R f?G
?
?
? ?
ta
? ?? ? ??
?
R(f?G
?)
? f?ta . If we choose ? = O ?
and G = argminG?G R
and nval = ? log ?G ? /? , the
nta
G
ci
i=1
? ?
? ? ?
?
??
??
ta
ta
ta
?
with probability at least 1 ? ?, E R fG? ? R (f ) = O E R f?G
? R (f ta ) where the
?
expectation is taken over T so and T ta .
Remark 1: This theorem implies that if no-transfer function (G (a, b) = b) is in G then we will
end up choosing a transformation function that has the same order of excess risk as using no-transfer
learning algorithm, thus avoiding negative transfer.
Remark 2: Note number of validation set is only logarithmically depending on the size of set of
transformation functions. Therefore, we only need to use a very small amount of data from the target
domain to do cross-validation.
6
Experiments
In this section we use robotics and neural imaging data to demonstrate the effectiveness of the
proposed framework. We conduct experiments on real-world data sets with the following procedures.
7
Only Target KS
Only Target KRR
Only Source KRR
Combined KS
Combined KRR
CDM
Offset KS
Offset KRR
Scale KS
Scale KRR
nta = 10
nta = 20
nta = 40
nta = 80
nta = 160
nta = 320
0.086 ? 0.022
0.080 ? 0.017
0.098 ? 0.017
0.092 ? 0.011
0.087 ? 0.025
0.105 ? 0.023
0.080 ? 0.026
0.146 ? 0.112
0.078 ? 0.022
0.102 ? 0.033
0.076 ? 0.010
0.078 ? 0.022
0.098 ? 0.017
0.084 ? 0.008
0.077 ? 0.015
0.074 ? 0.020
0.066 ? 0.023
0.066 ? 0.017
0.065 ? 0.013
0.095 ? 0.100
0.066 ? 0.008
0.063 ? 0.013
0.098 ? 0.017
0.077 ? 0.009
0.062 ? 0.009
0.064 ? 0.008
0.052 ? 0.006
0.053 ? 0.007
0.056 ? 0.009
0.057 ? 0.014
0.064 ? 0.007
0.050 ? 0.007
0.098 ? 0.017
0.075 ? 0.006
0.061 ? 0.005
0.060 ? 0.007
0.054 ? 0.006
0.048 ? 0.006
0.056 ? 0.005
0.052 ? 0.010
0.065 ? 0.006
0.048 ? 0.006
0.098 ? 0.017
0.074 ? 0.006
0.047 ? 0.003
0.053 ? 0.009
0.050 ? 0.003
0.043 ? 0.004
0.054 ? 0.008
0.044 ? 0.004
0.063 ? 0.005
0.040 ? 0.005
0.098 ? 0.017
0.067 ? 0.006
0.041 ? 0.004
0.056 ? 0.004
0.052 ? 0.004
0.041 ? 0.003
0.055 ? 0.004
0.042 ? 0.002
Table 1: 1 standard deviation intervals for the mean squared errors of various algorithms when
transferring from kin-8fm to kin-8nh. The values in bold are the smallest errors for each nta . Only
Source KS has much worse performance than other algorithms so we do not show its result here.
Directly training on the target data T ta (Only Target KS, Only Target KRR).
Only training on the source data T so (Only Source KS, Only Source KRR).
Training on the combined source and target data (Combined KS, Combined KRR).
The CDM algorithm proposed by Wang and Schneider [2014] with KRR (CDM).
The algorithm described in this paper with G(a, b) = (a + ?)b where ? is a hyper-parameter
(Scale KS, Scale KRR).
? The algorithm described in this paper with G(a, b) = ?a + b where ? is a hyper-parameter
(Offset KS, Offset KRR). ?
?
?
?
?
?
For the ?rst experiment, we vary the size of the target domain to study the effect of nta relative
to nso . We use two datasets from the ?kin? family in Delve [Rasmussen et al., 1996]. The two
datasets we use are ?kin-8fm? and ?kin-8nh?, both with 8 dimensional inputs. kin-8fm has fairly linear
output, and low noise. kin-8nh on the other hand has non-linear output, and high noise. We consider
the task of transfer learning from kin-8fm to kin-8nh. In this experiment, We set nso to 320, and
vary nta in {10, 20, 40, 80, 160, 320}. Hyper-parameters were picked using grid search with 10-fold
cross-validation on the target data (or source domain data when not using the target domain data).
Table 1 shows the mean squared errors on the target data. To better understand the results, we show a
box plot of the mean squared errors for nta = 40 onwards in Figure 2(a). The results for nta = 10
and nta = 20 have high variance, so we do not show them in the plot. We also omit the results of
Only Source KRR because of its poor performance. We note that our proposed algorithm outperforms
other methods across nearly all values of nta especially when nta is small. Only when there are as
many points in the target as in the source, does simply training on the target give the best performance.
This is to be expected since the primary purpose in doing transfer learning is to alleviate the problem
of lack of data in the target domain. Though quite comparable, the performance of the scale methods
was worse than the offset methods in this experiment. In general, we would use cross-validation to
choose between the two.
We now consider another real-world dataset where the covariates are fMRI images taken while
subjects perform a Stroop task [Stroop, 1935]. We use the dataset collected by Verstynen [2014]
which contains fMRI data of 28 subjects. A total of 120 trials were presented to each participant and
fMRI data was collected throughout the trials, and went through a standard post-processing scheme.
The result of this is a feature vector corresponding to each trial that describes the activity of brain
regions (voxels), and the goal is to use this to predict the response time.
To frame the problem in the transfer learning setting, we consider as source the data of all but one
subject. The goal is to predict on the remaining subject. We performed ?ve repetitions for each
algorithm by drawing nso = 300 data points randomly from the 3000 points in the source domain.
We used nta = 80 points from the target domain for training and cross-validation; evaluation was
done on the 35 remaining points in the target domain. Figure 2 (b) shows a box plot of the coeffecient
of determination values (R-squared) for the best performing algorithms. R-squared is de?ned as
1 ? SSres /SStot where SSres is the sum of squared residuals, and SStot is the total sum of squares.
Note that R-squared can be negative when predicting on unseen samples ? which were not used to ?t
the model ? as in our case. When positive, it indicates the proportion of explained variance in the
dependent variable (higher the better). From the plot, it is clear that Offset KRR and Only Target
KRR have the best performances on average and Offset KRR has smaller variance.
8
????
??????????????????
???
?????????
????
????
????
???
???
???
????
??
??
???
????????????????????????????????????????????
??????????????????????????????????????????????????
??????????????
???????????????
???????????
????????????
???
?????????
???
??????????????????????????????
??????????
????????
?????????
Figure 2: Box plots of experimental results on real datasets. Each box extends from the ?rst to third
quartile, and the horizontal lines in the middle are medians. For the robotics data, we report mean
squared error (lower the better) and for the fMRI data, we report R-squared (the higher the better).
For the ease of presentation, we only show results of algorithms with good performances.
Only Target KS
Only Target KRR
Only Source KS
Only Source KRR
Combined KS
Combined KRR
CDM
Offset KS
Offset KRR
Scale KS
Scale KRR
Mean
Median
-0.0096
0.1041
-0.4932
-0.8763
-0.7540
-0.5868
-3.1183
0.1190
0.1080
0.0017
0.0897
0.0444
0.1186
-0.5366
-0.9363
-0.2023
-0.0691
-3.4510
0.1081
0.1221
-0.0321
0.1107
Standard Deviation
0.1041
0.2361
0.4555
0.6265
1.5109
1.3223
2.6473
0.0612
0.0682
0.0632
0.1104
Table 2: Mean, median, and standard deviation for the coef?cient of determination (R-squared) of
various algorithms on the fMRI dataset.
Table 2 shows the full table of results for the fMRI task. Using only the source data produces large
negative R-squared, and while Only Target KRR does produce a positive mean R-squared, it comes
with a high variance. On the other hand, both Offset methods have low variance, showing consistent
performance. For this particular case, the Scale methods do not perform as well as the Offset methods,
and as has been noted earlier, in general we would use cross validation to select an appropriate
transfer function.
7
Conclusion and Future Works
In this paper, we proposed a general transfer learning framework for the HTL regression problem
when there is some data available from the target domain. Theoretical analysis shows it is possible to
achieve better statistical rate using transfer learning than standard supervised learning.
Now we list two future directions and how our results could be further improved. First, in many real
world applications, there is also a large amount of unlabeled data from the target domain available.
Combining our proposed framework with previous works for this scenario [Cortes and Mohri, 2014,
Huang et al., 2006] is a promising direction to pursue. Second, we only present upper bounds in
this paper. It is an interesting direction to obtain lower bounds for HTL and other transfer learning
scenarios.
8
Acknowledgements
S.S.D. and B.P. were supported by NSF grant IIS1563887 and ARPA-E Terra program. A.S. was
supported by AFRL grant FA8750-17-2-0212.
9
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6,258 | 6,661 | Controllable Invariance
through Adversarial Feature Learning
Qizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, Graham Neubig
Language Technologies Institute
Carnegie Mellon University
{qizhex, dzihang, yulund, hovy, gneubig}@cs.cmu.edu
Abstract
Learning meaningful representations that maintain the content necessary for a
particular task while filtering away detrimental variations is a problem of great
interest in machine learning. In this paper, we tackle the problem of learning
representations invariant to a specific factor or trait of data. The representation
learning process is formulated as an adversarial minimax game. We analyze the
optimal equilibrium of such a game and find that it amounts to maximizing the
uncertainty of inferring the detrimental factor given the representation while maximizing the certainty of making task-specific predictions. On three benchmark tasks,
namely fair and bias-free classification, language-independent generation, and
lighting-independent image classification, we show that the proposed framework
induces an invariant representation, and leads to better generalization evidenced by
the improved performance.
1
Introduction
How to produce a data representation that maintains meaningful variations of data while eliminating
noisy signals is a consistent theme of machine learning research. In the last few years, the dominant
paradigm for finding such a representation has shifted from manual feature engineering based on
specific domain knowledge to representation learning that is fully data-driven, and often powered by
deep neural networks [Bengio et al., 2013]. Being universal function approximators [Gybenko, 1989],
deep neural networks can easily uncover the complicated variations in data [Zhang et al., 2017],
leading to powerful representations. However, how to systematically incorporate a desired invariance
into the learned representation in a controllable way remains an open problem.
A possible avenue towards the solution is to devise a dedicated neural architecture that by construction
has the desired invariance property. As a typical example, the parameter sharing scheme and pooling
mechanism in modern deep convolutional neural networks (CNN) [LeCun et al., 1998] take advantage
of the spatial structure of image processing problems, allowing them to induce more generic feature
representations than fully connected networks. Since the invariance we care about can vary greatly
across tasks, this approach requires us to design a new architecture each time a new invariance
desideratum shows up, which is time-consuming and inflexible.
When our belief of invariance is specific to some attribute of the input data, an alternative approach is
to build a probabilistic model with a random variable corresponding to the attribute, and explicitly
reason about the invariance. For instance, the variational fair auto-encoder (VFAE) [Louizos et al.,
2016] employs the maximum mean discrepancy (MMD) to eliminate the negative influence of specific
?nuisance variables?, such as removing the lighting conditions of images to predict the person?s
identity. Similarly, under the setting of domain adaptation, standard binary adversarial cost [Ganin
and Lempitsky, 2015, Ganin et al., 2016] and central moment discrepancy (CMD) [Zellinger et al.,
2017] have been utilized to learn features that are domain invariant. However, all these invariance
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
inducing criteria suffer from a similar drawback, which is they are defined to measure the divergence
between a pair of distributions. Consequently, they can only express the invariance belief w.r.t. a
pair of values of the random variable at a time. When the attribute is a multinomial variable that
takes more than two values, combinatorial number of pairs (specifically, O(n2 )) have to be added to
express the belief that the representation should be invariant to the attribute. The problem is even
more dramatic when the attribute represents a structure that has exponentially many possible values
(e.g. the parse tree of a sentence) or when the attribute is simply a continuous variable.
Motivated by the aforementioned drawbacks and difficulties, in this work, we consider the problem of
learning a feature representation with the desired invariance. We aim at creating a unified framework
that is (1) generic enough such that it can be easily plugged into different models, and (2) more
flexible to express an invariance belief in quantities beyond discrete variables with limited value
choices. Specifically, inspired by the recent advancement of adversarial learning [Goodfellow et al.,
2014], we formulate the representation learning as a minimax game among three players: an encoder
which maps the observed data deterministically into a feature space, a discriminator which looks
at the representation and tries to identify a specific type of variation we hope to eliminate from the
feature, and a predictor which makes use of the invariant representation to make predictions as in
typical discriminative models. We provide theoretical analysis of the equilibrium condition of the
minimax game, and give an intuitive interpretation. On three benchmark tasks from different domains,
we show that the proposed approach not only improves upon vanilla discriminative approaches that do
not encourage invariance, but also outperforms existing approaches that enforce invariant features.
2
Related Work
As a specific case of our problem where s takes two values, domain adaption has attracted a large
amount of research interest. Domain adaptation aims to learn domain-invariant representations that
are transferable to other domains. For example, in image classification, adversarial training has
been shown to able to learn an invariant representation across domains [Ganin and Lempitsky, 2015,
Ganin et al., 2016, Bousmalis et al., 2016, Tzeng et al., 2017] and enables classifiers trained on the
source domain to be applicable to the target domain. Moment discrepancy regularizations can also
effectively remove domain specific information [Zellinger et al., 2017, Bousmalis et al., 2016] for
the same purpose. By learning language-invariant representations, classifiers trained on the source
language can be applied to the target language [Chen et al., 2016b, Xu and Yang, 2017].
Works targeting the development of fair, bias-free classifiers also aim to learn representations invariant
to ?nuisance variables? that could induce bias and hence makes the predictions fair, as data-driven
models trained using historical data easily inherit the bias exhibited in the data. Zemel et al. [2013]
proposes to regularize the `1 distance between representation distributions for data with different
nuisance variables to enforce fairness. The Variational Fair Autoencoder [Louizos et al., 2016] targets
the problem with a Variational Autoencoder [Kingma and Welling, 2014, Rezende et al., 2014]
approach with maximum mean discrepancy regularization.
Our work is also related to learning disentangled representations, where the aim is to separate different
influencing factors of the input data into different parts of the representation. Ideally, each part of
the learned representation can be marginally independent to the other. An early work by Tenenbaum
and Freeman [1997] propose a bilinear model to learn a representation with the style and content
disentangled. From information theory perspective, Chen et al. [2016a] augments standard generative
adversarial networks with an inference network, whose objective is to infer part of the latent code
that leads to the generated sample. This way, the information carried by the chosen part of the latent
code can be retained in the generative sample, leading to disentangled representation.
As we have discussed in Section 1, these methods bear the same drawback that the cost used to
regularize the representation is pairwise, which does not scale well as the number of values that the
attribute can take could be large. Louppe et al. [2016] propose an adversarial training framework to
learn representations independent to a categorical or continuous variable. A basic assumption in their
theoretical analysis is that the attribute is irrelevant to the prediction, making it unsuitable to analyze
the fairness classifications.
2
3
Adversarial Invariant Feature Learning
In this section, we formulate our problem and then present the proposed framework of learning
invariant features.
Given paired observations hx, yi, we are interested in the task of predicting the target y based on the
value of x using a discriminative approach, i.e. directly modeling the conditional distribution p(y | x).
As the input x can have highly complicated structure, we employ a dedicated model or algorithm to
extract an expressive representation h from x. In addition, we have access to some intrinsic attribute s
of x as well as a prior belief that the prediction result should be invariant to s. Thus, when we extract
the representation h from x, we want the representation h to preserve variations that are necessary to
predict y while eliminating information of s.
To achieve the aforementioned goal, we employ a deterministic encoder E to obtain the representation
by encoding x and s into h, namely, h = E(x, s). It should be noted here that we are using s as
an additional input. Intuitively, this can inform and guide the encoder to remove information about
undesired variations within the representation. For example, if we want to learn a representation
of image x that is invariant to the lighting condition s, the model can learn to ?brighten? the
representation if it knows the original picture is dark, and vice versa.
Given the obtained representation h, the target y is predicted by a predictor M , which effectively
models the distribution qM (y | h). By construction, instead of modeling p(y | x) directly, the
discriminative model we formulate captures the conditional distribution p(y | x, s) with additional
information coming from s.
Surely, feeding s into the encoder by no means guarantees the induced feature h will be invariant to s.
Thus, in order to enforce the desired invariance and eliminate variations of factor s from h, we set up
an adversarial game by introducing a discriminator D which inspects the representation h and ensure
that it is invariant to s. Concretely, the discriminator D is trained to predict s based on the encoded
representation h, which effectively maximizes the likelihood qD (s | h). Simultaneously, the encoder
fights to minimize the same likelihood of inferring the correct s by the discriminator. Intuitively, the
discriminator and the encoder form an adversarial game where the discriminator tries to detect an
attribute of the data while the encoder learns to conceal it.
Note that under our framework, in theory, s can be any type of data as long as it represents an attribute
of x. For example, s can be a real value scalar/vector, which may take many possible values, or a
complex sub-structure such as the parse tree of a natural language sentence. But in this paper, we
focus mainly on instances where s is a discrete label with multiple choices. We plan to extend our
framework to deal with continuous s and structured s in the future.
Formally, E, M and D jointly play the following minimax game:
min max J(E, M, D)
E,M
D
where
J(E, M, D) =
E
x,s,y?p(x,s,y)
[? log qD (s | h = E(x, s)) ? log qM (y | h = E(x, s))]
(1)
where ? is a hyper-parameter to adjust the strength of the invariant constraint, and p(x, s, y) is the
true underlying distribution that the empirical observations are drawn from.
Note that the problem of domain adaption can be seen as a special case of our problem, where s is a
Bernoulli variable representing the domain and the model only has access to the target y when s =
?source domain? during training.
4
Theoretical Analysis
In this section, we theoretically analyze, given enough capacity and training time, whether such a
minimax game will converge to an equilibrium where variations of y are preserved and variations of
s are removed. The theoretical analysis is done in a non-parametric limit, i.e., we assume a model
with infinite capacity.
3
Since both the discriminator and the predictor only use h which is transformed deterministically from
x and s, we can substitute x with h and define a joint distribution p?(h, s, y) of h, s and y as follows
Z
Z
Z
p?(h, s, y) =
p?(x, s, h, y)dx =
p(x, s, y)pE (h | x, s)dx =
p(x, s, y)?(E(x, s) = h)dx
x
x
x
Here, we have used the fact that the encoder is a deterministic transformation and thus the distribution
pE (h | x, s) is merely a delta function denoted by ?(?). Intuitively, h absorbs the randomness in x
and has an implicit distribution of its own. Also, note that the joint distribution p?(h, s, y) depends on
the transformation defined by the encoder.
Thus, we can equivalently rewrite objective (1) as
J(E, M, D) =
E
h,s,y?p(h,s,y)
?
[? log qD (s | h) ? log qM (y | h)]
(2)
To analyze the equilibrium condition of the new objective (2), we first deduce the optimal discriminator
D and the optimal predictor M for a given encoder E and then prove the global optimality of the
minimax game.
?
(s | h) = p?(s | h) and the
Claim 1. Given a fixed encoder E, the optimal discriminator outputs qD
?
optimal predictor corresponds to qM (y | h) = p?(y | h).
Proof. The proof uses the fact that the objective is functionally convex w.r.t. each distribution, and
by taking the variations we can obtain the stationary point for qD and qM as a function of q?. The
detailed proof is included in the supplementary material A.
?
?
Note that the optimal qD
(s | h) and qM
(y | h) given in Claim 1 are both functions of the encoder
?
?
E. Thus, by plugging qD
and qM
into the original minimax objective (2), it can be simplified as a
minimization problem only w.r.t. the encoder E with the following form:
min J(E) = min
E
E
[? log q?(s | h) ? log q?(y | h)]
E
h,s,y??
q (h,s,y)
= min ??H(?
q (s | h)) + H(?
q (y | h))
(3)
E
where H(?
q (s | h)) is the conditional entropy of the distribution q?(s | h). As we can see, the objective
(3) consists of two conditional entropies with different signs. Optimizing the first term amounts to
maximizing the uncertainty of inferring s based on h, which is essentially filtering out any information
of s from the representation. On the contrary, optimizing the second term leads to increasing the
certainty of predicting y based on h. Implicitly, the objective defines the equilibrium of the minimax
game.
? Firstly, for cases where the attribute s is entirely irrelevant to the prediction task, the two terms
can reach the optimum at the same time, leading to a win-win equilibrium. For example, with
the lighting condition of an image removed, we can still/better classify the identity of the people
in that image. With enough model capacity, the optimal equilibrium solution would be the same
regardless of the value of ?.
? However, there are cases where these two optimization objectives are competing. For example,
in fair classifications, sensitive factors such as gender and age may help the overall prediction
accuracies. Hence learning a fair/invariant representation is harmful to predictions. In this case,
the optimality of these two entropies cannot be achieved simultaneously, and ? defines the relative
strengths of the two objectives in the final equilibrium.
5
5.1
Parametric Instantiation of the Proposed Framework
Models
To show the general applicability of our framework, we experiment on three different tasks including
sentence generation, image classification and fair classifications. Due to the different natures of data
of x and y, here we present the specific model instantiations we use.
4
Sentence Generation We use multi-lingual machine translation as the testbed for sentence generation. Concretely, we have translation pairs between several source languages and a target language. x
is the source sentence to be translated and s is a scalar denoting which source language x belongs to.
y is the translated sentence for the target language.
Recall that s is used as an input of E to obtain a language-invariant representation. To make full
use of s, we employ separate encoders Encs for sentences in each language s. In other words,
h = E(s, x) = Encs (x) where each Encs is a different encoder. The representation of a sentence is
captured by the hidden states of an LSTM encoder [Hochreiter and Schmidhuber, 1997] at each time
step.
We employ a single LSTM predictor for different encoders. As often used in language generation,
the probability qM output by the predictor is parametrized by an autoregressive process, i.e.,
qM (y1:T | h) =
T
Y
qM (yt |y<t , h)
t=1
where we use an LSTM with attention model [Bahdanau et al., 2015] to compute qM (yt |y<t , h).
The discriminator is also parameterized as an LSTM which gives it enough capacity to deal with
input of multiple timesteps. qD (s | h) is instantiated with the multinomial distribution computed by a
softmax layer on the last hidden state of the discriminator LSTM.
Classification For our classification experiments, the input is either a picture or a feature vector.
All of the three players in the minimax game are constructed by feedforward neural networks. We
feed s to the encoder as an embedding vector.
5.2
Optimization
There are two possible approaches to optimize our framework in an adversarial setting. The first one
is similar to the alternating approach used in Generative Adversarial Nets (GANs) [Goodfellow et al.,
2014]. We can alternately train the two adversarial components while freezing the third one. This
approach has more control in balancing the encoder and the discriminator, which effectively avoids
saturation. Another method is to train all three components together with a gradient reversal layer
[Ganin and Lempitsky, 2015]. In particular, the encoder admits gradients from both the discriminator
and the predictor, with the gradient from the discriminator negated to push the encoder in the opposite
direction desired by the discriminator. Chen et al. [2016b] found the second approach easier to
optimize since the discriminator and the encoder are fully in sync being optimized altogether. Hence
we adopt the latter approach. In all of our experiments, we use Adam [Kingma and Ba, 2014] with a
learning rate of 0.001.
6
Experiments
In this section, we perform empirical experiments to evaluate the effectiveness of proposed framework.
We first introduce the tasks and corresponding datasets we consider. Then, we present the quantitative
results showing the superior performance of our proposed framework, and discuss some qualitative
analysis which verifies the learned representations have the desired invariance property.
6.1
Datasets
Our experiments include three tasks in different domains: (1) fair classification, in which predictions
should be unaffected by nuisance factors; (2) language-independent generation which is conducted
on the multi-lingual machine translation problem; (3) lighting-independent image classification.
Fair Classification For fair classification, we use three datasets to predict the savings, credit ratings
and health conditions of individuals with variables such as gender or age specified as ?nuisance
variable? that we would like to not consider in our decisions [Zemel et al., 2013, Louizos et al.,
2016]. The German dataset [Frank et al., 2010] is a small dataset with 1, 000 samples describing
whether a person has a good credit rating. The sensitive nuisance variable to be factored out is gender.
5
The Adult income dataset [Frank et al., 2010] has 45, 222 data points and the objective is to predict
whether a person has savings of over 50, 000 dollars with the sensitive factor being age. The task of
the health dataset1 is to predict whether a person will spend any days in the hospital in the following
year. The sensitive variable is also the age and the dataset contains 147, 473 entries. We follow the
same 5-fold train/validation/test splits and feature preprocessing used in [Zemel et al., 2013, Louizos
et al., 2016].
Both the encoder and the predictor are parameterized by single-layer neural networks. A three-layer
neural network with batch normalization [Ioffe and Szegedy, 2015] is employed for the discriminator.
We use a batch size of 16 and the number of hidden units is set to 64. ? is set to 1 in our experiments.
Multi-lingual Machine Translation For the multi-lingual machine translation task we use French
to English (fr-en) and German to English (de-en) pairs from IWSLT 2015 dataset [Cettolo et al., 2012].
There are 198, 435 pairs of fr-en sentences and 188, 661 pairs of de-en sentences in the training set. In
the test set, there are 4, 632 pairs of fr-en sentences and 7, 054 pairs of de-en sentences. We evaluate
BLEU scores [Papineni et al., 2002] using the standard Moses multi-bleu.perl script. Here, s
indicates the language of the source sentence.
We use the OpenNMT [Klein et al., 2017] in our multi-lingual MT experiments. The encoder is a
two-layer bidirectional LSTM with 256 units for each direction. The discriminator is a one-layer
single-directional LSTM with 256 units. The predictor is a two-layer LSTM with 512 units and
attention mechanism [Bahdanau et al., 2015]. We follow Johnson et al. [2016] and use Byte Pair
Encoding (BPE) subword units [Sennrich et al., 2016] as the cross-lingual input. Every model is run
for 20 epochs. ? is set to 8 and the batch size is set to 64.
Image Classification We use the Extended Yale B dataset [Georghiades et al., 2001] for our image
classification task. It comprises face images of 38 people under 5 different lighting conditions: upper
right, lower right, lower left, upper left, or the front. The variable s to be purged is the lighting
condition. The label y is the identity of the person. We follow Li et al. [2014], Louizos et al. [2016]?s
train/test split and no validation is used: 38 ? 5 = 190 samples are used for training and all other
1, 096 data points are used for testing.
We use a one-layer neural network for the encoder and a one-layer neural network for prediction. ? is
set to 2. The discriminator is a two-layer neural network with batch normalization. The batch size is
set to 16 and the hidden size is set to 100.
6.2
Results
Fair Classification The results on three fairness tasks are shown in Figure 1. We compare our model
with two prior works on learning fair representations: Learning Fair Representations (LFR) [Zemel
et al., 2013] and Variational Fair Autoencoder (VFAE) [Louizos et al., 2016]. Results of VAE and
directly using x as the representation are also shown.
We first study how much information about s is retained in the learned representation h by using a
logistic regression to predict factor s. In the top row, we see that s cannot be recognized from the
representations learned by three models targeting at fair representations. The accuracy of classifying
s is similar to the trivial baseline predicting the majority label shown by the black line.
The performance on predicting label y is shown in the second row. We see that LFR and VFAE
suffer on Adult and German datasets after removing information of s. In comparison, our model?s
performance does not suffer even when making fair predictions. Specifically, on German, our
model?s accuracy is 0.744 compared to 0.727 and 0.723 achieved by VFAE and LFR. On Adult, our
model?s accuracy is 0.844 while VFAE and LFR have accuracies of 0.813 and 0.823 respectively.
On the health dataset, all models? performances are barely better than the majority baseline. The
unsatisfactory performances of all models may be due to the extreme imbalance of the dataset, in
which 85% of the data has the same label.
We also investigate how fair representations would alleviate biases of machine learning models. We
measure the unbiasedness by evaluating models? performances on identifying minority groups. For
instance, suppose the task is to predict savings with the nuisance factor being age, with savings
1
www.heritagehealthprize.com
6
Adult
German
0.9
0.9
0.78
0.78
Majority
0.67
0.65
0.53
0.4
Health
0.9
Majority
0.8
0.65
0.65
0.53
0.53
LFR
VAE
VFAE
Majority
0.58
0.4
0.4
x
0.78
Ours
x
LFR
VAE
VFAE
x
Ours
LFR
VAE
VFAE
Ours
(a) Accuracy on predicting s. The closer the result is to the majority line, the better the model is in eliminating
the effect of nuisance variables.
Adult
Health
German
0.9
Majority
0.75
0.78
0.9
0.9
0.78
0.78
0.65
0.65
0.53
0.53
Majority
0.71
x
LFR
VAE
VFAE
x
Ours
0.65
0.53
0.4
0.4
Majority
0.84
LFR
VAE
VFAE
Ours
0.4
x
LFR
VAE
VFAE
Ours
(b) Accuracy on predicting y. High accuracy in predicting y is desireable.
x
0.9
Health
German
Adult
Ours
x
0.9
Ours
0.775
0.78
0.78
0.65
0.65
0.65
0.525
0.53
0.53
0.4
Overall
Biased categories
0.4
Overall
Biased categories
x
0.9
Ours
0.4
Overall
Biased categories
(c) Overall performance and performance on biased categories. Fair representations lead to high accuracy on
baised categories.
Figure 1: Fair classification results on different representations. x denotes directly using the observation x as the representation. The black lines in the first and the second row show the performance
of predicting the majority label. ?Biased categories? in the third row are explained in the fourth
paragraph of Section 6.2.
above a threshold of $50, 000 being adequate, otherwise being insufficient. If people of advanced
age generally have fewer savings, then a biased model would tend to predict insufficient savings for
those with an advanced age. In contrast, an unbiased model can better factor out age information and
recognize people that do not fit into these stereotypes.
Concretely, for groups pooled by each possible value of y, we seek for the minority s in each of these
groups and define the minority s as the biased category for the group. Then we first calculate the
accuracy on each biased category and report the average performance for all categories. We do not
compute the instance-level average performance since one category may hold the dominant amount
of data among all categories.
As shown in the third row of Figure 1, on German and Adult, we achieve higher accuracy on the
biased categories, even though our overall accuracy is similar to or lower than the baseline which
does not employ fairness constraints. Specifically, on Adult, our performance on the biased categories
is 0.788 while the baseline?s accuracy is 0.748. On German, our accuracy on biased categories is
0.676 while the baseline achieves 0.648. The results show that our model is able to learn a more
unbiased representation.
Multi-lingual Machine Translation The results of systems on multi-lingual machine translation
are shown in Table 1. We compare our model with attention based encoder-decoder trained on
bilingual data [Bahdanau et al., 2015] and multi-lingual data [Johnson et al., 2016]. The encoderdecoder trained on multi-lingual data employs a single encoder for both source languages. Firstly,
both multi-lingual systems outperform the bilingual encoder-decoder even though multi-lingual
systems use similar number of parameters to translate two languages, which shows that learning
7
Model
test (fr-en) test (de-en)
Bilingual Enc-Dec [Bahdanau et al., 2015]
35.2
27.3
Multi-lingual Enc-Dec [Johnson et al., 2016]
35.5
27.7
Our model
36.1
28.1
w.o. discriminator
35.3
27.6
w.o. separate encoders
35.4
27.7
Table 1: Results on multi-lingual machine translation.
Method
Accuracy of classifying s Accuracy of classifying y
Logistic regression
0.96
0.78
NN + MMD [Li et al., 2014]
0.82
VFAE [Louizos et al., 2016]
0.57
0.85
Ours
0.57
0.89
Table 2: Results on Extended Yale B dataset. A better representation has lower accuracy of classifying
factor s and higher accuracy of classifying label y
invariant representation leads to better generalization in this case. The better generalization may be
due to transferring statistical strength between data in two languages.
Comparing two multi-lingual systems, our model outperforms the baseline multi-lingual system on
both languages, where the improvement on French-to-English is 0.6 BLEU score. We also verify
the design decisions in our framework by ablation studies. Firstly, without the discriminator, the
model?s performance is worse than the standard multi-lingual system, which rules out the possibility
that the gain of our model comes from more parameters of separating encoders. Secondly, when we
do not employ separate encoders, the model?s performance deteriorates and it is more difficult to
learn a cross-lingual representation, which means that the encoder needs to have enough capacity to
reach the equilibrium in the minimax game. Intuitively, German and French have different grammars
and vocabulary, so it is hard to obtain a unified semantic representation by performing the same
operations.
Image Classification We report the results in Table 2 with two baselines [Li et al., 2014, Louizos
et al., 2016] that use MMD regularizations to remove lighting conditions. The advantage of factoring
out lighting conditions is shown by the improved accuracy 89% for classifying identities, while the
best baseline achieves an accuracy of 85%.
(a) Using the original image x as the representation
(b) Representation learned by our model
Figure 2: t-SNE visualizations of images in the Extended Yale B. The original pictures are clustered
by the lighting conditions, while the representation learned by our model is clustered by identities of
individuals
8
In terms of removing s, our framework can filter the lighting conditions since the accuracy of
classifying s drops from 0.96 to 0.57, as shown in Table 2. We also visualize the learned representation
by t-SNE [Maaten and Hinton, 2008] in comparison to the visualization of original pictures in Figure
2. We see that, without removing lighting conditions, the images are clustered based on the lighting
conditions. After removing information of lighting conditions, images are clustered according to the
identity of each person.
7
Conclusion
In sum, we propose a generic framework to learn representations invariant to a specified factor or trait.
We cast the representation learning problem as an adversarial game among an encoder, a discriminator,
and a predictor. We theoretically analyze the optimal equilibrium of the minimax game and evaluate
the performance of our framework on three tasks from different domains empirically. We show that
an invariant representation is learned, resulting in better generalization and improvements on the
three tasks.
Acknowledgement
We thank Shi Feng, Di Wang and Zhilin Yang for insightful discussions. This research was supported
in part by DARPA grant FA8750-12-2-0342 funded under the DEFT program.
9
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11
A
Supplementary Material: Proofs
The proof for Claim 1:
?
(s | h) = p?(s | h). The
Claim. Given a fixed encoder E, the optimal discriminator outputs qD
?
optimal predictor corresponds to qM
(y | h) = p?(y | h).
Proof. We first prove the optimal solution of the discriminator. With a fixed encoder, we have the
following optimization problem
min
qD
s.t.
? J(E, M, D)
X
qD (s | h) = 1, ?h
s
P
P
Then L = J(E, M, D) ? h ?(h)( s qD (s | h) ? 1) is the Lagrangian dual function of the above
optimization problem where ?(h) are the dual variables introduced for equality constraints.
The optimal D satisfies the following equation
0=
??
??
??
?L
? (s
?qD
| h)
?J
? ?(h)
0=? ?
?qD (s | h)
P
?(h, s, y)
yq
?(h) = ? ?
qD (s | h)
?
?(h)qD (s | h) = ??
q (s, h)
Summing w.r.t. s on both sides of the last line of Eqn. (4) and using the fact that
we get
?(h) = ??
q (h)
(4)
?
s qD (s
P
| h) = 1,
(5)
Substituting Eqn. 5 back into Eqn. 4, we can prove the optimal discriminator is
?
qD
(s | h) = q?(s | h)
?
Similarly, taking derivation w.r.t. qM (y | h) and setting it to 0, we can prove qM
(y | h) = q?(y |
h).
12
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6,259 | 6,662 | Convergence Analysis of Two-layer Neural Networks
with ReLU Activation
Yuanzhi Li
Computer Science Department
Princeton University
[email protected]
Yang Yuan
Computer Science Department
Cornell University
[email protected]
Abstract
In recent years, stochastic gradient descent (SGD) based techniques has become
the standard tools for training neural networks. However, formal theoretical understanding of why SGD can train neural networks in practice is largely missing.
In this paper, we make progress on understanding this mystery by providing a
convergence analysis for SGD on a rich subset of two-layer feedforward networks
with ReLU activations. This subset is characterized by a special structure called
?identity mapping?.?We prove that, if input follows from Gaussian distribution,
with standard O(1/ d) initialization of the weights, SGD converges to the global
minimum in polynomial number of steps. Unlike normal vanilla networks, the
?identity mapping? makes our network asymmetric and thus the global minimum is
unique. To complement our theory, we are also able to show experimentally that
multi-layer networks with this mapping have better performance compared with
normal vanilla networks.
Our convergence theorem differs from traditional non-convex optimization techniques. We show that SGD converges to optimal in ?two phases?: In phase I, the
gradient points to the wrong direction, however, a potential function g gradually
decreases. Then in phase II, SGD enters a nice one point convex region and converges. We also show that the identity mapping is necessary for convergence, as it
moves the initial point to a better place for optimization. Experiment verifies our
claims.
1
Introduction
Deep learning is the mainstream technique for many machine learning tasks, including image
recognition, machine translation, speech recognition, etc. [17]. Despite its success, the theoretical
understanding on how it works remains poor. It is well known that neural networks have great
expressive power [22, 7, 3, 8, 31]. That is, for every function there exists a set of weights on the
neural network such that it approximates the function everywhere. However, it is unclear how to
obtain the desired weights. In practice, the most commonly used method is stochastic gradient
descent based methods (e.g., SGD, Momentum [40], Adagrad [10], Adam [25]), but to the best of
our knowledge, there were no theoretical guarantees that such methods will find good weights.
In this paper, we give the first convergence analysis of SGD for two-layer feedforward network with
ReLU activations. For this basic network, it is known that even in the simplified setting where the
weights are initialized symmetrically and the ground truth forms orthonormal basis, gradient descent
might get stuck at saddle points [41].
Inspired by the structure of residual network (ResNet) [21], we add an extra identity mapping for
the hidden layer (see Figure 1). Surprisingly, we show that simply by adding this mapping, with the
standard initialization scheme and small step size, SGD always converges to the ground truth. In other
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
output
Take sum
ReLU(W> x)
output
Take sum
> x)
L ReLU((I + W)
I + W?
Identity
Link +x
W> x
I+W
Identity mapping
W> x
I
Easy for SGD
Seems hard
input x
O
input x
Figure 1: Vanilla network (left), with identity mapping (right)
Unknown
Figure 2: Illustration for our result.
words, the optimization becomes significantly easier, after adding the identity mapping. See Figure
2, based on our analysis, the region near the identity matrix I contains only one global minimum
without any saddle points or local minima, thus is easy for SGD to optimize. The role of the identity
mapping here, is to move the initial point to this easier region (better initialization).
Other than being feedforward and shallow, our network is different from ResNet in the sense that
our identity mapping skips one layer instead of two. However, as we will show in Section 5.1, the
skip-one-layer identity mapping already brings significant improvement to vanilla networks.
Formally, we consider the following function.
f (x, W) = kReLU((I + W)> x)k1
(1)
where ReLU(v) = max(v, 0) is the ReLU activation function. x ? Rd is the input vector sampled
from a Gaussian distribution, and W ? Rd?d is the weight matrix, where d is the number of input
units. Notice that I adds ei to column i of W, which makes f asymmetric in the sense that by
switching any two columns in W, we get different functions.
Following the standard setting [34, 41], we assume that there exists a two-layer teacher network with
weight W? . We train the student network using `2 loss:
L(W) = Ex [(f (x, W) ? f (x, W? ))2 ]
(2)
We will define a potential function g, and show that if g is small, the gradient points to partially
correct direction and we get closer to W? after every SGD step. However, g could be large and thus
gradient might point to the reverse direction. Fortunately, we also show that if g is large, by doing
SGD, it will keep decreasing until it is small enough while maintaining the weight W in a nice region.
We call the process of decreasing g as Phase I, and the process of approaching W? as Phase II. See
Figure 3 and simulations in Section 5.3.
Our two phases framework is fundamentally different from any type of local convergence, as in Phase
I, the gradient is pointing to the wrong direction to W? , so the path from W to W? is non-convex,
and SGD takes a long detour to arrive W? . This framework could be potentially useful for analyzing
other non-convex problems.
To support our theory, we have done a few other experiments and got interesting observations.
For example, as predicted by our theorem, we found that for multilayer feedforward network with
identity mappings, zero initialization performs as good as random initialization. At the first glance, it
contradicts the common belief ?random initialization is necessary to break symmetry?, but actually
the identity mapping itself serves as the asymmetric component. See Section 5.4.
Another common belief is that neural network has lots of local minima and saddle points [9], so
even if there exists a global minimum, we may not be able to arrive there. As a result, even when
the teacher network is shallow, the student network usually needs to be deeper, otherwise it will
underfit. However, both our theorem and our experiment show that if the shallow teacher network
is in a pretty large region near identity (Figure 2), SGD always converges to the global minimum
by initializing the weights I + W in this region, with equally shallow student network. By contrast,
wrong initialization gets stuck at local minimum and underfit. See Section 5.2.
2
Related Work
Expressivity. Even two-layer network has great expressive power. For example, two-layer network
with sigmoid activations could approximate any continuous function [22, 7, 3]. ReLU is the state-ofthe-art activation function [30, 13], and has great expressive power as well [29, 32, 31, 4, 26].
Learning. Most previous results on learning neural network are negative [39, 28, 38], or positive but
with algorithms other than SGD [23, 43, 37, 14, 15, 16], or with strong assumptions on the model
[1, 2]. [35] proved that with high probability, there exists a continuous decreasing path from random
initial point to the global minimum, but SGD may not follow this path. Recently, Zhong et al. showed
that with initialization point found using tensor decomposition, gradient descent could find the ground
truth for one hidden layer network [44].
Linear network and independent activation. Some previous works simplified the model by ignoring the activation functions and considering deep linear networks [36, 24] or deep linear residual
networks [19], which can only learn linear functions. Some previous results are based on independent
activation assumption that the activations of ReLU and the input are independent [5, 24].
Saddle points. It is observed that saddle point is not a big problem for neural networks [9, 18]. In
general, if the objective is strict-saddle [11], SGD could escape all saddle points.
2
Preliminaries
Denote x as the input vector in Rd . For now, we first consider x sampled from normal distribution
N (0, I). Denote W? = (w1? , ? ? ? , wn? ) ? Rd?d as the weights for the teacher network, W =
(w1 , ? ? ? , wn ) ? Rd?d as the weights for the student network, where wi? , wi ? Rd are column
vectors. f (x, W? ), f (x, W) are defined in (1), representing the teacher and student network.
We want to know whether a randomly initialized W will converge to W? , if we run SGD with l2
loss defined in (2). Alternatively, we can write the loss L(W) as
Ex [(?i ReLU(hei + wi , xi) ? ?i ReLU(hei + wi? , xi))2 ]
Taking derivative with respect to wj , we get
"
!
#
X
X
?
?L(W)j = 2Ex
ReLU(hei + wi , xi) ?
ReLU(hei + wi , xi) x1hej +wj ,xi?0
i
i
where 1e is the indicator function that equals 1 if the event e is true, and 0 otherwise. Here
?L(W) ? Rd?d , and ?L(W)j is its j-th column.
Denote ?i,j as the angle between ei + wi and ej + wj , ?i? ,j as the angle between ei + wi? and ej + wj .
v
Denote v? = kvk
. Denote I + W? and I + W? as the column-normalized version of I + W? and
2
I + W such that every column has unit norm. Since the input is from a normal distribution, one can
compute the expectation inside the gradient as follows.
Pd ? ?
Lemma 2.1 (Eqn (13) from [41]). If x ? N (0, I), then ??L(W)j =
i=1 2 (wi ? wi ) +
?
?
?
?
?
?
2 ? ?i ,j (ei + wi ) ? 2 ? ?i,j (ei + wi ) + kei + wi k2 sin ?i ,j ? kei + wi k2 sin ?i,j ej + wj
Remark. Although the gradient of ReLU is not well defined at the point of zero, if we assume input x
is from the Gaussian distribution, the loss function becomes smooth, and the gradient is well defined
everywhere.
Denote u ? Rd as the all one vector. Denote Diag(W) as the diagonal matrix of matrix W,
Diag(v) as a diagonal matrix whose main diagonal equals to the vector v. Denote Off-Diag(W) ,
W ? Diag(W). Denote [d] as the set {1, ? ? ? , d}. Throughout the paper, we abuse the notation of
inner product between matrices W, W? , ?L(W), such that h?L(W), Wi means the summation of
the entrywise products. kWk2 is the spectral norm of W, and kWkF is the Frobenius norm of W.
We define the potential function g and variables gj , Aj , A below, which will be useful in the proof.
Pd
Definition 2.2. We define the potential function g , i=1 (kei + wi? k2 ? kei + wi k2 ), and variable
P
gj , i6=j (kei + wi? k2 ? kei + wi k2 ).
3
W1
15
W6
10
5
W10
0
W?
?5
200
150
200
150
100
100
50
50
0
Figure 3: Phase I: W1 ? W6 , W may go to
the wrong direction but the potential is shrinking.
Phase II: W6 ? W10 , W gets closer to W? in
every step by one point convexity.
Definition 2.3. Denote Aj ,
wi? )ei
+
>
wi?
? (ei + wi )ei +
0
Figure 4: The function is one point strongly convex as every point?s negative gradient points to
the center, but not convex as any line between
the center and the red region is below surface.
?
? > ? (e + w )e + w > ), A
i
i i
i
i6=j ((ei + wi )ei + wi
>
>
>
?
?
wi ) = (I + W )I + W ? (I + W)I + W .
P
,
Pd
i=1 ((ei
+
In this paper, we consider the standard SGD with mini batch method for training the neural network.
Assume W0 is the initial point, and in step t > 0, we have the following updating rule:
Wt+1 = Wt ? ?t Gt
where the stochastic gradient Gt = ?L(Wt ) + Et with E[Et ] = 0 and kEt kF ? ?. Let G2 ,
6d? + ?, GF , 6d1.5 ? + ?, where ? is the upper bound of kW? k2 and kW0 k2 (defined later). As
we will see in Lemma C.2, they are the upper bound of kGt k2 and kGt kF respectively.
It?s clear that L is not convex, In order to get convergence guarantees, we need a weaker condition
called one point convexity.
Definition 2.4 (One point strongly convexity). A function f (x) is called ?-one point strongly convex
in domain D with respect to point x? , if ?x ? D, h??f (x), x? ? xi > ?kx? ? xk22 .
By definition, if a function f is strongly convex, it is also one point strongly convex in the entire space
with respect to the global minimum. However, the reverse is not necessarily true, e.g., see Figure
4. If a function is one point strongly convex, then in every step a positive fraction of the negative
gradient is pointing to the optimal point. As long as the step size is small enough, we will finally
arrive the optimal point, possibly by a winding path. See Figure 3 for illustration, where starting from
W6 (Phase II), we get closer to W? in every step. Formally, we have the following lemma.
Lemma 2.5. For function f (W), consider the SGD update Wt+1 = Wt ? ?Gt , where E[Gt ] =
?f (Wt ), E[kGt k2F ] ? G2 . Suppose for all t, Wt is always inside the ?-one point strongly convex
region with diameter D, i.e., kWt ? W? kF ? D. Then for any ? > 0 and any T such that
(1+?) log T
T G2
D2 ?2
, we have EkWT ? W? k2F ? (1+?)?2log
.
T ? log T ? (1+?)G
2 , if ? =
?T
T
The proof can be found in Appendix J. Lemma 2.5 uses fixed step size, so it easily fits the standard
practical scheme that shrinks ? by a factor of 10 after every a few epochs. For example, we may
apply Lemma 2.5 every time ? gets changed. Notice that our lemma does not imply that WT will
converge to W? . Instead, it only says WT will be sufficiently close to W? with small step size ?.
3
Main Theorem
Theorem 3.1 (Main Theorem). There exists constants ? > ?0 > 0 such that If x ? N (0, I),
kW0 k2 , kW? k2 ? ?0 , d ? 100, ? ? ? 2 , then SGD for L(W) will find the ground truth W? by
?2
two phases. In Phase I, by setting ? ? G
2 , the potential function will keep decreasing until it is
?
T log T ?
36d
1004 (1+?)G2F
, if we set ? =
2
1
16? steps.
(1+?) log T
,
?T
smaller than 197? 2 , which takes at most
In Phase II, for any ? > 0 and any T such that
we have EkWT ? W? k2F ?
1002 (1+?) log T G2F
9T
.
?
Remarks. Randomly initializing the weights with O(1/ d) is standard ?
in deep learning, see
[27, 12, 20]. It is also well known that if the entries are initialized with O(1/ d), the spectral norm
4
of the random matrix is O(1) [33]. So our result matches with the common practice. Moreover, as we
will show in Section 5.5, networks with small average spectral norm already have good performance.
Thus, our assumption kW? k2 = O(1) is reasonable. Notice that here we assume the spectral
norm
?
?
?
of W to be constant, which means the Frobenius norm kW kF could be as big as O( d).
The assumption that the input follows a Gaussian distribution is not necessarily true in practice
(Although this is a common assumption appeared in the previous papers [5, 41, 42], and also
considered plausible in [6]). We could easily generalize the analysis to rotation invariant distributions,
and potentially more general distributions (see Section 6). Moreover, previous analyses either ignore
the nonlinear activations and thus consider linear model [36, 24, 19], or directly [5, 24] or indirectly
[41]1 assume that the activations are independent. By contrast, in our model the ReLU activations
are highly correlated2 as kWk2 , kW? k2 = ?(1). As pointed out by [6], eliminating the unrealistic
assumptions on activation independence is the central problem of analyzing the loss surface of neural
network, which was not fully addressed by the previous analyses.
To prove the main theorem, we split the process and present the following two theorems, which will
be proved in Appendix C and D.
Theorem 3.2 (Phase I). There exists a constant ? > ?0 > 0 such that If kW0 k2 , kW? k2 ? ?0 ,
?2
2
d ? 100, ? ? G
2 , ? ? ? , then gt will keep decreasing by a factor of 1 ? 0.5?d for every step,
2
until gt1 ? 197? 2 for step t1 ?
1
kWT k2 ? 100
and gT ? 0.1.
1
16? .
After that, Phase II starts. That is, for every T > t1 , we have
Theorem 3.3 (Phase II). There exists a constant ? such that if kWk2 , kW? k2 ? ?, and g ? 0.1,
Pd
then h??L(W), W? ? Wi = j=1 h??L(W)j , wj? ? wj i > 0.03kW? ? Wk2F .
With these two theorems, we get the main theorem immediately.
Proof for Theorem 3.1. By Theorem 3.2, we know the statement for Phase I is true, and we will enter
1
phase II in 16?
steps. After entering Phase II, based on Theorem 3.3, we simply use Lemma 2.5 by
setting ? = 0.03, D =
4
?
d
50 , G
= GF to get the convergence guarantee.
Overview of the Proofs
General Picture. In many convergence analyses for non-convex functions, one would like to show
that L is one point strongly convex, and directly apply Lemma 2.5 to get the convergence result.
However, this is not true for 2-layer neural network, as the gradient may point to the wrong direction,
see Section 5.3.
So when is our L one point convex? Consider the following thought experiment: First, suppose
kWk2 , kW? k2 ? 0, we know kwi k2 , kwi? k2 also go to 0. Thus, ei + wi and ei + wi? are close to ei .
As a result, ?i,j , ?i? ,j ? ?2 , and ?i? ,i ? 0. Based on Lemma 2.1, this gives us a na?ve approximation
Pd
P
of the negative gradient, i.e., ??L(W)j ? ?2 (wj? ? wj ) + ?2 i=1 (wi? ? wi ) + ej + wj i6=j (kei +
wi? k2 ? kei + wi k2 ) .
Pd
While the first two terms ?2 (wj? ?wj ) and ?2 i=1 (wi? ?wi ) have positive inner product with W? ?W,
P
the last term gj = ej + wj i6=j (kei +wi? k2 ? kei +wi k2 ) can point to arbitrary direction. If the last
term is small, it can be covered by the first two terms, and L becomes one point strongly convex. So
Pd
we define a potential function closely related to the last term: g = i=1 (kei + wi? k2 ? kei + wi k2 ).
We show that if g is small enough, L is also one point strongly convex (Theorem 3.3).
?
However, from random initialization, g can be as large as of ?( d), which is too big to be covered.
Fortunately, we show that if g is big, it will gradually decrease simply by doing SGD on L. More
specifically, we introduce a two phases convergence analysis framework:
1
They assume input is Gaussian and the W? is orthonormal, which means the activations are independent in
teacher network.
P
2
Let ?i be the output of i-th ReLU unit, then in our setting, i,j Cov[?i , ?j ] can be as large as ?(d), which
is far from being independent.
5
h
Constant
Part
+
First
Order
+
Higher
Order
, W? ? W
i
?
2
?
2
?
2
? [?
2 ? O(g)]kW ? WkF ?1.3kW ? WkF ?0.085kW ? WkF
Lemma D.1
Lemma B.2
Lemma D.2 + Lemma D.3
Figure 5: Lower bounds of inner product using Taylor expansion
1. In Phase I, the potential function g is decreasing to a small value.
2. In Phase II, g remains small, so L is one point convex and thus W starts to converge to W? .
We believe that this framework could be helpful for other non-convex problems.
Technical difficulty: Phase I. Our key technical challenge is to show that in Phase I, the potential
function actually decreases to O(1) after polynomial number of iterations. However, we cannot show
this by merely looking at g itself. Instead, we introduce an auxiliary variable s = (W? ? W)u,
where u is the all one vector. By doing a careful calculation, we get their joint update rules (Lemma
C.3 and Lemma C.4):
?
?
st+1 ? st ? ??d
d?)
2 st + ?O( ?dgt +
gt+1 ? gt ? ?dgt + ?O(? dkst k2 + d? 2 )
Solving this dynamics, we can show that gt will approach to (and stay around) O(?), thus we enter
Phase II.
Technical difficulty: Phase II. Although the overall approximation in the thought experiment looks
simple, the argument is based on an over simplified assumption that ?i? ,j , ?i,j ? ?2 for i 6= j.
However, when W? has constant spectral norm, even when W is very close to W? , ?i,j ? could be
constantly far away from ?2 , which prevents us from applying this approximation directly. To get a
formal proof, we use the standard Taylor expansion and control the higher order terms. Specifically,
we write ?i? ,j as ?i? ,j = arccoshei + wi? , ej + wj i and expand arccos at point 0, thus,
?i? ,j =
?
? hei + wi? , ej + wj i + O(hei + wi? , ej + wj i3 )
2
However, even when W ? W? , the higher order term O(hei + wi? , ej + wj i3 ) still can be as large
as a constant, which is too big for us. Our trick here is to consider the ?joint Taylor expansion?:
?i? ,j ? ?i,j = hei + wi ? ei + wi? , ej + wj i + O(|hei + wi? , ej + wj i3 ? hei + wi , ej + wj i3 |)
As W approaches W? , |hei + wi? , ej + wj i3 ? hei + wi , ej + wj i3 | also tends to zero, therefore
our approximation has bounded error.
In the thought experiment, we already know that the constant part in the Taylor expansion of ?L(W)
is ?2 ? O(g)-one point convex. We show that after taking inner product with W? ? W, the first
order terms are lower bounded by (roughly) ?1.3kW? ? Wk2F and the higher order terms are lower
bounded by ?0.085kW? ? Wk2F . Adding them together, we can see that L(W) is one point convex
as long as g is small. See Figure 5.
Geometric Lemma. In order to get through the whole analysis, we need tight bounds on a few
common terms that appear everywhere. Instead of using na?ve algebraic techniques, we come up with
a nice geometric proof to get nearly optimal bounds. Due to space limit, we defer it to Appendix E.
5
Experiments
In this section, we present several simulation results to support our theory. Our code can be found in
the supplementary materials.
5.1
Importance of identity mapping
In this experiment, we compare the standard ResNet [21] and single skip model where identity
mapping skips only one layer. See Figure 6 for the single skip model. We also ran the vanilla network,
where the identity mappings are completely removed.
6
Single skip
9.01%
?
Vanilla
12.04%
ReLU
ResNet
6.97%
BatchNorm
Test Err
Convolution
input
Table 1: Test error of three 56-layer networks on
Cifar-10
output
Identity
Figure 6: Illustration of one block in single skip
model in Sec 5.1
10
20.0
Test (ResLink)
Test (Vanilla)
Test (3-Block)
Train (ResLink)
Train (Vanilla)
Train (3-Block)
loss
6
4
17.5
15.0
W (ResLink)
W-W* (ResLink)
W (Vanilla)
W-W* (Vanilla)
12.5
l2 norm
8
10.0
7.5
5.0
2
2.5
0
0
25
50
75
100
epochs
125
150
175
0.0
200
(a) Test Error, Train Error
0
(b)
25
50
75
100
epochs
125
150
175
200
?
kW ? WkF , kWkF
Figure 7: Verifying the global convergence
In this experiment, we choose Cifar-10 as the dataset, and all the networks have 56-layers. Other than
the identity mappings, all other settings are identical and default. We run the experiments for 5 times
and report the average test error. As we can see in Table 1, compared with vanilla network, by simply
using a single skip identity mapping, one can already improve the test error by 3.03%, and is 2.04%
close to the ResNet. So single skip identity mapping brings significant improvement on test accuracy.
5.2
Global minimum convergence
In this experiment, we verify our main theorem that for two-layer teacher network and student network
with identity mappings, as long as kW0 k2 , kW? k2 is small, SGD always converges to the global
minimum W? , thus gives almost 0 training error and test error. We consider three student networks.
The first one (ResLink) is defined using (2), the second one (Vanilla) is the same model without the
identity mapping. The last one (3-Block) is a three block network with each block containing a linear
layer (500 hidden nodes), a batch normalization and a ReLU layer. The teacher network always
shares the same structure as the student network.
The input dimension is 100. We generated a fixed W? for all the trials with kW? k2 ? 0.6, kW? kF ?
5.7. We generated a training set of size 100, 000, and test set of size 10, 000, sampled from a Gaussian
distribution. We use batch size 200, step size 0.001. We run ResLink for 5 times with random
initialization (kWk2 ? 0.6 and kWkF ? 5), and plot the curves by taking the average.
Figure 7(a) shows test error and training error of the three networks. Comparing Vanilla with 3-Block,
we find that 3-Block is more expressive, so its training error is smaller compared with vanilla network;
but it suffers from overfitting and has bigger test error. This is the standard overfitting vs underfitting
tradeoff. Surprisingly, with only one hidden layer, ResLink has both zero test error and training
error. If we look at Figure 7(b), we know the distance between W and W? converges to 0, meaning
ResLink indeed finds the global optimal in all 5 trials. By contrast, for vanilla network, which is
essentially the same network with different initialization, kW ? W? k2 does not converge to zero3 .
This is exactly what our theory predicted.
5.3
Verify the dynamics
In this experiment, we verify our claims on the dynamics. Based on the analysis, we construct a
1500 ? 1500 matrix W s.t. kWk2 ? 0.15, kWkF ? 5 , and set W? = 0. By plugging them into (2),
one can see that even in this simple case that W? = 0, initially the gradient is pointing to the wrong
direction, i.e., not one point convex. We then run SGD on W by using samples x from Gaussian
distribution, with batch size 300, step size 0.0001.
3
To make comparison meaningful, we set W ? I to be the actual weight for Vanilla as its identity mapping
is missing, which is why it has a much bigger initial norm.
7
25
16
P-I P-II
20
Distance to optimal
Inner product
Potential g
Loss
14
12
15
10
10
8
5
6
4
0
2
?5
?10
0
0
20
40
60
80
100
?2
(a) First 100 iterations
0
50000
100000 150000 200000 250000 300000 350000
(b) The entire process
Figure 8: Verifying the dynamics
Figure 8(a) shows the first 100 iterations. We can see that initially the inner product defined in
Definition 2.4 is negative, then after about 15 iterations, it turns positive, which means W is in the
one point strongly convex region. At the same time, the potential g keeps decreasing to a small value,
while the distance to optimal (which also equals to kWkF in this experiment) is not affected. They
precisely match with our description of Phase I in Theorem 3.2.
After that, we enter Phase II and slowly approach to W? , see Figure 8(b). Notice that the potential
g is always very small, the inner product is always positive, and the distance to optimal is slowly
decreasing. Again, they precisely match with our Theorem 3.3.
5.4
Zero initialization works
In this experiment, we used a simple 5-block neural network on MNIST, where every block contains
a 784 ? 784 feedforward layer, an identity mapping, and ?
a ReLU layer. Cross entropy criterion is
used. We compare zero initialization with standard O(1/ d) random initialization. We found that
for zero initialization, we can get 1.28% test error, while for random initialization, we can get 1.27%
test error. Both results were obtained by taking average among 5 runs and use step size 0.1, batch
size 256. If the identity mapping is removed, zero initialization no longer works.
5.5
Spectral norm of W?
We also applied the exact model f defined in (1) to distinguish two classes in MNIST. For any input
image x, We say it?s in class A if f (x, W) < TA,B , and in class B otherwise. Here TA,B is the
optimal threshold for the function f (x, 0) to distinguish A and B. If W = 0, we get 7% training
error for distinguish class 0 and class 1. However, it can be improved to 1% with kWk2 = 0.6.
We tried this experiment for all possible 45 pairs of classes in MNIST, and improve the average
training error from 34% (using W = 0) to 14% (using kWk2 = 0.6). Therefore our model with
kWk2 = ?(1) has reasonable expressive power, and is substantially different from just using the
identity mapping alone.
6
Discussions
The assumption that the input is Gaussian can be relaxed in several ways. For example, when
the distribution is N (0, ?) where k? ? Ik2 is bounded by a small constant, the same result holds
with slightly worse constants. Moreover, since the analysis relies Lemma 2.1, which is proved by
converting the original input space into polar space, it is easy to generalize the calculation to rotation
invariant distributions. Finally, for more general distributions, as long as we could explicitly compute
the expectation, which is in the form of O(W? ? W) plus certain potential function, our analysis
framework may also be applied.
There are many exciting open problems. For example, Our paper is the first one that gives solid
SGD analysis for neural network with nonlinear activations, without unrealistic assumptions like
independent activation assumption. It would be great if one could further extend it to multiple layers,
which would be a major breakthrough of understanding optimization for deep learning. Moreover,
our two phase framework could be applied to other non-convex problems as well.
8
Acknowledgement
The authors want to thank Robert Kleinberg, Kilian Weinberger, Gao Huang, Adam Klivans and
Surbhi Goel for helpful discussions, and the anonymous reviewers for their comments.
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6,260 | 6,663 | Doubly Accelerated
Stochastic Variance Reduced Dual Averaging Method
for Regularized Empirical Risk Minimization
Tomoya Murata
NTT DATA Mathematical Systems Inc. , Tokyo, Japan
[email protected]
Taiji Suzuki
Department of Mathematical Informatics
Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan
PRESTO, Japan Science and Technology Agency, Japan
Center for Advanced Integrated Intelligence Research, RIKEN, Tokyo, Japan
[email protected]
Abstract
We develop a new accelerated stochastic gradient method for efficiently solving
the convex regularized empirical risk minimization problem in mini-batch settings.
The use of mini-batches has become a golden standard in the machine learning
community, because the mini-batch techniques stabilize the gradient estimate and
can easily make good use of parallel computing. The core of our proposed method
is the incorporation of our new ?double acceleration? technique and variance reduction technique. We theoretically analyze our proposed method and show that
our method much improves the mini-batch efficiencies
of previous accelerated
?
stochastic methods, and essentially only needs size n mini-batches for achieving
the optimal iteration complexities for both non-strongly and strongly convex objectives, where n is the training set size. Further, we show that even in non-mini-batch
settings, our method achieves the best known convergence rate for non-strongly
convex and strongly convex objectives.
1
Introduction
We consider a composite convex optimization problem associated with regularized empirical risk
minimization, which often arises in machine learning. In particular, our goal is to minimize the
sum of finite smooth convex functions and a relatively simple (possibly) non-differentiable convex
function by using first order methods in mini-batch settings. The use of mini-batches is now a
golden standard in the machine learning community, because it is generally more efficient to execute
matrix-vector multiplications over a mini-batch than an equivalent amount of vector-vector ones each
over a single instance; and more importantly, mini-batch techniques can easily make good use of
parallel computing.
Traditional and effective methods for solving the abovementioned problem are the ?proximal gradient?
(PG) method and ?accelerated proximal gradient? (APG) method [10, 3, 20]. These methods are
well known to achieve linear convergence for strongly convex objectives. Particularly, APG achieves
optimal iteration complexities for both non-strongly and strongly convex objectives. However, these
methods need a per iteration cost of O(nd), where n denotes the number of components of the
finite sum, and d is the dimension of the solution space. In typical machine learning tasks, n and d
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
correspond to the number of instances and features respectively, which can be very large. Then, the
per iteration cost of these methods can be considerably high.
A popular alternative is the ?stochastic gradient descent? (SGD) method [19, 5, 17]. As the per
iteration cost of SGD is only O(d) in non-mini-batch settings, SGD is suitable for many machine
learning tasks. However, SGD only achieves sublinear rates and is ultimately slower than PG and
APG.
Recently, a number of stochastic gradient methods have been proposed; they use a variance reduction
technique that utilizes the finite sum structure of the problem (?stochastic averaged gradient? (SAG)
method [15, 16], ?stochastic variance reduced gradient" (SVRG) method [6, 22] and SAGA [4]).
Even though the per iteration costs of these methods are same as that of SGD, they achieve a linear
convergence for strongly convex objectives. Consequently, these methods dramatically improve the
total computational cost of PG. However, in size b mini-batch settings, the rate is essentially b times
worse than in non-mini-batch settings (the extreme situation is b = n which corresponds to PG). This
means that there is little benefit in applying mini-batch scheme to these methods.
More recently, several authors have proposed accelerated stochastic methods for the composite finite
sum problem (?accelerated stochastic dual coordinate ascent? (ASDCA) method [18], Universal
Catalyst (UC) [8], ?accelerated proximal coordinate gradient? (APCG) method [9], ?stochastic
primal-dual coordinate? (SPDC) method [23], and Katyusha [1]). ASDCA (UC), APCG, SPDC and
2
Katyusha essentially achieve the optimal total computational cost1 for strongly convex objectives
?
in non-mini-batch settings. However, in size b mini-batch settings, the rate is essentially b times
worse than that in non-mini-batch settings, and these methods need size O(n) mini-batches for
achieving the optimal iteration complexity3 , which is essentially the same as APG. In addition,
[12, 13] has proposed the ?accelerated mini-batch proximal stochastic variance reduced gradient?
(AccProxSVRG) method and its variant, the ?accelerated efficient mini-batch stochastic variance
reduced gradient? (AMSVRG) method. In non-mini-batch settings, AccProxSVRG only achieves the
same rate as SVRG. However, in mini-batch settings, AccProxSVRG significantly improves the minibatch efficiency of non-accelerated
variance reduction methods, and surprisingly, AccProxSVRG
?
essentially only needs size O( ?) mini-batches for achieving the optimal iteration complexity for
strongly convex objectives, where ? is the condition number of the problem. However, the necessary
size of mini-batches depends on the condition number and gradually increases when the condition
number increases and ultimately matches with O(n) for a large condition number.
Main contribution
We propose a new accelerated stochastic variance reduction method that achieves better convergence
than existing methods do, and it particularly takes advantages of mini-batch settings well; it is called
the ?doubly accelerated stochastic variance reduced dual averaging? (DASVRDA) method. We
describe the main feature of our proposed method below and list the comparisons of our method with
several preceding methods in Table 1.
Our method significantly improves the mini-batch?
efficiencies of state-of-the-art methods. As a
result, our method essentially only needs size O( n) mini-batches4 for achieving the optimal
iteration complexities for both non-strongly and strongly convex objectives.
1
More precisely, the rate of ASDCA (UC) is with extra log-factors, and near but worse than the one of APCG,
SPDC and Katyusha. This means that ASDCA (UC) cannot be optimal.
2
Katyusha also achieves a near optimal total computational cost for non-strongly convex objectives.
3
We refer to ?optimal iteration complexity? as the iteration complexity of deterministic Nesterov?s acceleration method [11].
p
p
4
Actually, when L/? ? n and L/??? n, our method needs size O(n ?/L) and O(n ?/L) mini-batches,
respectively, which are larger than O( n), but smaller than O(n). Achieving optimal iteration complexity for
solving high accuracy and bad conditioned problems is much more important than doing ones with low accuracy
and well-conditioned ones, because the former needs more overall computational cost than the latter.
2
Table 1: Comparisons of our method with SVRG (SVRG++ [2]), ASDCA (UC), APCG, SPDC,
Katyusha and AccProxSVRG. n is the number of components of the finite sum, d is the dimension of
the solution space, b is the mini-batch size, L is the smoothness parameter of the finite sum, ? is the
strong convexity parameter of objectives, and ? is accuracy. ?Necessary mini-batch size? indicates
thep
order of the necessary p
size of mini-batches for achieving the optimal iteration complexities
O( L/?log(1/?)) and O( L/?) for strongly and non-strongly convex objectives, respectively.
We regard one computation of a full gradient as n/b iterations in size b mini-batch settings, for
a fair comparison. ?Unattainable? implies that the algorithm cannot achieve the optimal iteration
e hides extra log-factors.
complexity even if it uses size n mini-batches. O
SVRG (++ )
ASDCA (UC)
APCG
SPDC
Katyusha
AccProxSVRG
DASVRDA
2
?-strongly convex
Total computational cost
Necessary mini-batch size
in size
L/? ? n
otherwise
settings
b mini-batch
1
bL
Unattainable Unattainable
O d n + ? log ?
q
e d n + nbL log 1
O
Unattainable Unattainable
?
q ?
log 1?
O(n)
O(n)
O d n + nbL
q ?
O d n + nbL
log 1?
O(n)
O(n)
?
q
O d n + nbL
log 1?
O(n)
O(n)
?
q
q
p?
L
1
L
n?b L
O
O n L
O d n + n?1
? +b
? log ?
?
p?
? qL
?
1
O n L
O d n + (b + n) ? log ?
O ( n)
Non-strongly convex
Total computational cost
Necessary mini-batch size
in size b mini-batch settings
L/? ? nlog2 (1/?)
otherwise
bL
1
O d nlog ? + ?
Unattainable
Unattainable
?
e d n+? nbL
O
Unattainable
Unattainable
?
No direct analysis
Unattainable
Unattainable
No direct analysis
q
O d nlog 1? + nbL
?
Unattainable
Unattainable
O(n)
O(n)
No direct analysis
? q
O d nlog 1? + (b + n) L?
Unattainable
?
O ( n)
Unattainable
p
e n ?
O
L
Preliminary
In this section, we formally describe the problem to be considered in this paper and the assumptions
for our theory.
pP
2
We use k ? k to denote the Euclidean L2 norm k ? k2 : kxk = kxk2 =
i xi . For natural number n,
[n] denotes set {1, . . . , n}.
In this paper, we consider the following composite convex minimization problem:
def
min {P (x) = F (x) + R(x)},
x?Rd
(1)
Pn
where F (x) = n1 i=1 fi (x). Here each fi : Rd ? R is a Li -smooth convex function and
R : Rd ? R is a relatively simple and (possibly) non-differentiable convex function. Problems
of this form often arise in machine learning and fall under regularized empirical risk minimization
(ERM). In ERM problems, we are given n training examples {(ai , bi )}ni=1 , where each ai ? Rd
is the feature vector of example i, and each bi ? R is the label of example i. Important examples
of ERM in our setting include linear regression and logistic regression with Elastic Net regularizer
R(x) = ?1 k ? k1 + (?2 /2)k ? k22 (?1 , ?2 ? 0).
We make the following assumptions for our analysis:
Assumption 1. There exists a minimizer x? of (1).
Assumption 2. Each fi is convex, and is Li -smooth, i.e.,
k?fi (x) ? ?fi (y)k ? Li kx ? yk (?x, y ? Rd ).
Assumption 3. Regularization function R is convex, and is relatively
simple, which means
that
computing the proximal mapping of R at y, proxR (y) = argmin x?Rd 21 kx ? yk2 + R(x) , takes
O(d) computational cost, for any y ? Rd .
We always consider Assumptions 1, 2 and 3 in this paper.
Assumption 4. There exists ? > 0 such that objective function P is ?-optimally strongly convex,
i.e., P has a minimizer and satisfies
?
kx ? x? k2 ? P (x) ? P (x? ) (?x ? Rd , ?x? ? argminx?Rd f (x)).
2
Note that the requirement of optimally strong convexity is weaker than the one of ordinary strong
convexity (for the definition of ordinary strong convexity, see [11]).
We further consider Assumption 4 when we deal with strongly convex objectives.
3
3
Our Approach: Double Acceleration
In this section, we provide high-level ideas of our main contribution called ?double acceleration.?
First, we consider deterministic PG (Algorithm 1) and (non-mini-batch) SVRG (Algorithm 2). PG
is an extension of the steepest descent to proximal settings. SVRG is a stochastic gradient method
using the variance reduction technique, which utilizes the finite sum structure of the problem, and
it achieves a faster convergence rate than PG does. As SVRG (Algorithm 2) matches with PG
(Algorithm 1) when the number of inner iterations m equals 1, SVRG can be seen as a generalization
of PG. The key element of SVRG is employing a simple but powerful technique called the variance
reduction technique for gradient estimate. The variance reduction of the gradient is realized by
setting gk = ?fik (xk?1 ) ? ?fik (e
x) + ?F (e
x) rather than vanilla stochastic gradient ?fik (xk?1 ).
Generally, stochastic gradient ?fik (xk?1 ) is an unbiased estimator of ?F (xk?1 ), but it may have
high variance. In contrast, gk is also unbiased, and one can show that its variance is ?reduced?; that
is, the variance converges to zero as xk?1 and x
e to x? .
Algorithm 1: PG (e
x0 , ?, S)
for s = 1 to S do
x
es = One Stage PG(e
xs?1 , ?).
end for P
S
return S1 s=1 x
es .
Algorithm 3: One Stage PG (e
x, ?)
x
e+ = prox?R (e
x ? ??F (e
x)).
return x
e+ .
Algorithm 4: One Stage SVRG (e
x, ?, m)
x0 = x
e.
for k = 1 to m do
Pick ik ? [1, n] randomly.
gk = ?fik (xk?1 ) ? ?fik (e
x) + ?F (e
x).
xk = prox?R (xk?1 ? ?gk ).
end for P
n
return n1 k=1 xk .
Algorithm 2: SVRG (e
x0 , ?, m, S)
for s = 1 to S do
x
es = One Stage SVRG(e
xs?1 , ?, m).
end for P
S
return S1 s=1 x
es .
Algorithm 5: APG (e
x0 , ?, S)
x
e?1 = x
e0 , ?e0 = 0.
for s = 1 to S do
es?1 +
?es = s+1 , yes = x
2
?es?1 ?1
(e
xs?1
?es
?x
es?2 ).
x
es = One Stage PG(e
ys , ?).
end for
return xS .
Next, we explain the method of accelerating SVRG and obtaining an even faster convergence rate
based on our new but quite natural idea ?outer acceleration.? First, we would like to remind you
that the procedure of deterministic APG is given as described in Algorithm 5. APG uses the famous
?momentum? scheme and achieves the optimal iteration complexity. Our natural idea is replacing
One Stage PG in Algorithm 5 with One Stage SVRG. With slight modifications, we can show
that this algorithm improves the rates of PG, SVRG and APG, and is optimal. We call this new
algorithm outerly accelerated SVRG. However, this algorithm has poor mini-batch
efficiency, because
?
in size b mini-batch settings, the rate of this algorithm is essentially b times worse than that of
non-mini-batch settings. State-of-the-art methods APCG, SPDC, and Katyusha also suffer from the
same problem in the mini-batch setting.
Now, we illustrate that for improving the mini-batch efficiency, using the ?inner acceleration? technique is beneficial. The author of [12] has proposed AccProxSVRG in mini-batch settings. AccProxSVRG applies the momentum scheme to One Stage SVRG, and we call this technique ?inner?
acceleration. He showed that the inner acceleration could significantly improve the mini-batch
efficiency of vanilla SVRG. This fact indicates that inner acceleration is essential to fully utilize
the mini-batch settings. However, AccProxSVRG is not a truly accelerated method, because in
non-mini-batch settings, the rate of AccProxSVRG is same as that of vanilla SVRG.
In this way, we arrive at our main high-level idea called ?double? acceleration, which involves
applying momentum scheme to both outer and inner algorithms. This enables us not only to lead to
4
the optimal total computational cost in non-mini-batch settings, but also to improving the mini-batch
efficiency of vanilla acceleration methods.
We have considered SVRG and its accelerations so far; however, we actually adopt stochastic variance
reduced dual averaging (SVRDA) rather than SVRG itself, because we can construct lazy update
rules of (innerly) accelerated SVRDA for sparse data. In Section G of supplementary material, we
briefly discuss a SVRG version of our proposed method and provide its convergence analysis.
4
Algorithm Description
In this section, we describe the concrete procedure of the proposed algorithm in detail.
4.1
DASVRDA for non-strongly convex objectives
Algorithm 6: DASVRDAns (e
x0 , ze0 , ?, {Li }ni=1 , m, b, S)
? = 1 Pn Li , Q = {qi } = L?i , ? =
x
e?1 = ze0 , ?e0 = 0, L
i=1
n
nL
for s =
1 to S do
e
?s = 1 ? ?1 s+2
es = x
es?1 +
2 , y
?es?1 ?1
(e
xs?1
?es
1
(1+ ?(m+1)
)L?
b
?x
es?2 ) +
?es?1
(e
zs?1
?es
.
?x
es?1 ).
(e
xs , zes ) = One Stage AccSVRDA(e
ys , x
es?1 , ?, m, b, Q).
end for
return x
eS .
Algorithm 7: One Stage AccSVRDA (e
y, x
e, ?, m, b, Q)
x0 = z0 = ye, g?0 = 0, ?0 = 12 .
for k = 1 to m do
Pick independently i1k , . . . ,ibk ? [1, n] according to Q, set Ik = {i`k }b`=1 .
?k =
k+1
2 ,
yk = 1 ?
1
?k
xk?1 +
1
?k zk?1 .
(?fi (yk ) ? ?fi (e
x)) + ?F (e
x), g?k = 1 ? ?1k g?k?1 + ?1k gk .
o
n
zk = argmin h?
gk , zi + R(z) + 2??k1?k?1 kz ? z0 k2 = prox??k ?k?1 R (z0 ? ??k ?k?1 g?k ) .
z?Rd
xk = 1 ? ?1k xk?1 + ?1k zk .
end for
return (xm , zm ).
gk =
1
b
1
i?Ik nqi
P
We provide details of the doubly accelerated SVRDA (DASVRDA) method for non-strongly convex
objectives in Algorithm 6. Our momentum step is slightly different from that of vanilla deterministic
accelerated methods: we not only add momentum term ((?es?1 ? 1)/?es )(e
xs?1 ? x
es?2 ) to the
e
e
current solution x
es?1 but also add term (?s?1 /?s )(e
zs?1 ? x
es?1 ), where zes?1 is the current more
?aggressively? updated solution rather than x
es?1 ; thus, this term also can be interpreted as momentum5 .
Then, we feed yes to One Stage Accelerated SVRDA (Algorithm 7) as an initial point. Note that
Algorithm 6 can be seen as a direct generalization of APG, because if we set m = 1, One Stage
Accelerated SVRDA is essentially the same as one iteration PG with initial point yes ; then, we can
see that zes = x
es , and Algorithm 6 essentially matches with deterministic APG. Next, we move
to One Stage Accelerated SVRDA (Algorithm 7). Algorithm 7 is essentially a combination of the
?accelerated regularized dual averaging? (AccSDA) method [21] with the variance reduction technique
of SVRG. It updates zk by using the weighted average of all past variance reduced gradients g?k
instead of only using a single variance reduced gradient gk . Note that for constructing variance
reduced gradient gk , we use the full gradient of F at x
es?1 rather than the initial point yes . The
5
This form also arises in Monotone APG [7]. In Algorithm 7, x
e = xm can be rewritten as (2/(m(m +
P
1)) m
e is updated more ?aggressively? than
k=1 kzk , which is a weighted average of zk ; thus, we can say that z
x
e. For the outerly accelerated SVRGP
(that is a combination of Algorithm 6 with vanilla SVRG, see section 3), ze
and x
e correspond to xm and (1/m) m
e is updated more
k=1 xk in [22], respectively. Thus, we can also see that z
?aggressively? than x
e.
5
Adoption of (Innerly) Accelerated SVRDA rather than (Innerly) Accelerated SVRG enables us to
construct lazy updates for sparse data (for more details, see Section E of supplementary material).
4.2
DASVRDA for strongly convex objectives
Algorithm 8: DASVRDAsc (?
x0 , ?, {Li }ni=1 , m, b, S, T )
for t = 1 to T do
x
?t = DASVRDAns (?
xt?1 , x
?t?1 , ?, {Li }ni=1 , m, b, S).
end for
return x
?T .
Algorithm 8 is our proposed method for strongly convex objectives. Instead of directly accelerating
the algorithm using a constant momentum rate, we restart Algorithm 6. Restarting scheme has several
advantages both theoretically and practically. First, the restarting scheme only requires the optimal
strong convexity of the objective instead of the ordinary strong convexity. Whereas, non-restarting
accelerated algorithms essentially require the ordinary strong convexity of the objective. Second,
for restarting algorithms, we can utilize adaptive restart schemes [14]. The adaptive restart schemes
have been originally proposed for deterministic cases. The schemes are heuristic but quite effective
empirically. The most fascinating property of these schemes is that we need not prespecify the
strong convexity parameter ?, and the algorithms adaptively determine the restart timings. [14]
have proposed two heuristic adaptive restart schemes: the function scheme and gradient scheme.
We can easily apply these ideas to our method, because our method is a direct generalization of
the deterministic APG. For the function scheme, we restart Algorithm 6 if P (e
xs ) > P (e
xs?1 ). For
the gradient scheme, we restart the algorithm if (e
ys ? x
es )> (e
ys+1 ? x
es ) > 0. Here yes ? x
es can be
interpreted as a ?one stage? gradient mapping of P at yes . As yes+1 ? x
es is the momentum, this scheme
can be interpreted such that we restart whenever the momentum and negative one Stage gradient
mapping form an obtuse angle (this means that the momentum direction seems to be ?bad?). We
numerically demonstrate the effectiveness of these schemes in Section 6.
Parameter tunings
For DASVRDAns , only learning rate ? needs to be tuned, because we can theoretically obtain the
optimal choice of ?, and we can naturally use m = n/b as a default epoch length (see Section 5). For
DASVRDAsc , both learning rate ? and fixed restart interval S need to be tuned.
5
Convergence Analysis of DASVRDA Method
In this section, we provide the convergence analysis of our algorithms. Unless otherwise specified,
serial computation is assumed. First, we consider the DASVRDAns algorithm.
e0 , ze0 ? Rd , ? ? 3, m ? N, b ? [n]
Theorem 5.1. Suppose that Assumptions 1, 2 and 3 hold. Let x
and S ? N. Then DASVRDAns (e
x0 , ze0 , ?, {Li }ni=1 , m, b, S) satisfies
4
E [P (e
xS ) ? P (x? )] ?
2
1 ? ?1 (S + 2)2
1
1?
?
2
2
(P (e
x0 ) ? P (x? )) +
ke
z0 ? x? k2
?(m + 1)m
!
.
The proof of Theorem 5.1 is found
p in the supplementary material (Section A). We can easily see that
the optimal choice of ? is (3+ 9 + 8b/(m + 1))/2 = O(1+b/m) (see Section B of supplementary
material). We denote this value as ?? . From Theorem 5.1, we obtain the following corollary:
Corollary 5.2. Suppose that Assumptions
e0 ? Rd , ??= ?? p
, m ? n/b and b ?
p 1, 2, and 3 hold. Let x
? x0 ? x? k2 /?),
[n]. If we appropriately choose S = O( (P (e
x0 ) ? P (x? ))/?+(1/m+1/ mb) Lke
ns
n
then a total computational cost of DASVRDA (e
x0 , ?? , {Li }i=1 , m, b, S) for E [P (e
xS ) ? P (x? )] ?
? is
!!
r
r
? x0 ? x? k2
? Lke
P (e
x0 ) ? P (x? )
O d n
+ b+ n
.
?
?
6
Remark. If we adopt a warm start scheme for DASVRDAns , we can further improve the rate to
!!
r
?
Lke
x0 ? x? k 2
P (e
x0 ) ? P (x? )
+ (b + n)
O d nlog
?
?
(see Section C and D of supplementary material).
Next, we analyze the DASVRDAsc algorithm for optimally strongly convex objectives. Combining
Theorem 5.1 with the optimal strong convexity of the objective function immediately yields the
following theorem, which implies that the DASVRDAsc algorithm achieves a linear convergence.
?0 ? Rd , ? = ?? , m ? N, b ? [n]
Theorem 5.3. Suppose that Assumptions 1, 2, 3 and 4 hold. Let x
def
and T ? N. Define ? = 4{(1 ? 1/?? )2 + 4/(?(m + 1)m?)}/{(1 ? 1/?? )2 (S + 2)2 }. If S is
sufficiently large such that ? ? (0, 1), then DASVRDAsc (?
x0 , ?? , {Li }ni=1 , m, b, S, T ) satisfies
E[P (?
xT ) ? P (x? )] ? ?T [P (?
x0 ) ? P (x? )].
From Theorem 5.3, we have the following corollary.
Corollary 5.4. Suppose that Assumptions 1, 2,p
3 and 4 hold. Let x
?0 ? Rd , ? = ?? , m ? n/b,
?
?
b ? [n]. There exists S = O(1 + (b/n + 1/ n) L/?), such that 1/log(1/?) = O(1). Moreover,
if we appropriately choose T = O(log(P (?
x0 ) ? P (x? )/?), then a total computational cost of
DASVRDAsc (?
x0 , ?? , {Li }ni=1 , m, b, S, T ) for E [P (?
xT ) ? P (x? )] ? ? is
? ?
?
O ?d ?n + b + n
?
s ?
?
P
(?
x
)
?
P
(x
)
L
0
?
? log
?.
?
?
?
sc
(?
x0 , ?? ,
Remark. Corollary 5.4 implies that if the mini-batch size b is O( n), DASVRDA
p
n
?
{Li }i=1 , n/b, b, S, T ) still achievesp
the total computational cost of O(d(n + nL/?)log(1/?)),
?
which is much better than O(d(n + nbL/?)log(1/?))
of APCG, SPDC, and Katyusha.
?
sc
Remark. Corollary 5.4 also implies that DASVRDA
only needs size O( n) mini-batches for
p
achieving the optimal iteration complexity O( L/?log(1/?)), when L/? ? n. Inp
contrast, APCG,
SPDC and Katyusha need size O(n) mini-batches and AccProxSVRG does O( L/?) ones for
achievingpthe optimal iteration complexity. Note that even when L/? ? n, our method only needs
size O(n ?/L) mini-batches 6 . This size is smaller than O(n) of APCG, SPDC, and Katyusha, and
the same as that of AccProxSVRG.
6
Numerical Experiments
In this section, we provide numerical experiments to demonstrate the performance of DASVRDA.
We numerically compare our method with several well-known stochastic gradient methods in minibatch settings: SVRG [22] (and SVRG++ [2]), AccProxSVRG [12], Universal Catalyst [8] , APCG
[9], and Katyusha [1]. The details of the implemented algorithms and their parameter tunings
are found in the supplementary material. In the experiments, we focus on the regularized logistic
regression problem for binary classification, with regularizer ?1 k ? k1 + (?2 /2)k ? k22 .
We used three publicly available data sets in the experiments. Their sizes n and dimensions d, and
common min-batch sizes b for all implemented algorithms are listed in Table 2.
Table 2: Summary of the data sets and mini-batch size used in our numerical experiments
Data sets
a9a
rcv1
sido0
n
32, 561
20, 242
12, 678
d
123
47, 236
4, 932
b
180
140
100
For regularization parameters, we used three settings (?1 , ?2 ) = (10?4 , 0), (10?4 , 10?6 ), and
(0, 10?6 ). For the former case, the objective is non-strongly convex, and for the latter two cases,
6
p
p
Note that the required size is O(n ?/L)(? O(n)), which is not O(n L/?) ? O(n).
7
(a) a9a, (?1 , ?2 ) = (10?4 , 0)
(b) a9a, (?1 , ?2 ) = (10?4 , 10?6 )
(c) a9a, (?1 , ?2 ) = (0, 10?6 )
(d) rcv1, (?1 , ?2 ) = (10?4 , 0)
(e) rcv1, (?1 , ?2 ) = (10?4 , 10?6 )
(f) rcv1, (?1 , ?2 ) = (0, 10?6 )
(g) sido0, (?1 , ?2 ) = (10?4 , 0)
(h) sido0, (?1 , ?2 ) = (10?4 , 10?6 )
(i) sido0, (?1 , ?2 ) = (0, 10?6 )
Figure 1: Comparisons on a9a (top), rcv1 (middle) and sido0 (bottom) data sets, for regularization
parameters (?1 , ?2 ) = (10?4 , 0) (left), (?1 , ?2 ) = (10?4 , 10?6 ) (middle) and (?1 , ?2 ) = (0, 10?6 )
(right).
the objectives are strongly convex. Note that for the latter two cases, the strong convexity of the
objectives is ? = 10?6 and is relatively small; thus, it makes acceleration methods beneficial.
Figure 1 shows the comparisons of our method with the different methods described above on several
settings. ?Objective Gap? means P (x) ? P (x? ) for the output solution x. ?Elapsed Time [sec]?
means the elapsed CPU time (sec). ?Restart_DASVRDA? means DASVRDA with heuristic adaptive
restarting (Section 4). We can observe the following from these results:
? Our proposed DASVRDA and Restart DASVRDA significantly outperformed all the other
methods overall.
? DASVRDA with the heuristic adaptive restart scheme efficiently made use of the local
strong convexities of non-strongly convex objectives and significantly outperformed vanilla
DASVRDA. For the other settings, the algorithm was still comparable to vanilla DASVRDA.
? UC+AccProxSVRG7 outperformed vanilla AccProxSVRG but was outperformed by our
methods overall.
7
Although there has been no theoretical guarantee for UC + AccProxSVRG, we thought that it was fair
to include experimental results about that because UC + AccProxSVRG gives better performances than the
vanilla AccProxSVRG. Through some theoretical analysis, we can prove that UC + AccProxSVRG also has the
similar rate and mini-batch efficiency to our proposed method, although these results are not obtained in any
literature. However, our proposed method is superior to this algorithm both theoretically and practically, because
the algorithm has several drawbacks due to the use of UC as follows. First, the algorithm has an additional
logarithmic factor in its convergence rate. This factor is generally not negligible and slows down its practical
performances. Second, the algorithm has more tuning parameters than our method. Third, the stopping criterion
of each sub-problem of UC is hard to be tuned.
8
? APCG sometimes performed unstably and was outperformed by vanilla SVRG. On sido0
data set, for Ridge Setting, APCG significantly outperformed all the other methods.
? Katyusha always outperformed vanilla SVRG, but was significantly outperformed by our
methods.
7
Conclusion
In this paper, we developed a new accelerated stochastic variance reduced gradient method for
regularized empirical risk minimization problems in mini-batch settings: DASVRDA. Wephave shown
?
that DASVRDA achieves
the total computational costs of O(d(nlog(1/?) + (b + n) L/?)) and
? p
O(d(n + (b + n) L/?)log(1/?)) in size b mini-batch settings for non-strongly and optimally
strongly convex objectives, respectively. In
? addition, DASVRDA essentially achieves the optimal
iteration complexities only with size O( n) mini-batches for both settings. In the numerical
experiments, our method significantly outperformed state-of-the-art methods, including Katyusha
and AccProxSVRG.
Acknowledgment
This work was partially supported by MEXT kakenhi (25730013, 25120012, 26280009 and
15H05707), JST-PRESTO and JST-CREST.
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6,261 | 6,664 | Langevin Dynamics with Continuous Tempering for
Training Deep Neural Networks
Nanyang Ye
University of Cambridge
Cambridge, United Kingdom
[email protected]
Zhanxing Zhu
Center for Data Science, Peking University
Beijing Institute of Big Data Research (BIBDR)
Beijing, China
[email protected]
Rafal K.Mantiuk
University of Cambridge
Cambridge, United Kingdom
[email protected]
Abstract
Minimizing non-convex and high-dimensional objective functions is challenging,
especially when training modern deep neural networks. In this paper, a novel approach is proposed which divides the training process into two consecutive phases
to obtain better generalization performance: Bayesian sampling and stochastic
optimization. The first phase is to explore the energy landscape and to capture
the ?fat? modes; and the second one is to fine-tune the parameter learned from the
first phase. In the Bayesian learning phase, we apply continuous tempering and
stochastic approximation into the Langevin dynamics to create an efficient and
effective sampler, in which the temperature is adjusted automatically according to
the designed ?temperature dynamics?. These strategies can overcome the challenge
of early trapping into bad local minima and have achieved remarkable improvements in various types of neural networks as shown in our theoretical analysis and
empirical experiments.
1
Introduction
Minimizing non-convex error functions over continuous and high-dimensional spaces has been a
primary challenge. Specifically, training modern deep neural networks presents severe difficulties,
mainly because of the large number of critical points with respect to the number of dimensions,
including various saddle points and local minima [9, 5]. In addition, the landscapes of the error
functions are theoretically and computationally impossible to characterize rigidly.
Recently, some researchers have attempted to investigate the landscapes of the objective functions
for several types of neural networks. Under some strong assumptions, previous works [21, 4, 12]
showed that there exists multiple, almost equivalent local minima for deep neural networks, using a
wide variety of theoretical analysis and empirical observations. Despite of the nearly equivalent local
minima during training, obtaining good generalization performance is often more challenging with
current stochastic gradient descent (SGD) or some of its variants. It was demonstrated in [22] that
deep network structures are sensitive to initialization and learning rates. And even networks without
nonlinear activation functions may have degenerate or hard to escape saddle points [12].
One important reason of the difficulty to achieve good generalization is, that SGD and some variants
may tend to trap into a certain local minima or flat regions with poor generalization property [25, 1, 13].
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In other words, most of existing optimization methods do not explore the landscapes of the error
functions efficiently and effectively. To increase the possibility of sufficient exploration of the
parameter space, [25] proposed to train multiple deep networks in parallel and made individual
networks explore by modulating their distance to the ensemble average.
Another kind of approaches attempt to tackle this issue through borrowing the idea of classical
simulated annealing or tempering [15, 6, 10]. The authors of [19] proposed to inject Gaussian
noise with annealed variance (corresponding to the annealed temperature in simulated annealing)
into the standard SGD to make the original optimization dynamics more ?stochastic?. In essence,
this approach is the same as a scalable Bayesian learning method, Stochastic Gradient Langevin
Dynamics (SGLD [24]) with decayed stepsizes. The Santa algorithm [1] incorporated a similar idea
into a more sophisticated stochastic gradient Markov Chain Monte Carlo (SG-MCMC) framework.
However, previous studies show that the efficiency and performance of these methods for training deep
neural networks is very sensitive to the annealing schedule of the temperature in these methods. Slow
annealing will lead to significantly slow optimization process as observed in the literature of simulated
annealing [10], while fast annealing hinders the exploration dramatically, leading to the optimizer
trapped in poor local minima too early. Unfortunately, searching for a suitable annealing schedule
for training deep neural network is hard and time-consuming according to empirical observations in
these works.
To facilitate more efficient and effective exploration for training deep networks, we divide the whole
training process into two phases: Bayesian sampling for exploration and optimization for fine-tuning.
The motivation of implementing a sampling phase is that sampling is theoretically capable of fully
exploring the parameter space and can provide a good initialization for optimization phase. This
strategy is motivated by the sharp minima theory [13] and its validity will be verified by our empirical
experiments.
Crucially, in the sampling phase, we employ the idea of continuous tempering [8, 17] in molecule
dynamics [20], and implement an extended stochastic gradient second-order Langevin dynamics with
smoothly varying temperatures. Importantly, the change of temperature is governed automatically
by a specifically designed dynamics coupled with the original Langevin dynamics. This is different
from the idea of simulated annealing adopted in [19, 1], in which the temperature is only allowed to
decrease according to a manually predefined schedule. Our ?temperature dynamics? is beneficial in
the sense that it increases the capability of exploring the energy landscapes and hopping between
different modes of the sampling distributions. Thus, it may avoid the problem of early trapping into
bad local minima that exists in other algorithms. We name our approach CTLD, abbreviated for
?Continuously Tempered Langevin Dynamics?. With support of extensive empirical evidence, we
demonstrated the efficiency and effectiveness of our proposed algorithm for training various types of
deep neural networks. To the best of our knowledge, this is the first attempt that adopts continuous
tempering into training modern deep networks and produces remarkable improvements over the
state-of-the-art techniques.
2
Preliminaries
The goal of training deep neural network is to minimize the objective function U (?) corresponding
to a non-convex model of interest, where ? 2 Rd are the model parameters. In a Bayesian setting,
the objective U (?) can be treated as the potential energy function, i.e., the negative log posterior,
PN
U (?) =
log 0 (?), where xi represents the i-th observed data point, 0 (?) is
i=1 log (xi |?)
the prior distribution for the model parameters and (xi |?) is the likelihood term for each observation.
In optimization scenario, the counterpart of the complete negative log likelihood is the loss function
and log 0 (?) is typically referred to as a regularization term.
2.1 Stochastic Gradient MCMC
In the scenario of Bayesian learning, obtaining the samples of a high-dimensional distribution is a
necessary procedure for many tasks. Classic dynamics offers such a way to sample the distribution.
The Hamiltonian in classic dynamics is H(?, r) = U (?)+ 12 rT r, the sum of the potential energy U (?)
and kinetic energy 12 rT r, where r 2 Rd is the momentum term Standard (second-order) Langevin dy2
namics1 with constant temperature Tc can be described by following stochastic differential equations
(SDEs),
p
1 dW
d? = rdt, dr = r? U (?)dt
rdt + 2
(1)
where r? U (?) is the gradient of the potential energy w.r.t. the configuration states ?, denotes
the friction coefficient, 1 = kB Tc with Boltzmann constant kB , and dW is the standard Wiener
process. In the context of this work for Markov Chain Monte Carlo (MCMC) and optimization theory,
we always assume = 1 for simplicity.
If we simulate the dynamics in Eqs (1), a well-known
stationary distribution can be achieved [20],
RR
(?, r) = exp ( H(?, r)) /Z, where Z =
exp ( H(?, r)) d?dr is the normalization constant for the probability density. The desired probability distribution Rassociated with the parameters
? can be obtained by marginalizing the joint distribution, (?) =
(?, r)dr / exp ( U (?)).
The MCMC procedures using the analogy of dynamics described by SDEs are often referred to as
dynamics-based MCMC methods.
However, in the ?Big Data? settings with large N , evaluating the full gradient term r? U (?) is
computationally expensive. The usage of stochastic approximation reduces the computational burden
dramatically, where a much smaller subset of the data, {xk1 , . . . , xkm }, is selected randomly to
approximate the full one,
? (?) =
U
m
NX
log (xkj |?)
m j=1
log
0 (?).
(2)
? (?) is an unbiased estimation of the true gradient. Then the
And the resulting stochastic gradient rU
stochastic gradient approximation can be used in the dynamics-based MCMC methods, often referred
to as SG-MCMC, such as [24, 3].
2.2
Simulated Annealing for Global Optimization
Simulated annealing (SA [15, 6, 10]) is a probabilistic technique for approximating the global
optimum of a given function U (?). A Brownian-type of diffusion algorithm was proposed [6] for
continuous optimization by discretizing the following SDE,
p
d? = rU (?)dt + 2 1 (t)dW,
(3)
1
where
(t) = kB T (t) decays as T (t) = c/ log(2 + t) with a sufficiently large constant c, to
ensure theoretical convergence. Unfortunately, this logarithmic annealing schedule is extremely
slow for optimization. In practice, the polynomial schedules are often adopted to accelerate the
optimization processes though without any theoretical guarantee, such as T (t) = c/(a + t)b , where
a > 0, b 2 (0.5, 1), c > 0 are hyperparameters. Recently, [19, 1] incorporated the simulated
annealing with this polynomial cooling schedule into the training of neural networks. A critical issue
behind these methods is that the generalization performance and efficiency of the optimization are
highly sensitive to the cooling schedule. Unfortunately, searching for a suitable annealing schedule
for training deep neural network is hard and time-consuming according to empirical observations in
these works.
These challenges motivate our work. We proposed to divide the whole optimization process into two
phases: Bayesian sampling based on stochastic gradient for parameter space exploration and standard
SGD with momentum for parameters optimization. The key step in the first phase is that we employ
a new tempering scheme to facilitate more effective exploration over the whole energy landscape.
Now, we will elaborate on the proposed approach.
3
Two Phases for Training Neural Networks
As mentioned in Section 1, the objective functions of deep networks contain multiple, nearly equivalent local minima. The key difference between these local minima is whether they are ?flat? or
?sharp?, i.e., lying in ?wide valleys? or ?stiff valleys?. A recent study by [13] showed that sharp
1
Standard Langevin dynamics is different from that used in SGLD [24], which is the first-order Langevin
dynamics, i.e., Brownian dynamics.
3
minima often lead to poorer generalization performance. Flat minimizers of the energy landscape
tend to generalize better due to their robustness to data perturbations, noise in the activations as well
as perturbations of the parameters. However, most of existing optimization methods lack the ability
to efficiently explore the flat minima, often trapping into sharp minima too early.
We consider this issue in a Bayesian way: the flat minima corresponds to ?fat? modes of the induced
probability distribution over ?, (?) / exp ( U (?)). Obviously, these fat modes own much more
probability mass than ?thin? ones since they are nearly as ?tall? as each other. Based on this simple
observation, we propose to implement a Bayesian sampling procedure before the optimization phase.
Bayesian learning is capable of exploring the energy landscape more thoroughly. Due to the large
probability mass, the sampler tends to capture the desired regions near the ?flat? minima. This
provides a good starting region for optimization phase to fine-tune the parameters learning.
When sampling the distribution (?), the multi-modality issue demands the samplers to transit
between isolated modes efficiently. To this end, we incorporate the continuous tempering and
stochastic approximation techniques into the Langevin dynamics to derive an efficient and effective
sampling process for training deep neural networks.
4
CTLD: Continuously Tempered Langevin Dynamics
Faced with high-dimensional and non-convex energy landscapes U (?), such as the error functions
in deep neural networks, the key challenge is how to efficiency and effectively explore the energy
landscapes. Inspired by the idea of continuous tempering [8, 17] in molecule dynamics, we incorporate
the ?temperature dynamics? and stochastic approximation into the Langevin dynamics in a principled
way to allow a more effective exploration of the energy landscape. The temperature in CTLD
evolves automatically governed by the embedded ?temperature dynamics?, which is different from
the predefined annealing schedules used in [19, 1].
The primary dynamics we use for Bayesian sampling is as follows,
q
d? = rdt, dr = r? U (?)dt
rdt + 2 ? 1 (?)dW
p
d? = r? dt, dr? = h(?, r, ?)dt
2 ? dW? ,
? r? dt +
(4)
where ? is the newly augmented variable to control the inverse temperature ?, ? is the corresponding
friction coefficient. Note that ? 1 (?) = kB T (?) = 1/g(?), depending on the augmented variable
? to dynamically adjust the temperature. The function g(?) plays the role as scaling the constant
temperature Tc . The dynamics of ? and ? are coupled through the function h(?, r, ?). Both of the
two functions will be described later.
It can be shown that if we simulate the SDEs described in Eq (4), the following stationary distribution
will be achieved [8],
(?, r, ?, r? ) / exp ( He (?, r, ?, r? )) ,
(5)
with the extended Hamiltonian and the coupling function h(?) as
He (?, r, ?, r? ) = g(?)H(?, r) + (?) + r?2 /2,
h(?, r, ?) =
@? g(?)H(?, r)
@? (?), (6)
where (?) is some confining potential to enforce additional properties of ?, discussed in Section 4.2. The proof of achievement of this stationary distribution (?, r, ?, r? ) is provided in the
Supplementary Material for completeness.
In order to allow the system to overcome the issue of muli-modality efficiently, the temperature
scaling function g(?) can be any convenient form that satisfies: 0 < g(?) ? 1 and being smooth.
This will allow the system to experience different temperature configurations smoothly. A simple
choice would be the following piecewise polynomial function, with z(?) = |?|0 ,
8
<1, if |?| ? ,
g(?) = 1 S 3z 2 (?) 2z 3 (?) , if < |?| < 0
(7)
:
0
1 S, if |?|
,
Figure 1 presents this temperature scaling function with = 0.4, 0 = 1.5 and S = 0.85. In this
case, ? 1 (?) 2 [1 S, 1]. Experiencing high temperature configurations continuously allows the
4
1
g( )
0.8
0.6
0.4
0.2
0
-2
-1
0
1
2
Figure 1: Temperature scaling function g(?).
sampler to explore the parameter space more ?wildly?, significantly alleviating the issue of trapping
into local minima or flat regions. Moreover, it can be easily seen that when g(?) = 1, we can recover
the desired distribution (?) / exp( U (?)).
4.1 Stochastic Approximation for CTLD
With large-scale datasets, we adopt the technique of stochastic approximation to estimate the full
potential term U (?) and its gradient rU (?), as shown in Eq. (2). One way to analyse the impact of
the stochastic approximation is to make use of the central limit theorem,
? (?) = U (?) + N 0, 2 (?) , r? U
? (?) = r? U (?) + N (0, ?(?))
U
(8)
The usage of stochastic approximation results in a noisy potential term and gradient. Simply plugging
in the the noisy estimation into the original dynamics will lead to a dynamics with additional
noise terms. To dissipate the introduced noise, we assume the covariance matrices, 2 (?) and
?(?), are available, and satisfy the positive semi-definiteness, 2 ? 1 (?)I ??(?) < 0 and
2 ? ?@? g(?) 2 (?) 0 with ? as the associated step size of numerical integration for the SDEs.
With ? small enough, this is always true since the introduced stochastic noise scales down faster than
the added noise. Then, we propose CTLD with stochastic approximation,
q
? (?)dt
d? = rdt, dr = r? U
rdt + 2 ? 1 (?)I ??(?)dW
(9)
p
2 (?)dW ,
? r, ?)dt
d? = r? dt, dr? = h(?,
r
dt
+
2
?@
g(?)
? ?
?
?
?
?
?
T
?
?
where the coupling function h(?, r, ?) = @? g(?) U (?) + r r/2
@? (?). Then the following
theorem to show the stationary distribution of the dynamics described in Eq. (9).
Theorem 1. (?, r, ?, r? ) / exp ( He (?, r, ?, r? )) is the stationary distribution of the dynamics
SDEs Eq. (9), when the variance terms 2 (?) and ?(?) are available.
The proof for this theorem is provided in the Supplementary Materials. In practical implementation
of simulating the r and r? of Eq. (9), we have
?
?
2? (t)
(t)
(t)
(t 1)
(t) (t)
(t) 2 ?
(t 1)
?
r = (1 ?
)r
r? U (? )? + N 0,
I (? ) ?(?
)
g(?(t 1) )
(10)
? (t) , r(t) , ?(t) )? (t) + N (0, 2? (t) ? (? (t) )2 ? 2 (?)),
r?(t) = (1 ? (t) ? )r?(t 1) + h(?
?
where ?(?)
and ? 2 (?) are the estimation of the noise variance terms. In Eq. (10), the noise introduced
by the stochastic approximation is compensated by multiplying (? (t) )2 . To avoid the estimation of the
?
variance terms, we often choose ? (t) = ? small enough and , ? large enough to make the ? 2 ?(?)
2 2
and ? ? (?) numerically negligible, and thus ignored in practical use.
4.2 Control of The Augmented Variable
It is expected that the distribution of experienced temperatures of the system should only depend on
the form of the scaling function g(?). This would help us achieve the desired temperature distribution,
thus resulting in a more controllable system. To this end, two strategies are shown in this part.
Firstly, we confine the augmented variable ? to be in the interval [
achieve this is to configure its gradient as a ?force well?:
?
0, if |?| ? 0
@? (?) =
C, otherwise,
5
0
, 0 ]. One simple choice to
(11)
where C is some appropriate constant. Intuitively, when the particle ? ?escapes? from the interval
[ 0 , 0 ], a force induced by @? (?) will ?pull? it back.
Secondly, we restrict the distribution of ? to be uniform over the specified range. Together with the
design of g(?), this restriction can guarantee the required percent of running time for sampling with
the original inverse temperature = 1, and the remaining for high temperatures. For example, in
case of g(?) in Eq.(7), the percent of simulation time for high temperatures is (1
/ 0 )100%.
An adaptive biasing method metadynamics [16] can be used to achieve a flat density across a bounded
range of ?. Metadynamics incorporates a history-dependent potential term to gradually fill the
minima of energy surface corresponding to ??s marginal density, resulting in a uniform distribution
of ?. In essence, metadynamics biases the extended Hamiltonian by an additional potential Vb (?),
Hm (?, r, ?, r? ) = g(?)H(?, r) + (?) + r?2 /2 + Vb (?)
(12)
(0)
Vb (?)
The bias potential term is initialized
= 0, and then updated by iteratively adding Gaussian
kernel terms,
?
?
(t+1)
(t)
Vb
(?) = Vb (?) + w exp
(? ?(t) )2 /(2 2 ) ,
(13)
where ?(t) is the value of the t-th time step during simulation, the magnitude w and variance term
2
are hyperparameters. To update the bias potential over the range [ 0 , ], we can discretize this
(t)
(t)
interval into K equal bins, { 0 , ?1 , . . . , ?K 1 , 0 } and in each time step update ? in each bin.
Thus, the force induced by the bias potential can be approximated by the difference between adjacent
bins divided by the length of each bin. The force h(? (t) , r(t) , ?t ) over the particle ? will be biased
due to the force induced by metadynamics,
? (t) , r(t) , ?(t) )
h(?
? (t) , r(t) , ?(t) )
h(?
(t)
Vb
(t)
(?k? +1 ) Vb
2 0 /K
(?k? )
(14)
where k ? denotes the bin index inside which ?(t) is located. Finally, we summarize CTLD in Alg. 1.
Algorithm 1: Continuously Tempered Langevin Dynamics
Input: m, ?, number of steps for sampling Ls , , ? ; metadynamics parameters: C, w,
(0)
(0)
Initialize ? (0) , r(0) ? N (0, I), ?(0) = 0, r? ? N (0, 1), and Vb (?(0) ) = 0.
for t = 1, 2, . . . do
? (? (t) );
Randomly sample a minibatch of the dataset with size m to obtain U
if t < Ls then
Sample ? ? N (0, I), ?? ? N (0, 1);
q
2?
? (? (t) )? +
? (t) = ? (t 1) + ?r(t 1) , r(t) = (1 ? )r(t 1) r? U
?
g(?(t 1) )
2
and K.
(t 1)
?(t) = ?(t 1) + ?r?
.
Update Vb (?) according to Eq. (13); Find the k ? indexing which bin ?(t) is located in.
(t)
(t)
? (t) , r(t) , ?(t) ) = @? g(?(t) )H(?
? (t) , r(t) ) @? (?(t) ) Vb (?k? +10) Vb (?k? )
h(?
2 /K
(t)
(t 1)
? (t) , r(t) , ?(t) )? + p2? ? ??
r? = (1 ? ? )r?
+ h(?
else
? (? (t) )?
? (t) = ? (t 1) + ?r(t 1) , r(t) = (1 ? )r(t 1) r? U
end if
end for
4.3
Connection with Other Methods
There is a direct relationship between the proposed method and SGD with momentum. In the
optimization phase, CTLD essentially implements SGD with momentum: as shown in SGHMC
[3], the learning rate in the SGD with momentum corresponds to ? 2 in our method, the momentum
coefficient the SGD is equivalent q
to 1 ? . The key difference appears in the Bayesian learning
phase, a dynamical diffusion term g(?2?
(t 1) ) ? is added to the update of the momentum to empower
the sampler/optimizer to explore the parameter space more thoroughly. This directly avoids the issue
of being stuck into poor local minima too early. CTLD introduces stochastic approximation and
temperature dynamics into the Langevin dynamics in a principled way. This distinguishes it from the
deterministic annealing schedules adopted in Santa [1] and SGLD/AnnealSGD [24, 19].
6
4.4
Parameter Settings, Computational Time and Convergence Analysis
Though there exists several hyperparameters in our method, in practice, the only parameters we
need to tune are the learning rate and the momentum (i.e. related to friction coefficients). Across
all the experiments, for other hyperparamters, including those in confining potential function (?)
and metadynamics, we fix them with our empirical formulae relying on the learning rate. See
Supplementary Materials for a thorough analysis on hyperparameter settings. Moreover, through our
sensitivity analysis for these hyperparameters, we find they are quite robust to algorithm performance
within our estimation range, as shown in Section 5.3. Therefore, practical users can just tune the
learning rate and momentum to use CTLD for training neural networks, which is as simple as SGD
with momentum.
Compared with SGD with momentum, our proposal CTLD only introduces an additional 1D augmented variable ?, and its computational cost in almost negligible, as shown in Supplementary Materials.
The convergence analysis of CTLD is also provided in the Supplementary Materials to demonstrate
its stability.
5
Experiments
To evaluate the proposed method, we conduct experiments on stacked denoising autoencoders and
character-level recurrent neural networks. The comparing methods include SGD with momentum,
RMSprop, Adam [14], AnnealSGD [19], Santa [1] and our proposal CTLD. The same parameter
initialization method ?Xavier? [7] is used except for character recurrent neural networks. The
hyperparameter settings for each compared method are implemented by grid search, provided in the
Supplementary Materials.
5.1
Stacked Denoising Autoencoders
Stacked denoising autoencoders (SdA) [23] have been proven to be useful in pre-training neural
networks for improved performance. We focus on the greedy layer-wise training procedure of SdAs.
Dropout layers are appended to each layer with a rate of 0.2 except for the first and last layer. We use
the training set of MNIST data consisting of 60000 training images for this task. The network is fully
connected, 784-500-500-2000-10. The learning curves of mean square errors (MSE) for each method
are shown in Figure. 2(a). The bumps in iteration 1, 2, 3 ? 105 is due to the switching to next layer
during training. Though CTLD in the sampling phase is not as fast as other methods, it can find the
regions of good minima, and fine-tune to the best results in final stage.
7
SGD-M
RMSProp
Adam
AnnealSGD
Santa
CLTD
10 -1
6
5
1/g( )
MSE
10 0
4
3
10
2
-2
1
0
1
2
Iteration
3
4
3
3.1
3.2
Iteration
10 5
3.3
10 5
Figure 2: (Left) Learning curves of SdAs; (Right) The evolution of the noise magnitude ? = 1/g(?) during
the training the final layer.
We also track the evolution of the augmented variable ? and plot the noise magnitude ?(?) = 1/g(?)
during the training the final layer, shown in the right panel of Fig. 2. We can observe that the behavior
of the magnitude of the noise term is dramatically different from the predefined decreasing schedule
used in Santa and AnnealSGD. The temperature dynamics introduced in CTLD adjusts the noise term
adaptively. This helps the system to explore the landscape of the loss function more thoroughly and
find the regions of good local minima with a higher probability.
5.2
Character Recurrent Neural Networks for Language Modeling
We test our method on the task of character prediction
is to
?Pusing LSTM networks. The objective
?
PN
Ti
1
i i
i
log p(xt |x1 , ..., xt 1 ; ?) , where ?
minimize the per-character perplexity, N i=1 exp
t=1
7
is a set of parameters for the model, xnt is the observed data and Ti is the length of i-th sentence. The
hidden units are set as LSTM units. We run the models with different methods on the Wikipedia
Hutter Prize 100MB dataset with a setting of 3-layer LSTM, 64 hidden layers, the same with the
original paper [11]. The training and test perplexity are shown in Fig. 3.
The best training and test perplexities are reached by our method CTLD, which also has the fastest
convergence speed. RMSProp and Adam converge very fast in the early iterations, but they seem to
be trapped in some poor local minima.
RMSProp
Adam
AnnealSGD
Santa
CTLD
Training Perplexity
2.6
2.5
RMSProp
Adam
AnnealSGD
Santa
CTLD
2.7
2.6
Test Perplexity
2.7
2.4
2.3
2.2
2.1
2.5
2.4
2.3
2.2
2.1
2
2
1.9
1.9
0
5
10
15
20
Epoch
0
5
10
15
20
Epoch
Figure 3: (Left) CharRNN on Wiki training set; (Right) CharRNN on Wiki test set. Note that SGD-M doest
not appear in the Figure because the training and test perplexities for SGD-M are much higher.
5.3
Sensitivity Analysis
Since there exist several hyperparameters in CTLD, we analyze the sensitivity of hyperparameter
settings within our estimation range (provided in Supplementary Materials). We implement the
character-level RNN on War and Peace by Leo Tolstoy instead of Wiki dataset, considering the
computational speed. The same model architecture is used as [11]. The learning rate is set as 2?10 4 ,
momentum as 0.7. We train the model for 50 epochs until full convergence. The results are shown in
Fig. 4. According to Fig. 4, the setting of hyperparameter w and is robust within our estimation
Figure 4: (Left) test perplexity versus w and
w = 20,
(S = 0.85); (Right) test perplexity versus of S ( with
= 0.04)
range. This also shows metadynamics performs quite stable for training neural networks. For the
sensitivity of S, with the increase of S, the range of temperature enlarges accordingly. Larger range of
temperature slightly enhances the ability of CTLD to explore the energy landscape, and leads to better
local minima. However, this improvement is quite limited, as shown in Fig. 4, demonstrating the
robustness of the hyperparameter S. Therefore, we can conclude that with the continuous tempering
scheme, our proposed method remains relatively stable under different hyperparameter settings.
Practical users only need tune the learning rate and momentum to use CTLD.
6
Conclusion & Future Directions
We propose CTLD, an effective and efficient approach for training modern deep neural networks.
It involves scalable Bayesian sampling combined with continuous tempering to capture the ?fat?
modes, and thus avoiding the issue of getting trapped into poor local minima too early. Extensive
theoretical and empirical evidence verify the superiority of our proposal over the existing methods.
Future directions includes theoretically analyzing the effects of metadynamics and hyperparameter
settings, and usage of high-order integrators and preconditioners to improve convergence speed.
8
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9
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6,262 | 6,665 | Efficient Online Linear Optimization
with Approximation Algorithms
Dan Garber
Technion - Israel Institute of Technology
[email protected]
Abstract
We revisit the problem of online linear optimization in case the set of feasible actions is accessible through an approximated linear optimization oracle with a factor
? multiplicative approximation guarantee. This setting is in particular interesting
since it captures natural online extensions of well-studied offline linear optimization problems which are NP-hard, yet admit efficient approximation algorithms.
The goal here is to minimize the ?-regret which is the natural extension of the
standard regret in online learning to this setting. We present new algorithms with
significantly improved oracle complexity for both the full information and bandit
variants of the problem. Mainly, for both variants, we present ?-regret bounds of
O(T 1/3 ), were T is the number of prediction rounds, using only O(log(T )) calls
to the approximation oracle per iteration, on average. These are the first results to
obtain both average oracle complexity of O(log(T )) (or even poly-logarithmic in
T ) and ?-regret bound O(T c ) for a constant c > 0, for both variants.
1
Introduction
In this paper we revisit the problem of Online Linear Optimization (OLO) [14], which is a specialized
case of Online Convex Optimization (OCO) [12] with linear loss functions, in case the feasible set of
actions is accessible through an oracle for approximated linear optimization with a multiplicative
approximation error guarantee. In the standard setting of OLO, a decision maker is repeatedly
required to choose an action, a vector in some fixed feasible set in Rd . After choosing his action,
the decision maker incurs loss (or payoff) given by the inner product between his selected vector
and a vector chosen by an adversary. This game between the decision maker and the adversary then
repeats itself. In the full information variant of the problem, after the decision maker receives his
loss (payoff) on a certain round, he gets to observe the vector chosen by the adversary. In the bandit
version of the problem, the decision maker only observes his loss (payoff) and does not get to observe
the adversary?s vector. The standard goal of the decision maker in OLO is to minimize a quantity
known as regret, which measures the difference between the average loss of the decision maker on
a game of T consecutive rounds (where T is fixed and known in advance), and the average loss
of the best feasible action in hindsight (i.e., chosen with knowledge of all actions of the adversary
throughout the T rounds) (in case of payoffs this difference is reversed). The main concern when
designing algorithms for choosing the actions of the decision maker, is guaranteeing that the regret
goes to zero as the length of the game T increases, as fast as possible (i.e., the rate of the regret in
terms of T ). It should be noted that in this paper we focus on the case in which the adversary is
oblivious (a.k.a. non-adaptive), which means the adversary chooses his entire sequence of actions for
the T rounds beforehand.
While there exist well known algorithms for choosing the decision maker?s actions which guarantee
optimal regret bounds in T , such as the celebrated Follow the Perturbed Leader (FPL) and Online
Gradient Descent (OGD) algorithms [14, 17, 12], efficient implementation of these algorithms hinges
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
on the ability to efficiently solve certain convex optimization problems (e.g., linear minimization
for FPL or Euclidean projection for OGD) over the feasible set (or the convex hull of feasible
points). However, when the feasible set corresponds for instance to the set of all possible solutions
to some NP-Hard optimization problem, no such efficient implementations are known (or even
widely believed to exist), and thus these celebrated regret-minimizing procedures cannot be efficiently
applied. Luckily, many NP-Hard linear optimization problems (i.e., the objective function to either
minimize or maximize is linear) admit efficient approximation algorithms with a multiplicative
approximation guarantee. Some examples include MAX-CUT (factor 0.87856 approximation due to
[9]) , M ETRIC TSP (factor 1.5 approximation due to [6]), M INIMUM W EIGHTED V ERTEX C OVER
(factor 2 approximation [4]), and W EIGHTED S ET C OVER (factor (log n + 1) approximation due to
[7]). It is thus natural to ask wether an efficient factor ? approximation algorithm for an NP-Hard
offline linear optimization problem could be used to construct, in a generic way, an efficient algorithm
for the online version of the problem. Note that in this case, even efficiently computing the best fixed
action in hindsight is not possible, and thus, minimizing regret via an efficient algorithm does not
seem likely (given an approximation algorithm we can however compute in hindsight a decision
that corresponds to at most (at least) ? times the average loss (payoff) of the best fixed decision in
hindsight).
In their paper [13], Kakade, Kalai and Ligett were the first to address this question in a fully generic
way. They showed that using only an ?-approximation oracle for the set of feasible actions, it
is possible, at a high level, to construct an online algorithm which achieves vanishing (expected)
?-regret, which is the difference between the average loss of the decision maker and ? times the
average loss of the best fixed point in hindsight (for loss minimization problems and ?
1; a
corresponding definition exists for payoff maximization problems and ? < 1). Concretely, [13]
showed that one can guarantee O(T 1/2 ) expected ?-regret in the full-information setting, which is
optimal, and O(T 1/3 ) in the bandit setting under the additional assumption of the availability of a
Barycentric Spanner (which we discuss in the sequel).
While the algorithm in [13] achieves an optimal ?-regret bound (in terms of T ) for the full information
setting, in terms of computational complexity, the algorithm requires, in worst case, to perform on
each round O(T ) calls to the approximation oracle, which might be prohibitive and render the
algorithm inefficient, since as discussed, in general, T is assumed to grow to infinity and thus the
dependence of the runtime on T is of primary interest. Similarly, their algorithm for the bandit setting
requires O(T 2/3 ) calls to the approximation oracle per iteration.
The main contribution of our work is in providing new low ?-regret algorithms for the full information
and bandit settings with significantly improved oracle complexities. A detailed comparison with [13]
is given in Table 1. Concretely, for the full-information setting, we show it is possible to achieve
O(T 1/3 ) expected ?-regret using only O(log(T )) calls to the approximation oracle per iteration,
on average, which significantly improves over the O(T ) bound of [13]1 . pWe also show a bound
of O(T 1/2 ) on the expected ?-regret (which is optimal) using only O( T log(T )) calls to the
oracle per iteration, on average, which gives nearly quadratic improvement over [13]. In the bandit
setting we show it is possible to obtain a O(T 1/3 ) bound on the expected ?-regret (same as in [13])
using only O(log(T )) calls to the oracle per iteration, on average, under the same assumption on the
availability of a Barycentric Spanner (BS). It is important to note that while there exist algorithms
1/2
?
?
for OLO with bandit feedback which guarantee O(T
) expected regret [1, 11] (where the O(?)
hides poly-logarithmic factors in T ), these require on each iteration to either solve to arbitrarily small
accuracy a convex optimization problem over the feasible set [1], or sample a point from the feasible
set according to a specified distribution [11], both of which cannot be implemented efficiently in our
setting. On the other-hand, as we formally show in the sequel, at a high level, using a BS (originally
introduced in [2]) simply requires to find a single set of d points from the feasible set which span the
entire space Rd (assuming this is possible, otherwise the set could be mapped to a lower dimensional
space). The process of finding these vectors can be viewed as a preprocessing step and thus can be
carried out offline. Moreover, as discussed in [13], for many NP-Hard problems it is possible to
compute a BS in polynomial time and thus even this preprocessing step is efficient. Importantly, [13]
shows that the approximation oracle by itself is not strong enough to guarantee non-trivial ?-regret in
the bandit setting, and hence this assumption on the availability of a BS seems reasonable. Since the
1
as we show in the appendix, even if we relax the algorithm of [13] to only guarantee O(T
will still require O(T 2/3 ) calls to the oracle per iteration, on average.
2
1/3
) ?-regret, it
Reference
KKL [13]
This paper (Thm. 4.1, 4.2)
This paper (Thm. 4.1)
?
full information
regret oracle complexity
T 1/2
T
T 1/3
log(T )
p
T 1/2
T log(T )
?
bandit information
regret oracle complexity
T 1/3
T 2/3
1/3
T
log(T )
-
Table 1: comparison of expected ? regret bounds and average number of calls to the approximation
oracle per iteration. In all bounds we give only the dependence on the length of the game T and
omit all other dependencies which we treat as constants. In the bandit setting we report the expected
number of calls to the oracle per iteration.
best general regret bound known using a BS is O(T 1/3 ), the ?-regret bound of our bandit algorithm
is the best achievable to date via an efficient algorithm.
Technically, the main challenge in the considered setting is that as discussed, we cannot readily apply
standard tools such as FPL and OGD. At a high level, in [13] it was shown that it is possible to
apply the OGD method by replacing the exact projection step of OGD with an iterative algorithm
which finds an infeasible point, but one that both satisfies the projection property required by OGD
and is dominated by a convex combination of feasible points for every relevant linear loss (payoff)
function. Unfortunately, in worst case, the number of queries to the approximation oracle required by
this so-called projection algorithm per iteration is linear in T . While our online algorithms are also
based on an application of OGD, our approach to computing the so-called projections is drastically
different than [13], and is based on a coupling of two cutting plane methods, one that is based on
the Ellipsoid method, and the other that resembles Gradient Descent. This approach might be of
independent interest and might prove useful to similar problems.
1.1
Additional related work
Kalai and Vempala [14] showed that approximation algorithms which have point-wise approximation
guarantee, such as the celebrated MAX-CUT algorithm of [9], could be used to instantiate their
Follow the Perturbed Leader framework to achieve low ?-regret. However this construction is far
from generic and requires the oracle to satisfy additional non-trivial conditions. This approach was
also used in [3]. In [14] it was also shown that FPL could be instantiated with a FPTAS to achieve low
?-regret, however the approximation factor in the FPTAS needs to be set to roughly (1 + O(T 1/2 )),
which may result in prohibitive running times even if a FPTAS for the underlying problem is available.
Similarly, in [8] it was shown that if the approximation algorithm is based on solving a convex
relaxation of the original, possibly NP-Hard, problem, this additional structure can be used with
the FPL framework to achieve low ?-regret efficiently. To conclude all of the latter works consider
specialized cases in which the approximation oracle satisfies additional non-trivial assumptions
beyond its approximation guarantee, whereas here, similarly to [13], we will be interested in a generic
as possible conversion from the offline problem to the online one, without imposing additional
structure on the offline oracle.
2
2.1
Preliminaries
Online linear optimization with approximation oracles
Let K, F be compact sets of points in Rd+ (non-negative orthant in Rd ) such that maxx2K kxk ?
R, maxf 2F kf k ? F , for some R > 0, F > 0 (throughout this work we let k ? k denote the standard
Euclidean norm), and for all x 2 K, f 2 F it holds that C x ? f 0, for some C > 0.
We assume K is accessible through an approximated linear optimization oracle OK : Rd+ ! K with
?
parameter ? > 0 such that:
OK (c) ? c ? ? minx2K x ? c if ? 1;
d
8c 2 R+ :
OK (c) 2 K and
OK (c) ? c ? maxx2K x ? c if ? < 1.
Here K is the feasible set of actions for the player, and F is the set of all possible loss/payoff vectors2 .
2
we note that both of our assumptions that K ? Rd+ , F ? Rd+ and that the oracle takes inputs from Rd+ are
made for ease of presentation and clarity, and since these naturally hold for many NP-Hard optimization problem
that are relevant to our setting. Nevertheless, these assumptions could be easily generalized as done in [13].
3
Since naturally a factor ? > 1 for the approximation oracle is reasonable only for loss minimization
problems, and a value ? < 1 is reasonable for payoff maximization problems, throughout this
work it will be convenient to use the value of ? to differentiate between minimization problems and
maximization problems.
Given a sequence of linear loss/payoff functions {f1 , ..., fT } 2 F T and a sequence of feasible points
{x1 , ...., xT } 2 KT , we define the ? regret of the sequence {xt }t2[T ] with respect to the sequence
8 PT
P
{ft }t2[T ] as
< T1 t=1 xt ? ft ? ? minx2K T1 Tt=1 x ? ft if ? 1;
? regret({(xt , ft )}t2[T ] ) :=
(1)
PT
PT
:
1
1
? ? maxx2K T t=1 x ? ft T t=1 xt ? ft if ? < 1.
When the sequences {xt }t2[T ] , {ft }t2[T ] are obvious from context we will simply write ? regret
without stating these sequences. Also, when the sequence {xt }t2[T ] is randomized we will use
E[? regret] to denote the expected ?-regret.
2.1.1
Online linear optimization with full information
In OLO with full information, we consider a repeated game of T prediction rounds, for a fixed
T , where on each round t, the decision maker is required to choose a feasible action xt 2 K.
After committing to his choice, a linear loss function ft 2 F is revealed, and the decision maker
incurs loss of xt ? ft . In the payoff version, the decision maker incurs payoff of xt ? ft . The
game then continues to the next round. The overall goal of the decision maker is to guarantee
that ? regret({(xt , ft )}t2[T ] ) = O(T c ) for some c > 0, at least in expectation (in fact using
randomization is mandatory since K need not be convex). Here we assume that the adversary is
oblivious (aka non-adaptive), i.e., the sequence of losses/payoffs f1 , ..., fT is chosen in advance
(before the first round), and does not depend on the actions of the decision maker.
2.1.2
Bandit feedback
The bandit version of the problem is identical to the full information setting with one crucial difference:
on each round t, after making his choice, the decision maker does not observe the vector ft , but only
the value of his loss/payoff, given by xt ? ft .
2.2
Additional notation
For any two sets S, K ? Rd and a scalar 2 R we define the sets S + K := {x + y | x 2 S, y 2 K},
S := { x | x 2 S}. We also denote by CH(K) the convex-hull of all points in a set K. For a
convex and compact set S ? Rd and a point x 2 Rd we define dist(x, S) := minz2S kz xk. We
let B(c, r) denote the Euclidean ball or radius r centered in c.
2.3
Basic algorithmic tools
We now briefly describe two very basic ideas that are essential for constructing our algorithms,
namely the extended approximation oracle and the online gradient descent without feasibility method.
These were already suggested in [13] to obtain their low ?-regret algorithms. We note that in the
appendix we describe in more detail the approach of [13] and discuss its shortcomings in obtaining
oracle-efficient algorithms.
2.3.1
The extended approximation oracle
As discussed, a key difficulty of our setting that prevents us from directly applying well studied
algorithms for OLO, is that essentially all standard algorithms require to exactly solve (or up to
arbitrarily small error) some linear/convex optimization problem over the convexification of the
feasible set CH(K). However, not only that our approximation oracle OK (?) cannot perform exact
minimization, even for ? = 1 it is applicable only with inputs in Rd+ , and hence cannot optimize in
all directions. A natural approach, suggested in [13], to overcome the approximation error of the
oracle OK (?), is to consider optimization with respect to the convex set CH(?K) (i.e. convex hull
of all points in K scaled by a factor of ?) instead of CH(K). Indeed, if we consider for instance
the case ?
1, it is straightforward to see that for any c 2 Rd+ , OK (c) ? c ? ? minx2K x ? c =
4
? minx2CH(K) x ? c = minx2CH(?K) x ? c. Thus, in a certain sense, OK (?) can optimize with respect
to CH(?K) for all directions in Rd+ , although the oracle returns points in the original set K.
The following lemma shows that one can easily extend the oracle OK (?) to optimize with respect to
all directions in Rd .
Lemma 2.1 (Extended approximation oracle). Given c 2 Rd write c = c+ + c where c+ equals
to c on all non-negative coordinates of c and zero everywhere else, and c equals c on all negative
? K : Rd !
coordinates and zero everywhere else. The extended approximation oracle is a mapping O
(K + B(0, (1 + ?)R), K) defined as:
?
(OK (c+ ) ?R?
c , OK (c+ ))
if ? 1;
?K (c) = (v, s) :=
O
(2)
(OK ( c ) R?
c+ , OK ( c )) if ? < 1,
? = v/kvk if kvk > 0 and v
? = 0 otherwise, and it satisfies
where for any vector v 2 Rd we denote v
the following three properties:
1. v ? c ? minx2?K x ? c
2. 8f 2 F: s ? f ? v ? f if ?
3. kvk ? (? + 2)R
1 and s ? f
v ? f if ? < 1
The proof is given in the appendix for completeness.
It is important to note that while the extended oracle provides solutions with values at least as low as
any point in CH(?K), still in general the output point v need not be in either K or CH(?K), which
means that it is not a feasible point to play in our OLO setting, nor does it allow us to optimize
over CH(?K). This is why we also need the oracle to output the feasible point s 2 K which
dominates v for any possible loss/payoff vector in F. While we will use the outputs v to solve a
certain optimization problem involving CH(?K), this dominance relation will be used to convert the
solutions to these optimization problems into feasible plays for our OLO algorithms.
2.3.2
Online gradient descent with and without feasibility
As in [13], our online algorithms will be based on the well known Online Gradient Descent method
(OGD) for online convex optimization, originally due to [17]. For a sequence of loss vectors
{f1 , ..., fT } ? Rd OGD produces a sequence of plays {x1 , ..., xT } ? S, for a convex and compact
set S ? Rd via the following updates: 8t 1 : yt+1
xt ?ft , xt+1
arg minx2S kx yt+1 k2 ,
where x1 is initialized to some arbitrary point in S and ? is some pre-determined step-size. The
obvious difficulty in applying OGD to online linear optimization over S = CH(?K) is the step of
computing xt+1 by projecting yt+1 onto the feasible set S, since as discussed, even with the extended
approximation oracle, one cannot exactly optimize over CH(?K). Instead we will consider a variant
of OGD which may produce infeasible points, i.e., outside of S, but which guarantees low regret
with respect to any point in S. This algorithm, which we refer to as online gradient descent without
feasibility, is given below (Algorithm 1).
Algorithm 1 Online Gradient Descent Without Feasibility
1: input: learning rate ? > 0
2: x1
some point in S
3: for t = 1 . . . T do
d
4:
play xt and
? receive loss/payoff vector ft 2 R
5:
yt+1
6:
find xt+1
7: end for
xt ?ft for losses
xt + ?ft for payoffs
2 Rd such that
8z 2 S :
kz
xt+1 k2 ? kz
yt+1 k2
(3)
Lemma 2.2. [Online gradient descent without feasibility] Fix ? > 0. Suppose Algorithm 1 is applied
for T rounds and let {ft }Tt=1 ? Rd be the sequence of observed loss/payoff vectors, and let {xt }Tt=1
5
be the sequence of points played by the algorithm. Then for any x 2 S it holds that
PT
PT
? PT
1
1
1
2
xk2 + 2T
for losses;
t=1 xt ? ft
t=1 x ? ft ? 2T ? kx1
t=1 kft k
T
T
1
T
PT
t=1
x ? ft
1
T
PT
t=1
x t ? ft ?
1
2T ? kx1
xk2 +
The proof is given in the appendix for completeness.
3
?
2T
PT
t=1
kft k2
for payoffs.
Oracle-efficient Computation of (infeasible) Projections onto CH(?K)
In this section we detail our main technical tool for obtaining oracle-efficient online algorithms,
i.e., our algorithm for computing projections, in the sense of Eq. (3), onto the convex set CH(?K).
Before presenting our projection algorithm, Algorithm 2 and detailing its theoretical guarantees,
we first present the main algorithmic building block in the algorithm, which is described in the
following lemma. Lemma 3.1 shows that for any point x 2 Rd , we can either find a near-by point
p which is a convex combination of points outputted by the extended approximation oracle (and
hence, p is dominated by a convex combination of feasible points in K for any vector in F, as
discussed in Section 2.3.1), or we can find a separating hyperplane that separates x from CH(?K)
with sufficiently large margin. We achieve this by running the well known Ellipsoid method [10, 5]
in a very specialized way. This application of the Ellipsoid method is similar in spirit to those in
[15, 16], which applied this idea to computing correlated equilibrium in games and algorithmic
mechanism design, though the implementation details and the way in which we apply this technique
are quite different.
The proof of the following lemma is given in the appendix.
Lemma 3.1 (Separation-or-Decomposition via the
Fix x 2 Rd , ? 2
? Ellipsoid method).
?
(0, (? + 2)R], and a positive integer N
cd2 ln (?+1)R+kxk
, where c is a positive univer?
sal constant. Consider an attempt to apply the Ellipsoid method for N iterations to the following
feasibility problem:
find w 2 Rd such that:
8z 2 ?K :
(x
z) ? w
?
and
kwk ? 1,
(4)
such that each iteration of the Ellipsoid method applies the following consecutive steps:
?K ( w), where w is the current iterate. If (x v) ? w < ?, use v x as a
1. (v, s)
O
separating hyperplane for the Ellipsoid method and continue to to the next iteration
2. if kwk > 1, use w as a separating hyperplane for the Ellipsoid method and continue to the
next iteration
3. otherwise (kwk ? 1 and (x
vector w.
?), declare Problem (4) feasible and return the
v) ? w
Then, if the Ellipsoid method terminates declaring Problem 4 feasible, the returned vector w is a
feasible solution to Problem (4). Otherwise (the Ellipsoid method completes N iterations without
declaring Problem (4) feasible), let (v1 , s1 ), ..., (vN , sN ) be the outputs of the extended approximation oracle gathered throughout the run of the algorithm, and let (a1 , ..., aN ) be an optimal solution
to the following convex optimization problem:
2
N
N
X
1 X
min
ai v i x
such that 8i 2 {1, ..., N } : ai 0,
ai = 1.
(5)
(a1 ,...,aN ) 2
i=1
i=1
Then the point p =
PN
i=1
ai vi satisfies kx
pk ? 3?.
We are now ready to present our algorithm for computing projections onto CH(?K) (in the sense of
Eq. (3)). Consider now an attempt to project a point y 2 Rd , and note that in particular, y itself is a
valid projection (again, in the sense of Eq. (3)), however, in general, it is not a feasible point nor is
it dominated by a convex combination of feasible points. When attempting to project y 2 Rd , our
algorithm continuously applies the separation-or-decomposition procedure described in Lemma 3.1.
6
In case the procedure returns a decomposition, then by Lemma 3.1, we have a point that is sufficiently
close to y and is dominated for any vector in F by a convex combination (given explicitly) of feasible
points in K. Otherwise, the procedure returns a separating hyperplane which can be used to to ?pull
y closer" to CH(?K) in a way that the resulting point still satisfies the projection inequality given in
Eq. (3), and the process then repeats itself. Since each time we obtain a hyperplane separating our
current iterate from CH(?K), we pull the current iterate sufficiently towards CH(?K), this process
must terminate. Lemma 3.2 gives exact bounds on the performance of the algorithm.
Algorithm 2 (infeasible) Projection onto CH(?K)
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
input: point y 2 Rd , tolerance ? > 0
?
y
y/ max{1, kyk/(?R)}
for t = 1 . . . do
call the S EPARATION - OR -D ECOMPOSTION procedure (Lemma 3.1) with parameters (?
y, ?)
if the procedure outputs a separating hyperplane w then
?
? ?w
y
y
else
let (a1 , ..., aN ), {(v1 , s1 ), ..., (vN , sN )} be the decomposition returned
? , (a1 , ..., aN ), {(v1 , s1 ), ..., (vN , sN )}
return y
end if
end for
Lemma 3.2. Fix y 2 Rd and ? 2 (0, (? + 2)R]. Algorithm 2 terminates after at most d?2 R2 /?2 e
? 2 Rd , a distribution (a1 , ..., aN ) and a set {(v1 , s1 ), ..., (vN , sN )}
iterations, returning a point y
outputted by the extended approximation oracle, where N is as defined in Lemma 3.1, such that
X
? k ? 3? for p :=
1. 8z 2 CH(?K) : k?
y zk2 ? ky zk2 ,
2. kp y
ai v i .
i2[N ]
? satisfies
Moreover, if the for loop was entered a total number of k times, then the final value of y
dist2 (?
y, CH(?K)) ? min{2?2 R2 , dist2 (y, CH(?K))
(k
1)?2 },
and the overall number of queries to the approximation oracle is O kd2 ln ((? + 1)R/?) .
It is important to note that the worst case iteration bound in Lemma 3.2 does not seem so appealing
for our purposes, since it depends polynomially on 1/?, and in our online algorithms naturally we
will need to take ? = O(T c ) for some c > 0, which seems to contradict our goal of achieving
poly-logarithmic in T oracle complexity, at least on average. However, as Lemma 3.2 shows, the
more iterations Algorithm 2 performs, the closer it brings its final iterate to the set CH(?K). Thus, as
we will show when analyzing the oracle complexity of our online algorithms, while a single call to
Algorithm 2 can be expensive, when calling it sequentially, where each input is a small perturbation
of the output of the previous call, the average number of iterations performed per such call cannot be
too high.
4
Efficient Algorithms for the Full Information and Bandit Settings
We now turn to present our online algorithms for the full-information and bandit settings together
with their regret bounds and oracle-complexity guarantees.
4.1
Algorithm for the full information setting
Our algorithm for the full-information setting, Algorithm 3, is given below.
Theorem 4.1. [Main Theorem] Fix ? > 0, ? 2 (0, (? + 2)R]. Suppose Algorithm 3 is applied for T
rounds and let {ft }Tt=1 ? F be the sequence of observed loss/payoff vectors, and let {st }Tt=1 be the
sequence of points played by the algorithm. Then it holds that
?
?
E ? regret {(st , ft )}t2[T ] ? ?2 R2 T 1 ? 1 + ?F 2 /2 + 3F ?,
and the average number of calls to the approximation oracle of K per iteration is upper bounded by
K(?, ?) := O
1 + ??RF + ? 2 F 2 ?
7
2
d2 ln ((? + 1)R/?) .
Algorithm 3 Online Gradient Descent with Infeasible Projections onto CH(?K)
1: input: learning rate ? > 0, projection error parameter ? > 0
?1
2: s1
some point in K, y
?s1
3: for t = 1 . . . T do
4:
play st and
? receive loss/payoff vector ft 2 F
y?t ?ft if ? 1
y?t + ?ft if ? < 1
? t+1 , a distribution
6:
call Algorithm 2 with inputs (yt+1 , ?) to obtain an approximated projection y
(a1 , ..., aN ) and {(v1 , s1 ), ..., (vN , sN )} ? Rd ? K, for some N 2 N.
7:
sample st+1 2 {s1 , ..., sN } according to distribution (a1 , ..., aN )
8: end for
5:
yt+1
In particular, setting ? = ?RT 2/3 /F , ? = ?RT 1/3 gives E [? regret] = O ?RF T 1/3 ,
1/2
K = O d2 ln ?+1
. Alternatively,? setting ? = ?RT
/F , ? = ?RT 1/2 gives
? T
?
p
E [? regret] = O ?RF T 1/2 , K = O
T d2 ln ?+1
.
? T
The proof is given in the appendix.
4.2
Algorithm for the bandit information setting
Our algorithm for the bandit setting follows from a very well known reduction from the bandit setting
to the full information setting, also applied in the bandit algorithm of [13]. The algorithm simply
simulates the full information algorithm, Algorithm 3, by providing it with estimated loss/payoff
vectors ?f1 , ..., ?fT instead of the true vectors f1 , ..., fT which are not available in the bandit setting.
This reduction is based on the use of a Barycentric Spanner (defined next) for the feasible set K.
As standard, we assume the points in K span the entire space Rd , otherwise we can reformulate the
problem in a lower-dimensional space, in which this assumption holds.
Definition 4.1 (Barycentric Spanner3 ). We say that a set of d vectors {q1 , ..., qd } ? Rd is a
Barycentric Spanner with parameter > 0 for a set S ? Rd , denoted by -BS(S), if it holds that
Pd
1
{q1 , ..., qd } ? S, and the matrix Q := i=1 qi q>
qi k ? .
i is not singular and maxi2[d] kQ
Importantly, as discussed in [13], the assumption on the availability of such a set -BS(K) seems
reasonable, since i) for many sets that correspond to the set of all possible solutions to some wellstudied NP-Hard optimization problem, one can still construct in poly(d) time a barycentric spanner
with = poly(d), ii) -BS(K) needs to be constructed only once and then stored in memory (overall
d vectors in Rd ), and hence its construction can be viewed as a pre-processing step, and iii) as
illustrated in [13], without further assumptions, the approximation oracle by itself is not sufficient to
guarantee nontrivial regret bounds in the bandit setting.
The algorithm and the proof of the following theorem are given in the appendix.
Theorem 4.2. Fix ? > 0, ? 2 (0, (? + 2)R], 2 (0, 1). Suppose Algorithm 5 is applied for T
rounds and let {ft }Tt=1 ? F be the sequence of observed loss/payoff vectors, and let {?st }Tt=1 be the
sequence of points played by the algorithm. Then it holds that
?
?
E ? regret {(?st , ft )}t2[T ] ? ?2 R2 ? 1 T 1 + ?d2 C 2 2 1 /2 + 3?F + C,
and the expected number of calls to the approximation oracle of K per iteration is upper bounded by
E [K(?, ?, )] := O
In particular, setting ? =
O (? dCR + ?RF + C)T
1 + ?? dCR + (?dC )2 /
, ? = ?RT
, E[K] = O d2 ln
?R
dC T
1/3
2/3
3
?
,
?+1
? T
1/3
2
d2 ln ((? + 1)R/?) .
= T
.
1/3
gives E [?
regret] =
this definition is somewhat different than the classical one given in [2], however it is equivalent to a
C-approximate barycentric spanner [2], with an appropriately chosen constant C( ).
8
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9
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6,263 | 6,666 | Geometric Descent Method for
Convex Composite Minimization
Shixiang Chen1 , Shiqian Ma2 , and Wei Liu3
1
Department of SEEM, The Chinese University of Hong Kong, Hong Kong
2
Department of Mathematics, UC Davis, USA
3
Tencent AI Lab, China
Abstract
In this paper, we extend the geometric descent method recently proposed by Bubeck,
Lee and Singh [1] to tackle nonsmooth and strongly convex composite problems.
We prove that our proposed algorithm, dubbed geometric
proximal gradient method
?
(GeoPG), converges with a linear rate (1 ? 1/ ?) and thus achieves the optimal
rate among first-order methods, where ? is the condition number of the problem.
Numerical results on linear regression and logistic regression with elastic net
regularization show that GeoPG compares favorably with Nesterov?s accelerated
proximal gradient method, especially when the problem is ill-conditioned.
1
Introduction
Recently, Bubeck, Lee and Singh proposed a geometric descent method (GeoD) for minimizing a
smooth and strongly convex function [1]. They showed that GeoD achieves the same optimal rate
as Nesterov?s accelerated gradient method (AGM) [2, 3]. In this paper, we provide an extension of
GeoD that minimizes a nonsmooth function in the composite form:
min F (x) := f (x) + h(x),
x?Rn
(1.1)
where f is ?-strongly convex and ?-smooth (i.e., ?f is Lipschitz continuous with Lipschitz constant
?), and h is a closed nonsmooth convex function with simple proximal mapping. Commonly seen
examples of h include `1 norm, `2 norm, nuclear norm, and so on.
If h vanishes, then the objective function of (1.1) becomes?smooth and strongly convex. In this case,
it is known that AGM converges with a linear rate (1 ? 1/ ?), which is optimal among all first-order
methods, where ? = ?/? is the condition number of the problem. However, AGM lacks a clear
geometric intuition, making it difficult to interpret. Recently, there has been much work on attempting
to explain AGM or designing new algorithms with the same optimal rate (see, [4, 5, 1, 6, 7]). In
particular, the GeoD method proposed in [1] has a clear geometric intuition that is in the flavor
of the ellipsoid method [8]. The follow-up work [9, 10] attempted to improve the performance of
GeoD by exploiting the gradient information from the past with a ?limited-memory? idea. Moreover,
Drusvyatskiy, Fazel and Roy [10] showed how to extend the suboptimal version of GeoD (with the
convergence rate (1 ? 1/?)) to solve the composite problem (1.1). However, it was not clear how to
extend the optimal version of GeoD to address (1.1), and the authors posed this as an open question.
In this paper, we settle this question by proposing a geometric proximal gradient (GeoPG) algorithm
which can solve the composite problem (1.1). We further show how to incorporate various techniques
to improve the performance of the proposed algorithm.
Notation. We use B(c, r2 ) = x|kx ? ck2 ? r2 to denote the ball with center c and radius r.
We use Line(x, y) to denote the line that connects x and y, i.e., {x + s(y ? x), s ? R}. For fixed
t ? (0, 1/?], we denote x+ := Proxth (x ? t?f (x)), where the proximal mapping Proxh (?) is
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
defined as Proxh (x) := argminz h(z) + 12 kz ? xk2 . The proximal gradient of F is defined as
Gt (x) := (x?x+ )/t. It should be noted that x+ = x?tGt (x). We also denote x++ := x?Gt (x)/?.
Note that both x+ and x++ are related to t, and we omit t whenever there is no ambiguity.
The rest of this paper is organized as follows. In Section 2, we briefly review the GeoD method for
solving smooth and strongly convex problems. In Section 3, we provide our GeoPG algorithm for
solving nonsmooth problem (1.1) and analyze its convergence rate. We address two practical issues of
the proposed method in Section 4, and incorporate two techniques: backtracking and limited memory,
to cope with these issues. In Section 5, we report some numerical results of comparing GeoPG with
Nesterov?s accelerated proximal gradient method in solving linear regression and logistic regression
problems with elastic net regularization. Finally, we conclude the paper in Section 6.
2
Geometric Descent Method for Smooth Problems
The GeoD method [1] solves (1.1) when h ? 0, in which the problem reduces to a smooth and
strongly convex problem min f (x). We denote its optimal solution and optimal value as x? and
f ? , respectively. Throughout this section, we fix t = 1/?, which together with h ? 0 implies that
x+ = x ? ?f (x)/? and x++ = x ? ?f (x)/?. We first briefly describe the basic idea of the
suboptimal GeoD. Since f is ?-strongly convex, the following inequality holds
?
f (x) + h?f (x), y ? xi + ky ? xk2 ? f (y), ?x, y ? Rn .
(2.1)
2
By letting y = x? in (2.1), it is easy to obtain
x? ? B x++ , k?f (x)k2 /?2 ? 2(f (x) ? f ? )/? , ?x ? Rn .
(2.2)
Note that the ?-smoothness of f implies
f (x+ ) ? f (x) ? k?f (x)k2 /(2?), ?x ? Rn .
(2.3)
Combining (2.2) and (2.3) yields x ? B x , (1 ? 1/?)k?f (x)k /? ? 2(f (x ) ? f )/? . As
a result, suppose that initially we have a ball B(x0 , R02 ) that contains x? , then it follows that
+
2
2
?
x? ? B x0 , R02 ? B x++
(2.4)
0 , (1 ? 1/?)k?f (x0 )k /? ? 2(f (x0 ) ? f )/? .
Some simple algebraic calculations show that the squared radius of the minimum enclosing ball
of the right hand side of (2.4) is no larger than R02 (1 ? 1/?), i.e., there exists some x1 ? Rn such
that x? ? B x1 , R02 (1 ? 1/?) . Therefore, the squared radius of the initial ball shrinks by a factor
(1 ? 1/?). Repeating this process yields a linear convergent sequence {xk } with the convergence
rate (1 ? 1/?): kxk ? x? k2 ? (1 ? 1/?)k R02 .
?
The optimal GeoD (with the linear convergence rate (1 ? 1/ ?)) maintains two balls containing
x? in each iteration, whose centers are ck and x++
k+1 , respectively. More specifically, suppose that in
the k-th iteration we have ck and xk , then ck+1 and xk+1 are obtained as follows. First, xk+1 is the
2
minimizer of f on Line(ck , x+
k ). Second, ck+1 (resp. Rk+1 ) is the center (resp. squared radius) of
the ball (given by Lemma 2.1) that contains
2
2
B ck , Rk2 ? k?f (xk+1 )k2 /(?2 ?) ? B x++
k+1 , (1 ? 1/?)k?f (xk+1 )k /? .
Calculating ck+1 and Rk+1 is easy and we refer to Algorithm 1 of [1] for details. By applying
Lemma 2.1 with xA = ck , rA = Rk , rB = k?f (xk+1 )k/?, = 1/? and ? = ?2 (f (x+
) ? f (x? )),
?
? kk 2
2
2
?
2
we obtain Rk+1 = (1 ? 1/ ?)Rk , which further implies kx ? ck k ? (1 ? 1/ ?) R0 , i.e., the
?
optimal GeoD converges with the linear rate (1 ? 1/ ?).
2
2
Lemma 2.1 (see [1, 10]). Fix centers xA , xB ? Rn and squared radii rA
, rB
> 0. Also fix ? (0, 1)
2
2
n
and suppose kxA ? xB k ? rB . There exists a new center c ? R such that for any ? > 0, we have
? 2
2
2
2
B(xA , rA
? rB
? ?) ? B xB , rB
(1 ? ) ? ? ? B c, (1 ? )rA
?? .
?
3
++
2
2
+
?
Geometric Descent Method for Nonsmooth Convex Composite Problems
Drusvyatskiy, Fazel and Roy [10] extended the suboptimal GeoD to solve the composite problem
(1.1). However, it was not clear how to extend the optimal GeoD to solve problem (1.1). We resolve
this problem in this section.
The following lemma is useful to our analysis. Its proof is in the supplementary material.
2
Lemma 3.1. Given point x ? Rn and step size t ? (0, 1/?], denote x+ = x ? tGt (x). The following
inequality holds for any y ? Rn :
t
?
F (y) ? F (x+ ) + hGt (x), y ? xi + kGt (x)k2 + ky ? xk2 .
(3.1)
2
2
3.1
GeoPG Algorithm
In this subsection, we describe our proposed geometric proximal gradient method (GeoPG) for
solving (1.1). Throughout Sections 3.1 and 3.2, t ? (0, 1/?] is a fixed scalar. The key observation for
designing GeoPG is that in the k-th iteration one has to find xk that lies on Line(x+
k?1 , ck?1 ) such
that the following two inequalities hold:
t
1
+
++
2
(3.2)
F (x+
? ck?1 k2 ? 2 kGt (xk )k2 .
k ) ? F (xk?1 ) ? kGt (xk )k , and kxk
2
?
Intuitively, the first inequality in (3.2) requires that there is a function value reduction on x+
k from
x+
,
and
the
second
inequality
requires
that
the
centers
of
the
two
balls
are
far
away
from
each
other
k?1
so that Lemma 2.1 can be applied.
The following lemma gives a sufficient condition for (3.2). Its proof is in the supplementary material.
Lemma 3.2. (3.2) holds if xk satisfies
+
+
hx+
(3.3)
k ? xk , xk?1 ? xk i ? 0, and hxk ? xk , xk ? ck?1 i ? 0.
Therefore, we only need to find xk such that (3.3) holds. To do so, we define the following functions
for given x, c (x 6= c) and t ? (0, ?]:
?t,x,c (z) = hz + ? z, x ? ci, ?z ? Rn , and ??t,x,c (s) = ?t,x,c x + s(c ? x) , ?s ? R.
The functions ?t,x,c (z) and ??t,x,c (s) have the following properties. Its proof can be found in the
supplementary material.
Lemma 3.3. (i) ?t,x,c (z) is Lipschitz continuous. (ii) ??t,x,c (s) strictly monotonically increases.
We are now ready to describe how to find xk such that (3.3) holds. This is summarized in Lemma 3.4.
Lemma 3.4. The following two ways find xk satisfying (3.3).
(i) If ??t,x+
(1) ? 0, then (3.3) holds by setting xk := ck?1 ; if ??t,x+ ,ck?1 (0) ? 0, then
k?1
(3.3) holds by setting xk := x+ ; if ?? +
(1) > 0 and ?? +
(0) < 0, then
k?1 ,ck?1
k?1
there exists s ? [0, 1] such that ??t,x+
t,xk?1 ,ck?1
k?1
t,xk?1 ,ck?1
,ck?1 (s) = 0. As a result, (3.3) holds by setting
+
xk := x+
k?1 + s(ck?1 ? xk?1 ).
(ii) If ??t,x+
?
(0) ? 0, then (3.3) holds by setting xk := x+
(0) < 0,
k?1 ; if ?t,x+
k?1 ,ck?1
?
then there exists s ? 0 such that ?t,x+ ,ck?1 (s) = 0. As a result, (3.3) holds by setting
k?1 ,ck?1
k?1
+
xk := x+
k?1 + s(ck?1 ? xk?1 ).
Proof. Case (i) directly follows from the Mean-Value Theorem. Case (ii) follows from the monotonicity and continuity of ??t,x+ ,ck?1 from Lemma 3.3.
k?1
It is indeed very easy to find xk satisfying the two cases in Lemma 3.4, since we are tackling a
univariate Lipschitz continuous function ??t,x,c (s) . Specifically, for case (i) of Lemma 3.4, we can
use the bisection method to find the zero of ??t,x+ ,ck?1 in the closed interval [0, 1]. In practice, we
k?1
found that the Brent-Dekker method [11, 12] performs much better than the bisection method, so
we use the Brent-Dekker method in our numerical experiments. For case (ii) of Lemma 3.4, we
can use the semi-smooth Newton method to find the zero of ??t,x+ ,ck?1 in the interval [0, +?).
k?1
In our numerical experiments, we implemented the global semi-smooth Newton method [13, 14]
and obtained very encouraging results. These two procedures are described in Algorithms 1 and 2,
respectively. Based on the discussions above, we know that xk generated by these two algorithms
satisfies (3.3) and hence (3.2).
We are now ready to present our GeoPG algorithm for solving (1.1) as in Algorithm 3.
3
Algorithm 1 : The first procedure for finding xk from given x+
k?1 and ck?1 .
+
+
+
1: if h(x+
k?1 ) ? xk?1 , xk?1 ? ck?1 i ? 0 then
2:
set xk := x+
k?1 ;
3: else if hc+
?
ck?1 , x+
k?1
k?1 ? ck?1 i ? 0 then
4:
set xk := ck?1 ;
5: else
use the Brent-Dekker method to find s ? [0, 1] such that ??t,x+
6:
k?1 ,ck?1
xk :=
7: end if
x+
k?1
+ s(ck?1 ?
(s) = 0, and set
x+
k?1 );
Algorithm 2 : The second procedure for finding xk from given x+
k?1 and ck?1 .
+
+
+
1: if h(x+
k?1 ) ? xk?1 , xk?1 ? ck?1 i ? 0 then
+
2:
set xk := xk?1 ;
3: else
use the global semi-smooth Newton method [13, 14] to find the root s ? [0, +?) of
4:
+
??t,x+ ,ck?1 (s), and set xk := x+
k?1 + s(ck?1 ? xk?1 );
k?1
5: end if
3.2
Convergence Analysis of GeoPG
We are now ready to present our main convergence result for GeoPG.
2
(x0 )k
Theorem 3.5. Given initial point x0 and step size t ? (0, 1/?], we set R02 = kGt ?
(1 ? ?t).
2
?
Suppose that sequence {(xk , ck , Rk )} is generated by Algorithm 3, and that x is the optimal
?
?
2
solution of (1.1)
? and F2 is the optimal objective value. For any k ? 0, one has x ? B(ck , Rk ) and
2
Rk+1 ? (1 ? ?t)Rk , and thus
?
?
?
?
(1 ? ?t)k R02 .
kx? ? ck k2 ? (1 ? ?t)k R02 , and F (x+
(3.4)
k+1 ) ? F ?
2
?
Note that when t = 1/?, (3.4) implies the linear convergence rate (1 ? 1/ ?).
Proof. We prove a stronger result by induction that for every k ? 0, one has
?
x? ? B ck , Rk2 ? 2(F (x+
k ) ? F )/? .
(3.5)
Let y = x? in (3.1). We have kx? ? x++ k2 ? (1 ? ?t)kGt (x)2 k/?2 ? 2(F (x+ ) ? F ? )/?, implying
x? ? B x++ , kGt (x)k2 (1 ? ?t)/?2 ? 2(F (x+ ) ? F ? )/? .
(3.6)
Setting x = x0 in (3.6) shows that (3.5) holds for k = 0. We now assume that (3.5) holds for some
k ? 0, and in the following we will prove that (3.5) holds for k + 1. Combining (3.5) and the first
inequality of (3.2) yields
?
x? ? B ck , Rk2 ? tkGt (xk+1 )k2 /? ? 2(F (x+
(3.7)
k+1 ) ? F )/? .
By setting x = xk+1 in (3.6), we obtain
+
2
2
?
x? ? B x++
k+1 , kGt (xk+1 )k (1 ? ?t)/? ? 2(F (xk+1 ) ? F )/? .
x++
k+1 ,
(3.8)
We now apply Lemma 2.1 to (3.7) and (3.8). Specifically, we set xB =
xA = ck , = ?t,
+
2
?
2
2
rA = Rk , rB = kGt (xk+1 )k/?, ? = ? (F (xk ) ? F ), and note that kxA ? xB k ? rB
because of
the second inequality of (3.2). Then Lemma 2.1 indicates that there exists ck+1 such that
?
?
x? ? B ck+1 , (1 ? 1/ ?)Rk2 ? 2(F (x+
(3.9)
k+1 ) ? F )/? ,
?
2
? (1 ? ?t)Rk2 . Note that ck+1 is the center of the minimum
i.e., (3.5) holds for k + 1 with Rk+1
enclosing ball of the intersection of the two balls in (3.7) and (3.8), and can be computed
in the
?
same?way as Algorithm 1 of [1]. From (3.9) we obtain that kx? ? ck+1 k2 ? (1?? ?t)Rk2 ?
? 2
?
?
(1 ? ?t)k+1 R02 . Moreover, (3.7) indicates that F (x+
?t)k R02 .
k+1 ) ? F ? 2 Rk ? 2 (1 ?
4
Algorithm 3 : GeoPG: geometric proximal gradient descent for convex composite minimization.
Require: Parameters ?, ?, initial point x0 and step size t ? (0, 1/?].
2
2
2
1: Set c0 = x++
0 , R0 = kGt (x0 )k (1 ? ?t)/? ;
2: for k = 1, 2, . . . do
3:
Use Algorithm 1 or 2 to find xk ;
2
4:
Set xA := x++
= xk ? Gt (xk )/?, and RA
= kGt (xk )k2 (1 ? ?t)/?2 ;
k
+
2
2
5:
Set xB := ck?1 , and RB = Rk?1 ? 2(F (xk?1 ) ? F (x+
k ))/?;
2
2
Compute B(ck , Rk2 ): the minimum enclosing ball of B(xA , RA
) ? B(xB , RB
), which can be
6:
done using Algorithm 1 in [1];
7: end for
4
4.1
Practical Issues
GeoPG with Backtracking
In practice, the Lipschitz constant ? may be unknown to us. In this subsection, we describe a
backtracking strategy for GeoPG in which ? is not needed. From the ?-smoothness of f , we have
f (x+ ) ? f (x) ? th?f (x), Gt (x)i + tkGt (x)k2 /2.
(4.1)
Note that inequality (3.1) holds because of (4.1), which holds when t ? (0, 1/?]. If ? is unknown,
we can perform backtracking on t such that (4.1) holds, which is a common practice for proximal
gradient method, e.g., [15?17]. Note that the key step in our analysis of GeoPG is to guarantee that
the two inequalities in (3.2) hold. According to Lemma 3.2, the second inequality in (3.2) holds as
long as we use Algorithm 1 or Algorithm 2 to find xk , and it does not need the knowledge of ?.
However, the first inequality in (3.2) requires t ? 1/?, because its proof in Lemma 3.2 needs (3.1).
Thus, we need to perform backtracking on t until (4.1) is satisfied, and use the same t to find xk by
Algorithm 1 or Algorithm 2. Our GeoPG algorithm with backtracking (GeoPG-B) is described in
Algorithm 4.
Algorithm 4 : GeoPG with Backtracking (GeoPG-B)
Require: Parameters ?, ? ? (0, 1), ? ? (0, 1), initial step size t0 > 0 and initial point x0 .
Repeat t0 := ?t0 until (4.1) holds for t = t0 ;
kGt0 (x0 )k2
2
Set c0 = x++
(1 ? ?t0 );
0 , R0 =
?2
for k = 1, 2, . . . do
if no backtracking was performed in the (k ? 1)-th iteration then
Set tk := tk?1 /?;
else
Set tk := tk?1 ;
end if
Compute xk by Algorithm 1 or Algorithm 2 with t = tk ;
tk
2
while f (x+
k ) > f (xk ) ? tk h?f (xk ), Gtk (xk )i + 2 kGtk (xk )k do
Set tk := ?tk (backtracking);
Compute xk by Algorithm 1 or Algorithm 2 with t = tk ;
end while
kG (x )k2
2
Set xA := x++
= xk ? Gtk (xk )/?, RA
= tk?2 k (1 ? ?tk );
k
+
2
2
Set xB := ck?1 , RB
= Rk?1
? ?2 (F (x+
k?1 ) ? F (xk ));
2
2
Compute B(ck , Rk2 ): the minimum enclosing ball of B(xA , RA
) ? B(xB , RB
);
end for
Note that the sequence {tk } generated in Algorithm 4 is uniformly bounded away from 0. This is
because (4.1) always holds when tk ? 1/?. As a result, we know tk ? tmin := mini=0,...,k ti ? ?/?.
It is easy to see that in the k-th iteration of Algorithm 4, x? is contained in two balls:
2
?
x? ? B ck?1 , Rk?1
? tk kGtk (xk )k2 /? ? 2(F (x+
k ) ? F )/?,
+
2
2
?
x? ? B x++
k , kGtk (xk )k (1 ? ?tk )/? ? 2(F (xk ) ? F )/? .
5
Therefore, we have the following convergence result for Algorithm 4, whose proof is similar to that
for Algorithm 3. We thus omit the proof for succinctness.
Theorem 4.1. Suppose that {(xk , ck , Rk , tk )} is generated by Algorithm 4. For any k ? 0, one
Qk
?
?
2
has x? ? B(ck , Rk2 ) and Rk+1
? (1 ? ?tk )Rk2 , and thus kx? ? ck k2 ? i=0 (1 ? ?ti )i R02 ?
?
(1 ? ?tmin )k R02 .
4.2
GeoPG with Limited Memory
The basic idea of GeoD is that in each iteration we maintain two balls B(y1 , r12 ) and B(y2 , r22 ) that
both contain x? , and then compute the minimum enclosing ball of their intersection, which is expected
to be smaller than both B(y1 , r12 ) and B(y2 , r22 ). One very intuitive idea that can possibly improve
the performance of GeoD is to maintain more balls from the past, because their intersection should be
smaller than the intersection of two balls. This idea has been proposed by [9] and [10]. Specifically,
Bubeck and Lee [9] suggested to keep all the balls from past iterations and then compute the minimum
enclosing ball of their intersection. For a given bounded set Q, the center of its minimum enclosing
ball is known as the Chebyshev center, and is defined as the solution to the following problem:
min max ky ? xk2 = min max kyk2 ? 2y > x + Tr(xx> ).
y
y
x?Q
x?Q
(4.2)
2
(4.2) is not easy to solve for a general set Q. However, when Q := ?m
i=1 B(yi , ri ), Beck [18] proved
that the relaxed Chebyshev center (RCC) [19], which is a convex quadratic program, is equivalent to
(4.2) if m < n. Therefore, we can solve (4.2) by solving a convex quadratic program (RCC):
min max kyk2 ?2y > x+Tr(4) = max min kyk2 ?2y > x+Tr(4) = max ?kxk2 +Tr(4),
y
(x,4)??
(x,4)??
y
(x,4)??
2
where ? = {(x, 4) : x ? Q, 4 xx> }. If Q = ?m
i=1 B(ci , ri ), then the dual of (4.3) is
min kC?k2 ?
m
X
i=1
?i kci k2 +
m
X
?i ri2 , s.t.
i=1
m
X
?i = 1,
?i ? 0, i = 1, . . . , m,
(4.3)
(4.4)
i=1
where C = [c1 , . . . , cm ] and ?i , i = 1, 2, . . . , m are the dual variables. Beck [18] proved that the
optimal solutions of (4.2) and (4.4) are linked by x? = C?? if m < n.
Now we can give our limited-memory GeoPG algorithm (L-GeoPG) as in Algorithm 5.
Algorithm 5 : L-GeoPG: Limited-memory GeoPG
Require: Parameters ?, ?, memory size m > 0 and initial point x0 .
2
2
2
2
1: Set c0 = x++
0 , r0 = R0 = kGt (x0 )k (1 ? 1/?)/? , and t = 1/?;
2: for k = 1, 2, . . . do
3:
Use Algorithm 1 or 2 to find xk ;
4:
Compute rk2 = kGt (xk )k2 (1 ? 1/?)/?2 ;
2
5:
Compute B(ck , Rk2 ): an enclosing ball of the intersection of B(ck?1 , Rk?1
) and Qk :=
++ 2
++ 2
k
k
?i=k?m+1 B(xi , ri ) (if k ? m, then set Qk := ?i=1 B(xi , ri )). This is done by setting
ck = C?? , where ?? is the optimal solution of (4.4);
6: end for
Remark 4.2. Backtracking can also be incorporated into L-GeoPG. We denote the resulting algorithm as L-GeoPG-B.
L-GeoPG has the same linear convergence rate as GeoPG, as we show in Theorem 4.3.
Theorem 4.3. Consider the L-GeoPG algorithm. For any k ? 0, one has x? ? B(ck , Rk2 ) and
?
?
2
Rk2 ? (1 ? 1/ ?)Rk?1
, and thus kx? ? ck k2 ? (1 ? 1/ ?)k R02 .
++ 2
2
Proof. Note that Qk := ?ki=k?m+1 B(x++
i , ri ) ? B(xk , rk ). Thus, the minimum enclosing ball
++ 2
2
2
of B(ck?1 , Rk?1 )? B(xk , rk ) is an enclosing ball of B(ck?1 , Rk?1
)?Qk . The proof then follows
from the proof of Theorem 3.5, and we omit it for brevity.
6
5
Numerical Experiments
In this section, we compare our GeoPG algorithm with Nesterov?s accelerated proximal gradient
(APG) method for solving two nonsmooth problems: linear regression and logistic regression, both
with elastic net regularization. Because of the elastic net term, the strong convexity parameter ? is
known. However, we assume that ? is unknown, and implement backtracking for both GeoPG and
APG, i.e., we test GeoPG-B and APG-B (APG with backtracking). We do not target at comparing
with other efficient algorithms for solving these two problems. Our main purpose here is to illustrate
the performance of this new first-order method GeoPG. Further improvement of this algorithm and
comparison with other state-of-the-art methods will be a future research topic.
The initial points were set to zero. To obtain the optimal objective function value F ? , we ran
APG-B and GeoPG-B for a sufficiently long time and the smaller function value returned by the two
algorithms is selected as F ? . APG-B was terminated if (F (xk ) ? F ? )/F ? ? tol, and GeoPG-B was
?
?
?8
terminated if (F (x+
is the accuracy tolerance. The parameters
k ) ? F )/F ? tol, where tol = 10
used in backtracking were set to ? = 0.5 and ? = 0.9. In GeoPG-B, we used Algorithm 2 to find xk ,
because we found that the performance of Algorithm 2 is slightly better than Algorithm 1 in practice.
In the experiments, we ran Algorithm 2 until the absolute value of ?? is smaller than 10?8 . The code
was written in Matlab and run on a standard PC with 3.20 GHz I5 Intel microprocessor and 16GB of
memory. In all figures we reported, the x-axis denotes the CPU time (in seconds) and y-axis denotes
?
?
(F (x+
k ) ? F )/F .
5.1
Linear regression with elastic net regularization
In this subsection, we compare GeoPG-B and APG-B in terms of solving linear regression with
elastic net regularization, a popular problem in machine learning and statistics [20]:
1
?
min
kAx ? bk2 + kxk2 + ?kxk1 ,
(5.1)
x?Rn 2p
2
where A ? Rp?n , b ? Rp , and ?, ? > 0 are the weighting parameters.
We conducted tests on two real datasets downloaded from the LIBSVM repository: a9a, RCV1. The
results are reported in Figure 1. In particular, we tested ? = 10?8 and ? = 10?3 , 10?4 , 10?5 . Note
that since ? is very small, the problems are very likely to be ill-conditioned. We see from Figure 1
that GeoPG-B is faster than APG-B on these real datasets, which indicates that GeoPG-B is preferable
than APG-B. In the supplementary material, we show more numerical results on varying ?, which
further confirm that GeoPG-B is faster than APG-B when the problems are more ill-conditioned.
2
2
10
10
?3
GeoPG?B: ? =10?3
GeoPG?B: ? =10
APG?B: ? =10?3
0
10
APG?B: ? =10?3
0
10
GeoPG?B: ? =10?4
GeoPG?B: ? =10?4
?4
?4
APG?B: ? =10
?2
10
APG?B: ? =10
GeoPG?B: ? =10?5
10
?5
APG?B: ? =10
?4
(F ?F*)/F*
10
APG?B: ? =10?5
?4
10
k
(Fk?F*)/F*
GeoPG?B: ? =10?5
?2
?6
10
?6
10
?8
10
?8
10
?10
10
?12
10
?10
0
10
20
30
40
10
50
0
CPU(s)
(a) Dataset a9a
5
10
15
CPU(s)
20
25
30
(b) Dataset RCV1
Figure 1: GeoPG-B and APG-B for solving (5.1) with ? = 10?8 .
5.2
Logistic regression with elastic net regularization
In this subsection, we compare the performance of GeoPG-B and APG-B in terms of solving the
following logistic regression problem with elastic net regularization:
p
?
1X
min
log 1 + exp(?bi ? a>
kxk2 + ?kxk1 ,
(5.2)
i x) +
x?Rn p
2
i=1
7
where ai ? Rn and bi ? {?1} are the feature vector and class label of the i-th sample, respectively,
and ?, ? > 0 are the weighting parameters.
We tested GeoPG-B and APG-B for solving (5.2) on the three real datasets a9a, RCV1 and Gisette
from LIBSVM, and the results are reported in Figure 2. In particular, we tested ? = 10?8 and
? = 10?3 , 10?4 , 10?5 . Figure 2 shows that with the same ?, GeoPG-B is much faster than APG-B.
More numerical results are provided in the supplementary material, which also indicate that GeoPG-B
is much faster than APG-B, especially when the problems are more ill-conditioned.
2
2
10
4
10
10
GeoPG?B: ? =10?3
GeoPG?B: ? =10?3
APG?B: ? =10?3
0
10
10
GeoPG?B: ? =10?4
10
?4
10
?6
APG?B: ? =10?4
GeoPG?B: ? =10?5
APG?B: ? =10?5
APG?B: ? =10
*
(Fk?F*)/F*
APG?B: ? =10
GeoPG?B: ? =10?4
?5
?4
?2
10
*
?5
0
10
GeoPG?B: ? =10?5
?2
(Fk?F )/F
10
(Fk?F*)/F*
APG?B: ? =10?4
GeoPG?B: ? =10?5
APG?B: ? =10?3
2
10
GeoPG?B: ? =10?4
APG?B: ? =10?4
?2
GeoPG?B: ? =10?3
APG?B: ? =10?3
0
10
?4
10
?6
10
10
?6
10
?8
?8
10
10
?10
10
?8
10
?10
0
10
20
30
40
50
CPU(s)
60
70
80
10
90
?10
0
1
2
3
4
CPU(s)
5
6
7
10
8
0
500
1000
1500
2000
CPU(s)
2500
3000
3500
Figure 2: GeoPG-B and APG-B for solving (5.2) with ? = 10?8 . Left: dataset a9a; Middle: dataset
RCV1; Right: dataset Gisette.
5.3
Numerical results of L-GeoPG-B
In this subsection, we test GeoPG with limited memory described in Algorithm 5 in solving (5.2)
on the Gisette dataset. Since we still need to use the backtracking technique, we actually tested
L-GeoPG-B. The results with different memory sizes m are reported in Figure 3. Note that m = 0
corresponds to the original GeoPG-B without memory. The subproblem (4.4) is solved using the
function ?quadprog? in Matlab. From Figure 3 we see that roughly speaking, L-GeoPG-B performs
better for larger memory sizes, and in most cases, the performance of L-GeoPG-B with m = 100
is the best among the reported results. This indicates that the limited-memory idea indeed helps
improve the performance of GeoPG.
2
2
10
4
10
memorysize=0
memorysize=5
memorysize=20
memorysize=100
0
10
10
memorysize=0
memorysize=5
memorysize=20
memorysize=100
0
10
memorysize=0
memorysize=5
memorysize=20
memorysize=100
2
10
0
?2
?6
?4
?2
10
*
10
(Fk?F )/F
?4
*
10
(Fk?F*)/F*
(Fk?F*)/F*
10
?2
10
10
?4
10
?6
10
10
?6
10
?8
?8
10
10
?10
10
?8
10
?10
0
50
100
150
CPU(s)
200
250
10
?10
0
50
100
150
200
CPU(s)
250
300
350
10
0
50
100
150
200
CPU(s)
250
300
350
400
Figure 3: L-GeoPG-B for solving (5.2) on the dataset Gisette with ? = 10?8 . Left: ? = 10?3 ;
Middle: ? = 10?4 ; Right: ? = 10?5 .
6
Conclusions
In this paper, we proposed a GeoPG algorithm for solving nonsmooth convex composite problems,
which is an extension of the recent method GeoD that can only handle smooth problems. We proved
that GeoPG enjoys the same optimal rate as Nesterov?s accelerated gradient method for solving
strongly convex problems. The backtracking technique was adopted to deal with the case when the
Lipschitz constant is unknown. Limited-memory GeoPG was also developed to improve the practical
performance of GeoPG. Numerical results on linear regression and logistic regression with elastic net
regularization demonstrated the efficiency of GeoPG. It would be interesting to see how to extend
GeoD and GeoPG to tackle non-strongly convex problems, and how to further accelerate the running
time of GeoPG. We leave these questions in future work.
Acknowledgements. Shiqian Ma is supported by a startup funding in UC Davis.
8
References
[1] S. Bubeck, Y.-T. Lee, and M. Singh. A geometric alternative to Nesterov?s accelerated gradient
descent. arXiv preprint arXiv:1506.08187, 2015.
[2] Y. E. Nesterov. A method for unconstrained convex minimization problem with the rate of
convergence O(1/k 2 ). Dokl. Akad. Nauk SSSR, 269:543?547, 1983.
[3] Y. E. Nesterov. Introductory lectures on convex optimization: A basic course. Applied
Optimization. Kluwer Academic Publishers, Boston, MA, 2004. ISBN 1-4020-7553-7.
[4] W. Su, S. Boyd, and E. J. Cand?s. A differential equation for modeling Nesterov?s accelerated
gradient method: Theory and insights. In NIPS, 2014.
[5] H. Attouch, Z. Chbani, J. Peypouquet, and P. Redont. Fast convergence of inertial dynamics
and algorithms with asymptotic vanishing viscosity. Mathematical Programming, 2016.
[6] L. Lessard, B. Recht, and A. Packard. Analysis and design of optimization algorithms via
integral quadratic constraints. SIAM Journal on Optimization, 26(1):57?95, 2016.
[7] A. Wibisono, A. Wilson, and M. I. Jordan. A variational perspective on accelerated methods in
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[8] R. G. Bland, D. Goldfarb, and M. J. Todd. The ellipsoid method: A survey. Operations
Research, 29:1039?1091, 1981.
[9] S. Bubeck and Y.-T. Lee. Black-box optimization with a politician. ICML, 2016.
[10] D. Drusvyatskiy, M. Fazel, and S. Roy. An optimal first order method based on optimal quadratic
averaging. SIAM Journal on Optimization, 2016.
[11] R. P. Brent. An algorithm with guaranteed convergence for finding a zero of a function. In
Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973.
[12] T. J. Dekker. Finding a zero by means of successive linear interpolation. In Constructive Aspects
of the Fundamental Theorem of Algebra. London: Wiley-Interscience, 1969.
[13] M. Gerdts, S. Horn, and S. Kimmerle. Line search globalization of a semismooth Newton
method for operator equations in Hilbert spaces with applications in optimal control. Journal of
Industrial And Management Optimization, 13(1):47?62, 2017.
[14] E. Hans and T. Raasch. Global convergence of damped semismooth Newton methods for L1
Tikhonov regularization. Inverse Problems, 31(2):025005, 2015.
[15] A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse
problems. SIAM J. Imaging Sciences, 2(1):183?202, 2009.
[16] K. Scheinberg, D. Goldfarb, and X. Bai. Fast first-order methods for composite convex
optimization with backtracking. Foundations of Computational Mathematics, 14(3):389?417,
2014.
[17] Y. E. Nesterov. Gradient methods for minimizing composite functions. Mathematical Programming, 140(1):125?161, 2013.
[18] A. Beck. On the convexity of a class of quadratic mappings and its application to the problem of
finding the smallest ball enclosing a given intersection of balls. Journal of Global Optimization,
39(1):113?126, 2007.
[19] Y. C. Eldar, A. Beck, and M. Teboulle. A minimax Chebyshev estimator for bounded error
estimation. IEEE Transactions on Signal Processing, 56(4):1388?1397, 2008.
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Royal Statistical Society, Series B, 67(2):301?320, 2005.
9
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6,264 | 6,667 | Diffusion Approximations for Online Principal
Component Estimation and Global Convergence
Chris Junchi Li
Mengdi Wang
Princeton University
Department of Operations Research and Financial Engineering, Princeton, NJ 08544
{junchil,mengdiw}@princeton.edu
Tong Zhang
Tencent AI Lab
Shennan Ave, Nanshan District, Shenzhen, Guangdong Province 518057, China
[email protected]
Abstract
In this paper, we propose to adopt the diffusion approximation tools to study the
dynamics of Oja?s iteration which is an online stochastic gradient descent method
for the principal component analysis. Oja?s iteration maintains a running estimate
of the true principal component from streaming data and enjoys less temporal
and spatial complexities. We show that the Oja?s iteration for the top eigenvector
generates a continuous-state discrete-time Markov chain over the unit sphere. We
characterize the Oja?s iteration in three phases using diffusion approximation and
weak convergence tools. Our three-phase analysis further provides a finite-sample
error bound for the running estimate, which matches the minimax information
lower bound for principal component analysis under the additional assumption of
bounded samples.
1
Introduction
In the procedure of Principal Component Analysis (PCA) we aim at learning the principal leading
eigenvector of the covariance matrix of a d-dimensional random vector Z from its independent
and identically distributed realizations Z1 , . . . , Zn . Let E[Z] = 0, and let the eigenvalues of ? be
?1 > ?2 ? ? ? ? ? ?d > 0, then the PCA problem can be formulated as minimizing the expectation of
a nonconvex function:
minimize ? w> E ZZ > w,
(1.1)
subject to kwk = 1, w ? Rd ,
where k ? k denotes the Euclidean norm. Since the eigengap ?1 ? ?2 is nonzero, the solution to
(1.1) is unique, denoted by w? . The classical method of finding the estimator of the first leading
eigenvector w? can be formulated as the solution to the empirical covariance problem as
n
>
X
b (n) w,
b (n) ? 1
b (n) = argmin ?w> ?
w
where ?
Z (i) Z (i) .
n i=1
kwk=1
b (n) denotes the empirical covariance matrix for the first n samples. The estimator w
b (n)
In words, ?
b (n) . Precisely, [43] shows that the
produced via this process provides a statistical optimal solution w
e (n) that is a function of the first n samples and w? has the following
angle between any estimator w
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Left: an objective function for the top-1 PCA, where we use both the radius and heatmap to
represent the function value at each point of the unit sphere. Right: A quiver plot on the unit sphere
denoting the directions of negative gradient of PCA objective.
minimax lower bound
h
i
d?1
e (n) , w? ) ? c ? ??2 ?
E sin2 ?(w
,
n
e (n) Z?M(? 2 ,d)
w
?
inf
sup
(1.2)
e (n) is taken over all principal eigenvector
where c is some positive constant. Here the infimum of w
2
estimators, and M(?? , d) is the collection of all d-dimensional subgaussian distributions with mean
zero and eigengap ?1 ? ?2 > 0 satisfying ?1 ?2 /(?1 ? ?2 )2 ? ??2 . Classical PCA method has time
complexity O(nd2 ) and space complexity O(d2 ). The drawback of this method is that, when the
data samples are high-dimensional, computing and storage of a large empirical covariance matrix can
be costly.
In this paper we concentrate on the streaming or online method for PCA that processes online data
and estimates the principal component sequentially without explicitly computing and storing the
b Over thirty years ago, Oja [30] proposed an online PCA iteration that
empirical covariance matrix ?.
can be regarded as a projected stochastic gradient descent method as
h
i
w(n) = ? w(n?1) + ?Z (n) (Z (n) )> w(n?1) .
(1.3)
Here ? is some positive learning rule or stepsize, and ? is defined as ?w = kwk?1 w for each
nonzero vector w, namely, ? projects any vector onto the unit sphere S d?1 = {w ? Rd | kwk = 1}.
Oja?s iteration enjoys a less expensive time complexity O(nd) and space complexity O(d) and
thereby has been used as an alternative method for PCA when both the dimension d and number of
samples n are large.
In this paper, we adopt the diffusion approximation method to characterize the stochastic algorithm
using Markov processes and its differential equation approximations. The diffusion process approximation is a fundamental and powerful analytic tool for analyzing complicated stochastic process. By
leveraging the tool of weak convergence, we are able to conduct a heuristic finite-sample analysis of
the Oja?s iteration and obtain a convergence rate which, by carefully choosing the stepsize ?, matches
the PCA minimax information lower bound. Our analysis involves the weak convergence theory for
Markov processes [11], which is believed to have a potential for a broader class of stochastic algorithms for nonconvex optimization, such as tensor decomposition, phase retrieval, matrix completion,
neural network, etc.
Our Contributions We provide a Markov chain characterization of the stochastic process {w(n) }
generated by the Oja?s iteration with constant stepsize. We show that upon appropriate scalings, the
iterates as a Markov process weakly converges to the solution of an ordinary differential equation
system, which is a multi-dimensional analogue to the logistic equations. Also locally around the
neighborhood of a stationary point, upon a different scaling the process weakly converges to the
multidimensional Ornstein-Uhlenbeck processes. Moreover, we identify from differential equation
approximations that the global convergence dynamics of the Oja?s iteration has three distinct phases:
2
Figure 2: A simulation plot of Oja?s method, marked with the three phases.
(i) The initial phase corresponds to escaping from unstable stationary points;
(ii) The second phase corresponds to fast deterministic crossing period;
(iii) The third phase corresponds to stable oscillation around the true principal component.
Lastly, this is the first work that analyze the global rate of convergence analysis of Oja?s iteration,
i.e., the convergence rate does not have any initialization requirements.
Related Literatures This paper is a natural companion to paper by the authors? recent work [23]
that gives explicit rate analysis using a discrete-time martingale-based approach. In this paper, we
provide a much simpler and more insightful heuristic analysis based on diffusion approximation
method under the additional assumption of bounded samples.
The idea of stochastic approximation for PCA problem can be traced back to Krasulina [19] published
almost fifty years ago. His work proposed an algorithm that is regarded as the stochastic gradient
descent method for the Rayleigh quotient. In contrast, Oja?s iteration can be regarded as a projected
stochastic gradient descent method. The method of using differential equation tools for PCA appeared
in the first papers [19, 31] to prove convergence result to the principal component, among which, [31]
also analyze the subspace learning for PCA. See also [16, Chap. 1] for a gradient flow dynamical
system perspective of Oja?s iteration.
The convergence rate analysis of the online PCA iteration has been very few until the recent big data
tsunami, when the need to handle massive amounts of data emerges. Recent works by [6, 10, 17, 34]
study the convergence of online PCA from different perspectives, and obtain some useful rate results.
Our analysis using the tools of diffusion approximations suggests a rate that is sharper than all existing
results, and our global convergence rate result poses no requirement for initialization.
More Literatures Our work is related to a very recent line of work [3, 13, 21, 33, 38?41] on
the global dynamics of nonconvex optimization with statistical structures. These works carefully
characterize the global geometry of the objective functions, and in special, around the unstable
stationary points including saddle points and local maximizers. To solve the optimization problem
various algorithms were used, including (stochastic) gradient method with random initialization or
noise injection as well as variants of Newton?s method. The unstable stationary points can hence be
avoided, enabling the global convergence to desirable local minimizers.
Our diffusion process-based characterization of SGD is also related to another line of work [8, 10, 24,
26, 37]. Among them, [10] uses techniques based on martingales in discrete time to quantify the global
3
convergence of SGD on matrix decomposition problems. In comparison, our techniques are based on
Stroock and Varadhan?s weak convergence of Markov chains to diffusion processes, which yield the
continuous-time dynamics of SGD. The rest of these results mostly focus on analyzing continuoustime dynamics of gradient descent or SGD on convex optimization problems. In comparison, we are
the first to characterize the global dynamics for nonconvex statistical optimization. In particular, the
first and second phases of our characterization, especially the unstable Ornstein-Uhlenbeck process,
are unique to nonconvex problems. Also, it is worth noting that, using the arguments of [26], we can
show that the diffusion process-based characterization admits a variational Bayesian interpretation of
nonconvex statistical optimization. However, we do not pursue this direction in this paper.
In the mathematical programming and statistics communities, the computational and statistical
aspects of PCA are often studied separately. From the statistical perspective, recent developments
have focused on estimating principal components for very high-dimensional data. When the data
dimension is much larger than the sample size, i.e., d n, classical method using decomposition of
the empirical convariance matrix produces inconsistent estimates [18, 29]. Sparsity-based methods
have been studied, such as the truncated power method studied by [45] and [44]. Other sparsity
regularization methods for high dimensional PCA has been studied in [2, 7, 9, 18, 25, 42, 43, 46], etc.
Note that in this paper we do not consider the high-dimensional regime and sparsity regularization.
From the computational perspective, power iterations or the Lanczos method are well studied. These
iterative methods require performing multiple products between vectors and empirical covariance
matrices. Such operation usually involves multiple passes over the data, whose complexity may scale
with the eigengap and dimensions [20, 28]. Recently, randomized algorithms have been developed to
reduce the computation complexity [12, 35, 36]. A critical trend today is to combine the computational
and statistical aspects and to develop algorithmic estimator that admits fast computation as well as
good estimation properties. Related literatures include [4, 5, 10, 14, 27].
Organization ?2 introduces the settings and distributional assumptions. ?3 briefly discusses the
Oja?s iteration from the Markov processes perspective and characterizes that it globally admits
ordinary differential equation approximation upon appropriate scaling, and also stochastic differential
equation approximation locally in the neighborhood of each stationary point. ?4 utilizes the weak
convergence results and provides a three-phase argument for the global convergence rate analysis,
which is near-optimal for the Oja?s iteration. Concluding remarks are provided in ?5.
2
Settings
In this section, we present the basic settings for the Oja?s iteration. The algorithm maintains a running
estimate w(n) of the true principal component w? , and updates it while receiving streaming samples
from exterior data source. We summarize our distributional assumptions.
Assumption 2.1. The random vectors Z ? Z (1) , . . . , Z (n) ? Rd are independent and identically
distributed and have the following properties:
(i) E[Z] = 0 and E ZZ > = ?;
(ii) ?1 > ?2 ? ? ? ? ? ?d > 0;
(iii) There is a constant B such that kZk2 ? B.
For the easiness of presentation, we transform the iterates w(n) and define the rescaled samples, as
follows. First we let the eigendecomposition of the covariance matrix be
? = E ZZ > = U?U> ,
where ? = diag(?1 , ?2 , . . . , ?d ) is a diagonal matrix with diagonal entries ?1 , ?2 , . . . , ?d , and U is
an orthogonal matrix consisting of column eigenvectors of ?. Clearly the first column of U is equal
to the principal component w? . Note that the diagonal decomposition might not be unique, in which
case we work with an arbitrary one. Second, let
Y (n) = U> Z (n) , v(n) = U> w(n) , v? = U> w? .
One can easily verify that
E[Y ] = 0,
E Y Y > = ?;
4
(2.1)
The principal component of the rescaled random variable Y , which we denote by v? , is equal to e1 ,
where {e1 , . . . , ed } is the canonical basis of Rd . By applying the orthonormal transformation U>
to the stochastic process {w(n) }, we obtain an iterative process {v(n) = U> w(n) } in the rescaled
space:
>
(n)
> (n)
> (n?1)
> (n)
(n)
> (n?1)
v =U w =? U w
+ ?U Z
Z
UU w
(2.2)
>
(n?1)
(n)
(n)
(n?1)
=? v
+ ?Y
Y
v
.
Moreover, the angle processes associated with {w(n) } and {v(n) } are equivalent, i.e.,
?(w(n) , w? ) = ?(v(n) , v? ).
Therefore it would be sufficient to study the rescaled iteration v
iteration Y (n) throughout the rest of this paper.
3
(2.3)
(n)
in (2.2) and the transformed
A Theory of Diffusion Approximation for PCA
In this section we show that the stochastic iterates generated by the Oja?s iteration can be approximated
by the solution of an ODE system upon appropriate scaling, as long as ? is small. To work on the
approximation we first observe that the iteration v(n) , n = 0, 1, . . . generated by (2.2) forms a
discrete-time, time-homogeneous Markov process that takes values on S d?1 . Furthermore, v(n)
holds strong Markov property.
3.1
Global ODE Approximation
To state our results on differential equation approximations, let us define a new process, which is
obtained by rescaling the time index n according to the stepsize ?
?1
Ve ? (t) ? v?,(bt? c) .
(3.1)
We add the superscript ? in the notation to emphasize the dependence of the process on ?. We will
show that Ve ? (t) converges weakly to a deterministic function V (t), as ? ? 0+ .
Furthermore, we can identify the limit V (t) as the closed-form solution to an ODE system. Under
Assumption 2.1 and using an infinitesimal generator analysis we have
?
Ve (t + ?) ? Ve ? (t) = O(B?).
It follows that, as ? ? 0+ , the infinitesimal conditional variance tends to 0:
h
i
? ?1 var Ve ? (t + ?) ? Ve ? (t) Ve ? (t) = v = O(B?),
and the infinitesimal
h mean is
i
?1
? E Ve ? (t + ?) ? Ve ? (t) Ve ? (t) = v = ? ? V > ?V V + O(B 2 ? 2 ).
Using the classical weak convergence to diffusion argument [11, Corollary 4.2 in ?7.4], we obtain
the following result.
Theorem 3.1. If v?,(0) converges weakly to some constant vector V o ? S d?1 as ? ? 0+ then the
?1
Markov process v?,(bt? c) converges weakly to the solution V = V (t) to the following ordinary
differential equation system
dV
= ? ? V > ?V V ,
(3.2)
dt
with initial values V (0) = V o .
We can straightforwardly check for sanity that the solution vector V (t) lies on the unit sphere
S d?1 , i.e., kV (t)k = 1 for all t ? 0. Written in coordinates V (t) = (V1 (t), . . . , Vd (t))> , the ODE
is expressed for k = 1, . . . , d
d
X
dVk
= Vk
(?k ? ?i )Vi2 .
dt
i=1
5
One can straightforwardly verify that the solution to (3.2) has
Vk (t) = (Z(t))
where Z(t) is the normalization function
Z(t) =
d
X
?1/2
Vk (0) exp(?k t),
(3.3)
2
(Vio ) exp(2?i t).
i=1
To understand the limit function given by (3.3), we note that in the special case where ?2 = ? ? ? = ?d
2
2
Z(t) = (V1o ) exp(2?1 t) + 1 ? (V1o ) exp(2?2 t),
and
2
2
(V1 (t)) =
(V1o ) exp(2?1 t)
.
2
2
(V1o ) exp(2?1 t) + 1 ? (V1o ) exp(2?2 t)
(3.4)
This is the formula of the logistic curve. Hence analogously, V (t) in (3.3) is namely the generalized
logistic curves.
3.2
Local Approximation by Diffusion Processes
The weak convergence to ODE theorem introduced in ?3.1 characterizes the global dynamics of the
Oja?s iteration. Such approximation explains many behaviors, but neglected the presence of noise that
plays a role in the algorithm. In this section we aim at understanding the Oja?s iteration via stochastic
differential equations (SDE). We refer the readers to [32] for more on basic concepts of SDE.
In this section, we instead show that under some scaling, the process admits an approximation
of multidimensional Ornstein-Uhlenbeck process within a neighborhood of each of the unstable
stationary points, both stable and unstable. Afterwards, we develop some weak convergence results
to give a rough estimate on the rate of convergence of the Oja?s iteration. For purposes of illustration
and brevity, we restrict ourselves to the case of starting point v(0) being the stationary point ek for
some k = 1, . . . , d, and denote an arbitrary vector xk to be a (d ? 1)-dimensional vector that keeps
all but the kth coordinate of x. Using theory from [11] we conclude the following theorem.
?,(0)
Theorem 3.2. Let k = 1, . . . , d be arbitrary. If ? ?1/2 vk
as ? ? 0+ , then the Markov process
converges weakly to some Uok ? Rd?1
?,(bt? ?1 c)
? ?1/2 vk
converges weakly to the solution of the multidimensional stochastic differential equation
1/2
dUk (t) = ?(?k Id?1 ? ?k )Uk dt + ?k ?k
dBk (t),
with initial values Uk (0) =
Uok .
(3.5)
Here Bk (t) is a standard (d ? 1)-dimensional Brownian motion.
1
The solution to (3.5) can be solved explicitly. We let for a matrix A ? Rn?n the matrix
expoP?
1/2
1/2
n
1/2
nentiation exp(A) as exp(A) = n=0 (1/n!)A . Also, let ?
= diag ?1 , . . . , ?d
for the
positive semidefinite diagonal matrix ? = diag(?1 , . . . , ?d ). The solution to (3.5) is hence
Z
1/2 t
exp (s ? t)(?k Id?1 ? ?k ) dBk (s),
Uk (t) = exp ?t(?k Id?1 ? ?k ) Uko + ?k ?k
0
which is known as the multidimensional Ornstein-Uhlenbeck process, whose behavior depends on
the matrix ?(?k Id?1 ? ?k ) and is discussed in details in ?4.
Before concluding this section, we emphasize that the weak convergence to diffusions results in
?3.1 and ?3.2 should be distinguished from the convergence of the Oja?s iteration. From a random
process theoretical perspective, the former one treats the weak convergence of finite dimensional
distributions of a sequence of rescaled processes as ? tends to 0, while the latter one charaterizes the
long-time behavior of a single realization of iterates generated by algorithm for a fixed ? > 0.
1
The reason we have a (d ? 1)-dimensional Ornstein-Uhlenbeck process is because the objective function
of PCA is defined on a (d ? 1)-dimensional manifold S d?1 and has d ? 1 independent variables.
6
4
Global Three-Phase Analysis of Oja?s Dynamics
Previously ?3.1 and ?3.2 develop the tools of weak convergence to diffusion under global and local
scalings. In this section, we apply these tools to analyze the dynamics of online PCA in three phases
in sequel. For purposes of illustration and brevity, we restrict ourselves to the case of starting point
v(0) that is near a saddle point ek . Let A? . B ? denotes lim sup??0+ A? /B ? ? 1, a.s., and
A? B ? when both A? . B ? and B ? . A? hold.
4.1
Phase I: Noise Initialization
In consideration of global convergence, we need to analyze the initial phase approximation by
considering restrictions on initialization: what happens if the starting point is in a small neighborhood
of the following set
Se = v ? S d?1 : v1 = 0 .
When thinking the sphere S d?1 as the globe with ?e1 as north and south poles, Se corresponds to
the equator of the globe. Therefore, all unstable stationary points (including saddle points and local
maximizers) e2 , . . . , ed lies on the equator Se .
4.2
Phase II: Deterministic Crossing
In Phase II, the iteration escapes from the neighborhood of equator Se and converges to a basin of
attraction of the local minimizer v? . From strong Markov property of the Oja?s iteration introduced
in the beginning of ?2, one can forget the iteration steps in Phase I and analyze the algorithm from its
(0)
output. Suppose we have an initial point v(0) that satisfies (v1 )2 = ?. If we fix ? ? (0, 1/2) and the
initial value V (0) = V o that satisfies V12 = ?. Theorem 3.1 concludes that the iteration moves in a
deterministic pattern.
4.3
Phase III: Convergence to Principal Component
In Phase III, the iteration fluctuates around the true principal component v? = e1 . We start our it(0)
eration from a neighborhood around the principal component, where v(0) has (v1 )2 = 1 ? ?.
Letting
(3.5) and taking the limit t ? ?,
we have the limit EkU1 (?)k2 =
k = 1 in
tr E U1 (t)U1 (t)> = (?1 /2) tr ?1 (?1 Id?1 ? ?1 )?1 . In terms of the iterations v(n) , rescaling
the Markov process along with some calculations gives as n ? ?, in very rough sense,
d
X
?1 ?k
?1
tr ?1 (?1 Id?1 ? ?1 )?1 = ? ?
.
2
2(?1 ? ?k )
k=2
(4.1)
The above display implies that there will be some nondiminishing fluctuations, variance being
proportional to the constant stepsize ?, as time goes to infinity or at stationarity. Therefore in terms
of angle, at stationarity the Markov process concentrates on O(? 1/2 ) scale around the origin.
E sin2 ?(v(n) , v? ) ? ? EkU1 (?)k2 = ? ?
4.4
Crossing Time Estimate
We turn to estimate the running time, namely the crossing time, that is required for the algorithm to
cross the corresponding region in different phases.
Phase I. In the first phase, the crossing time is the number of iterates required for the algorithm
to escape an O(1)-neighborhood of the equator Se . For illustrative purposes we only consider the
special case where v is ek the kth coordinate vector, which is a saddle point that has a negative
Hessian eigenvalue. In this situation, the SDE (3.5) in terms of the first coordinate U (t) of Uk
reduces to
1/2
dU (t) = (?1 ? ?k )U (t) dt + (?1 ?k )
dB(t),
(4.2)
with initial value U (0) = 0. Solution to (4.2) is known as unstable Ornstein-Uhlenbeck process [1]
and can be expressed explicitly in closed-form, as
Z t
1/2
U (t) = W ? exp ((?1 ? ?k )t) , where W ? ? (?1 ?k )
exp (?(?1 ? ?k )s) dB(s).
0
7
Rescaling the time back to the discrete algorithm, we let n = t? ?1 and obtain
(n)
v1
? 1/2 W ? exp (?(?1 ? ?k )n) .
(4.3)
?
In (4.3), the term W is approximately distributed as t = n? ? ?
1/2
?1 ?k
?
?,
W
2(?1 ? ?k )
where ? stands for a standard normal variable. We have
1/2
?1 ?k
(n)
1/2
?
1/2
v1 = ? W ?
? exp (?(?1 ? ?k )n) .
2(?1 ? ?k )
(4.4)
(n)
In order to have (v1 )2 = ? in (4.4), we have as ? ? 0+ the crossing time is approximately
?1/2
?1 ?d
?1 ?1
?1 ?1
?
?1
?1/2
?
N1 (?1 ? ?k ) ? log ?|?|
+ (?1 ? ?k ) ? log
.
2(?1 ? ?d )
(4.5)
Therefore we have whenever the smallest
eigenvalue
?
is
bounded
away
from
0,
then
asymptotically
d
?1
N1? 0.5 (?1 ? ?k ) ? ?1 log ? ?1 . This suggests that the noise helps initial location to diffuse
away from ek rapidly.
Phase II. We turn to estimate the crossing time N2? in Phase II. Simple calculation ensures the
existence of a constant T , that depends only on ? such that V12 (T ) ? 1 ? ?. Furthermore T has the
following bounds:
1??
1??
?
?1
?1
. N2 . (?1 ? ?2 ) log
.
(4.6)
(?1 ? ?d ) log
?
?
In special, the time it takes is asymptotically N2? . (?1 ? ?2 )?1 ? ?1 log ((1 ? ?)/?) iterates to have
(n)
v1 ? 1 ? ?. Theorem 3.1 indicates that when ? is positively small, the iterates needed for the first
coordinate squared to traverse from ? to 1 ? ? is of O(? ?1 ). This is substantiated by simulation
results [4] suggesting that the Oja?s iteration is fast from the warm initialization.
Phase III. To estimate the crossing time N3? or the number of iterates needed in Phase III, we restart
our counter and have from the approximation in Theorem 3.2 and (3.5) that
Z ?n
(n)
(0)
E(vk )2 = (vk )2 exp (?2(?1 ? ?k )?n) + ??1 ?k
exp (?2(?1 ? ?k )(t ? s)) ds
0
=??
d
X
k=2
??
d
X
k=2
?1 ?k
+
2(?1 ? ?k )
d
X
(0)
(vk )2 ? ? ?
k=2
?1 ?k
2(?1 ? ?k )
exp (?2?(?1 ? ?k )n)
?1 ?k
+ ? exp (?2?(?1 ? ?2 )n) .
2(?1 ? ?k )
Pd
In terms of the iterations v(n) , note the relationship E sin2 ?(v, e1 ) = k=2 vk2 = 1 ? v12 . The end
(0)
of Phase II implies that E sin2 ?(v(0) , e1 ) = 1 ? (v1 )2 = ?, and hence by setting
?
E sin2 ?(v(N3 ) , e1 ) = ? ?
d
X
k=2
?1 ?k
+ o(?),
2(?1 ? ?k )
+
we conclude that as ? ? 0
N3? (0.5 + o(1)) ? (?1 ? ?2 )?1 ? ?1 log ?? ?1 .
4.5
(4.7)
Finite-Sample Rate Bound
In this subsection we attempt to establish the global rate of convergence analysis using the crossing
time estimates in the previous subsection. Starting from v(0) = ek where k = 2, . . . , d is arbitrary,
the global convergence time N ? = N1? + N2? + N3? as ? ? 0+ such that, by choosing ? ? (0, 1/2)
as a small fixed constant,
?1
N ? (?1 ? ?2 ) ? ?1 log ? ?1 ,
8
with the following estimation on global convergence rate as in (4.1)
?
sin2 ?(v(N ) , v? ) = ? ?
d
X
k=2
?1 ?k
.
2(?1 ? ?k )
Given a fixed number of samples T , by choosing ? as
log T
(?1 ? ?2 )T
? )?
? = ?(T
(4.8)
? )
? )?1 log ?(T
? ) ?1 = N ?(T
we have T (?1 ? ?2 )?1 ?(T
. Plugging in ? in (4.8) we have, by the
angle-preserving property of coordinate transformation (2.3), that
E sin2 ?(w(N
?
?(T
)
)
, w? ) = E sin2 ?(v(N
?
?(T
)
)
, v? ) ?
d
X
k=2
?1 ?k
log T
?
.
2(?1 ? ?k ) (?1 ? ?2 )T
(4.9)
The finite sample bound in (4.9) is better than any existing results and matches the information lower
bound. Moreover, (4.9) implies that the rate in terms of sine-squared angle is sin2 ?(w(T ) , w? ) ?
C ? ?1 ?2 /(?1 ? ?2 )2 ? d log T /T, which matches the minimax information lower bound (up to a log T
factor), see for example, Theorem 3.1 of [43]. Limited by space, details about the rate comparison is
provided in the supplementary material.
5
Concluding Remarks
We make several concluding remarks on the global convergence rate estimations, as follows.
Crossing Time Comparison. From the crossing time estimates in (4.5), (4.6), (4.7) we conclude
(i) As ? ? 0+ we have N2? /N1? ? 0. This implies that the algorithm demonstrates the cutoff
phenomenon which frequently occur in discrete-time Markov processes [22]. In words,
the Phase II where the objective value in Rayleigh quotient drops from 1 ? ? to ? is an
asymptotically a phase of short time, compared to Phases I and III, so the convergence curve
occurs instead of an exponentially decaying curve.
(ii) As ? ? 0+ we have N3? /N1? 1. This suggests that for the high-d case that Phase I of
escaping from the equator consumes roughly the same iterations as in Phase III.
To summarize from above, the cold initialization iteration roughly takes twice the number of steps
than the warm initialization version which is consistent with the simulation discussions in [31].
Subspace Learning. In this work we primarily concentrates on the problem of finding the top1 eigenvector. It is believed that the problem of finding top-k eigenvectors, a.k.a. the subspace
PCA problem, can be analyzed using our approximation methods. This will involve a careful
characterization of subspace angles and is hence more complex and is left for further investigation.
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11
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6,265 | 6,668 | Avoiding Discrimination through Causal Reasoning
Niki Kilbertus??
[email protected]
Moritz Hardt?
[email protected]
Mateo Rojas-Carulla??
[email protected]
Dominik Janzing?
[email protected]
Giambattista Parascandolo??
[email protected]
Bernhard Sch?olkopf?
[email protected]
?
Max Planck Institute for Intelligent Systems
?
University of Cambridge
?
Max Planck ETH Center for Learning Systems
?
University of California, Berkeley
Abstract
Recent work on fairness in machine learning has focused on various statistical
discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute,
features, and outcome. While convenient to work with, observational criteria have
severe inherent limitations that prevent them from resolving matters of fairness
conclusively.
Going beyond observational criteria, we frame the problem of discrimination
based on protected attributes in the language of causal reasoning. This viewpoint shifts attention from ?What is the right fairness criterion?? to ?What do we
want to assume about our model of the causal data generating process?? Through
the lens of causality, we make several contributions. First, we crisply articulate
why and when observational criteria fail, thus formalizing what was before a matter of opinion. Second, our approach exposes previously ignored subtleties and
why they are fundamental to the problem. Finally, we put forward natural causal
non-discrimination criteria and develop algorithms that satisfy them.
1
Introduction
As machine learning progresses rapidly, its societal impact has come under scrutiny. An important
concern is potential discrimination based on protected attributes such as gender, race, or religion.
Since learned predictors and risk scores increasingly support or even replace human judgment, there
is an opportunity to formalize what harmful discrimination means and to design algorithms that
avoid it. However, researchers have found it difficult to agree on a single measure of discrimination.
As of now, there are several competing approaches, representing different opinions and striking
different trade-offs. Most of the proposed fairness criteria are observational: They depend only
on the joint distribution of predictor R, protected attribute A, features X, and outcome Y. For
example, the natural requirement that R and A must be statistically independent is referred to as
demographic parity. Some approaches transform the features X to obfuscate the information they
contain about A [1]. The recently proposed equalized odds constraint [2] demands that the predictor
R and the attribute A be independent conditional on the actual outcome Y. All three are examples
of observational approaches.
A growing line of work points at the insufficiency of existing definitions. Hardt, Price and Srebro [2]
construct two scenarios with intuitively different social interpretations that admit identical joint dis31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
tributions over (R, A, Y, X). Thus, no observational criterion can distinguish them. While there
are non-observational criteria, notably the early work on individual fairness [3], these have not yet
gained traction. So, it might appear that the community has reached an impasse.
1.1
Our contributions
We assay the problem of discrimination in machine learning in the language of causal reasoning.
This viewpoint supports several contributions:
? Revisiting the two scenarios proposed in [2], we articulate a natural causal criterion that
formally distinguishes them. In particular, we show that observational criteria are unable
to determine if a protected attribute has direct causal influence on the predictor that is not
mitigated by resolving variables.
? We point out subtleties in fair decision making that arise naturally from a causal perspective, but have gone widely overlooked in the past. Specifically, we formally argue for the
need to distinguish between the underlying concept behind a protected attribute, such as
race or gender, and its proxies available to the algorithm, such as visual features or name.
? We introduce and discuss two natural causal criteria centered around the notion of interventions (relative to a causal graph) to formally describe specific forms of discrimination.
? Finally, we initiate the study of algorithms that avoid these forms of discrimination. Under
certain linearity assumptions about the underlying causal model generating the data, an
algorithm to remove a specific kind of discrimination leads to a simple and natural heuristic.
At a higher level, our work proposes a shift from trying to find a single statistical fairness criterion
to arguing about properties of the data and which assumptions about the generating process are
justified. Causality provides a flexible framework for organizing such assumptions.
1.2
Related work
Demographic parity and its variants have been discussed in numerous papers, e.g., [1, 4?6]. While
demographic parity is easy to work with, the authors of [3] already highlighted its insufficiency as
a fairness constraint. In an attempt to remedy the shortcomings of demographic parity [2] proposed
two notions, equal opportunity and equal odds, that were also considered in [7]. A review of various
fairness criteria can be found in [8], where they are discussed in the context of criminal justice.
In [9, 10] it has been shown that imperfect predictors cannot simultaneously satisfy equal odds and
calibration unless the groups have identical base rates, i.e. rates of positive outcomes.
A starting point for our investigation is the unidentifiability result of [2]. It shows that observedvational criteria are too weak to distinguish two intuitively very different scenarios. However, the
work does not provide a formal mechanism to articulate why and how these scenarios should be
considered different. Inspired by Pearl?s causal interpretation of Simpson?s paradox [11, Section 6],
we propose causality as a way of coping with this unidentifiability result.
An interesting non-observational fairness definition is the notion of individual fairness [3] that assumes the existence of a similarity measure on individuals, and requires that any two similar individuals should receive a similar distribution over outcomes. More recent work lends additional support
to such a definition [12]. From the perspective of causality, the idea of a similarity measure is akin
to the method of matching in counterfactual reasoning [13, 14]. That is, evaluating approximate
counterfactuals by comparing individuals with similar values of covariates excluding the protected
attribute.
Recently, [15] put forward one possible causal definition, namely the notion of counterfactual fairness. It requires modeling counterfactuals on a per individual level, which is a delicate task. Even
determining the effect of race at the group level is difficult; see the discussion in [16]. The goal of
our paper is to assay a more general causal framework for reasoning about discrimination in machine
learning without committing to a single fairness criterion, and without committing to evaluating individual causal effects. In particular, we draw an explicit distinction between the protected attribute
(for which interventions are often impossible in practice) and its proxies (which sometimes can be
intervened upon).
2
Moreover, causality has already been employed for the discovery of discrimination in existing data
sets by [14, 17]. Causal graphical conditions to identify meaningful partitions have been proposed
for the discovery and prevention of certain types of discrimination by preprocessing the data [18].
These conditions rely on the evaluation of path specific effects, which can be traced back all the
way to [11, Section 4.5.3]. The authors of [19] recently picked up this notion and generalized
Pearl?s approach by a constraint based prevention of discriminatory path specific effects arising
from counterfactual reasoning. Our research was done independently of these works.
1.3
Causal graphs and notation
Causal graphs are a convenient way of organizing assumptions about the data generating process.
We will generally consider causal graphs involving a protected attribute A, a set of proxy variables P,
features X, a predictor R and sometimes an observed outcome Y. For background on causal graphs
see [11]. In the present paper a causal graph is a directed, acyclic graph whose nodes represent
random variables. A directed path is a sequence of distinct nodes V1 , . . . , Vk , for k ? 2, such
that Vi ? Vi+1 for all i ? {1, . . . , k ? 1}. We say a directed path is blocked by a set of nodes Z,
where V1 , Vk ?
/ Z, if Vi ? Z for some i ? {2, . . . , k ? 1}.1
A structural equation model is a set of equations Vi = fi (pa(Vi ), Ni ), for i ? {1, . . . , n},
where pa(Vi ) are the parents of Vi , i.e. its direct causes, and the Ni are independent noise variables. We interpret these equations as assignments. Because we assume acyclicity, starting from
the roots of the graph, we can recursively compute the other variables, given the noise variables.
This leads us to view the structural equation model and its corresponding graph as a data generating model. The predictor R maps inputs, e.g., the features X, to a predicted output. Hence we
model it as a childless node, whose parents are its input variables. Finally, note that given the noise
variables, a structural equation model entails a unique joint distribution; however, the same joint
distribution can usually be entailed by multiple structural equation models corresponding to distinct
causal structures.
2
Unresolved discrimination and limitations of observational criteria
To bear out the limitations of observational criteria, we turn to
Pearl?s commentary on claimed gender discrimination in Berkeley college admissions [11, Section 4.5.3]. Bickel [20] had shown
earlier that a lower college-wide admission rate for women than
for men was explained by the fact that women applied in more
competitive departments. When adjusted for department choice,
women experienced a slightly higher acceptance rate compared
with men. From the causal point of view, what matters is the direct effect of the protected attribute (here, gender A) on the decision (here, college admission R) that cannot be ascribed to a resolving variable such as department choice X, see Figure 1. We
shall use the term resolving variable for any variable in the causal
graph that is influenced by A in a manner that we accept as nondiscriminatory. With this convention, the criterion can be stated as
follows.
A
X
R
Figure 1: The admission decision R does not only directly depend on gender A, but
also on department choice X,
which in turn is also affected
by gender A.
Definition 1 (Unresolved discrimination). A variable V in a causal graph exhibits unresolved discrimination if there exists a directed path from A to V that is not blocked by a resolving variable
and V itself is non-resolving.
Pearl?s commentary is consistent with what we call the skeptic viewpoint. All paths from the protected attribute A to R are problematic, unless they are justified by a resolving variable. The presence of unresolved discrimination in the predictor R is worrisome and demands further scrutiny.
In practice, R is not a priori part of a given graph. Instead it is our objective to construct it as a
function of the features X, some of which might be resolving. Hence we should first look for unresolved discrimination in the features. A canonical way to avoid unresolved discrimination in R is to
only input the set of features that do not exhibit unresolved discrimination. However, the remaining
1
As it is not needed in our work, we do not discuss the graph-theoretic notion of d-separation.
3
features might be affected by non-resolving and resolving variables. In Section 4 we investigate
whether one can exclusively remove unresolved discrimination from such features. A related notion
of ?explanatory features? in a non-causal setting was introduced in [21].
The definition of unresolved discrimination in
a predictor has some interesting special cases
R?
R?
worth highlighting. If we take the set of resolving variables to be empty, we intuitively get a
causal analog of demographic parity. No diX1
X2
A
Y
A
Y
rected paths from A to R are allowed, but A
and R can still be statistically dependent. SimiX2
X1
larly, if we choose the set of resolving variables
to be the singleton set {Y } containing the true
outcome, we obtain a causal analog of equal- Figure 2: Two graphs that may generate the same
ized odds where strict independence is not nec- joint distribution for? the Bayes optimal unconessary. The causal intuition implied by ?the strained? predictor R . If X1 is a resolving variprotected attribute should not affect the predic- able, R exhibits unresolved discrimination in the
tion?, and ?the protected attribute can only af- right graph (along the red paths), but not in the left
fect the prediction when the information comes one.
through the true label?, is neglected by (conditional) statistical independences A ?? R, and A ?? R | Y , but well captured by only considering
dependences mitigated along directed causal paths.
We will next show that observational criteria are fundamentally unable to determine whether a predictor exhibits unresolved discrimination or not. This is true even if the predictor is Bayes optimal.
In passing, we also note that fairness criteria such as equalized odds may or may not exhibit unresolved discrimination, but this is again something an observational criterion cannot determine.
Theorem 1. Given a joint distribution over the protected attribute A, the true label Y , and some
features X1 , . . . , Xn , in which we have already specified the resolving variables, no observational
criterion can generally determine whether the Bayes optimal unconstrained predictor or the Bayes
optimal equal odds predictor exhibit unresolved discrimination.
All proofs for the statements in this paper are in the supplementary material.
The two graphs in Figure 2 are taken from [2], which we here reinterpret in the causal context to
prove Theorem 1. We point out that there is an established set of conditions under which unresolved
discrimination can, in fact, be determined from observational data. Note that the two graphs are
not Markov equivalent. Therefore, to obtain the same joint distribution we must violate a condition
called faithfulness.2 We later argue that violation of faithfulness is by no means pathological, but
emerges naturally when designing predictors. In any case, interpreting conditional dependences can
be difficult in practice [22].
3
Proxy discrimination and interventions
We now turn to an important aspect of our framework. Determining causal effects in general requires
modeling interventions. Interventions on deeply rooted individual properties such as gender or
race are notoriously difficult to conceptualize?especially at an individual level, and impossible to
perform in a randomized trial. VanderWeele et al. [16] discuss the problem comprehensively in an
epidemiological setting. From a machine learning perspective, it thus makes sense to separate the
protected attribute A from its potential proxies, such as name, visual features, languages spoken at
home, etc. Intervention based on proxy variables poses a more manageable problem. By deciding on
a suitable proxy we can find an adequate mounting point for determining and removing its influence
on the prediction. Moreover, in practice we are often limited to imperfect measurements of A in any
case, making the distinction between root concept and proxy prudent.
As was the case with resolving variables, a proxy is a priori nothing more than a descendant of A in
the causal graph that we choose to label as a proxy. Nevertheless in reality we envision the proxy
2
If we do assume the Markov condition and faithfulness, then conditional independences determine the
graph up to its so called Markov equivalence class.
4
to be a clearly defined observable quantity that is significantly correlated with A, yet in our view
should not affect the prediction.
Definition 2 (Potential proxy discrimination). A variable V in a causal graph exhibits potential
proxy discrimination, if there exists a directed path from A to V that is blocked by a proxy variable
and V itself is not a proxy.
Potential proxy discrimination articulates a causal criterion that is in a sense dual to unresolved
discrimination. From the benevolent viewpoint, we allow any path from A to R unless it passes
through a proxy variable, which we consider worrisome. This viewpoint acknowledges the fact that
the influence of A on the graph may be complex and it can be too restraining to rule out all but a few
designated features. In practice, as with unresolved discrimination, we can naively build an unconstrained predictor based only on those features that do not exhibit potential proxy discrimination.
Then we must not provide P as input to R; unawareness, i.e. excluding P from the inputs of R,
suffices. However, by granting R access to P , we can carefully tune the function R(P, X) to cancel
the implicit influence of P on features X that exhibit potential proxy discrimination by the explicit
dependence on P . Due to this possible cancellation of paths, we called the path based criterion potential proxy discrimination. When building predictors that exhibit no overall proxy discrimination,
we precisely aim for such a cancellation.
Fortunately, this idea can be conveniently expressed by an intervention on P , which is denoted
by do(P = p) [11]. Visually, intervening on P amounts to removing all incoming arrows of P in
the graph; algebraically, it consists of replacing the structural equation of P by P = p, i.e. we put
point mass on the value p.
Definition 3 (Proxy discrimination). A predictor R exhibits no proxy discrimination based on a
proxy P if for all p, p0
P(R | do(P = p)) = P(R | do(P = p0 )) .
(1)
The interventional characterization of proxy discrimination leads to a simple procedure to remove
it in causal graphs that we will turn to in the next section. It also leads to several natural variants
of the definition that we discuss in Section 4.3. We remark that Equation (1) is an equality of
probabilities in the ?do-calculus? that cannot in general be inferred by an observational method,
because it depends on an underlying causal graph, see the discussion in [11]. However, in some
cases, we do not need to resort to interventions to avoid proxy discrimination.
Proposition 1. If there is no directed path from a proxy to a feature, unawareness avoids proxy
discrimination.
4
Procedures for avoiding discrimination
Having motivated the two types of discrimination that we distinguish, we now turn to building
predictors that avoid them in a given causal model. First, we remark that a more comprehensive
treatment requires individual judgement of not only variables, but the legitimacy of every existing
path that ends in R, i.e. evaluation of path-specific effects [18, 19], which is tedious in practice.
The natural concept of proxies and resolving variables covers most relevant scenarios and allows for
natural removal procedures.
4.1
Avoiding proxy discrimination
While presenting the general procedure, we illustrate each step in the example shown in Figure 3.
A protected attribute A affects a proxy P as well as a feature X. Both P and X have additional
unobserved causes NP and NX , where NP , NX , A are pairwise independent. Finally, the proxy also
has an effect on the features X and the predictor R is a function of P and X. Given labeled training
data, our task is to find a good predictor that exhibits no proxy discrimination within a hypothesis
class of functions R? (P, X) parameterized by a real valued vector ?.
We now work out a formal procedure to solve this task under specific assumptions and simultaneously illustrate it in a fully linear example, i.e. the structural equations are given by
P = ?P A + NP ,
X = ?X A + ?P + NX ,
R? = ?P P + ?X X .
Note that we choose linear functions parameterized by ? = (?P , ?X ) as the hypothesis class
for R? (P, X).
5
NP
NX
A
P
G?
NP
X
P
G
R
NX
A
NE
X
E
R
G?
Figure 3: A template graph G? for proxy discrimination (left) with its intervened version G
(right). While from the benevolent viewpoint we
do not generically prohibit any influence from A
on R, we want to guarantee that the proxy P has
no overall influence on the prediction, by adjusting P ? R to cancel the influence along P ?
X ? R in the intervened graph.
NX
A
X
R
NE
NX
A
E
G
X
R
Figure 4: A template graph G? for unresolved
discrimination (left) with its intervened version G (right). While from the skeptical
viewpoint we generically do not want A to
influence R, we first intervene on E interrupting all paths through E and only cancel
the remaining influence on A to R.
We will refer to the terminal ancestors of a node V in a causal graph D, denoted by taD (V ), which
are those ancestors of V that are also root nodes of D. Moreover, in the procedure we clarify the
notion of expressibility, which is an assumption about the relation of the given structural equations
and the hypothesis class we choose for R? .
Proposition 2. If there is a choice of parameters ?0 such that R?0 (P, X) is constant with respect
to its first argument and the structural equations are expressible, the following procedure returns a
predictor from the given hypothesis class that exhibits no proxy discrimination and is non-trivial in
the sense that it can make use of features that exhibit potential proxy discrimination.
1. Intervene on P by removing all incoming arrows and replacing the structural equation for P
by P = p. For the example in Figure 3,
P = p,
X = ?X A + ?P + NX ,
R? = ?P P + ?X X .
(2)
2. Iteratively substitute variables in the equation for R? from their structural equations until only
root nodes of the intervened graph are left, i.e. write R? (P, X) as R? (P, g(taG (X))) for some
function g. In the example, ta(X) = {A, P, NX } and
R? = (?P + ?X ?)p + ?X (?X A + NX ) .
(3)
3. We now require the distribution of R? in (3) to be independent of p, i.e. for all p, p0
P((?P + ?X ?)p + ?X (?X A + NX )) = P((?P + ?X ?)p0 + ?X (?X A + NX )) .
(4)
We seek to write the predictor as a function of P and all the other roots of G separately. If our
hypothesis class is such that there exists ?? such that R? (P, g(ta(X))) = R??(P, g?(ta(X) \ {P })),
we call the structural equation model and hypothesis class specified in (2) expressible. In our
example, this is possible with ?? = (?P + ?X ?, ?X ) and g? = ?X A + NX . Equation (4) then
yields the non-discrimination constraint ?? = ?0 . Here, a possible ?0 is ?0 = (0, ?X ), which
simply yields ?P = ??X ?.
4. Given labeled training data, we can optimize the predictor R? within the hypothesis class as given
in (2), subject to the non-discrimination constraint. In the example
R? = ??X ?P + ?X X = ?X (X ? ?P ) ,
with the free parameter ?X ? R.
In general, the non-discrimination constraint (4) is by construction just P(R | do(P = p)) =
P(R | do(P = p0 )), coinciding with Definition 3. Thus Proposition 2 holds by construction of
the procedure. The choice of ?0 strongly influences the non-discrimination constraint. However, as
the example shows, it allows R? to exploit features that exhibit potential proxy discrimination.
6
A
P
G?
R
X
A
P
G
DAG
R
X
DAG
Figure 5: Left: A generic graph G? to describe proxy discrimination. Right: The graph corresponding
to an intervention on P . The circle labeled ?DAG? represents any sub-DAG of G? and G containing
an arbitrary number of variables that is compatible with the shown arrows. Dashed arrows can, but
do not have to be present in a given scenario.
4.2
Avoiding unresolved discrimination
We proceed analogously to the previous subsection using the example graph in Figure 4. Instead of
the proxy, we consider a resolving variable E. The causal dependences are equivalent to the ones in
Figure 3 and we again assume linear structural equations
E = ?E A + NE ,
X = ?X A + ?E + NX ,
R? = ?E E + ?X X .
Let us now try to adjust the previous procedure to the context of avoiding unresolved discrimination.
1. Intervene on E by fixing it to a random variable ? with P(?) = P(E), the marginal distribution
? see Figure 4. In the example we find
of E in G,
E = ?,
X = ?X A + ?E + NX ,
R? = ?E E + ?X X .
(5)
2. By iterative substitution write R? (E, X) as R? (E, g(taG (X))) for some function g, i.e. in the
example
R? = (?E + ?X ?)? + ?X ?X A + ?X NX .
(6)
3. We now demand the distribution of R? in (6) be invariant under interventions on A, which coin? Hence, in the example, for all a, a0
cides with conditioning on A whenever A is a root of G.
P((?E + ?X ?)? + ?X ?X a + ?X NX )) = P((?E + ?X ?)? + ?X ?X a0 + ?X NX )) .
(7)
Here, the subtle asymmetry between proxy discrimination and unresolved discrimination becomes
apparent. Because R? is not explicitly a function of A, we cannot cancel implicit influences of A
through X. There might still be a ?0 such that R?0 indeed fulfils (7), but there is no principled way for us to construct it. In the example, (7) suggests the obvious non-discrimination constraint ?X = 0. We can then proceed as before and, given labeled training data, optimize R? = ?E E
by varying ?E . However, by setting ?X = 0, we also cancel the path A ? E ? X ? R, even
though it is blocked by a resolving variable. In general, if R? does not have access to A, we can not
adjust for unresolved discrimination without also removing resolved influences from A on R? .
If, however, R? is a function of A, i.e. we add the term ?A A to R? in (5), the non-discrimination
constraint is ?A = ??X ?X and we can proceed analogously to the procedure for proxies.
4.3
Relating proxy discriminations to other notions of fairness
Motivated by the algorithm to avoid proxy discrimination, we discuss some natural variants of the
notion in this section that connect our interventional approach to individual fairness and other proposed criteria. We consider a generic graph structure as shown on the left in Figure 5. The proxy P
and the features X could be multidimensional. The empty circle in the middle represents any number of variables forming a DAG that respects the drawn arrows. Figure 3 is an example thereof. All
dashed arrows are optional depending on the specifics of the situation.
Definition 4. A predictor R exhibits no individual proxy discrimination, if for all x and all p, p0
P(R | do(P = p), X = x) = P(R | do(P = p0 ), X = x) .
A predictor R exhibits no proxy discrimination in expectation, if for all p, p0
E[R | do(P = p)] = E[R | do(P = p0 )] .
7
Individual proxy discrimination aims at comparing examples with the same features X, for different
values of P . Note that this can be individuals with different values for the unobserved non-feature
variables. A true individual-level comparison of the form ?What would have happened to me, if I
had always belonged to another group? is captured by counterfactuals and discussed in [15, 19].
For an analysis of proxy discrimination, we need the structural equations for P, X, R in Figure 5
P = f?P (pa(P )) ,
X = f?X (pa(X)) = fX (P, taG (X) \ {P }) ,
R = f?R (P, X) = fR (P, taG (R) \ {P }) .
For convenience, we will use the notation taGP (X) := taG (X) \ {P }. We can find fX , fR
from f?X , f?R by first rewriting the functions in terms of root nodes of the intervened graph, shown
on the right side of Figure 5, and then assigning the overall dependence on P to the first argument.
We now compare proxy discrimination to other existing notions.
Theorem 2. Let the influence of P on X be additive and linear, i.e.
X = fX (P, taGP (X)) = gX (taGP (X)) + ?X P
for some function gX and ?X ? R. Then any predictor of the form
R = r(X ? E[X | do(P )])
for some function r exhibits no proxy discrimination.
Note that in general E[X | do(P )] 6= E[X | P ]. Since in practice we only have observational data
? one cannot simply build a predictor based on the ?regressed out features? X
? := X ?
from G,
E[X | P ] to avoid proxy discrimination. In the scenario of Figure 3, the direct effect of P on X
along the arrow P ? X in the left graph cannot be estimated by E[X | P ], because of the common
confounder A. The desired interventional expectation E[X | do(P )] coincides with E[X | P ] only
if one of the arrows A ? P or A ? X is not present. Estimating direct causal effects is a hard
problem, well studied by the causality community and often involves instrumental variables [23].
? as a ?fair representation? of X, as it implicitly
This cautions against the natural idea of using X
neglects that we often want to remove the effect of proxies and not the protected attribute. Nevertheless, the notion agrees with our interventional proxy discrimination in some cases.
Corollary 1. Under the assumptions of Theorem 2, if all directed paths from any ancestor of P
? :=
to X in the graph G are blocked by P , then any predictor based on the adjusted features X
X ? E[X | P ] exhibits no proxy discrimination and can be learned from the observational distribution P(P, X, Y ) when target labels Y are available.
Our definition of proxy discrimination in expectation (4) is motivated by a weaker notion proposed
in [24]. It asks for the expected outcome to be the same across the different populations E[R | P =
p] = E[R | P = p0 ]. Again, when talking about proxies, we must be careful to distinguish conditional
and interventional expectations, which is captured by the following proposition and its corollary.
Proposition 3. Any predictor of the form R = ?(X ? E[X | do(P )]) + c for ?, c ? R exhibits no
proxy discrimination in expectation.
From this and the proof of Corollary 1 we conclude the following Corollary.
Corollary 2. If all directed paths from any ancestor of P to X are blocked by P , any predictor of
the form R = r(X ? E[X | P ]) for linear r exhibits no proxy discrimination in expectation and can
be learned from the observational distribution P(P, X, Y ) when target labels Y are available.
5
Conclusion
The goal of our work is to assay fairness in machine learning within the context of causal reasoning.
This perspective naturally addresses shortcomings of earlier statistical approaches. Causal fairness
criteria are suitable whenever we are willing to make assumptions about the (causal) generating
8
process governing the data. Whilst not always feasible, the causal approach naturally creates an
incentive to scrutinize the data more closely and work out plausible assumptions to be discussed
alongside any conclusions regarding fairness.
Key concepts of our conceptual framework are resolving variables and proxy variables that play
a dual role in defining causal discrimination criteria. We develop a practical procedure to remove
proxy discrimination given the structural equation model and analyze a similar approach for unresolved discrimination. In the case of proxy discrimination for linear structural equations, the
procedure has an intuitive form that is similar to heuristics already used in the regression literature.
Our framework is limited by the assumption that we can construct a valid causal graph. The removal
of proxy discrimination moreover depends on the functional form of the causal dependencies. We
have focused on the conceptual and theoretical analysis, and experimental validations are beyond
the scope of the present work.
The causal perspective suggests a number of interesting new directions at the technical, empirical,
and conceptual level. We hope that the framework and language put forward in our work will be a
stepping stone for future investigations.
9
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11
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6,266 | 6,669 | Nonparametric Online Regression
while Learning the Metric
Ilja Kuzborskij
EPFL
Switzerland
[email protected]
Nicol`o Cesa-Bianchi
Dipartimento di Informatica
Universit`a degli Studi di Milano
Milano 20135, Italy
[email protected]
Abstract
We study algorithms for online nonparametric regression that learn the directions
along which the regression function is smoother. Our algorithm learns the Mahalanobis metric based on the gradient outer product matrix G of the regression
function (automatically adapting to the effective rank of this matrix), while simultaneously bounding the regret ?on the same data sequence? in terms of the
spectrum of G. As a preliminary step in our analysis, we extend a nonparametric
online learning algorithm by Hazan and Megiddo enabling it to compete against
functions whose Lipschitzness is measured with respect to an arbitrary Mahalanobis
metric.
1
Introduction
An online learner is an agent interacting with an unknown and arbitrary environment over a sequence
of rounds. At each round t, the learner observes a data point (or instance) xt ? X ? Rd , outputs a
prediction ybt for the label yt ? R associated with that instance, and incurs some loss `t (b
yt ), which in
this paper is the square loss (b
yt ? yt )2 . At the end of the round, the label yt is given to the learner,
which he can use to reduce his loss in subsequent rounds. The performance of an online learner is
typically measured using the regret. This is defined as the amount by which the learner?s cumulative
loss exceeds the cumulative loss (on the same sequence of instances and labels) of any function f in
a given reference class F of functions,
T
X
RT (f ) =
`t (b
yt ) ? `t f (xt )
?f ? F .
(1)
t=1
Note that typical regret bounds apply to all f ? F and to all individual data sequences. However,
the bounds are allowed to scale with parameters arising from the interplay between f and the data
sequence.
In order to capture complex environments, the reference class of functions should be large. In this
work we focus on nonparametric classes F, containing all differentiable functions that are smooth
with respect to some metric on X . Our approach builds on the simple and versatile algorithm for
nonparametric online learning introduced in [6]. This algorithm has a bound on the regret RT (f ) of
order (ignoring logarithmic factors)
v
?
?
u d
uX
d
2
?1 + t
k?i f k ? T 1+d
?f ? F .
(2)
?
i=1
Here k?i f k? is the value of the partial derivative ?f (x) ?xi maximized over x ? X . The square
root term is the Lipschitz constant of f , measuring smoothness with respect to the Euclidean metric.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
However, in some directions f may be smoother than in others. Therefore, if we knew in advance the
set of directions along which the best performing reference functions f are smooth, we could use
this information to control regret better. In this paper we extend the algorithm from [6] and make it
adaptive to the Mahalanobis distance defined through an arbitrary positive definite matrix M with
d
spectrum (ui , ?i ) i=1 and unit spectral radius (?1 = 1). We prove a bound on the regret RT (f ) of
order (ignoring logarithmic factors)
v
?
?
u d
uX k?u f k2
?T
p
i
? ? 1+?T
? det? (M ) + t
?f ? F .
(3)
T
?i
i=1
Here ?T ? d is, roughly, the number of eigenvalues of M larger than a threshold shrinking
polynomially in T , and det? (M ) ? 1 is the determinant of M truncated at ?? (with ? ? ?T ). The
2
quantity k?ui f k? is defined like k?i f k? , but with the directional derivative ?f (x)> u instead of
the partial derivative. When the spectrum of M is light-tailed (so that ?T d and, simultaneously,
det? (M ) 1), with the smaller
eigenvalues ?i corresponding to eigenvectors in which f is smoother
2
(so that the ratios k?ui f k? ?i remain controlled), then our bound improves on (2). On the other
hand, when no preliminary knowledge about good f is available, we may run the algorithm with M
equal to the identity matrix and recover exactly the bound (2).
Given that the regret can be improved by informed choices of M , it is natural to ask whether some
kind of improvement is still possible when M is learned online, from the same data sequence on
which the regret is being measured. Of course, this question makes sense if the data tell us something
about the smoothness of the f against which we are measuring the regret. In the second part of the
paper we implement this idea by considering a scenario where instances are drawn i.i.d. from some
unknown distribution, labels are stochastically generated by some unknown regression function f0 ,
and we have no preliminary knowledge about the directions along which f0 is smoother.
In this stochastic scenario, the expected gradient outer product matrix G = E ?f0 (X)?f0 (X)>
2
provides a natural choice for the matrix M in our algorithm. Indeed, E ?f0 (X)> ui
= ?i
where u1 , . . . , ud are the eigenvectors of G while ?1 , . . . , ?d are the corresponding eigenvalues.
Thus, eigenvectors u1 , . . . ud capture the principal directions of variation for f . In fact, assuming
that the labels obey a statistical model Y = g(BX) + ? where ? is the noise and B ? Rk?d projects
X onto a k-dimensional subspace of X , one can show [21] that span(B) ? span(u1 , . . . , ud ). In
this sense, G is the ?best? metric, because it recovers the k-dimensional relevant subspace.
b of G.
When G is unknown, we run our algorithm in phases using a recently proposed estimator G
The estimator is trained on the same data sequence and is fed to the algorithm at the beginning of
each phase. Under mild assumptions on f0 , the noise in the labels, and the instance distribution, we
prove a high probability bound on the regret RT (f0 ) of order (ignoring logarithmic factors)
v
?
?
u d
?
eT
uX
?uj f0
+
?V f0
2
?
?
?1 + t
? T 1+?eT .
(4)
?j /?1
j=1
Observe that the rate at which the regret grows is the same as the one in (3), though now the effective
dimension parameter ?eT is larger than ?T by an amount related to the rate of convergence of the
b to those of G. The square root term is also similar to (3), but for the extra quantity
eigenvalues of G
k?V f0 k? , which accounts for the error in approximating the eigenvectors of G. More precisely,
k?V f0 k? is k?v f k? maximized over directions v in the span of V , where V contains those
eigenvectors of G that cannot be identified because their eigenvalues are too close to each other (we
come back to this issue shortly). Finally, we lose the dependence on the truncated determinant, which
is replaced here by its trivial upper bound 1.
The proof of (2) in [6] is based on the sequential construction of a sphere packing of X , where the
spheres are centered on adaptively chosen instances xt , and have radii shrinking polynomially with
time. Each sphere hosts an online learner, and each new incoming instance is predicted using the
learner hosted in the nearest sphere. Our variant of that algorithm uses an ellipsoid packing, and
computes distances using the Mahalanobis distance k?kM . The main new ingredient in the analysis
leading to (3) is our notion of effective dimension ?T (we call it the effective rank of M ), which
measures how fast the spectrum of M vanishes. The proof also uses an ellipsoid packing bound and
a lemma relating the Lipschitz constant to the Mahalanobis distance.
2
The proof of (4) is more intricate because G is only known up to a certain approximation. We use an
b recently proposed in [14], which is consistent under mild distributional assumptions
estimator G,
when f0 is continuously differentiable. The first source of difficulty is adjusting the notion of effective
rank (which the algorithm needs to compute) to compensate for the uncertainty in the knowledge of
the eigenvalues of G. A further problematic issue arises because we want to measure the smoothness
of f0 along the eigendirections of G, and so we need to control the convergence of the eigenvectors,
b converges to G in spectral norm. However, when two eigenvalues of G are close, then
given that G
b are strongly affected by the stochastic
the corresponding eigenvectors in the estimated matrix G
perturbation (a phenomenon known as hybridization or spectral leaking in matrix perturbation theory,
see [1, Section 2]). Hence, in our analysis we need to separate out the eigenvectors that correspond to
well spaced eigenvalues
from the
others. This lack of discrimination causes the appearance in the
regret of the extra term
?V f0
? .
2
Related works
Nonparametric estimation problems have been a long-standing topic of study in statistics, where one
is concerned with the recovery of an optimal function from a rich class under appropriate probabilistic
assumptions. In online learning, the nonparametric approach was investigated in [15, 16, 17] by
Vovk, who considered regression problems in large spaces and proved bounds on the regret. Minimax
rates for the regret were later derived in [13] using a non-constructive approach. The first explicit
online nonparametric algorithms for regression with minimax rates were obtained in [4].
The nonparametric online algorithm of [6] is known to have a suboptimal regret bound for Lipschitz
classes of functions. However, it is a simple and efficient algorithm, well suited to the design of
extensions that exhibit different forms of adaptivity to the data sequence. For example, the paper
[9] derived a variant that automatically adapts to the intrinsic dimension of the data manifold. Our
work explores an alternative direction of adaptivity, mainly aimed at taming the effect of the curse
of dimensionality in nonparametric prediction through the learning of an appropriate Mahalanobis
distance on the instance space. There is a rich literature on metric learning (see, e.g., the survey [2])
where the Mahalanobis metric k?kM is typically learned through minimization of the pairwise loss
function of the form `(M , x, x0 ). This loss is high whenever dissimilar pairs of x and x0 are close in
the Mahalanobis metric, and whenever similar ones are far apart in the same metric ?see, e.g., [19].
The works [5, 7, 18] analyzed generalization and consistency properties of online learning algorithms
employing pairwise losses.
In this work we are primarily interested in using a metric k?kM where M is close to the gradient
outer product matrix of the best model in the reference class of functions. As we are not aware
whether pairwise loss functions can indeed consistently recover such metrics, we directly estimate
the gradient outer product matrix. This approach to metric learning was mostly explored in statistics
?e.g., by locally-linear Empirical Risk Minimization on RKHS [12, 11], and through Stochastic
Gradient Descent [3]. Our learning approach combines ?in a phased manner? a Mahalanobis
metric extension of the algorithm by [6] with the estimator of [14]. Our work is also similar in spirit
to the ?gradient weights? approach of [8], which learns a distance based on a simpler diagonal matrix.
Preliminaries and notation. Let B(z, r) ? Rd be the ball of center z and radius r > 0 and let
B(r) = B(0, r). We assume instances x belong to X ? B(1) and labels y belong to Y ? [0, 1].
We consider the following online learning protocol with oblivious adversary. Given an unknown
sequence (x1 , y1 ), (x2 , y2 ), ? ? ? ? X ? Y of instances and labels, for every round t = 1, 2, . . .
1. the environment reveals instance xt ? X ;
2
2. the learner selects an action ybt ? Y and incurs the square loss `t ybt = ybt ? yt ;
3. the learner observes yt .
Given a positive definite
pd ? d matrix M , the norm kx ? zkM induced by M (a.k.a. Mahalanobis
distance) is defined by (x ? z)> M (x ? z).
Definition 1 (Covering and Packing Numbers). An ?-cover of a set S w.r.t. some metric ? is a set
{x01 , . . . , x0n } ? S such that for each x ? S there exists i ? {1, . . . , n} such that ?(x, x0i ) ? ?. The
covering number N (S, ?, ?) is the smallest cardinality of a ?-cover.
3
An ?-packing of a set S w.r.t. some metric ? is a set {x01 , . . . , x0m } ? S such that for any distinct
i, j ? {1, . . . , m}, we have ?(x0i , x0j ) > ?. The packing number M(S, ?, ?) is the largest cardinality
of a ?-packing.
It is well known that M(S, 2?, ?) ? N (S, ?, ?) ? M(S, ?, ?). For all differentiable f : X ? Y and
for any orthonormal basis V ? {u1 , . . . , uk } with k ? d we define
k?V f k? =
max
sup ?f (x)> v .
v ? span(V ) x?X
kvk = 1
If V = {u} we simply write k?u f k? .
In the following, M is a positive definite d?d matrix with eigenvalues ?1 ? ? ? ? ? ?d > 0 and eigenvectors u1 , . . . , ud . For each k = 1, . . . , d the truncated determinant is detk (M ) = ?1 ? ? ? ? ? ?k .
The kappa function for the matrix M is defined by
Figure 1: Quickly decreasing
n
o
2
spectrum of M implies slow
? 1+r
?(r, t) = max m : ?m ? t
, m = 1, . . . , d
(5)
growth of its effective rank in t.
for t ? 1 and r = 1, . . . , d.
Eigenvalues of M
1.0
?t = min {r : ?(r, t) ? r, r = 1, . . . , d} .
(6)
Since ?(d, t) ? d for all t ? 1, this is a well defined quantity.
Note that ?1 ? ?2 ? ? ? ? ? d. Also, ?t = d for all t ? 1
when M is the d ? d identity matrix. Note that the effective
rank ?t measures the number of eigenvalues that are larger than a
threshold that shrinks with t. Hence matrices M with extremely
light-tailed spectra cause ?t to remain small even when t grows
large. This behaviour is shown in Figure 1.
O
e
O
Throughout the paper, we use f = (g) and f = (g) to denote,
e
respectively, f = O(g) and f = O(g).
3
0.8
0.6
0.4
0.2
0.0
1
?t
Note that ?(r + 1, t) ? ?(r, t). Now define the effective rank of
M at horizon t by
10
9
8
7
6
5
4
3
2
1
0
2
3 4 5 6 7 8
Effective Rank of M
2000
4000
t
6000
8000
9
10
10000
Online nonparametric learning with ellipsoid packing
In this section we present a variant (Algorithm 1) of the online nonparametric regression algorithm
introduced in [6]. Since our analysis is invariant to rescalings of the matrix M , without loss of
generality we assume M has unit spectral radius (i.e., ?1 = 1). Algorithm 1 sequentially constructs
a packing of X using M -ellipsoids centered on a subset of the past observed instances. At each step
t, the label of the current instance xt is predicted using the average ybt of the labels of past instances
that fell inside the ellipsoid whose center xs is closest to xt in the Mahalanobis metric. At the end
of the step, if xt was outside of the closest ellipsoid, then a new ellipsoid is created with center xt .
The radii ?t of all ellipsoids are shrunk at rate t?1/(1+?t ) . Note that efficient (i.e., logarithmic in the
number of centers) implementations of approximate nearest-neighbor search for the active center xs
exist [10].
The core idea of the proof (deferred to the supplementary material) is to maintain a trade-off between
the regret contribution of the ellipsoids and an additional regret term due to the approximation of f
by the Voronoi partitioning. The regret contribution of each ellipsoid is logarithmic in the number of
predictions made. Since each instance is predicted by a single ellipsoid, if we ignore log factors the
overall regret contribution is equal to the number of ellipsoids, which is essentially controlled by the
packing number w.r.t. the metric defined by M . The second regret term is due to the fact that ?at
any point of time? the prediction of the algorithm is constant within the Voronoi cells of X induced
by the current centers (recall that we predict with nearest neighbor). Hence, we pay an extra term
equal to the radius of the ellipsoids times the Lipschitz constant which depends on the directional
Lipschitzness of f with respect to the eigenbasis of M .
Theorem 1 (Regret with Fixed Metric). Suppose Algorithm 1 is run with a positive definite matrix
M with eigenbasis u1 , . . . , ud and eigenvalues 1 = ?1 ? ? ? ? ? ?d > 0. Then, for any differentiable
4
Algorithm 1 Nonparametric online regression
Input: Positive definite d ? d matrix M .
1: S ? ?
2: for t = 1, 2, . . . do
1
3:
?t ? t? 1+?t
4:
Observe xt
5:
if S ? ? then
6:
S ? {t}, Tt ? ?
7:
end if
8:
s ? arg min kxt ? xs kM
. Centers
. Update radius
. Create initial ball
. Find active center
s?S
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
if Ts ? ? then
yt = 21
else
1 X
ybt ?
yt0
|Ts | 0
t ?Ts
end if
Observe yt
if kxt ? xs kM ? ?t then
Ts ? Ts ? {t}
else
S ? S ? {s}, Ts ? ?
end if
end for
. Predict using active center
. Update list for active center
. Create new center
f : X ? Y we have that
v
?
?
u d
uX k?u f k2
?T
p
e
O
i
? ? 1+?T
T
RT (f ) = ? det? (M ) + t
?i
i=1
where ? = ?(?T , T ) ? ?T ? d.
We first prove two technical lemmas about packings of ellipsoids.
0
Lemma 1 (Volumetric packing bound). Consider a pair of norms k ? k , k ? k and let B, B 0 ? Rd be
the corresponding unit balls. Then
vol B + 2? B 0
0
M(B, ?, k ? k ) ?
.
vol 2? B 0
Lemma 2 (Ellipsoid packing bound). If B is the unit Euclidean ball then
? !s s
n
o
Yp
p
8 2
?i
where
s = max i : ?i ? ?, i = 1, . . . , d .
M B, ?, k ? kM ?
?
i=1
The following lemma states that whenever f has bounded partial derivatives with respect to the
eigenbase of M , then f is Lipschitz with respect to k ? kM .
Lemma 3 (Bounded derivatives imply Lipschitzness in M -metric). Let f : X ? R be everywhere
differentiable. Then for any x, x0 ? X ,
v
u d
uX k?u f k2
i
?
f (x) ? f (x0 ) ? kx ? x0 k t
.
M
?
i
i=1
4
Learning while learning the metric
In this section, we assume instances xt are realizations of i.i.d. random variables X t drawn according
to some fixed and unknown distribution ? which has a continuous density on its support X . We also
5
assume labels yt are generated according to the noise model yt = f0 (xt ) + ?(xt ), where f0 is some
unknown regression function and ?(x) is a subgaussian zero-mean random variable for all x ? X .
We then simply write RT to denote the regret RT (f0 ). Note that RT is now a random variable which
we bound with high probability.
We now show how the nonparametric online learning algorithm (Algorithm 1) of Section 3 can be
combined with an algorithm that learns an estimate
n
X
bn = 1
b 0 (xt )?f
b 0 (xt )>
G
?f
(7)
n t=1
of the expected outer product gradient matrix G = E ?f0 (X)?f0 (X)> . The algorithm (described
in the supplementary material) is consistent under the following assumptions. Let X (? ) be X blown
up by a factor of 1 + ? .
Assumption 1.
1. There exists ?0 > 0 such that f0 is continuously differentiable on X (?0 ).
2. There exists G > 0 such that max k?f0 (x)k ? G.
x?X (?0 )
3. The distribution ?
is full-dimensional: there exists C? > 0 such that for all x ? X and
? > 0, ? B(x, ?) ? C? ?d .
b n is a consistent estimate of G.
In particular, the next lemma states that, under Assumption 1, G
Lemma 4 ([14, Theorem 1]). If Assumption 1 holds, then there exists a nonnegative and nonincreasing sequence {?n }n?1 such that for all n, the estimated gradient outerproduct (7) computed
b n ? G
? ?n with high probability with
with parameters ?n > 0, and 0 < ?n < ?0 satisfies
G
2
2
1
1/4
respect do the random draw of X 1 , . . . , X n . Moreover, if ?n = ? ?n , ?n = ? ln n d n? d ,
1
and ?n = O n? 2(d+1) then ?n ? 0 as n ? ?.
Our algorithm works in phases i = 1, 2, . . . where phase i has length n(i) = 2i . Let T (i) = 2i+1 ? 2
be the index of the last time step in phase i. The algorithm uses a nonincreasing regularization
c (0) = ? 0 I. During each phase i, the algorithm predicts the
sequence ? 0 ? ? 1 ? ? ? ? > 0. Let M
c (i ? 1)
M
c (i ? 1)k2 (where k ? k2 denotes the
data points by running Algorithm 1 with M = M
spectral norm). Simultaneously, the gradient outer product estimate (7) is trained over the same data
b
b T (i) is used
points. At the end of phase i, the current gradient outer product estimate G(i)
=G
c (i) = G(i)
b + ? T (i) I. Algorithm 1 is then restarted in phase i + 1 with
to form a new matrix M
c (i)
M
c (i)k2 . Note that the metric learning algorithm can be also implemented efficiently
M =M
through nearest-neighbor search as explained in [14].
Let ?1 ? ?2 ? ? ? ? ? ?d be the eigenvalues and u1 , . . . , ud be the eigenvectors of G. We define the
j-th eigenvalue separation ?j by
?j = min ?j ? ?k .
k6=j
For any ? > 0 define also V? ? uj : |?j ? ?k | ? ?, k 6= j and V?? = {u1 , . . . , ud } \ V? .
Our results are expressed in terms of the effective rank (6) of G at horizon T . However, in order to
account for the error in estimating the eigenvalues of G, we define the effective rank ?et with respect
to the following slight variant of the function kappa,
n
o
2
?
e(r, t) = max m : ?m + 2? t ? ?1 t? 1+r , m = 1, . . . , d
t ? 1 and r = 1, . . . , d.
c (i) be the estimated gradient outer product constructed at the end of phase i, and let ?
Let M
b1 (i) +
c (i), where
b 1 (i), . . . , u
b d (i) be the eigenvalues and eigenvectors of M
?(i) ? ? ? ? ? ?
bd (i) + ?(i) and u
we also write ?(i) to denote ? T (i) . We use ?
b to denote the kappa function with estimated eigenvalues
and ?b to denote the effective rank defined through ?
b. We start with a technical lemma.
?? 2
Lemma 5. Let ?d , ? > 0 and d ? 1. Then the derivative of F (t) = ?d + 2 T0 + t
t 1+d is
1/?
.
positive for all t ? 1 when T0 ? d+1
2?d
6
2(T0 +t)
?(d+1)
Proof. We have that F 0 (t) ? 0 if and only if t ?
t?
2?d (T0 + t)1+?
?(d + 1)
1 + (T0 + t)? ?d . This is implied by
or, equivalently, T0 ? A1/(1+?) t1/(1+?) ? t
where A = ?(d + 1)/(2?d ). The right-hand side A1/(1+?) t1/(1+?) ? t is a concave function of t.
Hence the maximum is found at the value of t where the derivative is zero, this value satisfies
A1/(1+?) ??/(1+?)
t
= 1 which solved for t gives t = A1/? (1 + ?)?(1+?)/? .
1+?
Substituting this value of t in A1/(1+?) t1/(1+?) ? t gives the condition T0 ? A1/? ?(1 + ?)?(1+?)/?
1/?
.
which is satisfied when T0 ? d+1
2?d
Theorem 2. Suppose Assumption 1 holds. If the algorithm is ran with a regularization sequence
1/?
? 0 = 1 and ? t = t?? for some ? > 0 such that ? t ? ?t for all t ? d + 1 2?d
and for
?1 ? ?2 ? ? ? ? > 0 satisfying Lemma 4, then for any given ? > 0
v
?
?
u d
?uj f0
+
?V ? f0
2
X
u
?
eT
e
O
?
?
?
? T 1+?eT
RT = ?1 + t
?j /?1
j=1
with high probability with respect to the random draw of X 1 , . . . , X T .
Note
that the asymptotic
notation is hiding terms that depend on 1/?, hence we can not zero out the
term
?V?? f0
? in the bound by taking ? arbitrarily small.
Proof. Pick the smallest i0 such that
T (i0 ) ?
d+1
2?d
1/?
(8)
(we need this condition in the proof). The total regret in phases 1, 2, . . . , i0 is bounded by
1/?
d + 1 2?d
= O(1). Let the value ?bT (i) at the end of phase i be denoted by ?b(i). By Theorem 1,
the regret RT (i + 1) of Algorithm 1 in each phase i + 1 > i0 is deterministically upper bounded by
v
?
2 ?
u d
uX
?u
?
?(i+1)
b
f
0
b
(i)
?
b
(i+1)
j
? ? (i+1) 1+?(i+1)
b
2
RT (i + 1) ? ?8 ln e2i+1 8 2
+ 4t
(9)
?
(i)
?
(i)
j
1
j=1
c (i)
M
c (i)k2 ? 1 for
where ?j (i) = ?
bj (i) + ?(i). Here we used the trivial upper bound det? M
2
all ? = 1, . . . , d. Now assume that ?
b1 (i) + ?(i) ? ?
bm (i) + ?(i) t 1+r for some m, r ? {1, . . . , d}
and for some t in phase i + 1. Hence, using Lemma 4 and ?t ? ? t , we have that
b ? G
? ?(i) with high probability.
max ?
bj (i) ? ?j ?
G(i)
(10)
2
j=1,...,d
where the first inequality is straightforward. Hence we may write
2
?1 ? ?1 ? ?(i) + ?(i) ? ?
b1 (i) + ?(i) ? ?
bm (i) + ?(i) t 1+r
2
? ?m + ?(i) + ?(i) t 1+r
2
? ?m + 2?(i) t 1+r .
Recall ?(i) = T (i)?? . Using Lemma 5, we observe that the derivative of
?? 2
F (t) = ?m + 2 T (i) + t
t 1+r
is positive for all t ? 1 when
T (i) ?
r+1
2?d
1/?
?
7
r+1
2?m
1/?
(using Lemma 4)
2
2
which is guaranteed by our choice (8). Hence, ?m + 2?(i) t 1+r ? ?m + 2? T ) T 1+r and so
2
?
b1 (i) + ?(i)
? t 1+r
?
bm (i) + ?(i)
implies
2
?1
? T 1+r .
?m + 2? T
Recalling the definitions of ?
e and ?
b, this in turn implies ?
b(r, t) ? ?
e(r, T ), which also gives ?bt ? ?eT
for all t ? T . Next, we bound the approximation error in each individual eigenvalue of G. By (10)
we obtain, for any phase i and for any j = 1, . . . , d,
?j + 2?(i) ? ?j + ?(i) + ?(i) ? ?
bj (i) + ?(i) ? ?j ? ?(i) + ?(i) ? ?j .
Hence, bound (9) implies
?
RT (i + 1) ? ?8 ln e2i+1 12?eT
v
2 ?
u
d
X
u
?
eT
?
f
0
b
u
j
? ? (i+1) 1+?eT
2
.
+ 4t ?1 + 2?(i)
?j
j=1
(11)
The error in approximating the eigenvectors of G is controlled via the following first-order eigenvector
approximation result from matrix perturbation theory [20, equation (10.2)], for any vector v of
constant norm,
c
X u>
k M (i) ? G uj >
>
c (i) ? G
2
b j (i) ? uj =
v u
v uk + o
M
2
?j ? ?k
k6=j
X 2?(i)
?
v > uk + o ?(i)2
?j ? ?k
(12)
k6=j
c
c
where we used u>
k M (i) ? G uj ? M (i) ? G 2 ? ?(i) + ?(i) ? 2?(i). Then for all j such
that uj ? V? ,
X 2?(i)
b j (i) ? uj =
?f0 (x)> u
?f0 (x)> uk + o ?(i)2
?j ? ?k
k6=j
?
2?(i) ?
d k?f0 (x)k2 + o ?(i)2 .
?
Note that the coefficients
c
u>
k M (i) ? G uj
?k =
+ o ?(i)2
?j ? ?k
k 6= j
b j (i) ? uj w.r.t. the orthonormal basis u1 , . . . , ud . Then,
are a subset of coordinate values of vector u
by Parseval?s identity,
X
2
4 ? kb
uj (i) ? uj k2 ?
?k2 .
k6=j
Therefore, it must be that
u> M
k c (i) ? G uj
max
? 2 + o ?(i)2 .
k6=j
?j ? ?k
For any j such that uj ? V?? , since ?j ? ?k ? ? for all uk ? V? , we may write
b j (i) ? uj
?f0 (x)> u
X
2?(i) X
?f0 (x)> uk + o ?(i)2
?
?f0 (x)> uk + 2 + o ?(i)2
?
?
uk ?V?
?
uk ?V?
?
2?(i) ?
d kP V? ?f0 (x)k2 + 2 + o ?(i)2
d
P V?? ?f0 (x)
2 + o ?(i)2
?
8
where P V? and P V?? are the orthogonal projections onto, respectively, V? and V?? . Therefore, we
have that
>
?u
b j (i) = sup ?f0 (x)> u
b j (i) ? uj + uj
b j f0 ? = sup ?f0 (x) u
x?X
x?X
b j (i) ? uj
? sup ?f0 (x)> uj + sup ?f0 (x)> u
x?X
x?X
?
2?(i) ?
?
?uj f0
? +
d k?V? f0 k? + 2 + o ?(i)2
d
?V?? f0
? + o ?(i)2
(13)
?
?
?
Letting ?? (i) = 2?(i)
d k?V? f0 k? + 2 + o ?(i)2
d
?V?? f0
? + o ?(i)2 we can upper
?
bound (11) as follows
v
?
?
u
d
?u f0
+ ?? (i) 2
X
?
eT
u
j
?
? 2(i+1) 1+?eT .
RT (i + 1) ? ?8 ln e2i+1 12?eT + 4t ?1 + 2?(i)
?j
j=1
Recall that, due to (10), the above holds at the end of each phase i + 1 with high probability. Now
O
observe that ?(i) = O 2??i and so ?? (i) = (k?V? f0 k? /?+
?V?? f0
? ). Hence, by summing
over phases i = 1, . . . , log2 T and applying the union bound,
dlog2 T e
RT =
X
RT (i)
i=1
v
?
u
d
?uj f0
+ ?? (i ? 1) 2 ?eT i
X
u
?
? 2 1+?eT
? ?8 ln eT 12d+ 4t ?1 + 2?(i ? 1)
?j
j=1
?
2 ?
d
?
f
+
?
f
?
X
uj 0 ?
V? 0 ?
e ?
O
? 1+?eT?eT
= ?1 +
?T
?j ?1
j=1
?
concluding the proof.
5
Conclusions and future work
We presented an efficient algorithm for online nonparametric regression which adapts to the directions
along which the regression function f0 is smoother. It does so by learning the Mahalanobis metric
through the estimation of the gradient outer product matrix E[?f0 (X)?f0 (X)> ]. As a preliminary
result, we analyzed the regret of a generalized version of the algorithm from [6], capturing situations
where one competes against functions with directional Lipschitzness with respect to an arbitrary
Mahalanobis metric. Our main result is then obtained through a phased algorithm that estimates the
gradient outer product matrix while running online nonparametric regression on the same sequence.
Both algorithms automatically adapt to the effective rank of the metric.
This work could be extended by investigating a variant of Algorithm 1 for classification, in which ball
radii shrink at a nonuniform rate, depending on the mistakes accumulated within each ball rather than
on time. This could lead to the ability of competing against functions f that are only locally Lipschitz.
In addition, it is conceivable that under appropriate assumptions, a fraction of the balls could stop
shrinking at a certain point when no more mistakes are made. This might yield better asymptotic
bounds than those implied by Theorem 1, because ?T would never attain the ambient dimension d.
Acknowledgments
Authors would like to thank S?ebastien Gerchinovitz and Samory Kpotufe for useful discussions on
this work. IK would like to thank Google for travel support. This work also was in parts funded
by the European Research Council (ERC) under the European Union?s Horizon 2020 research and
innovation programme (grant agreement no 637076).
9
References
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Gradient Outerproduct. In Conference on Uncertainty in Artificial Intelligence (UAI), 2014.
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10
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6,267 | 667 | Deriving Receptive Fields Using An
Optimal Encoding Criterion
Ralph Linsker
IBM T. J. Watson Research Center
P. O. Box 218, Yorktown Heights, NY 10598
Abstract
An information-theoretic optimization principle ('infomax') has
previously been used for unsupervised learning of statistical regularities in an input ensemble. The principle states that the inputoutput mapping implemented by a processing stage should be chosen so as to maximize the average mutual information between
input and output patterns, subject to constraints and in the presence of processing noise. In the present work I show how infomax,
when applied to a class of nonlinear input-output mappings, can
under certain conditions generate optimal filters that have additional useful properties: (1) Output activity (for each input pattern) tends to be concentrated among a relatively small number
of nodes. (2) The filters are sensitive to higher-order statistical
structure (beyond pairwise correlations). If the input features are
localized, the filters' receptive fields tend to be localized as well.
(3) Multiresolution sets of filters with subsampling at low spatial
frequencies - related to pyramid coding and wavelet representations
- emerge as favored solutions for certain types of input ensembles.
1
INTRODUCTION
In unsupervised network learning, the development of the connection weights is
influenced by statistical properties of the ensemble of input vectors, rather than by
the degree of mismatch between the network's output and some 'desired' output.
An implicit goal of such learning is that the network should transform the input
so that salient features present in the input are represented at the output in a
953
954
Linsker
more useful form. This is often done by reducing the input dimensionality in a way
that preserves the high-variance components of the input (e.g., principal component
analysis, Kohonen feature maps).
The principle of maximum information preservation ('infomax') is an unsupervised
learning strategy that states (Linsker 1988): From a set of allowed input-output
mappings (e.g., parametrized by the connection weights), choose a mapping that
maximizes the (ensemble-averaged) Shannon information that the output vector
conveys about the input vector, in the presence of noise. Such a mapping maximizes
the ensemble-averaged mutual information (MI) between input and output.
This paper (a) summarize earlier results on info max solutions for linear networks,
(b) identifies some limitations of these solutions (ways in which very different filter
sets are equally optimal from the infomax standpoint), and (c) shows how, by adding
a small nonlinearity to the network, one can remove these limitations and at the
same time improve the utility of the output representations. We show that infomax,
acting on the modified network, tends to favor sparsely coded representations and
(depending on the input ensemble) sets of filters that span multiple resolution scales
(related to wavelets and 'pyramid coding').
2
INFOMAX IN LINEAR NETWORKS
For definiteness and brevity, we consider a linear network having a particular type
of noise model and input statistical properties. For a more detailed discussion of
related models see (Linsker 1989).
Since the computation of the MI (which involves the output entropy) is in general
intractable for continuous-valued output vectors, previous work (and the present
paper) makes use of a surrogate MI, which we will call the 'as-if-Gaussian' MI. This
quantity is, by definition, computed as though the output vectors comprised a multivariate Gaussian distribution having the same mean and covariance as the actual
distribution of output vectors. Although expedient, this substitution has lacked a
principled justification. The Appendix shows that, under certain conditions, using
this 'surrogate MI' (and not the full MI) is indeed appropriate and justified.
=
Denote the input vector by S {Si} (Si is the activity at input node i), the output
vector by Z
{Zn}, the matrix of connection weights by C {Cni}, noise at the
input nodes by N
{Nd, and noise at the output nodes by v {vn }. Then our
processing model is, in matrix form, Z = C(S + N) + v. Assume that N and v are
Gaussian random variables, (S)
(N)
(v) 0, \fN T ) (Sv T ) (NvT) 0,
and, for the covariance matrices, (SsT)
Q, (N N ) TJI, (vv T ) {3I'. (Angle
brackets denote an ensemble average, superscript T denotes transpose, and I and
I' denote unit matrices on the input and output spaces, respectively.) In general,
MI = Hz - (Hzls) where Hz is the output entropy and HZls is the entropy of the
output for given S. Replacing MI by the 'as-if-Gaussian' MI means replacing Hz
by the expression for the entropy of a multivariate Gaussian distribution, which is
(apart from an irrelevant constant term) H~ = (1/2) lndet Q', where Q' (ZZT)
CQC T + TJCC T + (3I' is the output covariance. Note that, when S is fixed, Z =
CS+(CN +v) is a Gaussian distribution centered on CS, so that we have (Hzls) =
(1/2)lndetQII where Q" = ((CN + v)(CN + v)T) = TJCC T + {3I'. Therefore the
=
=
=
=
= =
=
=
=
=
=
=
=
=
=
Deriving Receptive Fields Using An Optimal Encoding Criterion
'as-if-Gaussian' MI is
(1)
MI' = (1/2)[lndetQ' -lndetQ"].
The variance of the output at node n (prior to adding noise lin) is Vn = ([C(S +
N)]~) = (CQC T + TJCCT)nn. We will constrain the dynamic range of each output
node (limiting the number of output values that can be discriminated from one
another in the presence of output noise) by requiring that Vn = 1 for each n.
Subject to this constra.int, we are to find a matrix C that maximizes MI'. For a
local Hebbian algorithm that accomplishes this maximization, see (Linsker 1992).
Here, in order to proceed analytically, we consider a special case of interest.
Suppose that the input statistics are shift-invariant, so that the covariance (SiSj)
is a function of (j - i). We then use a shift-invariant filter Ansatz, Cni C(i - n).
Infomax then determines the optimal filter gain as a function of spatial frequency;
i.e., the magnitude of the Fourier components c(k) of C(i - n). The derivation is
summarized below.
=
Denote by q(k), q'(k), and q"(k) the Fourier transforms of QU - i), Q'(m - n),
and Q"(m - n) respectively. Since Q' = CQCT + TJccf1' + (3I', therefore
q'(k) = [q(k) + TJ) I c(k) 12 +(3. Similarly, q"(k) = TJ I c(k) 12 +(3. We obtain
MI' = (1/2)~dlnq'(k) - Inq"(k)]. Each node's output variance Vn is equal to
V = (I/K)~dq(k)+TJ] I c(k) 12 where K is the number of terms in the sum over k.
To maximize MI' subject to the constraint on V we use the Lagrange multiplier
method; that is, we maximize MI" MI' + J.L(V - 1) with respect to each I c(k) 12.
!his yields an equation for each k that is quadratic in I c(k) 12. The unique solution
=
IS
2_
(TJ/{3) I c(k) I - -1
q(k)
+ 2[q(k) + TJ]{1 + [1-
2TJK 1/2
J.L{3q(k)] }
(2)
if the RHS is positive, and zero otherwise. The Lagrange multiplier J.L( < 0) is chosen
so that the {I c( k) I} satisfy V = 1.
Starting from a differently-stated goal (that of reducing redundancy subject to a
limit on information loss), which turns out to be closely related to infomax, (Atick
& Redlich 1990a) found an expression for the optimal filter gain that is the same
as that of Eq. 2 except for the choice of constraint.
Filter properties found using this approach are related to those found in early stages
of biological sensory processing. Smoothing and bandpass (contrast-enhancing)
filters emerge as infomax solutions (Linsker 1989, Atick & Redlich 1990a) in certain
cases, and good agreement with retinal contrast sensitivity measurements has been
found (Atick & Redlich 1990b).
Nonetheless, the value of the infomax solution Eq. 2 is limited in two important
ways. First, the phases of the {c(k)} are left undetermined. Any choice of phases is
equally good at maximizing MI' in a linear network. Thus the real-space response
function C(i - n), which determines the receptive field properties of the output
nodes, is nonunique (and indeed may be highly nonlocalized in space).
Second, it is useful to extend the solution Ansatz to allow a number of different filter
types a = 1, ... , A at each output site, while continuing to require that each type
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956
Linsker
satisfy the shift-invariance condition Cni(a) == C(i - n;a). For example, one may
want to model a topographic 'retinocortical' mapping in which each patch of cortex
(each 'site') contains multiple filter types, yet each patch carries out the same set of
processing functions on its input. For this Ansatz, one again obtains Eq. 2 (derivation omitted here), but with 1 c(k) 12 on the LHS replaced by ~ap(a)1 c(k; a) 12,
where c(k; a) is the F.T. of C(i - n; a), and p(a) is the fraction of the total number
of filters (at each site) that are of type a. The partitioning of the overall (sumsquared) gain among the multiple filter types is thus left undetermined.
The higher-order statistical structure of the input (beyond covariance) is not being
exploited by infomaxin the above analysis, because (1) the network is linear and (2)
only pairwise correlations among the output activities enter into MI'. We shall show
that if we make the network even mildly nonlinear, MI' is no longer independent of
the choice of phases or of the partitioning of gain among multiple filter types.
3
NETWORK WITH WEAK NONLINEARITY
We consider the weakly nonlinear input-output relation Zn = Un + fU~ + ~iCniNi +
lin, where Un ~iCniSi, for small f. This differs from the linear network analyzed
above by the term in U~. (For simplicity, terms nonlinear in the noise are not
included.) The cubic term increases the signal-to-noise ratio selectively when Un is
large in absolute value. We maximize MI' as defined in Eq. l.
=
Heuristically, the new term will cause infomax to favor solutions in which some
output nodes have large (absolute) activity values, over solutions in which all output
nodes have moderate activities. The output layer can thus encode information
about the input vector (e.g., signal the presence of a feature) via the high activity
of a small number of nodes, rather than via the particular activity values of many
nodes. This has several (interrelated) potential advantages. (1) The concentration
of activity among fewer nodes is a type of sparse coding. (2) The resulting output
representation may be more resistant to noise. (3) The presence of a feature can be
signaled to a later processing stage using fewer connections. (4) Since the particular
nodes that have high activity depend upon the input vector, this type of mapping
transforms a set of continuous-valued inputs at each site into a partially place-coded
representation. A model of this sort may thus be useful for understanding better
the formation of place-coded representations in biological systems.
3.1
MATHEMATICAL DETAILS
This section may be skipped without loss of continuity. In matrix form, U CS,
Wn - U~ for each n, and Z = U + f W + C N + II. Keeping terms through first
{ZZT} = CQCT + 1]CCT + (3I' + fF,
order in f, the output covariance is Q'
where F {WUT)+{UWT} . [As an aside, Fnm = (UnUm(U~+U~)} resembles the
covariance (UnUm), except that presentations having large U~ +U~ are given greater
weight in the ensemble average.] For shift-invariant input statistics and one filter
type Cni C( i-n), taking the Fourier transform yields q'(k) = [q(k)+1]]1 c(k; a) 12+
(3 + ff(k) where f(k) is the F.T. of F(m - n)
Fnm. So In det Q' = ~k lnq'(k) =
~ln{[q(k)+1]] 1c(k) 12 +(3}+f~g(k) where g(k) _ [f(k)/{[q(k) +1]]1 c(k;a) 12+{3}] .
Using a Lagrange multiplier as before, the quantity to be maximized is MI"
=
=
=
=
=
Deriving Receptive Fields Using An Optimal Encoding Criterion
~~J[~
~~ -+-11~
Figure 1: Breaking of phase degeneracy. See text for discussion.
MI"(e = 0)
+ (e/2)~g(k).
Now suppose there are multiple filter types a = 1, ... , A at each output site. For
each k define d(k) to be the A x A matrix whose elements are: d(k)a.b
[q(k) +
TJ]c(k; a)c*(k; b) + [f3/ p(a)]c5a.b where c5a.b is the Kronecker delta. Also define f(k) to
be the A x A matrix each of whose elements f(k)ab is the F.T. of F(m - n;a,b)
where F(m - n; a, b) = (Un(a)Wm(b)) + (Wn(a)Um(b)). Then the O{e) part of MI"
is: (e/2)~kTr{[d(k)]-lf(k)}. Note that [d(k)]-l is the inverse of the matrix d(k),
and that 'Tr' denotes the trace. [Outline of derivation: In the basis defined by the
Fourier harmonics, Q' is block diagonal (one A x A block for each k). So In det Q' =
~k Indetq'(k) where each q'(k) is an A x A matrix of the form q~(k) + eq~(k).
Expanding In det q' (k) through O( e) yields the stated result.]
=
The infomax calculation to lowest order in e [i.e., O( eO)] is the same as for the linear
network. Here, for simplicity, we determine the sum-squared gain, ~ap(a)1 c(k; a) 12 ,
as in the linear case; then seek to maximize the new term, of O(e), subject to
this constraint on the value of the sum-squared gain. How the nonlinear term
breaks phase and gain-apportionment degeneracies is of interest here; a small O( e)
correction to the sum-squared gain is not.
4
ILLUSTRATIVE RESULTS
Two examples will show how adding the nonlinear perturbative term to the network's output breaks a degeneracy among different filter solutions. In each case the
input space is a one-dimensional 'retina' with wraparound.
4.1
BREAKING THE PHASE DEGENERACY
In this example (see Figure 1) there is one filter type at each output site. We consider
two types of input ensembles: (1) Each input vector (Fig. la shows one example)
is drawn from a multivariate Gaussian distribution (so there is no higher-order
statistical structure beyond pairwise correlations). The input covariance matrix
Q(j - i) is a Gaussian function of the distance between the sites. (2) Each input
957
958
Linsker
vector is a random sum of Gaussian 'bumps': Si = E j aj [s( i - j) - so] where s( i - j)
is a Gaussian (shown in Fig. 1b for j=20j there are 64 nodes in all); So is the mean
value of s( i - j); and each aj is independently and randomly chosen (with constant
probability) to be 1 or O. This ensemble does have higher-order structure, with
each input presentation being characterized by the presence of localized features
(the bumps) at particular locations.
The infomax solution for I c(k) 12 is plotted versus spatial frequency k in Fig. 1c
for a particular choice of noise parameters (1], (3). As stated earlier, MI' for a linear
network is indifferent to the phases of the Fourier components {c(k)}. A particular
random choice of phases produces the real-space filter C(i - n) shown in Fig. Id,
which spans the entire 'retina.' Setting all phases to zero produces the localized
filter shown in Fig. 1?. If the Gaussian 'bump' of Fig. 1b is presented as input to a
network of filters each of which is a shifted version of Fig. 1d, the linear response of
the network (i.e., the convolution of the 'bump' with the filter) is shown in Fig. Ie.
Replacing the filter of Fig. 1d by that of Fig. 1?, but keeping the input the same,
produces the output response shown in Fig. 19.
The cubic nonlinearity causes MI' to be larger for the filter of Fig. 1? than for that of
Fig. 1d. Heuristically, if we focus on the diagonal elements of the output covariance
Q', the nonlinear term is 2?(U~). Maximizing MI' favors increasing this term (subject to a constraint on output variance) hence favors filter solutions for which the
Un distribution is non-Gaussian with a preponderance of large values. Projection
pursuit methods also use a measure of the non-Gaussianity of the output distribution to construct filters that extract 'interesting' features from high-dimensional
data (cf. Intrator 1992).
4.2
BREAKING THE PARTITIONING DEGENERACY FOR
MULTIPLE FILTER TYPES
In this example (see Fig. 2), the input ensemble comprises a set of self-similar
patterns (each is a sine-Gabor 'ripple' as in Fig. 2a) that are related by translation
and dilation (scale change over a factor of 80). Figure 2b shows the input power
spectrum vs. k; the scaling region goes as 11k. Figure 2c shows the infomax solution
for the gain I c(k;a) I vs. k when there is just one filter type. When the input SNR
is large (as in the scaling region) the infomax filters 'whiten' the output; note the
flat portion of the output power spectrum (Fig. 2d). [We modify the infomax
solution by extending the power-law form of I c(k) I to low k (dotted line in Figs.
2c,d). This avoids artifacts resulting from the rapid increase in I c(k) I, which is
in turn caused by our having omitted low-k patterns from the input ensemble for
reasons of numerical efficiency.] The dotted envelope curve in Figure 2e shows the
sum-squared gain Eap(a) I c(k) 12 when multiple filter types a are allowed. The
quantity plotted is just the square of that shown in Fig. 2c, but on a linear rather
than log-log plot (note values greater than 5 are cut off to save space).
The network nonlinearity has the following effect. We first allow two filter types
to share the overall gain. Optimizing MI' over various partitionings, we find that
info max favors a crossover between filter types at k ~ 400. Allowing three, then four,
filter types produces additional crossovers at lower k. For an Ansatz in which each
filter's share of the sum-squared gain is tapered linearly near its cutoff frequencies,
Deriving Receptive Fields Using An Optimal Encoding Criterion
the best solution found for each p( a) 1c( k) 12 is shown in Fig. 2e (semilog plot vs.
k ). Figure 2f plots the corresponding 1 c( k; a) 1 vs. k on a linear scale. Note that the
three lower-k filters appear roughly self-similar. (The peak in the highest-k filter is
an artifact due to the cutoff of the input ensemble at high k.) The four real-space
filters C(i-n; a) are plotted vs. (i-n) in Fig. 2g [phases chosen to make C(i-n; a)
antisymmetric] .
The resulting filters span multiple resolution scales. The density p( a) is less for the
lower-frequency filters (spatial subsampling). When more filter types are allowed,
the increase in MI' becomes progressively less. Although in our model the filters are
present with density p at each output site, a similar MIl is obtained if one spaces
adjacent filters of type a by a distance <X 1/ p( a). The resulting arrangement of
filters resembles the 'tiling' of the joint space and spatial-frequency domain that is
used in wavelet and 'pyramid coding' approaches to image processing. [The infomax
filters overlap, rather than disjointly tiling the (:I;, k) domain.]
Using the infomax method, the region of (:1;, k) space spanned by an optimal filter
has an aspect ratio that depends upon the relative distances - along the :I; and k axes
- over which the input feature is 'coherent' (possesses higher-order correlations).
One may thus be able to use infomax to predict relationships between statistical
measures of coherence in natural scenes and observed (:1;, k) aspect ratios for, e.g.,
orientation-selective cells. See (Field 1989) for a discussion of this issue that is not
based on infomax.
5
APPENDIX: HEURISTIC JUSTIFICATION FOR
USING A SURROGATE, 'AS-IF-GAUSSIAN,'
MUTUAL INFORMATION
The mutual information between input S and output Z is MI
!dSdZPszln(Psz/PsPz)
!dSPsKD(Pzls;Pz) where KD(Pzls;PZ)
! dZPzlsln(Pzls/PZ) is a Kullback divergence. So, maximizing MI means maximizing the average (over S) of KD(Pzls; Pz).
What does the KD represent? Suppose that the network has somehow learned the
distribution Pz. Before being presented with a particular input S, the network 'expects' an output vector drawn from Pz. The actual output response to S, however,
is a vector drawn from PZls. The KD measures the 'surprise' (i.e., the amount of
information gained) upon seeing the actual distribution PZls when one expected
Pz. Infomax maximizes this average 'surprise.'
However, the network cannot in general have access to the full distribution Pz,
which contains far too much information (including all higher-order statistics) to
be stored in the connections and nodes of the network. Let us suppose for definiteness that the system remembers only the mean and the covariance matrix of
Z. Define pff to be the multivariate Gaussian distribution that has the same mean
and covariance as Pz. Then we may think of the system as a priori 'expecting' the
output vector to be drawn from the distribution pff.
We accordingly modify the principle so that we maximize the averThis equals
age (over S) of KD(PzIS; pff) (note the superscript G).
959
960
Linsker
1000
100
10
0.1
10
10
1000
10
100
1000
(d)
0.1 I
o
100
10
100
1000
~
__-
(9)
,...=;;
'
80
=
Figure 2: Partitioning among multiple filter types. See text.
J dSPs JdZ PZls In( P Z1s / pi) = (- H zls) s - JdZ Pz In pi
(where H denotes entropy). Using a property of the Gaussian distribution, we have - J dZPz In pi =
- J dZpi InPff = He;. We conclude that the average of KD equals He; - (HZls)s,
which is exactly equal to the surrogate ' as-if-Gaussian' MI defined preceding Eq. 1.
This argument provides a principled justification for using the surrogate MI, when
the system has stored information about the output vectors' mean and covariance,
but not about higher-order statistics.
References
J. J. Atick & A. N. Redlich. (1990a) Towards a theory of early visual processing.
Neural Computation 2:308-320.
J. J. Atick & A. N. Redlich. (1990b) Quantitative tests of a theory of retinal
processing: contrast sensitivity curves. Inst. Adv. Study IASSNS-HEP-90/51.
D. J. Field. (1989) What the statistics of natural images tell us about visual coding.
In Proc. SPIE 1077:269-276.
N.lntrator. (1992) Feature extraction using an unsupervised neural network. Neural Computation 4:98-107.
R. Linsker. (1988) Self-organization in a perceptual network. Computer 21(3):105117.
R. Linsker. (1989) An application of the principle of maximum information preservation to linear systems. In D. S. Touretzky (ed.), Advances in Neural Information
Processing Systems 1, 186-194. San Mateo, CA: Morgan Kaufmann.
R. Linsker. (1992) Local synaptic learning rules suffice to maximize mutual information in a linear network. Neural Computation 4(5 ):691-702.
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6,268 | 6,670 | Recycling Privileged Learning
and Distribution Matching for Fairness
Novi Quadrianto?
Predictive Analytics Lab (PAL)
University of Sussex
Brighton, United Kingdom
[email protected]
Viktoriia Sharmanska
Department of Computing
Imperial College London
London, United Kingdom
[email protected]
Abstract
Equipping machine learning models with ethical and legal constraints is a serious
issue; without this, the future of machine learning is at risk. This paper takes a step
forward in this direction and focuses on ensuring machine learning models deliver
fair decisions. In legal scholarships, the notion of fairness itself is evolving and
multi-faceted. We set an overarching goal to develop a unified machine learning
framework that is able to handle any definitions of fairness, their combinations,
and also new definitions that might be stipulated in the future. To achieve our
goal, we recycle two well-established machine learning techniques, privileged
learning and distribution matching, and harmonize them for satisfying multi-faceted
fairness definitions. We consider protected characteristics such as race and gender
as privileged information that is available at training but not at test time; this
accelerates model training and delivers fairness through unawareness. Further, we
cast demographic parity, equalized odds, and equality of opportunity as a classical
two-sample problem of conditional distributions, which can be solved in a general
form by using distance measures in Hilbert Space. We show several existing
models are special cases of ours. Finally, we advocate returning the Pareto frontier
of multi-objective minimization of error and unfairness in predictions. This will
facilitate decision makers to select an operating point and to be accountable for it.
1
Introduction
Machine learning technologies have permeated everyday life and it is common nowadays that an
automated system makes decisions for/about us, such as who is going to get bank credit. As
more decisions in employment, housing, and credit become automated, there is a pressing need
for addressing ethical and legal aspects, including fairness, accountability, transparency, privacy,
and confidentiality, posed by those machine learning technologies [1, 2]. This paper focuses on
enforcing fairness in the decisions made by machine learning models. A decision is fair if [3, 4,
5]: i) it is not based on a protected characteristic [6] such as gender, marital status, or age (fair
treatment), ii) it does not disproportionately benefit or hurt individuals sharing a certain value of
their protected characteristic (fair impact), and iii) given the target outcomes, it enforces equal
discrepancies between decisions and target outcomes across groups of individuals based on their
protected characteristic (fair supervised performance).
The above three fairness definitions have been studied before, and several machine learning frameworks for addressing each one or a combination of them are available. We first note that one could
ensure fair treatment by simply ignoring protected characteristic features, i.e. fairness through
unawareness. However this poses a risk of unfairness by proxy as there are ways of predicting
?
Also with National Research University Higher School of Economics, Moscow, Russia.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
protected characteristic features from other features [7, 8]. Existing models guard against unfairness
by proxy by enforcing fair impact or fair supervised performance constraints in addition to the fair
treatment constraint. An example of the fair impact constraint is the 80% rule (see e.g. [3, 9, 10]) in
which positive decisions must be in favour of group B individuals at least 80% as often as in favour
of group A individuals for the case of a binary protected characteristic and a binary decision. Another
example of the fair impact constraint is a demographic parity in which positive decisions of group B
individuals must be at the same rate as positive decisions of group A individuals (see e.g. [11] and
earlier works [12, 13, 14]).
In contrast to the fair impact that only concerns about decisions of an automated system, the
fair supervised performance takes into account, when enforcing fairness, the discrepancy between
decisions (predictions) and target outcomes, which is compatible to the standard supervised learning
setting. Kleinberg et al. [15] show that fair impact and fair supervised performance are indeed
mutually exclusive measures of fairness. Examples of the fair supervised performance constraint
are equality of opportunity [4] in which the true positive rates (false negative rates) across groups
must match, and equalized odds [4] in which both the true positive rates and false positive rates must
match. Hardt et al. [4] enforce equality of opportunity or equalized odds by post-processing the
soft-outputs of an unfair classifier. The post-processing step consists of learning a different threshold
for a different group of individuals. The utilization of an unfair classifier as a building block of the
model is deliberate as the main goal of supervised machine learning models is to perform prediction
tasks for future data as accurately as possible. Suppose the target outcome is correlated with the
protected characteristic, Hardt et al.?s model will be able to learn the ideal predictor, which is not
unfair as it represents the target outcome [4]. However, Hardt et al.?s model needs to access the
value of the protected characteristic for future data. Situations where the protected characteristic
is unavailable due to confidentiality or is prohibited to be accessed due to the fair treatment law
requirement will make the model futile [5]. Recent work of Zafar et al. [5] propose de-correlation
constraints between supervised performance, e.g. true positive rate, and protected characteristics as a
way to achieve fair supervised performance. Zafar et al.?s model, however, will not be able to learn
the ideal predictor when the target outcome is indeed correlated with a protected characteristic.
This paper combines the benefits of Hardt et al.?s model [4] in its ability to learn the ideal predictor
and of Zafar et al.?s model [5] in not requiring the availability of protected characteristic for future
data at prediction time. To achieve this, we will be building upon recent advances in the use of
privileged information for training machine learning models [16, 17, 18, 19]. Privileged information
refers to features that can be used at training time but will not be available for future data at prediction
time. We propose to consider protected characteristics such as race, gender, or marital status as
privileged information. The privileged learning framework is remarkably suitable for incorporating
fairness, as it learns the ideal predictor and does not require protected characteristics for future data.
Therefore, this paper recycles the overlooked privileged learning framework, which is designed for
accelerating learning and improving prediction performance, for building a fair classification model.
Enforcing fairness using the privileged learning framework alone, however, might increase the risk
of unfairness by proxy. Our proposed model guards against this by explicitly adding fair impact
and/or fair supervised performance constraints into the privileged learning model. We recycle a
distribution matching measure for fairness. This measure can be instantiated for both fair impact
(e.g. demographic parity) and fair supervised performance (e.g. equalized odds and equality of
opportunity) constraints. Matching a distribution between function outputs (decisions) across different
groups will deliver fair impact, and matching a distribution between errors (discrepancies between
decisions and target outcomes) across different groups will deliver fair supervised performance. We
further show several existing methods are special cases of ours.
2
Related Work
There is much work on the topic of fairness in the machine learning context in addition to those that
have been embedded in the introduction. One line of research can be described in terms of learning
fair models by modifying feature representations of the data (e.g. [20, 10, 21]), class label annotations
([22]), or even the data itself ([23]). Another line of research is to develop classifier regularizers that
penalize unfairness (e.g. [13, 14, 24, 11, 5]). Our method falls into this second line of research. It
has also been emphasized that fair models could enforce group fairness definitions (covered in the
introduction) as well as individual fairness definitions. Dwork et al. and Joseph et al. [25, 26] define
2
an individual fairness as a non-preferential treatment towards an individual A if this individual is not
as qualified as another individual B; this is a continuous analog of fairness through unawareness [23].
On privileged learning Vapnik et al. [16] introduce privileged learning in the context of Support
Vector Machines (SVM) and use the privileged features to predict values of the slack variables. It
was shown that this procedure can provably reduce the amount of data needed for learning an optimal
hyperplane [16, 27, 19]. Additional features for training a classifier that will not necessarily be
available at prediction time, privileged information, are widespread. As an example, features from 3D
cameras and laser scanners are slow to acquire and expensive to store but have the potential to boost
the predictive capability of a trained 2D system. Many variants of privileged learning methods and
settings have been proposed such as, structured prediction [28], margin transfer [17], and Bayesian
privileged learning [18, 29]. Privileged learning has also been shown [30] to be intimately related
to Hinton et al.?s knowledge distillation [31] and Bucila et al.?s [32] model compression in which a
complex model is learnt and is then replicated by a simpler model.
On distribution matching Distribution matching has been explored in the context of domain adaptation (e.g. [33, 34]), transduction learning (e.g. [35]), and recently in privileged learning [36],
among others. The empirical Maximum Mean Discrepancy (MMD) [37] is commonly used as the
nonparametric metric that captures discrepancy between two distributions. In the domain adaptation
setting, Pan et al. [38] use the MMD metric to project data from target and related source domain into
a common subspace such that the difference between the distributions of source and target domain
data is reduced. A similar idea has been explored in the context of deep neural networks by Zhang
et al. [34], where they use the MMD metric to match both the distribution of the features and the
distribution of the labels given features in the source and target domains. In the transduction setting,
Quadrianto et al. [35] propose to minimize the mismatch between the distribution of function outputs
on the training data and on the target test data. Recently, Sharmanska et al. [36] devise a cross-dataset
transfer learning method by matching the distribution of classifier errors across datasets.
3
The Fairness Model
In this section, we will formalize the setup of a supervised binary classification task subject to fairness
constraints. Assume that we are given a set of N training examples, represented by feature vectors
X = {x1 , . . . , xN } ? X = Rd , their label annotation, Y = {y1 , . . . , yN } ? Y = {+1, ?1}, and
protected characteristic information also in the form of feature vectors, Z = {z1 , . . . , zN } ? Z,
where zn encodes the protected characteristics of sample xn . The task of interest is to infer a predictor
f for the label ynew of an un-seen instance xnew , given Y , X and Z. However, f cannot use the
protected characteristic Z at decision (prediction) time, as it will constitute an unfair treatment.
The availability of protected characteristic at training time can be used to enforce fair impact
and/or fair supervised performance constraints. We first describe how to deliver fair treatment via
privileged learning. We then detail distribution matching viewpoint of fair impact and fair supervised
performance. Frameworks of privileged learning and distribution matching are suitable for protected
characteristics with binary/multi-class/continuous values. In this paper, however, we focus on a single
protected characteristic admitting binary values as in existing work (e.g. [20, 4, 5]).
3.1
Fairness through Unawareness: Privileged Learning
In the privileged learning setting [16], we are given training triplets (x1 , x?1 , y1 ), . . . , (xN , x?N , yN )
where (xn , yn ) ? X ? Y is the standard training input-output pair and x?n ? X ? is additional
information about a training instance xn . This additional (privileged) information is only available
during training. In our earlier illustrative example in the related work, xn is for example a colour
feature from a 2D image while x?n is a feature from 3D cameras and laser scanners. There is no direct
limitation on the form of privileged information, i.e. it could be yet another feature representation
like shape features from the 2D image, or a completely different modality like 3D cameras in addition
to the 2D image, that is specific for each training instance. The goal of privileged learning is to use
x?n to accelerate the learning process of inferring an optimal (ideal) predictor in the data space X ,
i.e. f : X ? Y. The difference between accelerated and non-accelerated methods is in the rate of
?
convergence to the optimal predictor, e.g. 1/N cf.1/ N for margin-based classifiers [16, 19].
From the description above, it is apparent that both privileged learning model and fairness model aim
to use data, privileged feature x?n and protected characteristic zn respectively, that are available at
3
training time only. We propose to recycle privileged learning model for achieving fairness through
unawareness by taking protected characteristics as privileged information. For a single binary
protected characteristic zn , x?n is formed by concatenating xn and zn . This is because privileged
information has to be instance specific and richer than xn alone, and this is not the case when only a
single binary protected characteristic is used. By using privileged learning framework, the predictor
f is unaware of protected characteristic zn as this information is not used as an input to the predictor.
Instead, zn , together with xn , is used to distinguish between easy-to-classify and difficult-to-classify
data instances and subsequently to use this knowledge to accelerate the learning process of a predictor
f [16, 17]. Easiness and hardness can be defined, for example, based on the distance of data instance
to the decision boundary (margin) [16, 17, 19] or based on the steepness of the logistic likelihood
function [18]. Our specific choice of easiness and hardness definition is detailed in Section 3.3.
A direct advantage of approaching fairness from the privileged lens is the learning acceleration can
be used to limit the performance degradation of the fair model as it now has to trade-off two goals:
good prediction performance and respecting fairness constraints. An obvious disadvantage is an
increased risk of unfairness by proxy as knowledge of easy-to-classify and difficult-to-classify data
instances is based on protected characteristics. The next section describes a way to alleviate this
based on a distribution matching principle.
3.2
Demographic Parity, Equalized Odds, Equality of opportunity, and Beyond: Matching
Conditional Distributions
We have the following definitions for several fairness criteria [25, 4, 5]:
Definition A Demographic parity (fair impact): A binary decision model is fair if its decision
{+1, ?1} are independent of the protected characteristic z ? {0, 1}. A decision f? satisfies this
definition if
P (sign(f?(x)) = +1|z = 0) = P (sign(f?(x)) = +1|z = 1).
Definition B Equalized odds (fair supervised performance): A binary decision model is fair if its
decisions {+1, ?1} are conditionally independent of the protected characteristic z ? {0, 1} given
the target outcome y. A decision f? satisfies this definition if
P (sign(f?(x)) = +1|z = 0, y) = P (sign(f?(x)) = +1|z = 1, y), for y ? {+1, ?1}.
For the target outcome y = +1, the definition above requires that f? has equal true positive rates
across two different values of protected characteristic. It requires f? to have equal false positive rates
for the target outcome y = ?1.
Definition C Equality of opportunity (fair supervised performance): A binary decision model is fair
if its decisions {+1, ?1} are conditionally independent of the protected characteristic z ? {0, 1}
given the positive target outcome y. A decision f? satisfies this definition if
P (sign(f?(x)) = +1|z = 0, y = +1) = P (sign(f?(x)) = +1|z = 1, y = +1).
Equality of opportunity only constrains equal true positive rates across the two demographics.
All three fairness criteria rely on the definition that data across the two demographics should exhibit
similar behaviour, i.e. matching positive predictions, matching true positive rates, and matching false
positive rates. A natural pathway to inject these into any learning model is to use a distribution
matching framework. This matching assumption is well founded if we assume that both data
z=1
Xz=0 = {xz=0
, . . . , xz=0
, . . . , xz=1
1
Nz=0 } ? X and another data Xz=1 = {x1
Nz=1 } ? X are drawn
independently and identically distributed from the same distribution p(x) on a domain X . It therefore
follows that for any function (or set of functions) f the distribution of f (x) where x ? p(x) should
also behave in the same way across the two demographics. We know that this is not automatically
true if we get to choose f after seeing Xz=0 and Xz=1 . In order to allow us to draw on a rich body
of literature for comparing distributions, we cast the goal of enforcing distributional similarity across
two demographics as a two-sample problem.
3.2.1
Distribution matching
First, we denote the applications of our predictor f? : X ? R to data having protected
?
characteristic value zero by f?(XZ=0 ) := {f?(xz=0
), . . . , f?(xz=0
1
Nz=0 )}, likewise by f (XZ=1 ) :=
4
{f?(xz=1
), . . . , f?(xz=1
1
Nz=1 )} for value one. For enforcing the demographic parity criterion, we can
enforce the closeness between the distributions of f?(x). We can achieve this by minimizing:
D(f?(XZ=0 ), f?(XZ=1 )), the distance between the two distributions f?(XZ=0 ) and f?(XZ=1 ). (1)
For enforcing the equalized odds criterion, we need to minimize both
D(I[Y = +1]f?(XZ=0 ), I[Y = +1]f?(XZ=1 )) and D(I[Y = ?1]f?(XZ=0 ), I[Y = ?1]f?(XZ=1 )).
(2)
We make use of Iverson?s bracket notation: I[P ] = 1 when condition P is true and 0 otherwise.
The first will match true positive rates (and also false negative rates) across the two demographics
and the latter will match false positive rates (and also true negative rates). For enforcing equality of
opportunity, we just need to minimize
D(I[Y = +1]f?(XZ=0 ), I[Y = +1]f?(XZ=1 )).
(3)
To go beyond true positive rates and false positive rates, Zafar et al. [5] raise the potential of removing
unfairness by enforcing equal misclassification rates, false discovery rates, and false omission rates
across two demographics. False discovery and false omission rates, however, with their fairness
model are difficult to encode. In the distribution matching sense, those can be easily enforced by
minimizing
D(1 ? Y f?(XZ=0 ), 1 ? Y f?(XZ=1 )),
D(I[Y = +1] max(0, ?f?(XZ=0 )), I[Y = +1] max(0, ?f?(XZ=1 ))), and
D(I[Y = ?1] max(0, f?(XZ=0 )), I[Y = ?1] max(0, f?(XZ=1 )))
(4)
(5)
(6)
for misclassification, false omission, and false discovery rates, respectively.
Maximum mean discrepancy To avoid a parametric assumption on the distance estimate between
distributions, we use the Maximum Mean Discrepancy (MMD) criterion [37], a non-parametric
distance estimate. Denote by H a Reproducing Kernel Hilbert Space with kernel k defined on X . In
this case one can show [37] that whenever k is characteristic (or universal), the map
2
? :p ? ?[p] := Ex?p(x) [k(f?(x), ?)] with associated distance MMD2 (p, p0 ) := k?[p] ? ?[p0 ]k
characterizes a distribution uniquely. Examples of characteristic kernels [39] are Gaussian RBF, Laplacian and B2n+1 -splines.
With a this choice of kernel functions, the
MMD criterion matches infinitely many moments in the Reproducing Kernel Hilbert Space
(RKHS). We use an unbiased linear-time estimate of MMD as follows [37, Lemma 14]:
\2 = 1 PN k(f?(x2i?1 ), f?(x2i )) ? k(f?(x2i?1 ), f?(x2i )) ? k(f?(x2i ), f?(x2i?1 )) +
MMD
z=0
z=1
z=0
z=0
z=0
z=1
i
N
? 2i
k(f?(x2i?1
z=1 ), f (xz=1 )), with N := bmin(Nz=1 , Nz=0 )c.
3.2.2
Special cases
Before discussing a specific composition of privileged learning and distribution matching to achieve
fairness, we consider a number of special cases of matching constraint to show that many of existing
methods use this basic idea.
Mean matching for demographic parity Zemel et al. [20] balance the mapping from data to
one of C latent prototypes across the two demographics by imposing the following constraint:
PNz=0 ? z=0
PNz=1 ? z=1
1
1
? z=0
n=1 f (xn ; c) = Nz=1
n=1 f (xn ; c); ?c = 1, . . . , C, where f (xn ) is a softmax
Nz=0
function with C prototypes. Assuming a linear kernel k on this constraint is equivalent to requiring
that for each c
?[f?(xz=0
n ; c)] =
1
N
z=0
X
Nz=0
n=1
D
E
f?(xz=0
n ; c), ? =
1
N
z=1
X
Nz=1
n=1
D
E
? z=1
f?(xz=1
n ; c), ? = ?[f (xn ; c)].
Mean matching for equalized odds and equality of opportunity To ensure equal false positive
rates across the two demographics, Zafar et al. [5] add the following constraint to the training objective
5
PNz=0
PNz=1
min(0, I[yn = ?1]f?(xz=0
of a linear classifier f?(x) = hw, xi: n=1
n )) =
n=1 min(0, I[yn =
?1]f?(xz=1
)).
Again,
assuming
a
linear
kernel
k
on
this
constraint
is
equivalent
to requiring that
n
N
z=0 D
E
1 X
min(0, I[yn = ?1]f?(xz=0
?[min(0, I[yn = ?1]f?(xz=0
n ))] =
n )), ?
Nz=0 n=1
=
1
N
z=1
X
Nz=1
n=1
D
E
? z=1
min(0, I[yn = ?1]f?(xz=1
n )), ? = ?[min(0, I[yn = ?1]f (xn ))].
The min(?) function ensures that we only match false positive rates as without it both false positive
and true negative rates will be matched. Relying on means for matching both false positive and true
negative is not sufficient as the underlying distributions are multi-modal; it motivates the need for
distribution matching.
3.3
Privileged learning with fairness constraints
Here we describe the proposed model that recycles two established frameworks, privileged learning
and distribution matching, and subsequently harmonizes them for addressing fair treatment, fair
impact, fair supervised performance and beyond in a unified fashion. We use SVM? + [19], an
SVM-based classification method for privileged learning, as a building block. SVM? + modifies
the required distance of data instance to the decision boundary based on easiness/hardness of
that data instance in the privileged space X ? , a space that contains protected characteristic Z.
Easiness/hardness is reflected in the negative of the confidence, ?yn (hw? , x?n i + b? ) where w? and
b? are some parameters; the higher this value, the harder this data instance to be classified correctly
even in the rich privileged space. Injecting the distribution matching constraint, the final Distribution
Matching+ (DM+) optimization problem is now:
minimize 1/2
2
kwk`2
| {z }
+1/2?
w?Rd ,b?R
?
w? ?Rd ,b? ?R regularisation on model without
protected characteristic
+C
N
X
2
kw? k`2
| {z }
+C?
N
X
max (0, ?yn [hw? , x?n i + b? ]) +
n=1
regularisation on model with
protected characteristic
|
{z
}
hinge loss on model with protected characteristic
max (0, 1 ? yn [hw? , x?n i + b? ] ? yn [hw, xn i + b])
(7a)
n=1
|
subject to
{z
hinge loss on model without protected characteristic
but with margin dependent on protected characteristic
\2 (p , p ) ? ,
MMD
z=0 z=1
|
{z
}
}
(7b)
constraint for removing unfairness by proxy
where C, ?, ? and an upper-bound are hyper-parameters. Terms pz=0 and pz=1 are distributions over
appropriately defined fairness variables across the two demographics, e.g. f?(XZ=0 ) and f?(XZ=1 )
with f?(?) = hw, ?i + b for demographic parity and I[Y = +1]f?(XZ=0 ) and I[Y = +1]f?(XZ=1 )
for equality of opportunity. We have the following observations of the knowledge transfer from the
privileged space to the space X without protected characteristic (refer to the last term in (7a)):
? Very large positive value of the negative of the confidence in the space that includes protected
characteristic, ?yn [hw? , x?n i + b? ] >> 0 means xn , without protected characteristic, is expected
to be a hard-to-classify instance therefore its margin distance to the decision boundary is increased.
? Very large negative value of the negative of the confidence in the space that includes protected
characteristic, ?yn [hw? , x?n i + b? ] << 0 means xn , without protected characteristic, is expected
to be an easy-to-classify instance therefore its margin distance to the decision boundary is reduced.
The formulation in (7) is a multi-objective optimization with three competing goals: minimizing
empirical error (hinge loss), minimizing model complexity (`2 regularisation), and minimizing
prediction discrepancy across the two demographics (MMD). Each goal corresponds to a different
optimal solution and we have to accept a compromise in the goals. While solving a single-objective
optimization implies to search for a single best solution, a collection of solutions at which no goal
can be improved without damaging one of the others (Pareto frontier) [40] is sought when solving a
multi-objective optimization.
6
Multi-objective optimization We first note that the MMD fairness criteria will introduce nonconvexity to our optimization problem. For a non-convex multi-objective optimization, the Pareto
frontier may have non-convex portions. However, any Pareto optimal solution of a multi-objective
optimization can be obtained by solving the constraint problem for an upper bound (as in (7b))
regardless of the non-convexity of the Pareto frontier [40].
Alternatively, the Convex Concave Procedure (CCP) [41], can be used to find an approximate solution
of the problem in (7) by solving a succession of convex programs. CCP has been used in several
other algorithms enforcing fair impact and fair supervised performance to deal with non-convexity of
the objective function (e.g. [24, 5]). However, it was noted in [35] that for an objective function that
has an additive structure as in our DM+ model, it is better to use the non-convex objective directly.
4
Experiments
We experiment with two datasets: The ProPublica COMPAS dataset and the Adult income dataset.
ProPublica COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) has a
total of 5,278 data instances, each with 5 features (e.g., count of prior offences, charge for which the
person was arrested, race). The binary target outcome is whether or not the defendant recidivated
within two years. For this dataset, we follow the setting in [5] and consider race, which is binarized
as either black or white, as a protected characteristic. We use 4, 222 instances for training and 1, 056
instances for test. The Adult dataset has a total of 45, 222 data instances, each with 14 features (e.g.,
gender, educational level, number of work hours per week). The binary target outcome is whether
or not income is larger than 50K dollars. For this dataset, we follow [20] and consider gender as a
binary protected characteristic. We use 36, 178 instances for training and 9, 044 instances for test.
Methods We have two variants of our distribution matching framework: DM that uses SVM as the
base classifier coupled with the constraint in (7b), and DM+ ((7a) and (7b)). We compare our methods
with several baselines: support vector machine (SVM), logistic regression (LR), mean matching with
logistic regression as the base classifier (Zafar et al.) [5], and a threshold classifier method with
protected characteristic-specific thresholds on the output of a logistic regression model (Hardt et
al.) [4]. All methods but Hardt et al. do not use protected characteristics at prediction time.
Optimization procedure For our DM and DM+ methods, we identify at least three options on how
to optimize the multi-objective optimization problem in (7): using Convex Concave Procedure (CCP),
using Broyden-Fletcher-Goldfarb-Shanno gradient descent method with limited-memory variation
(L-BFGS), and using evolutionary multi-objective optimization (EMO). We discuss those options
\2 (p , p ) fairness constraint (7b)
in turn. First, we can express each additive term in the MMD
z=0 z=1
as a difference of two convex functions, find the convex upper bound of each term, and place the
convexified fairness constraint as part of the objective function. In our initial experiments, solving
(7) with CCP tends to ignore the fairness constraint, therefore we do not explore this approach
further. As mentioned earlier, the convex upper bounds on each of the additive terms in the MMD
constraint become increasingly loose as we move away from the current point of approximation. This
leads to the second optimization approach. We turn the constrained optimization problem into an
\2 (p , p ) term.
unconstrained one by introducing a non-negative weight CMMD to scale the MMD
z=0 z=1
We then solve this unconstrained problem using L-BFGS. The main challenge with this procedure
is the need to trade-off multiple competing goals by tuning several hyper-parameters, which will
be discussed in the next section. The CCP and L-BFGS procedures will only return one optimal
solution from the Pareto frontier. Third, to approximate the Pareto-optimal set, we can instead use
EMO procedures (e.g. Non-dominated Sorting Genetic Algorithm (NSGA) ? II and Strength Pareto
Evolutionary Algorithm (SPEA) - II). For the EMO, we also solve the unconstrained problem as
in the second approach, but we do not need to introduce a trade-off parameter for each term in the
objective function. We use the DEAP toolbox [42] for experimenting with EMO.
Model selection For the baseline Zafar et al., as in [5], we set the hyper-parameters ? and
? corresponding to the Penalty Convex Concave procedure to 5.0 and 1.2, respectively. Gaussian RBF kernel with a kernel width ? 2 is used for the MMD term. When solving DM (and
DM+) optimization problems with L-BFGS, the hyper-parameters C, CMMD , ? 2 , (and ?) are set
to 1000., 5000., 10., (and 1.) for both datasets. For DM+, we select ? over the range {1., 2., . . . , 10.}
7
Table 1: Results on multi-objective optimization which balances two main objectives: performance
accuracies and fairness criteria. Equal true positive rates are required for ProPublica COMPAS
dataset, and equal accuracies between two demographics z = 0 and z = 1 are required for Adult
dataset. The solver of Zafar et al. fails on the Adult dataset while enforcing equal accuracies
across the two demographics. Hardt et al.?s method does not enforce equal accuracies. SVM and
LR only optimize performance accuracies. The terms |Acc.z=0 - Acc.z=1 |, |TPRz=0 - TPRz=1 |, and
|FPRz=0 - FPRz=1 | denote accuracy, true positive rate, and false positive rate discrepancies in an
absolute term between the two demographics (the smaller the fairer). For ProPublica COMPAS
dataset, we boldface |TPRz=0 - TPRz=1 | since we enforce the equality of opportunity criterion on
this dataset. For Adult dataset, we boldface |Acc.z=0 - Acc.z=1 | since this is the fairness criterion.
ProPublica COMPAS dataset (Fairness Constraint on equal TPRs)
|Acc.z=0 - Acc.z=1 | |TPRz=0 - TPRz=1 |
LR
0.0151?0.0116
0.2504?0.0417
SVM
0.0172?0.0102
0.2573?0.0158
Zafar et al.
0.0174?0.0142
0.1144?0.0482
Hardt et al.?
0.0219?0.0191
0.0463?0.0185
DM (L-BFGS)
0.0457?0.0289
0.1169?0.0690
DM+ (L-BFGS)
0.0608?0.0259
0.1065?0.0413
DM (EMO Usr1)
0.0537?0.0121
0.1346?0.0360
DM (EMO Usr2)
0.0535?0.0213
0.1248?0.0509
?
use protected characteristics at prediction time.
|FPRz=0 - FPRz=1 |
0.1618?0.0471
0.1603?0.0490
0.1914?0.0314
0.0518?0.0413
0.0791?0.0395
0.0973?0.0272
0.1028?0.0481
0.0906?0.0507
Acc.
0.6652?0.0139
0.6367?0.0212
0.6118?0.0198
0.6547?0.0128
0.5931?0.0599
0.6089?0.0398
0.6261?0.0133
0.6148?0.0137
Adult dataset (Fairness Constraint on equal accuracies)
SVM
DM (L-BFGS)
DM+ (L-BFGS)
DM (EMO Usr1)
DM (EMO Usr2)
|Acc.z=0 - Acc.z=1 |
0.1136?0.0064
0.0640?0.0280
0.0459?0.0372
0.0388?0.0179
0.0482?0.0143
|TPRz=0 - TPRz=1 |
0.0964?0.0289
0.0804?0.0659
0.0759?0.0738
0.0398?0.0284
0.0302?0.0212
|FPRz=0 - FPRz=1 |
0.0694?0.0109
0.0346?0.0343
0.0368?0.0349
0.0398?0.0284
0.0135?0.0056
Acc.
0.8457?0.0034
0.8152?0.0068
0.8127?0.0134
0.8057?0.0108
0.8111?0.0122
using 5-fold cross validation. The selection process goes as follow: we first sort ? values according
to how well they satisfy the fairness criterion, we then select a ? value at a point before it yields
a lower incremental classification accuracy. As stated earlier, we do not need C, CMMD , ? 2 , ?, ?
hyper-parameters for balancing multiple terms in the objective function when using EMO for DM
and DM+. There are however several free parameters related to the evolutionary algorithm itself.
We use the NSGA ? II selection strategy with a polynomial mutation operator as in the the original
implementation [43], and the mutation probability is set to 0.5. We do not use any mating operator.
We use 500 individuals in a loop of 50 iterations (generations).
Results Experimental results over 5 repeats are presented in Table 1. In the ProPublica COMPAS
dataset, we enforce equality of opportunity |TPRz=0 - TPRz=1 |, i.e. equal true positive rates (Equation
(3)), as the fairness criterion (refer to the ProPublica COMPAS dataset in Table 1). Additionally, our
distribution matching methods, DM+ and DM also deliver a reduction in discrepancies between false
positive rates. We experiment with both L-BFGS and EMO optimization procedures. For EMO, we
simulate two decision makers choosing an operating point based on the visualization of Pareto frontier
in Figure 1 ? Right (shown as DM (EMO Usr1) and DM (EMO Usr2) in Table 1). For this dataset,
Usr1 has an inclination to be more lenient in being fair for a gain in accuracy in comparison to the
Usr2. This is actually reflected in the selection of the operating point (see supplementary material).
The EMO is run on the 60% of the training data, the selection is done on the remaining 40%, and the
reported results are on the separate test set based on the model trained on the 60% of the training data.
The method Zafar et al. achieves similar reduction rate in the fairness criterion to our distribution
matching methods. As a reference, we also include results of Hardt et al.?s method; it achieves
the best equality of opportunity measure with only a slight drop in accuracy performance w.r.t. the
unfair LR. It is important to note that Hardt et al.?s method requires protected characteristics at
test time. If we allow the usage of protected characteristics at test time, we should expect similar
reduction rate in fairness and accuracy measures for other methods [5].
8
SVM
0.25
|T P RZ=0 ? T P RZ=1 |
0.06
0.04
0.20
0.02
MMD0.00
0.15
0.02
0.04
0.06
0.08
0.10
0.05
0
Re 5 10
gu 15
lar 20 25
iza 30
tio 35
n 40
2.50
2.25
1.50
1.25
1.00
0.00
0.75
Loss
Hinge
2.00
1.75
0.35
0.40
0.45
0.50
Classification Error
0.55
0.60
Figure 1: Visualization of a Pareto frontier of our DM method for the ProPublica COMPAS dataset.
Left: In a 3D criterion space corresponding to the three objective functions: hinge loss, i.e.
2
\2 (p , p ).
max (0, 1 ? yn [hw, xn i + b]), regularization, i.e. kwk`2 , and MMD, i.e. MMD
z=0 z=1
Fairer models (smaller MMD values) are gained at the expense of model complexity (higher regularization and/or hinge loss values). Note that the unbiased estimate of MMD may be negative [37].
Right: The same Pareto frontier but in a 2D space of error and unfairness in predictions. Only the
first repeat is visualized; please refer to the supplementary material for the other four repeats, for the
Adult dataset, and of the DM+ method.
In the Adult dataset, we enforce equal accuracies |Acc.z=0 - Acc.z=1 | (Equation (4)) as the fairness
criterion (refer to the Adult dataset in Table 1). The method whereby a decision maker uses a
Pareto frontier visualization for choosing the operating point (DM (EMO Usr1)) reaches the smallest
discrepancy between the two demographics. In addition to equal accuracies (Equation (4)), our
distribution matching methods, DM+ and DM, also deliver a reduction in discrepancies between true
positive and false positive rates w.r.t. SVM (second and third column). In this dataset, Zafar et
al. falls into numerical problems when enforcing equal accuracies (vide our earlier discussion on
different optimization procedures, especially related to CCP). As observed in prior work [5, 20],
the methods that do not enforce fairness (equal accuracies or equal true positive rates), SVM and LR,
achieve higher classification accuracy compared to the methods that do enforce fairness: Zafar et
al., DM+, and DM. This can be seen in the last column of Table 1.
5
Discussion and Conclusion
We have proposed a unified machine learning framework that is able to handle any definitions of
fairness, e.g. fairness through unawareness, demographic parity, equalized odds, and equality of opportunity. Our framework is based on learning using privileged information and matching conditional
distributions using a two-sample problem. By using distance measures in Hilbert Space to solve the
two-sample problem, our framework is general and will be applicable for protected characteristics
with binary/multi-class/continuous values. The current work focuses on a single binary protected
characteristic. This corresponds to conditional distribution matching with a binary conditioning
variable. To generalize this to any type and multiple dependence of protected characteristics, we can
use the Hilbert Space embedding of conditional distributions framework of [44, 45].
We note that there are important factors external to machine learning models that are relevant to
fairness. However, this paper adopts the established approach of existing work on fair machine
learning. In particular, it is taken as given that one typically does not have any control over the data
collection process because there is no practical way of enforcing truth/un-biasedness in datasets that
are generated by others, such as banks, police forces, and companies.
Acknowledgments
NQ is supported by the UK EPSRC project EP/P03442X/1 ?EthicalML: Injecting Ethical and Legal
Constraints into Machine Learning Models? and the Russian Academic Excellence Project ?5-100?.
We gratefully acknowledge NVIDIA for GPU donation and Amazon for AWS Cloud Credits. We
also thank Kristian Kersting and Oliver Thomas for helpful discussions, Muhammad Bilal Zafar for
sharing his implementations of [4] and [5], and Sienna Quadrianto for supporting the work.
9
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12
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6,269 | 6,671 | Safe and Nested Subgame Solving for
Imperfect-Information Games
Noam Brown
Computer Science Department
Carnegie Mellon University
Pittsburgh, PA 15217
[email protected]
Tuomas Sandholm
Computer Science Department
Carnegie Mellon University
Pittsburgh, PA 15217
[email protected]
Abstract
In imperfect-information games, the optimal strategy in a subgame may depend
on the strategy in other, unreached subgames. Thus a subgame cannot be solved
in isolation and must instead consider the strategy for the entire game as a whole,
unlike perfect-information games. Nevertheless, it is possible to first approximate
a solution for the whole game and then improve it in individual subgames. This
is referred to as subgame solving. We introduce subgame-solving techniques that
outperform prior methods both in theory and practice. We also show how to adapt
them, and past subgame-solving techniques, to respond to opponent actions that
are outside the original action abstraction; this significantly outperforms the prior
state-of-the-art approach, action translation. Finally, we show that subgame solving
can be repeated as the game progresses down the game tree, leading to far lower
exploitability. These techniques were a key component of Libratus, the first AI to
defeat top humans in heads-up no-limit Texas hold?em poker.
1
Introduction
Imperfect-information games model strategic settings that have hidden information. They have a
myriad of applications including negotiation, auctions, cybersecurity, and physical security.
In perfect-information games, determining the optimal strategy at a decision point only requires
knowledge of the game tree?s current node and the remaining game tree beyond that node (the
subgame rooted at that node). This fact has been leveraged by nearly every AI for perfect-information
games, including AIs that defeated top humans in chess [6] and Go [28]. In checkers, the ability to
decompose the game into smaller independent subgames was even used to solve the entire game [26].
However, it is not possible to determine a subgame?s optimal strategy in an imperfect-information
game using only knowledge of that subgame, because the game tree?s exact node is typically unknown.
Instead, the optimal strategy may depend on the value an opponent could have received in some other,
unreached subgame. Although this is counter-intuitive, we provide a demonstration in Section 2.
Rather than rely on subgame decomposition, past approaches for imperfect-information games
typically solved the game as a whole upfront. For example, heads-up limit Texas hold?em, a relatively
simple form of poker with 1013 decision points, was essentially solved without decomposition [2].
However, this approach cannot extend to larger games, such as heads-up no-limit Texas hold?em?the
primary benchmark in imperfect-information game solving?which has 10161 decision points [15].
The standard approach to computing strategies in such large games is to first generate an abstraction
of the game, which is a smaller version of the game that retains as much as possible the strategic
characteristics of the original game [23, 25, 24]. For example, a continuous action space might
be discretized. This abstract game is solved and its solution is used when playing the full game
by mapping states in the full game to states in the abstract game. We refer to the solution of an
abstraction (or more generally any approximate solution to a game) as a blueprint strategy.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In heavily abstracted games, a blueprint strategy may be far from the true solution. Subgame solving
attempts to improve upon the blueprint strategy by solving in real time a more fine-grained abstraction
for an encountered subgame, while fitting its solution within the overarching blueprint strategy.
2
Coin Toss
In this section we provide intuition for why an imperfect-information subgame cannot be solved in
isolation. We demonstrate this in a simple game we call Coin Toss, shown in Figure 1a, which will
be used as a running example throughout the paper.
Coin Toss is played between players P1 and P2 . The figure shows rewards only for P1 ; P2 always
receives the negation of P1 ?s reward. A coin is flipped and lands either Heads or Tails with equal
probability, but only P1 sees the outcome. P1 then chooses between actions ?Sell? and ?Play.? The
Sell action leads to a subgame whose details are not important, but the expected value (EV) of
choosing the Sell action will be important. (For simplicity, one can equivalently assume in this
section that Sell leads to an immediate terminal reward, where the value depends on whether the
coin landed Heads or Tails). If the coin lands Heads, it is considered lucky and P1 receives an EV of
$0.50 for choosing Sell. On the other hand, if the coin lands Tails, it is considered unlucky and P1
receives an EV of ?$0.50 for action Sell. (That is, P1 must on average pay $0.50 to get rid of the
coin). If P1 instead chooses Play, then P2 may guess how the coin landed. If P2 guesses correctly,
then P1 receives a reward of ?$1. If P2 guesses incorrectly, then P1 receives $1. P2 may also forfeit,
which should never be chosen but will be relevant in later sections. We wish to determine the optimal
strategy for P2 in the subgame S that occurs after P1 chooses Play, shown in Figure 1a.
Figure 1: (a) The example game of Coin Toss. ?C? represents a chance node. S is a Player 2 (P2 ) subgame.
The dotted line between the two P2 nodes means that P2 cannot distinguish between them. (b) The public game
tree of Coin Toss. The two outcomes of the coin flip are only observed by P1 .
Were P2 to always guess Heads, P1 would receive $0.50 for choosing Sell when the coin lands Heads,
and $1 for Play when it lands Tails. This would result in an average of $0.75 for P1 . Alternatively,
were P2 to always guess Tails, P1 would receive $1 for choosing Play when the coin lands Heads,
and ?$0.50 for choosing Sell when it lands Tails. This would result in an average reward of $0.25 for
P1 . However, P2 would do even better by guessing Heads with 25% probability and Tails with 75%
probability. In that case, P1 could only receive $0.50 (on average) by choosing Play when the coin
lands Heads?the same value received for choosing Sell. Similarly, P1 could only receive ?$0.50 by
choosing Play when the coin lands Tails, which is the same value received for choosing Sell. This
would yield an average reward of $0 for P1 . It is easy to see that this is the best P2 can do, because
P1 can average $0 by always choosing Sell. Therefore, choosing Heads with 25% probability and
Tails with 75% probability is an optimal strategy for P2 in the ?Play? subgame.
Now suppose the coin is considered lucky if it lands Tails and unlucky if it lands Heads. That is,
the expected reward for selling the coin when it lands Heads is now ?$0.50 and when it lands Tails
is now $0.50. It is easy to see that P2 ?s optimal strategy for the ?Play? subgame is now to guess
Heads with 75% probability and Tails with 25% probability. This shows that a player?s optimal
strategy in a subgame can depend on the strategies and outcomes in other parts of the game. Thus,
one cannot solve a subgame using information about that subgame alone. This is the central challenge
of imperfect-information games as opposed to perfect-information games.
2
3
Notation and Background
In a two-player zero-sum extensive-form game there are two players, P = {1, 2}. H is the set of
all possible nodes, represented as a sequence of actions. A(h) is the actions available in a node and
P (h) ? P ? c is the player who acts at that node, where c denotes chance. Chance plays an action
a ? A(h) with a fixed probability. If action a ? A(h) leads from h to h0 , then we write h ? a = h0 . If
a sequence of actions leads from h to h0 , then we write h @ h0 . The set of nodes Z ? H are terminal
nodes. For each player i ? P, there is a payoff function ui : Z ? < where u1 = ?u2 .
Imperfect information is represented by information sets (infosets). Every node h ? H belongs
to exactly one infoset for each player. For any infoset Ii , nodes h, h0 ? Ii are indistinguishable to
player i. Thus the same player must act at all the nodes in an infoset, and the same actions must be
available. Let P (Ii ) and A(Ii ) be such that all h ? Ii , P (Ii ) = P (h) and A(Ii ) = A(h).
A strategy ?i (Ii ) is a probability vector over A(Ii ) for infosets where P (Ii ) = i. The probability of
action a is denoted by ?i (Ii , a). For all h ? Ii , ?i (h) = ?i (Ii ). A full-game strategy ?i ? ?i defines
a strategy for each player i infoset. A strategy profile ? is a tuple of strategies, one for each player.
The expected payoff for player i if all players play the strategy profile h?i , ??i i is ui (?i , ??i ), where
??i denotes the strategies in ? of all players other than i.
Q
Let ? ? (h) = h0 ?avh ?P (h0 ) (h0 , a) denote the probability of reaching h if all players play according
to ?. ?i? (h) is the contribution of player i to this probability (that is, the probability of reaching h if
?
chance and all players other than i always chose actions leading to h). ??i
(h) is the contribution of
?
0
all players, and chance, other than i. ? (h, h ) is the probability of reaching h0 given that h has been
reached, and 0 if h 6@ h0 . This papers focuses on perfect-recall games, where a player never forgets
past information. Thus, for every Ii , ?h, h0 ? Ii , ?i? (h) = ?i? (h0 ). We define ?i? (Ii ) = ?i? (h) for
h ? Ii . Also, Ii0 @ Ii if for some h0 ? Ii0 and some h ? Ii , h0 @ h. Similarly, Ii0 ? a @ Ii if h0 ? a @ h.
A Nash equilibrium [21] is a strategy profile ? ? where no player can improve by shifting to a different
?
?
strategy, so ? ? satisfies ?i, ui (?i? , ??i
) = max?i0 ??i ui (?i0 , ??i
). A best response BR(??i ) is a
strategy for player i that is optimal against ??i . Formally, BR(??i ) satisfies ui (BR(??i ), ??i ) =
max?i0 ??i ui (?i0 , ??i ). In a two-player zero-sum game, the exploitability exp(?i ) of a strategy ?i is
how much worse ?i does against an opponent best response than a Nash equilibrium strategy would
do. Formally, exploitability of ?i is ui (? ? ) ? ui (?i , BR(?i )), where ? ? is a Nash equilibrium.
P
The expected value of a node h when players play according to ? is vi? (h) = z?Z ? ? (h, z)ui (z) .
An infoset?s value is the weighted average of the values of the nodes in the infoset, where a node
P
is weighed by the player?s belief that she is in that node. Formally, vi? (Ii ) =
P
and vi? (Ii , a) =
h?Ii
?
??i
(h)vi? (h?a)
?
h?I ??i (h)
P
h?Ii
P
?
??i
(h)vi? (h)
? (h)
??i
h?Ii
. A counterfactual best response [20] CBR(??i ) is a best
i
response that also maximizes value in unreached infosets. Specifically, a counterfactual best response is a best response ?i with the additional condition that if ?i (Ii , a) > 0 then vi? (Ii , a) =
maxa0 vi? (Ii , a0 ). We further define counterfactual best response value CBV ??i (Ii ) as the value
player i expects to achieve by playing according to CBR(??i ), having already reached infoset Ii .
hCBR(??i ),??i i
hCBR(??i ),??i i
Formally, CBV ??i (Ii ) = vi
(Ii ) and CBV ??i (Ii , a) = vi
(Ii , a).
An imperfect-information subgame, which we refer to simply as a subgame in this paper, can in
most cases (but not all) be described as including all nodes which share prior public actions (that is,
actions viewable to both players). In poker, for example, a subgame is uniquely defined by a sequence
of bets and public board cards. Figure 1b shows the public game tree of Coin Toss. Formally, an
imperfect-information subgame is a set of nodes S ? H such that for all h ? S, if h @ h0 , then
h0 ? S, and for all h ? S and all i ? P, if h0 ? Ii (h) then h0 ? S. Define Stop as the set of
earliest-reachable nodes in S. That is, h ? Stop if h ? S and h0 6? S for any h0 @ h.
4
Prior Approaches to Subgame Solving
This section reviews prior techniques for subgame solving in imperfect-information games, which we
build upon. Throughout this section, we refer to the Coin Toss game shown in Figure 1a.
As discussed in Section 1, a standard approach to dealing with large imperfect-information games is
to solve an abstraction of the game. The abstract solution is a (probably suboptimal) strategy profile
3
in the full game. We refer to this full-game strategy profile as the blueprint. The goal of subgame
solving is to improve upon the blueprint by changing the strategy only in a subgame.
Figure 2: The blueprint strategy we refer to in the game of Coin Toss. The Sell action leads to a subgame that is
not displayed. Probabilities are shown for all actions. The dotted line means the two P2 nodes share an infoset.
The EV of each P1 action is also shown.
Assume that a blueprint strategy profile ? (shown in Figure 2) has already been computed for Coin
Toss in which P1 chooses Play 34 of the time with Heads and 12 of the time with Tails, and P2 chooses
Heads 12 of the time, Tails 14 of the time, and Forfeit 14 of the time after P1 chooses Play. The details
of the blueprint strategy in the Sell subgame are not relevant in this section, but the EV for choosing
the Sell action is relevant. We assume that if P1 chose the Sell action and played optimally thereafter,
then she would receive an expected payoff of 0.5 if the coin is Heads, and ?0.5 if the coin is Tails.
We will attempt to improve P2 ?s strategy in the subgame S that follows P1 choosing Play.
4.1 Unsafe Subgame Solving
We first review the most intuitive form of subgame solving, which we refer to as Unsafe subgame
solving [1, 11, 12, 9]. This form of subgame solving assumes both players played according to the
blueprint strategy prior to reaching the subgame. That defines a probability distribution over the
nodes at the root of the subgame S, representing the probability that the true game state matches that
node. A strategy for the subgame is then calculated which assumes that this distribution is correct.
In all subgame solving algorithms, an augmented subgame containing S and a few additional nodes
is solved to determine the strategy for S. Applying Unsafe subgame solving to the blueprint strategy
in Coin Toss (after P1 chooses Play) means solving the augmented subgame shown in Figure 3a.
Specifically, the augmented subgame consists of only an initial chance node and S. The initial chance
?
node reaches h ? Stop with probability P 0 ? (h)
? ? (h0 ) . The augmented subgame is solved and its
h ?Stop
strategy for P2 is used in S rather than the blueprint strategy.
Unsafe subgame solving lacks theoretical solution quality guarantees and there are many situations
where it performs extremely poorly. Indeed, if it were applied to the blueprint strategy of Coin Toss
then P2 would always choose Heads?which P1 could exploit severely by only choosing Play with
Tails. Despite the lack of theoretical guarantees and potentially bad performance, Unsafe subgame
solving is simple and can sometimes produce low-exploitability strategies, as we show later.
We now move to discussing safe subgame-solving techniques, that is, ones that ensure that the
exploitability of the strategy is no higher than that of the blueprint strategy.
(a) Unsafe subgame solving
(b) Resolve subgame solving
Figure 3: The augmented subgames solved to find a P2 strategy in the Play subgame of Coin Toss.
4
4.2 Subgame Resolving
In subgame Resolving [5], a safe strategy is computed for P2 in the subgame by solving the augmented
subgame shown in Figure 3b, producing an equilibrium strategy ? S . This augmented subgame differs
from Unsafe subgame solving by giving P1 the option to ?opt out? from entering S and instead
receive the EV of playing optimally against P2 ?s blueprint strategy in S.
Specifically, the augmented subgame for Resolving differs from unsafe subgame solving as follows.
For each htop ? Stop we insert a new P1 node hr , which exists only in the augmented subgame,
between the initial chance node and htop . The set of these hr nodes is Sr . The initial chance node
connects to each node hr ? Sr in proportion to the probability that player P1 could reach htop if P1
?
tried to do so (that is, in proportion to ??1
(htop )). At each node hr ? Sr , P1 has two possible actions.
0
0
Action aS leads to htop , while action aT leads to a terminal payoff that awards the value of playing
optimally against P2 ?s blueprint strategy, which is CBV ?2 (I1 (htop )). In the blueprint strategy of
Coin Toss, P1 choosing Play after the coin lands Heads results in an EV of 0, and 12 if the coin is
Tails. Therefore, a0T leads to a terminal payoff of 0 for Heads and 12 for Tails. After the equilibrium
strategy ? S is computed in the augmented subgame, P2 plays according to the computed subgame
strategy ?2S rather than the blueprint strategy when in S. The P1 strategy ?1S is not used.
Clearly P1 cannot do worse than always picking action a0T (which awards the highest EV P1 could
achieve against P2 ?s blueprint). But P1 also cannot do better than always picking a0T , because P2
could simply play according to the blueprint in S, which means action a0S would give the same EV to
P1 as action a0T (if P1 played optimally in S). In this way, the strategy for P2 in S is pressured to be
no worse than that of the blueprint. In Coin Toss, if P2 were to always choose Heads (as was the case
in Unsafe subgame solving), then P1 would always choose a0T with Heads and a0S with Tails.
Resolving guarantees that P2 ?s exploitability will be no higher than the blueprint?s (and may be
better). However, it may miss opportunities for improvement. For example, if we apply Resolving to
the example blueprint in Coin Toss, one solution to the augmented subgame is the blueprint itself, so
P2 may choose Forfeit 25% of the time even though Heads and Tails dominate that action. Indeed,
the original purpose of Resolving was not to improve upon a blueprint strategy in a subgame, but
rather to compactly store it by keeping only the EV at the root of the subgame and then reconstructing
the strategy in real time when needed rather than storing the whole subgame strategy.
Maxmargin subgame solving [20], discussed in Appendix A, can improve performance by definS
S
ing a margin M ? (I1 ) = CBV ?2 (I1 ) ? CBV ?2 (I1 ) for each I1 ? Stop and maximizing
S
minI1 ?Stop M ? (I1 ). Resolving only makes all margins nonnegative. However, Maxmargin does
worse in practice when using estimates of equilibrium values as discussed in Appendix C.
5
Reach Subgame Solving
All of the subgame-solving techniques described in Section 4 only consider the target subgame in
isolation, which can lead to suboptimal strategies. For example, Maxmargin solving applied to S
in Coin Toss results in P2 choosing Heads with probability 58 and Tails with 38 in S. This results in
P1 receiving an EV of ? 14 by choosing Play in the Heads state, and an EV of 14 in the Tails state.
However, P1 could simply always choose Sell in the Heads state (earning an EV of 0.5) and Play in
the Tails state and receive an EV of 38 for the entire game. In this section we introduce Reach subgame
solving, an improvement to past subgame-solving techniques that considers what the opponent could
have alternatively received from other subgames.1 For example, a better strategy for P2 would be
to choose Heads with probability 34 and Tails with probability 14 . Then P1 is indifferent between
choosing Sell and Play in both cases and overall receives an expected payoff of 0 for the whole game.
However, that strategy is only optimal if P1 would indeed achieve an EV of 0.5 for choosing Sell
in the Heads state and ?0.5 in the Tails state. That would be the case if P2 played according to the
blueprint in the Sell subgame (which is not shown), but in reality we would apply subgame solving to
the Sell subgame if the Sell action were taken, which would change P2 ?s strategy there and therefore
P1 ?s EVs. Applying subgame solving to any subgame encountered during play is equivalent to
applying it to all subgames independently; ultimately, the same strategy is played in both cases. Thus,
we must consider that the EVs from other subgames may differ from what the blueprint says because
subgame solving would be applied to them as well.
1
Other subgame-solving methods have also considered the cost of reaching a subgame [30, 14]. However,
those approaches are not correct in theory when applied in real time to any subgame reached during play.
5
Figure 4: Left: A modified game of Coin Toss with two subgames. The nodes C1 and C2 are public chance
nodes whose outcomes are seen by both P1 and P2 . Right: An augmented subgame for one of the subgames
according to Reach subgame solving. If only one of the subgames is being solved, then the alternative payoff
for Heads can be at most 1. However, if both are solved independently, then the gift must be split among the
subgames and must sum to at most 1. For example, the alternative payoff in both subgames can be 0.5.
As an example of this issue, consider the game shown in Figure 4 which contains two identical
subgames S1 and S2 where the blueprint has P2 pick Heads and Tails with 50% probability. The Sell
action leads to an EV of 0.5 from the Heads state, while Play leads to an EV of 0. If we were to solve
just S1 , then P2 could afford to always choose Tails in S1 , thereby letting P1 achieve an EV of 1
for reaching that subgame from Heads because, due to the chance node C1 , S1 is only reached with
50% probability. Thus, P1 ?s EV for choosing Play would be 0.5 from Heads and ?0.5 from Tails,
which is optimal. We can achieve this strategy in S1 by solving an augmented subgame in which the
alternative payoff for Heads is 1. In that augmented subgame, P2 always choosing Tails would be a
solution (though not the only solution).
However, if the same reasoning were applied independently to S2 as well, then P2 might always
choose Tails in both subgames and P1 ?s EV for choosing Play from Heads would become 1 while the
EV for Sell would only be 0.5. Instead, we could allow P1 to achieve an EV of 0.5 for reaching each
subgame from Heads (by setting the alternative payoff for Heads to 0.5). In that case, P1 ?s overall
EV for choosing Play could only increase to 0.5, even if both S1 and S2 were solved independently.
We capture this intuition by considering for each I1 ? Stop all the infosets and actions I10 ? a0 @ I1
that P1 would have taken along the path to I1 . If, at some I10 ? a0 @ I1 where P1 acted, there was a
different action a? ? A(I10 ) that leads to a higher EV, then P1 would have taken a suboptimal action
if they reached I1 . The difference in value between a? and a0 is referred to as a gift. We can afford
to let P1 ?s value for I1 increase beyond the blueprint value (and in the process lower P1 ?s value in
some other infoset in Stop ), so long as the increase to I1 ?s value is small enough that choosing actions
leading to I1 is still suboptimal for P1 . Critically, we must ensure that the increase in value is small
enough even when the potential increase across all subgames is summed together, as in Figure 4.2
A complicating factor is that gifts we assumed were present may actually not exist. For example, in
Coin Toss, suppose applying subgame solving to the Sell subgame results in P1 ?s value for Sell from
the Heads state decreasing from 0.5 to 0.25. If we independently solve the Play subgame, we have
no way of knowing that P1 ?s value for Sell is lower than the blueprint suggested, so we may still
assume there is a gift of 0.5 from the Heads state based on the blueprint. Thus, in order to guarantee a
theoretical result on exploitability that is as strong as possible, we use in our theory and experiments
a lower bound on what gifts could be after subgame solving was applied to all other subgames.
Formally, let ?2 be a P2 blueprint and let ?2?S be the P2 strategy that results from applying subgame solving independently to a set of disjoint subgames other than S. Since we do not want
?S
to compute ?2?S in order to apply subgame solving to S, let bg ?2 (I10 , a0 )c be a lower bound of
?S
?S
CBV ?2 (I10 ) ? CBV ?2 (I10 , a0 ) that does not require knowledge of ?2?S . In our experiments we
2
In this paper and in our experiments, we allow any infoset that descends from a gift to increase by the size
of the gift (e.g., in Figure 4 the gift from Heads is 0.5, so we allow P1 ?s value for Heads in both S1 and S2
to increase by 0.5). However, any division of the gift among subgames is acceptable so long as the potential
increase across all subgames (multiplied by the probability of P1 reaching that subgame) does not exceed the
original gift. For example in Figure 4 if we only apply Reach subgame solving to S1 , then we could allow the
Heads state in S1 to increase by 1 rather than just by 0.5. In practice, some divisions may do better than others.
The division we use in this paper (applying gifts equally to all subgames) did well in practice.
6
?S
use bg ?2 (I10 , a0 )c = maxa?Az (I10 )?{a0 } CBV ?2 (I10 , a) ? CBV ?2 (I10 , a0 ) where Az (I10 ) ? A(I10 )
is the set of actions leading immediately to terminal nodes. Reach subgame solving modifies the
augmented subgame in Resolving and Maxmargin by increasing the alternative payoff for infoset
P
?S
I1 ? Stop by I 0 ?a0 vI1 |P (I 0 )=P1 bg ?2 (I10 , a0 )c. Formally, we define a reach margin as
1
1
X
?S
S
S
Mr? (I1 ) = M ? (I1 ) +
bg ?2 (I10 , a0 )c
(1)
I10 ?a0 vI1 |P (I10 )=P1
?S
This margin is larger than or equal to the one for Maxmargin, because bg ?2 (I 0 , a0 )c is nonnegative.
We refer to the modified algorithms as Reach-Resolve and Reach-Maxmargin.
Using a lower bound on gifts is not necessary to guarantee safety. So long as we use a gift value
0
g ? (I10 , a0 ) ? CBV ?2 (I10 ) ? CBV ?2 (I10 , a0 ), the resulting strategy will be safe. However, using
a lower bound further guarantees a reduction to exploitability when a P1 best response reaches
with positive probability an infoset I1 ? Stop that has positive margin, as proven in Theorem 1. In
?S
practice, it may be best to use an accurate estimate of gifts. One option is to use g??2 (I10 , a0 ) =
?S
?
?2 0
?2 0
? 2 is the closest P1 can get to
?
?
CBV
(I1 ) ? CBV
(I1 , a0 ) in place of bg ?2 (I10 , a0 )c, where CBV
the value of a counterfactual best response while P1 is constrained to playing within the abstraction
that generated the blueprint. Using estimates is covered in more detail in Appendix C.
Theorem 1 shows that when subgames are solved independently and using lower bounds on gifts,
Reach-Maxmargin solving has exploitability lower than or equal to past safe techniques. The theorem
statement is similar to that of Maxmargin [20], but the margins are now larger (or equal) in size.
Theorem 1. Given a strategy ?2 in a two-player zero-sum game, a set of disjoint subgames S,
and a strategy ?2S for each subgame S ? S produced via Reach-Maxmargin solving using lower
bounds for gifts, let ?20 be the strategy that plays according to ?2S for each subgame S ? S, and ?2
elsewhere. Moreover, let ?2?S be the strategy that plays according to ?20 everywhere except for P2
BR(?20 )
nodes in S, where it instead plays according to ?2 . If ?1
(I1 ) > 0 for some I1 ? Stop , then
P
S
?2
exp(?20 ) ? exp(?2?S ) ? h?I1 ??1
(h)Mr? (I1 ).
So far the described techniques have guaranteed a reduction in exploitability over the blueprint by
setting the value of a0T equal to the value of P1 playing optimally to P2 ?s blueprint. Relaxing this
guarantee by instead setting the value of a0T equal to an estimate of P1 ?s value when both players
play optimally leads to far lower exploitability in practice. We discuss this approach in Appendix C.
6
Nested Subgame Solving
As we have discussed, large games must be abstracted to reduce the game to a tractable size. This is
particularly common in games with large or continuous action spaces. Typically the action space is
discretized by action abstraction so that only a few actions are included in the abstraction. While
we might limit ourselves to the actions we included in the abstraction, an opponent might choose
actions that are not in the abstraction. In that case, the off-tree action can be mapped to an action that
is in the abstraction, and the strategy from that in-abstraction action can be used. For example, in an
auction game we might include a bid of $100 in our abstraction. If a player bids $101, we simply
treat that as a bid of $100. This is referred to as action translation [13, 27, 7]. Action translation is
the state-of-the-art prior approach to dealing with this issue. It has been used, for example, by all the
leading competitors in the Annual Computer Poker Competition (ACPC).
In this section, we develop techniques for applying subgame solving to calculate responses to
opponent off-tree actions, thereby obviating the need for action translation. That is, rather than simply
treat a bid of $101 as $100, we calculate in real time a unique response to the bid of $101. This can
also be done in a nested fashion in response to subsequent opponent off-tree actions. Additionally,
these techniques can be used to solve finer-grained models as play progresses down the game tree.
We refer to the first method as the inexpensive method.3 When P1 chooses an off-tree action a,
a subgame S is generated following that action such that for any infoset I1 that P1 might be in,
I1 ? a ? Stop . This subgame may itself be an abstraction. A solution ? S is computed via subgame
solving, and ? S is combined with ? to form a new blueprint ? 0 in the expanded abstraction that now
includes action a. The process repeats whenever P1 again chooses an off-tree action.
3
Following our study, the AI DeepStack used a technique similar to this form of nested subgame solving [19].
7
To conduct safe subgame solving in response to off-tree action a, we could calculate CBV ?2 (I1 , a)
by defining, via action translation, a P2 blueprint following a and best responding to it [4]. However,
that could be computationally expensive and would likely perform poorly in practice because, as we
show later, action translation is highly exploitable. Instead, we relax the guarantee of safety and use
? ?2 (I1 ) for the alternative payoff, where CBV
? ?2 (I1 ) is P1 ?s counterfactual best response value
CBV
in I1 when constrained to playing in the blueprint abstraction (which excludes action a). In this case,
? ?2 (I1 ) approximates CBV ?2? (I1 ), where ? ? is an optimal
exploitability depends on how well CBV
2
4
P2 strategy (see Appendix C). In general, we find that only a small number of near-optimal actions
? ?2 (I1 ) to be close to CBV ?2? (I1 ). We can
need to be included in the blueprint abstraction for CBV
then approximate a near-optimal response to any opponent action, even in a continuous action space.
The ?inexpensive? approach cannot be combined with Unsafe subgame solving because the probability
of reaching an action outside of a player?s abstraction is undefined. Nevertheless, a similar approach
is possible with Unsafe subgame solving (as well as all the other subgame-solving techniques) by
starting the subgame solving at h rather than at h ? a. In other words, if action a taken in node h is
not in the abstraction, then Unsafe subgame solving is conducted in the smallest subgame containing
h (and action a is added to that abstraction). This increases the size of the subgame compared to the
inexpensive method because a strategy must be recomputed for every action a0 ? A(h) in addition to
a. We therefore call this method the expensive method. We present experiments with both methods.
7
Experiments
Our experiments were conducted on heads-up no-limit Texas hold?em, as well as two smaller-scale
poker games we call No-Limit Flop Hold?em (NLFH) and No-Limit Turn Hold?em (NLTH). The
description for these games can be found in Appendix G. For equilibrium finding, we used CFR+ [29].
Our first experiment compares the performance of the subgame-solving techniques when applied
to information abstraction (which is card abstraction in the case of poker). Specifically, we solve
NLFH with no information abstraction on the preflop. On the flop, there are 1,286,792 infosets for
each betting sequence; the abstraction buckets them into 200, 2,000, or 30,000 abstract ones (using a
leading information abstraction algorithm [8]). We then apply subgame solving immediately after the
flop community cards are dealt. We experiment with two versions of the game, one small and one
large, which include only a few of the available actions in each infoset. We also experimented on
abstractions of NLTH. In that case, we solve NLTH with no information abstraction on the preflop or
flop. On the turn, there are 55,190,538 infosets for each betting sequence; the abstraction buckets
them into 200, 2,000, or 20,000 abstract ones. We apply subgame solving immediately after the
turn community card is dealt. Table 1 shows the performance of each technique when using 30,000
buckets (20,000 for NLTH). The full results are presented in Appendix E. In all our experiments,
exploitability is measured in the standard units used in this field: milli big blinds per hand (mbb/h).
Small Flop Holdem Large Flop Holdem
Blueprint Strategy
91.28
41.41
Unsafe
5.514
396.8
Resolve
54.07
23.11
Maxmargin
43.43
19.50
Reach-Maxmargin
41.47
18.80
Reach-Maxmargin (no split)
25.88
16.41
Estimate
24.23
30.09
Estimate+Distributional
34.30
10.54
Reach-Estimate+Distributional
22.58
9.840
Reach-Estimate+Distributional (no split) 17.33
8.777
Table 1: Exploitability of the various subgame-solving techniques in nested subgame solving.
of the pseudo-harmonic action translation is also shown.
Turn Holdem
345.5
79.34
251.8
234.4
233.5
175.5
76.44
74.35
72.59
70.68
The performance
Estimate and Estimate+Distributional are techniques introduced in Appendix C. We use a normal
distribution in the Distributional subgame solving experiments, with standard deviation determined
by the heuristic presented in Appendix C.1.
Since subgame solving begins immediately after a chance node with an extremely high branching
factor (1, 755 in NLFH), the gifts for the Reach algorithms are divided among subgames inefficiently.
?
?
?
?
We estimate CBV ?2 (I1 ) rather than CBV ?2 (I1 , a) because CBV ?2 (I1 ) ? CBV ?2 (I1 , a) is a gift that
may be added to the alternative payoff anyway.
4
8
Many subgames do not use the gifts at all, while others could make use of more. In the experiments
we show results both for the theoretically safe splitting of gifts, as well as a more aggressive version
where gifts are scaled up by the branching factor of the chance node (1, 755). This weakens the
theoretical guarantees of the algorithm, but in general did better than splitting gifts in a theoretically
correct manner. However, this is not universally true. Appendix F shows that in at least one case,
exploitability increased when gifts were scaled up too aggressively. In all cases, using Reach subgame
solving in at least the theoretical safe method led to lower exploitability.
Despite lacking theoretical guarantees, Unsafe subgame solving did surprisingly well in most games.
However, it did substantially worse in Large NLFH with 30,000 buckets. This exemplifies its
variability. Among the safe methods, all of the changes we introduce show improvement over
past techniques. The Reach-Estimate + Distributional algorithm generally resulted in the lowest
exploitability among the various choices, and in most cases beat unsafe subgame solving.
The second experiment evaluates nested subgame solving, and compares it to action translation. In
order to also evaluate action translation, in this experiment, we create an NLFH game that includes 3
bet sizes at every point in the game tree (0.5, 0.75, and 1.0 times the size of the pot); a player can also
decide not to bet. Only one bet (i.e., no raises) is allowed on the preflop, and three bets are allowed on
the flop. There is no information abstraction anywhere in the game. We also created a second, smaller
abstraction of the game in which there is still no information abstraction, but the 0.75? pot bet is
never available. We calculate the exploitability of one player using the smaller abstraction, while
the other player uses the larger abstraction. Whenever the large-abstraction player chooses a 0.75?
pot bet, the small-abstraction player generates and solves a subgame for the remainder of the game
(which again does not include any subsequent 0.75? pot bets) using the nested subgame-solving
techniques described above. This subgame strategy is then used as long as the large-abstraction player
plays within the small abstraction, but if she chooses the 0.75? pot bet again later, then the subgame
solving is used again, and so on.
Table 2 shows that all the subgame-solving techniques substantially outperform action translation.
We did not test distributional alternative payoffs in this experiment, since the calculated best response
values are likely quite accurate. These results suggest that nested subgame solving is preferable to
action translation (if there is sufficient time to solve the subgame).
mbb/h
Randomized Pseudo-Harmonic Mapping 1,465
Resolve
150.2
Reach-Maxmargin (Expensive)
149.2
Unsafe (Expensive)
148.3
Maxmargin
122.0
Reach-Maxmargin
119.1
Table 2: Exploitability of the various subgame-solving techniques in nested subgame solving. The performance
of the pseudo-harmonic action translation is also shown.
We used the techniques presented in this paper to develop Libratus, an AI that competed against
four top human specialists in heads-up no-limit Texas hold?em. Heads-up no-limit Texas hold?em
has been the primary benchmark challenge for AI in imperfect-information games. The competition
involved 120,000 hands of poker and a prize pool of $200,000 split among the humans to incentivize
strong play. The AI decisively defeated the human team by 147 mbb / hand, with 99.98% statistical
significance. This was the first, and so far only, time an AI defeated top humans in no-limit poker.
8
Conclusion
We introduced a subgame-solving technique for imperfect-information games that has stronger
theoretical guarantees and better practical performance than prior subgame-solving methods. We
presented results on exploitability of both safe and unsafe subgame-solving techniques. We also
introduced a method for nested subgame solving in response to the opponent?s off-tree actions, and
demonstrated that this leads to dramatically better performance than the usual approach of action
translation. This is, to our knowledge, the first time that exploitability of subgame-solving techniques
has been measured in large games.
Finally, we demonstrated the effectiveness of these techniques in practice in heads-up no-limit Texas
Hold?em poker, the main benchmark challenge for AI in imperfect-information games. We developed
the first AI to reach the milestone of defeating top humans in heads-up no-limit Texas Hold?em.
9
9
Acknowledgments
This material is based on work supported by the National Science Foundation under grants IIS1718457, IIS-1617590, and CCF-1733556, and the ARO under award W911NF-17-1-0082, as well
as XSEDE computing resources provided by the Pittsburgh Supercomputing Center. The Brains vs.
AI competition was sponsored by Carnegie Mellon University, Rivers Casino, GreatPoint Ventures,
Avenue4Analytics, TNG Technology Consulting, Artificial Intelligence, Intel, and Optimized Markets,
Inc. We thank Kristen Gardner, Marcelo Gutierrez, Theo Gutman-Solo, Eric Jackson, Christian Kroer,
Tim Reiff, and the anonymous reviewers for helpful feedback.
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11
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6,270 | 6,672 | Unsupervised Image-to-Image Translation Networks
Ming-Yu Liu,
Thomas Breuel, Jan Kautz
NVIDIA
{mingyul,tbreuel,jkautz}@nvidia.com
Abstract
Unsupervised image-to-image translation aims at learning a joint distribution of
images in different domains by using images from the marginal distributions in
individual domains. Since there exists an infinite set of joint distributions that
can arrive the given marginal distributions, one could infer nothing about the joint
distribution from the marginal distributions without additional assumptions. To
address the problem, we make a shared-latent space assumption and propose an
unsupervised image-to-image translation framework based on Coupled GANs.
We compare the proposed framework with competing approaches and present
high quality image translation results on various challenging unsupervised image
translation tasks, including street scene image translation, animal image translation,
and face image translation. We also apply the proposed framework to domain
adaptation and achieve state-of-the-art performance on benchmark datasets. Code
and additional results are available in https://github.com/mingyuliutw/unit.
1
Introduction
Many computer visions problems can be posed as an image-to-image translation problem, mapping
an image in one domain to a corresponding image in another domain. For example, super-resolution
can be considered as a problem of mapping a low-resolution image to a corresponding high-resolution
image; colorization can be considered as a problem of mapping a gray-scale image to a corresponding
color image. The problem can be studied in supervised and unsupervised learning settings. In the
supervised setting, paired of corresponding images in different domains are available [8, 15]. In the
unsupervised setting, we only have two independent sets of images where one consists of images
in one domain and the other consists of images in another domain?there exist no paired examples
showing how an image could be translated to a corresponding image in another domain. Due to
lack of corresponding images, the UNsupervised Image-to-image Translation (UNIT) problem is
considered harder, but it is more applicable since training data collection is easier.
When analyzing the image translation problem from a probabilistic modeling perspective, the key
challenge is to learn a joint distribution of images in different domains. In the unsupervised setting,
the two sets consist of images from two marginal distributions in two different domains, and the task is
to infer the joint distribution using these images. The coupling theory [16] states there exist an infinite
set of joint distributions that can arrive the given marginal distributions in general. Hence, inferring
the joint distribution from the marginal distributions is a highly ill-posed problem. To address the
ill-posed problem, we need additional assumptions on the structure of the joint distribution.
To this end we make a shared-latent space assumption, which assumes a pair of corresponding images
in different domains can be mapped to a same latent representation in a shared-latent space. Based on
the assumption, we propose a UNIT framework that are based on generative adversarial networks
(GANs) and variational autoencoders (VAEs). We model each image domain using a VAE-GAN. The
adversarial training objective interacts with a weight-sharing constraint, which enforces a sharedlatent space, to generate corresponding images in two domains, while the variational autoencoders
relate translated images with input images in the respective domains. We applied the proposed
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
E1
Z : shared latent space
x
?1!1
1
x1
z
E1
D1
G1
E2
G1
T/F
x
?2!1
2
G2
x2
x1
X2
X1
x
?1!2
1
x2
(a)
z
E2
T/F
x
?2!2
2
G2
(b)
D2
Figure 1: (a) The shared latent space assumption. We assume a pair of corresponding images (x1 , x2 ) in
two different domains X1 and X2 can be mapped to a same latent code z in a shared-latent space Z. E1 and
E2 are two encoding functions, mapping images to latent codes. G1 and G2 are two generation functions,
mapping latent codes to images. (b) The proposed UNIT framework. We represent E1 E2 G1 and G2 using
CNNs and implement the shared-latent space assumption using a weight sharing constraint where the connection
weights of the last few layers (high-level layers) in E1 and E2 are tied (illustrated using dashed lines) and the
connection weights of the first few layers (high-level layers) in G1 and G2 are tied. Here, x
?1!1
and x
?2!2
1
2
1!2
2!1
are self-reconstructed images, and x
?1
and x
?2
are domain-translated images. D1 and D2 are adversarial
discriminators for the respective domains, in charge of evaluating whether the translated images are realistic.
Table 1: Interpretation of the roles of the subnetworks in the proposed framework.
Networks
Roles
{E1 , G1 }
{E1 , G2 }
{G1 , D1 }
{E1 , G1 , D1 }
VAE for X1 Image Translator X1 ! X2 GAN for X1 VAE-GAN [14]
{G1 , G2 , D1 , D2 }
CoGAN [17]
framework to various unsupervised image-to-image translation problems and achieved high quality
image translation results. We also applied it to the domain adaptation problem and achieved state-ofthe-art accuracies on benchmark datasets. The shared-latent space assumption was used in Coupled
GAN [17] for joint distribution learning. Here, we extend the Coupled GAN work for the UNIT
problem. We also note that several contemporary works propose the cycle-consistency constraint
assumption [29, 10], which hypothesizes the existence of a cycle-consistency mapping so that an
image in the source domain can be mapped to an image in the target domain and this translated image
in the target domain can be mapped back to the original image in the source domain. In the paper, we
show that the shared-latent space constraint implies the cycle-consistency constraint.
2
Assumptions
Let X1 and X2 be two image domains. In supervised image-to-image translation, we are given
samples (x1 , x2 ) drawn from a joint distribution PX1 ,X2 (x1 , x2 ). In unsupervised image-to-image
translation, we are given samples drawn from the marginal distributions PX1 (x1 ) and PX2 (x2 ). Since
an infinite set of possible joint distributions can yield the given marginal distributions, we could infer
nothing about the joint distribution from the marginal samples without additional assumptions.
We make the shared-latent space assumption. As shown Figure 1, we assume for any given pair
of images x1 and x2 , there exists a shared latent code z in a shared-latent space, such that we
can recover both images from this code, and we can compute this code from each of the two
images. That is, we postulate there exist functions E1? , E2? , G?1 , and G?2 such that, given a pair of
corresponding images (x1 , x2 ) from the joint distribution, we have z = E1? (x1 ) = E2? (x2 ) and
?
conversely x1 = G?1 (z) and x2 = G?2 (z). Within this model, the function x2 = F1!2
(x1 ) that
?
?
?
maps from X1 to X2 can be represented by the composition F1!2 (x1 ) = G2 (E1 (x1 )). Similarly,
?
?
x1 = F2!1
(x2 ) = G?1 (E2? (x2 )). The UNIT problem then becomes a problem of learning F1!2
?
?
?
and F2!1
. We note that a necessary condition for F1!2
and F2!1
to exist is the cycle-consistency
?
?
?
?
constraint [29, 10]: x1 = F2!1
(F1!2
(x1 )) and x2 = F1!2
(F2!1
(x2 )). We can reconstruct
the input image from translating back the translated input image. In other words, the proposed
shared-latent space assumption implies the cycle-consistency assumption (but not vice versa).
To implement the shared-latent space assumption, we further assume a shared intermediate representation h such that the process of generating a pair of corresponding images admits a form of
2
x1
.
(1)
x2
Consequently, we have G?1 ? G?L,1 G?H and G?2 ? G?L,2 G?H where G?H is a common high-level
generation function that maps z to h and G?L,1 and G?L,2 are low-level generation functions that map
h to x1 and x2 , respectively. In the case of multi-domain image translation (e.g., sunny and rainy
image translation), z can be regarded as the compact, high-level representation of a scene ("car in
front, trees in back"), and h can be considered a particular realization of z through G?H ("car/tree
occupy the following pixels"), and G?L,1 and G?L,2 would be the actual image formation functions
in each modality ("tree is lush green in the sunny domain, but dark green in the rainy domain").
?
?
?
?
Assuming h also allow us to represent E1? and E2? by E1? ? EH
EL,1
and E2? ? EH
EL,2
.
z!h
%
&
In the next section, we discuss how we realize the above ideas in the proposed UNIT framework.
3
Framework
Our framework, as illustrated in Figure 1, is based on variational autoencoders (VAEs) [13, 22, 14]
and generative adversarial networks (GANs) [6, 17]. It consists of 6 subnetworks: including two
domain image encoders E1 and E2 , two domain image generators G1 and G2 , and two domain
adversarial discriminators D1 and D2 . Several ways exist to interpret the roles of the subnetworks,
which we summarize in Table 1. Our framework learns translation in both directions in one shot.
VAE. The encoder?generator pair {E1 , G1 } constitutes a VAE for the X1 domain, termed VAE1 . For
an input image x1 2 X1 , the VAE1 first maps x1 to a code in a latent space Z via the encoder E1 and
then decodes a random-perturbed version of the code to reconstruct the input image via the generator
G1 . We assume the components in the latent space Z are conditionally independent and Gaussian with
unit variance. In our formulation, the encoder outputs a mean vector E?,1 (x1 ) and the distribution
of the latent code z1 is given by q1 (z1 |x1 ) ? N (z1 |E?,1 (x1 ), I) where I is an identity matrix. The
reconstructed image is x
?1!1
= G1 (z1 ? q1 (z1 |x1 )). Note that here we abused the notation since
1
we treated the distribution of q1 (z1 |x1 ) as a random vector of N (E?,1 (x1 ), I) and sampled from it.
Similarly, {E2 , G2 } constitutes a VAE for X2 : VAE2 where the encoder E2 outputs a mean vector
E?,2 (x2 ) and the distribution of the latent code z2 is given by q2 (z2 |x2 ) ? N (z2 |E?,2 (x2 ), I). The
reconstructed image is x
?2!2
= G2 (z2 ? q2 (z2 |x2 )).
2
Utilizing the reparameterization trick [13], the non-differentiable sampling operation can be reparameterized as a differentiable operation using auxiliary random variables. This reparameterization trick
allows us to train VAEs using back-prop. Let ? be a random vector with a multi-variate Gaussian
distribution: ? ? N (?|0, I). The sampling operations of z1 ? q1 (z1 |x1 ) and z2 ? q2 (z2 |x2 ) can be
implemented via z1 = E?,1 (x1 ) + ? and z2 = E?,2 (x2 ) + ?, respectively.
Weight-sharing. Based on the shared-latent space assumption discussed in Section 2, we enforce a
weight-sharing constraint to relate the two VAEs. Specifically, we share the weights of the last few
layers of E1 and E2 that are responsible for extracting high-level representations of the input images
in the two domains. Similarly, we share the weights of the first few layers of G1 and G2 responsible
for decoding high-level representations for reconstructing the input images.
Note that the weight-sharing constraint alone does not guarantee that corresponding images in two
domains will have the same latent code. In the unsupervised setting, no pair of corresponding images
in the two domains exists to train the network to output a same latent code. The extracted latent
codes for a pair of corresponding images are different in general. Even if they are the same, the same
latent component may have different semantic meanings in different domains. Hence, the same latent
code could still be decoded to output two unrelated images. However, we will show that through
adversarial training, a pair of corresponding images in the two domains can be mapped to a common
latent code by E1 and E2 , respectively, and a latent code will be mapped to a pair of corresponding
images in the two domains by G1 and G2 , respectively.
The shared-latent space assumption allows us to perform image-to-image translation. We can
translate an image x1 in X1 to an image in X2 through applying G2 (z1 ? q1 (z1 |x1 )). We term such
an information processing stream as the image translation stream. Two image translation streams exist
in the proposed framework: X1 ! X2 and X2 ! X1 . The two streams are trained jointly with the
two image reconstruction streams from the VAEs. Once we could ensure that a pair of corresponding
3
images are mapped to a same latent code and a same latent code is decoded to a pair of corresponding
images, (x1 , G2 (z1 ? q1 (z1 |x1 ))) would form a pair of corresponding images. In other words, the
?
composition of E1 and G2 functions approximates F1!2
for unsupervised image-to-image translation
?
discussed in Section 2, and the composition of E2 and G1 function approximates F2!1
.
GANs. Our framework has two generative adversarial networks: GAN1 = {D1 , G1 } and GAN2 =
{D2 , G2 }. In GAN1 , for real images sampled from the first domain, D1 should output true, while
for images generated by G1 , it should output false. G1 can generate two types of images: 1) images
from the reconstruction stream x
?1!1
= G1 (z1 ? q1 (z1 |x1 )) and 2) images from the translation
1
stream x
?2!1
=
G
(z
?
q
(z
|x
)).
Since
the reconstruction stream can be supervisedly trained, it
1 2
2 2 2
2
is suffice that we only apply adversarial training to images from the translation stream, x
?2!1
. We
2
apply a similar processing to GAN2 where D2 is trained to output true for real images sampled from
the second domain dataset and false for images generated from G2 .
Cycle-consistency (CC). Since the shared-latent space assumption implies the cycle-consistency
constraint (See Section 2), we could also enforce the cycle-consistency constraint in the proposed
framework to further regularize the ill-posed unsupervised image-to-image translation problem. The
resulting information processing stream is called the cycle-reconstruction stream.
Learning. We jointly solve the learning problems of the VAE1 , VAE2 , GAN1 and GAN2 for the
image reconstruction streams, the image translation streams, and the cycle-reconstruction streams:
min
max LVAE1 (E1 , G1 ) + LGAN1 (E1 , G1 , D1 ) + LCC1 (E1 , G1 , E2 , G2 )
E1 ,E2 ,G1 ,G2 D1 ,D2
LVAE2 (E2 , G2 ) + LGAN2 (E2 , G2 , D2 ) + LCC2 (E2 , G2 , E1 , G1 ).
(2)
VAE training aims for minimizing a variational upper bound In (2), the VAE objects are
LVAE1 (E1 , G1 ) =
LVAE2 (E2 , G2 ) =
1 KL(q1 (z1 |x1 )||p? (z))
1 KL(q2 (z2 |x2 )||p? (z))
2 Ez1 ?q1 (z1 |x1 ) [log pG1 (x1 |z1 )]
2 Ez2 ?q2 (z2 |x2 ) [log pG2 (x2 |z2 )].
(3)
(4)
where the hyper-parameters 1 and 2 control the weights of the objective terms and the KL
divergence terms penalize deviation of the distribution of the latent code from the prior distribution.
The regularization allows an easy way to sample from the latent space [13]. We model pG1 and pG2
using Laplacian distributions, respectively. Hence, minimizing the negative log-likelihood term is
equivalent to minimizing the absolute distance between the image and the reconstructed image. The
prior distribution is a zero mean Gaussian p? (z) = N (z|0, I).
In (2), the GAN objective functions are given by
LGAN1 (E1 , G1 , D1 ) =
LGAN2 (E2 , G2 , D2 ) =
0 Ex1 ?PX1 [log D1 (x1 )]
+
0 Ex2 ?PX2 [log D2 (x2 )] +
0 Ez2 ?q2 (z2 |x2 ) [log(1
0 Ez1 ?q1 (z1 |x1 ) [log(1
D1 (G1 (z2 )))]
(5)
D2 (G2 (z1 )))].
(6)
The objective functions in (5) and (6) are conditional GAN objective functions. They are used to
ensure the translated images resembling images in the target domains, respectively. The hyperparameter 0 controls the impact of the GAN objective functions.
We use a VAE-like objective function to model the cycle-consistency constraint, which is given by
LCC1 (E1 , G1 , E2 , G2 ) =
LCC2 (E2 , G2 , E1 , G1 ) =
+ 3 KL(q2 (z2 |x1!2
))||p? (z))
1
[log
p
(x
|z
)]
4 Ez2 ?q2 (z2 |x1!2
G
1
2
)
1
1
3 KL(q1 (z1 |x1 )||p? (z))
3 KL(q2 (z2 |x2 )||p? (z))
2!1
))||p? (z))
3 KL(q1 (z1 |x2
+
4 Ez1 ?q1 (z1 |x2!1
) [log pG2 (x2 |z1 )].
2
(7)
(8)
where the negative log-likelihood objective term ensures a twice translated image resembles the
input one and the KL terms penalize the latent codes deviating from the prior distribution in the
cycle-reconstruction stream (Therefore, there are two KL terms). The hyper-parameters 3 and 4
control the weights of the two different objective terms.
Inheriting from GAN, training of the proposed framework results in solving a mini-max problem
where the optimization aims to find a saddle point. It can be seen as a two player zero-sum game.
The first player is a team consisting of the encoders and generators. The second player is a team
consisting of the adversarial discriminators. In addition to defeating the second player, the first player
has to minimize the VAE losses and the cycle-consistency losses. We apply an alternating gradient
4
Accuracy
0.7
0.7
0.56
0.56
0.42
0.42
0.28
0.28
0.14
0
1
6 Dis
5 Dis
4 Dis
3 Dis
2
3
0.14
4
# of shared layers in gen.
(a)
Method
(b)
0
1000
= 10
11=10
=1
11=1
= 0.1
11=0.1
= 0.01
11=0.01
100
10
1
Accuracy
Weight 0.569?0.029
Sharing
Cycle 0.568?0.010
Consistenc
Proposed 0.600?0.015
22
(c)
(d)
Figure 2: (a) Illustration of the Map dataset. Left: satellite image. Right: map. We translate holdout satellite
images to maps and measure the accuracy achieved by various configurations of the proposed framework.
(b) Translation accuracy versus different network architectures. (c) Translation accuracy versus different
hyper-parameter values. (d) Impact of weight-sharing and cycle-consistency constraints on translation accuracy.
update scheme similar to the one described in [6] to solve (2). Specifically, we first apply a gradient
ascent step to update D1 and D2 with E1 , E2 , G1 , and G2 fixed. We then apply a gradient descent
step to update E1 , E2 , G1 , and G2 with D1 and D2 fixed.
Translation: After learning, we obtain two image translation functions by assembling a subset of the
subnetworks. We have F1!2 (x1 ) = G2 (z1 ? q1 (z1 |x1 )) for translating images from X1 to X2 and
F2!1 (x2 ) = G1 (z2 ? q2 (z2 |x2 )) for translating images from X2 to X1 .
4
Experiments
We first analyze various components of the proposed framework. We then present visual results on
challenging translation tasks. Finally, we apply our framework to the domain adaptation tasks.
Performance Analysis. We used ADAM [11] for training where the learning rate was set to 0.0001
and momentums were set to 0.5 and 0.999. Each mini-batch consisted of one image from the first
domain and one image from the second domain. Our framework had several hyper-parameters. The
default values were 0 = 10, 3 = 1 = 0.1 and 4 = 2 = 100. For the network architecture,
our encoders consisted of 3 convolutional layers as the front-end and 4 basic residual blocks [7] as
the back-end. The generators consisted of 4 basic residual blocks as the front-end and 3 transposed
convolutional layers as the back-end. The discriminators consisted of stacks of convolutional layers.
We used LeakyReLU for nonlinearity. The details of the networks are given in Appendix A.
We used the map dataset [8] (visualized in Figure 2), which contained corresponding pairs of images
in two domains (satellite image and map) useful for quantitative evaluation. Here, the goal was to
learn to translate between satellite images and maps. We operated in an unsupervised setting where
we used the 1096 satellite images from the training set as the first domain and 1098 maps from the
validation set as the second domain. We trained for 100K iterations and used the final model to
translate 1098 satellite images in the test set. We then compared the difference between a translated
satellite image (supposed to be maps) and the corresponding ground truth maps pixel-wisely. A pixel
translation was counted correct if the color difference was within 16 of the ground truth color value.
We used the average pixel accuracy over the images in the test set as the performance metric. We
could use color difference for measuring translation accuracy since the target translation function
was unimodal. We did not evaluate the translation from maps to images since the translation was
multi-modal, which was difficult to construct a proper evaluation metric.
In one experiment, we varied the number of weight-sharing layers in the VAEs and paired each
configuration with discriminator architectures of different depths during training. We changed the
number of weight-sharing layers from 1 to 4. (Sharing 1 layer in VAEs means sharing 1 layer for
E1 and E2 and, at the same time, sharing 1 layer for G1 and G2 .) The results were reported in
Figure 2(b). Each curve corresponded to a discriminator architecture of a different depth. The x-axis
denoted the number of weigh-sharing layers in the VAEs. We found that the shallowest discriminator
architecture led to the worst performance. We also found that the number of weight-sharing layer
had little impact. This was due to the use of the residual blocks. As tying the weight of one layer, it
effectively constrained the other layers since the residual blocks only updated the residual information.
In the rest of the experiments, we used VAEs with 1 sharing layer and discriminators of 5 layers.
5
We analyzed impact of the hyper-parameter values to the translation accuracy. For different weight
values on the negative log likelihood terms (i.e., 2 , 4 ), we computed the achieved translation
accuracy over different weight values on the KL terms (i.e., 1 , 3 ). The results were reported in
Figure 2(c). We found that, in general, a larger weight value on the negative log likelihood terms
yielded a better translation accuracy. We also found setting the weights of the KL terms to 0.1 resulted
in consistently good performance. We hence set 1 = 3 = 0.1 and 2 = 4 = 100.
We performed an ablation study measuring impact of the weight-sharing and cycle-consistency
constraints to the translation performance and showed the results in Figure 2(d). We reported average
accuracy over 5 trials (trained with different initialized weights.). We note that when we removed
the weight-sharing constraint (as a consequence, we also removed the reconstruction streams in the
framework), the framework was reduced to the CycleGAN architecture [29, 10]. We found the model
achieved an average pixel accuracy of 0.569. When we removed the cycle-consistency constraint
and only used the weight-sharing constraint1 , it achieved 0.568 average pixel accuracy. But when we
used the full model, it achieved the best performance of 0.600 average pixel accuracy. This echoed
our point that for the ill-posed joint distribution recovery problem, more constraints are beneficial.
Qualitative results. Figure 3 to 6 showed results of the proposed framework on various UNIT tasks.
Street images. We applied the proposed framework to several unsupervised street scene image
translation tasks including sunny to rainy, day to night, summery to snowy, and vice versa. For each
task, we used a set of images extracted from driving videos recorded at different days and cities. The
numbers of the images in the sunny/day, rainy, night, summery, and snowy sets are 86165, 28915,
36280, 6838, and 6044. We trained the network to translate street scene image of size 640?480. In
Figure 3, we showed several example translation results . We found that our method could generate
realistic translated images. We also found that one translation was usually harder than the other.
Specifically, the translation that required adding more details to the image was usually harder (e.g.
night to day). Additional results are available in https://github.com/mingyuliutw/unit.
Synthetic to real. In Figure 3, we showed several example results achieved by applying the proposed
framework to translate images between the synthetic images in the SYNTHIA dataset [23] and the
real images in the Cityscape dataset [2]. For the real to synthetic translation, we found our method
made the cityscape images cartoon like. For the synthetic to real translation, our method achieved
better results in the building, sky, road, and car regions than in the human regions.
Dog breed conversion. We used the images of Husky, German Shepherd, Corgi, Samoyed, and Old
English Sheep dogs in the ImageNet dataset to learn to translate dog images between different breeds.
We only used the head regions, which were extracted by a template matching algorithm. Several
example results were shown in Figure 4. We found our method translated a dog to a different breed.
Cat species conversion. We also used the images of house cat, tiger, lion, cougar, leopard, jaguar,
and cheetah in the ImageNet dataset to learn to translate cat images between different species. We
only used the head regions, which again were extracted by a template matching algorithm. Several
example results were shown in Figure 5. We found our method translated a cat to a different specie.
Face attribute. We used the CelebA dataset [18] for attribute-based face images translation. Each face
image in the dataset had several attributes, including blond hair, smiling, goatee, and eyeglasses. The
face images with an attribute constituted the 1st domain, while those without the attribute constituted
the 2nd domain. In Figure 6, we visualized the results where we translated several images that do not
have blond hair, eye glasses, goatee, and smiling to corresponding images with each of the individual
attributes. We found that the translated face images were realistic.
Domain Adaptation. We applied the proposed framework to the problem for adapting a classifier
trained using labeled samples in one domain (source domain) to classify samples in a new domain
(target domain) where labeled samples in the new domain are unavailable during training. Early
works have explored ideas from subspace learning [4] to deep feature learning [5, 17, 26].
We performed multi-task learning where we trained the framework to 1) translate images between
the source and target domains and 2) classify samples in the source domain using the features
extracted by the discriminator in the source domain. Here, we tied the weights of the high-level
layers of D1 and D2 . This allows us to adapt a classifier trained in the source domain to the target
domain. Also, for a pair of generated images in different domains, we minimized the L1 distance
1
We used this architecture in an earlier version of the paper.
6
Figure 3: Street scene image translation results. For each pair, left is input and right is the translated image.
Input
Old Eng.
Sheep Dog
Input
Cougar
Husky
German
Shepherd
Corgi
Input
Husky
Corgi
Input
Leopard
Figure 4: Dog breed translation results.
Input
Cheetah
+Blond Hair +Eyeglasses
Leopard
Lion
Tiger
Figure 5: Cat species translation results.
+Goatee
+Smiling
Input
+Blond Hair +Eyeglasses
Figure 6: Attribute-based face translation results.
7
+Goatee
+Smiling
Table 2: Unsupervised domain adaptation performance. The reported numbers are classification accuracies.
Method
SA [4]
DANN [5]
DTN [26]
CoGAN
UNIT (proposed)
SVHN! MNIST
MNIST! USPS
USPS! MNIST
0.5932
-
0.7385
-
0.8488
-
0.9565
0.9315
0.9053
0.9597
0.9358
between the features extracted by the highest layer of the discriminators, which further encouraged
D1 and D2 to interpret a pair of corresponding images in the same way. We applied the approach to
several tasks including adapting from the Street View House Number (SVHN) dataset [20] to the
MNIST dataset and adapting between the MNIST and USPS datasets. Table 2 reported the achieved
performance with comparison to the competing approaches. We found that our method achieved a
0.9053 accuracy for the SVHN!MNIST task, which was much better than 0.8488 achieved by the
previous state-of-the-art method [26]. We also achieved better performance for the MNIST$SVHN
task than the Coupled GAN approach, which was the state-of-the-art. The digit images had a small
resolution. Hence, we used a small network. We also found that the cycle-consistency constraint was
not necessary for this task. More details about the experiments are available in Appendix B.
5
Related Work
Several deep generative models were recently proposed for image generation including GANs [6],
VAEs [13, 22], and PixelCNN [27]. The proposed framework was based on GANs and VAEs but it
was designed for the unsupervised image-to-image translation task, which could be considered as a
conditional image generation model. In the following, we first review several recent GAN and VAE
works and then discuss related image translation works.
GAN learning is via staging a zero-sum game between the generator and discriminator. The quality
of GAN-generated images had improved dramatically since the introduction. LapGAN [3] proposed
a Laplacian pyramid implementation of GANs. DCGAN [21] used a deeper convolutional network.
Several GAN training tricks were proposed in [24]. WGAN [1] used the Wasserstein distance.
VAEs optimize a variational bound. By improving the variational approximation, better image
generation results were achieved [19, 12]. In [14], a VAE-GAN architecture was proposed to improve
image generation quality of VAEs. VAEs were applied to translate face image attribute in [28].
Conditional generative model is a popular approach for mapping an image from one domain to
another. Most of the existing works were based on supervised learning [15, 8, 9]. Our work differed
to the previous works in that we do not need corresponding images. Recently, [26] proposed the
domain transformation network (DTN) and achieved promising results on translating small resolution
face and digit images. In addition to faces and digits, we demonstrated that the proposed framework
can translate large resolution natural images. It also achieved a better performance in the unsupervised
domain adaptation task. In [25], a conditional generative adversarial network-based approach was
proposed to translate a rendering images to a real image for gaze estimation. In order to ensure
the generated real image was similar to the original rendering image, the L1 distance between
the generated and original image was minimized. We note that two contemporary papers [29, 10]
independently introduced the cycle-consistency constraint for the unsupervised image translation.
We showed that that the cycle-consistency constraint is a natural consequence of the proposed
shared-latent space assumption. From our experiment, we found that cycle-consistency and the
weight-sharing (a realization of the shared-latent space assumption) constraints rendered comparable
performance. When the two constraints were jointed used, the best performance was achieved.
6
Conclusion and Future Work
We presented a general framework for unsupervised image-to-image translation. We showed it
learned to translate an image from one domain to another without any corresponding images in two
domains in the training dataset. The current framework has two limitations. First, the translation
model is unimodal due to the Gaussian latent space assumption. Second, training could be unstable
due to the saddle point searching problem. We plan to address these issues in the future work.
8
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9
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6,271 | 6,673 | Coded Distributed Computing for Inverse Problems
Yaoqing Yang, Pulkit Grover and Soummya Kar
Carnegie Mellon University
{yyaoqing, pgrover, soummyak}@andrew.cmu.edu
Abstract
Computationally intensive distributed and parallel computing is often bottlenecked
by a small set of slow workers known as stragglers. In this paper, we utilize the
emerging idea of ?coded computation? to design a novel error-correcting-code
inspired technique for solving linear inverse problems under specific iterative
methods in a parallelized implementation affected by stragglers. Example machinelearning applications include inverse problems such as personalized PageRank and
sampling on graphs. We provably show that our coded-computation technique can
reduce the mean-squared error under a computational deadline constraint. In fact,
the ratio of mean-squared error of replication-based and coded techniques diverges
to infinity as the deadline increases. Our experiments for personalized PageRank
performed on real systems and real social networks show that this ratio can be
as large as 104 . Further, unlike coded-computation techniques proposed thus far,
our strategy combines outputs of all workers, including the stragglers, to produce
more accurate estimates at the computational deadline. This also ensures that the
accuracy degrades ?gracefully? in the event that the number of stragglers is large.
1
Introduction
The speed of distributed computing is often affected by a few slow workers known as the ?stragglers?
[1?4]. This issue is often addressed by replicating tasks across workers and using this redundancy to
ignore some of the stragglers. Recently, methods from error-correcting codes (ECC) have been used
for speeding up distributed computing [5?15], which build on classical works on algorithm-based
fault-tolerance [16]. The key idea is to treat stragglers as ?erasures? and use ECC to retrieve the result
after a subset of fast workers have finished. In some cases, (e.g. [6, 8] for matrix multiplications),
techniques that utilize ECC achieve scaling-sense speedups in average computation time compared to
replication. In this work, we propose a novel coding-inspired technique to deal with stragglers in
distributed computing of linear inverse problems using iterative solvers [17].
Existing techniques that use coding to deal with stragglers treat straggling workers as ?erasures?, that
is, they ignore computation results of the stragglers. In contrast, when using iterative methods for
linear inverse problems, even if the computation result at a straggler has not converged, the proposed
algorithm does not ignore the result, but instead combines it (with appropriate weights) with results
from other workers. This is in part because the results of iterative methods often converge gradually
to the true solutions. We use a small example shown in Fig. 1 to illustrate this idea. Suppose we
want to solve two linear inverse problems with solutions x?1 and x?2 . We ?encode the computation? by
adding an extra linear inverse problem with solution x?1 + x?2 (see Section 3), and distribute these
three problems to three workers. Using this method, the solutions x?1 and x?2 can be obtained from
the results of any combination of two fast workers that first return their solutions.
But what if we have a computational deadline, Tdl , by which only one worker converges? The natural
extension of existing strategies (e.g., [6]) will declare a failure because it needs at least two workers
to respond. However, our strategy does not require convergence: even intermediate results can be
utilized to estimate solutions. In other words, our strategy degrades gracefully as the number of
stragglers increases, or as the deadline is pulled earlier. Indeed, we show that it is suboptimal to
ignore stragglers as erasures, and design strategies that treat the difference from the optimal solution
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
General coded computation
r1
r2
E
n
c
o
d
e
r1
r2
r1+r2
Processor1
Processor2
(slow)
Processor3
(slow)
waitEforEtwoE
fastEworkers
*1
D
e
c
o
ignore!
d
x1*+x2*
e
x
x*2
Proposed coded method
fail
E
n
c
o
d
e
r1
r2
r1
r2
r1+r2
DeadlineETdl
Processor1
x1* +e1
Processor2
(slow)
x2* +e2
Processor3
(slow)
x1* +x2* +e3
D
e weightedE
c combination
o
d
e
Figure 1: A comparison between the existing scheme in [6] and the proposed algorithm.
as ?soft? additive noise (see Section 3). We use an algorithm that is similar to weighted least-squares
for decoding, giving each worker a weight based on its proximity to convergence. In this way, we can
expect to fully utilize the computation results from all workers and obtain better speedup.
Theoretically, we show that for a specified deadline time Tdl , under certain conditions on worker
speed distributions, the coded linear inverse solver using structured codes has smaller mean squared
error than the replication-based linear solver (Theorem 4.4). In fact, under more relaxed conditions
on worker speed distributions, when the computation time Tdl increases, the ratio of the mean-squared
error (MSE) of replication-based and coded linear solvers can get arbitrarily large (Theorem 4.5)!
For validation of our theory, we performed experiments to compare coded and replication-based
computation for a graph mining problem, namely personalized PageRank [18] using the classical
power-iteration method [19]. We conduct experiments on the Twitter and Google Plus social networks
under a deadline on computation time using a given number of workers on a real computation cluster
(Section 6). We observe that the MSE of coded PageRank is smaller than that of replication by
a factor of 104 at Tdl = 2 seconds. From an intuitive perspective, the advantage of coding over
replication is that coding utilizes the diversity of all heterogeneous workers, whereas replication
cannot (see section 7 for details). To compare with existing coded technique in [6], we adapt it to
inverse problems by inverting only the partial results from the fast workers. However, from our
experiments, if only the results from the fast workers are used, the error amplifies due to inverting an
ill-conditioned submatrix during decoding (Section 6). This ill-conditioning issue of real-number
erasure codes has also been recognized in a recent communication problem [20]. In contrast, our
novel way of combining all the partial results including those from the stragglers helps bypass the
difficulty of inverting an ill-conditioned matrix.
The focus of this work is on utilizing computations to deliver the minimal MSE in solving linear
inverse problems. Our algorithm does not reduce the communication cost. However, because each
worker performs sophisticated iterative computations in our problem, such as the power-iteration
computations, the time required for computation dominates that of communication (Section 5.2).
This is unlike some recent works (e.g.[21?24]) where communication costs are observed to dominate
because the per-processor computation is smaller.
Finally, we summarize our main contributions in this paper:
? We propose a coded computing algorithm for multiple instances of a linear inverse problem;
? We theoretically analyze the mean-squared error of coded, uncoded and replication-based
iterative linear solvers under a deadline constraint, and show scaling sense advantage of
coded solvers in theory and orders of magnitude smaller error in data experiments.
? This is the first work that treats stragglers as soft errors instead of erasures, which leads to
graceful degradation in the event that the number of stragglers is large.
2
2.1
System Model and Problem Formulation
Preliminaries on Solving Linear Systems using Iterative Methods
Consider the problem of solving k inverse problems with the same linear transform matrix M and
different inputs ri : Mxi = ri , i = 1, 2, . . . k. When M is a square matrix, the closed-form solution
is xi = M?1 ri . When M is a non-square matrix, the regularized least-square solution is xi =
(M> M + ?I)?1 M> ri , i = 1, 2, . . . k, with an appropriate regularization parameter ?. Since matrix
inversion is hard, iterative methods are often used. We now look at two ordinary iterative methods,
namely the Jacobian method [17] and the gradient descent method. For a square matrix M = D + L,
(l+1)
(l)
where D is diagonal, the Jacobian iteration is written as xi
= D?1 (ri ? Lxi ). Under certain
conditions of D and L ([17, p.115]), the computation result converges to the true solution. One
example is the PageRank algorithm discussed in Section 2.2. For the `2 -minimization problem with a
(l+1)
(l)
non-square M, the gradient descent method has the form xi
= ((1??)I?M> M)xi +M> ri ,
2
where is an appropriate step-size. We can see that both the Jacobian iteration and the gradient
descent iteration mentioned above have the form
(l+1)
(l)
xi
= Bxi + Kri , i = 1, 2, . . . k,
(1)
?
for two appropriate matrices B and K, which solves the following equation with true solution xi :
x?i = Bx?i + Kri , i = 1, 2, . . . k.
(2)
(l)
(l)
Therefore, subtracting (2) from (1), we have that the computation error ei = xi ? x?i satisfies
(l+1)
(l)
ei
= Bei .
(3)
For the iterative method to converge, we always assume the spectral radius ?(B) < 1 (see [17,
p.115]). We will study iterative methods that have the form (1) throughout this paper.
2.2
Motivating Applications of Linear Inverse Problems
Our coded computation technique requires solving multiple inverse problems with the same linear
transform matrix M. One such problem is personalized PageRank. For a directed graph, the
PageRank algorithm [19] aims to measure the nodes? importance by solving the linear problem
x = Nd 1N + (1 ? d)Ax, where d = 0.15 is called the ?teleport? probability, N is the number of
nodes and A is the column-normalized adjacency matrix. The personalized PageRank problem [18]
considers a more general equation x = dr + (1 ? d)Ax, for any possible vector r ? RN that satisfies
1> r = 1. Compared to PageRank [19], personalized PageRank [18] incorporates r as the preference
of different users or topics. A classical method to solve PageRank is power-iteration, which iterates
the computation x(l+1) = dr + (1 ? d)Ax(l) until convergence. This iterative method is the same
as (1), which is essentially the Jacobian method mentioned above. Another example application is
the sampling and recovery problem in the emerging field of graph signal processing [25, 26] as a
non-square system, which is discussed in Supplementary section 8.1.
2.3
Problem Formulation: Distributed Computing and the Straggler Effect
Consider solving k linear inverse problems Mxi = ri , i = 1, 2, . . . k in n > k workers using the
iterative method (1), where each worker solves one inverse problem. Due to the straggler effect, the
computation at different workers can have different speeds. The goal is to obtain minimal MSE in
solving linear inverse problems before a deadline time Tdl . Suppose after Tdl , the i-th worker has
completed li iterations in (1). Then, from (3), the residual error at the i-th worker is
(l )
(0)
ei i = Bli ei .
(4)
For our theoretical results, we sometimes need the following assumption.
Assumption 1. We assume that the optimal solutions x?i , i = 1, 2, . . . k, are i.i.d.
Denote by ?E and CE respectively the mean and the covariance of each x?i . Note that Assumption 1
is equivalent to the assumption that the inputs ri , i = 1, 2, . . . k are i.i.d., because ri and x?i are
related by the linear equation (2). For the personalized PageRank problem discussed above, this
assumption is reasonable because queries from different users or topics are unrelated. Assume we
(0)
have estimated the mean ?E beforehand and we start with the initial estimate xi = ?E . Then,
(0)
(0)
ei = xi ? x?i has mean 0N and covariance CE . We also try to extend our results for the case
when x?i ?s (or equivalently, ri ?s) are correlated. Since the extension is rather long and may hinder the
understanding of the main paper, we provide it in supplementary section 8.2 and section 8.5.
2.4
Preliminaries on Error Correcting Codes
We will use ?encode? and ?decode? to denote preprocessing and post-processing before and after
parallel computation. In this paper, the encoder multiplies the inputs to the parallel workers with a
?generator matrix? G and the decoder multiplies the outputs of the workers with a ?decoding matrix?
L (see Algorithm 1). We call a code an (n, k) code if the generator matrix has size k ? n. We often
use generator matrices G with orthonormal rows, which means Gk?n G>
n?k = Ik . An example of
such a matrix is the submatrix formed by any k rows of an n ? n orthonormal matrix (e.g., a Fourier
matrix). Under this assumption, Gk?n can be augmented toform an n ?
n orthonormal matrix using
Gk?n
another matrix H(n?k)?n , i.e. the square matrix Fn?n =
satisfies F> F = In .
H(n?k)?n
3
3
Coded Distributed Computing of Linear Inverse Problems
The proposed coded linear inverse algorithm (Algorithm 1) has three stages: (1) preprocessing
(encoding) at the central controller, (2) parallel computing at n > k parallel workers, and (3) postprocessing (decoding) at the central controller. As we show later in the analysis of computing error,
the entries trace(C(li )) in the diagonal matrix ? are the expected MSE at each worker prior to
decoding. The decoding matrix Lk?n in the decoding step (7) is chosen to be (G??1 G> )?1 G??1
to reduce the mean-squared error of the estimates of linear inverse solutions by assigning different
weights to different workers based on the estimated accuracy of their computation (which is what ?
provides). This particular choice of ? is inspired from the weighted least-square solution.
Algorithm 1 Coded Distributed Linear Inverse
Input: Input vectors [r1 , r2 , . . . , rk ], generator matrix Gk?n , the linear system matrices B and K
defined in (1).
Initialize (Encoding): Encode the input vectors and the initial estimates by multiplying G:
[s1 , s2 , . . . , sn ] = [r1 , r2 , . . . , rk ] ? G.
(0)
(0)
[y1 , y2 , . . . , yn(0) ]
(5)
(0)
(0)
(0)
[x1 , x2 , . . . , xk ]
=
? G.
(6)
Parallel Computing:
for i = 1 to n (in parallel) do
(0)
(0)
Send si and yi to the i-th worker. Execute the iterative method (1) with initial estimate yi
and input si at each worker.
end for
(l )
After a deadline time Tdl , collect all linear inverse results yi i from these n workers. The
(li )
superscript li in yi represents that the i-th worker finished li iterations. Denote by Y(Tdl ) the
(Tdl )
(l )
(l )
(l )
collection of all results YN ?n
= [y1 1 , y2 2 , . . . , yn n ].
Post Processing (decoding at the central controller):
Compute an estimate of the linear inverse solutions using the following matrix multiplication:
? > = L ? (Y(Tdl ) )> := (G??1 G> )?1 G??1 (Y(Tdl ) )> ,
X
(7)
? N ?k = [?
?2, . . . , x
? k ], the matrix ? is
where the estimate X
x1 , x
? = diag [trace(C(l1 )), . . . , trace(C(ln ))] ,
(8)
where the matrices C(li ), i = 1, . . . , n are defined as
C(li ) = Bli CE (B> )li .
(9)
In computation of ?, if trace(C(li )) are not available, one can use precomputed estimates of this
trace as discussed in Supplementary Section 8.9 with negligible computational complexity and
theoretically guaranteed accuracy.
3.1
Bounds on Performance of the Coded Linear Inverse Algorithm
Define l = [l1 , l2 , . . . ln ] as the vector of the number of iterations at all workers. E[?|l] denotes the
conditional expectation taken with respect to the randomness of the optimal solution x?i (see Assumption 1) conditioned on fixed iteration number li at each worker, i.e., E[X|l] = E[X|l1 , l2 , . . . ln ].
Define X?N ?k = [x?1 , x?2 , . . . x?k ] as the matrix composed of all the true solutions.
? ? X? , i.e., the error of the decoding result (7). Assuming that the
Theorem 3.1. Define E = X
solutions for each linear inverse problem are chosen i.i.d. (across all problems) according to a
distribution with covariance CE . Then, the error covariance of E satisfies
2
E[kEk |l] ? ?max (G> G)trace (G??1 G> )?1 ,
(10)
where the norm k?k is the Frobenius norm, ?max (G> G) is the maximum eigenvalue of G> G and
the matrix ? is defined in (8). Further, when G has orthonormal rows,
2
E[kEk |l] ? trace (G??1 G> )?1 ,
(11)
4
Proof overview. See supplementary Section 8.3 for the complete proof. Here we provide the main
intuition by analyzing a ?scalar version? of the linear inverse problem, in which case the matrix B is
equal to a scalar a. For B = a, the inputs and the initial estimates in (5) and (6) are vectors instead
of matrices. As we show in Supplementary Section 8.3, if we encode both the inputs and the initial
estimates using (5) and (6), we also ?encode? the error
(0)
(0)
(0)
(0)
(0)
[1 , 2 , . . . , (0)
n ] = [e1 , e2 , . . . , ek ] ? G =: E0 G,
(0)
(0)
(0)
(12)
(0)
where i = yi ? yi? is the initial error at the i-th worker, ei = xi ? x?i is the initial error of
(0) (0)
(0)
(0)
the i-th linear inverse problem, and E0 := [e1 , e2 , . . . ek ]. Suppose var[ei ] = ce , which is a
scalar version of CE after Assumption 1. From (4), the error satisfies:
(l )
(0)
i i = ali i , i = 1, 2, . . . n.
(13)
Denote by D = diag{al1 , al2 , . . . aln }. Therefore, from (12) and (13), the error before the decoding
step (7) can be written as
(l )
(l )
(0)
(0)
(0)
n)
[1 1 , 2 2 , . . . (l
n ] =[1 , 2 , . . . n ] ? D = E0 GD.
(14)
We can show (see Supplementary Section 8.3 for details) that after the decoding step (7), the error
vector is also multiplied by the decoding matrix L = (G??1 G> )?1 G??1 :
h
i>
(l ) (l )
n)
(15)
E> = L 1 1 , 2 2 , . . . (l
= LD> G> E>
n
0.
Thus,
2
>
E[kEk |l] =E[trace[E> E]|l] = trace[LD> G> E[E>
0 E0 |l]GDL ]
(a)
= trace[LD> G> ce Ik GDL> ] = ce trace[LD> G> GDL> ]
(16)
(b)
?ce ?max (G> G)trace[LD> DL> ] = ?max (G> G)trace[L(ce D> D)L> ]
(c)
(d)
= ?max (G> G)trace[L?L> ] = ?max (G> G)trace[(G??1 G> )?1 ],
(0)
(0)
(0)
(0)
where (a) holds because E0 := [e1 , e2 , . . . ek ] and var[ei ] = ce , (b) holds because G> G
?max (G> G)In , (c) holds because ce D> D = ?, which is from the fact that for a scalar linear system
matrix B = a, the entries in the ? matrix in (8) satisfy
trace(C(li )) = ali ce (a> )li = ce a2li ,
(17)
>
which is the same as the entries in the diagonal matrix ce D D. Finally, (d) is obtained by directly plugging in L := (G??1 G> )?1 G??1 . Finally, inequality 11 holds because when G has
orthonormal rows, ?(G> G) = 1.
Additionally, we note that in (10), the term trace (G??1 G> )?1 resembles the MSE of ordinary
weighted least-square solution, and the term ?max (G> G) represents the ?inaccuracy? due to using
the weighted least-square solution as the decoding result, because the inputs to different workers
become correlated by multiplying the i.i.d. inputs with matrix G (see (5)).
4
Comparison with Uncoded Schemes and Replication-based Schemes
Here, we often assume (we will state explicitly in the theorem) that the number of iterations li at
different workers are i.i.d.. We use Ef [?] to denote expectation on randomness of both the linear
inverse solutions x?i and the number of iterations li (this is different from the notation E[?|l]).
Assumption 2. Within time Tdl , the number of iterations of linear inverse computations (see (1)) at
each worker follows an i.i.d. distribution li ? f (l).
4.1
Comparison between the coded and uncoded linear inverse before a deadline
First, we compare the coded linear inverse scheme with an uncoded scheme, in which case we use
the first k workers to solve k linear inverse problems in (2) without coding. The following theorem
quantifies the overall mean-squared error of the uncoded scheme given l1 , l2 , . . . , lk . The proof is in
Supplementary Section 8.6.
5
2
h
i
(l )
(l )
2
Theorem 4.1. In the uncoded scheme, the error E kEuncoded k |l = E
[e1 1 . . . , ek k ]
l =
Pk
i=1 trace (C(li )). Further, when the i.i.d. Assumption 2 holds,
h
i
2
Ef kEuncoded k = kEf [trace(C(l1 ))].
(18)
Then, we compare the overall mean-squared error of coded and uncoded linear inverse algorithms.
Note that this comparison is not fair because the coded algorithm uses more workers than uncoded.
However, we still include Theorem 4.2 because we need it for the fair comparison between coded and
replication-based linear inverse. The proof is in Supplementary section 8.4.
Theorem 4.2. (Coded linear inverse beats uncoded) Suppose the i.i.d. Assumptions 1 and
2 hold and
Gk?n
suppose G is a k ? n submatrix of an n ? n Fourier transform matrix F, i.e., Fn?n =
.
H(n?k)?n
Then, expected error of the coded linear inverse is strictly less than that of uncoded:
h
i
h
i
2
2
>
Ef kEuncoded k ? Ef kEcoded k ? Ef [trace(J2 J?1
(19)
4 J2 )],
J1 J2
where J2 and J4 are the submatrices of F?F> := >
and the matrix ? is defined in
J2 J4 n?n
(8). That is, (J1 )k?k is G?G> , (J2 )k?(n?k) is G?H> , and (J4 )(n?k)?(n?k) is H?H> .
4.2
Comparison between the replication-based and coded linear inverse before a deadline
Consider an alternative way of doing linear inverse using n > k workers. In this paper, we only
consider the case when n ? k < k, i.e., the number of extra workers is only slightly bigger than
the number of problems (both in theory and in experiments). Since we have n ? k extra workers,
a natural way is to pick any (n ? k) linear inverse problems and replicate them using these extra
(n ? k) workers. After we obtain two computation results for the same equation, we use two
natural ?decoding? strategies for this replication-based linear inverse: (i) choose the worker with
1
2
higher number of iterations; (ii) compute the weighted average using weights w1w+w
and w1w+w
,
2
2
p
p
where w1 = 1/ trace(C(l1 )) and w2 = 1/ trace(C(l2 )), and l1 and l2 are the number of iterations
completed at the two workers (recall that trace(C(li )) represents the residual MSE at the i-th worker).
Theorem 4.3. The replication-based schemes satisfy the following lower bound on the MSE:
h
i
h
i
2
2
Ef kErep k >Ef kEuncoded k ? (n ? k)Ef [trace(C(l1 ))].
(20)
Proof overview. Here the goal is to obtain a lower bound on the MSE of replication-based linear
inverse and compare it with an upper bound on the MSE of coded linear inverse.
Note that if an extra worker is used to replicate the computation at the i-th worker, i.e., the linear
inverse problem with input ri is solved on two workers, the expected error of the result of the i-th
problem could at best reduced from Ef [trace(C(l1 ))] (see Thm. 4.1) to zero1 . Therefore, (n?k) extra
workers make the error decrease by at most (and strictly smaller than) (n ? k)Ef [trace(C(l1 ))].
Using this lower bound, we can provably show that coded linear inverse beats replication-based linear
inverse when certain conditions are satisfied. One crucial condition is that the distribution of the
random variable trace(C(l)) (i.e., the expected MSE at each worker) satisfies a ?variance heavy-tail?
property defined as follows.
Definition 1. The random variable trace(C(l)) is said to have a ??-variance heavy-tail? property if
varf [trace(C(l))] > ?E2f [trace(C(l))],
(21)
for some constant ? > 1. Notice that the term trace(C(l)) is essentially the remaining MSE after l
iterations at a single machine. Therefore, this property simply means the remaining error at a single
machine has large variance. For the coded linear inverse, we will use a ?Fourier code?, the generator
matrix G of which is a submatrix of a Fourier matrix. This particular choice of code is only for ease
of analysis in comparing coded linear inverse and replication-based linear inverse. In practice, the
code that minimizes mean-squared error should be chosen.
1
Although this is clearly a loose bound, it makes for convenient comparison with coded linear inverse.
6
Theorem 4.4. (Coded linear inverse beats replication) Suppose the i.i.d. Assumptions 1 and 2 hold?and
G is a k ? n submatrix of k rows of an n ? n Fourier matrix F. Further, suppose (n ? k) = o( n).
Then, the expected error of the coded linear inverse satisfies
i
h
ii var [trace(C(l ))]
1 h h
f
1
2
2
lim
Ef kEuncoded k ? Ef kEcoded k
?
.
(22)
n?? n ? k
Ef [trace(C(l1 ))]
Moreover, if the random variable trace(C(l)) satisfies the ?-variance heavy-tail property for ? > 1,
coded linear inverse outperforms replication-based linear inverse in the following sense,
lim
n??
1
1
1
lim
Ef kEuncoded k2 ? Ef kErep k2 <
Ef kEuncoded k2 ? Ef kEcoded k2 .
(n ? k)
? n?? (n ? k)
(23)
Proof overview. See Supplementary Section 8.7 for a complete and rigorous proof.
h Here we only
i
2
provide the main intuition behind the proof. From Theorem 4.2, we have Ef kEuncoded k ?
h
i
2
>
Ef kEcoded k ? Ef [trace(J2 J?1
4 J2 )]. Therefore, to prove (22), the main technical difficulty is
?1 >
to simplifythe term trace(J
2 J4 J2 ). For a Fourier matrix F, we are able to show that the matrix
J1 J2
F?F> = >
(see Theorem 4.2) is a Toeplitz matrix, which provides a good structure for
J2 J 4
us to study its behavior. Then, we use the Gershgorin circle theorem [27] (with some algebraic
manipulations) to show that the maximum eigenvalue of J4 satisfies ?max (J4 ) ? Ef [trace(C(l1 ))],
and separately using some algebraic manipulations, we show
trace(J2 J>
2 ) ? (n ? k)varf [trace(C(l1 ))],
>
?1 >
for large matrix size n. Since trace(J2 J?1
J2 ) =
4 J2 ) ? trace(J2 (?max (J4 ))
(24)
1
>
?max (J4 ) trace(J2 J2 ),
(n ? k)varf [trace(C(l1 ))]
,
(25)
Ef [trace(C(l1 ))]
for large n. Then, (22) can be proved by plugging (25) into (19). After that, we can combine (22),
(20) and the variance heavy-tail property to prove (23).
>
trace(J2 J?1
4 J2 ) ?
4.3
Asymptotic Comparison between Coded, Uncoded and Replication-based linear inverse
as the Deadline Tdl ? ?
Assumption 3. We assume the computation time of one power iteration is fixed at each worker for
each linear inverse computation, i.e., there exist n independent (not necessarily identically distributed)
random variables v1 , v2 , . . . vn such that li = d Tvdli e, i = 1, 2, . . . n.
The above assumption is validated in experiments in Supplementary Section 8.13.
The k-th order statistic of a sample is equal to its k-th smallest value. Suppose the order statistics
of the sequence v1 , v2 , . . . vn are vi1 < vi2 < . . . vin , where {i1 , i2 , . . . in } is a permutation of
{1, 2, . . . n}. Denote by [k] the set {1, 2, . . . k} and [n] the set {1, 2, . . . n}.
Theorem 4.5. (Error exponent comparison when Tdl ? ?) Suppose the i.i.d. Assumption 1 and
Assumption 3 hold. Suppose n ? k < k. Then, the error exponents of the coded and uncoded
computation schemes satisfy
1
2
1
2
log E[kEcoded k |l] ?
log
,
(26)
Tdl ??
Tdl
vi k
1?d
1
1
1
2
2
2
lim ?
log E[kEuncoded k |l] = lim ?
log E[kErep k |l] =
log
. (27)
Tdl ??
Tdl ??
Tdl
Tdl
maxi?[k] vi
1?d
lim ?
The error exponents satisfy coded>replication=uncoded. Here the expectation E[?|l] is only taken
with respect to the randomness of the linear inverse sequence xi , i = 1, 2, . . . k.
Proof overview. See Supplementary Section 8.8 for a detailed proof. The main intuition behind this
result is the following: when Tdl approaches infinity, the error of uncoded computation is dominated
7
by the slowest worker among the first k workers, which has per-iteration time maxi?[k] vi . For the
replication-based scheme, since the number of extra workers n?k < k, there is a non-zero probability
(which does not change with Tdl ) that the n ? k extra workers do not replicate the computation in
the slowest one among the first worker. Therefore, replication when n ? k < k does not improve
the error exponent, because the error is dominated by this slowest worker. For coded computation,
we show in Supplementary Section 8.8 that the slowest n ? k workers among the overall n workers
do not affect the error exponent, which means that the error is dominated by the k-th fastest worker,
which has per-iteration time vik . Since the k-th fastest worker among all n workers can not be slower
than the slowest one among the first (unordered) k workers, the error exponent of coded linear inverse
is larger than that of the uncoded and the replication-based linear inverse.
5
5.1
Analyzing the Computational Complexity
Encoding and decoding complexity
We first show that the encoding and decoding complexity of Algorithm 1 are in scaling-sense smaller
than that of the computation at each worker. This ensures that straggling comes from the parallel
workers, not the encoder or decoder. The proof of Theorem 5.1 is in Supplementary Section 8.10. In
our experiment on the Google Plus graph (See Section 6) for computing PageRank, the computation
time at each worker is 30 seconds and the encoding and decoding time at the central controller is
about 1 second.
Theorem 5.1. The computational complexity for the encoding and decoding is ?(nkN ), where N is
the number of rows in the matrix B and k, n depend on the number of available workers assuming
that each worker performs a single linear inverse computation. For a general dense matrix B, the
computational complexity of computing linear inverse at each worker is ?(N 2 l), where l is the
number of iterations in the specified iterative algorithm. The complexity of encoding and decoding is
smaller than that of the computation at each user for large B matrices (large N ).
5.2
Analysis on the cost of communication versus computation
In this work, we focus on optimizing the computation cost. However, what if the computation cost is
small compared to the overall cost, including the communication cost? If this is true, optimizing the
computation cost is not very useful. In Theorem 5.2 (proof appears in Supplementary Section 8.11),
we show that the computation cost is larger than the communication cost in the scaling-sense.
Theorem 5.2. The ratio between the number of operations (computation) and the number of bits trans? operations
mitted (communication) at the i-th worker is COSTcomputation /COSTcommunication = ?(li d)
?
per integer, where li is the number of iterations at the i-th worker, and d is the average number of
non-zeros in each row of the B matrix.
6
Experiments on Real Systems
We test the performance of the coded linear inverse algorithm for the PageRank problem on the
Twitter graph and the Google Plus graph from the SNAP datasets [28]. The Twitter graph has 81,306
nodes and 1,768,149 edges, and the Google Plus graph has 107,614 nodes and 13,673,453 edges. We
use the HT-condor framework in a cluster to conduct the experiments. The task is to solve k = 100
personalized PageRank problems in parallel using n = 120 workers. The uncoded algorithm picks
the first k workers and uses one worker for each PageRank problem. The two replication-based
schemes replicate the computation of the first n ? k PageRank problems in the extra n ? k workers
(see Section 4.2). The coded PageRank uses n workers to solve these k = 100 equations using
Algorithm 1. We use a (120, 100) code where the generator matrix is the submatrix composed of the
first 100 rows in a 120 ? 120 DFT matrix. The computation results are shown in the left two figures in
Fig. 2. Note that the two graphs are of different sizes so the computation in the two experiments take
different time. From Fig. 2, we can see that the mean-squared error of uncoded and replication-based
schemes is larger than that of coded computation by a factor of 104 for large deadlines.
We also compare Algorithm 1 with the coded computing algorithm proposed in [6]. As we discussed
in the Figure 1, the original coded technique in [6] ignores partial results and is suboptimal even in the
toy example of three workers. However, it has a natural extension to iterative methods, which will be
8
Average mean-squared error
Google Plus graph
10 0
10 0
Repetition-2
Google Plus graph
Twitter graph
Uncoded
Uncoded
Coded
Repetition-1
10 -5
Twitter graph
Binary
10 0
10 0
10 -5
Repetition-2
Extension of coded
Method in [Lee et.al.]
Gaussian
>10 4
DFT
Original Coded
Method in [Lee et.al.]
Repetition-1
10 -10
10 -10
Coded
10 -5
Sparse
Algorithm 1
0
10
20
30
Deadline Tdl (sec)
0.5
1
1.5
Deadline Tdl (sec)
2
0
10
20
30
Deadline Tdl (sec)
0
1
Deadline Tdl (sec)
2
Figure 2: From left to right: (1,2) Experimentally computed overall MSE of uncoded, replicationbased and coded personalized PageRank on the Twitter and Google Plus graph on a cluster with 120
workers. The ratio of MSE for repetition-based schemes and coded linear inverse increase as Tdl
increases. (3) Comparison between an extended version of the algorithm in [6] and Algorithm 1 on
the Google Plus graph. The figure shows that naively extending the general coded method using
matrix inverse introduces error amplification. (4) Comparison of different codes. In this experiment
the DFT-code out-performs the other candidates in MSE.
discussed in details later. The third figure in Fig. 2 shows the comparison between the performance
of Algorithm 1 and this extension of the algorithm from [6]. This extension uses the (unfinished)
partial results from the k fastest workers to retrieve the required PageRank solutions. More concretely,
suppose S ? [n] is the index set of the k fastest workers. Then, this extension retrieves the solutions to
the original k PageRank problems by solving the equation YS = [x?1 , x?2 , . . . , x?k ] ? GS , where YS is
composed of the (partial) computation results obtained from the fastest k workers and GS is the k ? k
submatrix composed of the columns in the generator matrix G with indexes in S. However, since
there is some remaining error at each worker (i.e., the computation results YS have not converged
yet), when conducting the matrix-inverse-based decoding from [6], the error is magnified due to the
large condition number of GS . This is why the algorithm in [6] should not be naively extended in the
coded linear inverse problem.
One question remains: what is the best code design for the coded linear inverse algorithm? Although
we do not have a concrete answer to this question, we have tested different codes (with different
generator matrices G) in the Twitter graph experiment, all using Algorithm 1. The results are shown
in the fourth figure in Fig. 2. The generator matrix used for the ?binary? curve has i.i.d. binary entries
in {?1, 1}. The generator matrix used for the ?sparse? curve has random binary sparse entries. The
generator matrix for the ?Gaussian? curve has i.i.d. standard Gaussian entries. In this experiment, the
DFT-code performs the best. However, finding the best code in general is a meaningful future work.
7
Conclusions
By studying coding for iterative algorithms designed for distributed inverse problems, we aim to
introduce new applications and analytical tools to the problem of coded computing with stragglers.
Since these iterative algorithms designed for inverse problems commonly have decreasing error
with time, the partial computation results at stragglers can provide useful information for the final
outputs. Note that this is unlike recent works on coding for multi-stage computing problems [29, 30],
where the computation error can accumulate with time and coding has to be applied repeatedly to
suppress this error accumulation. An important connection worth discussing is the diversity gain in
this coded computing problem. The distributed computing setting in this work resembles random
fading channels, which means coding can be used to exploit straggling diversity just as coding is
used in communication channels to turn diverse channel fading into an advantage. What makes
coding even more suitable in our setting is that the amount of diversity gain achieved here through
replication is actually smaller than that can be achieved by replication in fading channels. This
is because for two computers that solve the same equation Mxi = ri , the remaining error at the
slow worker is a deterministic multiple of the remaining error at the fast worker (see equation (3)).
Therefore, taking a weighted average of the two computation results through replication does not
reduce error as in independent fading channels. How diversity gain can be achieved here optimally is
worth deep investigation. Our next goals are two-fold: (1) extend the current method to solving a
single large-scale inverse problem, such as graph mining with graphs that exceed the memory of a
single machine; (2) carry out experiments on faster distributed systems such as Amazon EC2.
9
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6,272 | 6,674 | A Screening Rule for `1-Regularized
Ising Model Estimation
Zhaobin Kuang1 , Sinong Geng2 , David Page3
University of Wisconsin
[email protected] , [email protected] , [email protected]
Abstract
We discover a screening rule for `1 -regularized Ising model estimation. The simple
closed-form screening rule is a necessary and sufficient condition for exactly
recovering the blockwise structure of a solution under any given regularization
parameters. With enough sparsity, the screening rule can be combined with various
optimization procedures to deliver solutions efficiently in practice. The screening
rule is especially suitable for large-scale exploratory data analysis, where the
number of variables in the dataset can be thousands while we are only interested
in the relationship among a handful of variables within moderate-size clusters for
interpretability. Experimental results on various datasets demonstrate the efficiency
and insights gained from the introduction of the screening rule.
1
Introduction
While the field of statistical learning with sparsity [Hastie et al., 2015] has been steadily rising to
prominence ever since the introduction of the lasso (least absolute shrinkage and selection operator)
at the end of the last century [Tibshirani, 1996], it was not until the recent decade that various
screening rules debuted to further equip the ever-evolving optimization arsenals for some of the
most fundamental problems in sparse learning such as `1 -regularized generalized linear models
(GLMs, Friedman et al. 2010) and inverse covariance matrix estimation [Friedman et al., 2008].
Screening rules, usually in the form of an analytic formula or an optimization procedure that is
extremely fast to solve, can accelerate learning drastically by leveraging the inherent sparsity of many
high-dimensional problems. Generally speaking, screening rules can identify a significant portion of
the zero components of an optimal solution beforehand at the cost of minimal computational overhead,
and hence substantially reduce the dimension of the parameterization, which makes possible efficient
computation for large-scale sparse learning problems.
Pioneered by Ghaoui et al. 2010, various screening rules have emerged to speed up learning for
generative models (e.g. Gaussian graphical models) as well as for discriminative models (e.g. GLMs),
and for continuous variables (e.g. lasso) as well as for discrete variables (e.g. logistic regression,
support vector machines). Table 1 summarizes some of the iconic work in the literature, where, to the
best of our knowledge, screening rules for generative models with discrete variables are still notably
absent.
Contrasted with this notable absence is the ever stronger craving in the big data era for scaling
up the learning of generative models with discrete variables, especially in a blockwise structure
identification setting. For example, in gene mutation analysis [Wan et al., 2015, 2016], among tens of
thousands of sparse binary variables representing mutations of genes, we are interested in identifying
a handful of mutated genes that are connected into various blocks and exert synergistic effects on the
phenotype. While a sparse Ising model is a desirable choice, for such an application the scalability of
the model fails due to the innate N P-hardness [Karger and Srebro, 2001] of inference, and hence
learning, owing to the partition function. To date, even with modern approximation techniques, a
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Table 1: Screening rules in the literature at a glance
Discriminative Models
Generative Models
Continuous
Variables
Ghaoui et al. 2010, Tibshirani et al. 2012
Wang et al. 2013, Liu et al. 2013
Fercoq et al. 2015, Xiang et al. 2016
Banerjee et al. 2008 Honorio and Samaras 2010
Witten et al. 2011,Mazumder and Hastie 2012
Danaher et al. 2014, Luo et al. 2014
Yang et al. 2015
Discrete
Variables
Ghaoui et al. 2010, Tibshirani et al. 2012
Wang et al. 2014
?
typical application with sparse discrete graphical models usually involves only hundreds of variables
[Viallon et al., 2014, Barber et al., 2015, Vuffray et al., 2016].
Between the need for the scalability of high-dimensional Ising models and the absence of screening
rules that are deemed crucial to accelerated and scalable learning, we have a technical gap to bridge:
can we identify screening rules that can speed up the learning of `1 -regularized Ising models? The
major contribution of this paper is to give an affirmative answer to this question. Specifically, we
show the following.
? The screening rule is a simple closed-form formula that is a necessary and sufficient condition for
exact blockwise structure recovery of the solution with a given regularization parameter. Upon the
identification of blockwise structures, different blocks of variables can be considered as different
Ising models and can be solved separately. The various blocks can even be solved in parallel to
attain further efficiency. Empirical results on both simulated and real-world datasets demonstrate
the tremendous efficiency, scalability, and insights gained from the introduction of the screening
rule. Efficient learning of `1 -regularized Ising models from thousands of variables on a single
machine is hence readily attainable.
? As an initial attempt to fill in the vacancy illustrated in Table 1, our work is instructive to further
exploration of screening rules for other graphical models with discrete random variables, and
to combining screening rules with various optimization methods to facilitate better learning.
Furthermore, compared with its Gaussian counterpart, where screening rules are available (Table 1)
and learning is scalable [Hsieh et al., 2013], the proposed screening rule is especially valuable and
desperately needed to address the more challenging learning problem of sparse Ising models.
We defer all the proofs in the paper to the supplement and focus on providing intuition and interpretation of the technical results in the paper.
2
2.1
Notation and Background
Ising Models
>
Let X = [X1 , X2 , ? ? ? , Xp ] be a p ? 1 binary random vector, with Xi ? {?1, 1}, and i ?
{1, 2, ? ? ? , p} , V . Let
there be a dataset X with n independent and identically distributed samples
of X, denoted as X = x(1) , x(2) , ? ? ? , x(n) . Here, x(k) is a p?1 vector of assignments that realizes
(k)
X, where k ? {1, 2, ? ? ? , n}. We further use xi to denote the ith component of the k th sample in
the dataset. Let ? ? ? be a p ? p symmetric matrix whose diagonal entries are zeros. An Ising model
[Wan et al., 2016] with the parameterization ? is:
?
?
p?1 X
p
X
1
P? (x) =
exp ?
?ij xi xj ? ,
(1)
Z(?)
i=1 j>i
where ?ij represents the component of ? at the ith row and the j th column, and xi and xj represent
the ith and the j th components of x, respectively. Z(?) is a normalization constant, partition
function, that ensures
probabilitiessum up to one. The partition function is given as Z(?) =
P theP
P
p?1
p
x?{?1,1}p exp
i=1
j>i ?ij xi xj . Note that for ease of presentation, we consider Ising
models with only pairwise interaction/potential here. Generalization to Ising models with unary
potentials is given in Section 6.
2
2.2
Graphical Interpretation
With the notion of the probability given by an Ising model in (1), estimating an `1 -regularized Ising
? the penalized maximum likelihood estimator (MLE) under the lasso
model is defined as finding ?,
penalty:
n
?
1X
?? = arg max
log P? x(k) ? k?k1
? n
2
k=1
(2)
n p?1 p
?
1 XXX
(k) (k)
?ij xi xj + A(?) + k?k1 .
= arg min ?
?
n
2
i=1 j>i
k=1
Pp Pp
Here, A(?) = log Z(?) is the log-partition function; k?k1 = i=1 j=1 |?ij | is the lasso penalty
that encourages a sparse parameterization. ? ? 0 is a given regularization parameter. Using ?2 is
Pp?1 Pp
suggestive of the symmetry of ? so that ?2 k?k1 = ? i=1 j>i |?ij |, which echoes the summations
in the negative log-likelihood function. Note that ? corresponds to the adjacency matrix constructed
by the p components of X as nodes, and ?ij 6= 0 indicates that there is an edge between Xi and
Xj . We further denote a partition of V into L blocks as {C1 , C2 , ? ? ? , CL }, where Cl , Cl0 ? V ,
SL
Cl ? Cl0 = ?, l=1 Cl = V , l 6= l0 , and for all l, l0 ? {1, 2, ? ? ? , L}. Without loss of generality, we
assume that the nodes in different blocks are ordered such that if i ? Cl , j ? Cl0 , and l < l0 , then
i < j.
2.3
Blockwise Solutions
We introduce the definition of a blockwise parameterization:
Definition 1. We call ? blockwise with respect to the partition {C1 , C2 , ? ? ? , CL } if ?l and l0 ?
{1, 2, ? ? ? , L}, where l 6= l0 , and ?i ? Cl , ?j ? Cl0 , we have ?ij = 0.
When ? is blockwise, we can represent ? in a block diagonal fashion:
? = diag (?1 , ?2 , ? ? ? , ?L ) ,
(3)
where ?1 , ?2 , ? ? ? , and ?L are symmetric matrices that correspond to C1 , C2 , ? ? ? , and CL , respectively.
Note that if we can identify the blockwise structure of ?? in advance, we can solve each block
independently (See A.1). Since the size of each block could be much smaller than the size of the
original problem, each block could be much easier to learn compared with the original problem.
Therefore, efficient identification of blockwise structure could lead to substantial speedup in learning.
3
3.1
The Screening Rule
Main Results
The preparation in Section 2 leads to the discovery of the following strikingly simple screening rule
presented in Theorem 1.
Theorem 1. Let a partition of V, {C1 , C2 , ? ? ? , CL }, be given. Let the dataset X =
(1) (2)
Pn
(k) (k)
x , x , ? ? ? , x(n) be given. Define EX Xi Xj = n1 k=1 xi xj . A necessary and sufficient
condition for ?? to be blockwise with respect to the given partition is that
|EX Xi Xj | ? ?,
(4)
for all l and l0 ? {1, 2, ? ? ? , L}, where l 6= l0 , and for all i ? Cl , j ? Cl0 .
In terms of exact blockwise structure identification, Theorem 1 provides a foolproof (necessary and
sufficient) and yet easily checkable result by comparing the absolute second empirical moments
|EX Xi Xj |?s with the regularization parameter ?. We also notice the remarkable similarity between
the proposed screening rule and the screening rule for Gaussian graphical model blockwise structure
identification in Witten et al. 2011, Mazumder and Hastie 2012. In the Gaussian case, the screening
rule can be attained by simply replacing the second empirical moment matrix in (4) with the sample
3
Algorithm 1 Blockwise Minimization
Input: dataset X, regularization parameter ?.
?
Output: ?.
?i, j ? V such that j > i, compute the second empirical moments EX Xi Xj ?s .
Identify the partition {C1 , C2 , ? ? ? , CL } using the second empirical moments from the previous
step and according to Witten et al. [2011], Mazumder and Hastie [2012].
5: ?l ? L, perform blockwise optimization over Cl for ??l .
?
6: Ensemble ??l ?s according to (3) for ?.
?
7: Return ?.
1:
2:
3:
4:
covariance matrix. While the exact solution in the Gaussian case can be computed in polynomial time,
estimating an Ising model exactly in general is N P-hard. However, as a consequence of applying the
screening rule, the blockwise structure of an `1 -regularized Ising model can be determined as easily
as the blockwise structure of a Gaussian graphical model, despite the fact that within each block,
exact learning of a sparse Ising model could still be challenging.
Furthermore, the screening rule also provides us a principal approach to leverage sparsity for the gain
of efficiency: by increasing ?, the nodes of the Ising model will be shattered into smaller and smaller
blocks, according to the screening rule. Solving many Ising models with small blocks of variables is
amenable to both estimation algorithm and parallelism.
3.2
Regularization Parameters
The screening rule also leads to a significant implication to the range of regularization parameters in
which ?? 6= 0. Specifically, we have the following theorem.
Theorem 2. Let the dataset X = x(1) , x(2) , ? ? ? , x(n) be given, and let ? = ?max represent the
smallest regularization parameter such that ?? = 0 in (2). Then ?max = maxi,j?V,i6=j |EX Xi Xj | ? 1.
With ?max , one can decide the range of regularization parameters, [0, ?max ], that generates graphs
with nonempty edge sets, which is an important first step for pathwise optimization algorithms
(a.k.a. homotopy algorithms) that learn the solutions to the problem under a range of ??s. Furthermore,
the fact that ?max ? 1 for any given dataset X suggests that comparison across different networks
generated by different datasets is comprehensible. Finally, in Section 4, ?max will also help to
establish the connection between the screening rule for exact learning and some of the popular inexact
(alternative) learning algorithms in the literature.
3.3
Fully Disconnected Nodes
Another consequence of the screening rule is the necessary and sufficient condition that determines
the regularization parameter with which a node is fully disconnected from the remaining nodes:
Corollary 1. Let the dataset X = x(1) , x(2) , ? ? ? , x(n) be given. Xi is fully disconnected from
? where i ? V (i.e., ??ij = ??ji = 0, ?j ? V \ {i}), if and only if
the remaining nodes in ?,
? ? maxj?V \{i} |EX Xi Xj |.
In high-dimensional exploratory data analysis, it is usually the case that most of the variables are
fully disconnected [Danaher et al., 2014, Wan et al., 2016]. In this scenario, Corollary 1 provides a
regularization parameter threshold with which we can identify exactly the subset of fully disconnected
nodes. Since we can choose a threshold large enough to make any nodes fully disconnected, we can
discard a significant portion of the variables efficiently and flexibly at will with exact optimization
guarantees due to Corollary 1. By discarding the large portion of fully disconnected variables, the
learning algorithm can focus on only a moderate number of connected variables, which potentially
results in a substantial efficiency gain.
3.4
Blockwise Minimization
We conclude this section by providing the blockwise minimization algorithm in Algorithm 1 due
to the screening rule. Note that both the second empirical moments and the partition of V in the
4
algorithm can be computed in O(p2 ) operations [Witten et al., 2011, Mazumder and Hastie, 2012].
On the contrary, the complexity of the exact optimization of a block of variables grows exponentially
with respect to the maximal clique size of that block. Therefore, by encouraging enough sparsity,
the blockwise minimization due to the screening rule can provide remarkable speedup by not only
shrinking the size of the blocks in general but also potentially reducing the size of cliques within each
block via eliminating enough edges.
4
Applications to Inexact (Alternative) Methods
We now discuss the interplay between the screening rule and two popular inexact (alternative)
estimation methods: node-wise (NW) logistic regression [Wainwright et al., 2006, Ravikumar et al.,
2010] and the pseudolikelihood (PL) method [H?fling and Tibshirani, 2009]. In what follows, we
use ??NW and ??PL to denote the solutions given by the node-wise logistic regression method and the
pseudolikelihood method, respectively. NW can be considered as an asymmetric pseudolikelihood
NW
NW
method (i.e., ?i,j ? V such that i 6= j and ??ij
), while PL is a pseudolikelihood method that
6= ??ji
is similar to NW but imposes additional symmetric constraints on the parameterization (i.e., ?i,j ? V
PL
PL
where i 6= j, we have ??ij
= ??ji
).
Our incorporation of the screening rule to the inexact methods is straightforward: after using the
screening rule to identify different blocks in the solution, we use inexact methods to solve each block
for the solution. As shown in Section 3, when combined with exact optimization, the screening
rule is foolproof for blockwise structure identification. However, in general, when combined with
inexact methods, the proposed screening rule is not foolproof any more because the screening rule is
derived from the exact problem in (2) instead of the approximate problems such as NW and PL. We
provide a toy example in A.6 to illustrate mistakes made by the screening rule when combined with
inexact methods. Nonetheless, as we will show in this section, NW and PL are deeply connected to
the screening rule, and when given a large enough regularization parameter, the application of the
screening rule to NW and PL can be lossless in practice (see Section 5). Therefore, when applied to
NW and PL, the proposed screening rule can be considered as a strong rule (i.e., a rule that is not
foolproof but barely makes mistakes) and an optimal solution can be safeguarded by adjusting the
screened solution to optimality based on the KKT conditions of the inexact problem [Tibshirani et al.,
2012].
4.1
Node-wise (NW) Logistic Regression and the Pseudolikelihood (PL) Method
In NW, for each i ? V , we consider the conditional probability of Xi upon X\i , where X\i =
{Xt | t ? V \ {i}}. This is equivalent to solving p `1 -regularized logistic regression problems
separately, i.e., ?i ? V :
n
i
1 X h (k) (k)
(k)
NW
??\i
= arg min
?yi ?\i + log 1 + exp ?\i
+ ?
?\i
1 ,
(5)
?\i n
k=1
(k)
?\i
(k)
(k)
>
where
= ?\i
(2x\i ), yi = 1
(k)
unsuccessful event xi = ?1, and
(k)
represents a successful event xi
(k)
= 1, yi
= 0 represents an
>
?\i = ?i1 ?i2 ? ? ? ?i(i?1) ?i(i+1) ? ? ? ?ip ,
h
i>
(k)
(k)
(k)
(k)
(k)
(k)
x\i = xi1 xi2 ? ? ? xi(i?1) xi(i+1) ? ? ? xip
.
NW
Note that ??NW constructed from ??\i
?s is asymmetric, and ad hoc post processing techniques are used
to generate a symmetric estimation such as setting each pair of elements from ??NW in symmetric
positions to the one with a larger (or smaller) absolute value.
On the other hand, PL can be considered as solving all p `1 -regularized logistic regression problems
in (5) jointly with symmetric constraints over the parameterization [Geng et al., 2017]:
p
n
i ?
1 X X h (k) (k)
(k)
??PL = arg min
?yi ?i + log 1 + exp ?i
+ k?k1 ,
(6)
??? n
2
i=1
k=1
5
P
(k)
(k)
where ?i = j?V \{i} 2?min{i,j},max{i,j} xj .That is to say, if i < j, then ?min{i,j},max{i,j} = ?ij ;
if i > j, then ?min{i,j},max{i,j} = ?ji . Recall that ? in (6) defined in Section 2.1 represents a space
of symmetric matrices whose diagonal entries are zeros.
4.2
Regularization Parameters in NW and PL
Since the blockwise structure of a solution is given by the screening rule under a fixed regularization
parameter, the ranges of regularization parameters under which NW and PL can return nonzero
solutions need to be linked to the range [0, ?max ] in the exact problem. Theorem 3 and Theorem 4
establish such relationships for NW and PL, respectively.
Theorem 3. Let the dataset X = x(1) , x(2) , ? ? ? , x(n) be given, and let ? = ?NW
max represent the
NW
NW
?
smallest regularization parameter such that ?\i = 0 in (5), ?i ? V . Then ?max = ?max .
Theorem 4. Let the dataset X = x(1) , x(2) , ? ? ? , x(n) be given, and let ? = ?PL
max represent the
PL
PL
?
smallest regularization parameter such that ? = 0 in (6), then ?max = 2?max .
Let ? be the regularization parameter used in the exact problem. A strategy is to set the corresponding
?NW = ? when using NW and ?PL = 2? when using PL, based on the range of regularization parameters given in Theorem 3 and Theorem 4 for NW and PL. Since the magnitude of the regularization
parameter is suggestive of the magnitude of the gradient of the unregulated objective, the proposed
strategy leverages that the magnitudes of the gradients of the unregulated objectives for NW and
PL are roughly the same as, and roughly twice as large as, that of the unregulated exact objective,
respectively.
This observation has been made in the literature of binary pairwise Markov networks [H?fling and
Tibshirani, 2009, Viallon et al., 2014]. Here, by Theorem 3 and Theorem 4, we demonstrate that
this relationship is exactly true if the optimal parameterization is zero. H?fling and Tibshirani 2009
even further exploits this observation in PL for exact optimization. Their procedure can be viewed as
iteratively solving adjusted PL problems regularized by ?PL = 2? in order to obtain an exact solution
regularized by ?. The close quantitative correspondence between the derivatives of the inexact
objectives and that of the exact objective also provides insights into why combing the screening rule
with inexact methods does not lose much in practice.
4.3
Preservation for Fully Disconnectedness
While the screening rule is not foolproof when combined with NW and PL, it turns out that in terms
of identifying fully disconnected nodes, the necessary and sufficient condition in Corollary 1 can be
preserved when applying NW with caution, as shown in the following.
NW
Theorem 5. Let the dataset X = x(1) , x(2) , ? ? ? , x(n) be given. Let ??min
? ? denote a symmetric
NW
NW
?
?
matrix derived from ?
by setting each pair of elements from ?
in symmetric positions to the
one with a smaller absolute value. A sufficient condition for Xi to be fully disconnected from the
NW
remaining nodes in ??min
, where i ? V , is that ?NW ? maxj?V \{i} |EX Xi Xj |. Furthermore, when
NW
??\i
= 0, the sufficient condition is also necessary.
In practice, the utility of Theorem 5 is to provide us a lower bound for ? above which we can fully
NW
disconnect Xi (sufficiency). Moreover, if ??\i
= 0 also happens to be true, which is easily verifiable,
we can conclude that such a lower bound is tight (necessity).
5
Experiments
Experiments are conducted on both synthetic data and real world data. We will focus on efficiency in
Section 5.1 and discuss support recovery performance in Section 5.2. We consider three synthetic
networks (Table 2) with 20, 35, and 50 blocks of 20-node, 35-node, and 50-node subnetworks,
respectively. To demonstrate the estimation of networks with unbalanced-size subnetworks, we also
consider a 46-block network with power law degree distributed subnetworks of sizes ranging from 5
to 50. Within each network, the subnetwork is generated according to a power law degree distribution,
which mimics the structure of a biological network and is believed to be more challenging to recover
6
1000
50
1500
Methods
PL
NW
PL+screen
NW+screen
750
NW
PL+screen
NW+screen
1000
500
250
Runtime (s)
Runtime (s)
Runtime (s)
100
1250
Methods
PL
NW
PL+screen
NW+screen
Runtime (s)
Methods
Methods
PL
NW
PL+screen
NW+screen
1000
750
500
500
250
0
0
400
800
1200
1600
400
Sample Size
(a) Network 1
800
1200
1600
400
Sample Size
800
1200
Sample Size
(b) Network 2
(c) Network 3
1600
400
800
1200
1600
Sample Size
(d) Network 4
Figure 1: Runtime of pathwise optimization on networks in Table 2. Runtime plotted is the median
runtime over five trials. The experiments of the baseline method PL without screening can not be
fully conducted on larger networks due to high memory cost. NW: Node-wise logistic regression
without screening; NW+screen: Node-wise logistic regression with screening; PL: Pseudolikelihood
without screening; PL+screen: Pseudolikelihood with screening.
compared with other less complicated structures [Chen and Sharp, 2004, Peng et al., 2009, Danaher
et al., 2014]. Each edge of each network is associated with a weight first sampled from a standard
normal distribution, and then increased or decreased by 0.2 to further deviate from zero. For each
network, 1600 samples are generated via Gibbs sampling within each subnetwork. Experiments on
exact optimization are reported in B.2.
5.1
Pathwise Optimization
Pathwise optimization aims to compute solutions over a range of different ??s. Formally, we denote
the set of ??s used in (2) as ? = {?1 , ?2 , ? ? ? , ?? }, and without loss of generality, we assume that
?1 < ?2 < ? ? ? < ?? .
The introduction of the screening rule provides us insightful heuristics for the determination of ?.
We start by choosing a ?1 that reflects the sparse blockwise structural assumption on the data. To
achieve sparsity and avoid densely connected structures, we assume that the number of edges in the
ground truth network is O(p). This assumption coincides with networks generated according to a
power law degree distribution and hence is a faithful representation of the prior knowledge stemming
from many
biological problems. As a heuristic, we relax and apply the screening rule in (4) on each
of the p2 second empirical moments and choose ?1 such that the number of the absolute second
empirical moments that are greater than ?1 is about p log p. Given a ?1 chosen this way, one can
? 1 ) has by the screening rule. To encourage blockwise structures, we
check how many blocks ?(?
? 1 ) has more than one block. We then choose ??
magnify ?1 via ?1 ? 1.05?1 until the current ?(?
such that the number of absolute second empirical moments that are greater than ?? is about p. In
our experiments, we use an evenly spaced ? with ? = 25.
To estimate the networks in Table 2, we implement both NW and PL with and without screening
using glmnet [Friedman et al., 2010] in R as a building block for logistic regression according to
Ravikumar et al. 2010 and Geng et al. 2017. To generate a symmetric parameterization for NW, we
set each pair of elements from ?NW in symmetric positions to the element with a larger absolute value.
Given ?, we screen only at ?1 to identify various blocks. Each block is then solved separately in a
pathwise fashion under ? without further screening. The rationale of performing only one screening
is that starting from a ?1 chosen in the aforementioned way has provided us a sparse blockwise
structure that sets a significant portion of the parameterization to zeros; further screening over larger
??s hence does not necessarily offer more efficiency gain.
Figure 1 summarizes the runtime of pathwise optimization on the four synthetic networks in Table 2.
The experiments are conducted on a PowerEdge R720 server with two Intel(R) Xeon(R) E5-2620
CPUs and 128GB RAM. As many as 24 threads can be run in parallel. For robustness, each runtime
reported is the median runtime over five trials. When the sample size is less than 1600, each trial
uses a subset of samples (subsamples) that are randomly drawn from the original datasets without
replacement. As illustrated in Figure 1, the efficiency gain due to the screening rule is self-evident.
Both NW and PL benefit substantially from the application of the screening rule. The speedup is
more apparent with the increase of sample size as well as the increase of the dimension of the data. In
our experiments, we observe that even with arguably the state-of-the-art implementation [Geng et al.,
7
#blk
20
35
50
46
#nd/blk
20
35
50
5-50
TL#nd
400
1225
2500
1265
Table 2: Summary of the four synthetic networks used in the experiments. indx represents the index
of each network. #blk represents
the number of blocks each network has. #nd/blk represents the
number of nodes each block has.
TL#nd represents the total number
of nodes each network has.
1.00
Methods
900
0.75
Runtime (s)
1
2
3
4
AUC
indx
PL
NW
PL+screen
NW+screen
Mix
600
0.50
Methods
PL
NW
PL+screen
NW+screen
Mix
0.25
0.00
300
0
1
2
3
4
Network Index
(a) Edge recovery AUC
1
2
3
4
Network Index
(b) Model selection runtime
Figure 2: Model selection performance. Mix: provide PL
+screen with the regularization parameter chosen by the model
selection of NW+screen. Other legend labels are the same as
in Figure 1.
2017], PL without screening still has a significantly larger memory footprint compared with that of
NW. Therefore, the experiments for PL without screening are not fully conducted in Figure 1b,1c,
and 1d for networks with thousands of nodes. On the contrary, PL with the screening rule has a
comparable memory footprint with that of NW. Furthermore, as shown in Figure 1, after applying the
screening rule, PL also has a similar runtime with NW. This phenomenon demonstrates the utility of
the screening rule for effectively reducing the memory footprint of PL, making PL readily available
for large-scale problems.
5.2
Model Selection
Our next experiment performs model selection by choosing an appropriate ? from the regularization
parameter set ?. We leverage the Stability Approach to Regularization Selection (StARS, Liu et al.
2010) for this task. In a nutshell, StARS learns a set of various models, denoted as M, over ? using
many subsamples that are drawn randomly from the original dataset without replacement. It then
picks a ?? ? ? that strikes the best balance between network sparsity and edge selection stability
among the models in M. After the determination of ?? , it is used on the entire original dataset to
learn a model with which we compare the ground truth model and calculate its support recovery Area
Under Curve (AUC). Implementation details of model selection are provided in B.1.
In Figure 2, we summarize the experimental results of model selection, where 24 subsamples are used
for pathwise optimization in parallel to construct M. In Figure 2a, NW with and without screening
achieve the same high AUC values over all four networks, while the application of the screening
rule to NW provides roughly a 2x speedup, according to Figure 2b. The same AUC value shared by
the two variants of NW is due to the same ?? chosen by the model selection procedure. Even more
importantly, it is also because that under the same ?? , the screening rule is able to perfectly identify
the blockwise structure of the parameterization.
Due to high memory cost, the model selection for PL without screening (green bars in Figure 2)
is omitted in some networks. To control the memory footprint, the model selection for PL with
screening (golden bars in Figure 2) also needs to be carried out meticulously by avoiding small ??s
in ? that correspond to dense structures in M during estimation from subsamples. While avoiding
dense structures makes PL with screening the fastest among all (Figure 2b), it comes at the cost
of delivering the least accurate (though still reasonably effective) support recovery performance
(Figure 2a). To improve the accuracy of this approach, we also leverage the connection between
NW and PL by substituting 2??NW for the resultant regularization parameter from model selection
of PL, where ??NW is the regularization parameter selected for NW. This strategy results in better
performance in support recovery (purple bars in Figure 2a).
5.3
Real World Data
Our real world data experiment applies NW with and without screening to a real world gene mutation
dataset collected from 178 lung squamous cell carcinoma samples [Weinstein et al., 2013]. Each
sample contains 13,665 binary variables representing the mutation statuses of various genes. For ease
8
USP34
ASTN2
FBN2
CDH9
ALPK2
UNC13C
STAB2
KIAA1109
FN1
DYNC1H1
PLXNA4
ADAMTS20
MYH4
BAI3
VCAN
PDE4DIP
COL12A1
WDR17
PTPRT
HRNR
ELTD1
ZNF804A
TMEM132D
NRXN1
SYNE2
TPR
ZNF676
SCN1A
VPS13B
MAGEC1
COL6A6
RIMS2
UNC5D
ROS1
ANKRD30A
FAT1
THSD7B
CNTNAP2
MYH1
C20orf26
Figure 3: Connected components learned from lung squamous cell carcinoma mutation data. Genes in
red are (lung) cancer and other disease related genes [Uhl?n et al., 2015]. Mutation data are extracted
via the TCGA2STAT package [Wan et al., 2015] in R and the figure is rendered by Cytoscape.
of interpretation, we keep genes whose mutation rates are at least 10% across all samples, yielding
a subset of 145 genes in total. We use the model selection procedure introduced in Section 5.2 to
determine a ??NW with which we learn the gene mutation network whose connected components are
shown in Figure 3. For model selection, other than the configuration in B.1, we choose ? = 25. 384
trials are run in parallel using all 24 threads. We also choose ?1 such that about 2p log(p) absolute
second empirical moments are greater than ?1 . We choose ?? such that about 0.25p absolute second
empirical moments are greater than ?? .
In our experiment, NW with and without screening select the same ??NW , and generate the same
network. Since the dataset in question has a lower dimension and a smaller sample size compared with
the synthetic data, NW without screening is adequately efficient. Nonetheless, with screening NW is
still roughly 20% faster. This phenomenon once again indicates that in practice the screening rule
can perfectly identify the blockwise sparsity pattern in the parameterization and deliver a significant
efficiency gain. The genes in red in Figure 3 represent (lung) cancer and other disease related genes,
which are scattered across the seven subnetworks discovered by the algorithm. In our experiment, we
also notice that all the weights on the edges are positive. This is consistent with the biological belief
that associated genes tend to mutate together to cause cancer.
6
Generalization
With unary potentials, the `1 -regularized MLE for the Ising model is defined as:
?
?
p
p?1 X
p
n
X
X
X
?
1
(k)
(k) (k)
?
?? = arg min ?
?ii xi +
?ij xi xj ? + A(?) + k?k1,off ,
?
n
2
i=1
i=1 j>i
(7)
k=1
Pp Pp
where k?k1,off = i=1 j6=i |?ij |. Note that the unary potentials are not penalized, which is a
common practice [Wainwright et al., 2006, H?fling and Tibshirani, 2009, Ravikumar et al., 2010,
Viallon et al., 2014] to ensure a hierarchical parameterization. The screening rule here is to replace
(4) in Theorem 3 with:
|EX Xi Xj ? EX Xi EX Xj | ? ?.
(8)
Exhaustive justification, interpretation, and experiments are provided in Supplement C.
7
Conclusion
We have proposed a screening rule for `1 -regularized Ising model estimation. The simple closed-form
screening rule is a necessary and sufficient condition for exact blockwise structural identification.
Experimental results suggest that the proposed screening rule can provide drastic speedups for
learning when combined with various optimization algorithms. Future directions include deriving
screening rules for more general undirected graphical models [Liu et al., 2012, 2014b,a, Liu, 2014,
Liu et al., 2016], and deriving screening rules for other inexact optimization algorithms [Liu and
Page, 2013]. Further theoretical justifications regarding the conditions upon which the screening rule
can be combined with inexact algorithms to recover block structures losslessly are also desirable.
9
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6,273 | 6,675 | Improved Dynamic Regret for Non-degenerate
Functions
Lijun Zhang? , Tianbao Yang? , Jinfeng Yi? , Rong Jin? , Zhi-Hua Zhou?
National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China
?
Department of Computer Science, The University of Iowa, Iowa City, USA
?
AI Foundations Lab, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA
?
Alibaba Group, Seattle, USA
[email protected], [email protected], [email protected]
[email protected], [email protected]
?
Abstract
Recently, there has been a growing research interest in the analysis of dynamic
regret, which measures the performance of an online learner against a sequence
of local minimizers. By exploiting the strong convexity, previous studies have
shown that the dynamic regret can be upper bounded by the path-length of the
comparator sequence. In this paper, we illustrate that the dynamic regret can be
further improved by allowing the learner to query the gradient of the function
multiple times, and meanwhile the strong convexity can be weakened to other
non-degenerate conditions. Specifically, we introduce the squared path-length,
which could be much smaller than the path-length, as a new regularity of the
comparator sequence. When multiple gradients are accessible to the learner, we
first demonstrate that the dynamic regret of strongly convex functions can be upper
bounded by the minimum of the path-length and the squared path-length. We then
extend our theoretical guarantee to functions that are semi-strongly convex or selfconcordant. To the best of our knowledge, this is the first time that semi-strong
convexity and self-concordance are utilized to tighten the dynamic regret.
1
Introduction
Online convex optimization is a fundamental tool for solving a wide variety of machine learning
problems [Shalev-Shwartz, 2011]. It can be formulated as a repeated game between a learner and
an adversary. On the t-th round of the game, the learner selects a point xt from a convex set X
and the adversary chooses a convex function ft : X 7? R. Then, the function is revealed to the
learner, who incurs loss ft (xt ). The standard performance measure is the regret, defined as the
difference between the learner?s cumulative loss and the cumulative loss of the optimal fixed vector
in hindsight:
T
T
X
X
ft (x).
(1)
ft (xt ) ? min
t=1
x?X
t=1
Over the past decades, various online algorithms, such as the online gradient descent [Zinkevich,
2003], have been proposed to yield sub-linear regret under different scenarios [Hazan et al., 2007,
Shalev-Shwartz et al., 2007].
Though equipped with rich theories, the notion of regret fails to illustrate the performance of online algorithms in dynamic setting, as a static comparator is used in (1). To overcome this limitation, there has been a recent surge of interest in analyzing a more stringent metric?dynamic regret
[Hall and Willett, 2013, Besbes et al., 2015, Jadbabaie et al., 2015, Mokhtari et al., 2016, Yang et al.,
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
2016], in which the cumulative loss of the learner is compared against a sequence of local minimizers, i.e.,
RT? :=R(x?1 , . . . , x?T ) =
T
X
t=1
ft (xt ) ?
T
X
ft (x?t ) =
T
X
t=1
t=1
ft (xt ) ?
T
X
t=1
min ft (x)
(2)
x?X
where x?t ? argminx?X ft (x). A more general definition of dynamic regret is to evaluate the
difference of the cumulative loss with respect to any sequence of comparators u1 , . . . , uT ? X
[Zinkevich, 2003].
It is well-known that in the worst-case, it is impossible to achieve a sub-linear dynamic regret bound,
due to the arbitrary fluctuation in the functions. However, it is possible to upper bound the dynamic
regret in terms of certain regularity of the comparator sequence or the function sequence. A natural
regularity is the path-length of the comparator sequence, defined as
PT? := P(x?1 , . . . , x?T ) =
T
X
t=2
kx?t ? x?t?1 k
(3)
that captures the cumulative Euclidean norm of the difference between successive comparators.
? For
convex functions, the dynamic regret of online gradient descent can be upper bounded by O( T PT? )
[Zinkevich, 2003]. And when all the functions are strongly convex and smooth, the upper bound can
be improved to O(PT? ) [Mokhtari et al., 2016].
In the aforementioned results, the learner uses the gradient of each function only once, and performs
one step of gradient descent to update the intermediate solution. In this paper, we examine an
interesting question: is it possible to improve the dynamic regret when the learner is allowed to
query the gradient multiple times? Note that the answer to this question is no if one aims to promote
the static regret in (1), according to the results on the minimax regret bound [Abernethy et al., 2008a].
We however show that when coming to the dynamic regret, multiple gradients can reduce the upper
bound significantly. To this end, we introduce a new regularity?the squared path-length:
ST? := S(x?1 , . . . , x?T ) =
T
X
t=2
kx?t ? x?t?1 k2
(4)
are small. For example, when
which could be much?smaller than PT? when the local variations
?
kx?t ? x?t?1 k = ?(1/ T ) for all t ? [T ], we have PT? = ?( T ) but ST? = ?(1). We advance the
analysis of dynamic regret in the following aspects.
? When all the functions are strongly convex and smooth, we propose to apply gradient descent multiple times in each round, and demonstrate that the dynamic regret is reduced from
O(PT? ) to O(min(PT? , ST? )), provided the gradients of minimizers are small. We further
present a matching lower bound which implies our result cannot be improved in general.
? When all the functions are semi-strongly convex and smooth, we show that the standard
online gradient descent still achieves the O(PT? ) dynamic regret. And if we apply gradient descent multiple times in each round, the upper bound can also be improved to
O(min(PT? , ST? )), under the same condition as strongly convex functions.
? When all the functions are self-concordant, we establish a similar guarantee if both the
gradient and Hessian of the function can be queried multiple times. Specifically, we propose to apply the damped Newton method multiple times in each round, and prove an
O(min(PT? , ST? )) bound of the dynamic regret under appropriate conditions.1
Application to Statistical Learning Most studies of dynamic regret, including this paper do not
make stochastic assumptions on the function sequence. In the following, we discuss how to interpret our results when facing the problem of statistical learning. In this case, the learner receives a
sequence of losses ?(x? z1 , y1 ), ?(x? z2 , y2 ), . . ., where (zi , yi )?s are instance-label pairs sampled
from a unknown distribution, and ?(?, ?) measures the prediction error. To avoid the random fluctuation caused by sampling, we can set ft as the loss averaged over a mini-batch of instance-label pairs.
As a result, when the underlying distribution is stationary or drifts slowly, successive functions will
be close to each other, and thus the path-length and the squared path-length are expected to be small.
1
PT? and ST? are modified slightly when functions are semi-strongly convex or self-concordant.
2
2
Related Work
The static regret in (1) has been extensively studied in the literature [Shalev-Shwartz,
2011]. It has
?
been established that the static regret can be upper bounded by O( T ), O(log T ), and O(log T )
for convex functions, strongly convex functions, and exponentially concave functions, respectively
[Zinkevich, 2003, Hazan et al., 2007]. Furthermore, those upper bounds are proved to be minimax
optimal [Abernethy et al., 2008a, Hazan and Kale, 2011].
The notion of dynamic regret is introduced by Zinkevich [2003]. If we choose the online gradient
descent as the learner, the dynamic?regret with respect to any comparator sequence u1 , . . . , uT , i.e.,
R(u1 , . . . , uT ), is on the order of T P(u1 , . . . , uTp
). When a prior knowledge of PT? is available,
?
the dynamic regret RT can be upper bounded by O( T PT? ) [Yang et al., 2016]. If all the functions
are strongly convex and smooth, the upper bound of RT? can be improved to O(PT? ) [Mokhtari et al.,
2016]. The O(PT? ) rate is also achievable when all the functions are convex and smooth, and all the
minimizers x?t ?s lie in the interior of X [Yang et al., 2016].
Another regularity of the comparator sequence, which is similar to the path-length, is defined as
P ? (u1 , . . . , uT ) =
T
X
t=2
kut ? ?t (ut?1 )k
where ?t (?) is a dynamic model that predicts a reference point for the t-th round. The advantage
of this measure is that when the comparator sequence follows the dynamical model closely, it can
be much smaller than the path-length P(u1 , . . . , uT ). A novel algorithm named dynamic mirror
descent is?proposed to take ?t (ut?1 ) into account, and the dynamic regret R(u1 , . . . , uT ) is on the
order of T P ? (u1 , . . . , uT ) [Hall and Willett, 2013]. There are also some regularities defined in
terms of the function sequence, such as the functional variation [Besbes et al., 2015]
FT := F(f1 , . . . , fT ) =
T
X
max |ft (x) ? ft?1 (x)|
(5)
max k?ft (x) ? ?ft?1 (x)k2 .
(6)
t=2
x?X
or the gradient variation [Chiang et al., 2012]
GT := G(f1 , . . . , fT ) =
T
X
t=2
x?X
Under the condition that FT ? FT and Ft is given beforehand, a restarted online gradient descent
2/3 1/3
is developed
? by Besbes et al. [2015], and the dynamic regret is upper bounded by O(T FT ) and
O(log T T FT ) for convex functions and strongly convex functions, respectively.
The regularities mentioned above reflect different aspects of the learning problem, and are not directly comparable in general. Thus, it is appealing to develop an algorithm that adapts to the smaller
regularity of the problem. Jadbabaie et al. [2015] propose an adaptive algorithm based on the optimistic mirror descent [Rakhlin and Sridharan, 2013], such that the dynamic regret is given in terms
of all the three regularities (PT? , FT , and GT ). However, it relies on the assumption that the learner
can calculate each regularity incrementally.
In the setting of prediction with expert advice, the dynamic regret is also referred to as tracking
regret or shifting regret [Herbster and Warmuth, 1998, Cesa-bianchi et al., 2012]. The path-length
of the comparator sequence is named as shift, which is just the number of times the expert changes.
Another related performance measure is the adaptive regret, which aims to minimize the static regret
over any interval [Hazan and Seshadhri, 2007, Daniely et al., 2015]. Finally, we note that the study
of dynamic regret is similar to the competitive analysis in the sense that both of them compete
against an optimal offline policy, but with significant differences in their assumptions and techniques
[Buchbinder et al., 2012].
3
Online Learning with Multiple Gradients
In this section, we discuss how to improve the dynamic regret by allowing the learner to query the
gradient multiple times. We start with strongly convex functions, and then proceed to semi-strongly
convex functions, and finally investigate self-concordant functions.
3
Algorithm 1 Online Multiple Gradient Descent (OMGD)
Require: The number of inner iterations K and the step size ?
1: Let x1 be any point in X
2: for t = 1, . . . , T do
3:
Submit xt ? X and the receive loss ft : X 7? R
4:
z1t = xt
5:
for j = 1, . . . , K do
6:
zj+1
= ?X zjt ? ??ft (zjt )
t
7:
end for
8:
xt+1 = zK+1
t
9: end for
3.1
Strongly Convex and Smooth Functions
To be self-contained, we provide the definitions of strong convexity and smoothness.
Definition 1. A function f : X 7? R is ?-strongly convex, if
f (y) ? f (x) + h?f (x), y ? xi +
?
ky ? xk2 , ?x, y ? X .
2
Definition 2. A function f : X 7? R is L-smooth, if
L
ky ? xk2 , ?x, y ? X .
2
Example 1. The following functions are both strongly convex and smooth.
f (y) ? f (x) + h?f (x), y ? xi +
1. A quadratic form f (x) = x? Ax ? 2b? x + c where aI A bI, a > 0 and b < ?;
2. The regularized logistic loss f (x) = log(1 + exp(b? x)) + ?2 kxk2 , where ? > 0.
Following previous studies [Mokhtari et al., 2016], we make the following assumptions.
Assumption 1. Suppose the following conditions hold for each ft : X 7? R.
1. ft is ?-strongly convex and L-smooth over X ;
2. k?ft (x)k ? G, ?x ? X .
When the learner can query the gradient of each function only once, the most popular learning
algorithm is the online gradient descent:
xt+1 = ?X (xt ? ??ft (xt ))
where ?X (?) denotes the projection onto the nearest point in X . Mokhtari et al. [2016] have established an O(PT? ) bound of dynamic regret, as stated below.
Theorem 1. Suppose Assumption 1 is true. By setting ? ? 1/L in online gradient descent, we have
T
X
t=1
where ? =
q
1?
ft (xt ) ? ft (x?t ) ?
1
1
GPT? +
Gkx1 ? x?1 k
1??
1??
2?
1/?+? .
We now consider the setting that the learner can access the gradient of each function multiple times.
The algorithm is a natural extension of online gradient descent by performing gradient descent multiple times in each round. Specifically, in the t-th round, given the current solution xt , we generate
a sequence of solutions, denoted by z1t , . . . , zK+1
, where K is a constant independent from T , as
t
follows:
z1t = xt , zj+1
= ?X zjt ? ??ft (zjt ) , j = 1, . . . , K.
t
Then, we set xt+1 = zK+1
. The procedure is named as Online Multiple Gradient Descent (OMGD)
t
and is summarized in Algorithm 1.
4
By applying gradient descent multiple times, we are able to extract more information from each
function and therefore are more likely to obtain a tight bound for the dynamic regret. The following
theorem shows that the multiple accesses of the gradient indeed help improve the dynamic regret.
Theorem 2. Suppose Assumption 1 is true. By setting ? ? 1/L and K = ? 1/?+?
2? ln 4? in Algorithm 1, for any constant ? > 0, we have
?
?
?
T
?2GPT + 2Gkx1 ? x1 k,
X
P
?
T
? 2
ft (xt ) ? ft (xt ) ? min
? t=1 k?ft (xt )k + 2(L + ?)S ? + (L + ?)kx1 ? x? k2 .
t=1
T
1
2?
PT
k?ft (x?t )k2 is small, Theorem 2 can be simplified as follows.
PT
Corollary 3. Suppose t=1 k?ft (x?t )k2 = O(ST? ), from Theorem 2, we have
When
t=1
T
X
t=1
In particular, if x?t
as ? ? 0, implies
T
X
t=1
ft (xt ) ? ft (x?t ) = O (min(PT? , ST? )) .
belongs to the relative interior of X (i.e., ?ft (x?t ) = 0) for all t ? [T ], Theorem 2,
ft (xt ) ? ft (x?t ) ? min 2GPT? + 2Gkx1 ? x?1 k, 2LST? + Lkx1 ? x?1 k2 .
Compared to Theorem 1, the proposed OMGD improves the dynamic regret from O(PT? ) to
O (min (PT? , ST? )), when the gradients of minimizers are small. Recall the definitions of PT? and
ST? in (3) and (4), respectively. We can see that ST? introduces a square when measuring the difference between x?t and x?t?1 . In this way, if the local variations (kx?t ? x?t?1 k?s) are small, ST? can
be significantly smaller than PT? , as indicated below.
Example 2. Suppose kx?t ? x?t?1 k = T ?? for all t ? 1 and ? > 0, we have
ST? +1 = T 1?2? ? PT? +1 = T 1?? .
?
In particular, when ? = 1/2, we have ST? +1 = 1 ? PT? +1 = T .
ST? is also closely related to the gradient variation in (6). When all the x?t ?s belong to the relative
interior of X , we have ?ft (x?t ) = 0 for all t ? [T ] and therefore
GT ?
T
X
t=2
k?ft (x?t?1 ) ? ?ft?1 (x?t?1 )k2 =
T
X
t=2
k?ft (x?t?1 ) ? ?ft (x?t )k2 ? ?2 ST?
(7)
where the last inequality follows from the property of strongly convex functions [Nesterov, 2004].
The following corollary is an immediate consequence of Theorem 2 and the inequality in (7).
Corollary 4. Suppose Assumption 1 is true, and further assume all the x?t ?s belong to the relative
interior of X . By setting ? ? 1/L and K = ? 1/?+?
2? ln 4? in Algorithm 1, we have
T
X
t=1
ft (xt ) ?
ft (x?t )
? min
2GPT?
+ 2Gkx1 ?
x?1 k,
2LGT
? 2
+ Lkx1 ? x1 k .
?2
In Theorem 2, the number of accesses of gradients K is set to be a constant depending on the
condition number of the function. One may ask whether we can obtain a tighter bound by using a
larger K. Unfortunately, according to our analysis, even if we take K = ?, which means ft (?) is
minimized exactly, the upper bound can only be improved by a constant factor and the order remains
the same. A related question is whether we can reduce the value of K by adopting more advanced
optimization techniques, such as the accelerated gradient descent [Nesterov, 2004]. This is an open
problem to us, and will be investigated as a future work.
Finally, we prove that the O(ST? ) bound is optimal for strongly convex and smooth functions.
5
Theorem 5. For any online learning algorithm A, there always exists a sequence of strongly convex
and smooth functions f1 , . . . , fT , such that
T
X
t=1
ft (xt ) ? ft (x?t ) = ?(ST? )
where x1 , . . . , xT is the solutions generated by A.
Thus, the upper bound in Theorem 2 cannot be improved in general.
3.2
Semi-strongly Convex and Smooth Functions
During the analysis of Theorems 1 and 2, we realize that the proof is built upon the fact that ?when
the function is strongly convex and smooth, gradient descent can reduce the distance to the optimal
solution by a constant factor? [Mokhtari et al., 2016, Proposition 2]. From the recent developments
in convex optimization, we know that a similar behavior also happens when the function is semistrongly convex and smooth [Necoara et al., 2015, Theorem 5.2], which motivates the study in this
section.
We first introduce the definition of semi-strong convexity [Gong and Ye, 2014].
Definition 3. A function f : X 7? R is semi-strongly convex over X , if there exists a constant ? > 0
such that for any x ? X
?
2
(8)
f (x) ? min f (x) ? kx ? ?X ? (x)k
x?X
2
where X ? = {x ? X : f (x) ? minx?X f (x)} is the set of minimizers of f over X .
The semi-strong convexity generalizes several non-strongly convex conditions, such as the quadratic
approximation property and the error bound property [Wang and Lin, 2014, Necoara et al., 2015]. A
class of functions that satisfy the semi-strongly convexity is provided below [Gong and Ye, 2014].
Example 3. Consider the following constrained optimization problem
min
x?X ?Rd
f (x) = g(Ex) + b? x
where g(?) is strongly convex and smooth, and X is either Rd or a polyhedral set. Then, f : X 7? R
is semi-strongly convex over X with some constant ? > 0.
Based on the semi-strong convexity, we assume the functions satisfy the following conditions.
Assumption 2. Suppose the following conditions hold for each ft : X 7? R.
1. ft is semi-strongly convex over X with parameter ? > 0, and L-smooth;
2. k?ft (x)k ? G, ?x ? X .
When the function is semi-strongly convex, the optimal solution may not be unique. Thus, we need
to redefine PT? and ST? to account for this freedom. We define
PT? :=
T
X
t=2
T
2
X
?
?
max
?Xt? (x) ? ?Xt?1
max
?Xt? (x) ? ?Xt?1
(x)
(x)
, and ST? :=
x?X
t=2
x?X
where Xt? = {x ? X : ft (x) ? minx?X ft (x)} is the set of minimizers of ft over X .
In this case, we will use the standard online gradient descent when the learner can query the gradient
only once, and apply the online multiple gradient descent (OMGD) in Algorithm 1, when the learner
can access the gradient multiple times. Using similar analysis as Theorems 1 and 2, we obtain the
following dynamic regret bounds for functions that are semi-strongly convex and smooth.
Theorem 6. Suppose Assumption 2 is true. By setting ? ? 1/L in online gradient descent, we have
T
X
t=1
where ? =
q
1?
?
1/?+? ,
ft (xt ) ?
T
X
t=1
min ft (x) ?
x?X
? 1 = ?X1? (x1 ).
and x
6
?1k
Gkx1 ? x
GPT?
+
1??
1??
Thus, online gradient descent still achieves an O(PT? ) bound of the dynamic regret.
ln 4? in AlgoTheorem 7. Suppose Assumption 2 is true. By setting ? ? 1/L and K = ? 1/?+?
?
rithm 1, for any constant ? > 0, we have
?
?
?1k
T
T
?2GPT + 2Gkx1 ? x
X
X
ft (xt ) ?
min ft (x) ? min G?T
x?X
?
? 1 k2
+ 2(L + ?)ST? + (L + ?)kx1 ? x
t=1
t=1
2?
PT
? 1 = ?X1? (x1 ).
where G?T = max{x?t ?Xt? }Tt=1 t=1 k?ft (x?t )k2 , and x
Again, when the gradients of minimizers are small, in other words, G?T = O(ST? ), the proposed
OMGD improves the dynamic regret form O(PT? ) to O(min(PT? , ST? )).
3.3
Self-concordant Functions
We extend our previous results to self-concordant functions, which could be non-strongly convex
and even non-smooth. Self-concordant functions play an important role in interior-point methods
for solving convex optimization problems. We note that in the study of bandit linear optimization
[Abernethy et al., 2008b], self-concordant functions have been used as barriers for constraints. However, to the best of our knowledge, this is the first time that losses themselves are self-concordant.
The definition of self-concordant functions is given below [Nemirovski, 2004].
Definition 4. Let X be a nonempty open convex set in Rd and f be a C 3 convex function defined on
X . f is called self-concordant on X , if it possesses the following two properties:
1. f (xi ) ? ? along every sequence {xi ? X } converging, as i ? ?, to a boundary point
of X ;
2. f satisfies the differential inequality
3/2
|D3 f (x)[h, h, h]| ? 2 h? ?2 f (x)h
for all x ? X and all h ? Rd , where
?3
|t =t =t =0 f (x + t1 h1 + t2 h2 + t3 h3 ) .
?t1 ?t2 ?t3 1 2 3
Example 4. We provide some examples of self-concordant functions below
[Boyd and Vandenberghe, 2004, Nemirovski, 2004].
D3 f (x)[h1 , h2 , h3 ] =
1. The function f (x) = ? log x is self-concordant on (0, ?).
2. A convex quadratic form f (x) = x? Ax ? 2b? x + c where A ? Rd?d , b ? Rd , and c ? R,
is self-concordant on Rd .
3. If f : Rd 7? R is self-concordant, and A ? Rd?k , b ? Rd , then f (Ax + b) is selfconcordant.
Using the concept of self-concordance, we make the following assumptions.
Assumption 3. Suppose the following conditions hold for each ft : Xt 7? R.
1. ft is self-concordant on domain Xt ;
2. ft is non-degenerate on Xt , i.e., ?2 ft (x) ? 0, ?x ? Xt ;
3. ft attains its minimum on Xt , and denote x?t = argminx?Xt ft (x).
Our approach is similar to previous cases except for the updating rule of xt . Since we do not assume
functions are strongly convex, we need to take into account the second order structure when updating
the current solution xt . Thus, we assume the learner can query both the gradient and Hessian of each
function multiple times. Specifically, we apply the damped Newton method [Nemirovski, 2004] to
update xt , as follows:
i?1
h
1
j
2
)
?ft (zjt ), j = 1, . . . , K
?
f
(z
z1t = xt , zj+1
= zjt ?
t
t
t
j
1 + ?t (zt )
where
r
h
i?1
j
?t (zt ) = ?ft (zjt )? ?2 ft (zjt )
?ft (zjt ).
(9)
7
Algorithm 2 Online Multiple Newton Update (OMNU)
Require: The number of inner iterations K in each round
1: Let x1 be any point in X1
2: for t = 1, . . . , T do
3:
Submit xt ? X and the receive loss ft : X 7? R
4:
z1t = xt
5:
for j = 1, . . . , K do
6:
h
i?1
1
?ft (zjt )
?2 ft (zjt )
zj+1
= zjt ?
t
j
1 + ?t (zt )
where ?t (zjt ) is given in (9)
7:
end for
8:
xt+1 = zK+1
t
9: end for
Then, we set xt+1 = zK+1
. Since the damped Newton method needs to calculate the inverse of the
t
Hessian matrix, its complexity is higher than gradient descent. The procedure is named as Online
Multiple Newton Update (OMNU) and is summarized in Algorithm 2.
To analyze the dynamic regret of OMNU, we redefine the two regularities PT? and ST? as follows:
PT? :=
T
X
kx?t ? x?t?1 kt =
T q
X
ST? :=
T
X
kx?t ? x?t?1 k2t =
T
X
(x?t ? x?t?1 )? ?2 ft (x?t )(x?t ? x?t?1 )
p
t=2
t=2
t=2
(x?t ? x?t?1 )? ?2 ft (x?t )(x?t ? x?t?1 )
t=2
Compared to the definitions in (3) and (4), we introduce ?2 ft (x?t )
where khkt =
when measuring the distance between x?t and x?t?1 . When functions are strongly convex and smooth,
these definitions are equivalent up to constant factors. We then define a quantity to compare the
second order structure of consecutive functions:
n
?1/2 2
?1/2 o
(10)
? ft (x?t ) ?2 ft?1 (x?t?1 )
? = max ?max ?2 ft?1 (x?t?1 )
h? ?2 f
?
t (xt )h.
t=2,...,T
where ?max (?) computes the maximum eigenvalue of its argument. When all the functions are ?strongly convex and L-smooth, ? ? L/?. Then, we have the following theorem regarding the
dynamic regret of the proposed OMNU algorithm.
Theorem 8. Suppose Assumption 3 is true, and further assume
1
kx?t?1 ? x?t k2t ?
, ?t ? 2.
(11)
144
When t = 1, we choose K = O(1)(f1 (x1 ) ? f1 (x?1 ) + log log ?) in OMNU such that
kx2 ? x?1 k21 ?
1
.
144?
(12)
For t ? 2, we set K = ?log4 (16?)? in OMNU, then
T
X
1
1 ?
?
?
PT , 4ST + f1 (x1 ) ? f1 (x?1 ) + .
ft (xt ) ? ft (xt ) ? min
3
36
t=1
The above theorem again implies the dynamic regret can be upper bounded by O(min(PT? , ST? ))
when the learner can access the gradient and Hessian multiple times. From the first property of
self-concordant functions in Definition 4, we know that x?t must lie in the interior of Xt , and thus
?ft (x?t ) = 0 for all t ? [T ]. As a result, we do not need the additional assumption that the gradients
of minimizers are small, which has been used before to simplify Theorems 2 and 7.
Compared to Theorems 2 and 7, Theorem 8 introduces an additional condition in (11). This condition is required to ensure that xt lies in the feasible region of ft (?), otherwise, ft (xt ) can be infinity
8
and it is impossible to bound the dynamic regret. The multiple applications of damped Newton method can enforce xt to be close to x?t?1 . Combined with (11), we conclude that xt is also close to x?t .
Then, based on the property of the Dikin ellipsoid of self-concordant functions [Nemirovski, 2004],
we can guarantee that xt is feasible for ft (?).
4
Conclusion and Future Work
In this paper, we discuss how to reduce the dynamic regret of online learning by allowing the learner
to query the gradient/Hessian of each function multiple times. By applying gradient descent multiple
times in each round, we show that the dynamic regret can be upper bounded by the minimum of the
path-length and the squared path-length, when functions are strongly convex and smooth. We then
extend this theoretical guarantee to functions that are semi-strongly convex and smooth. We finally
demonstrate that for self-concordant functions, applying the damped Newton method multiple times
achieves a similar result.
In the current study, we upper bound the dynamic regret in terms of the path-length or the squared
path-length of the comparator sequence. As we mentioned before, there also exist some regularities
defined in terms of the function sequence, e.g., the functional variation [Besbes et al., 2015]. In the
future, we will investigate whether multiple accesses of gradient/Hessian can improve the dynamic
regret when measured by certain regularities of the function sequence. Another future work is to
extend our results to the more general dynamic regret
R(u1 , . . . , uT ) =
T
X
t=1
ft (xt ) ?
T
X
ft (ut )
t=1
where u1 , . . . , uT ? X is an arbitrary sequence of comparators [Zinkevich, 2003].
Acknowledgments
This work was partially supported by the NSFC (61603177, 61333014), JiangsuSF (BK20160658),
NSF (IIS-1545995), and the Collaborative Innovation Center of Novel Software Technology and
Industrialization. Jinfeng Yi is now at Tencent AI Lab, Bellevue, WA, USA.
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10
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6,274 | 6,676 | Learning Efficient Object Detection Models with
Knowledge Distillation
Guobin Chen1,2 Wongun Choi1 Xiang Yu1 Tony Han2 Manmohan Chandraker1,3
1
2
3
NEC Labs America
University of Missouri
University of California, San Diego
Abstract
Despite significant accuracy improvement in convolutional neural networks (CNN)
based object detectors, they often require prohibitive runtimes to process an image
for real-time applications. State-of-the-art models often use very deep networks
with a large number of floating point operations. Efforts such as model compression
learn compact models with fewer number of parameters, but with much reduced
accuracy. In this work, we propose a new framework to learn compact and fast object detection networks with improved accuracy using knowledge distillation [20]
and hint learning [34]. Although knowledge distillation has demonstrated excellent
improvements for simpler classification setups, the complexity of detection poses
new challenges in the form of regression, region proposals and less voluminous labels. We address this through several innovations such as a weighted cross-entropy
loss to address class imbalance, a teacher bounded loss to handle the regression
component and adaptation layers to better learn from intermediate teacher distributions. We conduct comprehensive empirical evaluation with different distillation
configurations over multiple datasets including PASCAL, KITTI, ILSVRC and
MS-COCO. Our results show consistent improvement in accuracy-speed trade-offs
for modern multi-class detection models.
1
Introduction
Recent years have seen tremendous increase in the accuracy of object detection, relying on deep
convolutional neural networks (CNNs). This has made visual object detection an attractive possibility
for domains ranging from surveillance to autonomous driving. However, speed is a key requirement in
many applications, which fundamentally contends with demands on accuracy. Thus, while advances
in object detection have relied on increasingly deeper architectures, they are associated with an
increase in computational expense at runtime. But it is also known that deep neural networks are
over-parameterized to aid generalization. Thus, to achieve faster speeds, some prior works explore
new structures such as fully convolutional networks, or lightweight models with fewer channels and
small filters [22, 25]. While impressive speedups are obtained, they are still far from real-time, with
careful redesign and tuning necessary for further improvements.
Deeper networks tend to have better performance under proper training, since they have ample
network capacity. Tasks such as object detection for a few categories might not necessarily need
that model capacity. In that direction, several works in image classification use model compression,
whereby weights in each layer are decomposed, followed by layer-wise reconstruction or fine-tuning
to recover some of the accuracy [9, 26, 41, 42]. This results in significant speed-ups, but there is
often a gap between the accuracies of original and compressed models, which is especially large
when using compressed models for more complex problems such as object detection. On the other
hand, seminal works on knowledge distillation show that a shallow or compressed model trained
to mimic the behavior of a deeper or more complex model can recover some or all of the accuracy
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
drop [3, 20, 34]. However, those results are shown only for problems such as classification, using
simpler networks without strong regularization such as dropout.
Applying distillation techniques to multi-class object detection, in contrast to image classification, is
challenging for several reasons. First, the performance of detection models suffers more degradation
with compression, since detection labels are more expensive and thereby, usually less voluminous.
Second, knowledge distillation is proposed for classification assuming each class is equally important,
whereas that is not the case for detection where the background class is far more prevalent. Third,
detection is a more complex task that combines elements of both classification and bounding box
regression. Finally, an added challenge is that we focus on transferring knowledge within the same
domain (images of the same dataset) with no additional data or labels, as opposed other works that
might rely on data from other domains (such as high-quality and low-quality image domains, or
image and depth domains).
To address the above challenges, we propose a method to train fast models for object detection with
knowledge distillation. Our contributions are four-fold:
? We propose an end-to-end trainable framework for learning compact multi-class object detection
models through knowledge distillation (Section 3.1). To the best of our knowledge, this is the first
successful demonstration of knowledge distillation for the multi-class object detection problem.
? We propose new losses that effectively address the aforementioned challenges. In particular, we
propose a weighted cross entropy loss for classification that accounts for the imbalance in the
impact of misclassification for background class as opposed to object classes (Section 3.2), a
teacher bounded regression loss for knowledge distillation (Section 3.3) and adaptation layers for
hint learning that allows the student to better learn from the distribution of neurons in intermediate
layers of the teacher (Section 3.4).
? We perform comprehensive empirical evaluation using multiple large-scale public benchmarks.
Our study demonstrates the positive impact of each of the above novel design choices, resulting in
significant improvement in object detection accuracy using compressed fast networks, consistently
across all benchmarks (Sections 4.1 ? 4.3).
? We present insights into the behavior of our framework by relating it to the generalization and
under-fitting problems (Section 4.4).
2
Related Works
CNNs for Detection. Deformable Part Model (DPM) [14] was the dominant detection framework
before the widespread use of Convolutional Neural Networks (CNNs). Following the success of
CNNs in image classification [27], Girshick et al. proposed RCNN [24] that uses CNN features
to replace handcrafted ones. Subsequently, many CNN based object detection methods have been
proposed, such as Spatial Pyramid Pooling (SPP) [19], Fast R-CNN [13], Faster-RCNN [32] and
R-FCN [29], that unify various steps in object detection into an end-to-end multi-category framework.
Model Compression. CNNs are expensive in terms of computation and memory. Very deep networks
with many convolutional layers are preferred for accuracy, while shallower networks are also widely
used where efficiency is important. Model compression in deep networks is a viable approach to
speed up runtime while preserving accuracy. Denil et al. [9] demonstrate that neural networks are
often over-parametrized and removing redundancy is possible. Subsequently, various methods [5, 7,
10, 15, 17, 30] have been proposed to accelerate the fully connected layer. Several methods based on
low-rank decomposition of the convolutional kernel tensor [10, 23, 28] are also proposed to speed
up convolutional layers. To compress the whole network, Zhang et al. [41, 42] present an algorithm
using asymmetric decomposition and additional fine-tuning. In similar spirit, Kim et al. [26] propose
one-shot whole network compression that achieves around 1.8 times improvement in runtime without
significant drop in accuracy. We will use methods presented in [26] in our experiments. Besides, a
pruning based approach has been proposed [18] but it is challenging to achieve runtime speed-up
with a conventional GPU implementation. Additionally, both weights and input activations can be the
quantized( [18]) and binarized ( [21, 31]) to lower the computationally expensive.
Knowledge Distillation. Knowledge distillation is another approach to retain accuracy with model
compression. Bucila et al. [3] propose an algorithm to train a single neural network by mimicking
the output of an ensemble of models. Ba and Caruana [2] adopt the idea of [3] to compress deep
2
Teacher
Detection
Hint
Regression
Ground Truth
Label
Weighted
Cross Entropy
Loss
Bounded
Regression
Loss
SoftMax &
SmoothL1
Loss
Classification
Regression
Classification
L2 Loss
Back Propagation
Student
Adaptation
Hint
Soft
Label
Distillation
Detection
Guided
Ground
Truth
Figure 1: The proposed learning skeme on visual object detection task using Faster-RCNN, which mainly
consists of region proposal network (RPN) and region classification network(RCN). The two networks both
use multi-task loss to jointly learn the classifier and bounding-box regressor. We employ the final output of the
teacher?s RPN and RCN as the distillation targets, and apply the intermediate layer outputs as hint. Red arrows
indicate the backpropagation pathways.
networks into shallower but wider ones, where the compressed model mimics the ?logits?. Hinton
et al. [20] propose knowledge distillation as a more general case of [3], which applies the prediction
of the teacher model as a ?soft label?, further proposing temperature cross entropy loss instead of L2
loss. Romero et al. [34] introduce a two-stage strategy to train deep networks. In their method, the
teacher?s middle layer provides ?hint? to guide the training of the student model.
Other researchers [16, 38] explore distillation for transferring knowledge between different domains,
such as high-quality and low-quality images, or RGB and depth images. In a draft manuscript
concurrent with our work, Shen et al. [36] consider the effect of distillation and hint frameworks
in learning a compact object detection model. However, they formulate the detection problem as
a binary classification task applied to pedestrians, which might not scale well to the more general
multi-category object detection setup. Unlike theirs, our method is designed for multi-category object
detection. Further, while they use external region proposals, we demonstrate distillation and hint
learning for both the region proposal and classification components of a modern end-to-end object
detection framework [32].
3
Method
In this work, we adopt the Faster-RCNN [32] as the object detection framework. Faster-RCNN
is composed of three modules: 1) A shared feature extraction through convolutional layers, 2) a
region proposal network (RPN) that generates object proposals, and 3) a classification and regression
network (RCN) that returns the detection score as well as a spatial adjustment vector for each object
proposal. Both the RCN and RPN use the output of 1) as features, RCN also takes the result of RPN
as input. In order to achieve highly accurate object detection results, it is critical to learn strong
models for all the three components.
3.1
Overall Structure
We learn strong but efficient student object detectors by using the knowledge of a high capacity teacher
detection network for all the three components. Our overall learning framework is illustrated in Figure
1. First, we adopt the hint based learning [34] (Sec.3.4) that encourages the feature representation of
a student network is similar to that of the teacher network. Second, we learn stronger classification
modules in both RPN and RCN using the knowledge distillation framework [3, 20]. In order to handle
severe category imbalance issue in object detection, we apply weighted cross entropy loss for the
distillation framework. Finally, we transfer the teacher?s regression output as a form of upper bound,
that is, if the student?s regression output is better than that of teacher, no additional loss is applied.
3
Our overall learning objective can be written as follows:
1 X RCN
1 X RCN
Lcls + ?
L
LRCN =
N i
N j reg
LRPN =
1 X RPN
1 X RPN
Lcls + ?
L
M i
M j reg
L = LRPN + LRCN + ?LHint
(1)
where N is the batch-size for RCN and M for RPN. Here, Lcls denotes the classifier loss function
that combines the hard softmax loss using the ground truth labels and the soft knowledge distillation
loss [20] of (2). Further, Lreg is the bounding box regression loss that combines smoothed L1 loss [13]
and our newly proposed teacher bounded L2 regression loss of (4). Finally, Lhint denotes the hint
based loss function that encourages the student to mimic the teacher?s feature response, expressed as
(6). In the above, ? and ? are hyper-parameters to control the balance between different losses. We
fix them to be 1 and 0.5, respectively, throughout the experiments.
3.2
Knowledge Distillation for Classification with Imbalanced Classes
Conventional use of knowledge distillation has been proposed for training classification networks,
where predictions of a teacher network are used to guide the training of a student model. Suppose we
have dataset {xi , yi }, i = 1, 2, ..., n where xi ? I is the input image and yi ? Y is its class label.
Let t be the teacher model, with Pt = softmax( ZTt ) its prediction and Zt the final score output. Here,
T is a temperature parameter (normally set to 1). Similarly, one can define Ps = softmax( ZTs ) for the
student network s. The student s is trained to optimize the following loss function:
Lcls = ?Lhard (Ps , y) + (1 ? ?)Lsof t (Ps , Pt )
(2)
where Lhard is the hard loss using ground truth labels used by Faster-RCNN, Lsof t is the soft loss
using teacher?s prediction and ? is the parameter to balance the hard and soft losses. It is known that
a deep teacher can better fit to the training data and perform better in test scenarios. The soft labels
contain information about the relationship between different classes as discovered by teacher. By
learning from soft labels, the student network inherits such hidden information.
In [20], both hard and soft losses are the cross entropy losses. But unlike simpler classification
problems, the detection problem needs to deal with a severe imbalance across different categories, that
is, the background dominates. In image classification, the only possible errors are misclassifications
between ?foreground? categories. In object detection, however, failing to discriminate between
background and foreground can dominate the error, while the frequency of having misclassification
between foreground categories is relatively rare. To address this, we adopt class-weighted cross
entropy as the distillation loss:
X
Lsof t (Ps , Pt ) = ?
wc Pt log Ps
(3)
where we use a larger weight for the background class and a relatively small weight for other classes.
For example, we use w0 = 1.5 for the background class and wi = 1 for all the others in experiments
on the PASCAL dataset.
When Pt is very similar to the hard label, with probability for one class very close to 1 and most
others very close to 0, the temperature parameter T is introduced to soften the output. Using higher
temperature will force t to produce softer labels so that the classes with near-zero probabilities will
not be ignored by the cost function. This is especially pertinent to simpler tasks, such as classification
on small datasets like MNIST. But for harder problems where the prediction error is already high, a
larger value of T introduces more noise which is detrimental to learning. Thus, lower values of T are
used in [20] for classification on larger datasets. For even harder problems such as object detection,
we find using no temperature parameter at all (equivalent to T = 1) in the distillation loss works the
best in practice (see supplementary material for an empirical study).
3.3
Knowledge Distillation for Regression with Teacher Bounds
In addition to the classification layer, most modern CNN based object detectors [26, 29, 32, 33] also
use bounding-box regression to adjust the location and size of the input proposals. Often, learning a
4
good regression model is critical to ensure good object detection accuracy [13]. Unlike distillation
for discrete categories, the teacher?s regression outputs can provide very wrong guidance toward the
student model, since the real valued regression outputs are unbounded. In addition, the teacher may
provide regression direction that is contradictory to the ground truth direction. Thus, instead of using
the teacher?s regression output directly as a target, we exploit it as an upper bound for the student to
achieve. The student?s regression vector should be as close to the ground truth label as possible in
general, but once the quality of the student surpasses that of the teacher with a certain margin, we
do not provide additional loss for the student. We call this the teacher bounded regression loss, Lb ,
which is used to formulate the regression loss, Lreg , as follows:
2
2
2
kRs ? yk2 , if kRs ? yk2 + m > kRt ? yk2
Lb (Rs , Rt , y) =
0,
otherwise
Lreg = LsL1 (Rs , yreg ) + ?Lb (Rs , Rt , yreg ),
(4)
where m is a margin, yreg denotes the regression ground truth label, Rs is the regression output of
the student network, Rt is the prediction of teacher network and ? is a weight parameter (set as 0.5 in
our experiments). Here, LsL1 is the smooth L1 loss as in [13]. The teacher bounded regression loss
Lb only penalizes the network when the error of the student is larger than that of the teacher. Note
that although we use L2 loss inside Lb , any other regression loss such as L1 and smoothed L1 can be
combined with Lb . Our combined loss encourages the student to be close to or better than teacher in
terms of regression, but does not push the student too much once it reaches the teacher?s performance.
3.4
Hint Learning with Feature Adaptation
Distillation transfers knowledge using only the final output. In [34], Romero et al. demonstrate that
using the intermediate representation of the teacher as hint can help the training process and improve
the final performance of the student. They use the L2 distance between feature vectors V and Z:
2
LHint (V, Z) = kV ? Zk2
(5)
where Z represent the intermediate layer we selected as hint in the teacher network and V represent
the output of the guided layer in the student network. We also evaluate the L1 loss:
LHint (V, Z) = kV ? Zk1
(6)
While applying hint learning, it is required that the number of neurons (channels, width and height)
should be the same between corresponding layers in the teacher and student. In order to match the
number of channels in the hint and guided layers, we add an adaptation after the guided layer whose
output size is the same as the hint layer. The adaptation layer matches the scale of neuron to make
the norm of feature in student close to teacher?s. A fully connected layer is used as adaptation layer
when both hint and guided layers are also fully connected layers. When the hint and guided layers are
convolutional layers, we use 1 ? 1 convolutions to save memory. Interestingly, we find that having
an adaptation layer is important to achieve effective knowledge transferring even when the number
of channels in the hint and guided layers are the same (see Sec. 4.3). The adaptation layer can also
match the difference when the norms of features in hint and guided layers are different. When the
hint or guided layer is convolutional and the resolution of hint and guided layers differs (for examples,
VGG16 and AlexNet), we follow the padding trick introduced in [16] to match the number of outputs.
4
Experiments
In this section, we first introduce teacher and student CNN models and datasets that are used in the
experiments. The overall results on various datasets are shown in Sec.4.1. We apply our methods to
smaller networks and lower quality inputs in Sec.4.2. Sec.4.3 describes ablation studies for three different components, namely classification/regression, distillation and hint learning. Insights obtained
for distillation and hint learning are discussed in Sec.4.4. We refer the readers to supplementary
material for further details.
Datasets We evaluate our method on several commonly used public detection datasets, namely,
KITTI [12], PASCAL VOC 2007 [11], MS COCO [6] and ImageNet DET benchmark (ILSVRC
2014) [35]. Among them, KITTI and PASCAL are relatively small datasets that contain less object
5
Student
Model Info
Teacher PASCAL [email protected] COCO@[.5,.95]
KITTI
ILSVRC
54.7
25.4
11.8
49.3
20.6
AlexNet 57.6 (+2.9) 26.5 (+1.2)
12.3 (+0.5)
51.4 (+2.1) 23.6 (+1.3)
Tucker
11M / 47ms
VGGM 58.2 (+3.5) 26.4 (+1.1)
12.2 (+0.4)
51.4 (+2.1) 23.9 (+1.6)
VGG16 59.4 (+4.7) 28.3 (+2.9)
12.6 (+0.8)
53.7 (+4.4) 24.4 (+2.1)
57.2
32.5
15.8
55.1
27.3
AlexNet 62M / 74ms
VGGM 59.2 (+2.0) 33.4 (+0.9)
16.0 (+0.2)
56.3 (+1.2) 28.7 (+1.4)
VGG16 60.1 (+2.9) 35.8 (+3.3)
16.9 (+1.1)
58.3 (+3.2) 30.1 (+2.8)
59.8
33.6
16.1
56.7
31.1
VGGM
80M / 86ms
VGG16 63.7 (+3.9) 37.2 (+3.6)
17.3 (+1.2)
58.6 (+2.3) 34.0 (+2.9)
VGG16 138M / 283ms
70.4
45.1
24.2
59.2
35.6
Table 1: Comparison of student models associated with different teacher models across four datasets, in terms
of mean Average Precision (mAP). Rows with blank (-) teacher indicate the model is without distillation, serving
as baselines. The second column reports the number of parameters and speed (per image, on GPU).
categories and labeled images, whereas MS COCO and ILSVRC 2014 are large scale datasets.
Since KITTI and ILSVRC 2014 do not provide ground-truth annotation for test sets, we use the
training/validation split introduced by [39] and [24] for analysis. For all the datasets, we follow the
PASCAL VOC convention to evaluate various models by reporting mean average precision (mAP) at
IoU = 0.5 . For MS COCO dataset, besides the PASCAL VOC metric, we also report its own metric,
which evaluates mAP averaged for IoU ? [0.5 : 0.05 : 0.95] (denoted as mAP[.5, .95]).
Models The teacher and student models defined in our experiments are standard CNN architectures,
which consist of regular convolutional layers, fully connected layers, ReLU, dropout layers and
softmax layers. We choose several popular CNN architectures as our teacher/student models, namely,
AlexNet [27], AlexNet with Tucker Decomposition [26], VGG16 [37] and VGGM [4]. We use two
different settings for the student and teacher pairs. In the first set of experiments, we use a smaller
network (that is, less parameters) as the student and use a larger one for the teacher (for example,
AlexNet as student and VGG16 as teacher). In the second set of experiments, we use smaller input
image size for the student model and larger input image size for the teacher, while keeping the
network architecture the same.
4.1
Overall Performance
Table1 shows mAP for four student models on four object detection databases, with different
architectures for teacher guidance. For student models without teacher?s supervision, we train them to
the best numbers we could achieve. Not surprisingly, larger or deeper models with more parameters
perform better than smaller or shallower models, while smaller models run faster than larger ones.
The performance of student models improves significantly with distillation and hint learning over all
different pairs and datasets, despite architectural differences between teacher and student. With a
fixed scale (number of parameters) of a student model, training from scratch or fine-tuning on its own
is not an optimal choice. Getting aid from a better teacher yields larger improvements approaching
the teacher?s performance. A deeper model as teacher leads to better student performance, which
suggests that the knowledge transferred from better teachers is more informative. Notice that the
Tucker model trained with VGG16 achieves significantly higher accuracy than the Alexnet in the
PASCAL dataset, even though the model size is about 5 times smaller. The observation may support
the hypothesis that CNN based object detectors are highly over-parameterized. On the contrary,
when the size of dataset is much larger, it becomes much harder to outperform more complicated
models. This suggests that it is worth having even higher capacity models for such large scale datasets.
Typically, when evaluating efficiency, we get 3 times faster from VGG16 as teacher to AlexNet as
student on KITTI dataset. For more detailed runtimes, please refer to supplementary material.
Further, similar to [38], we investigate another student-teacher mode: the student and teacher share
exactly the same network structure, while the input for student is down-scaled and the input for
teacher remains high resolution. Recent works [1] report that image resolution critically affects
object detection performance. On the other hand, downsampling the input size quadratically reduces
convolutional resources and speeds up computation. In Table 2, by scaling input sizes to half in
6
High-res teacher
Low-res baseline
Low-res distilled student
mAP
Speed
mAP
Speed
mAP
Speed
AlexNet
57.2
1,205 / 74 ms
53.2
726 / 47 ms 56.7(+3.5) 726 / 47 ms
Tucker
54.7
663 / 41 ms
48.6
430 / 29 ms 53.5(+4.9) 430 / 29 ms
Table 2: Comparison of high-resolution teacher model (trained on images with 688 pixels) and low-resolution
student model (trained on 344 pixels input), on PASCAL. We report mAP and speed (both CPU and GPU)
of different models. The speed of low-resolution models are about 2 times faster than the corresponding
high-resolution models, while achieving almost the same accuracy when our distillation method is used.
FLOPS(%)
20
25
30
37.5
45
Finetune
30.3
49.3
51.4
54.7
55.2
Distillation 35.5(+5.2) 55.4(+6.1) 56.8(+5.4) 59.4(+4.7) 59.5(+4.3)
Table 3: Compressed AlexNet performance evaluated on PASCAL. We compare the model fine-tuned with the
ground truth and the model trained with our full method. We vary the compression ratio by FLOPS.
PASCAL VOC dataset for the student and using the original resolution for the teacher, we get almost
the same accuracy as the high-resolution teacher while being about two times faster1 .
4.2
Speed-Accuracy Trade off in Compressed Models
It is feasible to select CNN models from a wide range of candidates to strike a balance between
speed and accuracy. However, off-the-shelf CNN models still may not meet one?s computational
requirements. Designing new models is one option. But it often requires significant labor towards
design and training. More importantly, trained models are often designed for specific tasks, but speed
and accuracy trade-offs may change when facing a different task, whereby one may as well train a
new model for the new task. In all such situations, distillation becomes an attractive option.
To understand the speed-accuracy trade off in object detection with knowledge distillation, we vary the
compression ratio (the ranks of weight matrices) of Alexnet with Tucker decomposition. We measure
the compression ratio using FLOPS of the CNN. Experiments in Table 3 show that the accuracy drops
dramatically when the network is compressed too much, for example, when compressed size is 20%
of original, accuracy drops from 57.2% to only 30.3%. However, for the squeezed networks, our
distillation framework is able to recover large amounts of the accuracy drop. For instance, for 37.5%
compression, the original squeezed net only achieves 54.7%. In contrast, our proposed method lifts
it up to 59.4% with a deep teacher (VGG16), which is even better than the uncompressed AlexNet
model 57.2%.
4.3
Ablation Study
As shown in Table 4, we compare different strategies for distillation and hint learning to highlight the
effectiveness of our proposed novel losses. We choose VGG16 as the teacher model and Tucker as
our student model for all the experiments in this section. Other choices reflect similar trends. Recall
that proposal classification and bounding box regression are the two main tasks in the Faster-RCNN
framework. Traditionally, classification is associated with cross entropy loss, denoted as CLS in
Table 4, while bounding box regression is regularized with L2 loss, denoted as L2.
To prevent the classes with small probability being ignored by the objective function, soft label with
high temperature, also named weighted cross entropy loss, is proposed for the proposal classification
task in Sec.3.2. We compare the weighted cross entropy loss defined in (3), denoted as CLS-W in
Table 4, with the standard cross entropy loss (CLS), to achieve slightly better performance on both
PASCAL and KITTI datasets.
For bounding box regression, directly parroting to teacher?s output will suffer from labeling noise. An
improvement is proposed through (4) in Sec.3.3, where the teacher?s prediction is used as a boundary
to guide the student. Such strategy, denoted as L2-B in Table 4, improves over L2 by 1.3%. Note that
a 1% improvement in object detection task is considered very significant, especially on large-scale
datasets with voluminous number of images.
1
Ideally, the convolutional layers should be about 4 times faster. However, due to the loading overhead and
the non-proportional consumption from other layers, this speed up drops to around 2 times faster.
7
Baseline L2 L2-B CLS CLS-W Hints Hints-A L2-B+CLS-W L2-B+CLS-W+Hints-A
PASCAL 54.7 54.6 55.9 57.4 57.7
56.9
58
58.4
59.4
KITTI
49.3 48.5 50.1 50.8 51.3
50.3 52.1
51.7
53.7
Table 4: The proposed method component comparison, i.e., bounded L2 for regression (L2-B, Sec.3.3) and
weighted cross entropy for classification (CLS-W, Sec.3.2) with respect to traditional methods, namely, L2
and cross entropy (CLS). Hints learning w/o adaptation layer (Hints-A and Hints) are also compared. All
comparisons take VGG16 as the teacher and Tucker as the student, with evaluations on PASCAL and KITTI.
Baseline Distillation Hint Distillation + Hint
Trainval
79.6
78.3
80.9
83.5
PASCAL
Test
54.7
58.4
58
59.4
Train
45.3
45.4
47.1
49.6
COCO
Val
25.4
26.1
27.8
28.3
Table 5: Performance of distillation and hint learning on different datasets with Tucker and VGG16 pair.
Moreover, we find that the adaptation layer proposed in Sec.3.4 is critical for hint learning. Even
if layers from teacher and student models have the same number of neurons, they are almost never
in the same feature space. Otherwise, setting the student?s subsequent structure to be the same as
teacher?s, the student would achieve identical results as the teacher. Thus, directly matching a student
layer to a teacher layer [2, 3] is unlikely to perform well. Instead, we propose to add an adaptation
layer to transfer the student layer feature space to the corresponding teacher layer feature space.
Thereby, penalizing the student feature from the teacher feature is better-defined since they lie in
the same space, which is supported by the results in Table 4. With adaptation layer, hint learning
(Hint-A) shows a 1.1% advantage over the traditional method (Hint). Our proposed overall method
(L2-B+CLS-W+Hint-A) outperforms the one without adaptive hint learning (L2-B+CLS-W) by 1.0%,
which again suggests the significant advantage of hint learning with adaptation.
4.4
Discussion
In this section, we provide further insights into distillation and hint learning. Table 5 compares the
accuracy of Tucker model learned with VGG16 on the trainval and testing split of the PASCAL and
COCO datasets. In general, distillation mostly improves the generalization capability of student,
while hint learning helps improving both the training and testing accuracy.
Distillation improves generalization: Similarly to the image classification case discussed in [20],
there also exists structural relationship among the labels in object detection task. For example,
?Car? shares more common visual characteristics with ?Truck? than with ?Person?. Such structural
information is not available in the ground truth annotations. Thus, injecting such relational information
learned with a high capacity teacher model to a student will help generalization capability of the
detection model. The result of applying the distillation only shows consistent testing accuracy
improvement in Table 5.
Hint helps both learning and generalization: We notice that the ?under-fitting? is a common
problem in object detection even with CNN based models (see low training accuracy of the baselines).
Unlike simple classification cases, where it is easy to achieve (near) perfect training accuracy [40],
the training accuracy of the detectors is still far from being perfect. It seems the learning algorithm
is suffering from the saddle point problem [8]. On the contrary, the hint may provide an effective
guidance to avoid the problem by directly having a guidance at an intermediate layer. Thereby,
the model learned with hint learning achieves noticeable improvement in both training and testing
accuracy.
Finally, by combining both distillation and hint learning, both training and test accuracies are
improved significantly compared to the baseline. Table 5 empirically verifies consistent trends on
both the PASCAL and MS COCO datasets for object detection. We believe that our methods can also
be extended to other tasks that also face similar generalization or under-fitting problems.
5
Conclusion
We propose a novel framework for learning compact and fast CNN based object detectors with
the knowledge distillation. Highly complicated detector models are used as a teacher to guide
the learning process of efficient student models. Combining the knowledge distillation and hint
8
framework together with our newly proposed loss functions, we demonstrate consistent improvements
over various experimental setups. Notably, the compact models trained with our learning framework
execute significantly faster than the teachers with almost no accuracy compromises at PASCAL
dataset. Our empirical analysis reveals the presence of under-fitting issue in object detector learning,
which could provide good insights to further advancement in the field.
Acknowledgments This work was conducted as part of Guobin Chen?s internship at NEC Labs
America in Cupertino.
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6,275 | 6,677 | One-Sided Unsupervised Domain Mapping
Sagie Benaim1 and Lior Wolf1,2
1
The Blavatnik School of Computer Science , Tel Aviv University, Israel
2
Facebook AI Research
Abstract
In unsupervised domain mapping, the learner is given two unmatched datasets
A and B. The goal is to learn a mapping GAB that translates a sample in A
to the analog sample in B. Recent approaches have shown that when learning
simultaneously both GAB and the inverse mapping GBA , convincing mappings
are obtained. In this work, we present a method of learning GAB without learning
GBA . This is done by learning a mapping that maintains the distance between
a pair of samples. Moreover, good mappings are obtained, even by maintaining
the distance between different parts of the same sample before and after mapping.
We present experimental results that the new method not only allows for one
sided mapping learning, but also leads to preferable numerical results over the
existing circularity-based constraint. Our entire code is made publicly available
at https://github.com/sagiebenaim/DistanceGAN.
1
Introduction
The advent of the Generative Adversarial Network (GAN) [6] technology has allowed for the
generation of realistic images that mimic a given training set by accurately capturing what is inside
the given class and what is ?fake?. Out of the many tasks made possible by GANs, the task of
mapping an image in a source domain to the analog image in a target domain is of a particular interest.
The solutions proposed for this problem can be generally separated by the amount of required
supervision. On the one extreme, fully supervised methods employ pairs of matched samples, one
in each domain, in order to learn the mapping [9]. Less direct supervision was demonstrated by
employing a mapping into a semantic space and requiring that the original sample and the analog
sample in the target domain share the same semantic representation [22].
If the two domains are highly related, it was demonstrated that just by sharing weights between the
networks working on the two domains, and without any further supervision, one can map samples
between the two domains [21, 13]. For more distant domains, it was demonstrated recently that by
symmetrically leaning mappings in both directions, meaningful analogs are obtained [28, 11, 27].
This is done by requiring circularity, i.e., that mapping a sample from one domain to the other and
then back, produces the original sample.
In this work, we go a step further and show that it is possible to learn the mapping between the
source domain and the target domain in a one-sided unsupervised way, by enforcing high crossdomain correlation between the matching pairwise distances computed in each domain. The new
constraint allows one-sided mapping and also provides, in our experiments, better numerical results
than circularity. Combining both of these constraints together often leads to further improvements.
Learning the new constraint requires comparing pairs of samples. While there is no real practical
reason not to do so, since training batches contain multiple samples, we demonstrate that similar
constraints can even be applied per image by computing the distance between, e.g., the top part of the
image and the bottom part.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
1.1
Related work
Style transfer These methods [5, 23, 10] typically receive as input a style image and a content image
and create a new image that has the style of the first and the content of the second. The problem of
image translation between domains differs since when mapping between domains, part of the content
is replaced with new content that matches the target domain and not just the style. However, the
distinction is not sharp, and many of the cross-domain mapping examples in the literature can almost
be viewed as style transfers. For example, while a zebra is not a horse in another style, the horse to
zebra mapping, performed in [28] seems to change horse skin to zebra skin. This is evident from the
stripped Putin example obtained when mapping the image of shirtless Putin riding a horse.
Generative Adversarial Networks GAN [6] methods train a generator network G that synthesizes
samples from a target distribution, given noise vectors, by jointly training a second network D. The
specific generative architecture we and others employ is based on the architecture of [18]. In image
mapping, the created image is based on an input image and not on random noise [11, 28, 27, 13, 22, 9].
Unsupervised Mapping The work that is most related to ours, employs no supervision except for
sample images from the two domains. This was done very recently [11, 28, 27] in image to image
translation and slightly earlier for translating between natural languages [24]. Note that [11] proposes
the ?GAN with reconstruction loss? method, which applies the cycle constraint in one side and trains
only one GAN. However, unlike our method, this method requires the recovery of both mappings and
is outperformed by the full two-way method.
The CoGAN method [13], learns a mapping from a random input vector to matching samples from
the two domains. It was shown in [13, 28] that the method can be modified in order to perform
domain translation. In CoGAN, the two domains are assumed to be similar and their generators (and
GAN discriminators) share many of the layers weights, similar to [21]. As was demonstrated in [28],
the method is not competitive in the field of image to image translation.
Weakly Supervised Mapping In [22], the matching between the source domain and the target
domain is performed by incorporating a fixed pre-trained feature map f and requiring f -constancy,
i.e, that the activations of f are the same for the input samples and for mapped samples.
Supervised Mapping When provided with matching pairs of (input image, output image) the
supervision can be performed directly. An example of such method that also uses GANs is [9], where
the discriminator D receives a pair of images where one image is the source image and the other is
either the matching target image (?real? pair) or a generated image (?fake? pair); The linking between
the source and the target image is further strengthened by employing the U-net architecture [19].
Domain Adaptation In this setting, we typically are given two domains, one having supervision in
the form of matching labels, while the second has little or no supervision. The goal is to learn to
label samples from the second domain. In [3], what is common to both domains and what is distinct
is separated thus improving on existing models. In [2], a transformation is learned, on the pixel
level, from one domain to another, using GANs. In [7], an unsupervised adversarial approach to
semantic segmentation, which uses both global and category specific domain adaptation techniques,
is proposed.
2
Preliminaries
In the problem of unsupervised mapping, the learning algorithm is provided with unlabeled datasets
from two domains, A and B. The first dataset includes i.i.d samples from the distribution pA and the
second dataset includes i.i.d samples from the distribution pB . Formally, given
i.i.d
{xi }m
i=1 such that xi ? pA
and
i.i.d
{xj }nj=1 such that xj ? pB ,
our goal is to learn a function GAB , which maps samples in domain A to analog samples in domain
B, see examples below. In previous work [11, 28, 27], it is necessary to simultaneously recover a
second function GBA , which similarly maps samples in domain B to analog samples in domain A.
Justification In order to allow unsupervised learning of one directional mapping, we introduce
the constraint that pairs of inputs x, x0 , which are at a certain distance from each other, are mapped
to pairs of outputs GAB (x), GAB (x0 ) with a similar distance, i.e., that the distances kx ? x0 k and
2
Figure 1: Each triplet shows the source handbag image, the target shoe as produced by CycleGAN?s [28] mapper GAB and the results of approximating GAB by a fixed nonnegative linear
transformation T , which obtains each output pixel as a linear combination of input pixels. The linear
transformation captures the essence of GAB showing that much of the mapping is achieved by a fixed
spatial transformation.
kGAB (x) ? GAB (x0 )k are highly correlated. As we show below, it is reasonable to assume that this
constraint approximately holds in many of the scenarios demonstrated by previous work on domain
translation. Although approximate, it is sufficient, since as was shown in [21], mapping between
domains requires only little supervision on top of requiring that the output distribution of the mapper
matches that of the target distribution.
Consider, for example, the case of mapping shoes to edges, as presented in Fig. 4. In this case, the
edge points are simply a subset of the image coordinates, selected by local image criterion. If image
x is visually similar to image x0 , it is likely that their edge maps are similar. In fact, this similarity
underlies the usage of gradient information in the classical computer vision literature. Therefore,
while the distances are expected to differ in the two domains, one can expect a high correlation.
Next, consider the case of handbag to shoe mapping (Fig. 4). Analogs tend to have the same
distribution of image colors in different image formations. Assuming that the spatial pixel locations
of handbags follow a tight distribution (i.e., the set of handbag images share the same shapes) and the
same holds for shoes, then there exists a set of canonical displacement fields that transform a handbag
to a shoe. If there was one displacement, which would happen to be a fixed permutation of pixel
locations, distances would be preserved. In practice, the image transformations are more complex.
To study whether the image displacement model is a valid approximation, we learned a nonnegative
2
?642
linear transformation T ? R64
that maps, one channel at a time, handbag images of size
+
64 ? 64 ? 3 to the output shoe images of the same size given by the CycleGAN method. T ?s columns
can be interpreted as weights that determine the spread of mass in the output image for each pixel
location in the input image. It was estimated by minimizing the squared error of mapping every
channel (R, G, or B) of a handbag image to the same channel in the matching shoe. Optimization
was done by gradient descent with a projection to the space of nonnegative matrices, i.e., zeroing the
negative elements of T at each iteration.
Sample mappings by the matrix T are shown in Fig. 1. As can be seen, the nonnegative linear
transformation approximates CycleGAN?s multilayer CNN GAB to some degree. Examining the
elements of T , they share some properties with permutations: the mean sum of the rows is 1.06 (SD
0.08) and 99.5% of the elements are below 0.01.
In the case of adding glasses or changing gender or hair color (Fig 3), a relatively minor image
modification, which does not significantly change the majority of the image information, suffices
in order to create the desired visual effect. Such a change is likely to largely maintain the pairwise
image distance before and after the transformation.
In the case of computer generated heads at different angles vs. rotated cars, presented in [11],
distances are highly correlated partly because the area that is captured by the foreground object is
a good indicator of the object?s yaw. When mapping between horses to zebras [28], the texture
of a horse?s skin is transformed to that of the zebra. In this case, most of the image information
is untouched and the part that is changed is modified by a uniform texture, again approximately
maintaining pairwise distances. In Fig 2(a), we compare the L1 distance in RGB space of pairs
of horse images to the distance of the samples after mapping by the CycleGAN Network [28] is
performed, using the public implementation. It is evident that the cross-domain correlation between
pairwise distances is high. We also looked at Cityscapes image and ground truth label pairs in
Fig 2(c), and found that there is high correlation between the distances. This is the also the case in
many other literature-based mappings between datasets we have tested and ground truth pairs.
While there is little downside to working with pairs of training images in comparison to working with
single images, in order to further study the amount of information needed for successful alignment,
we also consider distances between the two halves of the same image. We compare the L1 distance
3
(a)
(b)
(c)
(d)
Figure 2: Justifying the high correlation between distances in different domains. (a) Using the
CycleGAN model [28], we map horses to zebras and vice versa. Green circles are used for the
distance between two random horse images and the two corresponding translated zebra images.
Blue crosses are for the reverse direction translating zebra to horse images. The Pearson correlation
for horse to zebra translation is 0.77 (p-value 1.7e?113) and for zebra to horse it is 0.73 (p-value
8.0e?96). (b) As in (a) but using the distance between two halves of the same image that is either a
horse image translated to a zebra or vice-versa. The Pearson correlation for horse to zebra translation
is 0.91 (p-value 9.5e?23) and for zebra to horse it is 0.87 (p-value 9.7e?19). (c) Cityscapes images
and associated labels. Green circles are used for distance between two cityscapes images and the
two corresponding ground truth images The Pearson correlation is 0.65 (p-value 6.0e?16). (d) As in
(c) but using the distance between two halves of the same image. The Pearson correlation is 0.65
(p-value 1.4e?12).
between the left and right halves as computed on the input image to that which is obtained on the
generated image or the corresponding ground truth image. Fig. 1(b) and Fig. 1(d) presents the results
for horses to zebras translation and for Cityscapes image and label pairs, respectively. As can be seen,
the correlation is also very significant in this case.
From Correlations to Sum of Absolute Differences We have provided justification and empirical
evidence that for many semantic mappings, there is a high degree of correlations between the
pairwise distances in the two domains. In other words, let dk be a vector of centered and unit-variance
normalized pairwise distances in one domain and let d0k be the vector of normalized distances obtained
P
in the other domain by translating each image out of each pair between the domains, then dk d0k
should be high. When training the mapper GAB , the mean and variance used for normalization in
each domain are precomputed based on the training samples in each domain, which assumes that the
post mapping distribution of samples is similar to the training distribution.
P
The pairwise distances in the source domain dk are fixed and maximizing dk d0k causes pairwise
distances dk with large absolute value
the optimization. Instead, we propose to minimize
P to dominate
0
the sum of absolute
differences
|d
?
d
|,
which
spreads the error in distances uniformly. The
k
k
k
P
P
two losses ? dk d0k and k |dk ? d0k | are highly related and the negative correlation between them
was explicitly computed for simple distributions and shown to be very strong [1].
4
3
Unsupervised Constraints on the Learned Mapping
There are a few types of constraints suggested in the literature, which do not require paired samples.
First, one can enforce the distribution of GAB (x) : x ? pA , which we denote as GAB (pA ), to
be indistinguishable from that of pB . In addition, one can require that mapping from A to B and
back would lead to an identity mapping. Another constraint suggested, is that for every x ? B
GAB (x) = x. We review these constraints and then present the new constraints we propose.
Adversarial constraints Our training sets are viewed as two discrete distributions p?A and p?B that
are sampled from the source and target domain distributions pA and pB , respectively. For the learned
network GAB , the similarity between the distributions GAB (pA ) and pB is modeled by a GAN. This
involves the training of a discriminator network DB : B ? {0, 1}. The loss is given by:
LGAN (GAB , DB , p?A , p?B ) =ExB ?p?B [log DB (xB )] + ExA ?p?A [log(1 ? DB (GAB (xA ))]
This loss is minimized over GAB and maximized over DB . When both GAB and GBA are learned
simultaneously, there is an analog expression LGAN (GBA , DA , p?B , p?A ), in which the domains A and
B switch roles and the two losses (and four networks) are optimized jointly.
Circularity constraints In three recent reports [11, 28, 27], circularity loss was introduced for
image translation. The rationale is that given a sample from domain A, translating it to domain B and
then back to domain A should result in the identical sample. Formally, the following loss is added:
Lcycle (GAB , GBA , p?A ) = Ex?p?A kGBA (GAB (x)) ? xk1
The L1 norm employed above was found to be mostly preferable, although L2 gives similar results.
Since the circularity loss requires the recovery of the mappings in both directions, it is usually
employed symmetrically, by considering Lcycle (GAB , GBA , p?A ) + Lcycle (GBA , GAB , p?B ).
The circularity constraint is often viewed as a definite requirement for admissible functions GAB and
GBA . However, just like distance-based constraints, it is an approximate one. To see this, consider
the zebra to horse mapping example. Mapping a zebra to a horse means losing the stripes. The
inverse mapping, therefore, cannot be expected to recover the exact input stripes.
Target Domain Identity A constraint that has been used in [22] and in some of the experiments
in [28] states that GAB applied to samples from the domain B performs the identity mapping. We
did not experiment with this constraint and it is given here for completeness:
LT-ID (GAB , p?B ) = Ex?p?B kx ? GAB (x)k2
Distance Constraints The adversarial loss ensures that samples from the distribution of A are
translated to samples in the distribution of B. However, there are many such possible mappings.
Given a mapping for n samples of A to n samples of B, one can consider any permutation of the
samples in B as a valid mapping and, therefore, the space of functions mapping from A to B is very
large. Adding the circularity constraint, enforces the mapping from B to A to be the inverse of the
permutation that occurs from A to B, which reduces the amount of admissible permutations.
To further reduce this space, we propose a distance preserving map, that is, the distance between two
samples in A should be preserved in the mapping to B. We therefore consider the following loss,
which is the expectation of the absolute differences between the distances in each domain up to scale:
Ldistance (GAB , p?A ) = Exi ,xj ?p?A |
1
1
(kxi ? xj k1 ? ?A ) ?
(kGAB (xi ) ? GAB (xj )k1 ? ?B )|
?A
?B
where ?A , ?B (?A , ?B ) are the means (standard deviations) of pairwise distances in the training sets
from A and B, respectively, and are precomputed.
In practice, we compute the loss over pairs of samples that belong to the same minibatch during
training. Even for minibatches with 64 samples, as in DiscoGAN [11], considering all pairs is feasible.
If needed, for even larger mini-batches, one can subsample the pairs.
When the two mappings are simultaneously learned, Ldistance (GBA , p?B ) is similarly defined. In both
cases, the absolute difference of the L1 distances between the pairs in the two domains is considered.
5
In comparison to circularity, the distance-based constraint does not suffer from the model collapse
problem that is described in [11]. In this phenomenon, two different samples from domain A are
mapped to the same sample in domain B. The mapping in the reverse direction then generates an
average of the two original samples, since the sample in domain B should be mapped back to both the
first and second original samples in A. Pairwise distance constraints prevents this from happening.
Self-distance Constraints Whether or not the distance constraint is more effective than the circularity constraint in recovering the alignment, the distance based constraint has the advantage of being
one sided. However, it requires that pairs of samples are transfered at once, which, while having little
implications on the training process as it is currently done, might effect the ability to perform on-line
learning. Furthermore, the official CycleGAN [28] implementation employs minibatches of size one.
We, therefore, suggest an additional constraint, which employs one sample at a time and compares
the distances between two parts of the same sample.
Let L, R : Rh?w ? Rh?w/2 be the operators that given an input image return the left or right part of
it. We define the following loss:
L
?A )
self- (GAB , p
distance
1
(kL(x) ? R(x)k1 ? ?A )
?A
1
? (kL(GAB (x)) ? R(GAB (x))k1 ? ?B )|
?B
= Ex?p?A |
(1)
where ?A and ?A are the mean and standard deviation of the pairwise distances between the two
halves of the image in the training set from domain P
A, and similarly for ?B and ?B , e.g., given the
training set {xj }nj=1 ? B, ?B is precomputed as n1 j kL(xj ) ? R(xj )k1 .
3.1
Network Architecture and Training
When training the networks GAB , GBA , DB and DA , we employ the following loss, which is
minimized over GAB and GBA and maximized over DB and DA :
?1A LGAN (GAB , DB , p?A , p?B ) + ?1B LGAN (GBA , DA , p?B , p?A ) + ?2A Lcycle (GAB , GBA , p?A )+
?2B Lcycle (GBA , GAB , p?B ) + ?3A Ldistance (GAB , p?A ) + ?3B Ldistance (GBA , p?B )+
?4A Lself-distance (GAB , p?A ) + ?4B Lself-distance (GBA , p?B )
where ?iA , ?iB are trade-off parameters. We did not test the distance constraint and the self-distance
constraint jointly, so in every experiment, either ?3A = ?3B = 0 or ?4A = ?4A = 0. When
performing one sided mapping from A to B, only ?1A and either ?3A or ?4A are non-zero.
We consider A and B to be a subset of R3?s?s of images where s is either 64, 128 or 256, depending
on the image resolution. In order to directly compare our results with previous work and to employ
the strongest baseline in each dataset, we employ the generator and discriminator architectures of
both DiscoGAN [11] and CycleGAN [28].
In DiscoGAN, the generator is build of an encoder-decoder unit. The encoder consists of convolutional layers with 4 ? 4 filters followed by Leaky ReLU activation units. The decoder consists of
deconvolutional layers with 4 ? 4 filters followed by a ReLU activation units. Sigmoid is used for
the output layer and batch normalization [8] is used before the ReLU or Leaky ReLU activations.
Between 4 to 5 convolutional/deconvolutional layers are used, depending on the domains used in
A and B (we match the published code architecture per dataset). The discriminator is similar to the
encoder, but has an additional convolutional layer as the first layer and a sigmoid output unit.
The CycleGAN architecture for the generator is based on [10]. The generators consist of two 2stride convolutional layers, between 6 to 9 residual blocks depending on the image resolution and
two fractionally strided convolutions with stride 1/2. Instance normalization is used as in [10].
The discriminator uses 70 ? 70 PatchGANs [9]. For training, CycleGAN employs two additional
techniques. The first is to replace the negative log-likelihood by a least square loss [25] and the second
is to use a history of images for the discriminators, rather then only the last image generated [20].
6
Table 2:
Normalized
Table 1: Tradeoff weights for each experiment. RMSE between the angles Table 3: MNIST clasof source and translated sification on mapped
Experiment ?1A ?1B ?2A ?2B ?3A ?3B ?4A ?4B
images.
SHVN images.
DiscoGAN
Distance ?
Distance ?
Dist+Cycle
Self Dist ?
Self Dist ?
0.5
0.5
0
0.5
0.5
0
0.5 0.5 0.5 0
0
0
0
0
0
0 0.5 0
0
0
0.5 0
0
0 0.5 0
0
0.5 0.5 0.5 0.5 0.5 0
0
0
0
0
0
0 0.5 0
0.5 0
0
0
0
0 0.5
Method
car2car car2head
DiscoGAN
Distance
Dist.+Cycle
Self Dist.
0.306
0.135
0.098
0.117
0.137
0.097
0.273
0.197
Method
CycleGAN
Distance
Dist.+Cycle
Self Dist.
Accuracy
26.1%
26.8%
18.0%
25.2%
Table 4: CelebA mapping results using the VGG face descriptor.
Male ? Female
4
Blond ? Black
Method
Cosine
Similarity
Separation
Accuracy
DiscoGAN
Distance
Distance+Cycle
Self Distance
0.23
0.32
0.35
0.24
0.87
0.88
0.87
0.86
DiscoGAN
Distance
Distance+Cycle
Self Distance
0.22
0.26
0.31
0.24
0.86
0.87
0.89
0.91
Cosine
Similarity
Separation
Accuracy
Glasses ? Without
Cosine
Similarity
0.15
0.89
0.13
0.24
0.92
0.42
0.24
0.91
0.41
0.24
0.91
0.34
???? Other direction ????
0.14
0.91
0.10
0.22
0.96
0.30
0.22
0.95
0.30
0.19
0.94
0.30
Separation
Accuracy
0.84
0.79
0.82
0.80
0.90
0.89
0.85
0.81
Experiments
We compare multiple methods: the DiscoGAN or the CycleGAN baselines; the one sided mapping
using Ldistance (A ? B or B ? A); the combination of the baseline method with Ldistance ; the
self distance method. For DiscoGAN, we use a fixed weight configuration for all experiments, as
shown in Tab. 1. For CycleGAN, there is more sensitivity to parameters and while the general pattern
is preserved, we used different weight for the distance constraint depending on the experiment, digits
or horses to zebra.
Models based on DiscoGAN Datasets that were tested by DiscoGAN are evaluated here using this
architecture. In initial tests, CycleGAN is not competitive on these out of the box. The first set of
experiments maps rotated images of cars to either cars or heads. The 3D car dataset [4] consists of
rendered images of 3D cars whose degree varies at 15? intervals. Similarly, the head dataset, [17],
consists of 3D images of rotated heads which vary from ?70? to 70? . For the car2car experiment,
the car dataset is split into two parts, one of which is used for A and one for B (It is further split
into train and test set). Since the rotation angle presents the largest source of variability, and since
the rotation operation is shared between the datasets, we expect it to be the major invariant that the
network learns, i.e., a semantic mapping would preserve angles.
A regressor was trained to calculate the angle of a given car image based on the training data. Tab. 2
shows the Root Mean Square Error (RMSE) between the angle of source image and translated image.
As can be seen, the pairwise distance based mapping results in lower error than the DiscoGAN
one, combining both further improves results, and the self distance outperforms both DiscoGAN
and pairwise distance. The original DiscoGAN implementation was used, but due to differences
in evaluation (different regressors) these numbers are not compatible with the graph shown in
DiscoGAN.
For car2head, DiscoGAN?s solution produces mirror images and combination of DiscoGAN?s
circularity constraint with the distance constraint produces a solution that is rotated by 90? . We
consider these biases as ambiguities in the mapping and not as mistakes and, therefore, remove the
mean error prior to computing the RMSE. In this experiment, distance outperforms all other methods.
The combination of both methods is less competitive than both, perhaps since each method pulls
toward a different solution. Self distance, is worse than circularity in this dataset.
7
Another set of experiments arises from considering face images with and without a certain property.
CelebA [26, 14] was annotated for multiple attributes including the person?s gender, hair color, and
the existence of glasses in the image. Following [11] we perform mapping between two values of each
of these three properties. The results are shown in the supplementary material with some examples
in Fig. 3. It is evident that the DiscoGAN method (using the unmodified authors? implementation)
presents many more failure cases than our pair based method. The self-distance method was
implemented with the top and bottom image halves, instead of left to right distances, since faces are
symmetric. This method also seems to outperform DiscoGAN.
In order to evaluate how well the face translation was performed, we use the representation layer of
VGG faces [16] on the image in A and its output in B. One can assume that two images that match
will have many similar features and so the VGG representation will be similar. The cosine similarities,
as evaluated between input images and their mapped versions, are shown in Tab. 4. In all cases, the
pair-distance produces more similar input-output faces. Self-distance performs slightly worse than
pairs, but generally better than DiscoGAN. Applying circularity together with pair-distance, provides
the best results but requires, unlike the distance, learning both sides simultaneously.
While we create images that better match in the face descriptor metric, our ability to create images
that are faithful to the second distribution is not impaired. This is demonstrated by learning a linear
classifier between the two domains based on the training samples and then applying it to a set of
test image before and after mapping. The separation accuracy between the input test image and the
mapped version is also shown in Tab. 4. As can be seen, the separation ability of our method is
similar to that of DiscoGAN (it arises from the shared GAN terms).
We additionally perform a user study to asses the quality of our results. The user is first presented
with a set of real images from the dataset. Then, 50 random pairs of images are presented to a user for
a second, one trained using DiscoGAN and one using our method. The user is asked to decide which
image looks more realistic. The test was performed on 22 users. On shoes to handbags translation,
our translation performed better on 65% of the cases. For handbags to shoes, the score was 87%.
For male to female, both methods showed a similar realness score (51% to 49% of DiscoGAN?s).
We, therefore, asked a second question: given the face of a male, which of the two generated female
variants is a better fit to the original face. Our method wins 88% of the time.
In addition, in the supplementary material we compare the losses of the GAN discriminator for the
various methods and show that these values are almost identical. We also measure the losses of the
various methods during test, even if these were not directly optimized. For example, despite this
constraints not being enforced, the distance based methods seem to present a low circularity loss,
while DiscoGAN presents a relatively higher distance losses.
Sample results of mapping shoes to handbags and edges to shoes and vice versa using the DiscoGAN
baseline architecture are shown in Fig. 3. More results are shown in the supplementary. Visually, the
results of the distance-based approach seem better then DiscoGAN while the results of self-distance
are somewhat worse. The combination of DiscoGAN and distance usually works best.
Models based on CycleGAN Using the CycleGAN architecture we map horses to zebras, see
Fig. 4 and supplementary material for examples. Note that on the zebra to horse mapping, all methods
fail albeit in different ways. Subjectively, it seems that the distance + cycle method shows the most
promise in this translation.
In order to obtain numerical results, we use the baseline CycleGAN method as well as our methods
in order to translate from Street View House Numbers (SVHN) [15] to MNIST [12]. Accuracy is
then measured in the MNIST space by using a neural net trained for this task. Results are shown in
Tab. 3 and visually in the Supplementary. While the pairwise distance based method improves upon
the baseline method, there is still a large gap between the unsupervised and semi-supervised setting
presented in [22], which achieves much higher results. This can be explained by the large amount of
irrelevant information in the SVHN images (examples are shown in the supplementary). Combining
the distance based constraint with the circularity one does not work well on this dataset.
We additionally performed a qualitative evaluation using FCN score as in [28]. The FCN metric
evaluates the interoperability images by taking a generated cityscape image and generating a label
using semantic segmentation algorithm. The generated label can then be compared to the ground
truth label. FCN results are given as three measures: per-pixel accuracy, per-class accuracy and Class
8
Input
Disco GAN
Distance
Distance
+cycle
Self distance
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: Translations using various methods on the celebA dataset: (a,b) Male to and from Female.
(c,d) Blond to and from black hair. (e,f) With eyeglasses to from without eyeglasses.
Input
Disco/
CycleGAN
Distance
Distance
+cycle
Self distance
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: (a,b) Handbags to and from shoes. (c,d) Edges to/from shoes. (e,f) Horse to/from zebra.
IOU. Our distance GAN method is preferable on all three scores (0.53 vs. 0.52, 0.19 vs. 0.17, and
0.11 vs 0.11, respectively). The paired t-test p-values are 0.29, 0.002 and 0.42 respectively. In a user
study similar to the one for DiscoGAN above, our cityscapes translation scores 71% for realness
when comparing to CycleGAN?s. When looking at similarity to the ground truth image we score
68%.
5
Conclusion
We have proposed an unsupervised distance-based loss for learning a single mapping (without its
inverse), which empirically outperforms the circularity loss. It is interesting to note that the new
loss is applied to raw RGB image values. This is in contrast to all of the work we are aware of that
computes image similarity. Clearly, image descriptors or low-layer network activations can be used.
However, by considering only RGB values, we not only show the general utility of our method, but
also further demonstrate that a minimal amount of information is needed in order to form analogies
between two related domains.
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European
Union?s Horizon 2020 research and innovation programme (grant ERC CoG 725974). The authors
would like to thank Laurens van der Maaten and Ross Girshick for insightful discussions.
9
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6,276 | 6,678 | Deep Mean-Shift Priors for Image Restoration
Siavash A. Bigdeli
University of Bern
[email protected]
Meiguang Jin
University of Bern
[email protected]
Paolo Favaro
University of Bern
[email protected]
Matthias Zwicker
University of Bern, and University of Maryland, College Park
[email protected]
Abstract
In this paper we introduce a natural image prior that directly represents a Gaussiansmoothed version of the natural image distribution. We include our prior in a
formulation of image restoration as a Bayes estimator that also allows us to solve
noise-blind image restoration problems. We show that the gradient of our prior
corresponds to the mean-shift vector on the natural image distribution. In addition,
we learn the mean-shift vector field using denoising autoencoders, and use it in a
gradient descent approach to perform Bayes risk minimization. We demonstrate
competitive results for noise-blind deblurring, super-resolution, and demosaicing.
1
Introduction
Image restoration tasks, such as deblurring and denoising, are ill-posed problems, whose solution
requires effective image priors. In the last decades, several natural image priors have been proposed,
including total variation [27], gradient sparsity priors [12], models based on image patches [5], and
Gaussian mixtures of local filters [24], just to name a few of the most successful ideas. See Figure 1
for a visual comparison of some popular priors. More recently, deep learning techniques have been
used to construct generic image priors.
Here, we propose an image prior that is directly based on an estimate of the natural image probability
distribution. Although this seems like the most intuitive and straightforward idea to formulate a prior,
only few previous techniques have taken this route [20]. Instead, most priors are built on intuition or
statistics of natural images (e.g., sparse gradients). Most previous deep learning priors are derived
in the context of specific algorithms to solve the restoration problem, but it is not clear how these
priors relate to the probability distribution of natural images. In contrast, our prior directly represents
the natural image distribution smoothed with a Gaussian kernel, an approximation similar to using
a Gaussian kernel density estimate. Note that we cannot hope to use the true image probability
distribution itself as our prior, since we only have a finite set of samples from this distribution. We
show a visual comparison in Figure 1, where our prior is able to capture the structure of the underlying
image, but others tend to simplify the texture to straight lines and sharp edges.
We formulate image restoration as a Bayes estimator, and define a utility function that includes the
smoothed natural image distribution. We approximate the estimator with a bound, and show that
the gradient of the bound includes the gradient of the logarithm of our prior, that is, the Gaussian
smoothed density. In addition, the gradient of the logarithm of the smoothed density is proportional
to the mean-shift vector [8], and it has recently been shown that denoising autoencoders (DAEs) learn
such a mean-shift vector field for a given set of data samples [1, 4]. Hence we call our prior a deep
mean-shift prior, and our framework is an example of Bayesian inference using deep learning.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Input
Our prior
BM3D [9]
EPLL [39]
FoE [26]
SF [29]
Figure 1: Visualization of image priors using the method by Shaham et al. [30]: Our deep mean-shift
prior learns complex structures with different curvatures. Other priors prefer simpler structures like
lines with small curvature or sharp corners.
We demonstrate image restoration using our prior for noise-blind deblurring, super-resolution, and
image demosaicing, where we solve Bayes estimation using a gradient descent approach. We achieve
performance that is competitive with the state of the art for these applications. In summary, the main
contributions of this paper are:
? A formulation of image restoration as a Bayes estimator that leverages the Gaussian
smoothed density of natural images as its prior. In addition, the formulation allows us
to solve noise-blind restoration problems.
? An implementation of the prior, which we call deep mean-shift prior, that builds on denoising
autoencoders (DAEs). We rely on the observation that DAEs learn a mean-shift vector field,
which is proportional to the gradient of the logarithm of the prior.
? Image restoration techniques based on gradient-descent risk minimization with competitive
results for noise-blind image deblurring, super-resolution, and demosaicing. 1
2
Related Work
Image Priors. A comprehensive review of previous image priors is outside the scope of this paper.
Instead, we refer to the overview by Shaham et al. [30], where they propose a visualization technique
to compare priors. Our approach is most related to techniques that leverage CNNs to learn image
priors. These techniques build on the observation by Venkatakrishnan et al. [31] that many algorithms
that solve image restoration via MAP estimation only need the proximal operator of the regularization
term, which can be interpreted as a MAP denoiser [22]. Venkatakrishnan et al. [31] build on the
ADMM algorithm and propose to replace the proximal operator of the regularizer with a denoiser
such as BM3D [9] or NLM [5]. Unsurprisingly, this inspired several researchers to learn the proximal
operator using CNNs [6, 38, 33, 22]. Meinhardt et al. [22] consider various proximal algorithms
including the proximal gradient method, ADMM, and the primal-dual hybrid gradient method, where
in each case the proximal operator for the regularizer can be replaced by a neural network. They
show that no single method will produce systematically better results than the others.
In the proximal techniques the relation between the proximal operator of the regularizer and the
natural image probability distribution remains unclear. In contrast, we explicitly use the Gaussiansmoothed natural image distribution as a prior, and we show that we can learn the gradient of its
logarithm using a denoising autoencoder.
Romano et al. [25] designed a prior model that is also implemented by a denoiser, but that does not
build on a proximal formulation such as ADMM. Interestingly, the gradient of their regularization
term boils down to the residual of the denoiser, that is, the difference between its input and output,
which is the same as in our approach. However, their framework does not establish the connection
between the prior and the natural image probability distribution, as we do. Finally, Bigdeli and
Zwicker [4] formulate an energy function, where they use a Denoising Autoencoder (DAE) network
for the prior, as in our approach, but they do not address the case of noise-blind restoration.
Noise- and Kernel-Blind Deconvolution. Kernel-blind deconvolution has seen the most effort
recently, while we support the fully (noise and kernel) blind setting. Noise-blind deblurring is usually
1
The source code of the proposed method is available at https://github.com/siavashbigdeli/DMSP.
2
performed by first estimating the noise level and then restoration with the estimated noise. Jin et
al. [14] proposed a Bayes risk formulation that can perform deblurring by adaptively changing the
regularization without the need of the noise variance estimate. Zhang et al. [35, 36] explored a
spatially-adaptive sparse prior and scale-space formulation to handle noise- or kernel-blind deconvolution. These methods, however, are tailored specifically to image deconvolution. Also, they only
handle the noise- or kernel-blind case, but not fully blind.
3
Bayesian Formulation
We assume a standard model for image degradation,
n ? N (0, ?n2 ),
y = k ? ? + n,
(1)
where ? is the unknown image, k is the blur kernel, n is zero-mean Gaussian noise with variance ?n2 ,
and y is the observed degraded image. We restore an estimate x of the unknown image by defining
and maximizing an objective consisting of a data term and an image likelihood,
argmax ?(x) = data(x) + prior(x).
(2)
x
Our core contribution is to construct a prior that corresponds to the logarithm of the Gaussiansmoothed probability distribution of natural images. We will optimize the objective using gradient
descent, and leverage the fact that we can learn the gradient of the prior using a denoising autoencoder
(DAE). We next describe how we define our objective by formulating a Bayes estimator in Section 3.1,
then explain how we leverage DAEs to obtain the gradient of our prior in Section 3.2, describe our
gradient descent approach in Section 3.3, and finally our image restoration applications in Section 4.
3.1
Defining the Objective via a Bayes Estimator
A typical approach to solve the restoration problem is via a maximum a posteriori (MAP) estimate,
where one considers the posterior distribution of the restored image p(x|y) ? p(y|x)p(x), derives an
objective consisting of a sum of data and prior terms by taking the logarithm of the posterior, and
maximizes it (minimizes the negative log-posterior, respectively). Instead, we will compute a Bayes
estimator x for the restoration problem by maximizing the posterior expectation of a utility function,
Z
Ex? [G(?
x, x)] = G(?
x, x)p(y|?
x)p(?
x)d?
x
(3)
where G denotes the utility function (e.g., a Gaussian), which encourages its two arguments to be
similar. This is a generalization of MAP, where the utility is a Dirac impulse.
Ideally, we would like to use the true data distribution as the prior p(?
x). But we only have data
samples, hence we cannot learn this exactly. Therefore, we introduce a smoothed data distribution
Z
0
p (x) = E? [p(x + ?)] = g? (?)p(x + ?)d?,
(4)
where ? has a Gaussian distribution with zero-mean and variance ? 2 , which is represented by the
smoothing kernel g? . The key idea here is that it is possible to estimate the smoothed distribution
p0 (x) or its gradient from sample data. In particular, we will need the gradient of its logarithm, which
we will learn using denoising autoencoders (DAEs). We now define our utility function as
G(?
x, x) = g? (?
x ? x)
p0 (x)
.
p(?
x)
(5)
where we use the same Gaussian function g? with standard deviation ? as introduced for the smoothed
distribution p0 . This penalizes the estimate x if the latent parameter x
? is far from it. In addition, the
term p0 (x)/p(?
x) penalizes the estimate if its smoothed density is lower than the true density of the
latent parameter. Unlike the utility in Jin et al. [14], this approach will allow us to express the prior
directly using the smoothed distribution p0 .
By inserting our utility function into the posterior expected utility in Equation (3) we obtain
Z
Z
Ex? [G(?
x, x)] = g? ()p(y|x + ) g? (?)p(x + ?)d?d,
3
(6)
where the true density p(?
x) canceled out, as desired, and we introduced the variable substitution
=x
? ? x.
We finally formulate our objective by taking the logarithm of the expected utility in Equation (6),
and introducing a lower bound that will allow us to split Equation (6) into a data term and an image
likelihood. By exploiting the concavity of the log function, we apply Jensen?s inequality and get our
objective ?(x) as
Z
Z
log Ex? [G(?
x, x)] = log g? ()p(y|x + ) g? (?)p(x + ?)d?d
"
#
Z
Z
?
g? () log p(y|x + ) g? (?)p(x + ?)d? d
Z
=
Z
g? () log p(y|x + )d + log
|
{z
} |
Data term data(x)
g? (?)p(x + ?)d? = ?(x).
{z
}
(7)
Image likelihood prior(x)
Image Likelihood. We denote the image likelihood as
Z
prior(x) = log g? (?)p(x + ?)d?.
(8)
The key observation here is that our prior expresses the image likelihood as the logarithm of the
Gaussian-smoothed true natural image distribution p(x), which is similar to a kernel density estimate.
Data Term. Given that the degradation noise is Gaussian, we see that [14]
Z
|y ? k ? x|2
?2
data(x) = g? () log p(y|x + )d = ?
? M 2 |k|2 ? N log ?n + const, (9)
2
2?n
2?n
where M and N denote the number of pixels in x and y respectively. This will allow us to address
noise-blind problems as we will describe in detail in Section 4.
3.2
Gradient of the Prior via Denoising Autoencoders (DAE)
A key insight of our approach is that we can effectively learn the gradients of our prior in Equation (8)
using denoising autoencoders (DAEs). A DAE r? is trained to minimize [32]
LDAE = E?,x |x ? r? (x + ?)|2 ,
(10)
where the expectation is over all images x and Gaussian noise ? with variance ? 2 , and r? indicates
that the DAE was trained with noise variance ? 2 . Alain et al. [1] show that the output r? (x) of the
optimal DAE (by assuming unlimited capacity) is related to the true data distribution p(x) as
R
g? (?)p(x ? ?)?d?
E? [p(x ? ?)?]
r? (x) = x ?
=x? R
(11)
E? [p(x ? ?)]
g? (?)p(x ? ?)d?
where the noise has a Gaussian distribution g? with standard deviation ?. This is simply a continuous
formulation of mean-shift, and g? corresponds to the smoothing kernel in our prior, Equation (8).
To obtain the relation between the DAE and the desired gradient of our prior, we first rewrite the
numerator in Equation (11) using the Gaussian derivative definition to remove ?, that is
Z
Z
Z
g? (?)p(x ? ?)?d? = ?? 2 ?g? (?)p(x ? ?)d? = ?? 2 ? g? (?)p(x ? ?)d?,
(12)
where we used the Leibniz rule to interchange the ? operator with the integral. Plugging this back
into Equation (11), we have
R
Z
? 2 ? g? (?)p(x ? ?)d?
r? (x) = x + R
= x + ? 2 ? log g? (?)p(x ? ?)d?.
(13)
g? (?)p(x ? ?)d?
One can now see that the DAE error, that is, the difference r? (x) ? x between the output of the DAE
and its input, is the gradient of the image likelihood in Equation (8). Hence, a main result of our
approach is that we can write the gradient of our prior using the DAE error,
Z
1
? prior(x) = ? log g? (?)p(x + ?)d? = 2 r? (x) ? x .
(14)
?
4
1
T
t?1
2 K (Kx
?n
? y) ? ?priorsL (xt?1 ) 2. u
? = ??
u ? ?ut
NB:
1. ut =
NA:
1. ut = ?t K T (Kxt?1 ? y) ? ?priorsL (xt?1 )
4. v t = ?t xT (K t?1 xt?1 ? y) + M ? 2 k t?1
KE:
3. xt = xt?1 + u
?
2. u
? = ??
u ? ?ut
3. xt = xt?1 + u
?
5. v? = ?k v? ? ?k v t
6. k t = k t?1 + v?
Table 1: Gradient descent steps for non-blind (NB), noise-blind (NA), and kernel-blind (KE) image
deblurring. Kernel-blind deblurring involves the steps for (NA) and (KE) to update image and kernel.
3.3
Stochastic Gradient Descent
We consider the optimization as minimization of the negative of our objective ?(x) and refer to it as
gradient descent. Similar to Bigdeli and Zwicker [4], we observed that the trained DAE is overfitted
to noisy images. Because of the large gap in dimensionality between the embedding space and the
natural image manifold, the vast majority of training inputs (noisy images) for the DAE lie at a
distance very close to ? from the natural image manifold. Hence, the DAE cannot effectively learn
mean-shift vectors for locations that are closer than ? to the natural image manifold. In other words,
our DAE does not produce meaningful results for input images that do not exhibit noise close to the
DAE training ?.
To address this issue, we reformulate our prior to perform stochastic gradient descent steps that
include noise sampling. We rewrite our prior from Equation (8) as
Z
prior(x) = log g? (?)p(x + ?)d?
(15)
Z
Z
= log g?2 (?2 ) g?1 (?1 )p(x + ?1 + ?2 )d?1 d?2
(16)
"
#
Z
Z
?
g?2 (?2 ) log
g?1 (?1 )p(x + ?1 + ?2 )d?1 d?2 = priorL (x),
(17)
where ?12 + ?22 = ? 2 , we used the fact that two Gaussian convolutions are equivalent to a single
convolution with a Gaussian whose variance is the sum of the two, and we applied Jensen?s inequality
again. This leads to a new lower bound for the prior, which we call priorL (x). Note that the bound
proposed by Jin et al. [14] corresponds to the special case where ?1 = 0 and ?2 = ?.
We address our DAE overfitting issue by using the new lower bound priorL (x) with ?1 = ?2 = ??2 .
Its gradient is
Z
2
g ??2 (?2 ) r ??2 (x + ?2 ) ? (x + ?2 ) d?2 .
?priorL (x) = 2
(18)
?
In practice, computing the integral over ?2 is not possible at runtime. Instead, we approximate the
integral with a single noise sample, which leads to the stochastic evaluation of the gradient as
2
(19)
?priorsL (x) = 2 r ??2 (x + ?2 ) ? x ,
?
where ?2 ? N (0, ?2 ). This addresses the overfitting issue, since it means we add noise each time
before we evaluate the DAE. Given the stochastically sampled gradient of the prior, we apply a
gradient descent approach with momentum that consists of the following steps:
1. ut = ?? data(xt?1 ) ? ? priorsL (xt?1 )
2. u
? = ??
u ? ?ut
3. xt = xt?1 + u
?
(20)
where ut is the update step for x at iteration t, u
? is the running step, and ? and ? are the momentum
and step-size.
4
Image Restoration using the Deep Mean-Shift Prior
We next describe the detailed gradient descent steps, including the derivatives of the data term, for
different image restoration tasks. We provide a summary in Table 1. For brevity, we omit the role of
downsampling (required for super-resolution) and masking.
5
Method
FD [18]
EPLL [39]
RTF-6 [28]*
CSF [29]
DAEP [4]
IRCNN [38]
EPLL [39] + NE
EPLL [39] + NA
TV-L2 + NA
GradNet 7S [14]
Ours
Ours + NA
?n :
2.55
30.03
32.03
32.36
29.85
32.64
30.86
31.86
32.16
31.05
31.43
29.68
32.57
Levin [19]
5.10
7.65
28.40 27.32
29.79 28.31
26.34 21.43
28.13 27.28
30.07 28.30
29.85 28.83
29.77 28.28
30.25 28.96
29.14 28.03
28.88 27.55
29.45 28.95
30.21 29.00
10.2
26.52
27.20
17.33
26.70
27.15
28.05
27.16
27.85
27.16
26.96
28.29
28.23
2.55
24.44
25.38
25.70
24.73
25.42
25.60
25.36
25.57
24.61
25.57
25.69
26.00
Berkeley [2]
5.10
7.65
23.24 22.64
23.53 22.54
23.45 19.83
23.61 22.88
23.67 22.78
24.24 23.42
23.53 22.55
23.90 22.91
23.65 22.90
24.23 23.46
24.45 23.60
24.47 23.61
10.2
22.07
21.91
16.94
22.44
22.21
22.91
21.90
22.27
22.34
22.94
22.99
22.97
Table 2: Average PSNR (dB) for non-blind deconvolution on two datasets (*trained for ?n = 2.55).
Non-Blind Deblurring (NB). The gradient descent steps for non-blind deblurring with a known
kernel and degradation noise variance are given in Table 1, top row (NB). Here K denotes the Toeplitz
matrix of the blur kernel k.
Noise-Adaptive Deblurring (NA). When the degradation noise variance ?n2 is unknown, we can
solve Equation (9) for the optimal ?n2 (since it is independent of the prior), which gives
?n2 =
1
|y ? k ? x|2 + M ? 2 |k|2 .
N
(21)
By plugging this back into the equation, we get the following data term
data(x) = ?
N
log |y ? k ? x|2 + M ? 2 |k|2 ,
2
(22)
which is independent of the degradation noise variance ?n2 . We show the gradient descent steps in
?1
Table 1, second row (NA), where ?t = N |y ? Kxt?1 |2 + M ? 2 |k|2
adaptively scales the data
term with respect to the prior.
Noise- and Kernel-Blind Deblurring (NA+KE). Gradient descent in noise-blind optimization
includes an intuitive regularization for the kernel. We can use the objective in Equation (22) to
jointly optimize for the unknown image and the unknown kernel. The gradient descent steps to
update the image remain as in Table 1, second row (NA), and we take additional steps to update
the kernel estimate, as in Table 1, third row (KE). Additionally, we project the kernel by applying
t
k t = max(k t , 0) and k t = |kkt |1 after each step.
5
Experiments and Results
Our DAE uses the neural network architecture by Zhang et al. [37]. We generated training samples
by adding Gaussian noise to images from ImageNet [10]. We experimented with different noise
levels and found ?1 = 11 to perform well for all our deblurring and super-resolution experiments.
Unless mentioned, for image restoration we always take 300 iterations with step length ? = 0.1
and momentum ? = 0.9. The runtime of our method is linear in the number of pixels, and our
implementation takes about 0.2 seconds per iteration for one megapixel on an Nvidia Titan X (Pascal).
5.1
Image Deblurring: Non-Blind and Noise-Blind
In this section we evaluate our method for image deblurring using two datasets. Table 2 reports
the average PSNR for 32 images from the Levin et al. [19] and 50 images from the Berkeley [2]
segmentation dataset, where 10 images are randomly selected and blurred with 5 kernels as in Jin et
al. [14]. We highlight the best performing PSNR in bold and underline the second best value. The
6
Ground Truth
EPLL [39]
DAEP [4]
GradNet 7S [14]
Ours
Ours + NA
Figure 2: Visual comparison of our deconvolution results.
Ground Truth
Blurred with 1% noise
Ours (blind)
SSD Error Ratio
100
% Below Error Ratio
90
80
70
60
Sun et al.
Wipf and Zhang
Levin et al.
Babacan et al.
Log?TV PD
Log?TV MM
Ours
50
40
30
1
2
3
4
Figure 3: Performance of our method for fully (noise- and kernel-) blind deblurring on Levin?s set.
upper half of the table includes non-blind methods for deblurring. EPLL [39] + NE uses a noise
estimation step followed by non-blind deblurring. Noise-blind experiments are denoted by NA for
noise adaptivity. We include our results for non-blind (Ours) and noise-blind (Ours + NA). Our noise
adaptive approach consistently performs well in all experiments and on average we achieve better
results than the state of the art. Figure 2 provides a visual comparison of our results. Our prior is able
to produce sharp textures while also preserving the natural image structure.
5.2
Image Deblurring: Noise- and Kernel-Blind
We performed fully blind deconvolution with our method using Levin et al.?s [19] dataset. In this test,
we performed 1000 gradient descent iterations. We used momentum ? = 0.7 and step size ? = 0.3
for the unknown image and momentum ?k = 0.995 and step size ?k = 0.005 for the unknown
kernel. Figure 3 shows visual results of fully blind deblurring and performance comparison to state
of the art (last column). We compare the SSD error ratio and the number of images in the dataset
that achieves error ratios less than a threshold. Results for other methods are as reported by Perrone
and Favaro [23]. Our method can reconstruct all the blurry images in the dataset with errors ratios
less than 3.5. Note that our optimization performs end-to-end estimation of the final results and we
do not use the common two stage blind deconvolution (kernel estimation, followed by non-blind
deconvolution). Additionally our method uses a noise adaptive scheme where we do not assume
knowledge of the input noise level.
5.3
Super-resolution
To demonstrate the generality of our prior, we perform an additional test with single image superresolution. We evaluate our method on the two common datasets Set5 [3] and Set14 [34] for different
upsampling scales. Since these tests do not include degradation noise (?n = 0), we perform our
optimization with a rough weight for the prior and decrease it gradually to zero. We compare our
method in Table 3. The upper half of the table represents methods that are specifically trained for
super-resolution. SRCNN [11] and TNRD [7] have separate models trained for ?2, 3, 4 scales, and
we used the model for ?4 to produce the ?5 results. VDSR [16] and DnCNN-3 [37] have a single
model trained for ?2, 3, 4 scales, which we also used to produce ?5 results. The lower half of the
table represents general priors that are not designed specifically for super-resolution. Our method
performs on par with state of the art methods over all the upsampling scales.
7
Method
Bicubic
SRCNN [11]
TNRD [7]
VDSR [16]
DnCNN-3 [37]
DAEP [4]
IRCNN [38]
Ours
scale:
?2
31.80
34.50
34.62
34.50
35.20
35.23
35.07
35.16
Set5 [3]
?3
?4
28.67 26.73
30.84 28.60
31.08 28.83
31.39 29.19
31.58 29.30
31.44 29.01
31.26 29.01
31.38 29.16
?5
25.32
26.12
26.88
25.91
26.30
27.19
27.13
27.38
?2
28.53
30.52
30.53
30.72
30.99
31.07
30.79
30.99
Set14 [34]
?3
?4
25.92 24.44
27.48 25.76
27.60 25.92
27.81 26.16
27.93 26.25
27.93 26.13
27.68 25.96
27.90 26.22
?5
23.46
24.05
24.61
24.01
24.26
24.88
24.73
25.01
Table 3: Average PSNR (dB) for super-resolution on two datasets.
Matlab [21]
33.9
RTF [15]
37.8
Gharbi et al. [13]
38.4
Gharbi et al. [13] f.t.
38.6
SEM [17]
38.8
Ours
38.7
Table 4: Average PSNR (dB) in linear RGB space for demosaicing on the Panasonic dataset [15].
5.4
Demosaicing
We finally performed a demosaicing experiment on the dataset introduced by Khashabi et al. [15].
This dataset is constructed by taking RAW images from a Panasonic camera, where the images are
downsampled to construct the ground truth data. Due to the down sampling effect, in this evaluation
we train a DAE with ?1 = 3 noise standard deviation. The test dataset consists of 100 noisy images
captured by a Panasonic camera using a Bayer color filter array (RGGB). We initialize our method
with Matlab?s demosaic function [21]. To get even better initialization, we perform our initial
optimization with a large degradation noise estimate (?n = 2.5) and then perform the optimization
with a lower estimate (?n = 1). We summarize the quantitative results in Table 4. Our method
is again on par with the state of the art. Additionally, our prior is not trained for a specific color
filter array and therefore is not limited to a specific sub-pixel order. Figure 4 shows a qualitative
comparison, where our method produces much smoother results compared to the previous state of the
art.
Ground Truth
RTF [15]
Gharbi et al. [13]
SEM [17]
Ours
Figure 4: Visual comparison for demosaicing noisy images from the Panasonic data set [15].
6
Conclusions
We proposed a Bayesian deep learning framework for image restoration with a generic image prior
that directly represents the Gaussian smoothed natural image probability distribution. We showed that
we can compute the gradient of our prior efficiently using a trained denoising autoencoder (DAE).
Our formulation allows us to learn a single prior and use it for many image restoration tasks, such as
noise-blind deblurring, super-resolution, and image demosaicing. Our results indicate that we achieve
performance that is competitive with the state of the art for these applications. In the future, we would
like to explore generalizing from Gaussian smoothing of the underlying distribution to other types of
kernels. We are also considering multi-scale optimization where one would reduce the Bayes utility
support gradually to get a tighter bound with respect to maximum a posteriori. Finally, our approach
is not limited to image restoration and could be exploited to address other inverse problems.
8
Acknowledgments. MJ and PF acknowledge support from the Swiss National Science Foundation
(SNSF) on project 200021-153324.
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6,277 | 6,679 | Greedy Algorithms for Cone Constrained
Optimization with Convergence Guarantees
Francesco Locatello
MPI for Intelligent Systems - ETH Zurich
Michael Tschannen
ETH Zurich
[email protected]
[email protected]
Gunnar R?tsch
ETH Zurich
Martin Jaggi
EPFL
[email protected]
[email protected]
Abstract
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe
(FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the
linear span and the convex hull of a set of atoms, respectively. In this paper, we
consider the intermediate case of optimization over the convex cone, parametrized
as the conic hull of a generic atom set, leading to the first principled definitions
of non-negative MP algorithms for which we give explicit convergence rates and
demonstrate excellent empirical performance. In particular, we derive sublinear
(O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e?t )) on strongly convex objectives, in both cases for general sets
of atoms. Furthermore, we establish a clear correspondence of our algorithms
to known algorithms from the MP and FW literature. Our novel algorithms and
analyses target general atom sets and general objective functions, and hence are
directly applicable to a large variety of learning settings.
1
Introduction
In recent years, greedy optimization algorithms have attracted significant interest in the domains
of signal processing and machine learning thanks to their ability to process very large data sets.
Arguably two of the most popular representatives are Frank-Wolfe (FW) [12, 21] and Matching
Pursuit (MP) algorithms [34], in particular Orthogonal MP (OMP) [9, 49]. While the former targets
minimization of a convex function over bounded convex sets, the latter apply to minimization over a
linear subspace. In both cases, the domain is commonly parametrized by a set of atoms or dictionary
elements, and in each iteration, both algorithms rely on querying a so-called linear minimization
oracle (LMO) to find the direction of steepest descent in the set of atoms. The iterate is then updated
as a linear or convex combination, respectively, of previous iterates and the newly obtained atom
from the LMO. The particular choice of the atom set allows to encode structure such as sparsity and
non-negativity (of the atoms) into the solution. This enables control of the trade-off between the
amount of structure in the solution and approximation quality via the number of iterations, which
was found useful in a large variety of use cases including structured matrix and tensor factorizations
[50, 53, 54, 18].
In this paper, we target an important ?intermediate case? between the two domain parameterizations
given by the linear span and the convex hull of an atom set, namely the parameterization of the
optimization domain as the conic hull of a possibly infinite atom set. In this case, the solution
can be represented as a non-negative linear combination of the atoms, which is desirable in many
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
applications, e.g., due to the physics underlying the problem at hand, or for the sake of interpretability.
Concrete examples include unmixing problems [11, 16, 3], model selection [33], and matrix and
tensor factorizations [4, 24]. However, existing convergence analyses do not apply to the currently
used greedy algorithms. In particular, all existing MP variants for the conic hull case [5, 38, 52] are
not guaranteed to converge and may get stuck far away from the optimum (this can be observed in
the experiments in Section 6). From a theoretical perspective, this intermediate case is of paramount
interest in the context of MP and FW algorithms. Indeed, the atom set is not guaranteed to contain
an atom aligned with a descent direction for all possible suboptimal iterates, as is the case when the
optimization domain is the linear span or the convex hull of the atom set [39, 32]. Hence, while conic
constraints have been widely studied in the context of a manifold of different applications, none of
the existing greedy algorithms enjoys explicit convergence rates.
We propose and analyze new MP algorithms tailored for the minimization of smooth convex functions
over the conic hull of an atom set. Specifically, our key contributions are:
? We propose the first (non-orthogonal) MP algorithm for optimization over conic hulls
guaranteed to converge, and prove a corresponding sublinear convergence rate with explicit constants. Surprisingly, convergence is achieved without increasing computational
complexity compared to ordinary MP.
? We propose new away-step, pairwise, and fully corrective MP variants, inspired by variants
of FW [28] and generalized MP [32], respectively, that allow for different degrees of weight
corrections for previously selected atoms. We derive corresponding sublinear and linear (for
strongly convex objectives) convergence rates that solely depend on the geometry of the
atom set.
? All our algorithms apply to general smooth convex functions. This is in contrast to all prior
work on non-negative MP, which targets quadratic objectives [5, 38, 52]. Furthermore, if
the conic hull of the atom set equals its linear span, we recover both algorithms and rates
derived in [32] for generalized MP variants.
? We make no assumptions on the atom set which is simply a subset of a Hilbert space, in
particular we do not assume the atom set to be finite.
Before presenting our algorithms (Section 3) along with the corresponding convergence guarantees
(Section 4), we briefly review generalized MP variants. A detailed discussion of related work can
be found in Section 5 followed by illustrative experiments on a least squares problem on synthetic
data, and non-negative matrix factorization as well as non-negative garrote logistic regression as
applications examples on real data (numerical evaluations of more applications and the dependency
between constants in the rate and empirical convergence can be found in the supplementary material).
Notation. Given a non-empty subset A of some Hilbert space, let conv(A) be the convex hull
of A, and let lin(A) denote its linear span. Given a closed set A, we call its diameter diam(A) =
maxz1 ,z2 ?A kz1 ? z2 k and its radius radius(A) = maxz?A kzk. kxkA := inf{c > 0 : x ?
c ? conv(A)} is the atomic norm of x over a set A (also known as the gauge function of conv(A)).
We call a subset A of a Hilbert space symmetric if it is closed under negation.
2
Review of Matching Pursuit Variants
Let H be a Hilbert space with associated inner product hx, yi, ? x, y ? H. The inner product induces
the norm kxk2 := hx, xi, ? x ? H. Let A ? H be a compact set (the ?set of atoms? or dictionary)
and let f : H ? R be convex and L-smooth (L-Lipschitz gradient in the finite dimensional case).
If H is an infinite-dimensional Hilbert space, then f is assumed to be Fr?chet differentiable. The
generalized MP algorithm studied in [32], presented in Algorithm 1, solves the following optimization
problem:
min f (x).
(1)
x?lin(A)
In each iteration, MP queries a linear minimization oracle (LMO) solving the following linear
problem:
LMOA (y) := arg min hy, zi
(2)
z?A
for a given query y ? H. The MP update step minimizes a quadratic upper bound gxt (x) =
f (xt ) + h?f (xt ), x ? xt i + L2 kx ? xt k2 of f at xt , where L is an upper bound on the smoothness
2
constant of f with respect to a chosen norm k ? k. Optimizing this norm problem instead of f
directly allows for substantial efficiency gains in the case of complicated f . For symmetric A and for
f (x) = 12 ky ? xk2 , y ? H, Algorithm 1 recovers MP (Variant 0) [34] and OMP (Variant 1) [9, 49],
see [32] for details.
Approximate linear oracles. Solving the
Algorithm 1 Norm-Corrective Generalized Match- LMO defined in (2) exactly is often hard in
practice, in particular when applied to matrix
ing Pursuit
(or tensor) factorization problems, while ap1: init x0 ? lin(A), and S := {x0 }
proximate versions can be much more efficient.
2: for t = 0 . . . T
Algorithm 1 allows for an approximate LMO.
3:
Find zt := (Approx-)LMOA (?f (xt ))
For given quality parameter ? ? (0, 1] and
4:
S := S ? {zt }
given direction d ? H, the approximate LMO
5:
Let b := xt ? L1 ?f (xt )
? ? A such
for Algorithm 1 returns a vector z
6:
Variant 0:
that
2
Update xt+1 := arg min kz ? bk
?i ? ?hd, zi,
hd, z
(3)
z:=xt +?zt
??R
relative to z = LMOA (d) being an exact solu7:
Variant 1:
tion.
2
Update xt+1 := arg min kz ? bk
z?lin(S)
8:
Optional: Correction of some/all atoms z0...t Discussion and limitations of MP. The analysis of the convergence of Algorithm 1 in [32]
9: end for
critically relies on the assumption that the origin is in the relative interior of conv(A) with
respect to its linear span. This assumption originates from the fact that the convergence of MP- and
FW-type algorithms fundamentally depends on an alignment assumption of the search direction
returned by the LMO (i.e., zt in Algorithm 1) and the gradient of the objective at the current iteration
(see third premise in [39]). Specifically, for Algorithm 1, the LMO is assumed to select a descent
direction, i.e., h?f (xt ), zt i < 0, so that the resulting weight (i.e., ? for Variant 0) is always positive.
In this spirit, Algorithm 1 is a natural candidate to minimize f over the conic hull of A. However,
if the optimization domain is a cone, the alignment assumption does not hold as there may be
non-stationary points x in the conic hull of A for which minz?A h?f (x), zi = 0. Algorithm 1 is
therefore not guaranteed to converge when applied to conic problems. The same issue arises for
essentially all existing non-negative variants of MP, see, e.g., Alg. 2 in [38] and in Alg. 2 in [52]. We
now present modifications corroborating this issue along with the resulting MP-type algorithms for
conic problems and corresponding convergence guarantees.
3
Greedy Algorithms on Conic Hulls
The cone cone(A ? y) tangent to the convex set conv(A) at a point y is formed by the half-lines
emanating from y and intersecting conv(A) in at least one point distinct from y. Without loss of
generality we consider 0 ? A and assume the set cone(A) (i.e., y = 0) to be closed. If A is finite
P|A|
the cone constraint can be written as cone(A) := {x : x = i=1 ?i ai s.t. ai ? A, ?i ? 0 ?i}. We
consider conic optimization problems of the form:
min
f (x).
x?cone(A)
(4)
Note that if the set A is symmetric or if the origin is in the relative interior of conv(A) w.r.t. its linear
span then cone(A) = lin(A). We will show later how our results recover known MP rates when the
origin is in the relative interior of conv(A).
As a first algorithm to solve problems of the form (4), we present the Non-Negative Generalized
Matching Pursuit (NNMP) in Algorithm 2 which is an extension of MP to general f and non-negative
weights.
Discussion: Algorithm 2 differs from Algorithm 1 (Variant 0) in line 4, adding the iterationdependent atom ? kxxttkA to the set of possible search directions1 . We use the atomic norm for the
This additional direction makes sense only if xt 6= 0. Therefore, we set ? kxxttkA = 0 if xt = 0, i.e., no
direction is added.
1
3
Algorithm 2 Non-Negative Matching Pursuit
1: init x0 = 0 ? A
2: for t = 0 . . . T
?t := (Approx-)LMOA (?f (xt ))
3:
Find z
4:
zt = arg min n ?xt o h?f (xt ), zi
?t , kx
z? z
t kA
h??f (xt ),zt i
Lkzt k2
5:
? :=
6:
Update xt+1 := xt + ?zt
7: end for
Figure 1: Two dimensional example for TA (xt ) where
A = {a1 , a2 }, for three different iterates x0 , x1 and
x2 . The shaded area corresponds to TA (xt ) and the
white area to lin(A) \ TA (xt ).
normalization because it yields the best constant in the convergence rate. In practice, one can replace
it with the Euclidean norm, which is often much less expensive to compute. This iteration-dependent
additional search direction allows to reduce the weights of the atoms that were previously selected,
thus admitting the algorithm to ?move back? towards the origin while maintaining the cone constraint.
This idea is informally explained here and formally studied in Section 4.1.
Recall the alignment assumption of the search direction and the gradient of the objective at the current
iterate discussed in Section 2 (see also [39]). Algorithm 2 obeys this assumption. The intuition
behind this is the following. Whenever xt is not a minimizer of (4) and minz?A h?f (xt ), zi = 0,
the vector ? kxxttkA is aligned with ?f (xt ) (i.e., h?f (xt ), ? kxxttkA i < 0), preventing the algorithm
from stopping at a suboptimal iterate. To make this intuition more formal, let us define the set of
feasible descent directions of Algorithm 2 at a point x ? cone(A) as:
n
x o
TA (x) := d ? H : ?z ? A ? ?
s.t. hd, zi < 0 .
(5)
kxkA
If at some iteration t = 0, 1, . . . the gradient ?f (xt ) is not in TA (xt ) Algorithm 2 terminates as
minz?A hd, zi = 0 and hd, ?xt i ? 0 (which yields zt = 0). Even though, in general, not every
direction in H is a feasible descent direction, ?f (xt ) ?
/ TA only occurs if xt is a constrained
minimum of Equation 4:
? ? cone(A) and ?f (?
? is a solution to minx?cone(A) f (x).
Lemma 1. If x
x) 6? TA then x
Initializing Algorithm 2 with x0 = 0 guarantees that the iterates xt always remain inside cone(A)
even though this is not enforced explicitly (by convexity of f , see proof of Theorem 2 in Appendix D
for details).
Limitations of Algorithm 2: Let us call active the atoms which have nonzero weights in the
Pt?1
representation of xt = i=0 ?i zi computed by Algorithm 2. Formally, the set of active atoms is
defined as S := {zi : ?i > 0, i = 0, 1, . . . , t ? 1}. The main drawback of Algorithm 2 is that when
the direction ? kxxttkA is selected, the weight of all active atoms is reduced. This can lead to the
algorithm alternately selecting ? kxxttkA and an atom from A, thereby slowing down convergence in a
similar manner as the zig-zagging phenomenon well-known in the Frank-Wolfe framework [28]. In
order to achieve faster convergence we introduce the corrective variants of Algorithm 2.
3.1
Corrective Variants
To achieve faster (linear) convergence (see Section 4.2) we introduce variants of Algorithm 2, termed
Away-steps MP (AMP) and Pairwise MP (PWMP), presented in Algorithm 3. Here, inspired by the
away-steps and pairwise variants of FW [12, 28], instead of reducing the weights of the active atoms
uniformly as in Algorithm 2, the LMO is queried a second time on the active set S to identify the
direction of steepest ascent in S. This allows, at each iteration, to reduce the weight of a previously
selected atom (AMP) or swap weight between atoms (PWMP). This selective ?reduction? or ?swap
of weight? helps to avoid the zig-zagging phenomenon which prevent Algorithm 2 from converging
linearly.
At each iteration, Algorithm 3 updates the weights of zt and vt as ?zt = ?zt + ? and ?vt = ?vt ? ?,
respectively. To ensure that xt+1 ? cone(A), ? has to be clipped according to the weight which is
currently on vt , i.e., ?max = ?vt . If ? = ?max , we set ?vt = 0 and remove vt from S as the atom vt
is no longer active. If dt ? A (i.e., we take a regular MP step and not an away step), the line search
is unconstrained (i.e., ?max = ?).
4
For both algorithm variants, the second LMO query increases the computational complexity. Note
that an exact search on S is feasible in practice as |S| has at most t elements at iteration t.
Taking an additional computational burden allows to update the weights of all active atoms in the
spirit of OMP. This approach is implemented in the Fully Corrective MP (FCMP), Algorithm 4.
Algorithm 3 Away-steps (AMP) and Pairwise Algorithm 4 Fully Corrective Non-Negative
(PWMP) Non-Negative Matching Pursuit
Matching Pursuit (FCMP)
1: init x0 = 0 ? A, and S := {x0 }
2: for t = 0 . . . T
3:
Find zt := (Approx-)LMOA (?f (xt ))
4:
Find vt := (Approx-)LMOS (??f (xt ))
5:
S = S ? zt
6:
AMP: dt = arg mind?{zt ,?vt }h?f (xt ), di
7:
PWMP: dnt = zt ? vt
o
h??f (xt ),dt i
, ?max
Lkdt k2
? := min
(?max see text)
9:
Update ?zt , ?vt and S according to ?
(? see text)
10:
Update xt+1 := xt + ?dt
11: end for
8:
1: init x0 = 0 ? A, S = {x0 }
2: for t = 0 . . . T
3:
Find zt := (Approx-)LMOA (?f (xt ))
4:
S := S ? {zt }
5:
Variant 0:
xt+1 = arg min kx?(xt ? L1 ?f (xt ))k2
x?cone(S)
Variant 1:
xt+1 = arg minx?cone(S) f (x)
7:
Remove atoms with zero weights from S
8: end for
6:
At each iteration, Algorithm 4 maintains the set of active atoms S by adding zt and removing atoms
with zero weights after the update. In Variant 0, the algorithm minimizes the quadratic upper bound
gxt (x) on f at xt (see Section 2) imitating a gradient descent step with projection onto a ?varying?
target, i.e., cone(S). In Variant 1, the original objective f is minimized over cone(S) at each iteration,
which is in general more efficient than minimizing f over cone(A) using a generic solver for cone
constrained problems. For f (x) = 12 ky ? xk2 , y ? H, Variant 1 recovers Algorithm 1 in [52] and
the OMP variant in [5] which both only apply to this specific objective f .
3.2
Computational Complexity
cost per iteration convergence
k(t)
algorithm
We briefly discuss the computaNNMP
C
+
O(d)
O(1/t)
tional complexity of the algorithms
t
PWMP
C + O(d + td)
O e??k(t)
we introduced. For H = Rd , sums
?
3|A|!+1
and inner products have cost O(d).
AMP
C + O(d + td)
O e? 2 k(t)
t/2
Let us assume that each call of the
t
??k(t)
FCMP v. 0 C + O(d) + h0
O e
3|A|!+1
LMO has cost C on the set A and
??k(t)
FCMP v. 1 C + O(d) + h1
O e
t
O(td) on S. The variants 0 and 1
of FCMP solve a cone problem at
each iteration with cost h0 and h1 , Table 1: Computational complexity versus convergence rate (see Section 4) for strongly convex objectives
respectively. In general, h0 can be
much smaller than h1 . In Table 1
we report the cost per iteration for every algorithm along with the asymptotic convergence rates
derived in Section 4.
4
Convergence Rates
In this section, we present convergence guarantees for Algorithms 2, 3, and 4. All proofs are deferred
to the Appendix in the supplementary material. We write x? ? arg minx?cone(A) f (x) for an optimal
solution. Our rates will depend on the atomic norm of the solution and the iterates of the respective
algorithm variant:
? = max {kx? kA , kx0 kA . . . , kxT kA } .
(6)
If the optimum is not unique, we consider x? to be one of largest atomic norm. A more intuitive
and looser notion is to simply upper-bound ? by the diameter of the level set of the initial iterate
x0 measured by the atomic norm. Then, boundedness follows since the presented method is a
descent method (due to Lemma 1 and line search on the quadratic upper bound, each iteration strictly
5
decreases the objective and our method stops only at the optimum). This justifies the statement
f (xt ) ? f (x0 ). Hence, ? must be bounded for any sequence of iterates produced by the algorithm,
and the convergence rates presented in this section are valid as T goes to infinity. A similar notion to
measure the convergence of MP was established in [32]. All of our algorithms and rates can be made
affine invariant. We defer this discussion to Appendix B.
4.1
Sublinear Convergence
We now present the convergence results for the non-negative and Fully-Corrective Matching Pursuit
algorithms. Sublinear convergence of Algorithm 3 is addressed in Theorem 3.
Theorem 2. Let A ? H be a bounded set with 0 ? A, ? := max {kx? kA , kx0 kA , . . . , kxT kA , }
and f be L-smooth over ? conv(A ? ?A). Then, Algorithms 2 and 4 converge for t ? 0 as
4 2? L?2 radius(A)2 + ?0
?
f (xt ) ? f (x ) ?
,
?t + 4
where ? ? (0, 1] is the relative accuracy parameter of the employed approximate LMO (see Equation (3)).
Relation to FW rates. By rescaling A by a large enough factor ? > 0, FW with ? A as atom
set could in principle be used to solve (4). In fact, for large enough ? , only the constraints of (4)
become active when minimizing f over conv(? A). The sublinear convergence rate obtained with
this approach is up to constants identical to that in Theorem 2 for our MP variants, see [21]. However,
as the correct scaling is unknown, one has to either take the risk of choosing ? too small and hence
failing to recover an optimal solution of (4), or to rely on too large ? which can result in slow
convergence. In contrast, knowledge of ? is not required to run our MP variants.
Relation to MP rates. If A is symmetric, we have that lin(A) = cone(A) and it is easy to show
that the additional direction ? kxxtt k in Algorithm 2 is never selected. Therefore, Algorithm 2 becomes
equivalent to Variant 0 of Algorithm 1, while Variant 1 of Algorithm 1 is equivalent to Variant 0 of
Algorithm 4. The rate specified in Theorem 2 hence generalizes the sublinear rate in [32, Theorem 2]
for symmetric A.
4.2
Linear Convergence
We start by recalling some of the geometric complexity quantities that were introduced in the context
of FW and are adapted here to the optimization problem we aim to solve (minimization over cone(A)
instead of conv(A)).
Directional Width. The directional width of a set A w.r.t. a direction r ? H is defined as:
r
dirW (A, r) := max krk
,s ? v
s,v?A
(7)
Pyramidal Directional Width [28]. The Pyramidal Directional Width of a set A with respect to a
direction r and a reference point x ? conv(A) is defined as:
P dirW (A, r, x) := min dirW (S ? {s(A, r)}, r),
S?Sx
(8)
where Sx := {S | S ? A and x is a proper convex combination of all the elements in S} and
r
s(A, r) := maxs?A h krk
, si.
Inspired by the notion of pyramidal width in [28], which is the minimal pyramidal directional width
computed over the set of feasible directions, we now define the cone width of a set A where only
the generating faces (g-faces) of cone(A) (instead of the faces of conv(A)) are considered. Before
doing so we introduce the notions of face, generating face, and feasible direction.
Face of a convex set. Let us consider a set K with a k?dimensional affine hull along with a
point x ? K. Then, K is a k?dimensional face of conv(A) if K = conv(A) ? {y : hr, y ? xi =
0} for some normal vector r and conv(A) is contained in the half-space determined by r, i.e.,
hr, y ? xi ? 0, ? y ? conv(A). Intuitively, given a set conv(A) one can think of conv(A) being a
dim(conv(A))?dimensional face of itself, an edge on the border of the set a 1-dimensional face and
a vertex a 0-dimensional face.
6
Face of a cone and g-faces. Similarly, a k?dimensional face of a cone is an open and unbounded
set cone(A) ? {y : hr, y ? xi = 0} for some normal vector r and cone(A) is contained in the half
space determined by r. We can define the generating faces of a cone as:
g-faces(cone(A)) := {B ? conv(A) : B ? faces(cone(A))} .
Note that g-faces(cone(A)) ? faces(conv(A)) and conv(A) ? g-faces(cone(A)). Furthermore,
for each K ? g-faces(cone(A)), cone(K) is a k?dimensional face of cone(A).
We now introduce the notion of feasible directions. A direction d is feasible from x ? cone(A) if it
points inwards cone(A), i.e., if ?? > 0 s.t. x + ?d ? cone(A). Since a face of the cone is itself a
cone, if a direction is feasible from x ? cone(K) \ 0, it is feasible from every positive rescaling of x.
We therefore can consider only the feasible directions on the generating faces (which are closed and
bounded sets). Finally, we define the cone width of A.
Cone Width.
CWidth(A) :=
min
K?g-faces(cone(A))
x?K
r?cone(K?x)\{0}
P dirW (K ? A, r, x)
(9)
We are now ready to show the linear convergence of Algorithms 3 and 4.
Theorem 3. Let A ? H be a bounded set with 0 ? A and let the objective function f : H ? R be
both L-smooth and ?-strongly convex over ? conv(A ? ?A). Then, the suboptimality of the iterates
of Algorithms 3 and 4 decreases geometrically at each step in which ? < ?vt (henceforth referred to
as ?good steps?) as:
?t+1 ? (1 ? ?) ?t ,
(10)
2
?
where ? := ? 2 ?LCWidth(A)
diam(A)2 ? (0, 1], ?t := f (xt )?f (x ) is the suboptimality at step t and ? ? (0, 1]
is the relative accuracy parameter of the employed approximate LMO (3). For AMP (Algorithm 3),
? AMP = ?/2. If ? = 0 Algorithm 3 converges with rate O(1/k(t)) where k(t) is the number of
?good steps? up to iteration t.
Discussion. To obtain a linear convergence rate, one needs to upper-bound the number of ?bad
steps? t ? k(t) (i.e., steps with ? ? ?vt ). We have that k(t) = t for Variant 1 of FCMP (Algorithm 4),
k(t) ? t/2 for AMP (Algorithm 3) and k(t) ? t/(3|A|! + 1) for PWMP (Algorithm 3) and Variant 0
of FCMP (Algorithm 4). This yields a global linear convergence rate of ?t ? ?0 exp (??k(t)). The
bound for PWMP is very loose and only meaningful for finite sets A. However, it can be observed
in the experiments in the supplementary material (Appendix A) that only a very small fraction of
iterations result in bad PWMP steps in practice. Further note that Variant 1 of FCMP (Algorithm 4)
does not produce bad steps. Also note that the bounds on the number of good steps given above are
the same as for the corresponding FW variants and are obtained using the same (purely combinatorial)
arguments as in [28].
Relation to previous MP rates. The linear convergence of the generalized (not non-negative) MP
variants studied in [32] crucially depends on the geometry of the set which is characterized by the
Minimal Directional Width mDW(A):
mDW(A) :=
min maxh
d?lin(A) z?A
d6=0
d
, zi .
kdk
(11)
The following Lemma relates the Cone Width with the minimal directional width.
Lemma 4. If the origin is in the relative interior of conv(A) with respect to its linear span, then
cone(A) = lin(A) and CWidth(A) = mDW(A).
Now, if the set A is symmetric or, more generally, if cone(A) spans the linear space lin(A) (which
implies that the origin is in the relative interior of conv(A)), there are no bad steps. Hence, by
Lemma 4, the linear rate obtained in Theorem 3 for non-negative MP variants generalizes the one
presented in [32, Theorem 7] for generalized MP variants.
7
Relation to FW rates. Optimization over conic hulls with non-negative MP is more similar to FW
than to MP itself in the following sense. For MP, every direction in lin(A) allows for unconstrained
steps, from any iterate xt . In contrast, for our non-negative MPs, while some directions allow for
unconstrained steps from some iterate xt , others are constrained, thereby leading to the dependence
of the linear convergence rate on the cone width, a geometric constant which is very similar in spirit
to the Pyramidal Width appearing in the linear convergence bound in [28] for FW. Furthermore, as
for Algorithm 3, the linear rate of Away-steps and Pairwise FW holds only for good steps. We finally
relate the cone width with the Pyramidal Width [28]. The Pyramidal Width is defined as
PWidth(A) :=
min
K?faces(conv(A))
x?K
r?cone(K?x)\{0}
P dirW (K ? A, r, x).
We have CWidth(A) ? PWidth(A) as the minimization in the definition (9) of CWidth(A) is only
over the subset g-faces(cone(A)) of faces(conv(A)). As a consequence, the decrease per iteration
characterized in Theorem 3 is larger than what one could obtain with FW on the rescaled convex set
? A (see Section 4.1 for details about the rescaling). Furthermore, the decrease characterized in [28]
scales as 1/? 2 due to the dependence on 1/ diam(conv(A))2 .
5
Related Work
The line of recent works by [44, 46, 47, 48, 37, 32] targets the generalization of MP from the
least-squares objective to general smooth objectives and derives corresponding convergence rates
(see [32] for a more in-depth discussion). However, only little prior work targets MP variants with
non-negativity constraint [5, 38, 52]. In particular, the least-squares objective was addressed and
no rigorous convergence analysis was carried out. [5, 52] proposed an algorithm equivalent to our
Algorithm 4 for the least-squares case. More specifically, [52] then developed an acceleration heuristic,
whereas [5] derived a coherence-based recovery guarantee for sparse linear combinations of atoms.
Apart from MP-type algorithms, there is a large variety of non-negative least-squares algorithms,
e.g., [30], in particular also for matrix and tensor spaces. The gold standard in factorization problems
is projected gradient descent with alternating minimization, see [43, 4, 45, 23]. Other related works
are [40], which is concerned with the feasibility problem on symmetric cones, and [19], which
introduces a norm-regularized variant of problem (4) and solves it using FW on a rescaled convex
set. To the best of our knowledge, in the context of MP-type algorithms, we are the first to combine
general convex objectives with conic constraints and to derive corresponding convergence guarantees.
Boosting: In an earlier line of work, a flavor of the generalized MP became popular in the context
of boosting, see [35]. The literature on boosting is vast, we refer to [42, 35, 7] for a general overview.
Taking the optimization perspective given in [42], boosting is an iterative greedy algorithm minimizing
a (strongly) convex objective over the linear span of a possibly infinite set called hypothesis class.
The convergence analysis crucially relies on the assumption of the origin being in the relative interior
of the hypothesis class, see Theorem 1 in [17]. Indeed, Algorithm 5.2 of [35] might not converge
if the [39] alignment assumption is violated. Here, we managed to relax this assumption while
preserving essentially the same asymptotic rates in [35, 17]. Our work is therefore also relevant in
the context of (non-negative) boosting.
6
Illustrative Experiments
We illustrate the performance of the presented algorithms on three different exemplary tasks, showing
that our algorithms are competitive with established baselines across a wide range of objective functions, domains, and data sets while not being specifically tailored to any of these tasks (see Section 3.2
for a discussion of the computational complexity of the algorithms). Additional experiments targeting
KL divergence NMF, non-negative tensor factorization, and hyperspectral image unmixing can be
found in the appendix.
Synthetic data. We consider minimizing the least squares objective on the conic hull
of 100 unit-norm vectors sampled at random in the first orthant of R50 . We compare
the convergence of Algorithms 2, 3, and 4 with the Fast Non-Negative MP (FNNOMP)
of [52], and Variant 3 (line-search) of the FW algorithm in [32] on the atom set rescaled
by ? = 10kyk (see Section 4.1), observing linear convergence for our corrective variants.
8
Suboptimality
104
10
2
10
0
Synthetic data
PWMP (Alg. 3)
NNMP (Alg. 2)
FCMP (Alg. 4)
FW
AMP (Alg. 3)
FNNOMP
Figure 2 shows the suboptimality ?t , averaged over 20
realizations of A and y, as a function of the iteration t. As
expected, FCMP achieves fastest convergence followed
by PWMP, AMP and NNMP. The FNNOMP gets stuck
instead. Indeed, [52] only show that the algorithm terminates and not its convergence.
10-2
Non-negative matrix factorization. The second task
consists of decomposing a given matrix into the product
10
0
10
20
30
40
50
of two non-negative matrices as in Equation (1) of [20].
Iteration
We consider the intersection of the positive semidefinite
Figure 2: Synthetic data experiment.
cone and the positive orthant. We parametrize the set A as
the set of matrices obtained as an outer product of vectors
from A1 = {z ? Rk : zi ? 0 ? i} and A2 = {z ? Rd : zi ? 0 ? i}. The LMO is approximated
using a truncated power method [55], and we perform atom correction with greedy coordinate descent
see, e.g., [29, 18], to obtain a better objective value while maintaining the same (small) number of
atoms. We consider three different datasets: The Reuters Corpus2 , the CBCL face dataset3 and the
KNIX dataset4 . The subsample of the Reuters corpus we used is a term frequency matrix of 7,769
documents and 26,001 words. The CBCL face dataset is composed of 2,492 images of 361 pixels
each, arranged into a matrix. The KNIX dataset contains 24 MRI slices of a knee, arranged in a
matrix of size 262, 144 ? 24. Pixels are divided by their overall mean intensity. For interpretability
reasons, there is interest to decompose MRI data into non-negative factorizations [25]. We compare
PWMP and FCMP against the multiplicative (mult) and the alternating (als) algorithm of [4], and the
greedy coordinate descent (GCD) of [20]. Since the Reuters corpus is much larger than the CBCL
and the KNIX dataset we only used the GCD for which a fast implementation in C is available. We
report the objective value for fixed values of the rank in Table 2, showing that FCMP outperform all
the baselines across all the datasets. PWMP achieves smallest error on the Reuters corpus.
-4
Non-negative garrote. We consider the non-negative garrote which is a common approach to
model order selection [6]. We evaluate NNMP, PWMP, and FCMP in the experiment described
in [33], where the non-negative garrote is used to perform model order selection for logistic regression
(i.e., for a non-quadratic objective function). We evaluated training and test accuracy on 100 random
splits of the sonar dataset from the UCI machine learning repository. In Table 3 we compare the
median classification accuracy of our algorithms with that of the cyclic coordinate descent algorithm
(NNG) from [33].
Reuters
CBCL
CBCL
KNIX
K = 10 K = 10 K = 50 K = 10
mult
2.4241e3 1.1405e3 2.4471e03
als
2.73e3
3.84e3 2.7292e03
GCD
5.9799e5 2.2372e3
806
2.2372e03
PWMP 5.9591e5 2.2494e3 789.901 2.2494e03
FCMP
5.9762e5 2.2364e3 786.15 2.2364e03
Table 2: Objective value for least-squares non-negative
matrix factorization with rank K.
algorithm
7
training accuracy
test accuracy
NNMP 0.8345 ? 0.0242 0.7419 ? 0.0389
PWMP 0.8379 ? 0.0240 0.7419 ? 0.0392
FCMP 0.8345 ? 0.0238 0.7419 ? 0.0403
NNG
0.8069 ? 0.0518 0.7258 ? 0.0602
Table 3: Logistic Regression with non-negative
Garrote, median ? std. dev.
Conclusion
In this paper, we considered greedy algorithms for optimization over the convex cone, parametrized
as the conic hull of a generic atom set. We presented a novel formulation of NNMP along with a
comprehensive convergence analysis. Furthermore, we introduced corrective variants with linear
convergence guarantees, and verified this convergence rate in numerical applications. We believe that
the generality of our novel analysis will be useful to design new, fast algorithms with convergence
guarantees, and to study convergence of existing heuristics, in particular in the context of non-negative
matrix and tensor factorization.
2
http://www.nltk.org/book/ch02.html
http://cbcl.mit.edu/software-datasets/FaceData2.html
4
http://www.osirix-viewer.com/resources/dicom-image-library/
3
9
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12
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6,278 | 668 | Single-iteration Threshold Hamming
Networks
Eytan Ruppin
Isaac Meilijson
Moshe Sipper
School of Mathematical Sciences
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, 69978 Tel Aviv, Israel
Abstract
We analyze in detail the performance of a Hamming network classifying inputs that are distorted versions of one of its m stored
memory patterns. The activation function of the memory neurons
in the original Hamming network is replaced by a simple threshold
function. The resulting Threshold Hamming Network (THN) correctly classifies the input pattern, with probability approaching 1,
using only O(mln m) connections, in a single iteration . The THN
drastically reduces the time and space complexity of Hamming Network classifiers.
1
Introduction
Originally presented in (Steinbuch 1961, Taylor 1964) the Hamming network (HN)
has received renewed attention in recent years (Lippmann et. al. 1987, Baum et.
al. 1988). The HN calculates the Hamming distance between the input pattern
and each memory pattern, and selects the memory with the smallest distance. It
is composed of two subnets: The similarity subnet, consisting of an n-neuron input
layer connected with an m-neuron memory layer, calculates the number of equal bits
between the input and each memory pattern. The winner-take-all (WTA) subnet,
consisting of a fully connected m-neuron topology, selects the memory neuron that
best matches the input pattern.
564
Single-iteration Threshold Hamming Networks
The similarity subnet uses mn connections and performs a single iteration. The
WTA sub net has m 2 connections. With randomly generated input and memory
patterns, it converges in 8(m In(mn)) iterations (Floreen 1991). Since m is exponential in n, the space and time complexity of the network is primarily due to
the WTA subnet (Domany & Orland 1987). We analyze the performance of the
HN in the practical scenario where the input pattern is a distorted version of some
stored memory vector. We show that it is possible to replace the original activation function of thf' neurons in the memory layer by a simple threshold function,
and completely discard the WTA subnet. If the threshold is properly tuned, only
the neuron standing for the 'correct' memory is likely to be activated. The resulting Threshold Hamming Network (THN) will perform correctly (with probability
approaching 1) in a single iteration, using only O(m In m) connections instead of
the O( m 2 ) connections in the original HN. We identify the optimal threshold, and
measure its performance relative to the original HN.
2
The Threshold Hamming Network
e",
We examine a HN storing m + 1 memory patterns
1 ~ jJ ~ m + 1, each
being an n-dimensional vector of ?1. The input pattern x is generated by selecting
some memory pattern ~I-' (w.l.g., ~m+l), and letting each bit Xi be either ~f or
-~f with probabilities a and (I - a) respectively, where a > 0.5. To analyze this
HN, we use some tight. approximations to the binomial distribution. Due to space
considerations, their proofs are omitted.
Lemlna 1.
Let X,...., Bin(n,p) . If Xn are integers such that
P(X
> xn) ~
-
(1 - ~
limn-+oo~
= /3 E (p, 1), then
exp{ -n[/3ln /3 + (1 _ /3) In 1 - /3]}
)v1211'n/3(1
- /3)
1p
p
in the sense that the ratio between LHS and RHS converges to 1 as n
the special case p = ~, let G{/3) = In 2 + /31n/3 + (1- /3) In{1 - /3), then
~ 00.
P(X> x ),...., -:--_e7'xp,-,{=--;:-=n=G::::(/3:::::)==}=.
n ,...., (2 - ~ )V211'n/3(1 - 13)
Lenllna 2.
Let Xi ,...., Bin(n,~) be independent, 'Y E (0,1), and let
m
(1)
p
Xn
For
(2)
be as in Lemma 1. If
= (2 - ~)\h7rn/3(I- 13) (ln~) enG ({3),
(3)
then
(4)
Lenllna 3.
Let y,...., Bin(n,Q') with a >~, let (Xi) and 'Y be as in Lemma 2, and let
Let
Xn
be the integer closest to
T}
E (0,1).
nf3, where
/3=a_/a(l-a)z
n
1)
V
_~
2n
(5)
565
566
Meilijson, Ruppin, and Sipper
and zTj is the
T] -
quantile of the standard normal distribution, i.e.,
T]
= -1-
vf2;
jZ'I e-
x
2
/2dx
(6)
-00
Then, if Y and (Xd are independent
P (max(X 1 , X 2 , " ' , Xm) < Y) 2:: P(max(X1 , X 2 , " ' , Xm) <
as n
-+-
00,
~
Xn
Y) =>
,T]
(7)
for m as in (3).
Based on the above binomial probability approximations, we can now propose and
analyze a n-neuron Threshold Hamming Network (THN) that classifies the input
patterns with probability of error not exceeding f, when the input vector is generated
with an initial bit-similarity a: Let Xi be the similarity between the input vector
and the j'th memory pattern (1 ~ j < m), and let Y be the similarity with
the 'correct' memory pattern ~m+l. Choose, and T] so that ,T] 2:: 1 - f, e.g.,
T] = VI="f; determine f3 by (5) and m by (3). Discard the WTA subnet, and
simply replace the neurons of the memory layer by m neurons having a threshold
Xn , the integer closest to nf3. If any memory neuron with similarity at least Xn
is declared 'the winner', then, by Lemma 3, the probability of error is at most f,
where 'error' may be due to the existence of no winner, wrong winner, or multiple
wmners.
,=
3
The Hamming Network and an Optimal Threshold
Hamming Network
We now calculate the choice of the threshold Xn that maximizes the storage capacity m = men, f, a). Let </J (eI? denote the standard normal density (cumulative
distribution function), and let r = </J/(l- eI? denote the corresponding failure rate
function. Then,
Lenuna 4.
The opt.imal proportion between the two error probabilities is
1- ,
---1 - T]
r(zTj)
vna(1 - a) In
(8)
6 '
~ -r==~==~--~
which we will denote by 8.
Proof:
Let AI = max(X 1 ,X2 ,""Xm ), and let Y denote the similarity with the
'correct' memory pattern, as before.
We have seen that P(M < x) ~
exp{ -m J exp{ -nG({3)} 1 } . Since G'(f3) = In (1~ ~)' then by Taylor expansion
21!"n{3(1-{3)(2-~)
P(AI<x)=P(M<xo+x-xo)~exp{-m
I-'
exp{-n[G(f3 + X-XO)]}
n
1
V211' n f3(1 - (3)(2 - fj)
exp{ -nG(f3) - (x - xo) In (1~{3)}
exp{ -m
V211' n f3(1 _ (3)(2 _ ~)
}
=,
}~
(-..L )",0-"
l-fj
(9)
Single-iteration Threshold Hamming Networks
(in accordance with Gnedenko extreme-value distribution of type 1 (Leadbetter et.
al. 1983)). Similarly,
P(Y
< x) = exp{ln P(Y < Xo + x - xo)}
~(z)
x - Xo
P(Y < x o)ex P {?1>$(z) Jna(l- a)}
~
x- Xo
= (1-7])exp{r(z) Jna(1- a)}
(10)
where ~ is the standard normal density function, ?1> is the standard normal cumulative distribution function, ?1>$ = 1 - ?1> and r =
is the corresponding failure
rate function. The probability of correct recognition using a threshold x can now
be expressed as
-J.
P(M < x)P(Y ~ x)
= ,(6)"'0-"'(1- (1-7])exp{r(z)
x - Xo })
Jna(l- a)
(11)
We differentiate expression (11) with respect to Xo - x, and equate the derivative
at Xo = x to zero, to obtain the relation between , and 7] that yields the optimal
threshold, i.e., that which maximizes the probability of correct recognition. This
yields
1'(Z)
1 - 7]
exp{--}
(12)
Jna(l-a)ln~ 7]
,=
We now approximate
1- ,
~
- In ,
~
r(z)
Jna(l- a)ln
4
( 1 - 7] )
(13)
and thus the optimal proportion between the two error probabilities is
1 -;
-~
1 - 7]
r(z)
jna(1 - a) In ~
= {yo
(14)
o
Based on Lemma 4, if the desired probability of error is (, we choose
{Jf.
,=I-I+{Y'
_ 17] (1
t
+ {y)
(15)
We start with, = 7] = ..;r=f, obtain {3 from (5) and {y from (8), and recompute 7]
and, from (15). The limiting values of j3 and, in this iterative process give the
maximal capacity m and threshold x n .
We now compute the error probability t( m, n, a) of the original HN (with the WTA
subnet) for arbitrary tn, n and a, and compare it with (.
Lemma 5.
For arbitrary n, a and t, let m, {3", 7] and {y be as calculated above. Then, the
probability of error ((m, n, a) of the HN satisfies
((m,n,a)~r(I-{Y)
1- e- 61n 6
-.L
{yIn
1-{3
{y6
({y)lH(1+6
1+
(16)
567
568
Meiiijson, Ruppin, and Sipper
where
(17)
is the Gamma function.
Proof:
P(Y ~ M) = LP(Y ~ x)P(M = x) =
x
LP(Y ~ x)[P(M < x+ 1) - P(M < x)] ~
x
L P(Y ~ xo)e- (xo-x)ln 6
6
x
(18)
We now approximate this sum by the integral of the summand: let b
=~
and
c = 6ln ~. We have seen that the probability of incorrect performance of the
WTA subnet is equal to
L
P(Y :S M) ~
P(Y ~ xo)e-c(xo-x)[(P(M < xo))b(ro-r-l) - (P(M < xo))b(ro-r)] ~
x
Now we transform variables t
= bY In ~ to get the integral in the form
This is the convergent difference between two divergent Gamma function integrals.
~e perform inte~rat~on by parts to obtain a representation as an integr~l wi~h rK2
mstead of t-(1+ 2) m the mtegrand. For 0 ~ K2 < 1, the correspondmg mtegral
converges. The final result is then
(1 - 7])
1 - eC
C
c
In b
1
'Y
c
r(l - -)(1n -)1iib
(21 )
Hence, we have
1
1 - e -61n -1L
l-{J
P(Y ~ M) ~ (1-7])
-L r(l- 6)(ln _)6 ~
6ln l-f3
'Y
1 - e- 6ln 6
r(l - 6)
-L
6In 1 _/3
(1
(f6)6
6)1+6 f
+
(22)
Single-iteration Threshold Hamming Networks
% error -+
threshold , m
predicted
THN
predicted
HN
experimental
THN
experimental"
HN
2.46
0.144
2.552
0.103
!
133 , 145
= 1.0
= 1.552)
(1 - 'Y = 1.03
1 - T/
1.46)
(1 - 'Y
1 - T/
=
134 ? 346
3.4
=
0.272
135 , 825
4.714
0.494
(1 - 'Y = 1.776
1 - 11 = 2.991)
136 , 1970
6.346
=
4.152
0.485
(1 - 'Y = 1.606
1 - 11 = 2.576)
0.857
(1 - 'Y = 2.274
1 - T/
4.167)
0.253
3.468
(1 - 'Y = 1.373
1 - T/ = 2.168)
(1 - 'Y
1.37
1-T/=2.1l)
6.447
0.863
(1 - 'Y = 2.335
1 - T/
4.162)
=
Table 1: The performance of a HN and optimal THN: A comparison between calculated and experimental results (a = 0.7,n = 210).
as claimed. Expression (22) is presented as K(f, 8, (3)f, where K(f, 8, (3) is the factor
(:::; 1) by which the probability of error f of the THN should be multiplied in order
to get the probability of error of the original HN with the WTA subnet. For small
8, K is close to 1, however, as will be seen in the next section, K is typically larger.
4
Numerical results
The experimental results presented in table 1 testify to the accuracy of the HN and
THN calculations. Figure 1 presents the calculated error probabilities for various
values of input similarity a and memory capacity m, as a function of the input size
n. As is evident, the performance of the THN is worse than that of the HN, but due
to the exponential growth of m, it requires only a minor increment in n to obtain
a THN that performs as well as the original HN.
To examine the sensitivity of the THN network to threshold variation, we have fixed
a = 0.7, n = 210, m = 825, and let the threshold vary between 132 and 138. As we
can see in figure 2, the threshold 135 is indeed optimal, but the performance with
threshold values of 134 and 136 is practically identical. The magnitude of the two
error types varies considerably with the threshold value, but this variation has no
effect on the overall performance near the optimum. These two error probabilities
might as well be taken equal to each other.
Conclusion In this paper we analyzed in detail the performance of a Hamming
Network and a Threshold Hamming Network. Given a desired storage capacity and
performance, we described how to compute the corresponding minimal network size
required. The THN drastically reduces the time and connectivity requirements of
Hamming Network classifiers.
569
570
Meilijson, Ruppin, and Sipper
alpha=0.6,m=10 3
0.0001
0.0003
0.000
0.002
epsilon
(error
probability)
THN~
HN
0.007
-+-
0.14
0.37
800
1000
1200
1400
1600
n (network size)
1800
2000
2200
alpha=0 .7,m=10 6
0.0001
0.0003
0.000
epsilon
(error
probability)
THN~
HN
-+-
0.14
0.37
300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600
n (network size)
a1pha=0.8,m=10 9
0.0001....,.-----------r--------..,...---,
0.0003
0.000
THN~
HN
epsilon
(error
probability)
-+-
0.37
160
180
200
220
240
260
n (network size)
280
300
320
Figure 1: Probability of error as a function of network size: three networks are
depicted , displaying the performance at various values of (}' and m . For graphical
convenience, we have plotted log ~ versus n.
Single-iteration Threshold Hamming Networks
THN performance
10
epsilon ~
1 - gamma +1 - eta -e-
9
8
7
% error
6
5
4
3
2
1
0
132
133
134
135
threshold
Figure 2: Threshold sensitivity of the THN (a
136
137
138
= 0.7, n = 210, m = 825).
References
[1] K. Steinbuch. Dei lernmatrix. /(ybernetic, 1:36-45, 1961.
[2] \iV.K. Taylor. Cortico-thalamic organization and memory. Proc. of the Royal
Society of London B, 159:466-478, 1964.
[3] R.P. Lippmann, B. Gold, and M.L. Malpass. A comparison of Hamming and
Hopfield neural nets for pattern classification. Technical Report TR-769, MIT
Lincoln Laboratory, 1987.
[4] E.E. Baum, J. Moody, and F. Wilczek. Internal representations for associative
memory. Biological Cybernetics, 59:217-228, 1987.
[5] P. Floreen. The convergence of hamming memory networks. IEEE Trans. on
Neural Networks, 2(4):449-457, 1991.
[6] E. Domany and H. Orland. A maximum overlap neural network for pattern
recognition. Physics Letters A, 125:32-34,1987.
[7] M.R. Leadbetter, G. Lindgren, and H. Rootzen. Extremes and related properties of random sequences and processes. Springer-Verlag, Berlin-HeidelbergNew York , 1983.
571
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6,279 | 6,680 | A New Theory for Matrix Completion
Guangcan Liu?
Qingshan Liu?
Xiao-Tong Yuan?
School of Information & Control, Nanjing University of Information Science & Technology
NO 219 Ningliu Road, Nanjing, Jiangsu, China, 210044
{gcliu,qsliu,xtyuan}@nuist.edu.cn
Abstract
Prevalent matrix completion theories reply on an assumption that the locations of
the missing data are distributed uniformly and randomly (i.e., uniform sampling).
Nevertheless, the reason for observations being missing often depends on the unseen
observations themselves, and thus the missing data in practice usually occurs in a
nonuniform and deterministic fashion rather than randomly. To break through the
limits of random sampling, this paper introduces a new hypothesis called isomeric
condition, which is provably weaker than the assumption of uniform sampling and
arguably holds even when the missing data is placed irregularly. Equipped with
this new tool, we prove a series of theorems for missing data recovery and matrix
completion. In particular, we prove that the exact solutions that identify the target
matrix are included as critical points by the commonly used nonconvex programs.
Unlike the existing theories for nonconvex matrix completion, which are built
upon the same condition as convex programs, our theory shows that nonconvex
programs have the potential to work with a much weaker condition. Comparing to
the existing studies on nonuniform sampling, our setup is more general.
1
Introduction
Missing data is a common occurrence in modern applications such as computer vision and image
processing, reducing significantly the representativeness of data samples and therefore distorting
seriously the inferences about data. Given this pressing situation, it is crucial to study the problem
of recovering the unseen data from a sampling of observations. Since the data in reality is often
organized in matrix form, it is of considerable practical significance to study the well-known problem
of matrix completion [1] which is to fill in the missing entries of a partially observed matrix.
Problem 1.1 (Matrix Completion). Denote the (i, j)th entry of a matrix as [?]ij . Let L0 ? Rm?n be
an unknown matrix of interest. In particular, the rank of L0 is unknown either. Given a sampling of
the entries in L0 and a 2D index set ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n} consisting of the locations
of the observed entries, i.e., given
{[L0 ]ij |(i, j) ? ?}
and
?,
can we restore the missing entries whose indices are not included in ?, in an exact and scalable
fashion? If so, under which conditions?
Due to its unique role in a broad range of applications, e.g., structure from motion and magnetic
resonance imaging, matrix completion has received extensive attentions in the literatures, e.g., [2?13].
?
The work of Guangcan Liu is supported in part by national Natural Science Foundation of China (NSFC)
under Grant 61622305 and Grant 61502238, in part by Natural Science Foundation of Jiangsu Province of China
(NSFJPC) under Grant BK20160040.
?
The work of Qingshan Liu is supported by NSFC under Grant 61532009.
?
The work of Xiao-Tong Yuan is supported in part by NSFC under Grant 61402232 and Grant 61522308, in
part by NSFJPC under Grant BK20141003.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Left and Middle: Typical configurations for the locations of the observed entries. Right: A
real example from the Oxford motion database. The black areas correspond to the missing entries.
In general, given no presumption about the nature of matrix entries, it is virtually impossible to
restore L0 as the missing entries can be of arbitrary values. That is, some assumptions are necessary
for solving Problem 1.1. Based on the high-dimensional and massive essence of today?s data-driven
community, it is arguable that the target matrix L0 we wish to recover is often low rank [23]. Hence,
one may perform matrix completion by seeking a matrix with the lowest rank that also satisfies the
constraints given by the observed entries:
min rank (L) ,
L
s.t.
[L]ij = [L0 ]ij , ?(i, j) ? ?.
(1)
Unfortunately, this idea is of little practical because the problem above is NP-hard and cannot be
solved in polynomial time [15]. To achieve practical matrix completion, Cand?s and Recht [4]
suggested to consider an alternative that minimizes instead the nuclear norm which is a convex
envelope of the rank function [12]. Namely,
min kLk? ,
L
s.t.
[L]ij = [L0 ]ij , ?(i, j) ? ?,
(2)
where k ? k? denotes the nuclear norm, i.e., the sum of the singular values of a matrix. Rather
surprisingly, it is proved in [4] that the missing entries, with high probability, can be exactly restored
by the convex program (2), as long as the target matrix L0 is low rank and incoherent and the set ? of
locations corresponding to the observed entries is a set sampled uniformly at random. This pioneering
work provides people several useful tools to investigate matrix completion and many other related
problems. Those assumptions, including low-rankness, incoherence and uniform sampling, are now
standard and widely used in the literatures, e.g., [14, 17, 22, 24, 28, 33, 34, 36]. In particular, the
analyses in [17, 33, 36] show that, in terms of theoretical completeness, many nonconvex optimization
based methods are as powerful as the convex program (2). Unfortunately, these theories still depend
on the assumption of uniform sampling, and thus they cannot explain why there are many nonconvex
methods which often do better than the convex program (2) in practice.
The missing data in practice, however, often occurs in a nonuniform and deterministic fashion instead
of randomly. This is because the reason for an observation being missing usually depends on the
unseen observations themselves. For example, in structure from motion and magnetic resonance
imaging, typically the locations of the observed entries are concentrated around the main diagonal of
a matrix4 , as shown in Figure 1. Moreover, as pointed out by [19, 21, 23], the incoherence condition
is indeed not so consistent with the mixture structure of multiple subspaces, which is also a ubiquitous
phenomenon in practice. There has been sparse research in the direction of nonuniform sampling,
e.g., [18, 25?27, 31]. In particular, Negahban and Wainwright [26] studied the case of weighted
entrywise sampling, which is more general than the setup of uniform sampling but still a special
form of random sampling. Kir?ly et al. [18] considered deterministic sampling and is most related to
this work. However, they had only established conditions to decide whether a particular entry of the
matrix can be restored. In other words, the setup of [18] may not handle well the dependence among
the missing entries. In summary, matrix completion still starves for practical theories and methods,
although has attained considerable improvements in these years.
To break through the limits of the setup of random sampling, in this paper we introduce a new
hypothesis called isomeric condition, which is a mixed concept that combines together the rank and
coherence of L0 with the locations and amount of the observed entries. In general, isomerism (noun
4
This statement means that the observed entries are concentrated around the main diagonal after a permutation
of the sampling pattern ?.
2
of isomeric) is a very mild hypothesis and only a little bit more strict than the well-known oracle
assumption; that is, the number of observed entries in each row and column of L0 is not smaller than
the rank of L0 . It is arguable that the isomeric condition can hold even when the missing entries have
irregular locations. In particular, it is provable that the widely used assumption of uniform sampling
is sufficient to ensure isomerism, not necessary. Equipped with this new tool, isomerism, we prove a
set of theorems pertaining to missing data recovery [35] and matrix completion. For example, we
prove that, under the condition of isomerism, the exact solutions that identify the target matrix are
included as critical points by the commonly used bilinear programs. This result helps to explain the
widely observed phenomenon that there are many nonconvex methods performing better than the
convex program (2) on real-world matrix completion tasks. In summary, the contributions of this
paper mainly include:
We invent a new hypothesis called isomeric condition, which provably holds given the
standard assumptions of uniform sampling, low-rankness and incoherence. In addition,
we also exemplify that the isomeric condition can hold even if the target matrix L0 is not
incoherent and the missing entries are placed irregularly. Comparing to the existing studies
about nonuniform sampling, our setup is more general.
Equipped with the isomeric condition, we prove that the exact solutions that identify L0
are included as critical points by the commonly used bilinear programs. Comparing to the
existing theories for nonconvex matrix completion, our theory is built upon a much weaker
assumption and can therefore partially reveal the superiorities of nonconvex programs over
the convex methods based on (2).
We prove that the isomeric condition is sufficient and necessary for the column and row
projectors of L0 to be invertible given the sampling pattern ?. This result implies that
the isomeric condition is necessary for ensuring that the minimal rank solution to (1) can
identify the target L0 .
The rest of this paper is organized as follows. Section 2 summarizes the mathematical notations used
in the paper. Section 3 introduces the proposed isomeric condition, along with some theorems for
matrix completion. Section 4 shows some empirical results and Section 5 concludes this paper. The
detailed proofs to all the proposed theorems are presented in the Supplementary Materials.
2
Notations
Capital and lowercase letters are used to represent matrices and vectors, respectively, except that the
lowercase letters, i, j, k, m, n, l, p, q, r, s and t, are used to denote some integers, e.g., the location of
an observation, the rank of a matrix, etc. For a matrix M , [M ]ij is its (i, j)th entry, [M ]i,: is its ith row
and [M ]:,j is its jth column. Let ?1 and ?2 be two 1D index sets; namely, ?1 = {i1 , i2 , ? ? ? , ik } and
?2 = {j1 , j2 , ? ? ? , js }. Then [M ]?1 ,: denotes the submatrix of M obtained by selecting the rows with
indices i1 , i2 , ? ? ? , ik , [M ]:,?2 is the submatrix constructed by choosing the columns j1 , j2 , ? ? ? , js ,
and similarly for [M ]?1 ,?2 . For a 2D index set ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}, we imagine it
as a sparse matrix and, accordingly, define its ?rows?, ?columns? and ?transpose? as follows: The
ith row ?i = {j1 |(i1 , j1 ) ? ?, i1 = i}, the jth column ?j = {i1 |(i1 , j1 ) ? ?, j1 = j} and the
transpose ?T = {(j1 , i1 )|(i1 , j1 ) ? ?}.
The special symbol (?)+ is reserved to denote the Moore-Penrose pseudo-inverse of a matrix. More
T
precisely, for a matrix M with Singular Value Decomposition (SVD)5 M = UM ?M VM
, its pseudo?1 T
+
inverse is given by M = VM ?M UM . For convenience, we adopt the conventions of using
span{M } to denote the linear space spanned by the columns of a matrix M , using y ? span{M } to
denote that a vector y belongs to the space span{M }, and using Y ? span{M } to denote that all the
column vectors of a matrix Y belong to span{M }.
Capital letters U , V , ? and their variants (complements, subscripts, etc.) are reserved for left singular
vectors, right singular vectors and index set, respectively. For convenience, we shall abuse the
notation U (resp. V ) to denote the linear space spanned by the columns of U (resp. V ), i.e., the
column space (resp. row space). The orthogonal projection onto the column space U , is denoted by
PU and given by PU (M ) = U U T M , and similarly for the row space PV (M ) = M V V T . The same
In this paper, SVD always refers to skinny SVD. For a rank-r matrix M ? Rm?n , its SVD is of the form
T
UM ?M VM
, where UM ? Rm?r , ?M ? Rr?r and VM ? Rn?r .
5
3
notation is also used to represent a subspace of matrices (i.e., the image of an operator), e.g., we say
that M ? PU for any matrix M which satisfies PU (M ) = M . We shall also abuse the notation ?
to denote the linear space of matrices supported on ?. Then the symbol P? denotes the orthogonal
projection onto ?, namely,
[M ]ij , if (i, j) ? ?,
[P? (M )]ij =
0,
otherwise.
Similarly, the symbol P?? denotes the orthogonal projection onto the complement space of ?. That
is, P? + P?? = I, where I is the identity operator.
Three types of matrix norms are used in this paper, and they are all functions of the singular values:
1) The operator norm or 2-norm (i.e., largest singular value) denoted by kM k, 2) the Frobenius norm
(i.e., square root of the sum of squared singular values) denoted by kM kF and 3) the nuclear norm
or trace norm (i.e., sum of singular values) denoted by kM k? . The only used vector norm is the `2
norm, which is denoted by k ? k2 . The symbol | ? | is reserved for the cardinality of an index set.
3
Isomeric Condition and Matrix Completion
This section introduces the proposed isomeric condition and a set of theorems for matrix completion.
But most of the detailed proofs are deferred until the Supplementary Materials.
3.1 Isomeric Condition
In general cases, as aforementioned, matrix completion is an ill-posed problem. Thus, some assumptions are necessary for studying Problem 1.1. To eliminate the disadvantages of the setup of random
sampling, we define and investigate a so-called isomeric condition.
3.1.1
Definitions
For the ease of understanding, we shall begin with a concept called k-isomerism (or k-isomeric in
adjective form), which could be regarded as an extension of low-rankness.
Definition 3.1 (k-isomeric). A matrix M ? Rm?l is called k-isomeric if and only if any k rows of
M can linearly represent all rows in M . That is,
rank ([M ]?,: ) = rank (M ) , ?? ? {1, 2, ? ? ? , m}, |?| = k,
where | ? | is the cardinality of an index set.
In general, k-isomerism is somewhat similar to Spark [37] which defines the smallest linearly
dependent subset of the rows of a matrix. For a matrix M to be k-isomeric, it is necessary that
rank (M ) ? k, not sufficient. In fact, k-isomerism is also somehow related to the concept of
coherence [4, 21]. When the coherence of a matrix M ? Rm?l is not too high, the rows of M will
sufficiently spread, and thus M could be k-isomeric with a small k, e.g., k = rank (M ). Whenever
the coherence of M is very high, one may need a large k to satisfy the k-isomeric property. For
example, consider an extreme case where M is a rank-1 matrix with one row being 1 and everywhere
else being 0. In this case, we need k = m to ensure that M is k-isomeric.
While Definition 3.1 involves all 1D index sets of cardinality k, we often need the isomeric property
to be associated with a certain 2D index set ?. To this end, we define below a concept called
?-isomerism (or ?-isomeric in adjective form).
Definition 3.2 (?-isomeric). Let M ? Rm?l and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Suppose
that ?j 6= ? (empty set), ?1 ? j ? n. Then the matrix M is called ?-isomeric if and only if
rank [M ]?j ,: = rank (M ) , ?j = 1, 2, ? ? ? , n.
Note here that only the number of rows in M is required to coincide with the row indices included in
?, and thereby l 6= n is allowable.
Generally, ?-isomerism is less strict than k-isomerism. Provided that |?j | ? k, ?1 ? j ? n, a matrix
M is k-isomeric ensures that M is ?-isomeric as well, but not vice versa. For the extreme example
where M is nonzero at only one row, interestingly, M can be ?-isomeric as long as the locations of
the nonzero elements are included in ?.
With the notation of ?T = {(j1 , i1 )|(i1 , j1 ) ? ?}, the isomeric property could be also defined on
the column vectors of a matrix, as shown in the following definition.
4
Definition 3.3 (?/?T -isomeric). Let M ? Rm?n and ? ? {1, 2, ? ? ? , m}?{1, 2, ? ? ? , n}. Suppose
?i 6= ? and ?j 6= ?, ?i = 1, ? ? ? , m, j = 1, ? ? ? , n. Then the matrix M is called ?/?T -isomeric if
and only if M is ?-isomeric and M T is ?T -isomeric as well.
To solve Problem 1.1 without the imperfect assumption of missing at random, as will be shown later,
we need to assume that L0 is ?/?T -isomeric. This condition has excluded the unidentifiable cases
where any rows or columns of L0 are wholly missing. In fact, whenever L0 is ?/?T -isomeric, the
number of observed entries in each row and column of L0 has to be greater than or equal to the rank
of L0 ; this is consistent with the results in [20]. Moreover, ?/?T -isomerism has actually well treated
the cases where L0 is of high coherence. For example, consider an extreme case where L0 is 1 at only
one element and 0 everywhere else. In this case, L0 cannot be ?/?T -isomeric unless the nonzero
element is observed. So, generally, it is possible to restore the missing entries of a highly coherent
matrix, as long as the ?/?T -isomeric condition is obeyed.
3.1.2
Basic Properties
While its definitions are associated with a certain matrix, the isomeric condition is actually characterizing some properties of a space, as shown in the lemma below.
Lemma 3.1. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Denote the SVD of L0 as
U0 ?0 V0T . Then we have:
1. L0 is ?-isomeric if and only if U0 is ?-isomeric.
2. LT0 is ?T -isomeric if and only if V0 is ?T -isomeric.
Proof. It could be manipulated that
[L0 ]?j ,: = ([U0 ]?j ,: )?0 V0T , ?j = 1, ? ? ? , n.
Since ?0 V0T is row-wisely full rank, we have
rank [L0 ]?j ,: = rank [U0 ]?j ,: , ?j = 1, ? ? ? , n.
As a result, L0 is ?-isomeric is equivalent to that U0 is ?-isomeric. In a similar way, the second
claim is proved.
It is easy to see that the above lemma is still valid even when the condition of ?-isomerism is replaced
by k-isomerism. Thus, hereafter, we may say that a space is isomeric (k-isomeric, ?-isomeric or
?T -isomeric) as long as its basis matrix is isomeric. In addition, the isomeric property is subspace
successive, as shown in the next lemma.
Lemma 3.2. Let ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n} and U0 ? Rm?r be the basis matrix of a
Euclidean subspace embedded in Rm . Suppose that U is a subspace of U0 , i.e., U = U0 U0T U . If U0
is ?-isomeric then U is ?-isomeric as well.
Proof. By U = U0 U0T U and U0 is ?-isomeric,
rank [U ]?j ,: = rank ([U0 ]?j ,: )U0T U = rank U0T U
= rank U0 U0T U = rank (U ) , ?1 ? j ? n.
The above lemma states that, in one word, the subspace of an isomeric space is isomeric as well.
3.1.3
Important Properties
As aforementioned, the isometric condition is actually necessary for ensuring that the minimal rank
solution to (1) can identify L0 . To see why, let?s assume that U0 ? ?? 6= {0}, where we denote by
U0 ?0 V0T the SVD of L0 . Then one could construct a nonzero perturbation, denoted as ? ? U0 ? ?? ,
? 0 = L0 + ? to the problem in (1). Since ? ? U0 , we
and accordingly, obtain a feasible solution L
?
? 0 ) < rank (L0 ). Such
have rank(L0 ) ? rank (L0 ). Even more, it is entirely possible that rank(L
a case is unidentifiable in nature, as the global optimum to problem (1) cannot identify L0 . Thus,
5
to ensure that the global minimum to (1) can identify L0 , it is essentially necessary to show that
U0 ? ?? = {0} (resp. V0 ? ?? = {0}), which is equivalent to that the operator PU0 P? PU0 (resp.
PV0 P? PV0 ) is invertible (see Lemma 6.8 of the Supplementary Materials). Interestingly, the isomeric
condition is indeed a sufficient and necessary condition for the operators PU0 P? PU0 and PV0 P? PV0
to be invertible, as shown in the following theorem.
Theorem 3.1. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Let the SVD of L0 be
U0 ?0 V0T . Denote PU0 (?) = U0 U0T (?) and PV0 (?) = (?)V0 V0T . Then we have the following:
1. The linear operator PU0 P? PU0 is invertible if and only if U0 is ?-isomeric.
2. The linear operator PV0 P? PV0 is invertible if and only if V0 is ?T -isomeric.
The necessity stated above implies that the isomeric condition is actually a very mild hypothesis. In
general, there are numerous reasons for the target matrix L0 to be isomeric. Particularly, the widely
used assumptions of low-rankness, incoherence and uniform sampling are indeed sufficient (but not
necessary) to ensure isomerism, as shown in the following theorem.
Theorem 3.2. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Denote n1 = max(m, n)
and n2 = min(m, n). Suppose that L0 is incoherent and ? is a 2D index set sampled uniformly
at random, namely Pr((i, j) ? ?) = ?0 and Pr((i, j) ?
/ ?) = 1 ? ?0 . For any ? > 0, if ?0 > ?
is obeyed and rank (L0 ) < ?n2 /(c log n1 ) holds for some numerical constant c then, with high
probability at least 1 ? n?10
, L0 is ?/?T -isomeric.
1
It is worth noting that the isomeric condition can be obeyed in numerous circumstances other than
the case of uniform sampling plus incoherence. For example,
"
#
1 0 0
? = {(1, 1), (1, 2), (1, 3), (2, 1), (3, 1)} and L0 = 0 0 0 ,
0 0 0
where L0 is a 3?3 matrix with 1 at (1, 1) and 0 everywhere else. In this example, L0 is not incoherent
and the sampling is not uniform either, but it could be verified that L0 is ?/?T -isomeric.
3.2 Results
In this subsection, we shall show how the isomeric condition can take effect in the context of
nonuniform sampling, establishing some theorems pertaining to missing data recovery [35] as well
as matrix completion.
3.2.1
Missing Data Recovery
Before exploring the matrix completion problem, for the ease of understanding, we would like
to consider a missing data recovery problem studied by Zhang [35], which could be described as
follows: Let y0 ? Rm be a data vector drawn form some low-dimensional subspace, denoted as
y0 ? S0 ? Rm . Suppose that y0 contains some available observations in yb ? Rk and some missing
entries in yu ? Rm?k . Namely, after a permutation,
yb
y0 =
, yb ? Rk , yu ? Rm?k .
(3)
yu
Given the observations in yb , we seek to restore the unseen entries in yu . To do this, we consider the
prevalent idea that represents a data vector as a linear combination of the bases in a given dictionary:
y0 = Ax0 ,
(4)
where A ? Rm?p is a dictionary constructed in advance and x0 ? Rp is the representation of y0 .
Utilizing the same permutation used in (3), we can partition the rows of A into two parts according to
the indices of the observed and missing entries, respectively:
Ab
A=
, Ab ? Rk?p , Au ? R(m?k)?p .
(5)
Au
In this way, the equation in (4) gives that
yb = Ab x0
and
6
y u = Au x 0 .
As we now can see, the unseen data yu could be restored, as long as the representation x0 is retrieved
by only accessing the available observations in yb . In general cases, there are infinitely many
representations that satisfy y0 = Ax0 , e.g., x0 = A+ y0 , where (?)+ is the pseudo-inverse of a matrix.
Since A+ y0 is the representation of minimal `2 norm, we revisit the traditional `2 program:
min
x
1
2
kxk2 ,
2
s.t.
yb = Ab x,
(6)
where k ? k2 is the `2 norm of a vector. Under some verifiable conditions, the above `2 program
is indeed consistently successful in a sense as in the following: For any y0 ? S0 with an arbitrary
partition y0 = [yb ; yu ] (i.e., arbitrarily missing), the desired representation x0 = A+ y0 is the unique
minimizer to the problem in (6). That is, the unseen data yu is exactly recovered by firstly computing
the minimizer x? to problem (6) and then calculating yu = Au x? .
Theorem 3.3. Let y0 = [yb ; yu ] ? Rm be an authentic sample drawn from some low-dimensional
subspace S0 embedded in Rm , A ? Rm?p be a given dictionary and k be the number of available
observations in yb . Then the convex program (6) is consistently successful, provided that S0 ?
span{A} and the dictionary A is k-isomeric.
Unlike the theory in [35], the condition of which is unverifiable, our k-isomeric condition could be
verified in finite time. Notice, that the problem of missing data recovery is closely related to matrix
completion, which is actually to restore the missing entries in multiple data vectors simultaneously.
Hence, Theorem 3.3 can be naturally generalized to the case of matrix completion, as will be shown
in the next subsection.
3.2.2
Matrix Completion
The spirits of the `2 program (6) can be easily transferred to the case of matrix completion. Following (6), one may consider Frobenius norm minimization for matrix completion:
min
X
1
2
kXkF , s.t. P? (AX ? L0 ) = 0,
2
(7)
where A ? Rm?p is a dictionary assumed to be given. As one can see, the problem in (7) is equivalent
to (6) if L0 is consisting of only one column vector. The same as (6), the convex program (7) can
also exactly recover the desired representation matrix A+ L0 , as shown in the theorem below. The
difference is that we here require ?-isomerism instead of k-isomerism.
Theorem 3.4. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Suppose that A ? Rm?p
is a given dictionary. Provided that L0 ? span{A} and A is ?-isomeric, the desired representation
X0 = A+ L0 is the unique minimizer to the problem in (7).
Theorem 3.4 tells us that, in general, even when the locations of the missing entries are interrelated
and nonuniformly distributed, the target matrix L0 can be restored as long as we have found a proper
dictionary A. This motivates us to consider the commonly used bilinear program that seeks both A
and X simultaneously:
min
A,X
1
1
2
2
kAkF + kXkF , s.t. P? (AX ? L0 ) = 0,
2
2
(8)
where A ? Rm?p and X ? Rp?n . The problem above is bilinear and therefore nonconvex. So, it
would be hard to obtain a strong performance guarantee as done in the convex programs, e.g., [4, 21].
Interestingly, under a very mild condition, the problem in (8) is proved to include the exact solutions
that identify the target matrix L0 as the critical points.
Theorem 3.5. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Denote the rank and SVD
of L0 as r0 and U0 ?0 V0T , respectively. If L0 is ?/?T -isomeric then the exact solution, denoted by
(A0 , X0 ) and given by
1
1
A0 = U0 ?02 QT , X0 = Q?02 V0T , ?Q ? Rp?r0 , QT Q = I,
is a critical point to the problem in (8).
To exhibit the power of program (8), however, the parameter p, which indicates the number of
columns in the dictionary matrix A, must be close to the true rank of the target matrix L0 . This is
7
observed entries (%)
95
convex (nonuniform) nonconvex (nonuniform)
95
95
convex (uniform)
nonconvex (uniform)
95
75
75
75
75
55
55
55
55
35
35
35
35
15
15
15
15
1
1
1 15 35 55 75 95
rank(L0)
1
1 15 35 55 75 95
rank(L0)
1
1 15 35 55 75 95
rank(L0)
1 15 35 55 75 95
rank(L0)
Figure 2: Comparing the bilinear program (9) (p = m) with the convex method (2). The numbers
plotted on the above figures are the success rates within 20 random trials. The white and black points
mean ?succeed? and ?fail?, respectively. Here the success is in a sense that PSNR ? 40dB, where
PSNR standing for peak signal-to-noise ratio.
impractical in the cases where the rank of L0 is unknown. Notice, that the ?-isomeric condition
imposed on A requires
rank (A) ? |?j |, ?j = 1, 2, ? ? ? , n.
This, together with the condition of L0 ? span{A}, essentially need us to solve a low rank matrix
recovery problem [14]. Hence, we suggest to combine the formulation (7) with the popular idea of
nuclear norm minimization, resulting in a bilinear program that jointly estimates both the dictionary
matrix A and the representation matrix X by
min kAk? +
A,X
1
2
kXkF , s.t. P? (AX ? L0 ) = 0,
2
(9)
which, by coincidence, has been mentioned in a paper about optimization [32]. Similar to (8), the
program in (9) has the following theorem to guarantee its performance.
Theorem 3.6. Let L0 ? Rm?n and ? ? {1, 2, ? ? ? , m} ? {1, 2, ? ? ? , n}. Denote the rank and SVD
of L0 as r0 and U0 ?0 V0T , respectively. If L0 is ?/?T -isomeric then the exact solution, denoted by
(A0 , X0 ) and given by
2
1
A0 = U0 ?03 QT , X0 = Q?03 V0T , ?Q ? Rp?r0 , QT Q = I,
is a critical point to the problem in (9).
Unlike (8), which possesses superior performance only if p is close to rank (L0 ) and the initial
solution is chosen carefully, the bilinear program in (9) can work well by simply choosing p = m
and using A = I as the initial solution. To see why, one essentially needs to figure out the conditions
under which a specific optimization procedure can produce an optimal solution that meets an exact
solution. This requires extensive justifications and we leave it as future work.
4
Simulations
To verify the superiorities of the nonconvex matrix completion methods over the convex program (2),
we would like to experiment with randomly generated matrices. We generate a collection of m ? n
(m = n = 100) target matrices according to the model of L0 = BC, where B ? Rm?r0 and
C ? Rr0 ?n are N (0, 1) matrices. The rank of L0 , i.e., r0 , is configured as r0 = 1, 5, 10, ? ? ? , 90, 95.
Regarding the index set ? consisting of the locations of the observed entries, we consider two settings: One is to create ? by using a Bernoulli model to randomly sample a subset from
{1, ? ? ? , m} ? {1, ? ? ? , n} (referred to as ?uniform?), the other is as in Figure 1 that makes the
locations of the observed entries be concentrated around the main diagonal of a matrix (referred to as
?nonuniform?). The observation fraction is set to be |?|/(mn) = 0.01, 0.05, ? ? ? , 0.9, 0.95. For each
pair of (r0 , |?|/(mn)), we run 20 trials, resulting in 8000 simulations in total.
When p = m and the identity matrix is used to initialize the dictionary A, we have empirically found
that program (8) has the same performance as (2). This is not strange, because it has been proven
in [16] that kLk? = minA,X 21 (kAk2F + kXk2F ), s.t. L = AX. Figure 2 compares the bilinear
8
program (9) to the convex method (2). It can be seen that (9) works distinctly better than (2). Namely,
while handling the nonuniformly missing data, the number of matrices successfully restored by the
bilinear program (9) is 102% more than the convex program (2). Even for dealing with the missing
entries chosen uniformly at random, in terms of the number of successfully restored matrices, the
bilinear program (9) can still outperform the convex method (2) by 44%. These results illustrate that,
even in the cases where the rank of L0 is unknown, the bilinear program (9) can do much better than
the convex optimization based method (2).
5
Conclusion and Future Work
This work studied the problem of matrix completion with nonuniform sampling, a significant setting
not extensively studied before. To figure out the conditions under which exact recovery is possible,
we proposed a so-called isomeric condition, which provably holds when the standard assumptions
of low-rankness, incoherence and uniform sampling arise. In addition, we also exemplified that
the isomeric condition can be obeyed in the other cases beyond the setting of uniform sampling.
Even more, our theory implies that the isomeric condition is indeed necessary for making sure
that the minimal rank completion can identify the target matrix L0 . Equipped with the isomeric
condition, finally, we mathematically proved that the widely used bilinear programs can include the
exact solutions that recover the target matrix L0 as the critical points; this guarantees the recovery
performance of bilinear programs to some extend.
However, there still remain several problems for future work. In particular, it is unknown under which
conditions a specific optimization procedure for (9) can produce an optimal solution that exactly
restores the target matrix L0 . To do this, one needs to analyze the convergence property as well as
the recovery performance. Moreover, it is unknown either whether the isomeric condition suffices
for ensuring that the minimal rank completion can identify the target L0 . These require extensive
justifications and we leave them as future work.
Acknowledgment
We would like to thanks the anonymous reviewers and meta-reviewers for providing us many valuable
comments to refine this paper.
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10
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6,280 | 6,681 | Robust Hypothesis Test for Nonlinear Effect
with Gaussian Processes
Jeremiah Zhe Liu, Brent Coull
Department of Biostatistics
Harvard University
Cambridge, MA 02138
{zhl112@mail, bcoull@hsph}.harvard.edu
Abstract
This work constructs a hypothesis test for detecting whether an data-generating
function h : Rp ? R belongs to a specific reproducing kernel Hilbert space
H0 , where the structure of H0 is only partially known. Utilizing the theory of
reproducing kernels, we reduce this hypothesis to a simple one-sided score test
for a scalar parameter, develop a testing procedure that is robust against the misspecification of kernel functions, and also propose an ensemble-based estimator
for the null model to guarantee test performance in small samples. To demonstrate
the utility of the proposed method, we apply our test to the problem of detecting
nonlinear interaction between groups of continuous features. We evaluate the
finite-sample performance of our test under different data-generating functions and
estimation strategies for the null model. Our results reveal interesting connections
between notions in machine learning (model underfit/overfit) and those in statistical
inference (i.e. Type I error/power of hypothesis test), and also highlight unexpected
consequences of common model estimating strategies (e.g. estimating kernel
hyperparameters using maximum likelihood estimation) on model inference.
1
Introduction
We study the problem of constructing a hypothesis test for an unknown data-generating function h :
Rp ? R, when h is estimated with a black-box algorithm (specifically, Gaussian Process regression)
from n observations {yi , xi }ni=1 . Specifically, we are interested in testing for the hypothesis:
H0 : h ? H0
v.s.
Ha : h ? Ha
where H0 , Ha are the function spaces for h under the null and the alternative hypothesis. We assume
only partial knowledge about H0 . For example, we may assume H0 = {h|h(xi ) = h(xi,1 )} is the
space of functions depend only on x1 , while claiming no knowledge about other properties (linearity,
smoothness, etc) about h. We pay special attention to the setting where the sample size n is small.
This type of tests carries concrete significance in scientific studies. In areas such as genetics,
drug trials and environmental health, a hypothesis test for feature effect plays a critical role in
answering scientific questions of interest. For example, assuming for simplicity x2?1 = [x1 , x2 ]T ,
an investigator might inquire the effect of drug dosage x1 on patient?s biometric measurement y
(corresponds to the null hypothesis H0 = {h(x) = h(x2 )}), or whether the adverse health effect
of air pollutants x1 is modified by patients? nutrient intake x2 (corresponds to the null hypothesis
H0 = {h(x) = h1 (x1 ) + h2 (x2 )}). In these studies, h usually represents some complex, nonlinear
biological process whose exact mathematical properties are not known. Sample size in these studies
are often small (n ? 100 ? 200), due to the high monetary and time cost in subject recruitment and
the lab analysis of biological samples.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
There exist two challenges in designing such a test. The first challenge arises from the low interpretability of the blackbox model. It is difficult to formulate a hypothesis about feature effect in
? implicitly using a collection of basis functions
these models, since the blackbox models represents h
constructed from the entire feature vector x, rather than a set of model parameters that can be
interpreted in the context of some effect of interest. For example, consider testing for the interaction
effect between x1 and x2 . With linear model h(xi ) = xi1 ?1 + xi2 ?2 + xi1 xi2 ?3 , we can simply
represent the interaction effect using a single parameter
Pn ?3 , and test for H0 : ?3 = 0. On the other
hand, Gaussian process (GP) [16] models h(xi ) = j=1 k(xi , xj )?j using basis functions defined
by the kernel function k. Since k is an implicit function that takes the entire feature vector as input, it
is not clear how to represent the interaction effect in GP models. We address this challenge assuming
h belongs to a reproducing kernel Hilbert space (RKHS) governed by the kernel function k? , such
that H = H0 when ? = 0, and H = Ha otherwise. In this way, ? encode exactly the feature effect
of interest, and the null hypothesis h ? H0 can be equivalently stated as H0 : ? = 0. To test for the
hypothesis, we re-formulate the GP estimates as the variance components of a linear mixed model
(LMM) [13], and derive a variance component score test which requires only model estimates under
the null hypothesis.
Clearly, performance of the hypothesis test depends on the quality of the model estimate under the
null hypothesis, which give rise to the second challenge: estimating the null model when only having
partial knowledge about H0 . In the case of Gaussian process, this translates to only having partial
knowledge about the kernel function k0 . The performance of Gaussian process is sensitive to the
choices of the kernel function k(z, z0 ). In principle, the RKHS H generated by a proper kernel
function k(z, z0 ) should be rich enough so it contains the data-generating function h, yet restrictive
? does not overfit in small samples. Choosing a kernel function that is too restrictive
enough such that h
or too flexible will lead to either model underfit or overfit, rendering the subsequent hypothesis
tests not valid. We address this challenge by proposing an ensemble-based estimator for h we term
Cross-validated Kernel Ensemble (CVEK). Using a library of base kernels, CVEK learns a proper H
from data by directly minimizing the ensemble model?s cross-validation error, therefore guaranteeing
robust test performance for a wide range of data-generating functions.
The rest of the paper is structured as follows. After a brief review of Gaussian process and its
connection with linear mixed model in Section 2, we introduce the test procedure for general
hypothesis h ? H0 in Section 3, and its companion estimation procedure CVEK in Section 4. To
demonstrate the utility of the proposed test, in section 5, we adapt our test to the problem of detecting
nonlinear interaction between groups of continuous features, and in section 6 we conduct simulation
studies to evaluate the finite-sample performance of the interaction test, under different kernel
estimation strategies, and under a range of data-generating functions with different mathematical
properties. Our simulation study reveals interesting connection between notions in machine learning
and those in statistical inference, by elucidating the consequence of model estimation (underfit /
overfit) on the Type I error and power of the subsequent hypothesis test. It also cautions against the
use of some common estimation strategies (most notably, selecting kernel hyperparameters using
maximum likelihood estimation) when conducting hypothesis test in small samples, by highlighting
inflated Type I errors from hypothesis tests based on the resulting estimates. We note that the methods
and conclusions from this work is extendable beyond the Gaussian Process models, due to GP?s
connection to other blackbox models such as random forest [5] and deep neural network [19].
2
Background on Gaussian Process
Assume we observe data from n independent subjects. For the ith subject, let yi be a continuous
response, xi be the set of p continuous features that has nonlinear effect on yi . We assume that the
outcome yi depends on features xi through below data-generating model:
iid
yi |h = ? + h(xi ) + i
where i ? N (0, ?)
(1)
and h : Rp ? R follows the Gaussian process prior GP(0, k) governed by the positive definite kernel
function k, such that the function evaluated at the observed record follows the multivariate normal
(MVN) distribution:
h = [h(x1 ), . . . , h(xn )] ? M V N (0, K)
2
with covariance matrix Ki,j = k(xi , xj ). Under above construction, the predictive distribution of h
evaluated at the samples is also multivariate normal:
h|{yi , xi }ni=1 ? M V N (h? , K? )
h? = K(K + ?I)?1 (y ? u)
K? = K ? K(K + ?I)?1 K
To understand the impact of ? and k on h? , recall that Gaussian process can be understood as the
Bayesian version of the kernel machine regression, where h? equivalently arise from the below
optimization problem:
h? = argmin ||y ? ? ? h(x)||2 + ?||h||2H
h?Hk
where Hk is the RKHS generated by kernel function k. From this perspective, h? is the element
in a spherical ball in Hk that best approximates the observed data y. The norm of h? , ||h||2H , is
constrained by the tuning parameter ?, and the mathematical properties (e.g. smoothness, spectral
density, etc) of h? are governed by the kernel function k. It should be noticed that although h? may
arise from Hk , the probability of the Gaussian Process h ? Hk is 0 [14].
Gaussian Process as Linear Mixed Model
[13] argued that if define ? =
?2
? ,
y =?+h+
h? can arise exactly from a linear mixed model (LMM):
where
h ? N (0, ? K)
? N (0, ? 2 I)
(2)
Therefore ? can be treated as part of the LMM?s variance components parameters. If K is correctly specified, then the variance components parameters (?, ? 2 ) can be estimated unbiasedly by
maximizing the Restricted Maximum Likelihood (REML)[12]:
LREML (?, ?, ? 2 |K) = ?log|V| ? log|1T V?1 1| ? (y ? ?)T V?1 (y ? ?)
(3)
where V = ? K + ? 2 I, and 1 a n ? 1 vector whose all elements are 1. However, it is worth noting
2
that REML is a model-based procedure. Therefore improper estimates for ? = ?? may arise when
the family of kernel functions are mis-specified.
3
A recipe for general hypothesis h ? H0
The GP-LMM connection introduced in Section 2 opens up the arsenal of statistical tools from Linear
Mixed Model for inference tasks in Gaussian Process. Here, we use the classical variance component
test [12] to construct a testing procedure for the hypothesis about Gaussian process function:
H0 : h ? H0 .
(4)
We first translate above hypothesis into a hypothesis in terms of model parameters. The key of our
approach is to assume that h lies in a RKHS generated by a garrote kernel function k? (z, z0 ) [15],
which is constructed by including an extra garrote parameter
? to a given kernel function. When
? = 0, the garrote kernel function k0 (x, x0 ) = k? (x, x0 )
generates exactly H0 , the space of
?=0
functions under the null hypothesis. In order to adapt this general hypothesisio to their hypothesis of
interest, practitioners need only to specify the form of the garrote kernel so that H0 corresponds to
the null hypothesis. For example, If k? (x) = k(? ? x1 , x2 , . . . , xp ), ? = 0 corresponds to the null
hypothesis H0 : h(x) = h(x2 , . . . , xp ), i.e. the function h(x) does not depend on x1 . (As we?ll see
in section 5, identifying such k0 is not always straightforward). As a result, the general hypothesis
(4) is equivalent to
H0 : ? = 0.
(5)
We now construct a test statistic T?0 for (5) by noticing that the garrote parameter ? can be treated as a
variance component parameter in the linear mixed model. This is because the Gaussian process under
garrote kernel can be formulated into below LMM:
y =?+h+
where
h ? N (0, ? K? )
3
? N (0, ? 2 I)
where K? is the kernel matrix generated by k? (z, z0 ). Consequently, we can derive a variance
component test for H0 by calculating the square derivative of LREML with respect ? under H0 [12]:
h
i
? T V0?1 ?K0 V0?1 (y ? ?)
?
T?0 = ?? ? (y ? ?)
(6)
where V0 = ?
? 2 I + ??K0 . In this expression, K0 = K? , and ?K0 is the null derivative kernel
?=0
?
matrix whose (i, j)th entry is ??
k? (x, x0 ) .
?=0
As discussed previously, misspecifying the null kernel function k0 negatively impacts the performance
of the resulting hypothesis test. To better understand the mechanism at play, we express the test
?
? ? h:
statistic T?0 from (6) in terms of the model residual ? = y ? ?
h
i
??
T
?,
?
?K
(7)
?
T?0 =
0
?
?4
? = (?
where we have used the fact V0?1 (y ? ?)
? 2 )?1 (?) [10]. As shown, the test statistic T?0 is a
scaled quadratic-form statistic that is a function of the model residual. If k0 is too restrictive, model
estimates will underfit the data even under the null hypothesis, introducing extraneous correlation
among the ?i ?s, therefore leading to overestimated T?0 and eventually underestimated p-value under
the null. In this case, the test procedure will frequently reject the null hypothesis (i.e. suggest the
existence of nonlinear interaction) even when there is in fact no interaction, yielding an invalid test
due to inflated Type I error. On the other hand, if k0 is too flexible, model estimates will likely
overfit the data in small samples, producing underestimated residuals, an underestimated test statistic,
and overestimated p-values. In this case, the test procedure will too frequently fail to reject the
null hypothesis (i.e. suggesting there is no interaction) when there is in fact interaction, yielding an
insensitive test with diminished power.
The null distribution of T? can be approximated using a scaled chi-square distribution ??2? using
Satterthwaite method [20] by matching the first two moments of T :
? ? ? = E(T ),
with solution (see Appendix for derivation):
h
i
?
? = ?I?? / ?? ? tr V0?1 ?K0
2 ? ?2 ? ? = V ar(T )
h
i2
?? = ?? ? tr V0?1 ?K0
/(2 ? ?I?? )
where ?I?? and ?I?? are the submatrices of the REML information matrix. Numerically more accurate,
but computationally less efficient approximation methods are also available [2].
Finally, the p-value of this test is calculated by examining the tail probability of ?
? ?2?? :
p = P (?
??2?? > T?) = P (?2?? > T?/?
?)
A complete summary of the proposed testing procedure is available in Algorithm 1.
Algorithm 1 Variance Component Test for h ? H0
1: procedure VCT FOR I NTERACTION
Input: Null Kernel Matrix K0 , Derivative Kernel Matrix ?K0 , Data (y, X)
Output: Hypothesis Test p-value p
# Step 1: Estimate Null Model using REML
? ??, ?
2:
(?,
? 2 ) = argmin LREML (?, ?, ? 2 |K0 ) as in (3)
# Step 2: Compute Test Statistic and Null Distribution Parameters
? T V?1 ?K0 V?1 (y ? X?)
?
3:
T?0 = ?? ? (y ? X?)
0 i
0 h
h
i2
4:
?
? = ?I?? / ?? ? tr V?1 ?K0 , ?? = ?? ? tr V?1 ?K0
/(2 ? ?I?? )
0
0
# Step 3: Compute p-value and reach conclusion
5:
p = P (?
??2?? > T?) = P (?2?? > T?/?
?)
6: end procedure
4
In light of the discussion about model misspecification in Introduction section, we highlight the fact
that our proposed test (6) is robust against model misspecification under the alternative [12], since
the calculation of test statistics do not require detailed parametric assumption about k? . However, the
test is NOT robust against model misspecification under the null, since the expression of both test
statistic T?0 and the null distribution parameters (?
?, ??) still involve the kernel matrices generated by
k0 (see Algorithm 1). In the next section, we address this problem by proposing a robust estimation
procedure for the kernel matrices under the null.
4
Estimating Null Kernel Matrix using Cross-validated Kernel Ensemble
Observation in (7) motivates the need for a kernel estimation strategy that is flexible so that it does
not underfit under the null, yet stable so that it does not overfit under the alternative. To this end, we
propose estimating h using the ensemble of a library of fixed base kernels {kd }D
d=1 :
?
h(x)
=
D
X
? d (x)
ud h
u ? ? = {u|u ? 0, ||u||22 = 1},
(8)
d=1
? d is the kernel predictor generated by dth base kernel kd . In order to maximize model stability,
where h
? We term
the ensemble weights u are estimated to minimize the overall cross-validation error of h.
this method the Cross-Validated Kernel Ensemble (CVEK). Our proposed method belongs to the
well-studied family of algorithms known as ensembles of kernel predictors (EKP) [7, 8, 3, 4], but with
specialized focus in maximizing the algorithm?s cross-validation stability. Furthermore, in addition to
? CVEK will also produce the ensemble estimate of the kernel matrix
producing ensemble estimates h,
? 0 that is required by Algorithm 1. The exact algorithm proceeds in three stages as follows:
K
?
? ?1 y,
Stage 1: For each basis kernel in the library {kd }D
d=1 , we first estimate hd = Kd (Kd + ?d I)
th
? d is selected by minimizing the
the prediction based on d kernel, where the tunning parameter ?
leave-one-out cross validation (LOOCV) error [6]:
? d,? ) where Ad,? = Kd (Kd + ?I)?1 .
LOOCV(?|kd ) = (I ? diag(Ad,? ))?1 (y ? h
(9)
? d |kd .
We denote estimate the final LOOCV error for dth kernel ?d = LOOCV ?
D
Stage 2: Using the estimated LOOCV errors {?
d }D
d=1 , estimate the ensemble weights u = {ud }d=1
such that it minimizes the overall LOOCV error:
D
X
?
u = argmin ||
ud ?d ||2
where ? = {u|u ? 0, ||u||22 = 1},
u??
d=1
and produce the final ensemble prediction:
?=
h
D
X
d=1
u
?d hd =
D
X
?
u
?d Ad,?? d y = Ay,
d=1
? = PD u
where A
? d is the ensemble hat matrix.
d=1 ?d Ad,?
? estimate the ensemble kernel matrix K
? by solving:
Stage 3: Using the ensemble hat matrix A,
? K
? + ?I)?1 = A.
?
K(
? then K
? adopts
Specifically, if we denote UA and {?A,k }nk=1 the eigenvector and eigenvalues of A,
the form (see Appendix for derivation):
?
A,k
? = UA diag
K
UTA
1 ? ?A,k
5
Application: Testing for Nonlinear Interaction
Recall in Section 3, we assume that we are given a k? that generates exactly H0 . However, depending
on the exact hypothesis of interest, identifying such k0 is not always straightforward. In this section,
5
we revisit the example about interaction testing discussed in challenge 1 at the Introduction section,
and consider how to build a k0 for below hypothesis of interest
H0 : h(x) = h1 (x1 ) + h2 (x2 )
Ha : h(x) = h1 (x1 ) + h2 (x2 ) + h12 (x1 , x2 )
where h12 is the "pure interaction" function that is orthogonal to main effect functions h1 and h2 .
Recall as discussed previously, this hypothesis is difficult to formulate with Gaussian process models,
since the kernel functions k(x, x0 ) in general do not explicitly separate the main and the interaction
effect. Therefore rather than directly define k0 , we need to first construct H0 and Ha that corresponds
to the null and alternative hypothesis, and then identify the garrote kernel function k? such it generates
exactly H0 when ? = 0 and Ha when ? > 0.
We build H0 using the tensor-product construction of RKHS on the product domain (x1,i , x2,i ) ?
Rp1 ? Rp2 [9], due to this approach?s unique ability in explicitly characterizing the space of "pure
interaction" functions. Let 1 = {f |f ? 1} be the RKHS of constant functions, and H1 , H2
be the RKHS of centered functions for x1 x2 , respectively. We can then define the full space as
H = ?2m=1 (1 ? Hm ). H describes the space of functions that depends jointly on {x1 , x2 }, and
adopts below orthogonal decomposition:
H = (1 ? H1 ) ? (1 ? H2 )
n
o n
o
?
= 1 ? H1 ? H2 ? H1 ? H2 = 1 ? H12
? H12
?
where we have denoted H12
= H1 ? H2 and H12 = H1 ? H2 , respectively. We see that H12 is
indeed the space of?pure interaction" functions , since H12 contains functions on the product domain
?
Rp1 ? Rp2 , but is orthogonal to the space of additive main effect functions H12
. To summarize, we
have identified two function spaces H0 and Ha that has the desired interpretation:
?
H0 = H12
?
Ha = H12
? H12
We are now ready to identify the garrote kernel k? (x, x0 ). To this end, we notice that both H0 and
H12 are composite spaces built from basis RKHSs using direct sum and tensor product. If denote
km (xm , x0m ) the reproducing kernel associated with Hm , we can construct kernel functions for
composite spaces H0 and H12 as [1]:
k0 (x, x0 ) = k1 (x1 , x01 ) + k2 (x2 , x02 )
k12 (x, x0 ) = k1 (x1 , x01 ) k2 (x2 , x02 )
and consequently, the garrote kernel function for Ha :
k? (x, x0 ) = k0 (x, x0 ) + ? ? k12 (x, x0 ).
(10)
Finally, using the chosen form of the garrote kernel function, the (i, j)th element of the null derivative
?
kernel matrix K0 is ??
k? (x, x0 ) = k12 (x, x0 ), i.e. the null derivative kernel matrix ?K0 is simply
the kernel matrix K12 that corresponds to the interaction space. Therefore the score test statistic T?0
in (6) simplifies to:
? T V?1 K12 V?1 (y ? X?)
?
T?0 = ?? ? (y ? X?)
0
0
(11)
where V0 = ?
? 2 I + ??K0 .
6
Simulation Experiment
We evaluated the finite-sample performance of the proposed interaction test in a simulation study
that is analogous to a real nutrition-environment interaction study. We generate two groups of input
features (xi,1 , xi,2 ) ? Rp1 ? Rp2 independently from standard Gaussian distribution, representing
normalized data representing subject?s level of exposure to p1 environmental pollutants and the levels
of a subject?s intake of p2 nutrients during the study. Throughout the simulation scenarios, we keep
n = 100, and p1 = p2 = 5. We generate the outcome yi as:
yi = h1 (xi,1 ) + h2 (xi,2 ) + ? ? h12 (xi,1 , xi,2 ) + i
6
(12)
where h1 , h2 , h12 are sampled from RKHSs H1 , H2 and H1 ? H2 , generated using a ground-truth
kernel ktrue . We standardize all sampled functions to have unit norm, so that ? represents the strength
of interaction relative to the main effect.
For each simulation scenario, we first generated data using ? and ktrue as above, then selected a
kmodel to estimate the null model and obtain p-value using Algorithm 1. We repeated each scenario
1000 times, and evaluate the test performance using the empirical probability P? (p ? 0.05). Under
null hypothesis H0 : ? = 0, P? (p ? 0.05) estimates the test?s Type I error, and should be smaller or
equal to the significance level 0.05. Under alternative hypothesis Ha : ? > 0, P? (p ? 0.05) estimates
the test?s power, and should ideally approach 1 quickly as the strength of interaction ? increases.
In this study, we varied ktrue to produce data-generating functions h? (xi,1 , xi,2 ) with different
smoothness and complexity properties, and varied kmodel to reflect different common modeling
strategies for the null model in addition to using CVEK. We then evaluated how these two aspects
impact the hypothesis test?s Type I error and power.
Data-generating Functions
We sampled the data-generate function by using ktrue from Mat?rn kernel family [16]:
? ?
21?? ?
k(r|?, ?) =
2??||r|| K?
2??||r|| ,
where
r = x ? x0 ,
?(?)
with two non-negative hyperparameters (?, ?). For a function h sampled using Mat?rn kernel, ?
determines the function?s smoothness, since h is k-times mean square differentiable if and only
if ? > k. In the case of ? ? ?, Mat?rn kernel reduces to the Gaussian RBF kernel which is
infinitely differentiable. ? determines the function?s complexity, this is because in Bochner?s spectral
decomposition[16]
Z
T
k(r|?, ?) = e2?is r dS(s|?, ?),
(13)
? determines how much weight the spectral density S(s) puts on the slow-varying, low-frequency
basis functions. In this work, we vary ? ? { 32 , 52 , ?} to generate once-, twice, and infinitelydifferentiable functions, and vary ? ? {0.5, 1, 1.5} to generate functions with varying degree of
complexity.
Modeling Strategies
Polynomial Kernels is a family of simple parametric kernels that is equivalent to the polynomial
ridge regression model favored by statisticians due to its model interpretability. In this work, we
use the linear kernel klinear (x, x0 |p) = xT x0 and quadratic kernel kquad (x, x0 |p) = (1 + xT x0 )2
which are common choices from this family.
Gaussian RBF Kernels kRBF (x, x0 |?) = exp(??||x ? x0 ||2 ) is a general-purpose kernel family that
generates nonlinear, but infinitely differentiable (therefore very smooth) functions. Under this kernel,
we consider two hyperparameter selection strategies common in machine learning applications: RBFMedian where we set ? to the sample median of {||xi ? xj ||}i6=j , and RBF-MLE who estimates ?
by maximizing the model likelihood.
Mat?rn and Neural Network Kernels are two flexible kernels favored by machine learning practitioners for their expressiveness. Mat?rn kernels generates functions that are more flexible compared
to that of Gaussian RBF due to the relaxed smoothness constraint [17]. In order to investigate the
consequences of added flexibility relative to the true model, we use Matern 1/2, Matern 3/2 and
Matern 5/2, corresponding to Mat?rn
kernels with ? = 1/2,
3/2, and 5/2. Neural network kernels
?0
2??
xT x
[16] knn (x, x0 |?) = ?2 ? sin?1 ?
, on the other hand, represent a 1-layer
T
0T 0
? )(1+2??
(1+2??
x x
x
? )
x
Bayesian neural network with infinite hidden unit and probit link function, with ? being the prior
variance on hidden weights. Therefore knn is flexible in the sense that it is an universal approximator
for arbitrary continuous functions on the compact domain [11]. In this work, we denote NN 0.1, NN
1 and NN 10 to represent Bayesian networks with different prior constraints ? ? {0.1, 1, 10}.
Result
The simulation results are presented graphically in Figure 1 and documented in detail in the Appendix.
We first observe that for reasonably specified kmodel , the proposed hypothesis test always has the
7
1.0
0.8
0.6
0.0
0.2
0.4
Linear
Quadratic
RBF_MLE
RBF_Median
Matern 1/2
Matern 3/2
Matern 5/2
(b) ktrue = Mat?rn 3/2, ? = 1
(c) ktrue = Mat?rn 3/2, ? = 1.5
0.6
0.8
1.0
(a) ktrue = Mat?rn 3/2, ? = 0.5
0.0
0.2
0.4
NN 0.1
NN 1
NN 10
CVKE_RBF
CVKE_NN
(e) ktrue = Mat?rn 5/2, ? = 1
(f) ktrue = Mat?rn 5/2, ? = 1.5
0.0
0.2
0.4
0.6
0.8
1.0
(d) ktrue = Mat?rn 5/2, ? = 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(g) ktrue = Gaussian RBF, ? = 0.5 (h) ktrue = Gaussian RBF, ? = 1 (i) ktrue = Gaussian RBF, ? = 1.5
Figure 1: Estimated P? (p < 0.05) (y-axis) as a function of Interaction Strength ? ? [0, 1] (x-axis).
Skype Blue: Linear (Solid) and Quadratic (Dashed) Kernels, Black: RBF-Median (Solid) and RBFMLE (Dashed), Dark Blue: Mat?rn Kernels with ? = 12 , 32 , 25 , Purple: Neural Network Kernels
with ? = 0.1, 1, 10, Red: CVEK based on RBF (Solid) and Neural Networks (Dashed).
Horizontal line marks the test?s significance level (0.05). When ? = 0, P? should be below this line.
correct Type I error and reasonable power. We also observe that the complexity of the data-generating
function h? (12) plays a role in test performance, in the sense that the power of the hypothesis
tests increases as the Mat?rn ktrue ?s complexity parameter ? becomes larger, which corresponds to
functions that put more weight on the complex, fast-varying eigenfunctions in (13).
We observed clear differences in test performance from different estimation strategies. In general,
polynomial models (linear and quadratic) are too restrictive and appear to underfit the data under
both the null and the alternative, producing inflated Type I error and diminished power. On the other
hand, lower-order Mat?rn kernels (Mat?rn 1/2 and Mat?rn 3/2, dark blue lines) appear to be too
flexible. Whenever data are generated from smoother ktrue , Mat?rn 1/2 and 3/2 overfit the data and
produce deflated Type I error and severely diminished power, even if the hyperparameter ? are fixed
at true value. Therefore unless there?s strong evidence that h exhibits behavior consistent with that
described by these kernels, we recommend avoid the use of either polynomial or lower-order Mat?rn
kernels for hypothesis testing. Comparatively, Gaussian RBF works well for a wider range of ktrue ?s,
but only if the hyperparameter ? is selected carefully. Specifically, RBF-Median (black dashed line)
works generally well, despite being slightly conservative (i.e. lower power) when the data-generation
function is smooth and of low complexity. RBF-MLE (black solid line), on the other hand, tends to
underfit the data under the null and exhibits inflated Type I error, possibly because of the fact that ? is
not strongly identified when the sample size is too small [18]. The situation becomes more severe as
h? becomes rougher and more complex, in the moderately extreme case of non-differentiable h with
? = 1.5, the Type I error is inflated to as high as 0.238. Neural Network kernels also perform well
for a wide range of ktrue , and its Type I error is more robust to the specification of hyperparameters.
Finally, the two ensemble estimators CVEK-RBF (based on kRBF ?s with log(?) ? {?2, ?1, 0, 1, 2})
and CVEK-NN (based on kNN ?s with ? ? {0.1, 1, 10, 50}) perform as well or better than the nonensemble approaches for all ktrue ?s, despite being slightly conservative under the null. As compared
to CVEK-NN, CVEK-RBF appears to be slightly more powerful.
8
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9
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6,281 | 6,682 | Lower bounds on the robustness to adversarial
perturbations
Jonathan Peck1,2 , Joris Roels2,3 , Bart Goossens3 , and Yvan Saeys1,2
1
Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Ghent, 9000, Belgium
2
Data Mining and Modeling for Biomedicine, VIB Inflammation Research Center, Ghent, 9052, Belgium
3
Department of Telecommunications and Information Processing, Ghent University, Ghent, 9000, Belgium
Abstract
The input-output mappings learned by state-of-the-art neural networks are significantly discontinuous. It is possible to cause a neural network used for image
recognition to misclassify its input by applying very specific, hardly perceptible
perturbations to the input, called adversarial perturbations. Many hypotheses have
been proposed to explain the existence of these peculiar samples as well as several
methods to mitigate them, but a proven explanation remains elusive. In this work,
we take steps towards a formal characterization of adversarial perturbations by
deriving lower bounds on the magnitudes of perturbations necessary to change the
classification of neural networks. The proposed bounds can be computed efficiently,
requiring time at most linear in the number of parameters and hyperparameters
of the model for any given sample. This makes them suitable for use in model
selection, when one wishes to find out which of several proposed classifiers is
most robust to adversarial perturbations. They may also be used as a basis for
developing techniques to increase the robustness of classifiers, since they enjoy the
theoretical guarantee that no adversarial perturbation could possibly be any smaller
than the quantities provided by the bounds. We experimentally verify the bounds
on the MNIST and CIFAR-10 data sets and find no violations. Additionally, the
experimental results suggest that very small adversarial perturbations may occur
with non-zero probability on natural samples.
1
Introduction
Despite their big successes in various AI tasks, neural networks are basically black boxes: there is no
clear fundamental explanation how they are able to outperform the more classical approaches. This
has led to the identification of several unexpected and counter-intuitive properties of neural networks.
In particular, Szegedy et al. [2014] discovered that the input-output mappings learned by state-of-theart neural networks are significantly discontinuous. It is possible to cause a neural network used for
image recognition to misclassify its input by applying a very specific, hardly perceptible perturbation
to the input. Szegedy et al. [2014] call these perturbations adversarial perturbations, and the inputs
resulting from applying them to natural samples are called adversarial examples.
In this paper, we hope to shed more light on the nature and cause of adversarial examples by
deriving lower bounds on the magnitudes of perturbations necessary to change the classification of
neural network classifiers. Such lower bounds are indispensable for developing rigorous methods
that increase the robustness of classifiers without sacrificing accuracy. Since the bounds enjoy the
theoretical guarantee that no adversarial perturbation could ever be any smaller, a method which
increases these lower bounds potentially makes the classifier more robust. They may also aid model
selection: if the bounds can be computed efficiently, then one can use them to compare different
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
models with respect to their robustness to adversarial perturbations and select the model that scores
the highest in this regard without the need for extensive empirical tests.
The rest of the paper is organized as follows. Section 2 discusses related work that has been done
on the phenomenon of adversarial perturbations; Section 3 details the theoretical framework used
to prove the lower bounds; Section 4 proves lower bounds on the robustness of different families of
classifiers to adversarial perturbations; Section 5 empirically verifies that the bounds are not violated;
Section 6 concludes the paper and provides avenues for future work.
2
Related work
Since the puzzling discovery of adversarial perturbations, several hypotheses have been proposed to
explain why they exist, as well as a number of methods to make classifiers more robust to them.
2.1
Hypotheses
The leading hypothesis explaining the cause of adversarial perturbations is the linearity hypothesis by
Goodfellow et al. [2015]. According this view, neural network classifiers tend to act very linearly on
their input data despite the presence of non-linear transformations within their layers. Since the input
data on which modern classifiers operate is often very high in dimensionality, such linear behavior
can cause minute perturbations to the input to have a large impact on the output. In this vein, Lou
et al. [2016] propose a variant of the linearity hypothesis which claims that neural network classifiers
operate highly linearly on certain regions of their inputs, but non-linearly in other regions. Rozsa et al.
[2016] conjecture that adversarial examples exist because of evolutionary stalling: during training,
the gradients of samples that are classified correctly diminish, so the learning algorithm ?stalls? and
does not create significantly flat regions around the training samples. As such, most of the training
samples will lie close to some decision boundary, and only a small perturbation is required to push
them into a different class.
2.2
Proposed solutions
Gu and Rigazio [2014] propose the Deep Contractive Network, which includes a smoothness penalty
in the training procedure inspired by the Contractive Autoencoder. This penalty encourages the
Jacobian of the network to have small components, thus making the network robust to small changes
in the input. Based on their linearity hypothesis, Goodfellow et al. [2015] propose the fast gradient
sign method for efficiently generating adversarial examples. They then use this method as a regularizer
during training in an attempt to make networks more robust. Lou et al. [2016] use their ?local linearity
hypothesis? as the basis for training neural network classifiers using foveations, i.e. a transformation
which selects certain regions from the input and discards all other information. Rozsa et al. [2016]
introduce Batch-Adjusted Network Gradients (BANG) based on their idea of evolutionary stalling.
BANG normalizes the gradients on a per-minibatch basis so that even correctly classified samples
retain significant gradients and the learning algorithm does not stall.
The solutions proposed above provide attractive intuitive explanations for the cause of adversarial
examples, and empirical results seem to suggest that they are effective at eliminating them. However,
none of the hypotheses on which these methods are based have been formally proven. Hence, even
with the protections discussed above, it may still be possible to generate adversarial examples for
classifiers using techniques which defy the proposed hypotheses. As such, there is a need to formally
characterize the nature of adversarial examples. Fawzi et al. [2016] take a step in this direction by
deriving precise bounds on the norms of adversarial perturbations of arbitrary classifiers in terms of
the curvature of the decision boundary. Their analysis encourages to impose geometric constraints
on this curvature in order to improve robustness. However, it is not obvious how such constraints
relate to the parameters of the models and hence how one would best implement such constraints
in practice. In this work, we derive lower bounds on the robustness of neural networks directly in
terms of their model parameters. We consider only feedforward networks comprised of convolutional
layers, pooling layers, fully-connected layers and softmax layers.
2
3
Theoretical framework
The theoretical framework used in this paper draws heavily from Fawzi et al. [2016] and Papernot
et al. [2016]. In the following, k?k denotes the Euclidean norm and k?kF denotes the Frobenius norm.
We assume we want to train a classifier f : Rd ? {1, . . . , C} to correctly assign one of C different
classes to input vectors x from a d-dimensional Euclidean space. Let ? denote the probability measure
on Rd and let f ? be an oracle that always returns the correct label for any input. The distribution ?
is assumed to be of bounded support, i.e. Px?? (x ? X ) = 1 with X = {x ? Rd | kxk ? M } for
some M > 0.
Formally, adversarial perturbations are defined relative to a classifier f and an input x. A perturbation
? is called an adversarial perturbation of x for f if f (x + ?) 6= f (x) while f ? (x + ?) = f ? (x).
An adversarial perturbation ? is called minimal if no other adversarial perturbation ? for x and f
satisfies k?k < k?k. In this work, we will focus on minimal adversarial perturbations.
The robustness of a classifier f is defined as the expected norm of the smallest perturbation necessary
to change the classification of an arbitrary input x sampled from ?:
?adv (f ) = Ex?? [?adv (x; f )],
where
?adv (x; f ) = min {k?k | f (x + ?) 6= f (x)}.
??Rd
A multi-index is a tuple of non-negative integers, generally denoted by Greek letters such as ? and ?.
For a multi-index ? = (?1 , . . . , ?m ) and a function f we define
??f =
|?| = ?1 + ? ? ? + ?n ,
? |?| f
.
n
. . . ?x?
n
1
?x?
1
The Jacobian matrix of a function f : Rn ? Rm : x 7? [f1 (x), . . . , fm (x)]T is defined as
? ?f1
?f1 ?
. . . ?x
?x1
n
?
?
?
..
..
f = ? ...
?.
.
.
?x
?fm
?fm
. . . ?xn
?x1
3.1
Families of classifiers
The derivation of the lower bounds will be built up incrementally. We will start with the family
of linear classifiers, which are among the simplest. Then, we extend the analysis to Multi-Layer
Perceptrons, which are the oldest neural network architectures. Finally, we analyze Convolutional
Neural Networks. In this section, we introduce each of these families of classifiers in turn.
A linear classifier is a classifier f of the form
f (x) = arg max wi ? x + bi .
i=1,...,C
The vectors wi are called weights and the scalars bi are called biases.
A Multi-Layer Perceptron (MLP) is a classifier given by
f (x) = arg max softmax(hL (x))i ,
i=1,...,C
hL (x) = gL (VL hL?1 (x) + bL ),
..
.
h1 (x) = g1 (V1 x + b1 ).
An MLP is nothing more than a series of linear transformations Vl hl?1 (x) + bl followed by nonlinear activation functions gl (e.g. a ReLU [Glorot et al., 2011]). Here, softmax is the softmax
function:
exp(wi ? y + bi )
softmax(y)i = P
.
j exp(wj ? y + bj )
3
This function is a popular choice as the final layer for an MLP used for classification, but it is by no
means the only possibility. Note that having a softmax as the final layer essentially turns the network
into a linear classifier of the output of its penultimate layer, hL (x).
A Convolutional Neural Network (CNN) is a neural network that uses at least one convolution
operation. For an input tensor X ? Rc?d?d and a kernel tensor W ? Rk?c?q?q , the discrete
convolution of X and W is given by
q X
q
c X
X
(X ? W)ijk =
wi,n,m,l xn,m+s(q?1),l+s(q?1) .
n=1 m=1 l=1
Here, s is the stride of the convolution. The output of such a layer is a 3D tensor of size k ? t ? t
k
where t = d?q
s + 1. After the convolution operation, usually a bias b ? R is added to each of the
feature maps. The different components (W ? X)i constitute the feature maps of this convolutional
layer. In a slight abuse of notation, we will write W ? X + b to signify the tensor W ? X where each
of the k feature maps has its respective bias added in:
(W ? X + b)ijk = (W ? X)ijk + bi .
CNNs also often employ pooling layers, which perform a sort of dimensionality reduction. If we
write the output of a pooling layer as Z(X), then we have
zijk (X) = p({xi,n+s(j?1),m+s(k?1) | 1 ? n, m ? q}).
Here, p is the pooling operation, s is the stride and q is a parameter. The output tensor Z(X) has
dimensions c ? t ? t. For ease of notation, we assume each pooling operation has an associated
function I such that
zijk (X) = p({xinm | (n, m) ? I(j, k)}).
In the literature, the set I(j, k) is referred to as the receptive field of the pooling layer. Each receptive
field corresponds to some q ? q region in the input X. Common pooling operations include taking
the maximum of all inputs, averaging the inputs and taking an Lp norm of the inputs.
4
Lower bounds on classifier robustness
Comparing the architectures of several practical CNNs such as LeNet [Lecun et al., 1998], AlexNet
[Krizhevsky et al., 2012], VGGNet [Simonyan and Zisserman, 2015], GoogLeNet [Szegedy et al.,
2015] and ResNet [He et al., 2016], it would seem the only useful approach is a ?modular? one. If we
succeed in lower-bounding the robustness of some layer given the robustness of the next layer, we can
work our way backwards through the network, starting at the output layer and going backwards until
we reach the input layer. That way, our approach can be applied to any feedforward neural network
as long as the robustness bounds of the different layer types have been established. To be precise, if a
given layer computes a function h of its input y and if the following layer has a robustness bound of
? in the sense that any adversarial perturbation to this layer has a Euclidean norm of at least ?, then
we want to find a perturbation r such that
kh(y + r)k = kh(y)k + ?.
This is clearly a necessary condition for any adversarial perturbation to the given layer. Hence, any
adversarial perturbation q to this layer will satisfy kqk ? krk. Of course, the output layer of the
network will require special treatment. For softmax output layers, ? is the norm of the smallest
perturbation necessary to change the maximal component of the classification vector.
The obvious downside of this idea is that we most likely introduce cumulative approximation errors
which increase as the number of layers of the network increases. In turn, however, we get a flexible
and efficient framework which can handle any feedforward architecture composed of known layer
types.
4.1
Softmax output layers
We now want to find the smallest perturbation r to the input x of a softmax layer such that f (x+r) 6=
f (x). It can be proven (Theorem A.3) that any such perturbation satisfies
|(wc0 ? wc ) ? x + bc0 ? bc |
krk ? min
,
c0 6=c
kwc0 ? wc k
where f (x) = c. Moreover, there exist classifiers for which this bound is tight (Theorem A.4).
4
4.2
Fully-connected layers
To analyze the robustness of fully-connected layers to adversarial perturbations, we assume the next
layer has a robustness of ? (this will usually be the softmax output layer, however there exist CNNs
which employ fully-connected layers in other locations than just at the end [Lin et al., 2014]). We
then want to find a perturbation r such that
khL (x + r)k = khL (x)k + ?.
We find
Theorem 4.1. Let hL : Rd ? Rn be twice differentiable with second-order derivatives bounded by
M . Then for any x ? Rd ,
q
?
2
kJ (x)k + 2M n? ? kJ (x)k
?
krk ?
,
(1)
M n
where J (x) is the Jacobian matrix of hL at x.
The proof can be found in Appendix A. In Theorem A.5 it is proved that the assumptions on hL
are usually satisfied in practice. The proof of this theorem also yields an efficient algorithm for
approximating M , a task which otherwise might involve a prohibitively expensive optimization
problem.
4.3
Convolutional layers
The next layer of the network is assumed to have a robustness bound of ?, in the sense that any
adversarial perturbation Q to X must satisfy kQkF ? ?. We can now attempt to bound the norm of a
perturbation R to X such that
kReLU(W ? (X + R) + b)kF = kReLU(W ? X + b)kF + ?.
We find
Theorem 4.2. Consider a convolutional layer with filter tensor W ? Rk?c?q?q and stride s whose
input consists of a 3D tensor X ? Rc?d?d . Suppose the next layer has a robustness bound of ?, then
any adversarial perturbation to the input of this layer must satisfy
?
kRkF ?
.
(2)
kWkF
The proof of Theorem 4.2 can be found in Appendix A.
4.4
Pooling layers
To facilitate the analysis of the pooling layers, we make the following assumption which is satisfied
by the most common pooling operations (see Appendix B):
Assumption 4.3. The pooling operation satisfies
zijk (X + R) ? zijk (X) + zijk (R).
We have
Theorem 4.4. Consider a pooling layer whose operation satisfies Assumption 4.3. Let the input be of
size c ? d ? d and the receptive field of size q ? q. Let the output be of size c ? t ? t. If the robustness
bound of the next layer is ?, then the following bounds hold for any adversarial perturbation R:
? MAX or average pooling:
?
.
t
(3)
?
.
tq 2/p
(4)
kRkF ?
? Lp pooling:
kRkF ?
Proof can be found in Appendix A.
5
Figure 1: Illustration of LeNet architecture. Image taken from Lecun et al. [1998].
Table 1: Normalized summary of norms of adversarial perturbations found by FGS on MNIST and
CIFAR-10 test sets
Data set
Mean
Median
Std
Min
Max
MNIST
CIFAR-10
5
0.933448
0.0218984
0.884287
0.0091399
0.4655439
0.06103627
0.000023
0.0000012
3.306903
1.6975207
Experimental results
We tested the theoretical bounds on the MNIST and CIFAR-10 test sets using the Caffe [Jia et al.,
2014] implementation of LeNet [Lecun et al., 1998]. The MNIST data set [LeCun et al., 1998]
consists of 70,000 28 ? 28 images of handwritten digits; the CIFAR-10 data set [Krizhevsky and
Hinton, 2009] consists of 60,000 32 ? 32 RGB images of various natural scenes, each belonging to
one of ten possible classes. The architecture of LeNet is depicted in Figure 1. The kernels of the two
convolutional layers will be written as W1 and W2 , respectively. The output sizes of the two pooling
layers will be written as t1 and t2 . The function computed by the first fully-connected layer will be
denoted by h with Jacobian J . The last fully-connected layer has a weight matrix V and bias vector
b. For an input sample x, the theoretical lower bound on the adversarial robustness of the network
with respect to x is given by ?1 , where
q
?
2
kJ (x)k + 2M 500?6 ? kJ (x)k
0
0
|(vc ? vc ) ? x + bc ? bc |
?
,
?5 =
?6 = min
,
c0 6=c
kvc0 ? vc k
M 500
?5
?4
,
?4 =
,
?3 =
t2
kW2 kF
?3
?2
?2 =
.
,
?1 =
t1
kW1 kF
Because our method only computes norms and does not provide a way to generate actual adversarial
perturbations, we used the fast gradient sign method (FGS) [Goodfellow et al., 2015] to adversarially
perturb each sample in the test sets in order to assess the tightness of our theoretical bounds. FGS
linearizes the cost function of the network to obtain an estimated perturbation
? = ?sign?x L(x, ?).
Here, ? > 0 is a parameter of the algorithm, L is the loss function and ? is the set of parameters of
the network. The magnitudes of the perturbations found by FGS depend on the choice of ?, so we
had to minimize this value in order to obtain the smallest perturbations the FGS method could supply.
This was accomplished using a simple binary search for the smallest value of ? which still resulted in
misclassification. As the MNIST and CIFAR-10 samples have pixel values within the range [0, 255],
we upper-bounded ? by 100.
No violations of the bounds were detected in our experiments. Figure 2 shows histograms of the
norms of adversarial perturbations found by FGS and Table 1 summarizes their statistics. Histograms
of the theoretical bounds of all samples in the test set are shown in Figure 3; their statistics are
summarized in Table 2. Note that the statistics of Tables 1 and 2 have been normalized by dividing
them by the dimensionality of their respective data sets (i.e. 28 ? 28 for MNIST and 3 ? 32 ? 32
for CIFAR-10) to allow for a meaningful comparison between the two networks. Figure 4 provides
histograms of the per-sample log-ratio between the norms of the adversarial perturbations and their
corresponding theoretical lower bounds.
6
(a) MNIST
(b) CIFAR-10
Figure 2: Histograms of norms of adversarial perturbations found by FGS on MNIST and CIFAR-10
test sets
(a) MNIST
(b) CIFAR-10
Figure 3: Histograms of theoretical bounds on MNIST and CIFAR-10 test sets
Although the theoretical bounds on average deviate considerably from the perturbations found by
FGS, one has to take into consideration that the theoretical bounds were constructed to provide a
worst-case estimate for the norms of adversarial perturbations. These estimates may not hold for
all (or even most) input samples. Furthermore, the smallest perturbations we were able to generate
on the two data sets have norms that are much closer to the theoretical bound than their averages
(0.0179 for MNIST and 0.0000012 for CIFAR-10). This indicates that the theoretical bound is
not necessarily very loose, but rather that very small adversarial perturbations occur with non-zero
probability on natural samples. Note also that the FGS method does not necessarily generate minimal
perturbations even with the smallest choice of ?: the method depends on the linearity hypothesis and
uses a first-order Taylor approximation of the loss function. Higher-order methods may find much
smaller perturbations by exploiting non-linearities in the network, but these are generally much less
efficient than FGS.
There is a striking difference in magnitude between MNIST and CIFAR-10 of both the empirical
and theoretical perturbations: the perturbations on MNIST are much larger than the ones found for
Table 2: Normalized summary of theoretical bounds on MNIST and CIFAR-10 test sets
Data set
Mean
Median
Std
Min
Max
MNIST
CIFAR-10
7.274e?8
4.812e?13
6.547e?8
4.445e?13
4.229566e?8
2.605381e?13
7
4.073e?10
7.563e?15
2.932e?7
2.098e?12
(a) MNIST
(b) CIFAR-10
Figure 4: Histograms of the per-sample log-ratio between adversarial perturbation and lower bound
for MNIST and CIFAR-10 test sets. A higher ratio indicates a bigger deviation of the theoretical
bound from the empirical norm.
CIFAR-10. This result can be explained by the linearity hypothesis of Goodfellow et al. [2015].
The input samples of CIFAR-10 are much larger in dimensionality than MNIST samples, so the
linearity hypothesis correctly predicts that networks trained on CIFAR-10 are more susceptible to
adversarial perturbations due to the highly linear behavior these classifiers are conjectured to exhibit.
However, these differences may also be related to the fact that LeNet achieves much lower accuracy
on the CIFAR-10 data set than it does on MNIST (over 99% on MNIST compared to about 60% on
CIFAR-10).
6
Conclusion and future work
Despite attracting a significant amount of research interest, a precise characterization of adversarial
examples remains elusive. In this paper, we derived lower bounds on the norms of adversarial
perturbations in terms of the model parameters of feedforward neural network classifiers consisting
of convolutional layers, pooling layers, fully-connected layers and softmax layers. The bounds can be
computed efficiently and thus may serve as an aid in model selection or the development of methods
to increase the robustness of classifiers. They enable one to assess the robustness of a classifier
without running extensive tests, so they can be used to compare different models and quickly select
the one with highest robustness. Furthermore, the bounds enjoy a theoretical guarantee that no
adversarial perturbation could ever be smaller, so methods which increase these bounds may make
classifiers more robust. We tested the validity of our bounds on MNIST and CIFAR-10 and found no
violations. Comparisons with adversarial perturbations generated using the fast gradient sign method
suggest that these bounds can be close to the actual norms in the worst case.
We have only derived lower bounds for feedforward networks consisting of fully-connected layers,
convolutional layers and pooling layers. Extending this analysis to recurrent networks and other types
of layers such as Batch Normalization [Ioffe and Szegedy, 2015] and Local Response Normalization
[Krizhevsky et al., 2012] is an obvious avenue for future work.
It would also be interesting to quantify just how tight the above bounds really are. In the absence
of a precise characterization of adversarial examples, the only way to do this would be to generate
adversarial perturbations using optimization techniques that make no assumptions on their underlying
cause. Szegedy et al. [2014] use a box-constrained L-BFGS approach to generate adversarial examples
without any assumptions, so using this method for comparison could provide a more accurate picture
of how tight the theoretical bounds are. It is much less efficient than the FGS method, however.
The analysis presented here is a ?modular? one: we consider each layer in isolation, and derive bounds
on their robustness in terms of the robustness of the next layer. However, it may also be insightful to
study the relationship between the number of layers, the breadth of each layer and the robustness of
the network. Providing estimates on the approximation errors incurred by this layer-wise approach
could also be useful.
8
Finally, there is currently no known precise characterization of the trade-off between classifier robustness and accuracy. Intuitively, one might expect that as the robustness of the classifier increases, its
accuracy will also increase up to a point since it is becoming more robust to adversarial perturbations.
Once the robustness exceeds a certain threshold, however, we expect the accuracy to drop because the
decision surfaces are becoming too flat and the classifier becomes too insensitive to changes. Having
a precise characterization of this relationship between robustness and accuracy may aid methods
designed to protect classifiers against adversarial examples while also maintaining state-of-the-art
accuracy.
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phenomenon:1 ex:1 |
6,282 | 6,683 | Minimizing a Submodular Function from Samples
Eric Balkanski
Harvard University
[email protected]
Yaron Singer
Harvard University
[email protected]
Abstract
In this paper we consider the problem of minimizing a submodular function from
training data. Submodular functions can be efficiently minimized and are consequently heavily applied in machine learning. There are many cases, however, in
which we do not know the function we aim to optimize, but rather have access
to training data that is used to learn it. In this paper we consider the question of
whether submodular functions can be minimized when given access to its training
data. We show that even learnable submodular functions cannot be minimized
within any non-trivial approximation when given access to polynomially-many samples. Specifically, we show that there is a class of submodular functions with range
in [0, 1] such that, despite being PAC-learnable and minimizable in polynomial-time,
no algorithm can obtain an approximation strictly better than 1/2 o(1) using
polynomially-many samples drawn from any distribution. Furthermore, we show
that this bound is tight via a trivial algorithm that obtains an approximation of 1/2.
1
Introduction
For well over a decade now, submodular minimization has been heavily studied in machine learning
(e.g. [SK10, JB11, JLB11, NB12, EN15, DTK16]). This focus can be largely attributed to the fact
that if a set function f : 2N ! R is submodular, meaning it has the following property of diminishing
returns: f (S [ {a}) f (S)
f (T [ {a}) f (T ) for all S ? T ? N and a 62 T , then it can
be optimized efficiently: its minimizer can be found in time that is polynomial in the size of the
ground set N [GLS81, IFF01]. In many cases, however, we do not know the submodular function,
and instead learn it from data (e.g. [BH11, IJB13, FKV13, FK14, Bal15, BVW16]). The question
we address in this paper is whether submodular functions can be (approximately) minimized when
the function is not known but can be learned from training data.
An intuitive approach for optimization from training data is to learn a surrogate function from training
data that predicts the behavior of the submodular function well, and then find the minimizer of the
surrogate learned and use that as a proxy for the true minimizer we seek. The problem however, is that
this approach does not generally guarantee that the resulting solution is close to the true minimum of
the function. One pitfall is that the surrogate may be non-submodular, and despite approximating the
true submodular function arbitrarily well, the surrogate can be intractable to minimize. Alternatively,
it may be that the surrogate is submodular, but its minimum is arbitrarily far from the minimum of
the true function we aim to optimize (see examples in Appendix A).
Since optimizing a surrogate function learned from data may generally result in poor approximations,
one may seek learning algorithms that are guaranteed to produce surrogates whose optima wellapproximate the true optima and are tractable to compute. More generally, however, it is possible that
there is some other approach for optimizing the function from the training samples, without learning
a model. Therefore, at a high level, the question is whether a reasonable number of training samples
suffices to minimize a submodular function. We can formalize this as optimization from samples.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Optimization from samples. We will say that a class of functions F = {f : 2N ! [0, 1]} is
?-optimizable from samples over distribution D if for every f 2 F and 2 (0, 1), when given
poly(|N |) i.i.d. samples {(Si , f (Si ))}m
over the
i=1 where Si ? D, with probability at least 1
samples one can construct an algorithm that returns a solution S ? N s.t,
f (S)
min f (T ) ? ?.
T ?N
This framework was recently introduced in [BRS17] for the problem of submodular maximization
where the standard notion of approximation is multiplicative. For submodular minimization, since
the optimum may have zero value, the suitable measure is that of additive approximations for [0, 1]bounded functions, and the goal is to obtain a solution which is a o(1) additive approximation to the
minimum (see e.g. [CLSW16, EN15, SK10]). The question is then:
Can submodular functions be minimized from samples?
Since submodular functions can be minimized in polynomial-time, it is tempting to conjecture that
when the function is learnable it also has desirable approximation guarantees from samples, especially
in light of positive results in related settings of submodular maximization:
? Constrained maximization. For functions that can be maximized in polynomial time under
a cardinality constraint, like modular and unit-demand functions, there are polynomial
time algorithms that obtain an arbitrarily good approximation using polynomially-many
samples [BRS16, BRS17]. For general monotone submodular functions which are NPhard to maximize under cardinality constraints, there is no algorithm that can obtain a
reasonable approximation from polynomially-many samples [BRS17]. For the problem of
unconstrained minimization, submodular functions can be optimized in polynomial time;
? Unconstrained maximization. For unconstrained maximization of general submodular
functions, the problem is NP-hard to maximize (e.g. MAX-CUT) and one seeks constant
factor approximations. For this problem, there is an extremely simple algorithm that uses
no queries and obtains a good approximation: choose elements uniformly at random with
probability 1/2 each. This algorithm achieves a constant factor approximation of 1/4 for
general submodular functions. For symmetric submodular functions (i.e. f (S) = f (N \ S)),
this algorithm is a 1/2-approximation which is optimal, since no algorithm can obtain an
approximation ratio strictly better than 1/2 using polynomially-many value queries, even
for symmetric submodular functions [FMV11]. For unconstrained symmetric submodular
minimization, there is an appealing analogue: the empty set and the ground set N are
guaranteed to be minimizers of the function (see Section 2). This algorithm, of course, uses
no queries either. The parallel between these two problems seems quite intuitive, and it
is tempting to conjecture that like for unconstrained submodular maximization, there are
optimization from samples algorithms for general unconstrained submodular minimization
with good approximation guarantees.
Main result. Somewhat counter-intuitively, we show that despite being computationally tractable
to optimize, submodular functions cannot be minimized from samples to within a desirable guarantee,
even when these functions are learnable. In particular, we show that there is no algorithm for
minimizing a submodular function from polynomially-many samples drawn from any distribution
that obtains an additive approximation of 1/2 o(1), even when the function is PAC-learnable.
Furthermore, we show that this bound is tight: the algorithm which returns the empty set or ground
set each with probability 1/2 achieves at least a 1/2 approximation. Notice that this also implies that
in general, there is no learning algorithm that can produce a surrogate whose minima is close to the
minima of the function we aim to optimize, as otherwise this would contradict our main result.
Technical overview. At a high level, hardness results in optimization from samples are shown
by constructing a family of functions, where the values of functions in the family are likely to be
indistinguishable for the samples, while having very different optimizers. The main technical difficulty
is to construct a family of functions that concurrently satisfy these two properties (indistinguishability
and different optimizers), and that are also PAC-learnable. En route to our main construction, we
first construct a family of functions that are completely indistinguishable given samples drawn from
the uniform distribution, in which case we obtain a 1/2 o(1) impossibility result (Section 2). The
2
general result that holds for any distribution requires heavier machinery to argue about more general
families of functions where some subset of functions can be distinguished from others given samples.
Instead of satisfying the two desired properties for all functions in a fixed family, we show that these
properties hold for all functions in a randomized subfamily (Section 3.2). We then develop an efficient
learning algorithm for the family of functions constructed for the main hardness result (Section 3.3).
This algorithm builds multiple linear regression predictors and a classifier to direct a fresh set to the
appropriate linear predictor. The learning of the classifier and the linear predictors relies on multiple
observations about the specific structure of this class of functions.
1.1
Related work
The problem of optimization from samples was introduced in the context of constrained submodular
maximization [BRS17, BRS16]. In general, for maximizing a submodular function under a cardinality
constraint, no algorithm can obtain a constant factor approximation guarantee from any samples. As
discussed above, for special classes of submodular functions that can be optimized in polynomial time
under a cardinality constraint, and for unconstrained maximization, there are desirable optimization
from samples guarantees. It is thus somewhat surprising that submodular minimization, which is an
unconstrained optimization problem that is optimizable in polynomial time in the value query model,
is hard to optimize from samples. From a technical perspective the constructions are quite different.
In maximization, the functions constructed in [BRS17, BRS16] are monotone so the ground set
would be an optimal solution if the problem was unconstrained. Instead, we need to construct novel
non-monotone functions. In convex optimization, recent work shows a tight 1/2-inapproximability
for convex minimization from samples [BS17]. Although there is a conceptual connection between
that paper and this one, from a technical perspective these papers are orthogonal. The discrete
analogue of the family of convex functions constructed in that paper is not (even approximately) a
family of submodular functions, and the constructions are significantly different.
2
Warm up: the Uniform Distribution
As a warm up to our main impossibility result, we sketch a tight lower bound for the special case in
which the samples are drawn from the uniform distribution. At a high level, the idea is to construct a
function which considers some special subset of ?good? elements that make its value drops when a
set contains all such ?good? elements. When samples are drawn from the uniform distribution and
?good? elements are sufficiently rare, there is a relatively simple construction that obfuscates which
elements the function considers ?good?, which then leads to the inapproximability.
2.1
Hardness for uniform distribution
p
We construct a family of functions F where fi 2 F is defined in terms of a set Gi ? N of size n.
For each such function we call Gi the set of good elements, and Bi = N \ Gi its bad elements. We
denote the number of good and bad elements in a set S by gS and bS , dropping the subscripts (S and
i) when clear from context, so g = |Gi \ S| and b = |Bi \ S|. The function fi is defined as follows:
8
p
1
1
>
< +
? (g + b) if g < n
fi (S) := 2 2n
p
>
: 1 ?b
if g = n
2n
It is easy to verify that these functions are submodular with range in [0, 1] (see illustration in
Figure 1a). Given samples drawn uniformly at random (u.a.r.),
pit is impossible to distinguish good
and bad elements since with high probability (w.h.p.) g < n for all samples. Informally, this
implies that a good learner for F over the uniform distribution D is f 0 (S) = 1/2 + |S|/(2n).
Intuitively, F is not 1/2 o(1) optimizable from samples because if an algorithm cannot learn the
set of good elements Gi , then it cannot find S such fi (S) < 1/2 o(1) whereas the optimal solution
Si? = Gi has value fi (Gi ) = 0.
Theorem 1. Submodular functions f : 2N ! [0, 1] are not 1/2 o(1) optimizable from samples
drawn from the uniform distribution for the problem of submodular minimization.
3
Proof. The details for the derivation of concentration bounds are in Appendix B. Consider fk drawn
u.a.r. from F and let f ? = fk and G? = Gk . Since the samples are all drawn from the uniform
distribution, by standard application of the Chernoff bound we have that every set Si in the sample
respects |Si | ? 3n/4, w.p. 1 e ?(n) . For sets S1 , . . . , Sm , all of size at most 3n/4, when fj is
p
1/2
drawn u.a.r. from F we get that |Si \ Gj | < n, w.p. 1 e ?(n ) for all i 2 [m], again by
1/2
Chernoff, and since m = poly(n). Notice that this implies that w.p. 1 e ?(n ) for all i 2 [m]:
1 |Si |
fj (Si ) = +
2
2n
Now, let F 0 be the collection of all functions fj for which fj (Si ) = 1/2 + |Si |/(2n) on all sets
1/2
0
{Si }m
e ?(n ) )|F|. Thus, since f ? is drawn u.a.r.
i=1 . The argument above implies that |F | = (1
1/2
from F we have that f ? 2 F 0 w.p. 1 e ?(n ) , and we condition on this event.
Let S be the (possibly randomized) solution returned by the algorithm. Observe that S is independent
of f ? 2 F 0 . In other words, the algorithm cannot learn any information about which function in F 0
1/6
generates the samples. By Chernoff, if we fix S and choose f u.a.r. from F, then, w.p. 1 e ?(n ) :
1
f (S)
o(1).
2
0
1/2
1/6
|
Since |F
e ?(n ) , it is also the case that f ? (S) 1/2 o(1) w.p. 1 e ?(n ) over the
|F | = 1
choice of f ? 2 F 0 . By the probabilistic method and since all the events we conditioned on occur
with exponentially high probability, there exists f ? 2 F s.t. the value of the set S returned by the
algorithm is 1/2 o(1) whereas the optimal solution is f ? (G? ) = 0.
2.2
A tight upper bound
We now show that the result above is tight. In particular, by randomizing between the empty set and
the ground set we get a solution whose value is at most 1/2. In the case of symmetric functions, i.e.
f (S) = f (N \ S) for all S ? N , ; and N are minima since f (N ) + f (;) ? f (S) + f (N \ S) for
all S ? N as shown below.1 Notice, that this does not require any samples.
Proposition 2. The algorithm which returns the empty set ; or the ground N with probability 1/2
each is a 1/2 additive approximation for the problem of unconstrained submodular minimization.
Proof. Let S ? N , observe that
f (N \ S) f (;) f ((N \ S) [ S) f (S) = f (N )
where the inequality is by submodularity. Thus, we obtain
1
1
1
(f (N ) + f (;)) ? f (S) + f (N \ S) ? f (S) +
2
2
2
In particular, this holds for S 2 argminT ?N f (T ).
3
f (S)
1
.
2
General Distribution
In this section, we show our main result, namely that there exists a family of submodular functions
such that, despite being PAC-learnable for all distributions, no algorithm can obtain an approximation
better than 1/2 o(1) for the problem of unconstrained minimization.
The functions in this section build upon the previous construction, though are inevitably more involved
in order to achieve learnability and inapproximability on any distribution. The functions constructed
for the uniform distribution do not yield inapproximability for general distributions due to the fact that
the indistinguishability between two functions no longer holds when sets S of large size are sampled
with non-negligible probability. Intuitively, in the previous construction, once a set is sufficiently
large the good elements of the function can be distinguished from the bad ones. The main idea to get
around this issue is to introduce masking elements M . We construct functions such that, for sets S of
large size, good and bad elements are indistinguishable if S contains at least one masking element.
1
Although ; and N are trivial minima if f is symmetric, the problem of minimizing a symmetric submodular
function over proper nonempty subsets is non-trivial (see [Que98]).
4
1
f(S)
f(S)
1
0.5
0.5
m = |S|
b = |S|
g = |S|
b = |S|-1, m=1
g = |S|-1, m=1
b = |S|
g = |S|
0
0 ?n - 1 ?n
0
n ? ?n
|S|
(a) Uniform distribution
0
1 n1/4
?n
|S|
n/2
(b) General distribution
Figure 1: An illustration of the value of a set S of good (blue), bad (red), and masking (green)
elements as a function of |S| for the functions constructed. For the general distribution case, we
also illustrate the value of a set S of good (dark blue) and bad (dark red) elements when S also
contains at least one masking element.
The construction. Each function fi 2 F is defined in terms of a partition Pi of the ground set
into good, bad, andpmasking elements. The
p partitions we consider are Pi = (Gi , Bi , Mi ) with
|Gi | = n/2, |Bi | = n, and |Mi | = n/2
n. Again, when clear from context, we drop indices of
i and S and the number of good, bad, and masking elements in a set S are denoted by g, b, and m.
For such a given partition Pi , the function fi is defined as follows (see illustration in Figure 1b):
8
1
>
>
p ? (b + g)
Region X : if m = 0 and g < n1/4
>
>2 n
>
>
?
? 1 ?
?
1 < 1
fi (S) = +
p ? b + n1/4
? g n1/4
Region Y : if m = 0 and g n1/4
2 >
n
2
n
>
>
>
>
1
>
:1
? (b + g)
Region Z : otherwise
2 n
3.1
Submodularity
In the appendix, we prove that the functions fi constructed pas above are indeed submodular
(Lemma 10). By rescaling fi with an additive term of n1/4 /(2 n) = 1/(2n1/4 ), it can be easily
verified that its range is in [0, 1]. We use the non-normalized definition as above for ease of notation.
3.2
Inapproximability
We now show that F cannot be minimized within a 1/2 o(1) approximation given samples from
any distribution. We first define F M , which is a randomized subfamily of F. We then give a general
lemma that shows that if two conditions of indistinguishability and gap are satisfied then we obtain
inapproximability. We then show that these two conditions are satisfied for the subfamily F M .
A randomization over masking elements. Instead of considering a function f drawn u.a.r. from
F as in the uniform case, we consider functions f in a randomized subfamily of functions F M ? F
to obtain the indistinguishability and gap
the family of functions F, let M be a
p conditions. Given
uniformly random subset of size n/2
n and define F M ? F:
F M := {fi 2 F : (Gi , Bi , M )}.
Since masking elements are distinguishable from good and bad elements, they need to be the same
set of elements for each function in family F M to obtain indistinguishability of functions in F M .
The inapproximability lemma. In addition to this randomized subfamily of functions, another
main conceptual departure of the following inapproximability lemma from the uniform case is that
no assumption can be made about the samples, such as their size, since the distribution is arbitrary.
We denote by U (A) the uniform distribution over the set A.
5
Lemma 3. Let F be a family of functions and F 0 = {f1 , . . . , fp } ? F be a subfamily of functions
drawn from some distribution. Assume the following two conditions hold:
1/4
1. Indistinguishability. For all S ? N , w.p. 1 e ?(n
fi (S) = fj (S);
)
over F 0 : for every fi , fj 2 F 0 ,
2. ?-gap. Let Si? be a minimizer of fi , then w.p. 1 over F 0 : for all S ? N ,
E 0 [fi (S) fi (Si? )] ?;
fi ?U (F )
Then, F is not ?-minimizable from strictly less than e?(n
1/4
)
samples over any distribution D.
Note that the ordering of the quantifiers is crucial. The proof is deferred to the appendix, but the main
ideas are summarized as follows. We use a probabilistic argument to switch from the randomization
over F 0 to the randomization over S ? D and show that there exists a deterministic F ? F such that
fi (S) = fj (S) for all fi , fj 2 F w.h.p. over S ? D. By a union bound this holds for all samples
S. Thus, for such a family of functions F = {f1 , . . . , fp }, the choices of an algorithm that is given
samples from fi for i 2 [p] are independent of i. By the ?-gap condition, this implies that there exists
fi 2 F for which a solution S returned by the algorithm is at least ? away from fi (Si? ).
Indistinguishability and gap of F. We now show the indistinguishability and gap conditions, with
? = 1/2 o(1), which immediately imply a 1/2 o(1) inapproximability by Lemma 3. For the
indistinguishability, it suffices to show that good and bad elements are indistinguishable since the
masking elements are identical for all functions in F M . Good and bad elements are indistinguishable
since, w.h.p., a set S is not in region Y, which is the only region distinguishing good and bad elements.
Lemma 4. For all S ? N s.t. |S| < n1/4 : For all fi 2 F M ,
(
1
p
? (b + g)
if m = 0 (Region X )
1
fi (S) = + 21 n 1
2
otherwise (Region Z)
2
n ? (b + g)
1/4
and for all S ? N such that |S|
n1/4 , with probability 1 e ?(n ) over F M : For all fi 2 F M ,
1
fi (S) = 1
? (b + g)
(Region Z).
n
Proof. Let S ? N . If |S| < n1/4 , then the proof follows immediately from the definition of fi . If
|S| n1/4 , then, the number of masking elements m in S is m = |M \ S| for all fi 2 F M . We
1/4
then get m 1, for all fi 2 F M , with probability 1 e ?(n ) over F M by Chernoff bound. The
proof then follows again immediately from the definition of fi .
Next, we show the gap. The gap is since the good elements can be any subset of N \ M .
Lemma 5. Let Si? be a minimizer of fi . With probability 1 over F M , for all S ? N ,
1
E
[fi (S)]
o(1).
M
2
fi ?U (F )
Proof. Let S ? N and fi ? U (F M ). Note that the order of the quantifiers in the statement of the
lemma implies that S can be dependent on M , but that it is independent of i. There are three cases. If
m 1, then S is in region Z and fi (S) 1/2. If m = 0 and |S| ? n7/8 , then S is in region X or Y
and fi (S) 1/2 n7/8 /n = 12 o(1). Otherwise, m = 0 and |S| n7/8 . Since S is independent
of i, by Chernoff bound, we get
p
p
n/2 + n
n/2 + n
p
(1 o(1)) ? |S| ?
? b and
? g ? (1 + o(1)) ? |S|
n/2
n
with probability 1
1
fi (S)
+ (1
2
?(n1/4 )
. Thus S is in region Y and
p
1
n
1
n/2
p ? |S| (1 + o(1)) ?
p ? |S|
o(1)) p ?
n n/2 + n
2 n n/2 + n
e
Thus, we obtain Efi ?U (F M ) [fi (S)]
1
2
o(1).
6
1
2
o(1).
Combining the above three lemmas, we obtain the inapproximability result.
Lemma 6. The problem of submodular minimization cannot be approximated with a 1/2
additive approximation given poly(n) samples from any distribution D.
o(1)
Proof. For any set S ? N , observe that the number g + b of elements in S that are either good or
bad is the same for any two functions fi , fj 2 F M and for any F M . Thus, by Lemma 4, we obtain
the indistinguishability condition. Next, the optimal solution Si? = Gi of fi has value fi (Gi ) = o(1),
so by Lemma 5, we obtain the ?-gap condition with ? = 1/2 o(1). Thus F is not 1/2 o(1)
minimizable from samples from any distribution D by Lemma 3. The class of functions F is a class
of submodular functions by Lemma 10 (in Appendix C).
3.3
Learnability of F
We now show that every function in F is efficiently learnable from samples drawn from any distribution D. Specifically, we show that for any ?, 2 (0, 1) the functions are (?, ) PAC learnable
with the absolute loss function (or any Lipschitz loss function) using poly(1/?, 1/ , n) samples and
running time. At a high level, since each function fi is piecewise-linear over three different regions
Xi , Yi , and Zi , the main idea is to exploit this structure by first training a classifier to distinguish
between regions and then apply linear regression in different regions.
The learning algorithm. Since every function f 2 F is piecewise linear over three different
regions, there are three different linear functions fX , fY , fZ s.t. for every S ? N its value f (S)
can be expressed as fR (S) for some region R 2 {X , Y, Z}. The learning algorithm produces a
predictor f? by using a multi-label classifier and a set of linear predictors {fX? , fY? } [ {[i2M? fZ?i }.
? [ {[ ? Z?i },
The multi-label classifier creates a mapping from sets to regions, g : 2N ! {X? , Y}
i2M
?
?
?
s.t. X , Y, Z are approximated by X , Y, [i2M? Zi . Given a sample S ? D, using the algorithm then
retuns f?(S) = fg(S) (S). We give a formal description below (detailed description is in Appendix D).
Algorithm 1 A learning algorithm for f 2 F which combines classification and linear regression.
Input: samples S = {(Sj , f (Sj ))}j2[m]
? M
?)
(Z,
(;, ;)
for i = 1 to n do
?
Z?i
{S : ai 2 S, S 62 Z}
reg
fZ?i
ERM ({(Sj , f (Sj )) : Sj 2 Z?i })
linear regression
P
if (Sj ,f (Sj )):Sj 2Z?i |fZ?i (Sj ) f (Sj )| = 0 then
?
? [ {ai }
Z?
Z? [ Z?i , M
M
cla
? j ? m/2})
C
ERM ({(Sj , f (Sj )) : Sj 62 Z,
train a classifier for regions X , Y
?
? C(S) = 1}, {S : S 62 Z,
? C(S) = 1})
(X? , Y)
({S : S 62 Z,
8
p
if S2X?
<|S|/(2 n)
reg
?
?
return f
S 7! fY? (S) = ERM ({(Sj , f (Sj )) : Sj 2 Y, j > m/2}) if S2Y?
:
? })
?
fZ?i (S) : i = min({i0 : ai0 2 S \ M
if S2Z
Overview of analysis of the learning algorithm. There are two main challenges in training the
algorithm. The first is that the region X , Y, or Z that a sample (Sj , f (Sj )) belongs to is not known.
? Z? using the samples,
Thus, even before being able to train a classifier which learns the regions X? , Y,
we need to learn the region a sample Sj belongs to using f (Sj ). The second is that the samples SR
used for training a linear regression predictor fR over region R need to be carefully selected so that
SR is a collection of i.i.d. samples from the distribution S ? D conditioned on S 2 R (Lemma 20).
We first discuss the challenge of labeling samples with the region they belong to. Observe that for a
fixed masking element ai 2 M , f 2 F is linear over all sets S containing ai since these sets are all in
region Z. Thus, there must exist a linear regression predictor fZ?i = ERMreg (?) with zero empirical
loss over all samples Sj containing ai if ai 2 M (and thus Sj 2 Z). ERMreg (?) minimizes the
empirical loss on the input samples over the class of linear regression predictors with bounded norm
7
Figure 2: An illustration of the regions. The dots represent the samples, the corresponding full circles represent
? Z? learned by the
the regions X (red), Y (blue), and Z (green). The ellipsoids represent the regions X? , Y,
?
classifier. Notice that Z has no false negatives.
(Lemma 19). If fZ?i has zero empirical loss, we directly classify any set S containing ai as being in
? Next, for a sample (Sj , f (Sj )) not in Z,
? we can label these samples since Sj 2 X if and only if
Z.
p
f (Sj ) = |Sj |/(2 n). With these labeled samples S 0 , we train a binary classifier C = ERMcla (S 0 )
? ERMcla (S 0 ) minimizes the empirical loss on
that indicates if S s.t. S 62 Z? is in region X? or Y.
0
labeled samples S over the class of halfspaces w 2 Rn (Lemma 23).
Regarding the second challenge, we cannot use all samples Sj s.t. Sj 2 Y? to train a linear predictor
? so they are not a collection of i.i.d.
fY? for region Y? since these same samples were used to define Y,
?
samples from the distribution S ? D conditioned on S 2 Y. To get around this issue, we partition
the samples into two distinct collections, one to train the classifier C and one to train fY? (Lemma 24).
? we predict f ? (T ) where i is s.t. ai 2 T \ M
? (breaking ties lexicographically)
Next, given T 2 Z,
Zi
?
? (Lemma 22). Since we break
which performs well since fZ?i has zero empirical error for ai 2 M
ties lexicographically, f?Z?i must be trained over samples Sj such that ai 2 Sj and ai0 62 Sj for i0 s.t.
? to obtain i.i.d. samples from the same distribution as T ? D conditioned on T
i0 < i and ai0 2 M
being directed to f?Z?i (Lemma 21).
The analysis of the learning algorithm leads to the following main learning result.
Lemma 7. Let f? be the predictor returned by Algorithm 1, then w.p. 1
over m 2 O(n3 +
2
2
n (log(2n/ ))/? ) samples S drawn i.i.d. from any distribution D, ES?D [|f?(S) f (S)|] ? ?.
3.4
Main Result
We conclude this section with our main result which combines Lemmas 6 and 7.
Theorem 8. There exists a family of [0, 1]-bounded submodular functions F that is efficiently PAClearnable and that cannot be optimized from polynomially many samples drawn from any distribution
D within a 1/2 o(1) additive approximation for unconstrained submodular minimization.
4
Discussion
In this paper, we studied the problem of submodular minimization from samples. Our main result
is an impossibility, showing that even for learnable submodular functions it is impossible to find
a non-trivial approximation to the minimizer with polynomially-many samples, drawn from any
distribution. In particular, this implies that minimizing a general submodular function learned from
data cannot yield desirable guarantees. In general, it seems that the intersection between learning and
optimization is elusive, and a great deal still remains to be explored.
8
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9
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6,283 | 6,684 | Introspective Classification with Convolutional Nets
Long Jin
UC San Diego
[email protected]
Justin Lazarow
UC San Diego
[email protected]
Zhuowen Tu
UC San Diego
[email protected]
Abstract
We propose introspective convolutional networks (ICN) that emphasize the importance of having convolutional neural networks empowered with generative
capabilities. We employ a reclassification-by-synthesis algorithm to perform training using a formulation stemmed from the Bayes theory. Our ICN tries to iteratively:
(1) synthesize pseudo-negative samples; and (2) enhance itself by improving the
classification. The single CNN classifier learned is at the same time generative
? being able to directly synthesize new samples within its own discriminative
model. We conduct experiments on benchmark datasets including MNIST, CIFAR10, and SVHN using state-of-the-art CNN architectures, and observe improved
classification results.
1
Introduction
Great success has been achieved in obtaining powerful discriminative classifiers via supervised
training, such as decision trees [34], support vector machines [42], neural networks [23], boosting
[7], and random forests [2]. However, recent studies reveal that even modern classifiers like deep
convolutional neural networks [20] still make mistakes that look absurd to humans [11]. A common
way to improve the classification performance is by using more data, in particular ?hard examples?,
to train the classifier. Different types of approaches have been proposed in the past including
bootstrapping [31], active learning [37], semi-supervised learning [51], and data augmentation [20].
However, the approaches above utilize data samples that are either already present in the given
training set, or additionally created by humans or separate algorithms.
In this paper, we focus on improving convolutional neural networks by endowing them with synthesis
capabilities to make them internally generative. In the past, attempts have been made to build
connections between generative models and discriminative classifiers [8, 27, 41, 15]. In [44], a
self supervised boosting algorithm was proposed to train a boosting algorithm by sequentially
learning weak classifiers using the given data and self-generated negative samples; the generative via
discriminative learning work in [40] generalizes the concept that unsupervised generative modeling
can be accomplished by learning a sequence of discriminative classifiers via self-generated pseudonegatives. Inspired by [44, 40] in which self-generated samples are utilized, as well as recent success
in deep learning [20, 9], we propose here an introspective convolutional network (ICN) classifier and
study how its internal generative aspect can benefit CNN?s discriminative classification task. There is
a recent line of work using a discriminator to help with an external generator, generative adversarial
networks (GAN) [10], which is different from our objective here. We aim at building a single CNN
model that is simultaneously discriminative and generative.
The introspective convolutional networks (ICN) being introduced here have a number of properties. (1)
We introduce introspection to convolutional neural networks and show its significance in supervised
classification. (2) A reclassification-by-synthesis algorithm is devised to train ICN by iteratively
augmenting the negative samples and updating the classifier. (3) A stochastic gradient descent
sampling process is adopted to perform efficient synthesis for ICN. (4) We propose a supervised
formulation to directly train a multi-class ICN classifier. We show consistent improvement over
state-of-the-art CNN classifiers (ResNet [12]) on benchmark datasets in the experiments.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
2
Related work
Our ICN method is directly related to the generative via discriminative learning framework [40].
It also has connection to the self-supervised learning method [44], which is focused on density
estimation by combining weak classifiers. Previous algorithms connecting generative modeling
with discriminative classification [8, 27, 41, 15] fall in the category of hybrid models that are direct
combinations of the two. Some existing works on introspective learning [22, 3, 38] have a different
scope to the problem being tackled here. Other generative modeling schemes such as MiniMax
entropy [50], inducing features [6], auto-encoder [1], and recent CNN-based generative modeling
approaches [48, 47] are not for discriminative classification and they do not have a single model that
is both generative and discriminative. Below we discuss the two methods most related to ICN, namely
generative via discriminative learning (GDL) [40] and generative adversarial networks (GAN) [10].
Relationship with generative via discriminative learning (GDL) [40]
ICN is largely inspired by GDL and it follows a similar pipeline developed in [40]. However, there is
also a large improvement of ICN to GDL, which is summarized below.
? CNN vs. Boosting. ICN builds on top of convolutional neural networks (CNN) by explicitly
revealing the introspectiveness of CNN whereas GDL adopts the boosting algorithm [7].
? Supervised classification vs. unsupervised modeling. ICN focuses on the supervised classification task with competitive results on benchmark datasets whereas GDL was originally applied
to generative modeling and its power for the classification task itself was not addressed.
? SGD sampling vs. Gibbs sampling. ICN carries efficient SGD sampling for synthesis through
backpropagation which is much more efficient than the Gibbs sampling strategy used in GDL.
? Single CNN vs. Cascade of classifiers. ICN maintains a single CNN classifier whereas GDL
consists of a sequence of boosting classifiers.
? Automatic feature learning vs. manually specified features. ICN has greater representational
power due to the end-to-end training of CNN whereas GDL relies on manually designed features.
Comparison with Generative Adversarial Networks (GANs) [10]
Recent efforts in adversarial learning [10] are also very interesting and worth comparing with.
? Introspective vs. adversarial. ICN emphasizes being introspective by synthesizing samples
from its own classifier while GAN focuses on adversarial ? using a distinct discriminator to
guide the generator.
? Supervised classification vs. unsupervised modeling. The main focus of ICN is to develop a
classifier with introspection to improve the supervised classification task whereas GAN is mostly
for building high-quality generative models under unsupervised learning.
? Single model vs. two separate models. ICN retains a CNN discriminator that is itself a generator
whereas GAN maintains two models, a generator and a discriminator, with the discriminator in
GAN trained to classify between ?real? (given) and ?fake? (generated by the generator) samples.
? Reclassification-by-synthesis vs. minimax. ICN engages an iterative procedure, reclassificationby-synthesis, stemmed from the Bayes theory whereas GAN has a minimax objective function to
optimize. Training an ICN classifier is the same as that for the standard CNN.
? Multi-class formulation. In a GAN-family work [36], a semi-supervised learning task is devised
by adding an additional ?not-real? class to the standard k classes in multi-class classification; this
results in a different setting to the standard multi-class classification with additional model parameters. ICN instead, aims directly at the supervised multi-class classification task by maintaining
the same parameter setting within the softmax function without additional model parameters.
Later developments alongside GAN [35, 36, 49, 3] share some similar aspects to GAN, which
also do not achieve the same goal as ICN does. Since the discriminator in GAN is not meant to
perform the generic two-class/multi-class classification task, some special settings for semi-supervised
learning [10, 35, 49, 3, 36] were created. ICN instead has a single model that is both generative and
discriminative, and thus, an improvement to ICN?s generator leads to a direct means to ameliorate its
discriminator. Other work like [11] was motivated from an observation that adding small perturbations
to an image leads to classification errors that are absurd to humans; their approach is however taken
by augmenting positive samples from existing input whereas ICN is able to synthesize new samples
from scratch. A recent work proposed in [21] is in the same family of ICN, but [21] focuses on
unsupervised image modeling using a cascade of CNNs.
2
3
Method
The pipeline of ICN is shown in Figure 1, which has an immediate improvement over GDL [40]
in several aspects that have been described in the previous section. One particular gain of ICN is
its representation power and efficient sampling process through backpropagation as a variational
sampling strategy.
3.1 Formulation
We start the discussion by introducing the basic formulation and borrow the notation from [40].
Let x be a data sample (vector) and y ? {?1, +1} be its label, indicating either a negative or
a positive sample (in multi-class classification y ? {1, ..., K}). We study binary classification
first. A discriminative classifier computes p(y|x), the probability of x being positive or negative.
p(y = ?1|x) + p(y = +1|x) = 1. A generative model instead models p(y, x) = p(x|y)p(y), which
captures the underlying generation process of x for class y. In binary classification, positive samples
are of primary interest. Under the Bayes rule:
p(y = +1|x)p(y = ?1)
p(x|y = +1) =
p(x|y = ?1),
(1)
p(y = ?1|x)p(y = +1)
which can be further simplified when assuming equal priors p(y = +1) = p(y = ?1):
p(x|y = +1) =
p(y = +1|x)
p(x|y = ?1).
1 ? p(y = +1|x)
(2)
Reclassification Step:
training on the given training data + generated pseudo-negatives
Initial Classification
(given training data)
Convolutional Neural Networks
Classification
Final Classification
(given training data +
self-generated
pseudo-negatives)
Introspective Convolutional Networks:
?
?
Synthesis
Reclassification
Synthesis
Synthesis Step:
synthesize pseudo-negative samples
Figure 1: Schematic illustration of our reclassification-by-synthesis algorithm for ICN training. The top-left
figure shows the input training samples where the circles in red are positive samples and the crosses in blue are
the negatives. The bottom figures are the samples progressively self-generated by the classifier in the synthesis
steps and the top figures show the decision boundaries (in purple) progressively updated in the reclassification
steps. Pseudo-negatives (purple crosses) are gradually generated and help tighten the decision boundaries.
We make two interesting and important observations from Eqn. (2): 1) p(x|y = +1) is dependent
on the faithfulness of p(x|y = ?1), and 2) a classifier C to report p(y = +1|x) can be made
simultaneously generative and discriminative. However, there is a requirement: having an informative distribution for the negatives p(x|y = ?1) such that samples drawn x ? p(x|y = ?1)
3
have good coverage to the entire space of x ? Rm , especially for samples that are close to the
positives x ? p(x|y = +1), to allow the classifier to faithfully learn p(y = +1|x). There seems
to exist a dilemma. In supervised learning, we are only given a set of limited amount of training
data, and a classifier C is only focused on the decision boundary to separate the given samples and
the classification on the unseen data may not be accurate. This can be seen from the top left plot
in Figure 1. This motivates us to implement the synthesis part within learning ? make a learned
discriminative classifier generate samples that pass its own classification and see how different these
generated samples are to the given positive samples. This allows us to attain a single model that has
two aspects at the same time: a generative model for the positive samples and an improved classifier
for the classification.
Suppose we are given a training set S = {(xi , yi ), i = 1..n} and x ? Rm and y ? {?1, +1}. One
can directly train a discriminative classifier C, e.g. a convolutional neural networks [23] to learn
p(y = +1|x), which is always an approximation due to various reasons including insufficient training
samples, generalization error, and classifier limitations. Previous attempts to improve classification
by data augmentation were mostly done to add more positive samples [20, 11]; we instead argue
the importance of adding more negative samples to improve the classification performance. The
dilemma is that S = {(xi , yi ), i = 1..n} is limited to the given data. For clarity, we now use p? (x)
to represent p(x|y = ?1). Our goal is to augment the negative training set by generating confusing
pseudo-negatives to improve the classification (note that in the end pseudo-negative samples drawn
x ? p?
t (x) will become hard to distinguish from the given positive samples. Cross-validation
can be used to determine when using more pseudo-negatives is not reducing the validation error).
We call the samples drawn from x ? p?
t (x) pseudo-negatives (defined in [40]). We expand
t
0
S = {(xi , yi ), i = 1..n} by Set = S ? Spn
, where Spn
= ? and for t ? 1
t
Spn = {(xi , ?1), i = n + 1, ..., n + tl}.
t
Spn
includes all the pseudo-negative samples self-generated from our model up to time t. l indicates
the number of pseudo-negatives generated at each round. We define a reference distribution p?
r (x) =
U (x), where U (x) is a Gaussian distribution (e.g. N (0.0, 0.32 ) independently). We carry out
learning with t = 0...T to iteratively obtain qt (y = +1|x) and qt (y = ?1|x) by updating classifier
t
C t on Set = S ? Spn
. The initial classifier C 0 on Se0 = S reports discriminative probability
q0 (y = +1|x). The reason for using q is because it is an approximation to the true p due to limited
samples drawn in Rm . At each time t, we then compute
1 qt (y = +1|x) ?
p?
p (x),
(3)
t (x) =
Zt qt (y = ?1|x) r
R (y=+1|x) ?
where Zt = qqtt (y=?1|x)
pr (x)dx. Draw new samples xi ? p?
t (x) to expand the pseudo-negative
set:
t+1
t
Spn
= Spn
? {(xi , ?1), i = n + tl + 1, ..., n + (t + 1)l}.
(4)
We name the specific training algorithm for our introspective convolutional network (ICN) classifier
reclassification-by-synthesis, which is described in Algorithm 1. We adopt convolutional neural
networks (CNN) classifier to build an end-to-end learning framework with an efficient sampling
process (to be discussed in the next section).
3.2 Reclassification-by-synthesis
We present our reclassification-by-synthesis algorithm for ICN in this section. A schematic illustration
is shown in Figure 1. A single CNN classifier is being trained progressively which is simultaneously a
discriminator and a generator. With the pseudo-negatives being gradually generated, the classification
boundary gets tightened, and hence yields an improvement to the classifier?s performance. The
reclassification-by-synthesis method is described in Algorithm 1. The key to the algorithm includes
two steps: (1) reclassification-step, and (2) synthesis-step, which will be discussed in detail below.
3.2.1 Reclassification-step
t
The reclassification-step can be viewed as training a normal classifier on the training set Set = S ?Spn
0
t
where S = {(xi , yi ), i = 1..n} and Spn = ?. Spn = {(xi , ?1), i = n + 1, ..., n + tl} for t ? 1. We
use CNN as our base classifier. When training a classifier C t on Set , we denote the parameters to be
(0)
(1)
learned in C t by a high-dimensional vector Wt = (wt , wt ) which might consist of millions of
(1)
(0)
(0)
parameters. wt denotes the weights of the top layer combining the features ?(x; wt ) and wt
4
carries all the internal representations. Without loss of generality, we assume a sigmoid function for
the discriminative probability
(1)
qt (y|x; Wt ) = 1/(1 + exp{?ywt
(0)
? ?(x; wt )}),
(0)
(1)
(0)
where ?(x; wt ) defines the feature extraction function for x. Both wt and wt can be learned by
the standard stochastic gradient descent algorithm via backpropagation to minimize a cross-entropy
loss with an additional term on the pseudo-negatives:
L(Wt ) = ?
i=1..n
X
ln qt (yi |xi ; Wt ) ?
i=n+1..n+tl
X
ln qt (?1|xi ; Wt ).
(5)
t
(xi ,?1)?Spn
(xi ,yi )?S
Algorithm 1 Outline of the reclassification-by-synthesis algorithm for discriminative classifier
training.
Input: Given a set of training data S = {(xi , yi ), i = 1..n} with x ? Rm and y ? {?1, +1}.
0
Initialization: Obtain a reference distribution: p?
r (x) = U (x) and train an initial CNN binary classifier C
0
on S, q0 (y = +1|x). Spn
= ?. U (x) is a zero mean Gaussian distribution.
For t=0..T
1 qt (y=+1|x) ?
1. Update the model: p?
t (x) = Zt qt (y=?1|x) pr (x).
2. Synthesis-step: sample l pseudo-negative samples xi ? p?
t (x), i = n + tl + 1, ..., n + (t + 1)l from the
current model p?
t (x) using an SGD sampling procedure.
t+1
t
3. Augment the pseudo-negative set with Spn
= Spn
? {(xi , ?1), i = n + tl + 1, ..., n + (t + 1)l}.
t+1
t+1
on Set+1 = S ? Spn
, resulting in qt+1 (y = +1|x).
4. Reclassification-step: Update CNN classifier to C
5. t ? t + 1 and go back to step 1 until convergence (e.g. no improvement on the validation set).
End
3.2.2 Synthesis-step
In the reclassification step, we obtain qt (y|x; Wt ) which is then used to update p?
t (x) according to
Eqn. (3):
1 qt (y = +1|x; Wt ) ?
p (x).
(6)
p?
t (x) =
Zt qt (y = ?1|x; Wt ) r
?
In the synthesis-step, our goal is to draw fair samples from pt (x) (fair samples refer to typical
samples by a sampling process after convergence w.r.t the target distribution). In [40], various Markov
chain Monte Carlo techniques [28] including Gibbs sampling and Iterated Conditional Modes (ICM)
have been adopted, which are often slow. Motivated by the DeepDream code [32] and Neural
qt (y=+1|x;Wt )
Artistic Style work [9], we update a random sample x drawn from p?
r (x) by increasing qt (y=?1|x;Wt )
using backpropagation. Note that the partition function (normalization) Zt is a constant that is not
dependent on the sample x. Let
gt (x) =
qt (y = +1|x; Wt )
(1)
(0)
= exp{wt ? ?(x; wt )},
qt (y = ?1|x; Wt )
(7)
and take its ln, which is nicely turned into the logit of qt (y = +1|x; Wt )
(1)
ln gt (x) = wt
(0)
? ?(x; wt ).
(1)T
(8)
(0)
Starting from x drawn from p?
?(x; wt ) using stochastic gradient
r (x), we directly increase wt
ascent on x via backpropagation, which allows us to obtain fair samples subject to Eqn. (6). Gaussian
noise can be added to Eqn. (8) along the line of stochastic gradient Langevin dynamics [43] as
(1)
(0)
?x = ?(wt ? ?(x; wt )) + ?
2
where ? ? N (0, ) is a Gaussian distribution and is the step size that is annealed in the sampling
process.
Sampling strategies. When conducting experiments, we carry out several strategies using stochastic
gradient descent algorithm (SGD) and SGD Lagenvin including: i) early-stopping for the sampling
process after x becomes positive (aligned with contrastive divergence [4] where a short Markov chain
is simulated); ii) stopping at a large confidence for x being positive, and iii) sampling for a fixed,
large number of steps. Table 2 shows the results on these different options and no major differences
in the classification performance are observed.
5
Building connections between SGD and MCMC is an active area in machine learning [43, 5, 30]. In
[43], combining SGD and additional Gaussian noise under annealed stepsize results in a simulation
of Langevin dynamics MCMC. A recent work [30] further shows the similarity between constant
SGD and MCMC, along with analysis of SGD using momentum updates. Our progressively learned
discriminative classifier can be viewed as carving out the feature space on ?(x), which essentially
becomes an equivalent class for the positives; the volume of the equivalent class that satisfies
the condition is exponentially large, as analyzed in [46]. The probability landscape of positives
(equivalent class) makes our SGD sampling process not particularly biased towards a small limited
modes. Results in Figure 2 illustrates that large variation of the sampled/synthesized examples.
3.3 Analysis
t=? +
The convergence of p?
t (x) ? p (x) can be derived (see the supplementary material), inspired by
?
+
+
the proof from [40]: KL[p (x)||p?
t+1 (x)] ? KL[p (x)||pt (x)] where KL denotes the Kullback+
Leibler divergence and p(x|y = +1) ? p (x), under the assumption that classifier at t + 1 improves
over t.
Remark. Here we pay particular attention to the negative samples which live in a space that is
often much larger than the positive sample space. For the negative training samples, we have
yi = ?1 and xi ? Q? (x), where Q? (x) is a distribution on the given negative examples in
the original training set. Our reclassification-by-synthesis algorithm (Algorithm 1) essentially
PT ?1
constructs a mixture model p?(x) ? T1 t=0 p?
t (x) by sequentially generating pseudo-negative
samples to augment our training set. Our new distribution for augmented negative sample set thus
n
Tl
?
becomes Q?
?(x), where p?(x) encodes pseudo-negative samples that
new (x) ? n+T l Q (x) + n+T l p
are confusing and similar to (but are not) the positives. In the end, adding pseudo-negatives might
degrade the classification result since they become more and more similar to the positives. Crossvalidation can be used to decide when adding more pseudo-negatives is not helping the classification
task. How to better use the pseudo-negative samples that are increasingly faithful to the positives
is an interesting topic worth further exploring. Our overall algorithm thus is capable of enhancing
classification by self-generating confusing samples to improve CNN?s robustness.
3.4 Multi-class classification
One-vs-all. In the above section, we discussed the binary classification case. When dealing with
multi-class classification problems, such as MNIST and CIFAR-10, we will need to adapt our
proposed reclassification-by-synthesis scheme to the multi-class case. This can be done directly using
a one-vs-all strategy by training a binary classifier Ci using the i-th class as the positive class and
then combine the rest classes into the negative class, resulting in a total of K binary classifiers. The
training procedure then becomes identical to the binary classification case. If we have K classes,
then the algorithm will train K individual binary classifiers with
(0)1
< (wt
The prediction function is simply
(1)1
, wt
(0)K
), ..., (wt
(1)k
f (x) = arg max exp{wt
k
(1)K
, wt
)>.
(0)k
? ?(x; wt
)}.
The advantage of using the one-vs-all strategy is that the algorithm can be made nearly identical to
the binary case at the price of training K different neural networks.
Softmax function. It is also desirable to build a single CNN classifier to perform multi-class
classification directly. Here we propose a formulation to train an end-to-end multiclass classifier
directly. Since we are directly dealing with K classes, the pseudo-negative data set will be slightly
0
different and we introduce negatives for each individual class by Spn
= ? and:
t
Spn
= {(xi , ?k), k = 1, ..., K, i = n + (t ? 1) ? k ? l + 1, ..., n + t ? k ? l}
Suppose we are given a training set S = {(xi , yi ), i = 1..n} and x ? Rm and y ? {1, .., K}. We
want to train a single CNN classifier with
(0)
(1)1
Wt =< wt , wt
(1)K
, ..., wt
>
(0)
wt
(1)
where
denotes the internal feature and parameters for the single CNN, and wt k denotes the
top-layer weights for the k-th class. We therefore minimize an integrated objective function
L(Wt )=?(1??)
Pn
i=1
ln
(1)y
i ??(x ;w(0) )}
P
exp{wt
i
t
+? n+t?K?l
PK
i=n+1
(1)
(0)
exp{wt k ??(xi ;wt )}
k=1
6
(1)|y |
i ??(x ;w0 )})
i
t
ln(1+exp{wt
(9)
The first term in Eqn. (9) encourages a softmax loss on the original training set S. The second term
in Eqn. (9) encourages a good prediction on the individual pseudo-negative class generated for the
(1)|y |
k-th class (indexed by |yi | for wt i , e.g. for pseudo-negative samples belong to the k-th class,
|yi | = | ? k| = k). ? is a hyperparameter balancing the two terms. Note that we only need to build
(0)
a single CNN sharing wt for all the K classes. In particular, we are not introducing additional
model parameters here and we perform a direct K-class classification where the parameter setting is
identical to a standard CNN multi-class classification task; to compare, an additional ?not-real? class
is created in [36] and the classification task there [36] thus becomes a K + 1 class classification.
4
Experiments
Figure 2: Synthesized pseudo-negatives for the MNIST dataset by our ICN classifier. The top row shows some
training examples. As t increases, our classifier gradually synthesize pseudo-negative samples that become
increasingly faithful to the training samples.
We conduct experiments on three standard benchmark datasets, including MNIST, CIFAR-10 and
SVHN. We use MNIST as a running example to illustrate our proposed framework using a shallow
CNN; we then show competitive results using a state-of-the-art CNN classifier, ResNet [12] on
MNIST, CIFAR-10 and SVHN. In our experiments, for the reclassification step, we use the SGD
optimizer with mini-batch size of 64 (MNIST) or 128 (CIFAR-10 and SVHN) and momentum equal
to 0.9; for the synthesis step, we use the Adam optimizer [17] with momentum term ?1 equal to 0.5.
All results are obtained by averaging multiple rounds.
Training and test time. In general, the training time for ICN is around double that of the baseline
CNNs in our experiments: 1.8 times for MNIST dataset, 2.1 times for CIFAR-10 dataset and 1.7
times for SVHN dataset. The added overhead in training is mostly determined by the number of
generated pseudo-negative samples. For the test time, ICN introduces no additional overhead to the
baseline CNNs.
4.1 MNIST
We use the standard MNIST [24] dataset, which
consists of 55, 000 training, 5, 000 validation
and 10, 000 test samples. We adopt a simple
network, containing 4 convolutional layers, each
having a 5 ? 5 filter size with 64, 128, 256 and
512 channels, respectively. These convolutional
layers have stride 2, and no pooling layers are
used. LeakyReLU activations [29] are used after
each convolutional layer. The last convolutional
layer is flattened and fed into a sigmoid output
(in the one-vs-all case).
Table 1: Test errors on the MNIST dataset. We compare
our ICN method with the baseline CNN, Deep Belief
Network (DBN) [14], and CNN w/ Label Smoothing
(LS) [39]. Moreover, the two-step experiments combining CNN + GDL [40] and combining CNN + DCGAN
[35] are also reported, and see descriptions in text for
more details.
Method
One-vs-all (%) Softmax (%)
DBN
1.11
CNN (baseline)
0.87
0.77
CNN w/ LS
0.69
CNN + GDL
0.85
CNN + DCGAN
0.84
In the reclassification step, we run SGD (for 5
ICN-noise (ours)
0.89
0.77
t
epochs) on the current training data Se , includICN (ours)
0.78
0.72
ing previously generated pseudo-negatives. Our
initial learning rate is 0.025 and is decreased by
a factor of 10 at t = 25. In the synthesis step, we use the backpropagation sampling process as
discussed in Section 3.2.2. In Table 2, we compare different sampling strategies. Each time we
synthesize a fixed number (200 in our experiments) of pseudo-negative samples.
We show some synthesized pseudo-negatives from the MNIST dataset in Figure 2. The samples in
the top row are from the original training dataset. ICN gradually synthesizes pseudo-negatives, which
are increasingly faithful to the original data. Pseudo-negative samples will be continuously used
while improving the classification result.
7
Comparison of different sampling Table 2: Comparison of different sampling strategies in the
strategies. We perform SGD and SGD synthesis step in ICN.
Langevin (with injected Gaussians), and
Sampling Strategy
One-vs-all (%) Softmax (%)
try several options via backpropagation
SGD (option 1)
0.81
0.72
for the sampling strategies. Option 1:
SGD Langevin (option 1)
0.80
0.72
0.78
0.72
early-stopping once the generated sam- SGD (option 2)
SGD Langevin (option 2)
0.78
0.74
ples are classified as positive; option 2:
0.81
0.75
stopping at a high confidence for sam- SGD (option 3)
SGD
Langevin
(option
3)
0.80
0.73
ples being positive; option 3: stopping
after a large number of steps. Table 2
shows the results and we do not observe significant differences in these choices.
Ablation study. We experiment using random noise as synthesized pseudo-negatives in an ablation
study. From Table 1, we observe that our ICN outperforms the CNN baseline and the ICN-noise
method in both one-vs-all and softmax cases.
Figure 3: MNIST test error against the number of training examples (std dev. of the test error is also displayed).
The effect of ICN is more clear when having fewer training examples.
Effects on varying training sizes. To better understand the effectiveness of our ICN method, we
carry out an experiment by varying the number of training examples. We use training sets with
different sizes including 500, 2000, 10000, and 55000 examples. The results are reported in Figure 3.
ICN is shown to be particularly effective when the training set is relatively small, since ICN has the
capability to synthesize pseudo-negatives by itself to aid training.
Comparison with GDL and GAN. GDL [40] focuses on unsupervised learning; GAN [10] and
DCGAN [35] show results for unsupervised learning and semi-supervised classification. To apply
GDL and GAN to the supervised classification setting, we design an experiment to perform a two-step
implementation. For GDL, we ran the GDL code [40] and obtained the pseudo-negative samples
for each individual digit; the pseudo-negatives are then used as augmented negative samples to
train individual one-vs-all CNN classifiers (using an identical CNN architecture to ICN for a fair
comparison), which are combined to form a multi-class classifier in the end. To compare with
DCGAN [35], we follow the same procedure: each generator trained by DCGAN [35] using the
TensorFlow implementation [16] was used to generate positive samples, which are then augmented
to the negative set to train the individual one-vs-all CNN classifiers (also using an identical CNN
architecture to ICN), which are combined to create the overall multi-class classifier. CNN+GDL
achieves a test error of 0.85% and CNN+DCGAN achieves a test error of 0.84% on the MNIST
dataset, whereas ICN reports an error of 0.78% using the same CNN architecture. As the supervised
learning task was not directly specified in DCGAN [35], some care is needed to design the optimal
setting to utilize the generated samples from DCGAN in the two-step approach (we made attempts to
optimize the results). GDL [40] can be made into a discriminative classifier by utilizing the given
negative samples first but boosting [7] with manually designed features was adopted which may not
produce competitive results as CNN classifier does. Nevertheless, the advantage of ICN being an
integrated end-to-end supervised learning single-model framework can be observed.
To compare with generative model based deep learning approach, we report the classification result
of DBN [14] in Table 1. DBN achieves a test error of 1.11% using the softmax function. We also
compare with Label Smoothing (LS), which has been used in [39] as a regularization technique by
encouraging the model to be less confident. In LS, for a training example with ground-truth label,
the label distribution is replaced with a mixture of the original ground-truth distribution and a fixed
distribution. LS achieves a test error of 0.69% in the softmax case.
8
In addition, we also adopt ResNet-32 [13] (using the softmax function) as another baseline CNN
model, which achieves a test error of 0.50% on the MNIST dataset. Our ResNet-32 based ICN
achieves an improved result of 0.47%.
Robustness to external adversarial examples. To show the improved robustness of ICN in dealing
with confusing and challenging examples, we compare the baseline CNN with our ICN classifier on
adversarial examples generated using the ?fast gradient sign? method from [11]. This ?fast gradient
sign? method (with = 0.25) can cause a maxout network to misclassify 89.4% of adversarial
examples generated from the MNIST test set [11]. In our experiment, we set = 0.125. Starting
with 10, 000 MNIST test examples, we first determine those which are correctly classified by the
baseline CNN in order to generate adversarial examples from them. We find that 5, 111 generated
adversarial examples successfully fool the baseline CNN, however, only 3, 134 of these examples
can fool our ICN classifier, which is a 38.7% reduction in error against adversarial examples. Note
that the improvement is achieved without using any additional training data, nor knowing a prior
about how these adversarial examples are generated by the specific ?fast gradient sign method? [11].
On the contrary, of the 2, 679 adversarial examples generated from the ICN classifier side that fool
ICN using the same method, 2, 079 of them can still fool the baseline CNN classifier. This two-way
experiment shows the improved robustness of ICN over the baseline CNN.
4.2
CIFAR-10
Table 3: Test errors on the CIFAR-10 dataset. In both oneThe CIFAR-10 dataset [18] consists of vs-all and softmax cases, ICN shows improvement over the
60, 000 color images of size 32 ? 32. This baseline ResNet model. The result of convolutional DBN is
set of 60, 000 images is split into two sets, from [19].
50, 000 images for training and 10, 000 images for testing. We adopt ResNet [13] as
Method
One-vs-all (%) Softmax (%)
our baseline model [45]. For data augmen- w/o Data Augmentation
Convolutional DBN
21.1
tation, we follow the standard procedure
ResNet-32 (baseline)
13.44
12.38
in [26, 25, 13] by augmenting the dataset
12.65
ResNet-32 w/ LS
by zero-padding 4 pixels on each side; we
ResNet-32
+
DCGAN
12.99
also perform cropping and random flipping.
ICN-noise (ours)
13.28
11.94
The results are reported in Table 3. In
12.94
11.46
ICN (ours)
both one-vs-all and softmax cases, ICN
w/
Data
Augmentation
outperforms the baseline ResNet classifiers.
ResNet-32 (baseline)
6.70
7.06
Our proposed ICN method is orthogonal to
ResNet-32 w/ LS
6.89
many existing approaches which use vari6.75
ResNet-32 + DCGAN
ous improvements to the network structures
ICN-noise (ours)
6.58
6.90
in order to enhance the CNN performance.
ICN (ours)
6.52
6.70
We also compare ICN with Convolutional
DBN [19], ResNet-32 w/ Label Smoothing (LS) [39] and ResNet-32+DCGAN [35] methods as
described in the MNIST experiments. LS is shown to improve the baseline but is worse than our ICN
method in most cases except for the MNIST dataset.
4.3
SVHN
Table 4: Test errors on the SVHN dataset.
We use the standard SVHN [33] dataset. We combine the
training data with the extra data to form our training set
and use the test data as the test set. No data augmentation
has been applied. The result is reported in Table 4. ICN is
shown to achieve competitive results.
5
Method
ResNet-32 (baseline)
ResNet-32 w/ LS
ResNet-32 + DCGAN
ICN-noise (ours)
ICN (ours)
Softmax (%)
2.01
1.96
1.98
1.99
1.95
Conclusion
In this paper, we have proposed an introspective convolutional nets (ICN) algorithm that performs
internal introspection. We observe performance gains within supervised learning using state-of-the-art
CNN architectures on standard machine learning benchmarks.
Acknowledgement This work is supported by NSF IIS-1618477, NSF IIS-1717431, and a Northrop
Grumman Contextual Robotics grant. We thank Saining Xie, Weijian Xu, Fan Fan, Kwonjoon Lee,
Shuai Tang, and Sanjoy Dasgupta for helpful discussions.
9
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11
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6,284 | 6,685 | Label Distribution Learning Forests
Wei Shen1,2 , Kai Zhao1 , Yilu Guo1 , Alan Yuille2
Key Laboratory of Specialty Fiber Optics and Optical Access Networks,
Shanghai Institute for Advanced Communication and Data Science,
School of Communication and Information Engineering, Shanghai University
2
Department of Computer Science, Johns Hopkins University
1
{shenwei1231,zhaok1206,gyl.luan0,alan.l.yuille}@gmail.com
Abstract
Label distribution learning (LDL) is a general learning framework, which assigns
to an instance a distribution over a set of labels rather than a single label or multiple
labels. Current LDL methods have either restricted assumptions on the expression
form of the label distribution or limitations in representation learning, e.g., to
learn deep features in an end-to-end manner. This paper presents label distribution
learning forests (LDLFs) - a novel label distribution learning algorithm based on
differentiable decision trees, which have several advantages: 1) Decision trees
have the potential to model any general form of label distributions by a mixture
of leaf node predictions. 2) The learning of differentiable decision trees can be
combined with representation learning. We define a distribution-based loss function
for a forest, enabling all the trees to be learned jointly, and show that an update
function for leaf node predictions, which guarantees a strict decrease of the loss
function, can be derived by variational bounding. The effectiveness of the proposed
LDLFs is verified on several LDL tasks and a computer vision application, showing
significant improvements to the state-of-the-art LDL methods.
1
Introduction
Label distribution learning (LDL) [6, 11] is a learning framework to deal with problems of label
ambiguity. Unlike single-label learning (SLL) and multi-label learning (MLL) [26], which assume an
instance is assigned to a single label or multiple labels, LDL aims at learning the relative importance
of each label involved in the description of an instance, i.e., a distribution over the set of labels. Such
a learning strategy is suitable for many real-world problems, which have label ambiguity. An example
is facial age estimation [8]. Even humans cannot predict the precise age from a single facial image.
They may say that the person is probably in one age group and less likely to be in another. Hence it is
more natural to assign a distribution of age labels to each facial image (Fig. 1(a)) instead of using a
single age label. Another example is movie rating prediction [7]. Many famous movie review web
sites, such as Netflix, IMDb and Douban, provide a crowd opinion for each movie specified by the
distribution of ratings collected from their users (Fig. 1(b)). If a system could precisely predict such a
rating distribution for every movie before it is released, movie producers can reduce their investment
risk and the audience can better choose which movies to watch.
Many LDL methods assume the label distribution can be represented by a maximum entropy model [2]
and learn it by optimizing an energy function based on the model [8, 11, 28, 6]. But, the exponential
part of this model restricts the generality of the distribution form, e.g., it has difficulty in representing
mixture distributions. Some other LDL methods extend the existing learning algorithms, e.g, by
boosting and support vector regression, to deal with label distributions [7, 27], which avoid making
this assumption, but have limitations in representation learning, e.g., they do not learn deep features
in an end-to-end manner.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: The real-world data which are suitable to be modeled by label distribution learning. (a)
Estimated facial ages (a unimodal distribution). (b) Rating distribution of crowd opinion on a movie
(a multimodal distribution).
In this paper, we present label distribution learning forests (LDLFs) - a novel label distribution
learning algorithm inspired by differentiable decision trees [20]. Extending differentiable decision
trees to deal with the LDL task has two advantages. One is that decision trees have the potential
to model any general form of label distributions by mixture of the leaf node predictions, which
avoid making strong assumption on the form of the label distributions. The second is that the split
node parameters in differentiable decision trees can be learned by back-propagation, which enables
a combination of tree learning and representation learning in an end-to-end manner. We define a
distribution-based loss function for a tree by the Kullback-Leibler divergence (K-L) between the
ground truth label distribution and the distribution predicted by the tree. By fixing split nodes, we
show that the optimization of leaf node predictions to minimize the loss function of the tree can
be addressed by variational bounding [19, 29], in which the original loss function to be minimized
gets iteratively replaced by a decreasing sequence of upper bounds. Following this optimization
strategy, we derive a discrete iterative function to update the leaf node predictions. To learn a forest,
we average the losses of all the individual trees to be the loss for the forest and allow the split nodes
from different trees to be connected to the same output unit of the feature learning function. In this
way, the split node parameters of all the individual trees can be learned jointly. Our LDLFs can be
used as a (shallow) stand-alone model, and can also be integrated with any deep networks, i.e., the
feature learning function can be a linear transformation and a deep network, respectively. Fig. 2
illustrates a sketch chart of our LDLFs, where a forest consists of two trees is shown.
We verify the effectiveness of our model on several LDL tasks, such as crowd opinion prediction on
movies and disease prediction based on human genes, as well as one computer vision application, i.e.,
facial age estimation, showing significant improvements to the state-of-the-art LDL methods. The
label distributions for these tasks include both unimodal distributions (e.g., the age distribution in
Fig. 1(a)) and mixture distributions (the rating distribution on a movie in Fig. 1(b)). The superiority
of our model on both of them verifies its ability to model any general form of label distributions
Figure 2: Illustration of a label distribution learning forest. The top circles denote the output units
of the function f parameterized by ?, which can be a feature vector or a fully-connected layer of
a deep network. The blue and green circles are split nodes and leaf nodes, respectively. Two index
function ?1 and ?2 are assigned to these two trees respectively. The black dash arrows indicate the
correspondence between the split nodes of these two trees and the output units of function f . Note
that, one output unit may correspond to the split nodes belonging to different trees. Each tree has
independent leaf node predictions q (denoted by histograms in leaf nodes). The output of the forest
is a mixture of the tree predictions. f (?; ?) and q are learned jointly in an end-to-end manner.
2
2
Related Work
Since our LDL algorithm is inspired by differentiable decision trees, it is necessary to first review
some typical techniques of decision trees. Then, we discuss current LDL methods.
Decision trees. Random forests or randomized decision trees [16, 1, 3, 4], are a popular ensemble
predictive model suitable for many machine learning tasks. In the past, learning of a decision tree was
based on heuristics such as a greedy algorithm where locally-optimal hard decisions are made at each
split node [1], and thus, cannot be integrated into in a deep learning framework, i.e., be combined
with representation learning in an end-to-end manner.
The newly proposed deep neural decision forests (dNDFs) [20] overcomes this problem by introducing
a soft differentiable decision function at the split nodes and a global loss function defined on a tree.
This ensures that the split node parameters can be learned by back-propagation and leaf node
predictions can be updated by a discrete iterative function.
Our method extends dNDFs to address LDL problems, but this extension is non-trivial, because
learning leaf node predictions is a constrained convex optimization problem. Although a step-size
free update function was given in dNDFs to update leaf node predictions, it was only proved to
converge for a classification loss. Consequently, it was unclear how to obtain such an update function
for other losses. We observed, however, that the update function in dNDFs can be derived from
variational bounding, which allows us to extend it to our LDL loss. In addition, the strategies used in
LDLFs and dNDFs to learning the ensemble of multiple trees (forests) are different: 1) we explicitly
define a loss function for forests, while only the loss function for a single tree was defined in dNDFs;
2) we allow the split nodes from different trees to be connected to the same output unit of the feature
learning function, while dNDFs did not; 3) all trees in LDLFs can be learned jointly, while trees in
dNDFs were learned alternatively. These changes in the ensemble learning are important, because as
shown in our experiments (Sec. 4.4), LDLFs can get better results by using more trees, but by using
the ensemble strategy proposed in dNDFs, the results of forests are even worse than those for a single
tree.
To sum up, w.r.t. dNDFs [20], the contributions of LDLFs are: first, we extend from classification [20]
to distribution learning by proposing a distribution-based loss for the forests and derive the gradient to
learn splits nodes w.r.t. this loss; second, we derived the update function for leaf nodes by variational
bounding (having observed that the update function in [20] was a special case of variational
bounding); last but not the least, we propose above three strategies to learning the ensemble of
multiple trees, which are different from [20], but we show are effective.
Label distribution learning. A number of specialized algorithms have been proposed to address the
LDL task, and have shown their effectiveness in many computer vision applications, such as facial
age estimation [8, 11, 28], expression recognition [30] and hand orientation estimation [10].
Geng et al. [8] defined the label distribution for an instance as a vector containing the probabilities
of the instance having each label. They also gave a strategy to assign a proper label distribution
to an instance with a single label, i.e., assigning a Gaussian or Triangle distribution whose peak
is the single label, and proposed an algorithm called IIS-LLD, which is an iterative optimization
process based on a two-layer energy based model. Yang et al. [28] then defined a three-layer energy
based model, called SCE-LDL, in which the ability to perform feature learning is improved by
adding the extra hidden layer and sparsity constraints are also incorporated to ameliorate the model.
Geng [6] developed an accelerated version of IIS-LLD, called BFGS-LDL, by using quasi-Newton
optimization. All the above LDL methods assume that the label distribution can be represented by a
maximum entropy model [2], but the exponential part of this model restricts the generality of the
distribution form.
Another way to address the LDL task, is to extend existing learning algorithms to deal with label
distributions. Geng and Hou [7] proposed LDSVR, a LDL method by extending support vector
regressor, which fit a sigmoid function to each component of the distribution simultaneously by a
support vector machine. Xing et al. [27] then extended boosting to address the LDL task by additive
weighted regressors. They showed that using the vector tree model as the weak regressor can lead to
better performance and named this method AOSO-LDLLogitBoost. As the learning of this tree model
is based on locally-optimal hard data partition functions at each split node, AOSO-LDLLogitBoost is
unable to be combined with representation learning. Extending current deep learning algorithms to
3
address the LDL task is an interesting topic. But, the existing such a method, called DLDL [5], still
focuses on maximum entropy model based LDL.
Our method, LDLFs, extends differentiable decision trees to address LDL tasks, in which the predicted
label distribution for a sample can be expressed by a linear combination of the label distributions
of the training data, and thus have no restrictions on the distributions (e.g., no requirement of the
maximum entropy model). In addition, thanks to the introduction of differentiable decision functions,
LDLFs can be combined with representation learning, e.g., to learn deep features in an end-to-end
manner.
3
Label Distribution Learning Forests
A forest is an ensemble of decision trees. We first introduce how to learn a single decision tree by
label distribution learning, then describe the learning of a forest.
3.1
Problem Formulation
Let X = Rm denote the input space and Y = {y1 , y2 , . . . , yC } denote the complete set of labels,
where C is the number of possible label values. We consider a label distribution learning (LDL)
problem, where for each input sample x ? X , there is a label distribution d = (dxy1 , dyx2 , . . . , dyxC )> ?
RC . Here dyxc expresses the probability of the sample x having the c-th label yc and thus has the
PC
constraints that dyxc ? [0, 1] and c=1 dyxc = 1. The goal of the LDL problem is to learn a mapping
function g : x ? d between an input sample x and its corresponding label distribution d.
Here, we want to learn the mapping function g(x) by a decision tree based model T . A decision
tree consists of a set of split nodes N and a set of leaf nodes L. Each split node n ? N defines
a split function sn (?; ?) : X ? [0, 1] parameterized by ? to determine whether a sample is sent
to the left or right subtree. Each leaf node ` ? L holds a distribution q` = (q`1 , q`2 , . . . , q`C )>
PC
over Y, i.e, q`c ? [0, 1] and c=1 q`c = 1. To build a differentiable decision tree, following [20],
we use a probabilistic split function sn (x; ?) = ?(f?(n) (x; ?)), where ?(?) is a sigmoid function,
?(?) is an index function to bring the ?(n)-th output of function f (x; ?) in correspondence with
split node n, and f : x ? RM is a real-valued feature learning function depending on the sample x
and the parameter ?, and can take any form. For a simple form, it can be a linear transformation
of x, where ? is the transformation matrix; For a complex form, it can be a deep network to
perform representation learning in an end-to-end manner, then ? is the network parameter. The
correspondence between the split nodes and the output units of function f , indicated by ?(?) that is
randomly generated before tree learning, i.e., which output units from ?f ? are used for constructing a
tree is determined randomly. An example to demonstrate ?(?) is shown in Fig. 2. Then, the probability
of the sample x falling into leaf node ` is given by
Y
l
r
p(`|x; ?) =
sn (x; ?)1(`?Ln ) (1 ? sn (x; ?))1(`?Ln ) ,
(1)
n?N
where 1(?) is an indicator function and Lln and Lrn denote the sets of leaf nodes held by the left and
right subtrees of node n, Tnl and Tnr , respectively. The output of the tree T w.r.t. x, i.e., the mapping
function g, is defined by
X
g(x; ?, T ) =
p(`|x; ?)q` .
(2)
`?L
3.2
Tree Optimization
Given a training set S = {(xi , di )}N
i=1 , our goal is to learn a decision tree T described in Sec. 3.1
which can output a distribution g(xi ; ?, T ) similar to di for each sample xi . To this end, a
straightforward way is to minimize the Kullback-Leibler (K-L) divergence between each g(xi ; ?, T )
and di , or equivalently to minimize the following cross-entropy loss:
R(q, ?; S) = ?
N C
N C
X
1 X X yc
1 X X yc
dxi log(gc (xi ; ?, T )) = ?
dxi log
p(`|xi ; ?)q`c , (3)
N i=1 c=1
N i=1 c=1
`?L
4
where q denote the distributions held by all the leaf nodes L and gc (xi ; ?, T ) is the c-th output unit
of g(xi ; ?, T ). Learning the tree T requires the estimation of two parameters: 1) the split node
parameter ? and 2) the distributions q held by the leaf nodes. The best parameters (?? , q? ) are
determined by
(?? , q? ) = arg min R(q, ?; S).
(4)
?,q
To solve Eqn. 4, we consider an alternating optimization strategy: First, we fix q and optimize
?; Then, we fix ? and optimize q. These two learning steps are alternatively performed, until
convergence or a maximum number of iterations is reached (defined in the experiments).
3.2.1
Learning Split Nodes
In this section, we describe how to learn the parameter ? for split nodes, when the distributions held
by the leaf nodes q are fixed. We compute the gradient of the loss R(q, ?; S) w.r.t. ? by the chain
rule:
N
?R(q, ?; S) X X ?R(q, ?; S) ?f?(n) (xi ; ?)
=
,
(5)
??
?f?(n) (xi ; ?)
??
i=1
n?N
where only the first term depends on the tree and the second term depends on the specific type of the
function f?(n) . The first term is given by
C
gc (xi ; ?, Tnl )
gc (xi ; ?, Tnr )
1 X yc
?R(q, ?; S)
=
dxi sn (xi ; ?)
? 1 ? sn (xi ; ?)
, (6)
?f?(n) (xi ; ?)
N c=1
gc (xi ; ?, T )
gc (xi ; ?, T )
P
P
where gc (xi ; ?, Tnl ) = `?Lln p(`|xi ; ?)q`c and g c (xi ; ?, Tnr ) = `?Lrn p(`|xi ; ?)q`c . Note that,
let Tn be the tree rooted at the node n, then we have gc (xi ; ?, Tn ) = gc (xi ; ?, Tnl ) + gc (xi ; ?, Tnr ).
This means the gradient computation in Eqn. 6 can be started at the leaf nodes and carried out in a
bottom up manner. Thus, the split node parameters can be learned by standard back-propagation.
3.2.2
Learning Leaf Nodes
Now, fixing the parameter ?, we show how to learn the distributions held by the leaf nodes q, which
is a constrained optimization problem:
min R(q, ?; S), s.t., ?`,
q
C
X
q`c = 1.
(7)
c=1
Here, we propose to address this constrained convex optimization problem by variational bounding [19, 29], which leads to a step-size free and fast-converged update rule for q. In variational
bounding, an original objective function to be minimized gets replaced by its bound in an iterative
manner. A upper bound for the loss function R(q, ?; S) can be obtained by Jensen?s inequality:
R(q, ?; S) = ?
N C
X
1 X X yc
dxi log
p(`|xi ; ?)q`c
N i=1 c=1
`?L
??
where ?` (q`c , xi ) =
1
N
N X
C
X
i=1 c=1
p(`|xi ;?)q`c
gc (xi ;?,T )
?(q, q
?) = ?
dyxci
X
?` (?
q`c , xi ) log
`?L
p(`|x ; ?)q
i
`c
,
?` (?
q`c , xi )
(8)
. We define
N C
p(`|x ; ?)q
1 X X yc X
i
`c
dxi
?` (?
q`c , xi ) log
.
N i=1 c=1
?` (?
q`c , xi )
(9)
`?L
? ) is an upper bound for R(q, ?; S), which has the property that for any q and q
Then ?(q, q
?,
?(q, q
?) ? R(q, ?; S), and ?(q, q) = R(q, ?; S). Assume that we are at a point q(t) corresponding
to the t-th iteration, then ?(q, q(t) ) is an upper bound for R(q, ?; S). In the next iteration, q(t+1)
is chosen such that ?(q(t+1) , q) ? R(q(t) , ?; S), which implies R(q(t+1) , ?; S) ? R(q(t) , ?; S).
5
Consequently, we can minimize ?(q, q
?) instead of R(q, ?; S) after ensuring that R(q(t) , ?; S) =
(t)
(t)
?(q , q
?), i.e., q
? = q . So we have
q(t+1) = arg min ?(q, q(t) ), s.t., ?`,
q
C
X
q`c = 1,
(10)
c=1
which leads to minimizing the Lagrangian defined by
?(q, q(t) ) = ?(q, q(t) ) +
X
?` (
`?L
where ?` is the Lagrange multiplier. By setting
?` =
(t+1)
Note that, q`c
(t+1)
? [0, 1] and
distributions held by the leaf nodes. The starting
(0)
distribution: q`c = C1 .
3.3
q`c ? 1),
(11)
c=1
??(q,q(t) )
?q`c
N C
1 X X yc
(t)
(t+1)
d ?` (q`c , xi ) and q`c
N i=1 c=1 xi
satisfies that q`c
C
X
= 0, we have
PN yc
(t)
dxi ?` (q`c , xi )
.
= PC i=1
PN yc
(t)
?
(q
,
x
)
d
x
`
i
i
c=1
i=1
`c
(12)
(t+1)
= 1. Eqn. 12 is the update scheme for
c=1 q`c
(0)
point q` can be simply initialized by the uniform
PC
Learning a Forest
A forest is an ensemble of decision trees F = {T1 , . . . , TK }. In the training stage, all trees in the
forest F use the same parameters ? for feature learning function f (?; ?) (but correspond to different
output units of f assigned by ?, see Fig. 2), but each tree has independent leaf node predictions
q. The loss function for a forest is given by averaging the loss functions for all individual trees:
PK
1
RF = K
k=1 RTk , where RTk is the loss function for tree Tk defined by Eqn. 3. To learn ? by
fixing the leaf node predictions q of all the trees in the forest F, based on the derivation in Sec. 3.2
and referring to Fig. 2, we have
N K
?f?k (n) (xi ; ?)
1 XX X
?RF
?RTk
=
,
??
K i=1
?f?k (n) (xi ; ?)
??
(13)
k=1 n?Nk
where Nk and ?k (?) are the split node set and the index function of Tk , respectively. Note that,
the index function ?k (?) for each tree is randomly assigned before tree learning, and thus split
nodes correspond to a subset of output units of f . This strategy is similar to the random subspace
method [17], which increases the randomness in training to reduce the risk of overfitting.
As for q, since each tree in the forest F has its own leaf node predictions q, we can update them
independently by Eqn. 12, given by ?. For implementational convenience, we do not conduct this
update scheme on the whole dataset S but on a set of mini-batches B. The training procedure of a
LDLF is shown in Algorithm. 1.
Algorithm 1 The training procedure of a LDLF.
Require: S: training set, nB : the number of mini-batches to update q
Initialize ? randomly and q uniformly, set B = {?}
while Not converge do
while |B| < nB do
Randomly select a mini-batch B from S
UpdateS
? by computing gradient (Eqn. 13) on B
B=B B
end while
Update q by iterating Eqn. 12 on B
B = {?}
end while
In the testing stage, the output of the forest F is given by averaging the predictions from all the
PK
1
individual trees: g(x; ?, F) = K
k=1 g(x; ?, Tk ).
6
4
Experimental Results
Our realization of LDLFs is based on ?Caffe? [18]. It is modular and implemented as a standard
neural network layer. We can either use it as a shallow stand-alone model (sLDLFs) or integrate it
with any deep networks (dLDLFs). We evaluate sLDLFs on different LDL tasks and compare it with
other stand-alone LDL methods. As dLDLFs can be learned from raw image data in an end-to-end
manner, we verify dLDLFs on a computer vision application, i.e., facial age estimation. The default
settings for the parameters of our forests are: tree number (5), tree depth (7), output unit number of
the feature learning function (64), iteration times to update leaf node predictions (20), the number of
mini-batches to update leaf node predictions (100), maximum iteration (25000).
4.1
Comparison of sLDLFs to Stand-alone LDL Methods
We compare our shallow model sLDLFs with other state-of-the-art stand-alone LDL methods.
For sLDLFs, the feature learning function f (x, ?) is a linear transformation of x, i.e., the i-th
output unit fi (x, ?i ) = ?i> x, where ?i is the i-th column of the transformation matrix ?. We
used 3 popular LDL datasets in [6], Movie, Human Gene and Natural Scene1 . The samples
in these 3 datasets are represented by numerical descriptors, and the ground truths for them are
the rating distributions of crowd opinion on movies, the diseases distributions related to human
genes and label distributions on scenes, such as plant, sky and cloud, respectively. The label
distributions of these 3 datasets are mixture distributions, such as the rating distribution shown in
Fig. 1(b). Following [7, 27], we use 6 measures to evaluate the performances of LDL methods,
which compute the average similarity/distance between the predicted rating distributions and the real
rating distributions, including 4 distance measures (K-L, Euclidean, S?rensen, Squared ?2 ) and two
similarity measures (Fidelity, Intersection).
We evaluate our shallow model sLDLFs on these 3 datasets and compare it with other state-of-the-art
stand-alone LDL methods. The results of sLDLFs and the competitors are summarized in Table 1.
For Movie we quote the results reported in [27], as the code of [27] is not publicly available. For the
results of the others two, we run code that the authors had made available. In all case, following [27, 6],
we split each dataset into 10 fixed folds and do standard ten-fold cross validation, which represents
the result by ?mean?standard deviation? and matters less how training and testing data get divided.
As can be seen from Table 1, sLDLFs perform best on all of the six measures.
Table 1: Comparison results on three LDL datasets [6]. ??? and ??? indicate the larger and the smaller
the better, respectively.
Dataset
Method
K-L ?
Euclidean ?
S?rensen ?
Squared ?2 ?
Fidelity ?
Intersection ?
Movie
sLDLF (ours)
AOSO-LDLogitBoost [27]
LDLogitBoost [27]
LDSVR [7]
BFGS-LDL [6]
IIS-LDL [11]
0.073?0.005
0.086?0.004
0.090?0.004
0.092?0.005
0.099?0.004
0.129?0.007
0.133?0.003
0.155?0.003
0.159?0.003
0.158?0.004
0.167?0.004
0.187?0.004
0.130?0.003
0.152?0.003
0.155?0.003
0.156?0.004
0.164?0.003
0.183?0.004
0.070?0.004
0.084?0.003
0.088?0.003
0.088?0.004
0.096?0.004
0.120?0.005
0.981?0.001
0.978?0.001
0.977?0.001
0.977?0.001
0.974?0.001
0.967?0.001
0.870?0.003
0.848?0.003
0.845?0.003
0.844?0.004
0.836?0.003
0.817?0.004
sLDLF (ours)
LDSVR [7]
BFGS-LDL [6]
IIS-LDL [11]
0.228?0.006
0.245?0.019
0.231?0.021
0.239?0.018
0.085?0.002
0.099?0.005
0.076?0.006
0.089?0.006
0.212?0.002
0.229?0.015
0.231?0.012
0.253?0.009
0.179?0.004
0.189?0.021
0.211?0.018
0.205?0.012
0.948?0.001
0.940?0.006
0.938?0.008
0.944?0.003
0.788?0.002
0.771?0.015
0.769?0.012
0.747?0.009
sLDLF (ours)
LDSVR [7]
BFGS-LDL [6]
IIS-LDL [11]
0.534?0.013
0.852?0.023
0.856?0.061
0.879?0.023
0.317?0.014
0.511?0.021
0.475?0.029
0.458?0.014
0.336?0.010
0.492?0.016
0.508?0.026
0.539?0.011
0.448?0.017
0.595?0.026
0.716?0.041
0.792?0.019
0.824?0.008
0.813?0.008
0.722?0.021
0.686?0.009
0.664?0.010
0.509?0.016
0.492?0.026
0.461?0.011
Human Gene
Natural Scene
4.2
Evaluation of dLDLFs on Facial Age Estimation
In some literature [8, 11, 28, 15, 5], age estimation is formulated as a LDL problem. We conduct
facial age estimation experiments on Morph [24], which contains more than 50,000 facial images
from about 13,000 people of different races. Each facial image is annotated with a chronological age.
To generate an age distribution for each face image, we follow the same strategy used in [8, 28, 5],
which uses a Gaussian distribution whose mean is the chronological age of the face image (Fig. 1(a)).
The predicted age for a face image is simply the age having the highest probability in the predicted
1
We download these datasets from http://cse.seu.edu.cn/people/xgeng/LDL/index.htm.
7
label distribution. The performance of age estimation is evaluated by the mean absolute error (MAE)
between predicted ages and chronological ages. As the current state-of-the-art result on Morph
is obtain by fine-tuning DLDL [5] on VGG-Face [23], we also build a dLDLF on VGG-Face, by
replacing the softmax layer in VGGNet by a LDLF. Following [5], we do standard 10 ten-fold cross
validation and the results are summarized in Table. 2, which shows dLDLF achieve the state-of-the-art
performance on Morph. Note that, the significant performance gain between deep LDL models
(DLDL and dLDLF) and non-deep LDL models (IIS-LDL, CPNN, BFGS-LDL) and the superiority
of dLDLF compared with DLDL verifies the effectiveness of end-to-end learning and our tree-based
model for LDL, respectively.
Table 2: MAE of age estimation comparison on Morph [24].
Method
IIS-LDL [11]
CPNN [11]
BFGS-LDL [6]
DLDL+VGG-Face [5]
dLDLF+VGG-Face (ours)
MAE
5.67?0.15
4.87?0.31
3.94?0.05
2.42?0.01
2.24?0.02
As the distribution of gender and ethnicity is very unbalanced in Morph, many age estimation methods [13, 14, 15] are evaluated on a subset of Morph, called Morph_Sub for short, which consists of
20,160 selected facial images to avoid the influence of unbalanced distribution. The best performance
reported on Morph_Sub is given by D2LDL [15], a data-dependent LDL method. As D2LDL used
the output of the ?fc7? layer in AlexNet [21] as the face image features, here we integrate a LDLF
with AlexNet. Following the experiment setting used in D2LDL, we evaluate our dLDLF and the
competitors, including both SLL and LDL based methods, under six different training set ratios (10%
to 60%). All of the competitors are trained on the same deep features used by D2LDL. As can be
seen from Table 3, our dLDLFs significantly outperform others for all training set ratios.
Note that, the generated age distri- Figure 3: MAE of age estimation comparison on
butions are unimodal distributions Morph_Sub.
and the label distributions used in
Training set ratio
Method
Sec. 4.1 are mixture distributions.
10%
20%
30%
40%
50%
60%
The proposed method LDLFs achieve
AAS [22]
4.9081
4.7616
4.6507
4.5553
4.4690
4.4061
the state-of-the-art results on both of
LARR [12]
4.7501
4.6112
4.5131
4.4273
4.3500
4.2949
IIS-ALDL [9]
4.1791
4.1683
4.1228
4.1107
4.1024
4.0902
them, which verifies that our model
D2LDL [15]
4.1080
3.9857
3.9204
3.8712
3.8560
3.8385
has the ability to model any general
dLDLF (ours)
3.8495
3.6220
3.3991
3.2401
3.1917
3.1224
form of label distributions.
4.3
Time Complexity
Let h and sB be the tree depth and the
batch size, respectively. Each tree has 2h?1 ? 1 split nodes and 2h?1 leaf nodes. Let D = 2h?1 ? 1.
For one tree and one sample, the complexity of a forward pass and a backward pass are O(D +
D + 1?C) = O(D?C) and O(D + 1?C + D?C) = O(D?C), respectively. So for K trees and
nB batches, the complexity of a forward and backward pass is O(D?C?K?nB ?sB ). The complexity of an iteration to update leaf nodes are O(nB ?sB ?K?C?D + 1) = O(D?C?K?nB ?sB ).
Thus, the complexity for the training procedure (one epoch, nB batches) and the testing procedure
(one sample) are O(D?C?K?nB ?sB ) and O(D?C?K), respectively. LDLFs are efficient: On
Morph_Sub (12636 training images, 8424 testing images), our model only takes 5250s for training
(25000 iterations) and 8s for testing all 8424 images.
4.4
Parameter Discussion
Now we discuss the influence of parameter settings on performance. We report the results of rating
prediction on Movie (measured by K-L) and age estimation on Morph_Sub with 60% training set
ratio (measured by MAE) for different parameter settings in this section.
Tree number. As a forest is an ensemble model, it is necessary to investigate how performances
change by varying the tree number used in a forest. Note that, as we discussed in Sec. 2, the
ensemble strategy to learn a forest proposed in dNDFs [20] is different from ours. Therefore, it is
necessary to see which ensemble strategy is better to learn a forest. Towards this end, we replace our
ensemble strategy in dLDLFs by the one used in dNDFs, and name this method dNDFs-LDL. The
corresponding shallow model is named by sNDFs-LDL. We fix other parameters, i.e., tree depth and
8
output unit number of the feature learning function, as the default setting. As shown in Fig. 4 (a), our
ensemble strategy can improve the performance by using more trees, while the one used in dNDFs
even leads to a worse performance than one for a single tree.
Observed from Fig. 4, the performance of LDLFs can be improved by using more trees, but the
improvement becomes increasingly smaller and smaller. Therefore, using much larger ensembles
does not yield a big improvement (On Movie, the number of trees K = 100: K-L = 0.070 vs K = 20:
K-L = 0.071). Note that, not all random forests based methods use a large number of trees, e.g.,
Shotton et al. [25] obtained very good pose estimation results from depth images by only 3 decision
trees.
Tree depth. Tree depth is another important parameter for decision trees. In LDLFs, there is an
implicit constraint between tree depth h and output unit number of the feature learning function ? :
? ? 2h?1 ? 1. To discuss the influence of tree depth to the performance of dLDLFs, we set ? = 2h?1
and fix tree number K = 1, and the performance change by varying tree depth is shown in Fig. 4 (b).
We see that the performance first improves then decreases with the increase of the tree depth. The
reason is as the tree depth increases, the dimension of learned features increases exponentially, which
greatly increases the training difficulty. So using much larger depths may lead to bad performance
(On Movie, tree depth h = 18: K-L = 0.1162 vs h = 9: K-L = 0.0831).
Figure 4: The performance change of age estimation on Morph_Sub and rating prediction on Movie
by varying (a) tree number and (b) tree depth. Our approach (dLDLFs/sLDLFs) can improve the
performance by using more trees, while using the ensemble strategy proposed in dNDFs (dNDFsLDL/sNDFs-LDL) even leads to a worse performance than one for a single tree.
5
Conclusion
We present label distribution learning forests, a novel label distribution learning algorithm inspired by
differentiable decision trees. We defined a distribution-based loss function for the forests and found
that the leaf node predictions can be optimized via variational bounding, which enables all the trees
and the feature they use to be learned jointly in an end-to-end manner. Experimental results showed
the superiority of our algorithm for several LDL tasks and a related computer vision application, and
verified our model has the ability to model any general form of label distributions.
Acknowledgement. This work was supported in part by the National Natural Science Foundation of
China No. 61672336, in part by ?Chen Guang? project supported by Shanghai Municipal Education
Commission and Shanghai Education Development Foundation No. 15CG43 and in part by ONR
N00014-15-1-2356.
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pages 1247?1250, 2015.
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6,285 | 6,686 | Unsupervised learning of object frames by dense
equivariant image labelling
James Thewlis1
Hakan Bilen2
1
Andrea Vedaldi1
2
Visual Geometry Group
University of Oxford
{jdt,vedaldi}@robots.ox.ac.uk
School of Informatics
University of Edinburgh
[email protected]
Abstract
One of the key challenges of visual perception is to extract abstract models of 3D
objects and object categories from visual measurements, which are affected by
complex nuisance factors such as viewpoint, occlusion, motion, and deformations.
Starting from the recent idea of viewpoint factorization, we propose a new approach
that, given a large number of images of an object and no other supervision, can
extract a dense object-centric coordinate frame. This coordinate frame is invariant
to deformations of the images and comes with a dense equivariant labelling neural
network that can map image pixels to their corresponding object coordinates.
We demonstrate the applicability of this method to simple articulated objects
and deformable objects such as human faces, learning embeddings from random
synthetic transformations or optical flow correspondences, all without any manual
supervision.
1
Introduction
Humans can easily construct mental models of complex 3D objects and object categories from
visual observations. This is remarkable because the dependency between an object?s appearance
and its structure is tangled in a complex manner with extrinsic nuisance factors such as viewpoint,
illumination, and articulation. Therefore, learning the intrinsic structure of an object from images
requires removing these unwanted factors of variation from the data.
The recent work of [37] has proposed an unsupervised approach to do so, based on on the concept
of viewpoint factorization. The idea is to learn a deep Convolutional Neural Network (CNN) that
can, given an image of the object, detect a discrete set of object landmarks. Differently from
traditional approaches to landmark detection, however, landmarks are neither defined nor supervised
manually. Instead, the detectors are learned using only the requirement that the detected points must
be equivariant (consistent) with deformations of the input images. The authors of [37] show that this
constraint is sufficient to learn landmarks that are ?intrinsic? to the objects and hence capture their
structure; remarkably, due to the generalization ability of CNNs, the landmark points are detected
consistently not only across deformations of a given object instance, which are observed during
training, but also across different instances. This behaviour emerges automatically from training on
thousands of single-instance correspondences.
In this paper, we take this idea further, moving beyond a sparse set of landmarks to a dense model
of the object structure (section 3). Our method relates each point on an object to a point in a low
dimensional vector space in a way that is consistent across variation in motion and in instance identity.
This gives rise to an object-centric coordinate system, which allows points on the surface of an object
to be indexed semantically (figure 1). As an illustrative example, take the object category of a face
and the vector space R3 . Our goal is to semantically map out the object such that any point on a
face, such as the left eye, lives at a canonical position in this ?label space?. We train a CNN to learn
the function that projects any face image into this space, essentially ?coloring? each pixel with its
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: Dense equivariant image labelling. Left: Given an image x of an object or object category
and no other supervision, our goal is to find a common latent space Z, homeomorphic to a sphere,
which attaches a semantically-consistent coordinate frame to the object points. This is done by
learning a dense labelling function that maps image pixels to their corresponding coordinate in the
Z space. This mapping function is equivariant (compatible) with image warps or object instance
variations. Right: An equivariant dense mapping learned in an unsupervised manner from a large
dataset of faces. (Results of S IMPLE network, Ldist , ? = 0.5)
corresponding label. As a result of our learning formulation, the label space has the property of being
locally smooth: points nearby in the image are nearby in the label space. In an ideal case, we could
imagine the surface of an object to be mapped to a sphere.
In order to achieve these results, we contribute several technical innovations (section 3.2). First,
we show that, in order to learn a non-trivial object coordinate frame, the concept of equivariance
must be complemented with the one of distinctiveness of the embedding. Then, we propose a CNN
implementation of this concept that can explicitly express uncertainty in the labelling of the object
points. The formulation is used in combination with a probabilistic loss, which is augmented with a
robust geometric distance to encourage better alignment of the object features.
We show that this framework can be used to learn meaningful object coordinate frames in a purely
unsupervised manner, by analyzing thousands of deformations of visual objects. While [37] proposed
to use Thin Plate Spline image warps for training, here we also consider simple synthetic articulated
objects having frames related by known optical flow (section 4).
We conclude the paper with a summary of our finding (section 5).
2
Related Work
Learning the structure of visual objects. Modeling the structure of visual objects is a widelystudied (e.g. [6, 7, 11, 39, 12]) computer vision problem with important applications such as facial
landmark detection and human body pose estimation. Much of this work is supervised and aimed
at learning detectors of objects or their parts, often using deep learning. A few approaches such as
spatial transformer networks [20] can learn geometric transformations without explicit geometric
supervision, but do not build explicit geometric models of visual objects.
More related to our work, WarpNet [21] and geometric matching networks [34] learn a neural network
that predicts Thin Plate Spline [3] transformations between pairs of images of an object, including
synthetic warps. Deep Deformation Network [42] improves WarpNet by using a Point Transformer
Network to refine the computed landmarks, but it requires manual supervision. None of these works
look at the problem of learning an invariant geometric embedding for the object.
Our work builds on the idea of viewpoint factorization (section 3.1), recently introduced in [37, 31].
However, we extend [37] in several significant ways. First, we construct a dense rather than discrete
embedding, where all pixels of an object are mapped to an invariant object-centric coordinate instead
of just a small set of selected landmarks. Second, we show that the equivariance constraint proposed
in [37] is not quite enough to learn such an embedding; it must be complemented with the concept of
a distinctive embedding (section 3.1). Third, we introduce a new neural network architecture and
corresponding training objective that allow such an embedding to be learned in practice (section 3.2).
Optical/semantic flow. A common technique to find correspondences between temporally related
video frames is optical flow [18]. The state-of-the-art methods [14, 38, 19] typically employ convolu2
tional neural networks to learn pairwise dense correspondences between the same object instances
at subsequent frames. The SIFT Flow method [25] extends the between-instance correspondences
to cross-instance mappings by matching SIFT features [27] between semantically similar object
instances. Learned-Miller [24] extends the pairwise correspondences to multiple images by posing
a problem of alignment among the images of a set. Collection Flow [22] and Mobahi et al. [29]
project objects onto a low-rank space that allow for joint alignment. FlowWeb [50], and Zhou et
al. [49] construct fully connected graphs to maximise cycle consistency between each image pair and
synthethic data as an intermediary by training a CNN. In our experiments (section 4) flow is known
from synthetic warps or motion, but our work could build on any unsupervised optical flow method.
Unsupervised learning. Classical unsupervised learning methods such as autoencoders [4, 2, 17]
and denoising autoencoders aim to learn useful feature representations from an input by simply
reconstructing it after a bottleneck. Generative adversarial networks [16] target producing samples of
realistic images by training generative models. These models when trained joint with image encoders
are also shown to learn good feature representations [9, 10]. More recently several studies have
emerged that train neural networks by learning auxiliary or pseudo tasks. These methods exploit
typically some existing information in input as ?self-supervision? without any manual labeling by
removing or perturbing some information from an input and requiring a network to reconstruct it.
For instance, Doersch et al. [8], and Noroozi and Favaro [30] train a network to predict the relative
locations of shuffled image patches. Other self-supervised tasks include colorizing images [44],
inpainting [33], ranking frames of a video in temporally correct order [28, 13]. More related to our
approach, Agrawal et al. [1] use egomotion as supervisory signal to learn feature representations in a
Siamese network by predicting camera transformations from image pairs, [32] learn to group pixels
that move together in a video. [48, 15] use a warping-based loss to learn depth from video.
3
Method
This section discusses our method in detail, first introducing the general idea of dense equivariant
labelling (section 3.1), and then presenting a concrete implementation of the latter using a novel deep
CNN architecture (section 3.2).
3.1
Dense equivariant labelling
Consider a 3D object S ? R3 or a class of such objects S that are topologically isomorphic to
a sphere Z ? R3 (i.e. the objects are simple closed surfaces without holes). We can construct a
homeomorphism p = ?S (q) mapping points of the sphere q ? Z to points p ? S of the objects.
Furthermore, if the objects belong to the same semantic category (e.g. faces), we can assume that
these isomorphisms are semantically consistent, in the sense that ?S 0 ? ?S?1 : S ? S 0 maps points of
object S to semantically-analogous points in object S 0 (e.g. for human faces the right eye in one face
should be mapped to the right eye in another [37]).
While this construction is abstract, it shows that we can endow the object (or object category) with
a spherical reference system Z. The authors of [37] build on this construction to define a discrete
system of object landmarks by considering a finite number of points zk ? Z. Here, we take the
geometric embedding idea more literally and propose to explicitly learn a dense mapping from images
of the object to the object-centric coordinate space Z. Formally, we wish to learn a labelling function
? : (x, u) 7? z that takes a RGB image x : ? ? R3 , ? ? R3 and a pixel u ? ? to the object point
z ? Z which is imaged at u (figure 1).
Similarly to [37], this mapping must be compatible or equivariant with image deformations. Namely,
let g : ? ? ? be a deformation of the image domain, either synthetic or due to a viewpoint change
or other motion. Furthermore, let gx = x ? g ?1 be the action of g on the image (obtained by inverse
warp). Barring occlusions and boundary conditions, pixel u in image x must receive the same label
as pixel gu in image gx, which results in the invariance constraint:
?x, u :
?(x, u) = ?(gx, gu).
(1)
Equivalently, we can view the network as a functional x 7? ?(x, ?) that maps the image to a
corresponding label map. Since the label map is an image too, g acts on it by inverse warp.1 Using
1
In the sense that g?(x, ?) = ?(x, ?) ? g ?1 .
3
this, the constraint (1) can be rewritten as the equivariance relation g?(x, ?) = ?(gx, ?). This can be
visualized by noting that the label image deforms in the same way as the input image, as show for
example in figure 3.
For learning, constraint (1) can be incorporated in a loss function as follows:
Z
1
2
L(?|?) =
k?(x, u) ? ?(gx, gu)k du.
|?| ?
However, minimizing this loss has the significant drawback that a global optimum is obtained by
simply setting ?(x, u) = const. The reason for this issue is that (1) is not quite enough to learn a
useful object representation. In order to do so, we must require the labels not only to be equivariant,
but also distinctive, in the sense that
?(x, u) = ?(gx, v)
?
v = gu.
We can encode this requirement as a loss in different ways. For example, by using the fact that points
?(x, u) are on the unit sphere, we can use the loss:
Z
1
2
0
kgu ? argmaxv h?(x, u), ?(gx, v)ik du.
(2)
L (?|x, g) =
|?| ?
By doing so, the labels ?(x, u) must be able to discriminate between different object points, so that a
constant labelling would receive a high penalty.
Relationship with learning invariant visual descriptors. As an alternative to loss (2), we could
have used a pairwise loss2 to encourage the similarity h?(x, u), ?(x0 , gu)i of the labels assigned
to corresponding pixels u and gu to be larger than the similarity h?(x, u), ?(x0 , v)i of the labels
assigned to pixels u and v that do not correspond. Formally, this would result in a pairwise loss
similar to the ones often used to learn invariant visual descriptors for image matching. The reason
why our method learns an object representation instead of a generic visual descriptor is that the
dimensionality of the label space Z is just enough to represent a point on a surface. If we replace
Z with a larger space such as Rd , d 2, we can expect ?(x, u) to learn to extract generic visual
descriptors like SIFT instead. This establishes an interesting relationship between visual descriptors
and object-specific coordinate vectors and suggests that it is possible to transition between the two by
controlling their dimensionality.
3.2
Concrete learning formulation
In this section we introduce a concrete implementation of our method (figure 2). For the mapping
?, we use a CNN that receives as input an image tensor x ? RH?W ?C and produces as output a
label tensor z ? RH?W ?L . We use the notation ?u (x) to indicate the L-dimensional label vector
extracted at pixel u from the label image computed by the network.
The dimension of the label vectors is set to L = 3 (instead of L = 2) in order to allow the network
to express uncertainty about the label assigned to a pixel. The network can do so by modulating
the norm of ?u (x). In fact, correspondences are expressed probabilistically by computing the inner
product of label vectors followed by the softmax operator. Formally, the probability that pixel v in
image x0 corresponds to pixel u in image x is expressed as:
0
eh?u (x),?v (x )i
p(v|u; x, x0 , ?) = P h? (x),? (x0 )i .
u
z
ze
(3)
In this manner, a shorter vector ?u results in a more diffuse probability distribution.
Next, we wish to define a loss function for learning ? from data. To this end, we consider a triplet
? = (x, x0 , g), where x0 = gx is an image that corresponds to x up to transformation g (the nature
2
Formally, this is achieved by the loss
Z
n
o
1
L00 (?|x, g) =
max 0, max ?(u, v) + h?(x, u), ?(gx, v)i ? h?(x, u), ?(gx, gu)i du,
v
|?| ?
where ?(u, v) ? 0 is an error-dependent margin.
4
Optical
flow
Figure 2: Unsupervised dense correspondence network. From left to right: The network ? extracts
label maps ?u (x) and ?v (x0 ) from the image pair x and x0 . An optical flow module (or ground truth
for synthetic transformation) computes the warp (correspondence field) g such that x0 = gx. Then
the label of each point u in the first image is correlated to each point v in the second, obtaining a
number of score maps. The loss evaluates how well the score maps predict the warp g.
of the data is discussed below). We then assess the performance of the network ? on the triplet ?
using two losses. The first loss is the negative log-likelihood of the ground-truth correspondences:
1 X
Llog (?|x, x0 , g) = ?
log p(gu|u; x, x0 , ?).
(4)
HW u
This loss has the advantage that it explicitly learns (3) as the probability of a match. However, it is
not sensitive to the size of a correspondence error v ? gu. In order to address this issue, we also
consider the loss
1 XX
Ldist (?|x, x0 , g) =
kv ? guk?2 p(v|u; x, x0 , ?).
(5)
HW u v
Here ? > 0 is an exponent used to control the robustness of the distance measure, which we set to
? = 0.5, 1.
Nework details. We test two architecture. The first one, denoted S IMPLE, is the same as [47, 37]
and is a chain (5, 20)+ , (2, mp), ?2 , (5, 48)+ , (3, 64)+ , (3, 80)+ , (3, 256)+ , (1, 3) where (h, c) is
a bank of c filters of size h ? h, + denotes ReLU, (h, mp) is h ? h max-pooling, ?s is
s? downsampling. Better performance can be obtained by increasing the support of the filters in the network; for this, we consider a second network D ILATIONS (5, 20)+ , (2, mp), ?2
, (5, 48)+ , (5, 64, 2)+ , (3, 80, 4)+ , (3, 256, 2)+ , (1, 3) where (h, c, d) is a filter with ?d dilation [41].
3.3
Learning from synthetic and true deformations
Losses (4) and (5) learn from triplets ? = (x, x0 , g). Here x0 can be either generated synthetically by
applying a random transformation g to a natural image x [37, 21], or it can be obtained by observing
image pairs (x, x0 ) containing true object deformations arising from a viewpoint change or an object
motion or deformation.
The use of synthetic transformations enables training even on static images and was considered
in [37], who showed it to be sufficient to learn meaningful landmarks for a number of real-world
object such as human and cat faces. Here, in addition to using synthetic deformations, we also
consider using animated image pairs x and x0 . In principle, the learning formulation can be modified
so that knowledge of g is not required; instead, images and their warps can be compared and aligned
directly based on the brightness constancy principle. In our toy video examples we obtain g from the
rendering engine, but it can in theory be obtained using an off-the-shelf optical flow algorithm which
would produce a noisy version of g.
4
Experiments
This section assesses our unsupervised method for dense object labelling on two representative tasks:
two toy problems (sections 4.1 and 4.2) and human and cat faces (section 4.3).
5
Figure 3: Roboarm equivariant labelling. Top: Original video frames of a simple articulated object.
Middle and bottom: learned labels, which change equivariantly with the arm, learned using Llog and
Ldist , respectively. Different colors denote different points of the spherical object frame.
4.1
Roboarm example
In order to illustrate our method we consider a toy problem consisting of a simple articulated object,
namely an animated robotic arm (figure 3) created using a 2D physics engine [36]. We do so for two
reasons: to show that the approach is capable of labelling correctly deformable/articulated objects and
to show that the spherical model Z is applicable also to thin objects, that have mainly a 1D structure.
Dataset details. The arm is anchored to the bottom left corner and is made up of colored capsules
connected with joints having reasonable angle limits to prevent unrealistic contortion and selfocclusion. Motion is achieved by varying the gravity vector, sampling each element from a Gaussian
with standard deviation 15 m s?2 every 100 iterations. Frames x of size 90 ? 90 pixels and the
corresponding flow fields g : x 7? x0 are saved every 20 iterations. We also save the positions of the
capsule centers. The final dataset has 23999 frames.
Learning. Using the correspondences ? = (x, x0 , g) provided by the flow fields, we use our method
to learn an object centric coordinate frame Z and its corresponding labelling function ?u (x). We
test learning ? using the probabilistic loss (4) and distance-based loss (5). In the loss we ignore
areas with zero flow, which automatically removes the background. We use the S IMPLE network
architecture (section 3.2).
Results. Figure 3 provides some qualitative results, showing by means of colormaps the labels ?u (x)
associated to different pixels of each input image. It is easy to see that the method attaches consistent
labels to the different arm elements. The distance-based loss produces a more uniform embedding, as
may be expected. The embeddings are further visualized in Figure 4 by projecting a number of video
frames back to the learned coordinate spaces Z. It can be noted that the space is invariant, in the sense
that the resulting figure is approximately the same despite the fact that the object deforms significantly
in image space. This is true for both embeddings, but the distance-based ones are geometrically more
consistent.
2
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2
1
1
1
0
0
0
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-1
-1
-2
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-4
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4
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Figure 4: Invariance of the object-centric coordinate space for Roboarm. The plot projects
frames 3,6,9 of figure 3 on the object-centric coordinate space Z, using the embedding functions
learned by means of the probabilistic (top) and distance (bottom) based losses. The sphere is then
unfolded, plotting latitude and longitude (in radians) along the vertical and horizontal axes.
6
Log
Dist 0.5
Dist 1
Ground truth
Figure 5: Left: Embedding spaces of different dimension. Spherical embedding (from the 3D
embedding function ?u (x) ? R3 ) learned using the distance loss compared to a circular embedding
with one dimension less. Right: Capsule center prediction for different losses.
Predicting capsule centers. We evaluate quantitatively the ability of our object frames to localise the
capsule centers. If our assumption is correct and a coordinate system intrinsic to the object has been
learned, then we should expect there to be a specific 3-vector in Z corresponding to each center, and
our job is to find these vectors. Various strategies could be used, such as averaging the object-centric
coordinates given to the centers over the training set, but we choose to incorporate the problem into
the learning framework. This is done using the negative log-likelihood in much the same way as
(4), limiting our vectors u to the centers. This is done as an auxiliary layer with no backpropagation
to the rest of the network, so that the embedding remains unsupervised. The error reported is the
Euclidean distance as a percentage of the image width.
Results are given for the different loss functions used for unsupervised training in Table 1 and
visualized in Figure 5 right, showing that the object centers can be located to a high degree of
accuracy. The negative log likelihood performs best while the two losses incorporating distance
perform similarly.
We also perform experiments varying the dimensionality L of the label space Z (Table 2). Perhaps
most interestingly, given the almost one-dimensional nature of the arm, is the case of L = 2, which
would correspond to an approximately circular space (since the length of vectors is used to code
for uncertainty). As seen in the right of Figure 5 left, the segments are represented almost perfectly
on the boundary of a circle, with the exception of the bifurcation which it is unable to accurately
represent. This is manifested by the light blue segment trying, and failing, to be in two places at once.
Unsupervised Loss
Error
Llog
0.97 %
Ldist , ? = 1
1.13 %
Ldist , ? = 0.5
1.14 %
Table 1: Predicting capsule centers.
Error as percent of image width.
4.2
Textured sphere example
Descriptor Dimension
2
3
5
20
Error
1.29 %
1.14 %
1.16 %
1.28 %
Table 2: Descriptor dimension
(Ldist , ? = 0.5). L>3 shows no improvement,
suggesting L=3 is the natural manifold of the arm.
The experiment of Figure 6 tests the ability of the method to understand a complete rotation of a 3D
object, a simple textured sphere. Despite the fact that the method is trained on pairs of adjacent video
frames (and corresponding optical flow), it still learns a globally-consistent embedding. However,
this required switching from from the S IMPLE to the D ILATIONS architecture (section 3.2).
4.3
Faces
After testing our method on a toy problem, we move to a much harder task and apply our method to
generate an object-centric reference frame Z for the category of human faces. In order to generate
an image pair and corresponding flow field for training we warp each face synthetically using Thin
Plate Spline warps in a manner similar to [37]. We train our models on the extensive CelebA [26]
dataset of over 200k faces as in [37], excluding MAFL [47] test overlap from the given training split.
It has annotations of the eyes, nose and mouth corners. Note that we do not use these to train our
model. We also use AFLW [23], testing on 2995 faces [47, 40, 46] with 5 landmarks. Like [37] we
use 10,122 faces for training. We additionally evaluate qualitatively on a dataset of cat faces [45],
using 8609 images for training.
Qualitative assessment. We find that for network S IMPLE the negative log-likelihood loss, while
performing best for the simple example of the arm, performs poorly on faces. Specifically, this model
7
Figure 6: Sphere equivariant labelling. Top: video frames of a rotating textured sphere. Middle:
learned dense labels, which change equivariantly with the sphere. Bottom: re-projection of the video
frames on the object frame (also spherical). Except for occlusions, the reprojections are approximately
invariant, correctly mapping the blue and orange sides to different regions of the label space
fails to disambiguate the left and right eye, as shown in Figure 9 (right). The distance-based loss (5)
produces a more coherent embedding, as seen in Figure 9 (left). Using D ILATIONS this problem
disappears, giving qualitatively smooth and unambiguous labels for both the distance loss (Figure 7)
and the log-likelihood loss (Figure 8). For cats our method is able to learn a consistent object frame
despite large variations in appearance (Figure 8).
Figure 7: Faces. D ILATIONS network with Ldist , ? = 0.5. Top: Input images, Middle: Predicted
dense labels mapped to colours, Bottom: Image pixels mapped to label sphere and flattened.
Figure 8: Cats. D ILATIONS network with Llog . Top: Input images, Middle: Labels mapped to
colours, Bottom: Images mapped to the spherical object frames.
8
Figure 9: Annotated landmark prediction from the shown unsupervised label maps (S IMPLE
network). Left: Trained with Ldist , ? = 0.5, Right: Failure to disambiguate eyes with Llog .
(Prediction: green, Ground truth: Blue)
Regressing semantic landmarks. We would like to quantify the accuracy of our model in terms of
ability to consistently locate manually annotated points, specifically the eyes, nose, and mouth corners
given in the CelebA dataset. We use the standard test split for evaluation of the MAFL dataset [47],
containing 1000 images. We also use the MAFL training subset of 19k images for learning to predict
the ground truth landmarks, which gives a quantitative measure of the consistency of our object frame
for detecting facial features. These are reported as Euclidean error normalized as a percentage of
inter-ocular distance.
In order to map the object frame to the semantic landmarks, as in the case of the robot arm centers, we
learn the vectors zk ? Z corresponding to the position of each point in our canonical reference space
and then, for any given image, find the nearest z and its corresponding pixel location u. We report
the localization performance of this model in Table 3 (?Error Nearest?). We empirically validate
that with the S IMPLE network the negative log-likelihood is not ideal for this task (Figure 9) and
we obtain higher performance for the robust distance with power 0.5. However, after switching to
D ILATIONS to increase the receptive field both methods perform comparably.
The method of [37] learns to regress P ground truth coordinates based on M > P unsupervised
landmarks. By regressing from multiple points it is not limited to integer pixel coordinates. While we
are not predicting landmarks as network output, we can emulate this method by allowing multiple
points in our object coordinate space to be predictive for a single ground truth landmark. We learn
one regressor per ground truth point, each formulated as a linear regressor R2M ? R2 on top of
coordinates from M = 50 learned intermediate points. This allows the regression to say which points
in Z are most useful for predicting each ground truth point.
We also report results after unsupervised finetuning of a CelebA network to the more challenging
AFLW followed by regressor training on AFLW. As shown in Tables 3 and 4, we outperform other
unsupervised methods on both datasets, and are comparable to fully supervised methods.
Network
Unsup. Loss
S IMPLE
S IMPLE
S IMPLE
Llog
Ldist , ? = 1
Ldist , ? = 0.5
Llog
Ldist , ? = 0.5
Error
Nearest
Error
Regress
?
7.94 %
7.18 %
D ILATIONS
5.83 %
D ILATIONS
5.87 %
[37]
6.67 %
Table 3: Nearest neighbour and regression
landmark prediction on MAFL
5
75.02 %
14.57 %
13.29 %
11.05 %
10.53 %
Method
Error
RCPR [5]
11.6 %
Cascaded CNN [35]
8.97 %
CFAN [43]
10.94 %
TCDCN [47]
7.65 %
RAR [40]
7.23 %
Unsup. Landmarks [37]
10.53 %
D ILATIONS Ldist , ? = 0.5
8.80 %
Table 4: Comparison with supervised and unsupervised methods on AFLW
Conclusions
Building on the idea of viewpoint factorization, we have introduce a new method that can endow
an object or object category with an invariant dense geometric embedding automatically, by simply
observing a large dataset of unlabelled images. Our learning framework combines in a novel way
the concept of equivariance with the one of distinctiveness. We have also proposed a concrete
implementation using novel losses to learn a deep dense image labeller. We have shown empirically
that the method can learn a consistent geometric embedding for a simple articulated synthetic robotic
arm as well as for a 3D sphere model and real faces. The resulting embeddings are invariant to
deformations and, importantly, to intra-category variations.
Acknowledgments: This work acknowledges the support of the AIMS CDT (EPSRC EP/L015897/1) and ERC
677195-IDIU. Clipart: FreePik.
9
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6,286 | 6,687 | Compression-aware Training of Deep Networks
Mathieu Salzmann
EPFL - CVLab
Lausanne, Switzerland
[email protected]
Jose M. Alvarez
Toyota Research Institute
Los Altos, CA 94022
[email protected]
Abstract
In recent years, great progress has been made in a variety of application domains
thanks to the development of increasingly deeper neural networks. Unfortunately,
the huge number of units of these networks makes them expensive both computationally and memory-wise. To overcome this, exploiting the fact that deep networks
are over-parametrized, several compression strategies have been proposed. These
methods, however, typically start from a network that has been trained in a standard manner, without considering such a future compression. In this paper, we
propose to explicitly account for compression in the training process. To this end,
we introduce a regularizer that encourages the parameter matrix of each layer to
have low rank during training. We show that accounting for compression during
training allows us to learn much more compact, yet at least as effective, models
than state-of-the-art compression techniques.
1
Introduction
With the increasing availability of large-scale datasets, recent years have witnessed a resurgence of
interest for Deep Learning techniques. Impressive progress has been made in a variety of application
domains, such as speech, natural language and image processing, thanks to the development of new
learning strategies [15, 53, 30, 45, 26, 3] and of new architectures [31, 44, 46, 23]. In particular, these
architectures tend to become ever deeper, with hundreds of layers, each of which containing hundreds
or even thousands of units.
While it has been shown that training such very deep architectures was typically easier than smaller
ones [24], it is also well-known that they are highly over-parameterized. In essence, this means that
equally good results could in principle be obtained with more compact networks. Automatically
deriving such equivalent, compact models would be highly beneficial in runtime- and memorysensitive applications, e.g., to deploy deep networks on embedded systems with limited hardware
resources. As a consequence, many methods have been proposed to compress existing architectures.
An early trend for such compression consisted of removing individual parameters [33, 22] or entire
units [36, 29, 38] according to their influence on the output. Unfortunately, such an analysis of
individual parameters or units quickly becomes intractable in the presence of very deep networks.
Therefore, currently, one of the most popular compression approaches amounts to extracting low-rank
approximations either of individual units [28] or of the parameter matrix/tensor of each layer [14].
This latter idea is particularly attractive, since, as opposed to the former one, it reduces the number of
units in each layer. In essence, the above-mentioned techniques aim to compress a network that has
been pre-trained. There is, however, no guarantee that the parameter matrices of such pre-trained
networks truly have low-rank. Therefore, these methods typically truncate some of the relevant
information, thus resulting in a loss of prediction accuracy, and, more importantly, do not necessarily
achieve the best possible compression rates.
In this paper, we propose to explicitly account for compression while training the initial deep network.
Specifically, we introduce a regularizer that encourages the parameter matrix of each layer to have
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
low rank in the training loss, and rely on a stochastic proximal gradient descent strategy to optimize
the network parameters. In essence, and by contrast with methods that aim to learn uncorrelated units
to prevent overfitting [5, 54, 40], we seek to learn correlated ones, which can then easily be pruned
in a second phase. Our compression-aware training scheme therefore yields networks that are well
adapted to the following post-processing stage. As a consequence, we achieve higher compression
rates than the above-mentioned techniques at virtually no loss in prediction accuracy.
Our approach constitutes one of the very few attempts at explicitly training a compact network
from scratch. In this context, the work of [4] has proposed to learn correlated units by making
use of additional noise outputs. This strategy, however, is only guaranteed to have the desired
effect for simple networks and has only been demonstrated on relatively shallow architectures. In
the contemporary work [51], units are coordinated via a regularizer acting on all pairs of filters
within a layer. While effective, exploiting all pairs can quickly become cumbersome in the presence
of large numbers of units. Recently, group sparsity has also been employed to obtain compact
networks [2, 50]. Such a regularizer, however, acts on individual units, without explicitly aiming to
model their redundancies. Here, we show that accounting for interactions between the units within a
layer allows us to obtain more compact networks. Furthermore, using such a group sparsity prior in
conjunction with our compression-aware strategy lets us achieve even higher compression rates.
We demonstrate the benefits of our approach on several deep architectures, including the 8-layers
DecomposeMe network of [1] and the 50-layers ResNet of [23]. Our experiments on ImageNet and
ICDAR show that we can achieve compression rates of more than 90%, thus hugely reducing the
number of required operations at inference time.
2
Related Work
It is well-known that deep neural networks are over-parametrized [13]. While, given sufficient
training data, this seems to facilitate the training procedure, it also has two potential drawbacks. First,
over-parametrized networks can easily suffer from overfitting. Second, even when they can be trained
successfully, the resulting networks are expensive both computationally and memory-wise, thus
making their deployment on platforms with limited hardware resources, such as embedded systems,
challenging. Over the years, much effort has been made to overcome these two drawbacks.
In particular, much progress has been made to reduce overfitting, for example by devising new
optimization strategies, such as DropOut [45] or MaxOut [16]. In this context, other works have
advocated the use of different normalization strategies, such as Batch Normalization [26], Weight
Normalization [42] and Layer Normalization [3]. Recently, there has also been a surge of methods
aiming to regularize the network parameters by making the different units in each layer less correlated.
This has been achieved by designing new activation functions [5], by explicitly considering the
pairwise correlations of the units [54, 37, 40] or of the activations [9, 52], or by constraining the
weight matrices of each layer to be orthonormal [21].
In this paper, we are more directly interested in addressing the second drawback, that is, the large
memory and runtime required by very deep networks. To tackle this, most existing research has
focused on pruning pre-trained networks. In this context, early works have proposed to analyze the
saliency of individual parameters [33, 22] or units [36, 29, 38, 34], so as to measure their impact on
the output. Such a local analysis, however, quickly becomes impractically expensive when dealing
with networks with millions of parameters.
As a consequence, recent works have proposed to focus on more global methods, which analyze
larger groups of parameters simultaneously. In this context, the most popular trend consists of
extracting low-rank approximations of the network parameters. In particular, it has been shown that
individual units can be replaced by rank 1 approximations, either via a post-processing step [28, 46]
or directly during training [1, 25]. Furthermore, low-rank approximations of the complete parameter
matrix/tensor of each layer were computed in [14], which has the benefit of reducing the number of
units in each layer. The resulting low-rank representation can then be fine-tuned [32], or potentially
even learned from scratch [47], given the rank of each layer in the network. With the exception of
this last work, which assumes that the ranks are known, these methods, however, aim to approximate
a given pre-trained model. In practice, however, the parameter matrices of this model might not have
low rank. Therefore, the resulting approximations yield some loss of accuracy and, more importantly,
2
will typically not correspond to the most compact networks. Here, we propose to explicitly learn a
low-rank network from scratch, but without having to manually define the rank of each layer a priori.
To this end, and in contrast with the above-mentioned methods that aim to minimize correlations,
we rather seek to maximize correlations between the different units within each layer, such that
many of these units can be removed in a post-processing stage. In [4], additional noise outputs were
introduced in a network to similarly learn correlated filters. This strategy, however, is only justified for
simple networks and was only demonstrated on relatively shallow architectures. The contemporary
work [51] introduced a penalty during training to learn correlated units. This, however, was achieved
by explicitly computing all pairwise correlations, which quickly becomes cumbersome in very deep
networks with wide layers. By contrast, our approach makes use of a low-rank regularizer that can
effectively be optimized by proximal stochastic gradient descent.
Our approach belongs to the relatively small group of methods that explicitly aim to learn a compact
network during training, i.e., not as a post-processing step. Other methods have proposed to make
use of sparsity-inducing techniques to cancel out individual parameters [49, 10, 20, 19, 35] or
units [2, 50, 55]. These methods, however, act, at best, on individual units, without considering the
relationships between multiple units in the same layer. Variational inference [17] has also been used
to explicitly compress the network. However, the priors and posteriors used in these approaches
will typically zero out individual weights. Our experiments demonstrate that accounting for the
interactions between multiple units allows us to obtain more compact networks.
Another line of research aims to quantize the weights of deep networks [48, 12, 18]. Note that,
in a sense, this research direction is orthogonal to ours, since one could still further quantize our
compact networks. Furthermore, with the recent progress in efficient hardware handling floating-point
operations, we believe that there is also high value in designing non-quantized compact networks.
3
Compression-aware Training of Deep Networks
In this section, we introduce our approach to explicitly encouraging compactness while training a
deep neural network. To this end, we propose to make use of a low-rank regularizer on the parameter
matrix in each layer, which inherently aims to maximize the compression rate when computing a
low-rank approximation in a post-processing stage. In the following, we focus on convolutional neural
networks, because the popular visual recognition models tend to rely less and less on fully-connected
layers, and, more importantly, the inference time of such models is dominated by the convolutions in
the first few layers. Note, however, that our approach still applies to fully-connected layers.
To introduce our approach, let us first consider the l-th layer of a convolutional network, and denote
H
W
its parameters by ?l ? RKl ?Cl ?dl ?dl , where Cl and Kl are the number of input and output channels,
H
W
respectively, and dl and dl are the height and width of each convolutional kernel. Alternatively,
these parameters can be represented by a matrix ??l ? RKl ?Sl with Sl = Cl dlH dlW . Following [14], a
network can be compacted via a post-processing step performing a singular value decomposition of
??l and truncating the 0, or small, singular values. In essence, after this step, the parameter matrix
can be approximated as ??l ? Ul MlT , where Ul is a Kl ? rl matrix representing the basis kernels, with
rl ? min(Kl , Sl ), and Ml is an Sl ? rl matrix that mixes the activations of these basis kernels.
By making use of a post-processing step on a network trained in the usual way, however, there is
no guarantee that, during training, many singular values have become near-zero. Here, we aim to
explicitly account for this post-processing step during training, by seeking to obtain a parameter
matrix such that rl << min(Kl , Sl ). To this end, given N training input-output pairs (xi , yi ), we
formulate learning as the regularized minimization problem
min
?
1 N
? `(yi , f (xi , ?)) + r(?) ,
N i=1
(1)
where ? encompasses all network parameters, `(?, ?) is a supervised loss, such as the cross-entropy,
and r(?) is a regularizer encouraging the parameter matrix in each layer to have low rank.
Since explicitly minimizing the rank of a matrix is NP-hard, following the matrix completion
literature [7, 6], we make use of a convex relaxation in the form of the nuclear norm. This lets us
3
write our regularizer as
L
r(?) = ? ? k??l k? ,
(2)
l=1
where ? is a hyper-parameter setting the influence of the regularizer, and the nuclear norm is defined
rank(??l ) j
as k??l k? = ?
? , with ? j the singular values of ??l .
j=1
l
l
In practice, to minimize (1), we make use of proximal stochastic gradient descent. Specifically,
this amounts to minimizing the supervised loss only for one epoch, with learning rate ?, and then
applying the proximity operator of our regularizer. In our case, this can be achieved independently
for each layer. For layer l, this proximity operator corresponds to solving
?l? = argmin
??l
1 ?
k?l ? ??l k2F + ?k??l k? ,
2?
(3)
where ??l is the current estimate of the parameter matrix for layer l. As shown in [6], the solution to
this problem can be obtained by soft-thresholding the singular values of ??l , which can be written as
?l? = Ul ?l (??)VlT ,
rank(??l )
where ?l (??) = diag([(?l1 ? ??)+ , . . . , (?l
? ??)+ ]),
(4)
Ul and Vl are the left - and right-singular vectors of ??l , and (?)+ corresponds to taking the maximum
between the argument and 0.
3.1
Low-rank and Group-sparse Layers
While, as shown in our experiments, the low-rank solution discussed above significantly reduces
the number of parameters in the network, it does not affect the original number of input and output
channels Cl and Kl . By contrast, the group-sparsity based methods [2, 50] discussed in Section 2
cancel out entire units, thus reducing these numbers, but do not consider the interactions between
multiple units in the same layer, and would therefore typically not benefit from a post-processing
step such as the one of [14]. Here, we propose to make the best of both worlds to obtain low-rank
parameter matrices, some of whose units have explicitly been removed.
To this end, we combine the sparse group Lasso regularizer used in [2] with the low-rank one
described above. This lets us re-define the regularizer in (1) as
!
L
L
p Kl
n
r(?) = ? (1 ? ?)?l Pl ? k?l k2 + ??l k?l k1 + ? ? k??l k? ,
(5)
l=1
n=1
l=1
?ln
where Kl is the number of units in layer l,
denotes the vector of parameters for unit n in layer
l, Pl is the size of this vector (the same for all units in a layer), ? ? [0, 1] balances the influence
of sparsity terms on groups vs. individual parameters, and ?l is a layer-wise hyper-parameter. In
practice, following [2], we use only two different values of ?l ; one for the first few layers and one for
the remaining ones.
To learn our model with this new regularizer consisting of two main terms, we make use of the
incremental proximal descent approach proposed in [39], which has the benefit of having a lower
memory footprint than parallel proximal methods. The proximity operator for the sparse group Lasso
regularizer also has a closed form solution derived in [43] and provided in [2].
3.2
Benefits at Inference
Once our model is trained, we can obtain a compact network for faster and more memory-efficient
inference by making use of a post-processing step. In particular, to account for the low rank of the
parameter matrix of each layer, we make use of the SVD-based approach of [14]. Specifically, for
each layer l, we compute the SVD of the parameter matrix as ??l = U? l ?? l V?l and only keep the rl singular
values that are either non-zero, thus incurring no loss, or larger than a pre-defined threshold, at some
H W
potential loss. The parameter matrix can then be represented as ??l = Ul Ml , with Ul ? RCl dl dl ?rl and
Ml = ?l Vl ? Rrl ?Kl ) . In essence, every layer is decomposed into two layers. This incurs significant
memory and computational savings if rl (Cl dlH dlW + Kl ) << (Cl dlH dlW Kl ).
4
Furthermore, additional savings can be achieved when using the sparse group Lasso regularizer
discussed in Section 3.1. Indeed, in this case, the zeroed-out units can explicitly be removed, thus
yielding only K? l filters, with K? l < Kl . Note that, except for the first layer, units have also been
removed from the previous layer, thus reducing Cl to a lower C?l . Furthermore, thanks to our low-rank
regularizer, the remaining, non-zero, units will form a parameter matrix that still has low rank, and
can thus also be decomposed. This results in a total of rl (C?l dlH dlW + K? l ) parameters.
In our experiments, we select the rank rl based on the percentage el of the energy (i.e., the sum of
singular values) that we seek to capture by our low-rank approximation. This percentage plays an
important role in the trade-off between runtime/memory savings and drop of prediction accuracy. In
our experiments, we use the same percentage for all layers.
4
Experimental Settings
Datasets: For our experiments, we used two image classification datasets: ImageNet [41] and
ICDAR, the character recognition dataset introduced in [27]. ImageNet is a large-scale dataset
comprising over 15 million labeled images split into 22, 000 categories. We used the ILSVRC2012 [41] subset consisting of 1000 categories, with 1.2 million training images and 50, 000 validation
images. The ICDAR dataset consists of 185,639 training samples combining real and synthetic
characters and 5,198 test samples coming from the ICDAR2003 training set after removing all
non-alphanumeric characters. The images in ICDAR are split into 36 categories. The use of ICDAR
here was motivated by the fact that it is fairly large-scale, but, in contrast with ImageNet, existing
architectures haven?t been heavily tuned to this data. As such, one can expect our approach consisting
of training a compact network from scratch to be even more effective on this dataset.
Network Architectures: In our experiments, we make use of architectures where each kernel
in the convolutional layers has been decomposed into two 1D kernels [1], thus inherently having
rank-1 kernels. Note that this is orthogonal to the purpose of our low-rank regularizer, since, here,
we essentially aim at reducing the number of kernels, not the rank of individual kernels. The
decomposed layers yield even more compact architectures that require a lower computational cost for
training and testing while maintaining or even improving classification accuracy. In the following, a
convolutional layer refers to a layer with 1D kernels, while a decomposed layer refers to a block of
two convolutional layers using 1D vertical and horizontal kernels, respectively, with a non-linearity
and batch normalization after each convolution.
Let us consider a decomposed layer consisting of C and K input and output channels, respectively.
Let v? and h? T be vectors of length d v and d h , respectively, representing the kernel size of each 1D
feature map. In this paper, we set d h = d v ? d. Furthermore, let ?(?) be a non-linearity, and xc denote
the c-th input channel of the layer. In this setting, the activation of the i-th output channel fi can be
written as
L
C
fi = ?(bhi + ? h? Til ? [?(bvl + ? v?lc ? xc )]),
(6)
c=1
l=1
where L is the number of vertical filters, corresponding to the number of input channels for the
horizontal filters, and bvl and bhl are biases.
We report results with two different models using such decomposed layers: DecomposeMe [1] and
ResNets [23]. In all cases, we make use of batch-normalization after each convolutional layer 1 . We
rely on rectified linear units (ReLU) [31] as non-linearities, although some initial experiments suggest
that slightly better performance can be obtained with exponential linear units [8]. For DecomposeMe,
we used two different Dec8 architectures, whose specific number of units are provided in Table 1. For
residual networks, we used a decomposed ResNet-50, and empirically verified that the use of 1D
kernels instead of the standard ones had no significant impact on classification accuracy.
Implementation details: For the comparison to be fair, all models, including the baselines, were
trained from scratch on the same computer using the same random seed and the same framework.
More specifically, we used the torch-7 multi-gpu framework [11].
1
We empirically found the use of batch normalization after each convolutional layer to have more impact with
our low-rank regularizer than with group sparsity or with no regularizer, in which cases the computational cost
can be reduced by using a single batch normalization after each decomposed layer.
5
Dec256
8
Dec512
8
Dec512
3
1v
32/11
32/11
48/9
1h
64/11
64/11
96/9
2v
128/5
128/5
160/9
2h
192/5
192/5
256/9
3v
256/3
256/3
512/8
3h
384/3
384/3
512/8
4v
256/3
256/3
?
4h
256/3
256/3
?
5v
256/3
512/3
?
5h
256/3
512/3
?
6v
256/3
512/3
?
6h
256/3
512/3
?
7v
256/3
512/3
?
7h
256/3
512/3
?
8v
256/3
512/3
?
8h
256/3
512/3
?
Table 1: Different DecomposeMe architectures used on ImageNet and ICDAR. Each entry represents the number of filters and their dimension.
Layer / Conf?
1v to 2h
3v to 8h
0
?
?
1
0.0127
0.0127
2
0.051
0.051
3
0.204
0.204
4
0.255
0.255
5
0.357
0.357
6
0.051
0.357
7
0.051
0.408
8
0.051
0.510
9
0.051
0.255
10
0.051
0.765
11
0.153
0.51
Table 2: Sparse group Lasso hyper-parameter configurations. The first row provides ? for the
first four convolutional layers, while the second one shows ? for the remaining layers. The first five
configurations correspond to using the same regularization penalty for all the layers, while the later
ones define weaker penalties on the first two layers, as suggested in [2].
For ImageNet, training was done on a DGX-1 node using two-P100 GPUs in parallel. We used
stochastic gradient descent with a momentum of 0.9 and a batch size of 180 images. The models were
trained using an initial learning rate of 0.1 multiplied by 0.1 every 20 iterations for the small models
512
(Dec256
8 in Table 1) and every 30 iterations for the larger models (Dec8 in Table 1). For ICDAR, we
trained each network on a single TitanX-Pascal GPU for a total of 55 epochs with a batch size of
256 and 1,000 iterations per epoch. We follow the same experimental setting as in [2]: The initial
learning rate was set to an initial value of 0.1 and multiplied by 0.1. We used a momentum of 0.9.
For DecomposeMe networks, we only performed basic data augmentation consisting of using random
crops and random horizontal flips with probability 0.5. At test time, we used a single central crop.
For ResNets, we used the standard data augmentation advocated for in [23]. In practice, in all models,
we also included weight decay with a penalty strength of 1e?4 in our loss function. We observed
empirically that adding this weight decay prevents the weights to overly grow between every two
computations of the proximity operator.
In terms of hyper-parameters, for our low-rank regularizer, we considered four values: ? ? {0, 1, 5, 10}.
For the sparse group Lasso term, we initially set the same ? to every layer to analyze the effect of
combining both types of regularization. Then, in a second experiment, we followed the experimental
set-up proposed in [2], where the first two decomposed layers have a lower penalty. In addition, we
set ? = 0.2 to favor promoting sparsity at group level rather than at parameter level. The sparse group
Lasso hyper-parameter values are summarized in Table 2.
Computational cost: While a convenient measure of computational cost is the forward time,
this measure is highly hardware-dependent. Nowadays, hardware is heavily optimized for current
architectures and does not necessarily reflect the concept of any-time-computation. Therefore, we
focus on analyzing the number of multiply-accumulate operations (MAC). Let a convolution be
defined as fi = ?(bi + ?Cj=1 Wi j ? x j ), where each Wi j is a 2D kernel of dimensions d H ? dW and
i ? [1, . . . K]. Considering a naive convolution algorithm, the number of MACs for a convolutional
layer is equal to PCKd h dW where P is the number of pixels in the output feature map. Therefore,
it is important to reduce CK whenever P is large. That is, reducing the number of units in the first
convolutional layers has more impact than in the later ones.
5
Experimental Results
Parameter sensitivity and comparison to other methods on ImagNet: We first analyze the
effect of our low-rank regularizer on its own and jointly with the sparse group Lasso one on MACs
and accuracy. To this end, we make use of the Dec256
8 model on ImageNet, and measure the impact
of varying both ? and ? in Eq. 5. Note that using ? = ? = 0 corresponds to the standard model,
and ? = 0 and ? 6= 0 to the method of [2]. Below, we report results obtained without and with the
post-processing step described in Section 3.2. Note that applying such a post-processing on the
standard model corresponds to the compression technique of [14]. Fig. 1 summarizes the results of
this analysis.
6
(a)
(b)
(c)
Figure 1: Parameter sensitivity for Dec256
on ImageNet. (a) Accuracy as a function of the
8
regularization strength. (b) MACs directly after training. (c) MACS after the post-processing step
of Section 3.2 for el = {100%, 80%}. In all the figures, isolated points represent the models trained
without sparse group Lasso regularizer. The red point corresponds to the baseline, where no low-rank
or sparsity regularization was applied. The specific sparse group Lasso hyper-parameters for each
configuration Conf? are given in Table 2.
Figure 2: Effect of the low-rank regularizer on its own on Dec256
8 on ImageNet. (Left) Number
of units per layer. (Right) Effective rank per layer for (top) el =100% and (bottom) el =80%. Note that,
on its own, our low-rank regularizer already helps cancel out entire units, thus inherently performing
model selection.
In Fig. 1(a), we can observe that accuracy remains stable for a wide range of values of ? and ? . In
fact, there are even small improvements in accuracy when a moderate regularization is applied.
Figs. 1(b,c) depict the MACs without and with applying the post-processing step discussed in
Section 3.2. As expected, the MACs decrease as the weights of the regularizers increase. Importantly,
however, Figs. 1(a,b) show that several models can achieve a high compression rate at virtually no
loss in accuracy. In Fig. 1(c), we provide the curves after post-processing with two different energy
percentages el = {100%, 80%}. Keeping all the energy tends to incur an increase in MAC, since
the inequality defined in Section 3.2 is then not satisfied anymore. Recall, however, that, without
post-processing, the resulting models are still more compact than and as accurate as the baseline one.
With el = 80%, while a small drop in accuracy typically occurs, the gain in MAC is significantly
larger. Altogether, these experiments show that, by providing more compact models, our regularizer
lets us consistently reduce the computational cost over the baseline.
Interestingly, by looking at the case where Conf? = 0 in Fig. 1(b), we can see that we already
significantly reduce the number of operations when using our low-rank regularizer only, even without
post-processing. This is due to the fact that, even in this case, a significant number of units are
automatically zeroed-out. Empirically, we observed that, for moderate values of ?, the number of
zeroed-out singular values corresponds to complete units going to zero. This can be observed in
Fig. 2(left), were we show the number of non-zero units for each layer. In Fig. 2(right), we further
show the effective rank of each layer before and after post-processing.
7
hyper-params
# Params
top-1
Baseline
?
3.7M
88.6%
[14]
el = 90%
3.6M
88.5%
[2]
?
525K
89.6%
Ours
? = 15, el = 90%
728K
88.8%
Ours+[2]
? = 15, el = 90%
318K
89.7%
Ours+[2]
? = 15, el = 100%
454K
90.5%
Table 3: Comparison to other methods on ICDAR.
Imagenet
Dec512
8 -el =80%
Dec512
8 -el =100%
Top-1
66.8
67.6
Low-Rank appx.
Params MAC
-53.5
-46.2
-21.1
-4.8
no SVD
Params MAC
-39.5
-25.3
-39.5
-25.3
ICDAR
Dec512
3 -el =80%
Dec512
3 -el =100%
Top-1
89.6
90.8
Low-Rank appx.
Params MAC
-91.9
-92.9
-85.3
-86.8
no SVD
Params MAC
-89.2
-81.6
-89.2
-81.6
512
Table 4: Accuracy and compression rates for Dec512
8 models on ImageNet (left) and Dec3 on
ICDAR (right). The number of parameters and MACs are given in % relative to the baseline model
(i.e., without any regularizer). A negative value indicates reduction with respect to the baseline. The
accuracy of the baseline is 67.0 for ImageNet and 89.3 for ICDAR.
Dec256
8 -? = 1
Dec256
8 -? = 5
Dec256
8 -? = 10
el = 80%
97.33
88.33
85.78
el = 100%
125.44
119.27
110.35
no SVD
94.60
90.55
91.36
baseline (? = 0)
94.70
94.70
94.70
Table 5: Forward time in milliseconds (ms) using a Titan X (Pascal). We report the average over
50 forward passes using a batch size of 256. A large batch size minimizes the effect of memory
overheads due to non-hardware optimizations.
Comparison to other approaches on ICDAR: We now compare our results with existing approaches on the ICDAR dataset. As a baseline, we consider the Dec512
3 trained using SGD and L2
regularization for 75 epochs. For comparison, we consider the post-processing approach in [14]
with el = 90%, the group-sparsity regularization approach proposed in [2] and three different instances of our model. First, using ? = 15, no group-sparsity and el = 90%. Then, two instances
combining our low-rank regularizer with group-sparsity (Section 3.1) with el = 90% and el = 100%.
In this case, the models are trained for 55 epochs and then reloaded and fine tuned for 20 more
epochs. Table 3 summarizes these results. The comparison with [14] clearly evidences the benefits
of our compression-aware training strategy. Furthermore, these results show the benefits of further
combining our low-rank regularizer with the groups-sparsity one of [2].
In addition, we also compare our approach with L1 and L2 regularizers on the same dataset and with
the same experimental setup. Pruning the weights of the baseline models with a threshold of 1e ? 4
resulted in 1.5M zeroed-out parameters for the L2 regularizer and 2.8M zeroed-out parameters for
the L1 regularizer. However, these zeroed out weights are sparsely located within units (neurons).
Applying our post-processing step (low-rank approximation with el = 100%) to these results yielded
models with 3.6M and 3.2M parameters for L2 and L1 regularizers, respectively. The top-1 accuracy
for these two models after post-processing was 87% and 89%, respectively. Using a stronger L1
regularizer resulted in lower top-1 accuracy. By comparison, our approach yields a model with 3.4M
zeroed-out parameters after post-processing and a top-1 accuracy of 90%. Empirically, we found the
benefits of our approach to hold for varying regularizer weights.
Results with larger models: In Table 4, we provide the accuracies and MACs for our approach and
the baseline on ImageNet and ICDAR for Dec512
8 models. Note that using our low-rank regularizer
yields more compact networks than the baselines for similar or higher accuracies. In particular, for
ImageNet, we achieve reductions in parameter number of more than 20% and more than 50% for
el = 100% and el = 80%, respectively. For ICDAR, these reductions are around 90% in both cases.
We now focus on our results with a ResNet-50 model on ImageNet. For post-processing we used
el = 90% for all these experiments which resulted in virtually no loss of accuracy. The baseline
corresponds to a top-1 accuracy of 74.7% and 18M parameters. Applying the post-processing step on
this baseline resulted in a compression rate of 4%. By contrast, our approach with low-rank yields
a top-1 accuracy of 75.0% for a compression rate of 20.6%, and with group sparsity and low-rank
8
Baseline
r5
r15
r25
r35
r45
r55
r65
Epoch
reload
?
5
15
25
35
45
55
65
Num. parameters
Total no SVD
3.7M
?
3.2M
3.71M
210K
2.08M
218K
1.60M
222K
1.52M
324K
1.24M
388K
1.24M
414K
1.23M
accuracy
top-1
88.4%
89.8%
90.0%
90.0%
89.0%
90.1%
89.2%
87.7%
Total
train-time
1.69h
1.81h
0.77h
0.88h
0.99h
1.12h
1.26h
1.36h
Figure 3: Forward-Backward training time in milliseconds when varying the reload epoch for
Dec512
3 on ICDAR. (Left) Forward-backward time per batch in milliseconds (with a batch size of
32). (Right) Summary of the results of each experiment. Note that we could reduce the training time
from 1.69 hours (baseline) to 0.77 hours by reloading the model at the 15th epoch. This corresponds
to a relative training-time speed up of 54.5% and yields a 2% improvement in top-1 accuracy.
jointly, a top-1 accuracy of 75.2% for a compression rate of 27%. By comparison, applying [2] to the
same model yields an accuracy of 74.5% for a compression rate of 17%.
Inference time: While MACs represent the number of operations, we are also interested in the
inference time of the resulting models. Table 5 summarizes several representative inference times
for different instances of our experiments. Interestingly, there is a significant reduction in inference
time when we only remove the zeroed-out neurons from the model. This is a direct consequence
of the pruning effect, especially in the first layers. However, there is no significant reduction in
inference time when post-processing our model via a low-rank decomposition. The main reason for
this is that modern hardware is designed to compute convolutions with much fewer operations than
a naive algorithm. Furthermore, the actual computational cost depends not only on the number of
floating point operations but also on the memory bandwidth. In modern architectures, decomposing a
convolutional layer into a convolution and a matrix multiplication involves (with current hardware)
additional intermediate computations, as one cannot reuse convolutional kernels. Nevertheless, we
believe that our approach remains beneficial for embedded systems using customized hardware, such
as FPGAs.
Additional benefits at training time: So far, our experiments have demonstrated the effectiveness
of our approach at test time. Empirically, we found that our approach is also beneficial for training,
by pruning the network after only a few epochs (e.g., 15) and reloading and training the pruned
network, which becomes much more efficient. Specifically, Table 3 summarizes the effect of varying
the reload epoch for a model relying on both low-rank and group-sparsity. We were able to reduce the
training time (with a batch size of 32 and training for 100 epochs) from 1.69 to 0.77 hours (relative
speedup of 54.5%). The accuracy also improved by 2% and the number of parameters reduced from
3.7M (baseline) to 210K (relative 94.3% reduction). We found this behavior to be stable across a
wide range of regularization parameters. If we seek to maintain accuracy compared to the baseline,
we found that we could achieve a compression rate of 95.5% (up to 96% for an accuracy drop of
0.5%), which corresponds to a training time reduced by up to 60%.
6
Conclusion
In this paper, we have proposed to explicitly account for a post-processing compression stage when
training deep networks. To this end, we have introduced a regularizer in the training loss to encourage
the parameter matrix of each layer to have low rank. We have further studied the case where this
regularizer is combined with a sparsity-inducing one to achieve even higher compression. Our
experiments have demonstrated that our approach can achieve higher compression rates than state-ofthe-art methods, thus evidencing the benefits of taking compression into account during training. The
SVD-based technique that motivated our approach is only one specific choice of compression strategy.
In the future, we will therefore study how regularizers corresponding to other such compression
mechanisms can be incorporated in our framework.
9
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12
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6,287 | 6,688 | Multiscale Semi-Markov Dynamics for
Intracortical Brain-Computer Interfaces
Daniel J. Milstein ?
[email protected]
John D. Simeral ? ?
[email protected]
Jason L. Pacheco ?
Leigh R. Hochberg ? ? ?
[email protected]
[email protected]
Beata Jarosiewicz k ? ??
[email protected]
Erik B. Sudderth ?? ?
[email protected]
Abstract
Intracortical brain-computer interfaces (iBCIs) have allowed people with tetraplegia
to control a computer cursor by imagining the movement of their paralyzed arm
or hand. State-of-the-art decoders deployed in human iBCIs are derived from a
Kalman filter that assumes Markov dynamics on the angle of intended movement,
and a unimodal dependence on intended angle for each channel of neural activity.
Due to errors made in the decoding of noisy neural data, as a user attempts to
move the cursor to a goal, the angle between cursor and goal positions may change
rapidly. We propose a dynamic Bayesian network that includes the on-screen goal
position as part of its latent state, and thus allows the person?s intended angle of
movement to be aggregated over a much longer history of neural activity. This
multiscale model explicitly captures the relationship between instantaneous angles
of motion and long-term goals, and incorporates semi-Markov dynamics for motion
trajectories. We also introduce a multimodal likelihood model for recordings
of neural populations which can be rapidly calibrated for clinical applications.
In offline experiments with recorded neural data, we demonstrate significantly
improved prediction of motion directions compared to the Kalman filter. We derive
an efficient online inference algorithm, enabling a clinical trial participant with
tetraplegia to control a computer cursor with neural activity in real time. The
observed kinematics of cursor movement are objectively straighter and smoother
than prior iBCI decoding models without loss of responsiveness.
1
Introduction
Paralysis of all four limbs from injury or disease, or tetraplegia, can severely limit function, independence, and even sometimes communication. Despite its inability to effect movement in muscles,
neural activity in motor cortex still modulates according to people?s intentions to move their paralyzed
arm or hand, even years after injury [Hochberg et al., 2006, Simeral et al., 2011, Hochberg et al.,
?
Department of Computer Science, Brown University, Providence, RI, USA.
Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA.
?
School of Engineering, Brown University, Providence, RI, USA; and Department of Neurology, Massachusetts General Hospital, Boston, MA, USA.
?
Rehabilitation R&D Service, Department of Veterans Affairs Medical Center, Providence, RI, USA; and
Brown Institute for Brain Science, Brown University, Providence, RI, USA.
?
Department of Neurology, Harvard Medical School, Boston, MA, USA.
k
Department of Neuroscience, Brown University, Providence, RI, USA.
??
Present affiliation: Dept. of Neurosurgery, Stanford University, Stanford, CA, USA.
??
Department of Computer Science, University of California, Irvine, CA, USA.
?
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Figure 1: A microelectrode array (left) is implanted in the motor cortex (center) to record electrical activity.
Via this activity, a clinical trial participant (right, lying on his side in bed) then controls a computer cursor with
an iBCI. A cable connected to the electrode array via a transcutaneous connector (gray box) sends neural signals
to the computer for decoding. Center drawing from Donoghue et al. [2011] and used with permission of the
author. The right image is a screenshot of a video included in the supplemental material that demonstrates real
time decoding via our MSSM model.
2012, Collinger et al., 2013]. Intracortical brain-computer interfaces (iBCIs) utilize neural signals
recorded from implanted electrode arrays to extract information about movement intentions. They
have enabled individuals with tetraplegia to control a computer cursor to engage in tasks such as
on-screen typing [Bacher et al., 2015, Jarosiewicz et al., 2015, Pandarinath et al., 2017], and to regain
volitional control of their own limbs [Ajiboye et al., 2017].
Current iBCIs are based on a Kalman filter that assumes the vector of desired cursor movement
evolves according to Gaussian random walk dynamics, and that neural activity is a Gaussian-corrupted
linear function of this state [Kim et al., 2008]. In Sec. 2, we review how the Kalman filter is applied
to neural decoding, and studies of the motor cortex by Georgopoulos et al. [1982] that justify its use.
In Sec. 3, we improve upon the Kalman filter?s linear observation model by introducing a flexible,
multimodal likelihood inspired by more recent research [Amirikian and Georgopulos, 2000]. Sec. 4
then proposes a graphical model (a dynamic Bayesian network [Murphy, 2002]) for the relationship
between the angle of intended movement and the intended on-screen goal position. We derive an
efficient inference algorithm via an online variant of the junction tree algorithm [Boyen and Koller,
1998]. In Sec. 5, we use recorded neural data to validate the components of our multiscale semiMarkov (MSSM) model, and demonstrate significantly improved prediction of motion directions in
offline analysis. Via a real time implementation of the inference algorithm on a constrained embedded
system, we then evaluate online decoding performance as a participant in the BrainGate21 iBCI pilot
clinical trial uses the MSSM model to control a computer cursor with his neural activity.
2
Neural decoding via a Kalman filter
The Kalman filter is the current state-of-the-art for iBCI decoding. There are several configurations
of the Kalman filter used to enable cursor control in contemporary iBCI systems [Pandarinath et al.,
2017, Jarosiewicz et al., 2015, Gilja et al., 2015] and there is no broad consensus in the iBCI field on
which is most suited for clinical use. In this paper, we focus on the variant described by Jarosiewicz
et al. [2015].
Participants in the BrainGate2 clinical trial receive one or two microelectrode array implants in the
motor cortex (see Fig. 1). The electrical signals recorded by this electrode array are then transformed
(via signal processing methods designed to reduce noise) into a D-dimensional neural activity vector
zt ? RD , sampled at 50 Hz. From the sequence of neural activity, the Kalman filter estimates the
latent state xt ? R2 , a vector pointing in the intended direction of cursor motion. The Kalman filter
assumes a jointly Gaussian model for cursor dynamics and neural activity,
xt | xt?1 ? N (Axt?1 , W ),
zt | xt ? N (b + Hxt , Q),
2?2
(1)
2?2
with cursor dynamics A ? R , process noise covariance W ? R , and (typically non-diagonal)
observation covariance Q ? RD?D . At each time step, the on-screen cursor?s position is moved by
the estimated latent state vector (decoder output) scaled by a constant, the speed gain.
The function relating neural activity to some measurable quantity of interest is called a tuning curve.
A common model of neural activity in the motor cortex assumes that each neuron?s activity is highest
1
Caution: Investigational Device. Limited by Federal Law to Investigational Use.
2
for some preferred direction of motion, and lowest in the opposite direction, with intermediate activity
often resembling a cosine function. This cosine tuning model is based on pioneering studies of
the motor cortex of non-human primates [Georgopoulos et al., 1982], and is commonly used (or
implicitly assumed) in iBCI systems because of its mathematical simplicity and tractability.
Expressing the inner product between vectors via the cosine of the angle between them, the expected
neural activity of the j th component of Eq. (1) can be written as
hj2
T
E[ztj | xt ] = bj + hj xt = bj + ||xt || ? ||hj || ? cos ?t ? atan
,
(2)
hj1
where ?t is the intended angle of movement at timestep t, bj is the baseline activity rate for channel j,
and hj is the j th row of the observation matrix H = (hT1 , . . . , hTD )T . If xt is further assumed
to be a unit vector (a constraint not enforced by the Kalman filter), Eq. (2) simplifies to hTj xt =
mj cos(?t ? pj ), where mj is the modulation of the tuning curve and pj specifies the angular location
of the peak of the cosine tuning curve (the preferred direction). Thus, cosine tuning models are linear.
To collect labeled training data for decoder calibration, the participant is asked to attempt to move
a cursor to prompted target locations. We emphasize that although the clinical random target
task displays only one target at a time, this target position is unknown to the decoder. Labels are
constructed for the neural activity patterns by assuming that at each 20ms time step, the participant
intends to move the cursor straight to the target [Jarosiewicz et al., 2015, Gilja et al., 2015]. These
labeled data are used to fit the observation matrix H and neuron baseline rates (biases) b via ridge
regression. The observation noise covariance Q is estimated as the empirical covariance of the
residuals. The state dynamics matrix A and process covariance matrix W may be tuned to adjust the
responsiveness of the iBCI system.
3
Flexible tuning likelihoods
The cosine tuning model reviewed in the previous section has several shortcomings. First, motor
cortical neurons that have unimodal tuning curves often have narrower peaks that are better described
by von Mises distributions [Amirikian and Georgopulos, 2000]. Second, tuning can be multimodal.
Third, neural features used for iBCI decoding may capture the pooled activity of several neurons,
not just one [Fraser et al., 2009]. While bimodal von Mises models were introduced by Amirikian
and Georgopulos [2000], up to now iBCI decoders based on von Mises tuning curves have only
employed unimodal mean functions proportional to a single von Mises density [Koyama et al., 2010].
In contrast, we introduce a multimodal likelihood proportional to an arbitrary number of regularly
spaced von Mises densities and incorporate this likelihood into an iBCI decoder. Moreover, we can
efficiently fit parameters of this new likelihood via ridge regression. Computational efficiency is
crucial to allow rapid calibration in clinical applications.
Let ?t ? [0, 2?) denote the intended angle of cursor movement at time t. The flexible tuning likelihood
captures more complex neural activity distributions via a regression model with nonlinear features:
zt | ?t ? N b + wT ?(?t ), Q ,
?k (?t ) = exp [ cos (?t ? ?k )] .
(3)
The features are a set of K von Mises basis functions ?(?) = (?1 (?), . . . , ?K (?))T . Basis functions
?k (x) are centered on a regular grid of angles ?k , and have tunable concentration .
Using human neural data recorded during cued target tasks, we compare regression fits for the flexible
tuning model to the standard cosine tuning model (Fig. 2). In addition to providing better fits for
channels with complex or multimodal activity, the flexible tuning model also provides good fits to
apparently cosine-tuned signals. This leads to higher predictive likelihoods for held-out data, and as
we demonstrate in Sec. 5, more accurate neural decoding algorithms.
4
Multiscale Semi-Markov Dynamical Models
The key observation underlying our multiscale dynamical model is that the sampling rate used for
neural decoding (typically around 50 Hz) is much faster than the rate that the goal position changes
(under normal conditions, every few seconds). In addition, frequent but small adjustments of cursor
aim angle are required to maintain a steady heading. State-of-the-art Kalman filter approaches to iBCIs
3
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Spike power measurement
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Spike power measurement
Spike power measurement
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Figure 2: Flexible tuning curves. Each panel shows the empirical mean and standard deviation (red) of
example neural signals recorded from a single intracortical electrode while a participant is moving within
45 degrees of a given direction in a cued target task. These signals can violate the assumptions of a cosine
tuning model (black), as evident in the left two examples. The flexible regression likelihood (cyan) captures
neural activity with varying concentration (left) and multiple tuning directions (center), as well as cosine-tuned
signals (right). Because neural activity from individual electrodes is very noisy (the standard deviation within
each angular bin exceeds the change in mean activity across angles), information from multiple electrodes is
aggregated over time for effective decoding.
are incapable of capturing these multiscale dynamics since they assume first-order Markov dependence
across time and do not explicitly represent goal position. To cope with this, hyperparameters of the
linear Gaussian dynamics must be tuned to simultaneously remain sensitive to frequent directional
adjustments, but not so sensitive that cursor dynamics are dominated by transient neural activity.
Our proposed MSSM decoder, by contrast, explicitly represents goal position in addition to cursor
aim angle. Through the use of semi-Markov dynamics, the MSSM enables goal position to evolve
at a different rate than cursor angle while allowing for a high rate of neural data acquisition. In this
way, the MSSM can integrate across different timescales to more robustly infer the (unknown) goal
position and the (also unknown) cursor aim. We introduce the model in Sec. 4.1 and 4.2. We derive
an efficient decoding algorithm, based on an online variant of the junction tree algorithm, in Sec. 4.3.
4.1
Modeling Goals and Motion via a Dynamic Bayesian Network
The MSSM directed graphical model (Fig. 3) uses a structured latent state representation, sometimes
referred to as a dynamic Bayesian network [Murphy, 2002]. This factorization allows us to discretize
latent state variables, and thereby support non-Gaussian dynamics and data likelihoods. At each time
t we represent discrete cursor aim ?t as 72 values in [0, 2?) and goal position gt as a regular grid
of 40 ? 40 = 1600 locations (see Fig. 4). Each cell of the grid is small compared to elements of a
graphical interface. Cursor aim dynamics are conditioned on goal position and evolve according to a
smoothed von Mises distribution:
vMS(?t | gt , pt ) , ?/2? + (1 ? ?)vonMises(?t | a(gt , pt ), ?
? ).
(4)
?1
Here, a(g, p) = tan ((gy ? py )/(gx ? px )) is the angle from the cursor p = (px , py ) to the goal
g = (gx , gy ), and the concentration parameter ?
? encodes the expected accuracy of user aim. Neural
activity from some participants has short bursts of noise during which the learned angle likelihood is
inaccurate; the outlier weight 0 < ? < 1 adds robustness to these noise bursts.
4.2
Multiscale Semi-Markov Dynamics
The first-order Markov assumption made by existing iBCI decoders (see Eq. (1)) imposes a geometric
decay in state correlation over time. For example, consider a scalar Gaussian state-space model:
xt = ?xt?1 + v, v ? N (0, ? 2 ). For time lag k > 0, the covariance between two states cov(xt , xt+k )
decays as ? ?k . This weak temporal dependence is highly problematic in the iBCI setting due to the
mismatch between downsampled sensor acquisition rates used for decoding (typically around 50Hz,
or 20ms per timestep) and the time scale at which the desired goal position changes (seconds).
We relax the first-order Markov assumption via a semi-Markov model of state dynamics [Yu, 2010].
Semi-Markov models, introduced by Levy [1954] and Smith [1955], divide the state evolution into
contiguous segments. A segment is a contiguous series of timesteps during which a latent variable is
unchanged. The conditional distribution over the state at time xt depends not only on the previous
state xt?1 , but also on a duration dt which encodes how long the state is to remain unchanged:
4
Goal
reconsideration
counter
Goal position
Aim adjustment
counter
Angle of aim
Observation
Cursor position
Multiscale Semi-Markov Model
Junction Tree for Online Decoding
Figure 3: Multiscale semi-Markov dynamical model. Left: The multiscale directed graphical model of how
goal positions gt , angles of aim ?t , and observed cursor positions pt evolve over three time steps. Dashed nodes
are counter variables enabling semi-Markov dynamics. Right: Illustration of the junction tree used to compute
marginals for online decoding, as in Boyen and Koller [1998]. Dashed edges indicate cliques whose potentials
depend on the marginal approximations at time t ? 1. The inference uses an auxiliary variable rt , a(gt , pt ),
the angle from the cursor to the current goal, to reduce computation and allow inference to operate in real time.
p(xt | xt?1 , dt ). Duration is modeled via a latent counter variable, which is drawn at the start of
each segment and decremented deterministically until it reaches zero, at which point it is resampled.
In this way the semi-Markov model is capable of integrating information over longer time horizons,
and thus less susceptible to intermittent bursts of sensor noise.
We define separate semi-Markov dynamical models for the goal position and the angle of intended
movement. As detailed in the supplement, in experiments our duration distributions were uniform,
with parameters informed by knowledge about typical trajectory durations and reaction times.
Goal Dynamics A counter ct encodes the temporal evolution of the semi-Markov dynamics on
goal positions: ct is drawn from a discrete distribution p(c) at the start of each trajectory, and then
decremented deterministically until it reaches zero. (During decoding we do not know the value of
the counter, and maintain a posterior probability distribution over its value.) The goal position gt
remains unchanged until the goal counter reaches zero, at which point with probability ? we resample
a new goal, and we keep the same goal with the remaining probability 1 ? ?:
(
1,
ct = ct?1 ? 1, ct?1 > 0, Decrement
p(ct ), ct?1 = 0,
Sample new counter
p(ct | ct?1 ) =
(5)
0,
Otherwise
?
1,
ct?1 > 0, gt = gt?1 , Goal position unchanged
?
? 1
? G + (1 ? ?), ct?1 = 0, gt = gt?1 , Sample same goal position
p(gt | ct?1 , gt?1 ) =
(6)
1
ct?1 = 0, gt 6= gt?1 , Sample new goal position
?
? ?G,
0,
Otherwise
Cursor Angle Dynamics We define similar semi-Markov dynamics for the cursor angle via an aim
counter bt . Once the counter reaches zero, we sample a new aim counter value from the discrete
distribution p(b), and a new cursor aim angle from the smoothed von Mises distribution of Eq. (4):
(
1,
bt = bt?1 ? 1, bt?1 > 0, Decrement
p(bt ), bt?1 = 0,
Sample new counter
p(bt | bt?1 ) =
(7)
0,
Otherwise
?t?1
bt?1 > 0, Keep cursor aim
p(?t | bt?1 , ?t?1 , pt , gt ) =
(8)
vMS(?t | gt , pt ) bt?1 = 0, Sample new cursor aim
4.3
Decoding via Approximate Online Inference
Efficient decoding is possible via an approximate variant of the junction tree algorithm [Boyen and
Koller, 1998]. We approximate the full posterior at time t via a partially factorized posterior:
p(gt , ct , ?t , bt | z1...t ) ? p(gt , ct | z1...t )p(?t , bt | z1...t ).
(9)
5
Goal Positions
0s
0.5s
1s
1.5s
2s
Figure 4: Decoding goal positions. The MSSM represents goal position via a regular grid of 40 ? 40 locations
(upper left). For one real sequence of recorded neural data, the above panels illustrate the motion of the cursor
(white dot) to the user?s target (red circle). Panels show the marginal posterior distribution over goal positions at
0.5s intervals (25 discrete time steps of graphical model inference). Yellow goal states have highest probability,
dark blue goal states have near-zero probability. Note the temporal aggregation of directional cues.
Here p(gt , ct | z1...t ) is the marginal on the goal position and goal counter, and p(?t , bt | z1...t ) is the
marginal on the angle of aim and the aim counter. Note that in this setting goal position gt and cursor
aim ?t , as well as their respective counters ct and bt , are unknown and must be inferred from neural
data. At each inference step we use the junction tree algorithm to compute state marginals at time t,
conditioned on the factorized posterior approximation from time step t ? 1 (see Fig. 3). Boyen and
Koller [1998] show that this technique has bounded approximation error over time, and Murphy and
Weiss [2001] show this as a special case of loopy belief propagation.
Detailed inference equations are derived in the supplemental material. Given G goal positions
and A discrete angle states, each temporal update for our online decoder requires O(GA + A2 )
operations. In contrast, the exact junction tree algorithm would require O(G2 A2 ) operations; for
practical numbers of goals G, realtime implementation of this exact decoder is infeasible.
Figure 4 shows several snapshots of the marginal posterior
over
h
i goal position. At each time the
?pt
MSSM decoder moves the cursor along the vector E kggtt ?p
, computed by taking an average of
tk
the directions needed to get to each possible goal, weighted by the inferred probability that each goal
is the participant?s true target. This vector is smaller in magnitude when the decoder is less certain
about the direction in which the intended goal lies, which has the practical benefit of allowing the
participant to slow down near the goal.
5
Experiments
We evaluate all decoders under a variety of conditions and a range of configurations for each decoder.
Controlled offline evaluations allow us to assess the impact of each proposed innovation. To analyze
the effects of our proposed likelihood and multiscale dynamics in isolation, we construct a baseline
hidden Markov model (HMM) decoder using the same discrete representation of angles as the
MSSM, and either cosine-tuned or flexible likelihoods. Our findings show that the offline decoding
performance of the MSSM is superior in all respects to baseline models.
We also evaluate the MSSM decoder in two online clinical research sessions, and compare headto-head performance with the Kalman filter. Previous studies have tested the Kalman filter under a
variety of responsive parameter configurations and found a tradeoff between slow, smooth control
versus fast, meandering control [Willett et al., 2016, 2017]. Through comparisons to the Kalman, we
demonstrate that the MSSM decoder maintains smoother and more accurate control at comparable
speeds. These realtime results are preliminary since we have yet to evaluate the MSSM decoder on
other clinical metrics such as communication rate.
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Figure 5: Offline decoding. Mean squared error of angular prediction for a variety of decoders, where each
decoder processes the same sets of recorded data. We analyze 24 minutes (eight 3-minute blocks) of neural data
recorded from participant T9 on trial days 546 and 552. We use one block for testing and the remainder for
training, and average errors across the choice of test block. On the left, we report errors over all time points. On
the right, we report errors on time points during which the cursor was outside a fixed distance from the target.
For both analyses, we exclude the initial 1s after target acquisition, during which the ground truth is unreliable.
To isolate preprocessing effects, the plots separately report the Kalman without preprocessing (?raw?). Dynamics
effects are isolated by separately evaluating HMM dynamics (?HMM?), and likelihood effects are isolated by
separately evaluating flexible likelihood and cosine tuning in each configuration. ?KalmanBC? denotes the
Kalman filter with an additional kinematic bias-correction heuristic [Jarosiewicz et al., 2015].
5.1
Offline evaluation
We perform offline analysis using previously recorded data from two historical sessions of iBCI use
with a single participant (T9). During each session the participant is asked to perform a cued target
task in which a target appears at a random location on the screen and the participant attempts to move
the cursor to the target. Once the target is acquired or after a timeout (10 seconds), a new target is
presented at a different location. Each session is composed of several 3 minute segments or blocks.
To evaluate the effect of each innovation we compare to an HMM decoder. This HMM baseline
isolates the effect of our flexible likelihood since, like the Kalman filter, it does not model goal
positions and assumes first-order Markov dynamics. Let ?t be the latent angle state at time t and
T
x(?) = (cos(?), sin(?)) the corresponding unit vector. We implement a pair of HMM decoders for
cosine tuning and our proposed flexible tuning curves,
zt | ?t ? N b + wT ?(?t ), Q
|
{z
}
zt | ?t ? N (b + Hx(?t ), Q),
{z
}
|
Cosine HMM
Flexible HMM
Here, ?(?) are the basis vectors defined in Eq. (3). The state ?t is discrete, taking one of 72 angular
values equally spaced in [0, 2?), the same discretization used by the MSSM. Continuous densities
are appropriately normalized. Unlike the linear Gaussian state-space model, the HMMs constrain
latent states to be valid angles (equivalently, unit vectors) rather than arbitrary vectors in R2 .
We analyze decoder accuracy within each session using a leave-one-out approach. Specifically, we
test the decoder on each held-out block using the remaining blocks in the same session for training.
We report MSE of the predicted cursor direction, using the unit vector from the cursor to the target
as ground truth, and normalizing decoder output vectors. We used the same recorded data for each
decoder. See the supplement for further details.
Figure 5 summarizes the findings of the offline comparisons for a variety of decoder configurations.
First, we evaluate the effect of preprocessing the data by taking the square root, applying a low-pass
IIR filter, and clipping the data outside a 5? threshold, where ? is the empirical standard deviation of
training data. This preprocessing significantly improves accuracy for all decoders. The MSSM model
compares favorably to all configurations of the Kalman decoders. The majority of benefit comes from
the semi-Markov dynamical model, but additional gains are observed when including the flexible
tuning likelihood. Finally, it has been observed that the Kalman decoder is sensitive to outliers for
which Jarosiewicz et al. [2015] propose a correction to avoid biased estimates. We test the Kalman
filter with and without this correction.
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Figure 6: Realtime decoding. A realtime comparison of the Kalman filter and MSSM with flexible likelihoods
from two sessions with clinical trial participant T10. Left: Box plots of squared error between unit vectors from
cursor to target and normalized (unit vector) decoder output for each four-minute comparison block in a session.
MSSM errors are consistently smaller. Right: Two metrics that describe the smoothness of cursor trajectories,
introduced in MacKenzie et al. [2001] and commonly used to quantify iBCI performance [Kim et al., 2008,
Simeral et al., 2011]. The task axis for a trajectory is the straight line from the cursor?s starting position at the
beginning of a trajectory to a goal. Orthogonal directional changes measure the number of direction changes
towards or away from the goal, and movement direction changes measure the number of direction changes
towards or away from the task axis. The MSSM shows significantly fewer direction changes according to both
metrics.
5.2
Realtime evaluation
Next, we examined whether the MSSM method was effective for realtime iBCI control by a clinical
trial participant. On two different days, a clinical trial participant (T10) completed six four-minute
comparison blocks. In these blocks, we alternated using an MSSM decoder with flexible likelihoods
and novel preprocessing, or a standard Kalman decoder. As with the Kalman decoding described in
Jarosiewicz et al. [2015], we used the Kalman filter in conjunction with a bias correcting postprocessing heuristic. We used the feature selection method proposed by Malik et al. [2015] to select D = 60
channels of neural data, and used these same 60 channels for both decoders.
Jarosiewicz et al. [2015] selected the timesteps of data to use for parameter learning by taking the
first two seconds of each trajectory after a 0.3s reaction time. For both decoders, we instead selected
all timesteps in which the cursor was a fixed distance from the cued goal because we found this
alternative method lead to improvements in offline decoding. Both methods for selecting subsets of
the calibration data are designed to compensate for the fact that vectors from cursor to target are not a
reliable estimator for participants? intended aim when the cursor is near the target.
Decoding accuracy. Figure 6 shows that our MSSM decoder had less directional error than the
configuration of the Kalman filter that we compared to. We confirmed the statistical significance
of this result using a Wilcoxon rank sum test. To accommodate the Wilcoxon rank sum test?s
independence assumption, we divided the data into individual trajectories from a starting point
towards a goal, that ended either when the cursor reached the goal or at a timeout (10 seconds). We
then computed the mean squared error of each trajectory, where the squared error is the squared
Euclidean distance between the normalized (unit vector) decoded vectors and the unit vectors from
cursor to target. Within each session, we compared the distributions of these mean squared errors for
trajectories between decoders (p < 10?6 for each session). MSSM also performed better than the
Kalman on metrics from MacKenzie et al. [2001] that measure the smoothness of cursor trajectories
(see Fig. 6).
Figure 7 shows example trajectories as the cursor moves toward its target via the MSSM decoder or
the (bias-corrected) Kalman decoder. Consistent with the quantitative error metrics, the trajectories
produced by the MSSM model were smoother and more direct than those of the Kalman filter,
especially as the cursor approached the goal. The distance ratio (the ratio of the length of the
trajectory to the line from the starting position to the goal) averaged 1.17 for the MSSM decoder and
1.28 for the Kalman decoder, a significant difference (Wilcoxon rank sum test, p < 10?6 ). Some
trajectories for both decoders are shown in Figure 7. Videos of cursor movement under both decoding
algorithms, and additional experimental details, are included in the supplemental material.
Decoding speed. We controlled for speed by configuring both decoders to average the same fast
speed determined in collaboration with clinical research engineers familiar with the participant?s
8
Near Goal
Goal
Figure 7: Examples of realtime decoding trajectories. Left: 20 randomly selected trajectories for the Kalman
decoder, and 20 trajectories for the MSSM decoder. The trajectories are aligned so that the starting position is at
the origin and rotated so the goal position is on the positive, horizontal axis. The MSSM decoder exhibits fewer
abrupt direction changes. Right: The empirical probability of instantaneous angle of movement, after rotating all
trajectories from the realtime data (24 minutes of iBCI use with each decoder). The MSSM distribution (shown
as translucent cyan) is more peaked at zero degrees, corresponding to direct motion towards the goal.
preferred cursor speed. For each decoder, we collected a block of data in which the participant used
that decoder to control the cursor. For each of these blocks, we computed the trimmed mean of
the speed, and then linearly extrapolated the speed gain needed for the desired speed. Although
such an extrapolation is approximate, the average times to acquire a target with each decoder at the
extrapolated speed gains were within 6% of each other: 2.6s for the Kalman decoder versus 2.7s for
the MSSM decoder. This speed discrepancy is dominated by the relative performance improvement
of MSSM over Kalman: the Kalman had a 30.7% greater trajectory mean squared error, 249% more
orthogonal direction changes, and 224% more movement direction changes.
This approach to evaluating decoder performance differs from that suggested by Willett et al. [2016],
which discusses the possibility of optimizing the speed gain and other decoder parameters to minimize
target acquisition time. In contrast, we matched the speed of both decoders and evaluated decoding
error and smoothness. We did not extensively tune the dynamics parameters for either decoder,
instead relying on the Kalman parameters in everyday use by T10. For MSSM we tried two values of
?, which controls the sampling of goal states (6), and chose the remaining parameters offline.
6
Conclusion
We introduce a flexible likelihood model and multiscale semi-Markov (MSSM) dynamics for cursor
control in intracortical brain-computer interfaces. The flexible tuning likelihood model extends the
cosine tuning model to allow for multimodal tuning curves and narrower peaks. The MSSM dynamic
Bayesian network explicitly models the relationship between the goal position, the cursor position,
and the angle of intended movement. Because the goal position changes much less frequently than
the angle of intended movement, a decoder?s past knowledge of the goal position stays relevant for
longer, and the MSSM model can use longer histories of neural activity to infer the direction of
desired movement.
To create a realtime decoding algorithm, we derive an online variant of the junction tree algorithm
with provable accuracy guarantees. We demonstrate a significant improvement over the Kalman
filter in offline experiments with neural recordings, and demonstrate promising preliminary results in
clinical trial tests. Future work will further evaluate the suitability of this method for clinical use.
We hope that the MSSM graphical model will also enable further advances in iBCI decoding, for
example by encoding the structure of a known user interface in the set of latent goals.
Author contributions DJM, JLP, and EBS created the flexible tuning likelihood and the multiscale
semi-Markov dynamics. DJM derived the inference (decoder), wrote software implementations of
these methods, and performed data analyses. DJM, JLP, and EBS designed offline experiments. DJM,
BJ, and JDS designed clinical research sessions. LRH is the sponsor-investigator of the BrainGate2
pilot clinical trial. DJM, JLP, and EBS wrote the manuscript with input from all authors.
9
Acknowledgments
The authors thank Participants T9 and T10 and their families, Brian Franco, Tommy Hosman, Jessica
Kelemen, Dave Rosler, Jad Saab, and Beth Travers for their contributions to this research. Support
for this study was provided by the Office of Research and Development, Rehabilitation R&D Service,
Department of Veterans Affairs (B4853C, B6453R, and N9228C), the National Institute on Deafness
and Other Communication Disorders of National Institutes of Health (NIDCD-NIH: R01DC009899),
MGH-Deane Institute, and The Executive Committee on Research (ECOR) of Massachusetts General
Hospital. The content is solely the responsibility of the authors and does not necessarily represent the
official views of the National Institutes of Health, or the Department of Veterans Affairs or the United
States Government. CAUTION: Investigational Device. Limited by Federal Law to Investigational
Use.
Disclosure: Dr. Hochberg has a financial interest in Synchron Med, Inc., a company developing
a minimally invasive implantable brain device that could help paralyzed patients achieve direct
brain control of assistive technologies. Dr. Hochberg?s interests were reviewed and are managed by
Massachusetts General Hospital, Partners HealthCare, and Brown University in accordance with their
conflict of interest policies.
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11
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6,288 | 6,689 | PredRNN: Recurrent Neural Networks for Predictive
Learning using Spatiotemporal LSTMs
Yunbo Wang
School of Software
Tsinghua University
[email protected]
Jianmin Wang
School of Software
Tsinghua University
[email protected]
Mingsheng Long?
School of Software
Tsinghua University
[email protected]
Zhifeng Gao
School of Software
Tsinghua University
[email protected]
Philip S. Yu
School of Software
Tsinghua University
[email protected]
Abstract
The predictive learning of spatiotemporal sequences aims to generate future images
by learning from the historical frames, where spatial appearances and temporal variations are two crucial structures. This paper models these structures by presenting a
predictive recurrent neural network (PredRNN). This architecture is enlightened by
the idea that spatiotemporal predictive learning should memorize both spatial appearances and temporal variations in a unified memory pool. Concretely, memory
states are no longer constrained inside each LSTM unit. Instead, they are allowed
to zigzag in two directions: across stacked RNN layers vertically and through all
RNN states horizontally. The core of this network is a new Spatiotemporal LSTM
(ST-LSTM) unit that extracts and memorizes spatial and temporal representations
simultaneously. PredRNN achieves the state-of-the-art prediction performance on
three video prediction datasets and is a more general framework, that can be easily
extended to other predictive learning tasks by integrating with other architectures.
1
Introduction
As a key application of predictive learning, generating images conditioned on given consecutive
frames has received growing interests in machine learning and computer vision communities. To
learn representations of spatiotemporal sequences, recurrent neural networks (RNN) [17, 27] with
the Long Short-Term Memory (LSTM) [9] have been recently extended from supervised sequence
learning tasks, such as machine translation [22, 2], speech recognition [8], action recognition [28, 5]
and video captioning [5], to this spatiotemporal predictive learning scenario [21, 16, 19, 6, 25, 12].
1.1
Why spatiotemporal memory?
In spatiotemporal predictive learning, there are two crucial aspects: spatial correlations and temporal
dynamics. The performance of a prediction system depends on whether it is able to memorize
relevant structures. However, to the best of our knowledge, the state-of-the-art RNN/LSTM predictive
learning methods [19, 21, 6, 12, 25] focus more on modeling temporal variations (such as the object
moving trajectories), with memory states being updated repeatedly over time inside each LSTM unit.
Admittedly, the stacked LSTM architecture is proved powerful for supervised spatiotemporal learning
?
Corresponding author: Mingsheng Long
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
(such as video action recognition [5, 28]). Two conditions are met in this scenario: (1) Temporal
features are strong enough for classification tasks. In contrast, fine-grained spatial appearances prove
to be less significant; (2) There are no complex visual structures to be modeled in the expected
outputs so that spatial representations can be highly abstracted. However, spatiotemporal predictive
leaning does not satisfy these conditions. Here, spatial deformations and temporal dynamics are
equally significant to generating future frames. A straightforward idea is that if we hope to foretell
the future, we need to memorize as many historical details as possible. When we recall something
happened before, we do not just recall object movements, but also recollect visual appearances from
coarse to fine. Motivated by this, we present a new recurrent architecture called Predictive RNN
(PredRNN), which allows memory states belonging to different LSTMs to interact across layers (in
conventional RNNs, they are mutually independent). As the key component of PredRNN, we design
a novel Spatiotemporal LSTM (ST-LSTM) unit. It models spatial and temporal representations in a
unified memory cell and convey the memory both vertically across layers and horizontally over states.
PredRNN achieves the state-of-the-art prediction results on three video datasets. It is a general and
modular framework for predictive learning and is not limited to video prediction.
1.2
Related work
Recent advances in recurrent neural network models provide some useful insights on how to predict
future visual sequences based on historical observations. Ranzato et al. [16] defined a RNN
architecture inspired from language modeling, predicting the frames in a discrete space of patch
clusters. Srivastava et al. [21] adapted the sequence to sequence LSTM framework. Shi et al. [19]
extended this model to further extract visual representations by exploiting convolutions in both
input-to-state and state-to-state transitions. This Convolutional LSTM (ConvLSTM) model has
become a seminal work in this area. Subsequently, Finn et al. [6] constructed a network based on
ConvLSTMs that predicts transformations on the input pixels for next-frame prediction. Lotter et al.
[12] presented a deep predictive coding network where each ConvLSTM layer outputs a layer-specific
prediction at each time step and produces an error term, which is then propagated laterally and
vertically in the network. However, in their settings, the predicted next frame always bases on the
whole previous ground truth sequence. By contrast, we predict sequence from sequence, which is
obviously more challenging. Patraucean et al. [15] and Villegas et al. [25] brought optical flow
into RNNs to model short-term temporal dynamics, which is inspired by the two-stream CNNs [20]
designed for action recognition. However, the optical flow images are hard to use since they would
bring in high additional computational costs and reduce the prediction efficiency. Kalchbrenner
et al. [10] proposed a Video Pixel Network (VPN) that estimates the discrete joint distribution
of the raw pixel values in a video using the well-established PixelCNNs [24]. But it suffers from
high computational complexity. Besides the above RNN architectures, other deep architectures are
involved to solve the visual predictive learning problem. Oh et al. [14] defined a CNN-based action
conditional autoencoder model to predict next frames in Atari games. Mathieu et al. [13] successfully
employed generative adversarial networks [7, 4] to preserve the sharpness of the predicted frames.
In summary, these existing visual prediction models yield different shortcomings due to different
causes. The RNN-based architectures [21, 16, 19, 6, 25, 12] model temporal structures with LSTMs,
but their predicted images tend to blur due to a loss of fine-grained visual appearances. In contrast,
CNN-based networks [13, 14] predict one frame at a time and generate future images recursively,
which are prone to focus on spatial appearances and relatively weak in capturing long-term motions.
In this paper, we explore a new RNN framework for predictive learning and present a novel LSTM
unit for memorizing spatiotemporal information simultaneously.
2
2.1
Preliminaries
Spatiotemporal predictive learning
Suppose we are monitoring a dynamical system (e.g. a video clip) of P measurements over time,
where each measurement (e.g. a RGB channel) is recorded at all locations in a spatial region
represented by an M ? N grid (e.g. video frames). From the spatial view, the observation of these P
measurements at any time can be represented by a tensor X ? RP ?M ?N . From the temporal view,
the observations over T time steps form a sequence of tensors X1 , X2 , . . . , XT . The spatiotemporal
predictive learning problem is to predict the most probable length-K sequence in the future given the
2
previous length-J sequence including the current observation:
Xbt+1 , . . . , Xbt+K =
arg max p (Xt+1 , . . . , Xt+K |Xt?J+1 , . . . , Xt ) .
(1)
Xt+1 ,...,Xt+K
Spatiotemporal predictive learning is an important problem, which could find crucial and high-impact
applications in various domains: video prediction and surveillance, meteorological and environmental
forecasting, energy and smart grid management, economics and finance prediction, etc. Taking video
prediction as an example, the measurements are the three RGB channels, and the observation at
each time step is a 3D video frame of RGB image. Another example is radar-based precipitation
forecasting, where the measurement is radar echo values and the observation at every time step is a
2D radar echo map that can be visualized as an RGB image.
2.2
Convolutional LSTM
Compared with standard LSTMs, the convolutional LSTM (ConvLSTM) [19] is able to model the
spatiotemporal structures simultaneously by explicitly encoding the spatial information into tensors,
overcoming the limitation of vector-variate representations in standard LSTM where the spatial
information is lost. In ConvLSTM, all the inputs X1 , . . . , Xt , cell outputs C1 , . . . , Ct , hidden state
H1 , . . . , Ht , and gates it , ft , gt , ot are 3D tensors in RP ?M ?N , where the first dimension is either the
number of measurement (for inputs) or the number of feature maps (for intermediate representations),
and the last two dimensions are spatial dimensions (rows M and columns N ). To get a better picture
of the inputs and states, we may imagine them as vectors standing on a spatial grid. ConvLSTM
determines the future state of a certain cell in the M ? N grid by the inputs and past states of its
local neighbors. This can easily be achieved by using convolution operators in the state-to-state and
input-to-state transitions. The key equations of ConvLSTM are shown as follows:
gt = tanh(Wxg ? Xt + Whg ? Ht?1 + bg )
it = ?(Wxi ? Xt + Whi ? Ht?1 + Wci Ct?1 + bi )
ft = ?(Wxf ? Xt + Whf ? Ht?1 + Wcf Ct?1 + bf )
Ct = ft Ct?1 + it gt
ot = ?(Wxo ? Xt + Who ? Ht?1 + Wco Ct + bo )
Ht = ot tanh(Ct ),
(2)
where ? is sigmoid activation function, ? and denote the convolution operator and the Hadamard
product respectively. If the states are viewed as the hidden representations of moving objects, then a
ConvLSTM with a larger transitional kernel should be able to capture faster motions while one with a
smaller kernel can capture slower motions [19]. The use of the input gate it , forget gate ft , output
gate ot , and input-modulation gate gt controls information flow across the memory cell Ct . In this
way, the gradient will be prevented from vanishing quickly by being trapped in the memory.
The ConvLSTM network adopts the encoder-decoder RNN architecture that is proposed in [23] and
extended to video prediction in [21]. For a 4-layer ConvLSTM encoder-decoder network, input
frames are fed into the the first layer and future video sequence is generated at the fourth one. In this
process, spatial representations are encoded layer-by-layer, with hidden states being delivered from
bottom to top. However, the memory cells that belong to these four layers are mutually independent
and updated merely in time domain. Under these circumstances, the bottom layer would totally ignore
what had been memorized by the top layer at the previous time step. Overcoming these drawbacks of
this layer-independent memory mechanism is important to the predictive learning of video sequences.
3
PredRNN
In this section, we give detailed descriptions of the predictive recurrent neural network (PredRNN).
Initially, this architecture is enlightened by the idea that a predictive learning system should memorize
both spatial appearances and temporal variations in a unified memory pool. By doing this, we make
the memory states flow through the whole network along a zigzag direction. Then, we would like
to go a step further to see how to make the spatiotemporal memory interact with the original long
short-term memory. Thus we make explorations on the memory cell, memory gate and memory
fusion mechanisms inside LSTMs/ConvLSTMs. We finally derive a novel Spatiotemporal LSTM
(ST-LSTM) unit for PredRNN, which is able to deliver memory states both vertically and horizontally.
3
3.1
Spatiotemporal memory flow
X? t
X? t+1
l=4
l=4
M t?1
, H t?1
W4
W3
M tl=4 , H tl=4
W4
l=3
l=3
M t?1
, H t?1
X? t+2
l=2
l=2
M t?1
, H t?1
W2
M ,H
l=3
l=3
M t+1
, H t+1
W3
M tl=2 , H tl=2
W2
l=1
t?1
l=1
t?1
l=4
l=4
M t+1
, H t+1
W4
M tl=3 , H tl=3
W3
X? t+1
l=2
l=2
M t+1
, H t+1
W2
l=1
t
M ,H
l=1
t
l=1
t+1
M ,H
W1
W1
W1
Xt?1
Xt
Xt+1
l=1
t+1
l=4
l=4
Ct?1
, H t?1
l=3
l=3
Ct?1
, H t?1
l=2
l=2
Ct?1
, H t?1
l=1
l=1
Ct?1
, H t?1
W4
W3
W2
W1
Ctl=4 , H tl=4
H tl=3 l=3 l=3
Ct , H t
H tl=2 l=2 l=2
Ct , H t
H tl=1 l=1 l=1
Ct , H t
l=4
l=4
M t?2
, H t?2
Xt
Figure 1: Left: The convolutional LSTM network with a spatiotemporal memory flow. Right: The
conventional ConvLSTM architecture. The orange arrows denote the memory flow direction for all
memory cells.
For generating spatiotemporal predictions, PredRNN initially exploits convolutional LSTMs (ConvLSTM) [19] as basic building blocks. Stacked ConvLSTMs extract highly abstract features layer-bylayer and then make predictions by mapping them back to the pixel value space. In the conventional
ConvLSTM architecture, as illustrated in Figure 1 (right), the cell states are constrained inside each
ConvLSTM layer and be updated only horizontally. Information is conveyed upwards only by hidden
states. Such a temporal memory flow is reasonable in supervised learning, because according to the
study of the stacked convolutional layers, the hidden representations can be more and more abstract
and class-specific from the bottom layer upwards. However, we suppose in predictive learning,
detailed information in raw input sequence should be maintained. If we want to see into the future,
we need to learn from representations extracted at different-level convolutional layers. Thus, we
apply a unified spatiotemporal memory pool and alter RNN connections as illustrated in Figure 1
(left). The orange arrows denote the feed-forward directions of LSTM memory cells. In the left
figure, a unified memory is shared by all LSTMs which is updated along a zigzag direction. The key
equations of the convolutional LSTM unit with a spatiotemporal memory flow are shown as follows:
gt = tanh(Wxg ? Xt 1{l=1} + Whg ? Htl?1 + bg )
it = ?(Wxi ? Xt 1{l=1} + Whi ? Htl?1 + Wci ? Ml?1
+ bi )
t
ft = ?(Wxf ? Xt 1{l=1} + Whf ? Htl?1 + Wcf ? Ml?1
+ bf )
t
Mlt = ft Ml?1
+ it gt
t
(3)
ot = ?(Wxo ? Xt 1{l=1} + Who ? Htl?1 + Wco ? Mlt + bo )
Htl = ot tanh(Mlt ).
The input gate, input modulation gate, forget gate and output gate no longer depend on the hidden
states and cell states from the previous time step at the same layer. Instead, as illustrated in Figure
1 (left), they rely on hidden states Htl?1 and cell states Ml?1
t (l ? {1, ..., L}) that are updated by
the previous layer at current time step. Specifically, the bottom LSTM unit (l = 1) receives state
L
values from the top layer at the previous time step: Htl?1 = Ht?1
, Ml?1
= ML
t
t?1 . The four layers
in this figure have different sets of input-to-state and state-to-state convolutional parameters, while
they maintain a spatiotemporal memory cell and update its states separately and repeatedly as the
information flows through the current node. Note that, we replace the notation for memory cell from C
to M to emphasize that it flows in the zigzag direction in PredRNN, instead of the horizontal direction
in standard recurrent networks. Different from ConvLSTM that uses Hadamard product for state
transitions in the gates, we adopt convolution operators ? for finer-grained memory transitions.
4
3.2
Spatiotemporal LSTM
Input Gate
Output Gate
it
?
gt
Ctl
Input Modulation Gate
?
l
Ct?1
H
l
t?1
Xt
Forget Gate
ft
ot
Original Temporal
Memory
Spatiotemporal
Memory
M tl
?
X? t +1
W4
W4
M tl=3
?
H tl
W3
it?
Xt ?1
Ctl=2
W2
H tl=2
H tl=1
W1
l=4
M t?1
W4
W3
H tl=3
W2
W1
M tl?1
Ctl=3
H tl=2
M tl=1
ft?
Ctl=4 , H tl=4
W3
W2
X? t +2
H tl=3
M tl=2
gt?
?
X? t
Ctl=1
H tl=1
Xt
W1
M tl=4
Xt +1
Figure 2: ST-LSTM (left) and PredRNN (right). The orange circles in the ST-LSTM unit denotes the
differences compared with the conventional ConvLSTM. The orange arrows in PredRNN denote the
spatiotemporal memory flow, namely the transition path of spatiotemporal memory Mtl in the left.
In this section, we present the predictive recurrent neural network (PredRNN), by replacing convolutional LSTMs with a novel spatiotemporal long short-term memory (ST-LSTM) unit (see Figure 2).
In the architecture presented in the previous sub-section, the spatiotemporal memory cells are updated
in a zigzag direction, information is delivered first upwards across layers then forwards over time. We
wonder what would happen if memory cell states are passed in these two directions simultaneously.
With the aid of ST-LSTMs, the above PredRNN model with a spatiotemporal memory flow evolves
into our ultimate architecture. The equations of ST-LSTM are shown as follows:
l
gt = tanh(Wxg ? Xt + Whg ? Ht?1
+ bg )
l
it = ?(Wxi ? Xt + Whi ? Ht?1
+ bi )
l
ft = ?(Wxf ? Xt + Whf ? Ht?1
+ bf )
l
Ctl = ft Ct?1
+ it gt
0
gt0 = tanh(Wxg
? Xt + Wmg ? Ml?1
+ b0g )
t
0
i0t = ?(Wxi
? Xt + Wmi ? Ml?1
+ b0i )
t
(4)
0
ft0 = ?(Wxf
? Xt + Wmf ? Ml?1
+ b0f )
t
Mlt = ft0 Ml?1
+ i0t gt0
t
l
ot = ?(Wxo ? Xt + Who ? Ht?1
+ Wco ? Ctl + Wmo ? Mlt + bo )
Htl = ot tanh(W1?1 ? [Ctl , Mlt ]).
Two memory cells are maintained: Ctl is the standard temporal cell that is delivered from the previous
node at t ? 1 to the current time step within each LSTM unit. Mtl is the spatiotemporal memory we
described in the current section, which is conveyed vertically from the l ? 1 layer to the current node
at the same time step. For the bottom ST-LSTM layer where l = 1, Ml?1
= ML
t
t?1 , as described in
the previous subsection. We construct another set of gate structures for Mtl , while maintaining the
original gates for Ctl in standard LSTMs. At last, the final hidden states of this node rely on the fused
spatiotemporal memory. We concatenate these memory derived from different directions together and
then apply a 1 ? 1 convolution layer for dimension reduction, which makes the hidden state Htl of the
same dimensions as the memory cells. Different from simple memory concatenation, the ST-LSTM
unit uses a shared output gate for both memory types to enable seamless memory fusion, which can
effectively model the shape deformations and motion trajectories in the spatiotemporal sequences.
5
4
Experiments
Our model is demonstrated to achieve the state-of-the-art performance on three video prediction
datasets including both synthetic and natural video sequences. Our PredRNN model is optimized
with a L1 + L2 loss (other losses have been tried, but L1 + L2 loss works best). All models are
trained using the ADAM optimizer [11] with a starting learning rate of 10?3 . The training process is
stopped after 80, 000 iterations. Unless otherwise specified, the batch size of each iteration is set to 8.
All experiments are implemented in TensorFlow [1] and conducted on NVIDIA TITAN-X GPUs.
4.1
Moving MNIST dataset
Implementation We generate Moving MNIST sequences with the method described in [21]. Each
sequence consists of 20 consecutive frames, 10 for the input and 10 for the prediction. Each frame
contains two or three handwritten digits bouncing inside a 64 ? 64 grid of image. The digits were
chosen randomly from the MNIST training set and placed initially at random locations. For each
digit, we assign a velocity whose direction is randomly chosen by a uniform distribution on a unit
circle, and whose amplitude is chosen randomly in [3, 5). The digits bounce-off the edges of image
and occlude each other when reaching the same location. These properties make it hard for a model to
give accurate predictions without learning the inner dynamics of the movement. With digits generated
quickly on the fly, we are able to have infinite samples size in the training set. The test set is fixed,
consisting of 5,000 sequences. We sample digits from the MNIST test set, assuring the trained model
has never seen them before. Also, the model trained with two digits is tested on another Moving
MNIST dataset with three digits. Such a test setup is able to measure PredRNN?s generalization and
transfer ability, because no frames containing three digits are given throughout the training period.
As a strong competitor, we include the latest state-of-the-art VPN model [10]. We find it hard to
reproduce VPN?s experimental results on Moving MNIST since it is not open source, thus we adopt
its baseline version that uses CNNs instead of PixelCNNs as its decoder and generate each frame in
one pass. We observe that the total number of hidden states has a strong impact on the final accuracy
of PredRNN. After a number of trials, we present a 4-layer architecture with 128 hidden states in each
layer, which yields a high prediction accuracy using reasonable training time and memory footprint.
Table 1: Results of PredRNN with spatiotemporal memory M, PredRNN with ST-LSTMs, and
state-of-the-art models. We report per-frame MSE and Cross-Entropy (CE) of generated sequences
averaged across the Moving MNIST test sets. Lower MSE or CE denotes better prediction accuracy.
Model
FC-LSTM [21]
ConvLSTM Enc.-Dec. (128 ? 4) [19]
CDNA [6]
DFN [3]
VPN baseline [10]
PredRNN with spatiotemporal memory M
PredRNN + ST-LSTM (128 ? 4)
MNIST-2
(CE/frame)
MNIST-2
(MSE/frame)
MNIST-3
(MSE/frame)
483.2
367.0
346.6
285.2
110.1
118.5
97.0
118.3
103.3
97.4
89.0
70.0
74.0
56.8
162.4
142.1
138.2
130.5
125.2
118.2
93.4
Results As an ablation study, PredRNN only with a zigzag memory flow reduces the per-frame
MSE to 74.0 on the Moving MNIST-2 test set (see Table 1). By replacing convolutional LSTMs
with ST-LSTMs, we further decline the sequence MSE from 74.0 down to 56.8. The corresponding
frame-by-frame quantitative comparisons are presented in Figure 3. Compared with VPN, our model
turns out to be more accurate for long-term predictions, especially on Moving MNIST-3. We also
use per-frame cross-entropy likelihood as another evaluation metric on Moving MNIST-2. PredRNN
with ST-LSTMs significantly outperforms all previous methods, while PredRNN with spatiotemporal
memory M performs comparably with VPN baseline.
A qualitative comparison of predicted video sequences is given in Figure 4. Though VPN?s generated
frames look a bit sharper, its predictions gradually deviate from the correct trajectories, as illustrated
in the first example. Moreover, for those sequences that digits are overlapped and entangled, VPN has
difficulties in separating these digits clearly while maintaining their individual shapes. For example,
6
140
180
120
160
Mean Square Error
Mean Square Error
in the right figure, digit ?8? loses its left-side pixels and is predicted as ?3? after overlapping. Other
baseline models suffer from a severer blur effect, especially for longer future time steps. By contrast,
PredRNN?s results are not only sharp enough but also more accurate for long-term motion predictions.
100
80
60
ConvLSTM Enc.-Dec.
CDNA
DFN
VPN baseline
PredRNN + ST-LSTM
40
20
0
1
2
3
4
5 6 7
time steps
(a) MNIST-2
8
9
140
120
100
80
ConvLSTM Enc.-Dec.
CDNA
DFN
VPN baseline
PredRNN + ST-LSTM
60
40
20
1
10
2
3
4
5 6 7
time steps
(b) MNIST-3
8
9
10
Figure 3: Frame-wise MSE comparisons of different models on the Moving MNIST test sets.
Input frames
Ground truth
PredRNN
VPN
baseline
CDNA
ConvLSTM
Enc.-Dec.
Figure 4: Prediction examples on the Moving MNIST-2 test set.
4.2
KTH action dataset
Implementation The KTH action dataset [18] contains six types of human actions (walking,
jogging, running, boxing, hand waving and hand clapping) performed several times by 25 subjects in
four different scenarios: outdoors, outdoors with scale variations, outdoors with different clothes and
indoors. All video clips were taken over homogeneous backgrounds with a static camera in 25fps
frame rate and have a length of four seconds in average. To make the results comparable, we adopt
the experiment setup in [25] that video frames are resized into 128 ? 128 pixels and all videos are
divided with respect to the subjects into a training set (persons 1-16) and a test set (persons 17-25).
All models, including PredRNN as well as the baselines, are trained on the training set across all six
action categories by generating the subsequent 10 frames from the last 10 observations, while the the
presented prediction results in Figure 5 and Figure 6 are obtained on the test set by predicting 20 time
steps into the future. We sample sub-clips using a 20-frame-wide sliding window with a stride of 1 on
the training set. As for evaluation, we broaden the sliding window to 30-frame-wide and set the stride
to 3 for running and jogging, while 20 for the other categories. Sub-clips for running, jogging, and
walking are manually trimmed to ensure humans are always present in the frame sequences. In the
end, we split the database into a training set of 108,717 sequences and a test set of 4,086 sequences.
Results We use the Peak Signal to Noise Ratio (PSNR) and the Structural Similarity Index Measure (SSIM) [26] as metrics to evaluate the prediction results and provide frame-wise quantitative
comparisons in Figure 5. A higher value denotes a better prediction performance. The value of SSIM
ranges between -1 and 1, and a larger score means a greater similarity between two images. PredRNN
consistently outperforms the comparison models. Specifically, the Predictive Coding Network [12]
always exploits the whole ground truth sequence before the current time step to predict the next
7
frame. Thus, it cannot make sequence predictions. Here, we make it predict the next 20 frames by
feeding the 10 ground truth frames and the recursively generated frames in all previous time steps.
The performance of MCnet [25] deteriorates quickly for long-term predictions. Residual connections
of MCnet convey the CNN features of the last frame to the decoder and ignore the previous frames,
which emphasizes the spatial appearances while weakens temporal variations. By contrast, results of
PredRNN in both metrics remain stable over time, only with a slow and reasonable decline. Figure 6
visualizes a sample video sequence from the KTH test set. The ConvLSTM network [19] generates
blurred future frames, since it fails to memorize the detailed spatial representations. MCnet [25]
produces sharper images but is not able to forecast the movement trajectory accurately. Thanks
to ST-LSTMs, PredRNN memorizes detailed visual appearances as well as long-term motions. It
outperforms all baseline models and shows superior predicting power both spatially and temporally.
1
ConvLSTM Enc.-Dec.
MCnet + Res.
Predictive Coding Network
PredRNN + ST-LSTMs
30
Structural Similarity
Peak Signal Noise Ratio
35
25
ConvLSTM Enc.-Dec.
MCnet + Res.
Predictive Coding Network
PredRNN + ST-LSTMs
0.9
0.8
0.7
20
2
4
6
0.6
8 10 12 14 16 18 20
time steps
(a) Frame-wise PSNR
2
4
6
8 10 12 14 16 18 20
time steps
(b) Frame-wise SSIM
Figure 5: Frame-wise PSNR and SSIM comparisons of different models on the KTH action test set.
A higher score denotes a better prediction accuracy.
Input sequence
t=3
t=6
Ground truth and predictions
t=9
t=12
t=15
t=18
t=21
t=24
t=27
t=30
PredRNN
MCnet + Res.
ConvLSTM Enc.-Dec.
Predictive Coding Network
Figure 6: KTH prediction samples. We predict 20 frames into the future by observing 10 frames.
4.3
Radar echo dataset
Predicting the shape and movement of future radar echoes is a real application of predictive learning
and is the foundation of precipitation nowcasting. It is a more challenging task because radar echoes
are not rigid. Also, their speeds are not as fixed as moving digits, their trajectories are not as periodical
as KTH actions, and their shapes may accumulate, dissipate or change rapidly due to the complex
atmospheric environment. Modeling spatial deformation is significant for the prediction of this data.
8
Implementation We first collect the radar echo dataset by adapting the data handling method
described in [19]. Our dataset consists of 10,000 consecutive radar observations, recorded every 6
minutes in Guangzhou, China. For preprocessing, we first map the radar intensities to pixel values,
and represent them as 100 ? 100 gray-scale images. Then we slice the consecutive images with a
20-frame-wide sliding window. Thus, each sequence consists of 20 frames, 10 for the input, and 10
for forecasting. The total 9,600 sequences are split into a training set of 7,800 samples and a test
set of 1,800 samples. The PredRNN model consists of two ST-LSTM layers with 128 hidden states
each. The convolution filters inside ST-LSTMs are set to 3 ? 3. After prediction, we transform the
resulted echo intensities into colored radar maps, as shown in Figure 7, and then calculate the amount
of precipitation at each grid cell of these radar maps using Z-R relationships. Since it would bring in
an additional systematic error to rainfall prediction and makes final results misleading, we do not take
them into account in this paper, but only compare the predicted echo intensity with the ground truth.
Results Two baseline models are considered. The ConvLSTM network [19] is the first architecture
that models sequential radar maps with convolutional LSTMs, but its predictions tend to blur and
obviously inaccurate (see Figure 7). As a strong competitor, we also include the latest state-of-the-art
VPN model [10]. The PixelCNN-based VPN predicts an image pixel by pixel recursively, which
takes around 15 minutes to generate a radar map. Given that precipitation nowcasting has a high
demand on real-time computing, we trade off both prediction accuracy and computation efficiency
and adopt VPN?s baseline model that uses CNNs as its decoders and generates each frame in one pass.
Table 2 shows that the prediction error of PredRNN is significantly lower than VPN baseline. Though
VPN generates more accurate radar maps for the near future, it suffers from a rapid decay for the
long term. Such a phenomenon results from a lack of strong LSTM layers to model spatiotemporal
variations. Furthermore, PredRNN takes only 1/5 memory space and training time as VPN baseline.
Table 2: Quantitative results of different methods on the radar echo dataset.
Model
ConvLSTM Enc.-Dec. [19]
VPN baseline [10]
PredRNN
MSE/frame
Training time/100 batches
Memory usage
68.0
60.7
44.2
105 s
539 s
117 s
1756 MB
11513 MB
2367 MB
Input frames
Ground truth
PredRNN
ConvLSTM
Enc.-Dec.
VPN
baseline
Figure 7: A prediction example on the radar echo test set.
5
Conclusions
In this paper, we propose a novel end-to-end recurrent network named PredRNN for spatiotemporal
predictive learning that models spatial deformations and temporal variations simultaneously. Memory
states zigzag across stacked LSTM layers vertically and through all time states horizontally. Furthermore, we introduce a new spatiotemporal LSTM (ST-LSTM) unit with a gate-controlled dual
memory structure as the key building block of PredRNN. Our model achieves the state-of-the-art
performance on three video prediction datasets including both synthetic and natural video sequences.
9
Acknowledgments
This work was supported by the National Key R&D Program of China (2016YFB1000701), National
Natural Science Foundation of China (61772299, 61325008, 61502265, 61672313) and TNList Fund.
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6,289 | 669 | Hidden Markov Model Induction by Bayesian
Model Merging
Andreas Stolcke*'**
*Computer Science Division
University of California
Berkeley, CA 94720
[email protected]
Stephen Omohundro"
**International Computer Science Institute
1947 Center Street, Suite 600
Berkeley, CA 94704
[email protected]
Abstract
This paper describes a technique for learning both the number of states and the
topology of Hidden Markov Models from examples. The induction process starts
with the most specific model consistent with the training data and generalizes
by successively merging states. Both the choice of states to merge and the
stopping criterion are guided by the Bayesian posterior probability. We compare
our algorithm with the Baum-Welch method of estimating fixed-size models, and
find that it can induce minimal HMMs from data in cases where fixed estimation
does not converge or requires redundant parameters to converge.
1 INTRODUCTION AND OVERVIEW
Hidden Markov Models (HMMs) are a well-studied approach to the modelling of sequence
data. HMMs can be viewed as a stochastic generalization of finite-state automata, where
both the transitions between states and the generation of output symbols are governed by
probability distributions. HMMs have been important in speech recognition (Rabiner &
Juang, 1986), cryptography, and more recently in other areas such as protein classification
and alignment (Haussler, Krogh, Mian & SjOlander, 1992; Baldi, Chauvin, Hunkapiller &
McClure, 1993).
Practitioners have typically chosen the HMM topology by hand, so that learning the HMM
from sample data means estimating only a fixed number of model parameters. The standard
approach is to find a maximum likelihood (ML) or maximum a posteriori probability (MAP)
estimate of the HMM parameters. The Baum-Welch algorithm uses dynamic programming
11
12
Stokke and Omohundro
to approximate these estimates (Baum, Petrie, Soules & Weiss, 1970).
A more general problem is to additionally find the best HMM topology. This includes
both the number of states and the connectivity (the non-zero transitions and emissions).
One could exhaustively search the model space using the Baum-Welch algorithm on fully
connected models of varying sizes, picking the model size and topology with the highest
posterior probability. (Maximum likelihood estimation is not useful for this comparison
since larger models usually fit the data better.) This approach is very costly and BaumWelch may get stuck at sub-optimal local maxima. Our comparative results later in the
paper show that this often occurs in practice. The problem can be somewhat alleviated by
sampling from several initial conditions, but at a further increase in computational cost.
The HMM induction method proposed in this paper tackles the structure learning problem
in an incremental way. Rather than estimating a fixed-size model from scratch for various
sizes, the model size is adjusted as new evidence arrives. There are two opposing tendencies
in adjusting the model size and structure. Initially new data adds to the model size, because
the HMM has to be augmented to accommodate the new samples. If enough data of a similar
structure is available, however, the algorithm collapses the shared structure, decreasing the
model size. The merging of structure is also what drives generalization, Le., creates HMMs
that generate data not seen during training.
Beyond being incremental, our algorithm is data-driven, in that the samples themselves
completely determine the initial model shape. Baum-Welch estimation, by comparison,
uses an initially random set of parameters for a given-sized HMM and iteratively updates
them until a point is found at which the sample likelihood is locally maximal. What seems
intuitively troublesome with this approach is that the initial model is completely uninformed
by the data. The sample data directs the model formation process only in an indirect manner
as the model approaches a meaningful shape.
2 HIDDEN MARKOV MODELS
For lack of space we cannot give a full introduction to HMMs here; see Rabiner & Juang
(1986) for details. Briefly, an HMM consists of states and transitions like a Markov chain.
In the discrete version considered here, it generates strings by performing random walks
between an initial and a final state, outputting symbols at every state in between. The
probability P(xlM) that a model M generates a string x is determined by the conditional
probabilities of making a transition from one state to another and the probability of emitting
each symbol from each state. Once these are given, the probability of a particular path
through the model generating the string can be computed as the product of all transition
and emission probabilities along the path. The probability of a string x is the sum of the
probabilities of all paths generating x.
For example, the model M3 in Figure 1 generates the strings ab, abab, ababab, ... with
2 2 2
. I
pro b ab I?l??
lUes 3'
3!' 3!' ... , respective
y.
3 HMM INDUCTION BY STATE MERGING
3.1
MODEL MERGING
Omohundro (1992) has proposed an approach to statistical model inference in which initial
Hidden Markov Model Induction by Bayesian Model Merging
models simply replicate the data and generalize by similarity. As more data is received,
component models are fit from more complex model spaces. This allows the formation of
arbitrarily complex models without overfitting along the way. The elementary step used in
modifying the overall model is a merging of sub-models, collapsing the sample sets for the
corresponding sample regions. The search for sub-models to merge is guided by an attempt
to sacrifice as little of the sample likelihood as possible as a result of the merging process.
This search can be done very efficiently if (a) a greedy search strategy can be used, and (b)
likelihood computations can be done locally for each sub-model and don't require global
recomputation on each model update.
3.2 STATE MERGING IN HMMS
We have applied this general approach to the HMM learning task. We describe the algorithm
here mostly by presenting an example. The details are available in Stolcke & Omohundro
(1993).
To obtain an initial model from the data, we first construct an HMM which produces
exactly the input strings. The start state has as many outgoing transitions as there are
strings and each string is represented by a unique path with one state per sample symbol.
The probability of entering these paths from the start state is uniformly distributed. Within
each path there is a unique transition arc whose probability is 1. The emission probabilities
are 1 for each state to produce the corresponding symbol.
As an example, consider the regular language (abt and two samples drawn from it, the
strings ab and abab. The algorithm constructs the initial model Mo depicted in Figure 1.
This is the most specific model accounting for the observed data. It assigns each sample
a probability equal to its relative frequency, and is therefore a maximum likelihood model
for the data.
Learning from the sample data means generalizing from it. This implies trading off
model likelihood against some sort of bias towards 'simpler' models, expressed by a prior
probability distribution over HMMs. Bayesian analysis provides a formal basis for this
tradeoff. Bayes' rule tells us that the posterior model probability P(Mlx) is proportional
to the product of the model prior P(M) and the likelihood of the data P(xlM). Smaller or
simpler models will have a higher prior and this can outweigh the drop in likelihood as long
as the generalization is conservative and keeps the model close to the data. The choice of
model priors is discussed in the next section.
The fundamental idea exploited here is that the initial model Mo can be gradually transformed into the generating model by repeatedly merging states. The intuition for this
heuristic comes from the fact that if we take the paths that generate the samples in an actual
generating HMM M and 'unroll' them to make them completely disjoint, we obtain Mo.
The iterative merging process, then, is an attempt to undo the unrolling, tracing a search
through the model space back to the generating model.
Merging two states q] and q2 in this context means replacing q] and q2 by a new state r with
a transition distribution that is a weighted mixture of the transition probabilities of q], q2,
and with a similar mixture distribution for the emissions. Transition probabilities into q] or
q2 are added up and redirected to r. The weights used in forming the mixture distributions
are the relative frequencies with which q] and q2 are visited in the current model.
Repeatedly performing such merging operations yields a sequence of models Mo, M J ,
13
14
Stokke and Omohundro
Mo:
a
b
log L(xIMo) =
- 1. 39
log L(xIMJ) = log L(xIMo)
a
b
Figure I: Sequence of models obtained by merging samples {ab, abab}. All transitions without
special annotations have probability 1; Output symbols appear above their respective states and also
carry an implicit probability of 1. For each model the log likelihood is given.
M2 ? ..., along which we can search for the MAP model. To make the search for M efficient,
we use a greedy strategy: given Mi. choose a pair of states for merging that maximizes
P(Mi+llX)?
Continuing with the previous example, we find that states 1 and 3 in Mo can be merged
without penalizing the likelihood. This is because they have identical outputs and the loss
due to merging the outgoing transitions is compensated by the merging of the incoming
transitions. The .5/.5 split is simply transferred to outgoing transitions of the merged state.
The same situation obtains for states 2 and 4 once 1 and 3 are merged. From these two
first merges we get model M. in Figure 1. By convention we reuse the smaller of two state
indices to denote the merged state.
At this point the best merge turns out to be between states 2 and 6, giving model M2.
However, there is a penalty in likelihood, which decreases to about .59 of its previous
value. Under all the reasonable priors we considered (see below), the posterior model
probability still increases due to an increase in the prior. Note that the transition probability
ratio at state 2 is now 2/1, since two samples make use of the first transition, whereas only
one takes the second transition.
Finally, states 1 and 5 can be merged without penalty to give M3, the minimal model that
generates (ab)+. Further merging at this point would reduce the likelihood by three orders
of magnitude. The resulting decrease in the posterior probability tells the algorithm to stop
Hidden Markov Model Induction by Bayesian Model Merging
at this point.
3.3 MODEL PRIORS
As noted previously, the likelihoods P(XIMj ) along the sequence of models considered by
the algorithm is monotonically decreasing. The prior P(M) must account for an overall
increase in posterior probability, and is therefore the driving force behind generalization.
As in the work on Bayesian learning of classification trees by Buntine (1992), we can split
the prior P(M) into a term accounting for the model structure, P(Ms), and a term for the
adjustable parameters in a fixed structure P(MpIMs).
We initially relied on the structural prior only, incorporating an explicit bias towards smaller
models. Size here is some function of the number of states and/or transitions, IMI. Such a
prior can be obtained by making P(Ms ) <X e- 1M1 , and can be viewed as a description length
prior that penalizes models according to their coding length (Rissanen, 1983; Wallace &
Freeman, 1987). The constants in this "MOL" term had to be adjusted by hand from
examples of 'desirable' generalization.
For the parameter prior P(MpIMs), it is standard practice to apply some sort of smoothing
or regularizing prior to avoid overfitting the model parameters. Since both the transition
and the emission probabilities are given by multinomial distributions it is natural to use a
Dirichlet conjugate prior in this case (Berger, 1985). The effect of this prior is equivalent
to having a number of 'virtual' samples for each of the possible transitions and emissions
which are added to the actual samples when it comes to estimating the most likely parameter
settings. In our case, the virtual samples made equal use of all potential transitions and
emissions, adding bias towards uniform transition and emission probabilities.
We found that the Dirichlet priors by themselves produce an implicit bias towards smaller
models, a phenomenon that can be explained as follows. The prior alone results in a model
with uniform, flat distributions. Adding actual samples has the effect of putting bumps into
the posterior distributions, so as to fit the data. The more samples are available, the more
peaked the posteriors will get around the maximum likelihood estimates of the parameters,
increasing the MAP value. In estimating HMM parameters, what counts is not the total
number of samples, but the number of samples per state, since transition and emission
distributions are local to each state. As we merge states, the available evidence gets shared
by fewer states, thus allowing the remaining states to produce a better fit to the data.
This phenomenon is similar, but not identical, to the Bayesian 'Occam factors' that prefer
models with fewer parameter (MacKay, 1992). Occam factors are a result of integrating
the posterior over the parameter space, something which we do not do because of the
computational complications it introduces in HMMs (see below).
3.4 APPROXIMATIONS
At each iteration step, our algorithm evaluates the posterior resulting from every possible
merge in the current HMM. To keep this procedure feasible, a number of approximations
are incorporated in the implementation that don't seem to affect its qualitative properties.
? For the purpose of likelihood computation, we consider only the most likely path
through the model for a given sample string (the Viterbi path). This allows us to
15
16
Stokke and Omohundro
express the likelihood in product form, computable from sufficient statistics for each
transition and emission.
? We assume the Viterbi paths are preserved by the merging operation, that is, the
paths previously passing through the merged states now go through the resulting new
state. This allows us to update the sufficient statistics incrementally, and means only
O(number of states) likelihood terms need to be recomputed.
? The posterior probability of the model structure is approximated by the posterior of the
MAP estimates for the model parameters. Rigorously integrating over all parameter
values is not feasible since varying even a single parameter could change the paths of
all samples through the HMM.
? Finally, it has to be kept in mind that our search procedure along the sequence of
merged models finds only local optima, since we stop as soon as the posterior starts to
decrease. A full search of the space would be much more costly. However, we found
a best-first look-ahead strategy to be sufficient in rare cases where a local maximum
caused a problem. In those cases we continue merging along the best-first path for a
fixed number of steps (typically one) to check whether the posterior has undergone
just a temporary decrease.
4 EXPERIMENTS
We have used various artificial finite-state languages to test our algorithm and compare its
performance to the standard Baum-Welch algorithm.
Table 1 summarizes the results on the two sample languages ac? a u bc? band a+ b+a+ b+.
The first of these contains a contingency between initial and final symbols that can be hard
for learning algorithms to uncover.
We used no explicit model size prior in our experiments after we found that the Dirichlet
prior was very robust in giving just the the right amount of bias toward smaller models.!
Summarizing the results, we found that merging very reliably found the generating model
structure from a very small number of samples. The parameter values are determined by
the sample set statistics.
The Baum-Welch algorithm, much like a backpropagation network, may be sensitive to its
random initial parameter settings. We therefore sampled from a number of initial conditions.
Interestingly, we found that Baum-Welch has a good chance of settling into a suboptimal
HMM structure, especially if the number of states is the minimal number required for the
target language. It proved much easier to estimate correct language models when extra
states were provided. Also, increasing the sample size helped it converge to the target
model.
5 RELATED WORK
Our approach is related to several other approaches in the literature.
The concept of state merging is implicit in the notion of state equivalence classes, which is
fundamental to much of automata theory (Hopcroft & Ullman, 1979) and has been applied
I The
number of 'virtual' samples per transition/emission was held constant at 0.1 throughout.
Hidden Markov Model Induction by Bayesian Model Merging
(a)
(b)
Method
Merging
Merging
Baum-Welch
(10 trials)
Baum-Welch
(l0 trials)
Baum-Welch
Baum-Welch
Sample
8 m.p.
20 random
8 m.p.
Method
Merging
Baum-Welch
(3 trials)
Merging
Baum-We1ch
(3 trials)
Sample
5m.p.
5m.p.
20 random
8 m.p.
20 random
10 random
10 random
Entropy
2.295
2.087
2.087
2.773
2.087
2.775
2.384
2.085
Cross-entropy
2.188 ? .020
2.158 ? .033
2.894 ? .023 (best)
4.291 ? .228 (worst)
2.105 ? .031 (best)
2.825 ? .031 (worst)
3.914 ? .271
2.155 ? .032
Entropy
2.163
3.545
3.287
5.009
5.009
6.109
Cross-entropy
7.678 ? .158
8.963 ? .161 (best)
59.663 ? .007 (worst)
5.623 ? .074
5.688 ? .076 (best)
8.395 ? .137 (worst)
Language
ac'a v bc'b
ac'a v bc'b
(a v b)c'(a v b)
(a v b)c'(a v b)
ac'a v bc'b
(a v b)c'(a v b)
ac'a v bc'b
ac'a v bc'b
Language
a+b+a+b+
(a+b+t
(a+b+t
a+b+a+b+
a+b+a+b+
(a+b+t
n
6
6
6
6
6
6
10
10
n
4
4
4
4
4
4
Table 1: Results for merging and Baum-Welch on two regular languages: (a) ac'a v bc'b and (b)
a+b+a+b+. Samples were either the top most probable (m.p.) ones from the target language, or a
set of randomly generated ones. 'Entropy' is the average negative log probability on the training
set, whereas 'cross-entropy' refers to the empirical cross-entropy between the induced model and
the generating model (the lower, the better generalization). n denotes the final number of model
states for merging, or the fixed model size for Baum-Welch. For Baum-Welch, both best and worst
performance over several initial conditions is listed.
to automata learning as well (Angluin & Smith, 1983).
Tomita (1982) is an example of finite-state model space search guided by a (nonprobabilistic) goodness measure.
Horning (1969) describes a Bayesian grammar induction procedure that searches the model
space exhaustively for the MAP model. The procedure provably finds the globally optimal
grammar in finite time, but is infeasible in practice because of its enumerative character.
The incremental augmentation of the HMM by merging in new samples has some of the
flavor of the algorithm used by Porat & Feldman (1991) to induce a finite-state model from
positive-only, ordered examples.
Haussler et al. (1992) use limited HMM 'surgery' (insertions and deletions in a linear
HMM) to adjust the model size to the data, while keeping the topology unchanged.
6 FURTHER RESEARCH
We are investigating several real-world applications for our method. One task is the
construction of unified multiple-pronunciation word models for speech recognition. This
is currently being carried out in collaboration with Chuck Wooters at ICSI, and it appears
that our merging algorithm is able to produce linguistically adequate phonetic models.
Another direction involves an extension of the model space to stochastic context-free
grammars, for which a standard estimation method analogous to Baum-Welch exists (Lari
17
18
Stokke and Omohundro
& Young, 1990). The notions of sample incorporation and merging carry over to this domain
(with merging now involving the non-terminals of the CFO), but need to be complemented
with a mechanism that adds new non-terminals to create hierarchical structure (which we
call chunking).
Acknowledgements
We would like to thank Peter Cheeseman, Wray Buntine, David Stoutamire, and Jerry
Feldman for helpful discussions of the issues in this paper.
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Mich., pp. 105-108.
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the Royal Statistical Society, Series B 49(3),240-265 .
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6,290 | 6,690 | Detrended Partial Cross Correlation
for Brain Connectivity Analysis
Jaime S Ide?
Yale University
New Haven, CT 06519
[email protected]
Fabio A Cappabianco
Federal University of Sao Paulo
S.J. dos Campos, 12231, Brazil
[email protected]
Fabio A Faria
Federal University of Sao Paulo
S.J. dos Campos, 12231, Brazil
[email protected]
Chiang-shan R Li
Yale University
New Haven, CT
chiang-shan.li-yale.edu
Abstract
Brain connectivity analysis is a critical component of ongoing human connectome
projects to decipher the healthy and diseased brain. Recent work has highlighted
the power-law (multi-time scale) properties of brain signals; however, there remains
a lack of methods to specifically quantify short- vs. long- time range brain connections. In this paper, using detrended partial cross-correlation analysis (DPCCA),
we propose a novel functional connectivity measure to delineate brain interactions
at multiple time scales, while controlling for covariates. We use a rich simulated
fMRI dataset to validate the proposed method, and apply it to a real fMRI dataset
in a cocaine dependence prediction task. We show that, compared to extant methods, the DPCCA-based approach not only distinguishes short and long memory
functional connectivity but also improves feature extraction and enhances classification accuracy. Together, this paper contributes broadly to new computational
methodologies in understanding neural information processing.
1
Introduction
Brain connectivity is crucial to understanding the healthy and diseased brain states [15, 1]. In recent
years, investigators have pursued the construction of human connectomes and made large datasets
available in the public domain [23, 24]. Functional Magnetic Resonance Imaging (fMRI) has been
widely used to examine complex processes of perception and cognition. In particular, functional
connectivity derived from fMRI signals has proven to be effective in delineating biomarkers for many
neuropsychiatric conditions [15].
One of the challenges encountered in functional connectivity analysis is the precise definition of
nodes and edges of connected brain regions [21]. Functional nodes can be defined based on activation
maps or with the use of functional or anatomical atlases. Once nodes are defined, the next step is to
estimate the weights associated with the edges. Traditionally, these functional connectivity weights
are measured using correlation-based metrics. Previous simulation studies have shown that they can
be quite successful, outperforming higher-order statistics (e.g. linear non-gaussian acyclic causal
models) and lag-based approaches (e.g. Granger causality) [20].
On the other hand, very few studies have investigated the power-law cross-correlation properties
(equivalent to multi-time scale measures) of brain connectivity. Recent research suggested that fMRI
?
Corresponding author: Department of Psychiatry, 34 Park St. S110. New Haven CT 06519.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
signals have power-law properties (e.g. their power-spectrum follows a power law) [8, 3] and that the
deviations from the typical range of power-exponents have been noted in neuropsychiatric disorders
[11]. For instance, in [3], using wavelet-based multivariate methods, authors observed that scale-free
properties are characteristic not only of univariate fMRI signals but also of pairwise cross-temporal
dynamics. Moreover, they found an association between the magnitude of scale-free dynamics and
task performance. We hypothesize that power-law correlation measures may capture additional
dimensions of brain connectivity not available from conventional analyses and thus enhance clinical
prediction.
In this paper, we aim to answer three key open questions: (i) whether and how brain networks are
cross-correlated at different time scales with long-range dependencies (?long-memory? process,
equivalent to power-law in the frequency domain); (ii) how to extract the intrinsic association between
two regions controlling for the influence of other interconnected regions; and (iii) whether multi-time
scale connectivity measures can improve clinical prediction. We address the first two questions
by using the detrended partial cross-correlation analsyis (DPCCA) coefficient [25], a measure that
quantifies correlations on multiple time scales between non-stationary time series, as is typically the
case with task-related fMRI signals. DPCCA is an extension of detrended cross-correlation analysis
[17, 13], and has been successfully applied to analyses of complex systems, including climatological
[26] and financial [18] data. Unlike methods based on filtering particular frequency bands, DPCCA
directly informs correlations across multiple time scales, and unlike wavelet-based approaches (e.g.
cross wavelet transformation and wavelet transform coherence [2]), DPCCA has the advantage of
estimating pairwise correlations controlling for the influence of other regions. This is critical because
brain regions and thus fMRI signals thereof are highly interconnected. To answer the third question,
we use the correlation profiles, generated from DPCCA, as input features for different machine
learning methods in classification tasks and compare the performance of DPCCA-based features with
all other competing features.
In Section 2, we describe the simulated and real data sets used in this study, and show how features
of the classification task are extracted from the fMRI signals. In Section 3, we provide further details
about DPCCA (Section 3.1), and present the proposed multi-time scale functional connectivity measure (Section 3.2). In Section 4, we describe core experiments designed to validate the effectiveness
of DPCCA in brain connectivity analysis and clinical prediction. We demonstrate that DPCCA
(i) detects connectivity at multiple-time scales while controlling for covariates (Sections 4.1 and
4.3), (ii) accurately identifies functional connectivity in well-known gold-standard simulated data
(Section 4.2), and (iii) improves classification accuracy of cocaine dependence with fMRI data of
seventy-five cocaine dependent and eighty-eight healthy control individuals (Section 4.4). In Section
5, we conclude by highlighting the significance of the study as well as the limitations and future
work.
2
2.1
Material and Methods
Simulated dataset: NetSim fMRI data
We use fMRI simulation data - NetSim [20] - previously developed for the evaluation of network
modeling methods. Simulating rich and realistic fMRI time series, NetSim is comprised of twentyeight different brain networks, with different levels of complexity. These signals are generated
using dynamic causal modeling (DCM [6]), a generative network model aimed to quantify neuronal
interactions and neurovascular dynamics, as measured by the fMRI signals. NetSim graphs have 5
to 50 nodes organized with ?small-world? topology, in order to reflect real brain networks. NetSim
signals have 200 time points (mostly) sampled with repetition time (TR) of 3 seconds. For each
network, 50 separate realizations (?subjects?) are generated. Thus, we have a total of 1400 synthetic
dataset for testing. Finally, once the signals are generated, white noise of standard deviation 0.1-1%
is added to reproduce the scan thermal noise.
2.2
Real-world dataset: Cocaine dependence prediction
Seventy-five cocaine dependent (CD) and eighty-eight healthy control (HC) individuals matched in
age and gender participated in this study. CD were recruited from the local, greater New Haven area
in a prospective study and met criteria for current cocaine dependence, as diagnosed by the Structured
Clinical Interview for DSM-IV. They were drug-free while staying in an inpatient treatment unit.
2
The Human Investigation committee at Yale University School of Medicine approved the study, and
all subjects signed an informed consent prior to participation. In the MR scanner, they performed a
simple cognitive control paradigm called stop-signal task [14]. FMRI data were collected with 3T
Siemens Trio scanner. Each scan comprised four 10-min runs of the stop signal task. Functional
blood oxygenation level dependent (BOLD) signals were acquired with a single-shot gradient echo
echo-planar imaging (EPI) sequence, with 32 axial slices parallel to the AC-PC line covering the
whole brain: TR=2000 ms, TE=25 ms, bandwidth=2004 Hz/pixel, flip angle=85? , FOV=220?220
mm2 , matrix=66?64, slice thickness=4 mm and no gap. A high-resolution 3D structural image
(MPRAGE; 1 mm resolution) was also obtained for anatomical co-registration. Three hundred
images were acquired in each session. Functional MRI data was pre-processed with standard pipeline
using Statistical Parametric Mapping 12 (SPM12) (Wellcome Department of Imaging Neuroscience,
University College London, U.K.).
2.2.1
Brain activation
We constructed general linear models and localized brain regions responding to conflict (stop signal)
anticipation (encoded by the probability P(stop)) at the group level [10]. The regions responding to
P(stop) comprised the bilateral parietal cortex, the inferior frontal gyrus (IFG) and the right middle
frontal gyrus (MFG); and regions responding to motor slowing bilateral insula, the left precentral
cortex (L.PC), and the supplementary motor area (SMA) (Fig. 1(a))2 . These regions of interest (ROIs)
were used as masks to extract average activation time courses for functional connectivity analyses.
2.2.2
Functional connectivity
We analyzed the frontoparietal circuit involved in conflict anticipation and response adjustment using
a standard Pearson correlation analysis and multivariate Granger causality analysis or mGCA [19]. In
Fig. 1(b), we illustrate fifteen correlation coefficients derived from the six ROIs for each individual
CD and HC as shown in Fig. 1(a). According to mGCA, connectivities from bilateral parietal to L.PC
and SMA were disrupted in CD (Fig. 1(b)). These findings offer circuit-level evidence of altered
cognitive control in cocaine addiction.
Figure 1: Disrupted frontoparietal circuit
in cocaine addicts. The frontoparietal
circuit included six regions responding
to Bayesian conflict anticipation (?S?)
and regions of motor slowing (?RT?): (a)
CD and HC shared connections (orange
arrows). (b) Connectivity strengths between nodes in the frontoparietal circuit.
We show connectivity strengths between
nodes for each individual subject in CD
(red line) and HC (blue line) groups.
(a)
3
(b)
A Novel Measure of Brain Functional Connectivity
3.1
Detrended partial cross-correlation analysis (DPCCA)
Detrended partial cross-correlation is a novel measure recently proposed by [25]. DPCCA combines
the advantages of detrended cross-correlation analysis (DCCA) [17] and standard partial correlation.
Given two time series {x(a) }, {x(b) } ? Xt , where Xt ? IRm , t = 1, 2, ..., N time points, DPCCA is
given by Equation 1:
?DP CCA (a, b; s) = p
2
?Ca,b (s)
,
Ca,a (s).Cb,b (s)
(1)
Peak MNI coordinates for IFG:[39,53,-1], MFG:[42,23,38],bilateral insula:[-33,17,8] and [30,20,2], L.PC:[36,-13,56], and SMA:[-9,-1,50] in mm.
3
where s is the time scale and each term Ca,b (s) is obtained by inverting the matrix ?(s), e.g.
C(s) =??1 (s). The coefficient ?a,b ? ?(s) is the so called DCCA coefficient [13]. The DCCA
coefficient is an extension of the detrented cross correlation analysis [17] combined with detrended
fluctuation analysis (DFA) [12].
Given two time series {x}, {y} ? Xt (indices omitted for the sake of simplicity) with N time
points and time scale s, DCCA coefficient is given by Equation 2:
2
FDCCA
(s)
?(s) =
FDF A,x (s)FDF A,y (s)
,
(2)
where the numerator and denominator are the average of detrended covariances and variances of the
N ? s + 1 windows (partial sums), respectively, as described in Equations 3-4:
PN ?s+1
2
FDCCA
(s)
j=1
=
(3)
N ?s
PN ?s+1
2
FDF
A,x (s)
2
fDCCA
(s, j)
2
fDF
A,x (s, j)
j=1
=
N ?s
.
(4)
The partial sums (profiles) are obtained with sliding windows across the integrated time series
Pt
Pt
Xt = i=1 xi and Yt = i=1 yi . For each time window j with size s, detrended covariances
and variances are computed according to Equations 5-6:
Pj+s?1
2
fDCCA
(s, j)
=
t=j
d
[
(Xt ? X
t,j )(Yt ? Yt,j )
s?1
Pj+s?1
2
fDF
A,x (s, j)
=
t=j
2
[
(Xt ? X
t,j )
s?1
,
,
(5)
(6)
d
[
where X
t,j and Yt,j are polynomial fits of time trends. We used a linear fit as originally proposed
[13], but higher order fits could also be used [25]. DCCA can be used to measure power-law
cross-correlations. However, we focus on DCCA coefficient as a robust measure to detect pairwise
cross-correlation in multiple time scales, while controlling for covariates. Importantly, DPCCA
quantifies correlations among time series with varying levels of non-stationarity [13].
3.2
DPCCA for functional connectivity analysis
In this section, we propose the use of DPCCA as a novel measure of brain functional connectivity.
First, we show in simulation experiments that the measure satisfies desired connectivity properties.
Further, we define the proposed connectivity measure. Although these properties are expected by
mathematical definition of DPCCA, it is critical to confirm its validity on real fMRI data. Additionally,
it is necessary to establish the statistical significance of the computed measures at the group level.
3.2.1
Desired properties
Given real fMRI signals, the measure should accurately detect the time scale in which the pairwise
connections occur, while controlling for the covariates. To verify this, we create synthetic data by
combining real fMRI signals and sinusoidal waves (Fig. 2). To simplify, we assume additive property
of signals and sinusoidal waves reflecting the time onset of the connections. For each simulation, we
randomly sample 100 sets of time series or ?subjects?.
a) Distinction of short and long memory connections. Given two fMRI signals {xA }, {xB }, we
derive three pairs with known connectivity profiles: short-memory {XA = xA + sin(T1 ) + e},
{XB = xB + sin(T1 ) + e}, long-memory {XA = xA + sin(T2 ) + e}, {XB = xB +
sin(T2 ) + e} and mixed {XA = xA + sin(T1 ) + sin(T2 ) + e}, {XB = xB + sin(T1 ) +
sin(T2 ) + e}, where T1 << T2 and e is a Gaussian signal to simulate measurement noise. We
hypothesize that the two nodes A and B are functionally connected at time scales T1 and T2 .
4
b) Control for covariates. Given three fMRI signals {xA }, {xB }, {xC }, we derive three signals
with known connectivity {XAC = xA +xC +sin(T )+e}, {XBC = xB +xC +sin(T )+e},
{XC = xC + e}, where e is the measurement noise. We hypothesize that the two nodes A and B
are functionally connected mostly at scale T, once the mutual influence of node C is controlled.
Figure 2: Illustration of synthetic fMRI signals generated by combining real fMRI signals and sinusoidal waves. (a) Original fMRI
signals, (b) original signals with sin(T =
10s) and sin(T = 30s) waves added.
(a)
3.2.2
(b)
Statistical significance
Given two nodes and their time series, we assume that they are functionally connected if the
max |?DP CCA |, within a time range srange , is significantly greater than the null distribution.
Empirical null distributions are estimated from the original data by randomly shuffling time series
across different subjects and nodes, as proposed in [20]. In this way, we generate realistic distributions
of connectivity weights occurring by chance. Since we have a multivariate measure, the null dataset
is always generated with the same number of nodes as the tested network. Multiple comparisons are
controlled by estimating the false discovery rate. Importantly, the null distribution is also computed on
max |?DP CCA | within the time range srange . We use a srange from 6 to 18 seconds, assuming
that functional connections transpire in this range. Thus, we allow connections with different
time-scales. We use this binary definition of functional connectivity for the current approach to be
comparable with other methods, but it is also possible to work with the whole temporal profile of
?DP CCA (s), as is done in the classification experiment (Section 4.4). To keep the same statistical
criteria, we also generate null distributions for all the other connectivity measures.
3.2.3
DPCCA + Canonical correlation analysis
As further demonstrated by simulation results (Table 1), DPCCA alone has lower true positive rate
(TPR) compared to other competing methods, likely because of its restrictive statistical thresholds. In
order to increase the sensitivity of DPCCA, we augmented the method by including an additional
canonical correlation analysis (CCA) [7]. CCA was previously used in fMRI in different contexts
to detect brain activations [5], functional connectivity [27], and for multimodal information fusion
[4]. In short, given two sets of multivariate time series {XA (t) ? IRm , t = 1, 2, ..., N } and
{XB (t) ? IRn , t = 1, 2, ..., N }, where m and n are the respective dimensions of the two sets
A and B, and N is the number of time points, CCA seeks the linear transformations u and v so that
the correlation between the linear combinations XA (t)u and XB (t)v is maximized. In this work,
we propose the use of CCA to define the existence of a true connection, in addition to the DPCCA
connectivity results. The proposed method is summarized in Algorithm 1. With CCA (Lines 8-14),
we identify the nodes that are strongly connected after linear transformations. In Line 18, we use
CCA to inform DPCCA in terms of positive connections.
4
4.1
Experiments and Results
Connectivity properties: Controlling time scales and covariates
In Figure 3, we observe that DPCCA successfully captured the time scales of the correlations
between time series {XA }, {XB }, despite the noisy nature of fMRI signals. For instance, it
distinguished between short and long-memory connections, represented using T1 = 10s and
T2 = 30s, respectively (Figs. 3a-c). Importantly, it clearly detected the peak connection at 10s after
controlling for the influence of covariate signal XC (Fig. 3f). Further, unlike DPCCA, the original
DCCA method did not rule out the mutual influence of XC with peak at 30s (Fig. 3e).
5
Algorithm 1 DPCCA+CCA
Input: Time series {Xt ? IRm , t = 1, 2, ..., N }, where m is the number of vectors and N is
the number of time points; time range srange with k values
Output: Connectivity matrix F C : [m ? m] and associated matrices
1: Step: DPCCA(Xt )
. Compute pairwise DPCCA
2:
for pair of vectors {x(a) }, {x(b) } ? Xt do
3:
for s in srange do
4:
Compute the coefficient ?DP CCA (a, b; s)
. Equation(1)
5:
F C[a, b] ? max |?DP CCA | in srange
6:
P [a, b] ? statistical significance of F C[a, b] given the null empirical distribution
7:
return F C and P
. Matrix of connection weights and p-values
8: Step: CCA(Xt )
. Compute CCA connectivity
9:
for x(a) ? Xt do
10:
for x(b) ? Xt , b 6= a do
11:
rCCA [a, b] ? (1? CCA between {x(a) }, {x(c) }, c 6= a, b)
. Effect of
excluding node b
12:
indexcon ? k-means(rCCA [a])
. Split connections into binary groups
13:
CCA[a, indexcon ] ? 1
14:
return CCA
. CCA is a binary connectivity matrix
15: Step: DPCCA+CCA(P,CCA)
. Augment DPCCA with CCA results
16:
for pair of nodes {a, b} do
17:
F C ? [a, b] ? 1, if P [a, b] < 0.05
. DPCCA significant connections
. Fill missing connections
18:
F C ? [a, b] ? max(F C ? [a, b], CCA[a, b])
19:
return F C ? , F C and P
. F C ? is a binary matrix
Figure 3: DPCCA temporal profiles
among the synthetic signals (details
in Section 3.2.1). (a)-(c): DPCCA
with peak at T=10s and T=30s, and
mixed. (d) DPCCA of the original fMRI signals used to generate
the synthetics signals. (e) Temporal
profile obtained with DCCA without partial correlation. (f) DPCCA
peak at T=10s after controlling for
XC . Dashed lines are the 95% confidence interval of DPCCA for the
empirical null distribution.
4.2
Simulated networks: Improved connectivity accuracy
The goal of this experiment is to validate the proposed methods in an extensive dataset designed
to test functional connectivity methods. In this dataset, ground truth networks are known with the
architectures aimed to reflect real brain networks. We use the full NetSim dataset comprised of
28 different brain circuits and 50 subjects. For each sample of time series, we compute the partial
correlation (parCorr) and the regularized inverse covariance (ICOV), reported as the best performers
in [20], as well as the proposed DPCCA and DPCCA+CCA methods. For each measure, we construct
empirical null distributions, as described in Section 3.2.2, and generate the binary connectivity matrix
using threshold ? = 0.05. To evaluate their connectivity accuracy, given the ground truth networks,
we compute the true positive and negative rates (TPR and TNR, respectively) and the balanced
N R)
accuracy BAcc= (T P R+T
.
2
Using NetSim fMRI data as the testing benchmark, we observed that the proposed DPCCA+CCA
method provided more accurate functional connectivity results than the best methods reported in the
original paper [20]. Results are summarized in Table 1. Here we use the balanced accuracy (BAcc)
6
as the evaluation metric, since it is a straightforward way to quantify both true positive and negative
connections.
Table 1: Comparison of functional connectivity methods using NetSim dataset. Mean and standard
deviation of balanced accuracy (BAcc), true positive rate (TPR) and true negative rate (TNR) are
reported. ParCorr: partial correlation, ICOV: regularized inverse covariance, DPCCA: detrended
cross correlation analysis, DPCCA+CCA: DPCCA augmented with CCA. DPCCA+CCA balanced
accuracy is significantly higher than the best competing method ICOV (Wilcoxon signed paired test,
Z=3.35 and p=8.1e-04).
Metrics
Mean
Std
4.3
ParCorr
BAcc
TPR TNR
0.834 0.866 0.804
0.096 0.129 0.188
Functional connectivity measures
ICOV
DPCCA
BAcc TPR
TNR
BAcc TPR
TNR
0.841 0.866 0.817 0.846 0.835 0.855
0.095 0.131 0.181 0.095 0.150 0.177
DPCCA+CCA
BAcc TPR
TNR
0.859 0.893 0.824
0.091 0.081 0.169
Real-world dataset: Learning connectivity temporal profiles
We use unsupervised methods to (i) learn representative temporal profiles of connectivity from
DPCCAF ull , and (ii) perform dimensionality reduction. The use of temporal profiles may capture
additional information (such as short- and long-memory connectivity). However, it increases the
feature set dimensionality, imposing additional challenges on classifier training, particularly with
small dataset. The first natural choice for this task is principal component analysis (PCA), which can
represent original features by their linear combination. Additionally, we use two popular non-linear
dimensionality reduction methods Isomap [22] and autoencoders [9]. With Isomap, we attempt to
learn the intrinsic geometry (manifold) of the temporal profile data. With autoencoders, we seek to
represent the data using restricted Boltzmann machines stacked into layers.
In Figure 4, we show some representative correlation profiles obtained by computing DPPCA
among frontoparietal regions (circuit presented in Fig. 1), and the first three principal components.
Interestingly, PCA seemed to learn some of the characteristic temporal profiles. For instance, as
expected, the first components captured the main trend, while the second components captured some
of the short (task-related) and long (resting-state) memory connectivity trends (Figs.4a-b).
Figure 4: Illustration of some
DPCCA profiles and their principal components. IFG: inferior frontal gyrus, SMA: supplementary motor area, PC:
premotor cortex. Explained
variances of the components
are also reported.
4.4
Real-world dataset: Cocaine dependence prediction
The classification task consists of predicting the class membership, cocaine dependence (CD) and
healthy control (HC), given each individual?s fMRI data. After initial preprocessing (Section 2.2), we
extract average time series within the frontoparietal circuit of 6 regions 3 (Figure 1), and compute
the different cross-correlation measures. These coefficients are used as features to train and test
(leave-one-out cross-validation) a set of popular classifiers available in scikit-learn toolbox [16]
(version 0.18.1), including k-nearest neighbors (kNN), support vector machine (SVM), multilayer
perceptron (MLP), Gaussian processes (GP), naive Bayes (NB) and the ensemble method Adaboost
(Ada). For the DPCCA coefficients, we test both peak values DPCCAmax as well as the rich
temporal profiles DPCCAF ull . Finally, we also include the brain activation maps (Section 2.2.1) as
feature set, thus allowing comparison with popular fMRI classification softwares such as PRONTO
(http://www.mlnl.cs.ucl.ac.uk/pronto/). Features are summarized in Table 2.
3
Although these regions are obtained from the whole-group, no class information is used to avoid inflated
classification rates.
7
Table 2: Features used in the cocaine dependence classification task.
Type
Name
P(stop)
UPE
Corr
ParCorr
ICOV
DPCCAmax
DPCCAF ull
DPCCAIso
DPCCAAutoE
DPCCAP CA
Activation
Connectivity
Size
1042
1042
15
15
15
15
270
135-180
30-45
135-180
Description
Brain regions responding to anticipation of stop signals
Brain regions responding to unsigned prediction error of P(stop)
Pearson cross-correlation among the six frontoparietal regions
Partial cross-correlation among the six frontoparietal regions
Regularized inverse covariance among the six frontoparietal regions
Maximum DPCCA within the range 6-40 seconds
Temporal profile of DPCCA within the range 6-40 seconds
Isomap with 9-12 components and 30 neighbors
Autoencoders with 2-3 hidden layers, 5-20 neurons, batch=100, epoch=1000
PCA with 9-12 components
Classification results are summarized in Table 3 and Figure 5. We used the area under curve (AUC)
as an evaluation metric in order to consider both sensitivity and specificity of the classifiers, as well as
balanced accuracy (BAcc). Here we tested all features described in Table 2, including the DPCCA full
profiles after dimensionality reduction (Isomap, autoencoders and PCA). Activation maps produced
poor classification results (P(stop): 0.525?0.048 and UPE: 0.509?0.032), comparable to the results
obtained with PRONTO software using the same features (accuracy 0.556).
Features
Corr
ParCorr
ICOV
DPCCAmax
DPCCAF ull
DPCCAIso
DPCCAAutoE
DPCCAP CA
Mean AUC
(? std)
0.757 (? 0.041)
0.901 (? 0.034)
0.900 (? 0.030)
0.906 (? 0.019)
0.899 (? 0.028)
0.902 (? 0.030)
0.815 (? 0.149)
0.928 (? 0.035)
Mean BAcc
(? std)
0.674 (? 0.037)
0.848 (? 0.025)
0.838 (? 0.023)
0.831 (? 0.022)
0.820 (? 0.052)
0.827 (? 0.068)
0.813 (? 0.106)
0.844 (? 0.064)
Top classifier
(AUC / BAcc)
GP / NB
GP / Ada
GP / SVM
GP / Ada
GP / GP
GP / MLP
SVM / kNN5
Ada / NB
Accuracy
(AUC / BAcc)
0.794 / 0.710
0.948 / 0.875
0.948 / 0.858
0.929 / 0.857
0.957 / 0.874
0.954 / 0.894
0.939 / 0.863
0.963 / 0.911
Table 3: Comparison of classification results for different
features. The DPCCA features combined with PCA produced
the top classifiers according to both criteria (0.963/0.911).
However, DPCCAP CA is not statistically better than ParCorr
or ICOV (Wilcoxon signed paired test, p>0.05). See Figure 5
for accuracy across different classification methods.
5
Figure 5: Comparison of classification results for different features and methods (described in
Section 4.4).
Conclusions
In summary, as a multi-time scale approach to characterize brain connectivity, the proposed method
(DPCCA+CCA) (i) identified connectivity peak-times (Fig. 3), (ii) produced higher connectivity
accuracy than the best competing method ICOV (Table 1), and (iii) distinguished short/long memory
connections between brain regions involved in cognitive control (IFC&SMA and SMA&PC) (Fig.
4). Second, using the connectivity weights as features, DPCCA measures combined with PCA
produced the highest individual accuracies (Table 3). However, it was not statistically different
from the second best feature (ParCorr) across different classifiers. Further separate test set would be
necessary to identify the best classifiers. We performed extensive experiments with a large simulated
fMRI dataset to validate DPCCA as a promising functional connectivity analytic. On the other
hand, our conclusions on clinical prediction (classification task) are still limited to one case. Finally,
further optimization of Isomap and autoencoders methods could improve the learning of connectivity
temporal profiles produced by DPCCA.
Acknowledgments
Supported by FAPESP (2016/21591-5), CNPq (408919/2016-7), NSF (BCS1309260) and NIH
(AA021449, DA023248).
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6,291 | 6,691 | Contrastive Learning for Image Captioning
Bo Dai
Dahua Lin
Department of Information Engineering, The Chinese University of Hong Kong
[email protected]
[email protected]
Abstract
Image captioning, a popular topic in computer vision, has achieved substantial
progress in recent years. However, the distinctiveness of natural descriptions is
often overlooked in previous work. It is closely related to the quality of captions,
as distinctive captions are more likely to describe images with their unique aspects.
In this work, we propose a new learning method, Contrastive Learning (CL), for
image captioning. Specifically, via two constraints formulated on top of a reference
model, the proposed method can encourage distinctiveness, while maintaining the
overall quality of the generated captions. We tested our method on two challenging
datasets, where it improves the baseline model by significant margins. We also
showed in our studies that the proposed method is generic and can be used for
models with various structures.
1
Introduction
Image captioning, a task to generate natural descriptions of images, has been an active research
topic in computer vision and machine learning. Thanks to the advances in deep neural networks,
especially the wide adoption of RNN and LSTM, there has been substantial progress on this topic
in recent years [23, 24, 15, 19]. However, studies [1, 3, 2, 10] have shown that even the captions
generated by state-of-the-art models still leave a lot to be desired. Compared to human descriptions,
machine-generated captions are often quite rigid and tend to favor a ?safe? (i.e. matching parts of
the training captions in a word-by-word manner) but restrictive way. As a consequence, captions
generated for different images, especially those that contain objects of the same categories, are
sometimes very similar [1], despite their differences in other aspects.
We argue that distinctiveness, a property often overlooked in previous work, is significant in natural
language descriptions. To be more specific, when people describe an image, they often mention or
even emphasize the distinctive aspects of an image that distinguish it from others. With a distinctive
description, someone can easily identify the image it is referring to, among a number of similar
images. In this work, we performed a self-retrieval study (see Section 4.1), which reveals the lack of
distinctiveness affects the quality of descriptions.
From a technical standpoint, the lack of distinctiveness is partly related to the way that the captioning
model was learned. A majority of image captioning models are learned by Maximum Likelihood
Estimation (MLE), where the probabilities of training captions conditioned on corresponding images
are maximized. While well grounded in statistics, this approach does not explicitly promote distinctiveness. Specifically, the differences among the captions of different images are not explicitly taken
into account. We found empirically that the resultant captions highly resemble the training set in a
word-by-word manner, but are not distinctive.
In this paper, we propose Contrastive Learning (CL), a new learning method for image captioning,
which explicitly encourages distinctiveness, while maintaining the overall quality of the generated
captions. Specifically, it employs a baseline, e.g. a state-of-the-art model, as a reference. During
learning, in addition to true image-caption pairs, denoted as (I, c), this method also takes as input
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
mismatched pairs, denoted as (I, c/ ), where c/ is a caption describing another image. Then, the
target model is learned to meet two goals, namely (1) giving higher probabilities p(c|I) to positive
pairs, and (2) lower probabilities p(c/ |I) to negative pairs, compared to the reference model. The
former ensures that the overall performance of the target model is not inferior to the reference; while
the latter encourages distinctiveness.
It is noteworthy that the proposed learning method (CL) is generic. While in this paper, we focused on
models based on recurrent neural networks [23, 15], the proposed method can also generalize well to
models based on other formulations, e.g. probabilistic graphical models [4, 9]. Also, by choosing the
state-of-the-art model as the reference model in CL, one can build on top of the latest advancement in
image captioning to obtain improved performances.
2
Related Work
Models for Image Captioning The history of image captioning can date back to decades ago.
Early attempts are mostly based on detections, which first detect visual concepts (e.g. objects and
their attributes) [9, 4] followed by template filling [9] or nearest neighbor retrieving for caption
generation [2, 4]. With the development of neural networks, a more powerful paradigm, encoderand-decoder, was proposed by [23], which then becomes the core of most state-of-the-art image
captioning models. It uses a CNN [20] to represent the input image with a feature vector, and applies
a LSTM net [6] upon the feature to generate words one by one.
Based on the encoder-and-decoder, many variants are proposed, where attention mechanism [24] appears to be the most effective add-on. Specifically, attention mechanism replaces the feature vector
with a set of feature vectors, such as the features from different regions [24] , and those under different
conditions [27]. It also uses the LSTM net to generate words one by one, where the difference is that
at each step, a mixed guiding feature over the whole feature set, will be dynamically computed. In
recent years, there are also approaches combining attention mechanism and detection. Instead of
doing attention on features, they consider the attention on a set of detected visual concepts, such as
attributes [25] and objects [26].
Despite of the specific structure of any image captioning model, it is able to give p(c|I), the probability
of a caption conditioned on an image. Therefore, all image captioning models can be used as the
target or the reference in CL method.
Learning Methods for Image Captioning Many state-of-the-art image captioning models adopt
Maximum Likelihood Estimation (MLE) as their learning method, which maximizes the conditional
log-likelihood of the training samples, as:
X
Ti
X
(t)
(t?1)
ln p(wi |Ii , wi
(1)
, ..., wi , ?),
(1)
(ci ,Ii )?D t=1
(1)
(2)
(T )
where ? is the parameter vector, Ii and ci = (wi , wi , ..., wi i ) are a training image and its
caption. Although effective, some issues, including high resemblance in model-gerenated captions,
are observed [1] on models learned by MLE.
Facing these issues, alternative learning methods are proposed in recent years. Techniques of
reinforcement learning (RL) have been introduced in image captioning by [19] and [14]. RL sees the
procedure of caption generation as a procedure of sequentially sampling actions (words) in a policy
space (vocabulary). The rewards in RL are defined to be evaluation scores of sampled captions. Note
that distinctiveness has not been considered in both approaches, RL and MLE.
Prior to this work, some relevant ideas have been explored [21, 16, 1]. Specifically, [21, 16] proposed
an introspective learning (IL) approach that learns the target model by comparing its outputs on (I, c)
and (I/ , c). Note that IL uses the target model itself as a reference. On the contrary, the reference
model in CL provides more independent and stable indications about distinctiveness. In addition,
(I/ , c) in IL is pre-defined and fixed across the learning procedure, while the negative sample in
CL, i.e. (I, c/ ), is dynamically sampled, making it more diverse and random. Recently, Generative
Adversarial Networks (GAN) was also adopted for image captioning [1], which involves an evaluator
that may help promote the distinctiveness. However, this evaluator is learned to directly measure the
2
A man performing stunt
in the air at skate park
A man doing a trick
on a skateboard
Self Retrieval
Self Retrieval
(a) Nondistinctive Caption
(b) Distinctive Caption
Figure 1: This figure illustrates respectively a nondistinctive and distinctive captions of an image, where the
nondistinctive one fails to retrieve back the original image in self retrieval task.
Self Retrieval Top-K Recall
Method
Neuraltalk2 [8]
AdaptiveAttention [15]
AdaptiveAttention + CL
Captioning
1
5
50
500
ROUGE_L
CIDEr
0.02
0.10
0.32
0.32
0.96
1.18
3.02
11.76
11.84
27.50
78.46
80.96
0.652
0.689
0.695
0.827
1.004
1.029
Table 1: This table lists results of self retrieval and captioning of different models. The results are reported on
standard MSCOCO test set. See sec 4.1 for more details.
distinctiveness as a parameterized approximation, and the approximation accuracy is not ensured in
GAN. In CL, the fixed reference provides stable bounds about the distinctiveness, and the bounds are
supported by the model?s performance on image captioning. Besides that, [1] is specifically designed
for models that generate captions word-by-word, while CL is more generic.
3
Background
Our formulation is partly inspired by Noise Contrastive Estimation (NCE) [5]. NCE is originally
introduced for estimating probability distributions, where the partition functions can be difficult or
even infeasible to compute. To estimate a parametric distribution pm (.; ?), which we refer to as the
target distribution, NCE employs not only the observed samples X = (x1 , x2 , ..., xTm ), but also
the samples drawn from a reference distribution pn , denoted as Y = (y1 , y2 , ..., yTn ). Instead of
estimating pm (.; ?) directly, NCE estimates the density ratio pm /pn by training a classifier based on
logistic regression.
Specifically, let U = (u1 , ..., uTm +Tn ) be the union of X and Y . A binary class label Ct is assigned
to each ut , where Ct = 1 if ut ? X and Ct = 0 if ut ? Y . The posterior probabilities for the class
labels are therefore
P (C = 1|u, ?) =
pm (u; ?)
,
pm (u; ?) + ?pn (u)
P (C = 0|u, ?) =
?pn (u)
,
pm (u; ?) + ?pn (u)
(2)
where ? = Tn /Tm . Let G(u; ?) = ln pm (u; ?) ? ln pn (u) and h(u, ?) = P (C = 1|u, ?), then we
can write
1
h(u; ?) = r? (G(u; ?)), with r? (z) =
.
(3)
1 + ? exp(?z)
The objective function of NCE is the joint conditional log-probabilities of Ct given the samples U ,
which can be written as
L(?; X, Y ) =
Tm
X
ln[h(xt ; ?)] +
t=1
Tn
X
ln[1 ? h(yt ; ?)].
(4)
t=1
Maximizing this objective with respect to ? leads to an estimation of G(?; ?), the logarithm of the
density ratio pm /pn . As pn is a known distribution, pm (: |?) can be readily derived.
4
Contrastive Learning for Image Captioning
Learning a model by characterizing desired properties relative to a strong baseline is a convenient
and often quite effective way in situations where it is hard to describe these properties directly.
Specifically, in image captioning, it is difficult to characterize the distinctiveness of natural image
descriptions via a set of rules, without running into the risk that some subtle but significant points are
3
missed. Our idea in this work is to introduce a baseline model as a reference, and try to enhance the
distinctiveness on top, while maintaining the overall quality of the generated captions.
In the following we will first present an empirical study on the correlation between distinctiveness of
its generated captions and the overall performance of a captioning model. Subsequently, we introduce
the main framework of Contrastive Learning in detail.
4.1
Empirical Study: Self Retrieval
In most of the existing learning methods of image captioning, models are asked to generate a caption
that best describes the semantics of a given image. In the meantime, distinctiveness of the caption,
which, on the other hand, requires the image to be the best matching among all images for the caption,
has not been explored. However, distinctiveness is crucial for high-quality captions. A study by Jas
[7] showed that specificity is common in human descriptions, which implies that image descriptions
often involve distinctive aspects. Intuitively, a caption satisfying this property is very likely to contain
key and unique content of the image, so that the original image could easily be retrieved when the
caption is presented.
To verify this intuition, we conducted an empirical study which we refer to as self retrieval. In this
experiment, we try to retrieve the original image given its model-generated caption and investigate topk recalls, as illustrated in Figure 1. Specifically, we randomly sampled 5, 000 images (I1 , I2 , ..., I5000 )
from standard MSCOCO [13] test set as the experiment benchmark. For an image captioning model
pm (:, ?), we first ran it on the benchmark to get corresponding captions (c1 , c2 , ..., c5000 ) for the
images. After that, using each caption ct as a query, we computed the conditional probabilities
(pm (ct |I1 ), pm (ct |I2 ), ..., pm (ct |I5000 )), which were used to get a ranked list of images, denoted by
rt . Based on all ranked lists, we can compute top-k recalls, which is the fraction of images within
top-k positions of their corresponding ranked lists. The top-k recalls are good indicators of how well
a model captures the distinctiveness of descriptions.
In this experiment, we compared three different models, including Neuraltalk2 [8] and AdaptiveAttention [15] that are learned by MLE, as well as AdaptiveAttention learned by our method. The
top-k recalls are listed in Table 1, along with overall performances of these models in terms of
Rouge [12] and Cider [22]. These results clearly show that the recalls of self retrieval are positively
correlated to the performances of image captioning models in classical captioning metrics. Although
most of the models are not explicitly learned to promote distinctiveness, the one with better recalls
of self retrieval, which means the generated-captions are more distinctive, performs better in the
image captioning evaluation. Such positive correlation clearly demonstrates the significance of
distinctiveness to captioning performance.
4.2
Contrastive Learning
In Contrastive Learning (CL), we learn a target image captioning model pm (:; ?) with parameter ?
by constraining its behaviors relative to a reference model pn (:; ?) with parameter ?. The learning
procedure requires two sets of data: (1) the observed data X, which is a set of ground-truth imagecaption pairs ((c1 , I1 ), (c2 , I2 ), ..., (cTm , ITm )), and is readily available in any image captioning
dataset, (2) the noise set Y , which contains mismatched pairs ((c/1 , I1 ), (c/2 , I2 ), ..., (c/Tn , ITn )),
and can be generated by randomly sampling c/t ? C/It for each image It , where C/It is the set of
all ground-truth captions except captions of image It . We refer to X as positive pairs while Y as
negative pairs.
For any pair (c, I), the target model and the reference model will respectively give their estimated
conditional probabilities pm (c|I, ?) and pn (c|I, ?). We wish that pm (ct |It , ?) is greater than
pn (ct |It , ?) for any positive pair (ct , It ), and vice versa for any negative pair (c/t , It ). Following this intuition, our initial attempt was to define D((c, I); ?, ?), the difference between pm (c|I, ?)
and pn (c|I, ?), as
D((c, I); ?, ?) = pm (c|I, ?) ? pn (c|I, ?),
(5)
and set the loss function to be:
L0 (?; X, Y, ?) =
Tm
X
D((ct , It ); ?, ?) ?
t=1
Tn
X
t=1
4
D((c/t , It ); ?, ?).
(6)
In practice, this formulation would meet with several difficulties. First, pm (c|I, ?) and pn (c|I, ?) are
very small (? 1e-8), which may result in numerical problems. Second, Eq (6) treats easy samples,
hard samples, and mistaken samples equally. This, however, is not the most effective way. For
example, when D((ct , It ); ?, ?) 0 for some positive pair, further increasing D((ct , It ); ?, ?)
is probably not as effective as updating D((ct0 , It0 ); ?, ?) for another positive pair, for which
D((ct0 , It0 ); ?, ?) is much smaller.
To resolve these issues, we adopted an alternative formulation inspired by NCE (sec 3), where we
replace the difference function D((c, I); ?, ?) with a log-ratio function G((c, I); ?, ?):
G((c, I); ?, ?) = ln pm (c|I, ?) ? ln pn (c|I, ?),
(7)
and further use a logistic function r? (Eq(3)) after G((c, I); ?, ?) to saturate the influence of easy
samples. Following the notations in NCE, we let ? = Tn /Tm , and turn D((c, I); ?, ?) into:
h((c, I); ?, ?) = r? (G((c, I); ?, ?))).
(8)
Note that h((c, I); ?, ?) ? (0, 1). Then, we define our updated loss function as:
L(?; X, Y, ?) =
Tm
X
ln[h((ct , It ); ?, ?)] +
t=1
Tn
X
ln[1 ? h((c/t , It ); ?, ?)].
(9)
t=1
For the setting of ? = Tn /Tm , we choose ? = 1, i.e. Tn = Tm , to ensure balanced influences from
both positive and negative pairs. This setting consistently yields good performance in our experiments.
Furthermore, we copy X for K times and sample K different Y s, in order to involve more diverse
negative pairs without overfitted to them. In practice we found K = 5 is sufficient to make the
learning stable. Finally, our objective function is defined to be
K
1 1 X
J(?) =
L(?; X, Yk , ?).
(10)
K Tm
k=1
Note that J(?) attains its upper bound 0 if positive and negative pairs can be perfectly distinguished,
namely, for all t, h((ct , It ); ?, ?) = 1 and h((c/t , It ); ?, ?) = 0. In this case, G((ct , It ); ?, ?) ? ?
and G((c/t , It ); ?, ?) ? ??, which indicates the target model will give higher probability p(ct |It )
and lower probability p(c/t |It ), compared to the reference model. Towards this goal, the learning
process would encourage distinctiveness by suppressing negative pairs, while maintaining the overall
performance by maximizing the probability values on positive pairs.
4.3
Discussion
Maximum Likelihood Estimation (MLE) is a popular learning method in the area of image captioning
[23, 24, 15]. The objective of MLE is to maximize only the probabilities of ground-truth imagecaption pairs, which may lead to some issues [1], including high resemblance in generated captions.
While in CL, the probabilities of ground-truth pairs are indirectly ensured by the positive constraint
(the first term in Eq(9)), and the negative constraint (the second term in Eq(9)) suppresses the
probabilities of mismatched pairs, forcing the target model to also learn from distinctiveness.
Generative Adversarial Network (GAN) [1] is a similar learning method that involves an auxiliary
model. However, in GAN the auxiliary model and the target model follow two opposite goals, while
in CL the auxiliary model and the target model are models in the same track. Moreover, in CL the
auxiliary model is stable across the learning procedure, while itself needs careful learning in GAN.
It?s worth noting that although our CL method bears certain level of resemblance with Noise Contrastive Estimation (NCE) [5]. The motivation and the actual technical formulation of CL and NCE
are essentially different. For example, in NCE the logistic function is a result of computing posterior
probabilities, while in CL it is explicitly introduced to saturate the influence of easy samples.
As CL requires only pm (c|I) and pn (c|I), the choices of the target model and the reference model
can range from models based on LSTMs [6] to models in other formats, such as MRFs [4] and
memory-networks [18]. On the other hand, although in CL, the reference model is usually fixed
across the learning procedure, one can replace the reference model with the latest target model
periodically. The reasons are (1) ?J(?) 6= 0 when the target model and the reference model are
identical, (2) latest target model is usually stronger than the reference model, (3) and a stronger
reference model can provide stronger bounds and lead to a stronger target model.
5
COCO Online Testing Server C5
Method
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
Google NIC [23]
Hard-Attention[24]
AdaptiveAttention [15]
AdpativeAttention + CL (Ours)
PG-BCMR [14]
0.713
0.705
0.735
0.742
0.754
0.542
0.528
0.569
0.577
0.591
0.407
0.383
0.429
0.436
0.445
0.309
0.277
0.323
0.326
0.332
0.254
0.241
0.258
0.260
0.257
0.530
0.516
0.541
0.544
0.550
0.943
0.865
1.001
1.010
1.013
ATT-FCN? [26]
MSM? [25]
AdaptiveAttention? [15]
Att2in? [19]
0.731
0.739
0.746
-
0.565
0.575
0.582
-
0.424
0.436
0.443
-
0.316
0.330
0.335
0.344
0.250
0.256
0.264
0.268
0.535
0.542
0.550
0.559
0.943
0.984
1.037
1.123
COCO Online Testing Server C40
Method
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
Google NIC [23]
Hard-Attention [24]
AdaptiveAttention [15]
AdaptiveAttention + CL (Ours)
PG-BCMR [14]
0.895
0.881
0.906
0.910
-
0.802
0.779
0.823
0.831
-
0.694
0.658
0.717
0.728
-
0.587
0.537
0.607
0.617
-
0.346
0.322
0.347
0.350
-
0.682
0.654
0.689
0.695
-
0.946
0.893
1.004
1.029
-
ATT-FCN? [26]
MSM? [25]
AdaptiveAttention? [15]
Att2in? [19]
0.900
0.919
0.918
-
0.815
0.842
0.842
-
0.709
0.740
0.740
-
0.599
0.632
0.633
-
0.335
0.350
0.359
-
0.682
0.700
0.706
-
0.958
1.003
1.051
-
Table 2: This table lists published results of state-of-the-art image captioning models on the online COCO
testing server. ? indicates ensemble model. "-" indicates not reported. In this table, CL improves the base model
(AdaptiveAttention [15]) to gain the best results among all single models on C40.
5
5.1
Experiment
Datasets
We use two large scale datasets to test our contrastive learning method. The first dataset is MSCOCO
[13], which contains 122, 585 images for training and validation. Each image in MSCOCO has 5
human annotated captions. Following splits in [15], we reserved 2, 000 images for validation. A
more challenging dataset, InstaPIC-1.1M [18], is used as the second dataset, which contains 648, 761
images for training, and 5, 000 images for testing. The images and their ground-truth captions
are acquired from Instagram, where people post images with related descriptions. Each image in
InstaPIC-1.1M is paired with 1 caption. This dataset is challenging, as its captions are natural posts
with varying formats. In practice, we reserved 2, 000 images from the training set for validation.
On both datasets, non-alphabet characters except emojis are removed, and alphabet characters are
converted to lowercases. Words and emojis that appeared less than 5 times are replaced with UNK.
And all captions are truncated to have at most 18 words and emojis. As a result, we obtained a
vocabulary of size 9, 567 on MSCOCO, and a vocabulary of size 22, 886 on InstaPIC-1.1M.
5.2
Settings
To study the generalization ability of proposed CL method, we tested it on two different image
captioning models, namely Neuraltalk2 [8] and AdaptiveAttention [15]. Both models are based
on encoder-and-decoder [23], where no attention mechanism is used in the former, and an adaptive
attention component is used in the latter.
For both models, we have pretrained them by MLE, and use the pretrain checkpoints as initializations.
In all experiments except for the experiment on model choices, we choose the same model and use
the same initialization for target model and reference model. In all our experiments, we fixed the
learning rate to be 1e-6 for all components, and used Adam optimizer. Seven evaluation metrics have
been selected to compare the performances of different models, including Bleu-1,2,3,4 [17], Meteor
[11], Rouge [12] and Cider [22]. All experiments for ablation studies are conducted on the validation
set of MSCOCO.
6
Two people on a tennis court
playing tennis
Two tennis players shaking
AA + CL
hands on a tennis court
A fighter jet flying through a
blue sky
A fighter jet flying over a lush
green field
A row of boats on a river near
a river
A row of boats docked in a
river
A bathroom with a toilet and a
sink
A bathroom with a red toilet
and red walls
Three clocks are mounted to
the side of a building
Three three clocks with three
AA + CL
different time zones
Two people on a yellow
yellow and yellow motorcycle
Two people riding a yellow
motorcycle in a forest
A baseball player pitching a
ball on top of a field
A baseball game in progress
with pitcher throwing the ball
A bunch of lights hanging
from a ceiling
A bunch of baseballs bats
hanging from a ceiling
AA
AA
Figure 2: This figure illustrates several images with captions generated by different models, where AA represents
AdaptiveAttention [15] learned by MLE, and AA + CL represents the same model learned by CL. Compared to
AA, AA + CL generated more distinctive captions for these images.
Method
Google NIC [23]
Hard-Attention [24]
CSMN [18]
AdaptiveAttention [15]
AdaptiveAttention + CL (Ours)
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
0.055
0.106
0.079
0.065
0.072
0.019
0.015
0.032
0.026
0.028
0.007
0.000
0.015
0.011
0.013
0.003
0.000
0.008
0.005
0.006
0.038
0.026
0.037
0.029
0.032
0.081
0.140
0.120
0.093
0.101
0.004
0.049
0.133
0.126
0.144
Table 3: This table lists results of different models on the test split of InstaPIC-1.1M [18], where CL improves
the base model (AdaptiveAttention [15]) by significant margins, achieving the best result on Cider.
5.3
Results
Overall Results We compared our best model (AdaptiveAttention [15] learned by CL) with stateof-the-art models on two datasets. On MSCOCO, we submitted the results to the online COCO
testing server. The results along with other published results are listed in Table 2. Compared to
MLE-learned AdaptiveAttention, CL improves the performace of it by significant margins across
all metrics. While most of state-of-the-art results are achieved by ensembling multiple models, our
improved AdaptiveAttention gains competitive results as a single model. Specifically, on Cider, CL
improves AdaptiveAttention from 1.003 to 1.029, which is the best single-model result on C40 among
all published ones. In terms of Cider, if we use MLE, we need to combine 5 models to get 4.5% boost
on C40 for AdaptiveAttention. Using CL, we improve the performance by 2.5% with just a single
model. On InstaPIC-1.1M, CL improves the performance of AdaptiveAttention by 14% in terms of
Cider, which is the state-of-the-art. Some qualitative results are shown in Figure 2. It?s worth noting
that the proposed learning method can be used with stronger base models to obtain better results
without any modification.
Compare Learning Methods Using AdaptiveAttention learned by MLE as base model and initialization, we compared our CL with similar learning methods, including CL(P) and CL(N) that
Method
AdaptiveAttention [15] (Base)
Base + IL [21]
Base + GAN [1]
Base + CL(P)
Base + CL(N)
Base + CL(Full)
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
0.733
0.706
0.629
0.735
0.539
0.755
0.572
0.544
0.437
0.573
0.411
0.598
0.433
0.408
0.290
0.437
0.299
0.460
0.327
0.307
0.190
0.334
0.212
0.353
0.260
0.253
0.212
0.262
0.246
0.271
0.540
0.530
0.458
0.545
0.479
0.559
1.042
1.004
0.700
1.059
0.603
1.142
Table 4: This table lists results of a model learned by different methods. The best result is obtained by the one
learned with full CL, containing both the positive constraint and negative constraint.
7
Target Model
Reference Model
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
NT
NT
NT
NT
AA
0.697
0.708
0.716
0.525
0.536
0.547
0.389
0.399
0.411
0.291
0.300
0.311
0.238
0.242
0.249
0.516
0.524
0.533
0.882
0.905
0.956
AA
AA
AA
0.733
0.755
0.572
0.598
0.433
0.460
0.327
0.353
0.260
0.271
0.540
0.559
1.042
1.142
Table 5: This table lists results of different model choices on MSCOCO. In this table, NT represents Neuraltalk2
[8], and AA represents AdaptiveAttention [15]. "-" indicates the target model is learned using MLE.
Run
B-1
B-2
B-3
B-4
METEOR
ROUGE_L
CIDEr
0
1
2
0.733
0.755
0.756
0.572
0.598
0.598
0.433
0.460
0.460
0.327
0.353
0.353
0.260
0.271
0.272
0.540
0.559
0.559
1.042
1.142
1.142
Table 6: This table lists results of periodical replacement of the reference in CL. In run 0, the model is learned
by MLE, which are used as both the target and the reference in run 1. In run 2, the reference is replaced with the
best target in run 1.
respectively contains only the positive constraint and the negative constraint in CL. We also compared
with IL [21], and GAN [1]. The results on MSCOCO are listed in Table 4, where (1) among IL, CL
and GAN, CL improves performance of the base model, while both IL and GAN decrease the results.
This indicates the trade-off between learning distinctiveness and maintaining overall performance is
not well settled in IL and GAN. (2) comparing models learned by CL(P), CL(N) and CL, we found
using the positive constraint or the negative constraint alone is not sufficient, as only one source of
guidance is provided. While CL(P) gives the base model lower improvement than full CL, CL(N)
downgrades the base model, indicating overfits on distinctiveness. Combining CL(P) and CL(N),
CL is able to encourage distinctiveness while also emphasizing on overall performance, resulting in
largest improvements on all metrics.
Compare Model Choices To study the generalization ability of CL, AdaptiveAttention and Neuraltalk2 are respectively chosen as both the target and the reference in CL. In addition, AdaptiveAttention learned by MLE, as a better model, is chosen to be the reference, for Neuraltalk2. The
results are listed in Table 5, where compared to models learned by MLE, both AdaptiveAttention
and Neuraltalk2 are improved after learning using CL. For example, on Cider, AdaptiveAttention
improves from 1.042 to 1.142, and Neuraltalk2 improves from 0.882 to 0.905. Moreover, by using
a stronger model, AdaptiveAttention, as the reference, Neuraltalk2 improves further from 0.905 to
0.956, which indicates stronger references empirically provide tighter bounds on both the positive
constraint and the negative constraint.
Reference Replacement As discussed in sec 4.3, one can periodically replace the reference with
latest best target model, to further improve the performance. In our study, using AdaptiveAttention
learned by MLE as a start, each run we fix the reference model util the target saturates its performance
on the validation set, then we replace the reference with latest best target model and rerun the learning.
As listed in Table 6, in second run, the relative improvements of the target model is incremental,
compared to its improvement in the first run. Therefore, when learning a model using CL, with a
sufficiently strong reference, the improvement is usually saturated in the first run, and there is no
need, in terms of overall performance, to replace the reference multiple times.
6
Conclusion
In this paper, we propose Contrastive Learning, a new learning method for image captioning. By
employing a state-of-the-art model as a reference, the proposed method is able to maintain the
optimality of the target model, while encouraging it to learn from distinctiveness, which is an
important property of high quality captions. On two challenging datasets, namely MSCOCO and
InstaPIC-1.1M, the proposed method improves the target model by significant margins, and gains
state-of-the-art results across multiple metrics. On comparative studies, the proposed method extends
well to models with different structures, which clearly shows its generalization ability.
8
Acknowledgment This work is partially supported by the Big Data Collaboration Research grant
from SenseTime Group (CUHK Agreement No.TS1610626), the General Research Fund (GRF) of
Hong Kong (No.14236516) and the Early Career Scheme (ECS) of Hong Kong (No.24204215).
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10
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6,292 | 6,692 | Safe Model-based Reinforcement Learning with
Stability Guarantees
Felix Berkenkamp
Department of Computer Science
ETH Zurich
[email protected]
Matteo Turchetta
Department of Computer Science,
ETH Zurich
[email protected]
Angela P. Schoellig
Institute for Aerospace Studies
University of Toronto
[email protected]
Andreas Krause
Department of Computer Science
ETH Zurich
[email protected]
Abstract
Reinforcement learning is a powerful paradigm for learning optimal policies from
experimental data. However, to find optimal policies, most reinforcement learning
algorithms explore all possible actions, which may be harmful for real-world systems. As a consequence, learning algorithms are rarely applied on safety-critical
systems in the real world. In this paper, we present a learning algorithm that
explicitly considers safety, defined in terms of stability guarantees. Specifically,
we extend control-theoretic results on Lyapunov stability verification and show
how to use statistical models of the dynamics to obtain high-performance control
policies with provable stability certificates. Moreover, under additional regularity
assumptions in terms of a Gaussian process prior, we prove that one can effectively
and safely collect data in order to learn about the dynamics and thus both improve
control performance and expand the safe region of the state space. In our experiments, we show how the resulting algorithm can safely optimize a neural network
policy on a simulated inverted pendulum, without the pendulum ever falling down.
1
Introduction
While reinforcement learning (RL, [1]) algorithms have achieved impressive results in games, for
example on the Atari platform [2], they are rarely applied to real-world physical systems (e.g., robots)
outside of academia. The main reason is that RL algorithms provide optimal policies only in the
long-term, so that intermediate policies may be unsafe, break the system, or harm their environment.
This is especially true in safety-critical systems that can affect human lives. Despite this, safety in RL
has remained largely an open problem [3].
Consider, for example, a self-driving car. While it is desirable for the algorithm that drives the
car to improve over time (e.g., by adapting to driver preferences and changing environments), any
policy applied to the system has to guarantee safe driving. Thus, it is not possible to learn about the
system through random exploratory actions, which almost certainly lead to a crash. In order to avoid
this problem, the learning algorithm needs to consider its ability to safely recover from exploratory
actions. In particular, we want the car to be able to recover to a safe state, for example, driving at a
reasonable speed in the middle of the lane. This ability to recover is known as asymptotic stability
in control theory [4]. Specifically, we care about the region of attraction of the closed-loop system
under a policy. This is a subset of the state space that is forward invariant so that any state trajectory
that starts within this set stays within it for all times and converges to a goal state eventually.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this paper, we present a RL algorithm for continuous state-action spaces that provides these kind
of high-probability safety guarantees for policies. In particular, we show how, starting from an initial,
safe policy we can expand our estimate of the region of attraction by collecting data inside the safe
region and adapt the policy to both increase the region of attraction and improve control performance.
Related work Safety is an active research topic in RL and different definitions of safety exist [5, 6].
Discrete Markov decision processes (MDPs) are one class of tractable models that have been analyzed.
In risk-sensitive RL, one specifies risk-aversion in the reward [7]. For example, [8] define risk as
the probability of driving the agent to a set of known, undesirable states. Similarly, robust MDPs
maximize rewards when transition probabilities are uncertain [9, 10]. Both [11] and [12] introduce
algorithms to safely explore MDPs so that the agent never gets stuck without safe actions. All these
methods require an accurate probabilistic model of the system.
In continuous state-action spaces, model-free policy search algorithms have been successful. These
update policies without a system model by repeatedly executing the same task [13]. In this setting, [14] introduces safety guarantees in terms of constraint satisfaction that hold in expectation.
High-probability worst-case safety guarantees are available for methods based on Bayesian optimization [15] together with Gaussian process models (GP, [16]) of the cost function. The algorithms
in [17] and [18] provide high-probability safety guarantees for any parameter that is evaluated on
the real system. These methods are used in [19] to safely optimize a parametric control policy on a
quadrotor. However, resulting policies are task-specific and require the system to be reset.
In the model-based RL setting, research has focused on safety in terms of state constraints. In
[20, 21], a priori known, safe global backup policies are used, while [22] learns to switch between
several safe policies. However, it is not clear how one may find these policies in the first place.
Other approaches use model predictive control with constraints, a model-based technique where the
control actions are optimized online. For example, [23] models uncertain environmental constraints,
while [24] uses approximate uncertainty propagation of GP dynamics along trajectories. In this
setting, robust feasability and constraint satisfaction can be guaranteed for a learned model with
bounded errors using robust model predictive control [25]. The method in [26] uses reachability
analysis to construct safe regions in the state space. The theoretical guarantees depend on the solution
to a partial differential equation, which is approximated.
Theoretical guarantees for the stability exist for the more tractable stability analysis and verification
under a fixed control policy. In control, stability of a known system can be verified using a Lyapunov
function [27]. A similar approach is used by [28] for deterministic, but unknown dynamics that are
modeled as a GP, which allows for provably safe learning of regions of attraction for fixed policies.
Similar results are shown in [29] for stochastic systems that are modeled as a GP. They use Bayesian
quadrature to compute provably accurate estimates of the region of attraction. These approaches do
not update the policy.
Our contributions We introduce a novel algorithm that can safely optimize policies in continuous
state-action spaces while providing high-probability safety guarantees in terms of stability. Moreover,
we show that it is possible to exploit the regularity properties of the system in order to safely learn
about the dynamics and thus improve the policy and increase the estimated safe region of attraction
without ever leaving it. Specifically, starting from a policy that is known to stabilize the system
locally, we gather data at informative, safe points and improve the policy safely based on the improved
model of the system and prove that any exploration algorithm that gathers data at these points reaches
a natural notion of full exploration. We show how the theoretical results transfer to a practical
algorithm with safety guarantees and apply it to a simulated inverted pendulum stabilization task.
2
Background and Assumptions
We consider a deterministic, discrete-time dynamic system
xt+1 = f (xt , ut ) = h(xt , ut ) + g(xt , ut ),
(1)
with states x ? X ? Rq and control actions u ? U ? Rp and a discrete time index t ? N. The true
dynamics f : X ? U ? X consist of two parts: h(xt , ut ) is a known, prior model that can be
obtained from first principles, while g(xt , ut ) represents a priori unknown model errors. While the
model errors are unknown, we can obtain noisy measurements of f (x, u) by driving the system to
the state x and taking action u. We want this system to behave in a certain way, e.g., the car driving
2
on the road. To this end, we need to specify a control policy ? : X ? U that, given the current state,
determines the appropriate control action that drives the system to some goal state, which we set as
the origin without loss of generality [4]. We encode the performance requirements of how to drive
the system to the origin through a positive cost r(x, u) that is associated with states and actions and
has r(0, 0) = 0. The policy aims to minimize the cumulative, discounted costs for each starting state.
The goal is to safely learn about the dynamics from measurements and adapt the policy for performance, without encountering system failures. Specifically, we define the safety constraint on the
state divergence that occurs when leaving the region of attraction. This means that adapting the
policy is not allowed to decrease the region of attraction and exploratory actions to learn about the
dynamics f (?) are not allowed to drive the system outside the region of attraction. The region of
attraction is not known a priori, but is implicitly defined through the system dynamics and the choice
of policy. Thus, the policy not only defines performance as in typical RL, but also determines safety
and where we can obtain measurements.
Model assumptions In general, this kind of safe learning is impossible without further assumptions.
For example, in a discontinuous system even a slight change in the control policy can lead to drastically
different behavior. Moreover, to expand the safe set we need to generalize learned knowledge about
the dynamics to (potentially unsafe) states that we have not visited. To this end, we restrict ourselves
to the general and practically relevant class of models that are Lipschitz continuous. This is a typical
assumption in the control community [4]. Additionally, to ensure that the closed-loop system remains
Lipschitz continuous when the control policy is applied, we restrict policies to the rich class of
L? -Lipschitz continuous functions ?L , which also contains certain types of neural networks [30].
Assumption 1 (continuity). The dynamics h(?) and g(?) in (1) are Lh - and Lg Lipschitz continuous
with respect to the 1-norm. The considered control policies ? lie in a set ?L of functions that
are L? -Lipschitz continuous with respect to the 1-norm.
To enable safe learning, we require a reliable statistical model. While we commit to GPs for the
exploration analysis, for safety any suitable, well-calibrated model is applicable.
Assumption 2 (well-calibrated model). Let ?n (?) and ?n (?) denote the posterior mean and covariance matrix functions of the statistical model of the dynamics (1) conditioned on n noisy measurements.
1/2
With ?n (?) = trace(?n (?)), there exists a ?n > 0 such that with probability at least (1 ? ?) it holds
for all n ? 0, x ? X , and u ? U that kf (x, u) ? ?n (x, u)k1 ? ?n ?n (x, u).
This assumption ensures that we can build confidence intervals on the dynamics that, when scaled by
an appropriate constant ?n , cover the true function with high probability. We introduce a specific
statistical model that fulfills both assumptions under certain regularity assumptions in Sec. 3.
Lyapunov function To satisfy the specified safety constraints for safe learning, we require a tool
to determine whether individual states and actions are safe. In control theory, this safety is defined
through the region of attraction, which can be computed for a fixed policy using Lyapunov functions [4]. Lyapunov functions are continuously differentiable functions v : X ? R?0 with v(0) = 0
and v(x) > 0 for all x ? X \ {0}. The key idea behind using Lyapunov functions to show stability
of the system (1) is similar to that of gradient descent on strictly quasiconvex functions: if one can
show that, given a policy ?, applying the dynamics f on the state maps it to strictly smaller values
on the Lyapunov function (?going downhill?), then the state eventually converges to the equilibrium
point at the origin (minimum). In particular, the assumptions in Theorem 1 below imply that v is
strictly quasiconvex within the region of attraction if the dynamics are Lipschitz continuous. As a
result, the one step decrease property for all states within a level set guarantees eventual convergence
to the origin.
Theorem 1 ([4]). Let v be a Lyapunov function, f Lipschitz continuous dynamics, and ? a policy. If
v(f (x, ?(x))) < v(x) for all x within the level set V(c) = {x ? X \ {0} | v(x) ? c}, c > 0, then
V(c) is a region of attraction, so that x0 ? V(c) implies xt ? V(c) for all t > 0 and limt?? xt = 0.
It is convenient to characterize the region of attraction through a level set of the Lyapunov function,
since it replaces the challenging test for convergence with a one-step decrease condition on the
Lyapunov function. For the theoretical analysis in this paper, we assume that a Lyapunov function is
given to determine the region of attraction. For ease of notation, we also assume ?v(x)/?x 6= 0 for
all x ? X \ 0, which ensures that level sets V(c) are connected if c > 0. Since Lyapunov functions
are continuously differentiable, they are Lv -Lipschitz continuous over the compact set X .
3
In general, it is not easy to find suitable Lyapunov functions. However, for physical models, like the
prior model h in (1), the energy of the system (e.g., kinetic and potential for mechanical systems) is a
good candidate Lyapunov function. Moreover, it has recently been shown that it is possible to compute
suitable Lyapunov functions [31, 32]. In our experiments, we exploit the fact that value functions in
RL are Lyapunov functions if the costs are strictly positive away from the origin. This follows directly
from the definition of the value function, where v(x) = r(x, ?(x)) + v(f (x, ?(x)) ? v(f (x, ?(x))).
Thus, we can obtain Lyapunov candidates as a by-product of approximate dynamic programming.
Initial safe policy Lastly, we need to ensure that there exists a safe starting point for the learning
process. Thus, we assume that we have an initial policy ?0 that renders the origin of the system in (1)
asymptotically stable within some small set of states S0x . For example, this policy may be designed
using the prior model h in (1), since most models are locally accurate but deteriorate in quality as
state magnitude increases. This policy is explicitly not safe to use throughout the state space X \ S0x .
3
Theory
In this section, we use these assumptions for safe reinforcement learning. We start by computing the
region of attraction for a fixed policy under the statistical model. Next, we optimize the policy in order
to expand the region of attraction. Lastly, we show that it is possible to safely learn about the dynamics
and, under additional assumptions about the model and the system?s reachability properties, that this
approach expands the estimated region of attraction safely. We consider an idealized algorithm that is
amenable to analysis, which we convert to a practical variant in Sec. 4. See Fig. 1 for an illustrative
run of the algorithm and examples of the sets defined below.
Region of attraction We start by computing the region of attraction for a fixed policy. This is an
extension of the method in [28] to discrete-time systems. We want to use the Lyapunov decrease condition in Theorem 1 to guarantee safety for the statistical model of the dynamics. However, the posterior
uncertainty in the statistical model of the dynamics means that one step predictions about v(f (?)) are
uncertain too. We account for this by constructing high-probability confidence intervals on v(f (x, u)):
Qn (x, u) := [v(?n?1 (x, u)) ? Lv ?n ?n?1 (x, u)]. From Assumption 2 together with the Lipschitz
property of v, we know that v(f (x, u)) is contained in Qn (x, u) with probability at least (1 ? ?). For
our exploration analysis, we need to ensure that safe state-actions cannot become unsafe; that is, an
initial set of safe set S0 remains safe (defined later). To this end, we intersect the confidence intervals:
Cn (x, u) := Cn?1 ? Qn (x, u), where the set C is initialized to C0 (x, u) = (??, v(x) ? L?v ? )
when (x, u) ? S0 and C0 (x, u) = R otherwise. Note that v(f (x, u)) is contained in Cn (x, u) with
the same (1 ? ?) probability as in Assumption 2. The upper and lower bounds on v(f (?)) are defined
as un (x, u) := max Cn (x, u) and ln (x, u) := min Cn (x, u).
Given these high-probability confidence intervals, the system is stable according to Theorem 1 if
v(f (x, u)) ? un (x) < v(x) for all x ? V(c). However, it is intractable to verify this condition
directly on the continuous domain without additional, restrictive assumptions about the model.
Instead, we consider a discretization of the state space X? ? X into cells, so that kx ? [x]? k1 ? ?
holds for all x ? X . Here, [x]? denotes the point in X? with the smallest l1 distance to x. Given this
discretization, we bound the decrease variation on the Lyapunov function for states in X? and use the
Lipschitz continuity to generalize to the continuous state space X .
Theorem 2. Under Assumptions 1 and 2 with L?v := Lv Lf (L? + 1) + Lv , let X? be a discretization of X such that kx ? [x]? k1 ? ? for all x ? X . If, for all x ? V(c) ? X? with c > 0, u = ?(x),
and for some n ? 0 it holds that un (x, u) < v(x) ? L?v ?, then v(f (x, ?(x))) < v(x) holds for all
x ? V(c) with probability at least (1 ? ?) and V(c) is a region of attraction for (1) under policy ?.
The proof is given in Appendix A.1. Theorem 2 states that, given confidence intervals on the statistical
model of the dynamics, it is sufficient to check the stricter decrease condition in Theorem 2 on the
discretized domain X? to guarantee the requirements for the region of attraction in the continuous
domain in Theorem 1. The bound in Theorem 2 becomes tight as the discretization constant ?
and |v(f (?)) ? un (?)| go to zero. Thus, the discretization constant trades off computation costs for
accuracy, while un approaches v(f
? (?)) as we obtain more measurement data and the posterior model
uncertainty about the dynamics, ?n ?n decreases. The confidence intervals on v(f (x, ?(x)) ? v(x)
and the corresponding estimated region of attraction (red line) can be seen in the bottom half of Fig. 1.
Policy optimization So far, we have focused on estimating the region of attraction for a fixed policy.
Safety is a property of states under a fixed policy. This means that the policy directly determines
4
?30
?15
policy ?0
? (x, u)
S30
V(c15 )
u0
?L?v ?
l0
state x
(a) Initial safe set (in red).
state x
(b) Exploration: 15 data points.
state x
?v(x, ?(x))
v(x)
action u
D30
(c) Final policy after 30 evaluations.
Figure 1: Example application of Algorithm 1. Due to input constraints, the system becomes unstable
for large states. We start from an initial, local policy ?0 that has a small, safe region of attraction (red
lines) in Fig. 1(a). The algorithm selects safe, informative state-action pairs within Sn (top, white
shaded), which can be evaluated without leaving the region of attraction V(cn ) (red lines) of the
current policy ?n . As we gather more data (blue crosses), the uncertainty in the model decreases
(top, background) and we use (3) to update the policy so that it lies within Dn (top, red shaded) and
fulfills the Lyapunov decrease condition. The algorithm converges to the largest safe set in Fig. 1(c).
It improves the policy without evaluating unsafe state-action pairs and thereby without system failure.
which states are safe. Specifically, to form a region of attraction all states in the discretizaton X?
within a level set of the Lyapunov function need to fulfill the decrease condition in Theorem 2 that
depends on the policy choice. The set of all state-action pairs that fulfill this decrease condition is
given by
Dn = (x, u) ? X? ? U | un (x, u) ? v(x) < ?L?v ? ,
(2)
see Fig. 1(c) (top, red shaded). In order to estimate the region of attraction based on this set, we
need to commit to a policy. Specifically, we want to pick the policy that leads to the largest possible
region of attraction according to Theorem 2. This requires that for each discrete state in X? the
corresponding state-action pair under the policy must be in the set Dn . Thus, we optimize the policy
according to
?n , cn =
argmax c,
???L ,c?R>0
such that for all x ? V(c) ? X? : (x, ?(x)) ? Dn .
(3)
The region of attraction that corresponds to the optimized policy ?n according to (3) is given
by V(cn ), see Fig. 1(b). It is the largest level set of the Lyapunov function for which all state-action
pairs (x, ?n (x)) that correspond to discrete states within V(cn ) ? X? are contained in Dn . This
means that these state-action pairs fulfill the requirements of Theorem 2 and V(cn ) is a region of
attraction of the true system under policy ?n . The following theorem is thus a direct consequence
of Theorem 2 and (3).
Theorem 3. Let R?n be the true region of attraction of (1) under the policy ?n . For any ? ? (0, 1),
we have with probability at least (1 ? ?) that V(cn ) ? R?n for all n > 0.
Thus, when we optimize the policy subject to the constraint in (3) the estimated region of attraction is
always an inner approximation of the true region of attraction. However, solving the optimization
problem in (3) is intractable in general. We approximate the policy update step in Sec. 4.
Collecting measurements Given these stability guarantees, it is natural to ask how one might obtain
data points in order to improve the model of g(?) and thus efficiently increase the region of attraction.
This question is difficult to answer in general, since it depends on the property of the statistical model.
In particular, for general statistical models it is often not clear whether the confidence intervals
contract sufficiently quickly. In the following, we make additional assumptions about the model and
reachability within V(cn ) in order to provide exploration guarantees. These assumptions allow us to
highlight fundamental requirements for safe data acquisition and that safe exploration is possible.
5
We assume that the unknown model errors g(?) have bounded norm in a reproducing kernel Hilbert
space (RKHS, [33]) corresponding to a differentiable
kernel k, kg(?)kk ? Bg . These are a class of
P?
well-behaved functions of the form g(z) = i=0 ?i k(zi , z) defined through representer points zi
and weights ?i that decay sufficiently fast with i. This assumption ensures
p that g satisfies the
Lipschitz property in Assumption 1, see [28]. Moreover, with ?n = Bg + 4? ?n + 1 + ln(1/?) we
can use GP models for the dynamics that fulfill Assumption 2 if the state if fully observable and the
measurement noise is ?-sub-Gaussian (e.g., bounded in [??, ?]), see [34]. Here ?n is the information
capacity. It corresponds to the amount of mutual information that can be obtained about g from nq
measurements, a measure of the size of the function class encoded by the model. The information
capacity has a sublinear dependence on n for common kernels and upper bounds can be computed
efficiently [35]. More details about this model are given in Appendix A.2.
In order to quantify the exploration properties of our algorithm, we consider a discrete action
space U? ? U. We define exploration as the number of state-action pairs in X? ? U? that we can
safely learn about without leaving the true region of attraction. Note that despite this discretization,
the policy takes values on the continuous domain. Moreover, instead of using the confidence intervals
directly as in (3), we consider an algorithm that uses the Lipschitz constants to slowly expand the safe
set. We use this in our analysis to quantify the ability to generalize beyond the current safe set. In
practice, nearby states are sufficiently correlated under the model to enable generalization using (2).
Suppose we are given a set S0 of state-action pairs about which we can learn safely. Specifically, this
means that we have a policy such that, for any state-action pair (x, u) in S0 , if we apply action u in
state x and then apply actions according to the policy, the state converges to the origin. Such a set
can be constructed using the initial policy ?0 from Sec. 2 as S0 = {(x, ?0 (x)) | x ? S0x }. Starting
from this set, we want to update the policy to expand the region of attraction according to Theorem 2.
To this end, we use the confidence intervals on v(f (?)) for states inside S0 to determine state-action
pairs that fulfill the decrease condition. We thus redefine Dn for the exploration analysis to
[
0
Dn =
z ? X? ? U? | un (x, u) ? v(x) + L?v kz0 ? (x, u)k1 < ?L?v ? . (4)
(x,u)?Sn?1
This formulation is equivalent to (2), except that it uses the Lipschitz constant to generalize safety.
Given Dn , we can again find a region of attraction V(cn ) by committing to a policy according to (3).
In order to expand this region of attraction effectively we need to decrease the posterior model
uncertainty about the dynamics of the GP by collecting measurements. However, to ensure safety
as outlined in Sec. 2, we are not only restricted to states within V(cn ), but also need to ensure that
the state after taking an action is safe; that is, the dynamics map the state back into the region of
attraction V(cn ). We again use the Lipschitz constant in order to determine this set,
[
Sn =
z0 ? V(cn ) ? X? ? U? | un (z) + Lv Lf kz ? z0 k1 ? cn }.
(5)
z?Sn?1
The set Sn contains state-action pairs that we can safely evaluate under the current policy ?n without
leaving the region of attraction, see Fig. 1 (top, white shaded).
What remains is to define a strategy for collecting data points within Sn to effectively decrease model
uncertainty. We specifically focus on the high-level requirements for any exploration scheme without
committing to a specific method. In practice, any (model-based) exploration strategy that aims to
decrease model uncertainty by driving the system to specific states may be used. Safety can be
ensured by picking actions according to ?n whenever the exploration strategy reaches the boundary
of the safe region V(cn ); that is, when un (x, u) > cn . This way, we can use ?n as a backup policy
for exploration.
The high-level goal of the exploration strategy is to shrink the confidence intervals at state-action
pairs Sn in order to expand the safe region. Specifically, the exploration strategy should aim to visit
state-action pairs in Sn at which we are the most uncertain about the dynamics; that is, where the
confidence interval is the largest:
(xn , un ) = argmax un (x, u) ? ln (x, u).
(6)
(x,u)?Sn
As we keep collecting data points according to (6), we decrease the uncertainty about the dynamics
for different actions throughout the region of attraction and adapt the policy, until eventually we
6
Algorithm 1 S AFE LYAPUNOV L EARNING
1: Input: Initial safe policy ?0 , dynamics model GP(?(z), k(z, z0 ))
2: for all n = 1, . . . do
3:
Compute policy ?n via SGD on (7)
4:
cn = argmaxc c, such that ?x ? V(cn ) ? X? : un (x, ?n (x)) ? v(x) < ?L?v ?
5:
Sn = {(x, u) ? V(cn ) ? U? | un (x, u) ? cn }
6:
Select (xn , un ) within Sn using (6) and drive system there with backup policy ?n
7:
Update GP with measurements f (xn , un ) + n
have gathered enough information in order to expand it. While (6) implicitly assumes that any
state within V(cn ) can be reached by the exploration policy, it achieves the high-level goal of any
exploration algorithm that aims to reduce model uncertainty. In practice, any safe exploration scheme
is limited by unreachable parts of the state space.
We compare the active learning scheme in (6) to an oracle baseline that starts from the same initial
safe set S0 and knows v(f (x, u)) up to accuracy within the safe set. The oracle also uses knowledge
about the Lipschitz constants and the optimal policy in ?L at each iteration. We denote the set that this
baseline manages to determine as safe with R (S0 ) and provide a detailed definition in Appendix A.3.
Theorem 4. Assume ?-sub-Gaussian measurement noise and that the model error g(?)
in (1) has RKHSpnorm smaller than Bg .
Under the assumptions of Theorem 2,
with ?n = Bg + 4? ?n + 1 + ln(1/?), and with measurements collected according to (6), let
?
0 )|+1)
n? be the smallest positive integer so that ? 2 n? ? ? Cq(|R(S
where C = 8/ log(1 + ? ?2 ).
L2v 2
n? n
Let R? be the true region of attraction of (1) under a policy ?. For any > 0, and ? ? (0, 1), the
following holds jointly with probability at least (1 ? ?) for all n > 0:
(i) V(cn ) ? R?n
(ii) f (x, u) ? R?n ?(x, u) ? Sn .
(iii) R (S0 ) ? Sn ? R0 (S0 ).
Theorem 4 states that, when selecting data points according to (6), the estimated region of attraction V(cn ) is (i) contained in the true region of attraction under the current policy and (ii) selected
data points do not cause the system to leave the region of attraction. This means that any exploration
method that considers the safety constraint (5) is able to safely learn about the system without leaving
the region of attraction. The last part of Theorem 4, (iii), states that after a finite number of data
points n? we achieve at least the exploration performance of the oracle baseline, while we do not
classify unsafe state-action pairs as safe. This means that the algorithm explores the largest region
of attraction possible for a given Lyapunov function with residual uncertaint about v(f (?)) smaller
than . Details of the comparison baseline are given in the appendix. In practice, this means that any
exploration method that manages to reduce the maximal uncertainty about the dynamics within Sn is
able to expand the region of attraction.
An example run of repeatedly evaluating (6) for a one-dimensional state-space is shown in Fig. 1. It
can be seen that, by only selecting data points within the current estimate of the region of attraction,
the algorithm can efficiently optimize the policy and expand the safe region over time.
4
Practical Implementation and Experiments
In the previous section, we have given strong theoretical results on safety and exploration for an
idealized algorithm that can solve (3). In this section, we provide a practical variant of the theoretical
algorithm in the previous section. In particular, while we retain safety guarantees, we sacrifice
exploration guarantees to obtain a more practical algorithm. This is summarized in Algorithm 1.
The policy optimization problem in (3) is intractable to solve and only considers safety, rather
than a performance metric. We propose to use an approximate policy update that that maximizes
approximate performance while providing stability guarantees. It proceeds by optimizing the policy
first and then computes the region of attraction V(cn ) for the new, fixed policy. This does not
impact safety, since data is still only collected inside the region of attraction. Moreover, should the
optimization fail and the region of attraction decrease, one can always revert to the previous policy,
which is guaranteed to be safe.
7
V(c50 )
angle [rad]
angular velocity [rad/s]
0.30
5
V(c0 )
0
unsafe region
?5
safely optimized policy
initial policy
0.25
0.20
0.15
?0
0.10
?50
0.05
0.00
?1.0
?0.5
0.0
0.5
1.0
0.0
angle [rad]
0.5
1.0
1.5
time [s]
(a) Estimated safe set.
(b) State trajectory (lower is better).
Figure 2: Optimization results for an inverted pendulum. Fig. 2(a) shows the initial safe set (yellow)
under the policy ?0 , while the green region represents the estimated region of attraction under the
optimized neural network policy. It is contained within the true region of attraction (white). Fig. 2(b)
shows the improved performance of the safely learned policy over the policy for the prior model.
In our experiments, we use approximate dynamic programming [36] to capture the performance of the
policy. Given a policy ?? with parameters ?, we compute an estimate of the cost-to-go J?? (?) for the
mean dynamics ?n based on the cost r(x, u) ? 0. At each state, J?? (x) is the sum of ?-discounted
rewards encountered when following the policy ?? . The goal is to adapt the parameters of the policy
for minimum cost as measured by J?? , while ensuring that the safety constraint on the worst-case
decrease on the Lyapunov function in Theorem 2 is not violated. A Lagrangian formulation to this
constrained optimization problem is
X
?n = argmin
r(x, ?? (x)) + ?J?? (?n?1 (x, ?? (x)) + ? un (x, ?? (x)) ? v(x) + L?v ? , (7)
?? ??L
x?X?
where the first term measures long-term cost to go and ? ? 0 is a Lagrange multiplier for the safety
constraint from Theorem 2. In our experiments, we use the value function as a Lyapunov function
candidate, v = J with r(?, ?) ? 0, and set ? = 1. In this case, (7) corresponds to an high-probability
upper bound on the cost-to-go given the uncertainty in the dynamics. This is similar to worst-case
performance formulations found in robust MDPs [9, 10], which consider worst-case value functions
given parametric uncertainty in MDP transition model. Moreover, since L?v depends on the Lipschitz
constant of the policy, this simultaneously serves as a regularizer on the parameters ?.
To verify safety, we use the GP confidence intervals ln and un directly, as in (2). We also use
confidence to compute Sn for the active learning scheme, see Algorithm 1, Line 5. In practice, we
do not need to compute the entire set Sn to solve (3), but can use a global optimization method or
even a random sampling scheme within V(cn ) to find suitable state-actions. Moreover, measurements
for actions that are far away from the current policy are unlikely to expand V(cn ), see Fig. 1(c). As
we optimize (7) via gradient descent, the policy changes only locally. Thus, we can achieve better
data-efficiency by restricting the exploratory actions u with (x, u) ? Sn to be close to ?n , u ?
[?n (x) ? u
?, ?n (x) + u
?] for some constant u
?.
Computing the region of attraction by verifying the stability condition on a discretized domain suffers
from the curse of dimensionality. However, it is not necessary to update policies in real time. In
particular, since any policy that is returned by the algorithm is provably safe within some level set,
any of these policies can be used safely for an arbitrary number of time steps. To scale this method to
higher-dimensional system, one would have to consider an adaptive discretization for the verification
as in [27].
Experiments A Python implementation of Algorithm 1 and the experiments based on TensorFlow [37] and GPflow [38] is available at https://github.com/befelix/safe_learning.
We verify our approach on an inverted pendulum benchmark problem. The true, continuous-time
dynamics are given by ml2 ?? = gml sin(?) ? ??? + u, where ? is the angle, m the mass, g the
gravitational constant, and u the torque applied to the pendulum. The control torque is limited, so that
the pendulum necessarily falls down beyond a certain angle. We use a GP model for the discrete-time
dynamics, where the mean dynamics are given by a linearized and discretized model of the true
8
dynamics that considers a wrong, lower mass and neglects friction. As a result, the optimal policy for
the mean dynamics does not perform well and has a small region of attraction as it underactuates the
system. We use a combination of linear and Mat?rn kernels in order to capture the model errors that
result from parameter and integration errors.
For the policy, we use a neural network with two hidden layers and 32 neurons with ReLU activations
each. We compute a conservative estimate of the Lipschitz constant as in [30]. We use standard
approximate dynamic programming with a quadratic, normalized cost r(x, u) = xT Qx + uT Ru,
where Q and R are positive-definite, to compute the cost-to-go J?? . Specifically, we use a piecewiselinear triangulation of the state-space as to approximate J?? , see [39]. This allows us to quickly
verify the assumptions that we made about the Lyapunov function in Sec. 2 using a graph search. In
practice, one may use other function approximators. We optimize the policy via stochastic gradient
descent on (7).
The theoretical confidence intervals for the GP model are conservative. To enable more data-efficient
learning, we fix ?n = 2. This corresponds to a high-probability decrease condition per-state, rather
than jointly over the state space. Moreover, we use local Lipschitz constants of the Lyapunov function
rather than the global one. While this does not affect guarantees, it greatly speeds up exploration.
For the initial policy, we use approximate dynamic programming to compute the optimal policy for
the prior mean dynamics. This policy is unstable for large deviations from the initial state and has poor
performance, as shown in Fig. 2(b). Under this initial, suboptimal policy, the system is stable within
a small region of the state-space Fig. 2(a). Starting from this initial safe set, the algorithm proceeds to
collect safe data points and improve the policy. As the uncertainty about the dynamics decreases, the
policy improves and the estimated region of attraction increases. The region of attraction after 50 data
points is shown in Fig. 2(a). The resulting set V(cn ) is contained within the true safe region of the
optimized policy ?n . At the same time, the control performance improves drastically relative to the
initial policy, as can be seen in Fig. 2(b). Overall, the approach enables safe learning about dynamic
systems, as all data points collected during learning are safely collected under the current policy.
5
Conclusion
We have shown how classical reinforcement learning can be combined with safety constraints in terms
of stability. Specifically, we showed how to safely optimize policies and give stability certificates
based on statistical models of the dynamics. Moreover, we provided theoretical safety and exploration
guarantees for an algorithm that can drive the system to desired state-action pairs during learning. We
believe that our results present an important first step towards safe reinforcement learning algorithms
that are applicable to real-world problems.
Acknowledgments
This research was supported by SNSF grant 200020_159557, the Max Planck ETH Center for
Learning Systems, NSERC grant RGPIN-2014-04634, and the Ontario Early Researcher Award.
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12
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6,293 | 6,693 | Online Multiclass Boosting
Young Hun Jung
Jack Goetz
Department of Statistics
University of Michigan
Ann Arbor, MI 48109
{yhjung, jrgoetz, tewaria}@umich.edu
Ambuj Tewari
Abstract
Recent work has extended the theoretical analysis of boosting algorithms to multiclass problems and to online settings. However, the multiclass extension is in the
batch setting and the online extensions only consider binary classification. We fill
this gap in the literature by defining, and justifying, a weak learning condition for
online multiclass boosting. This condition leads to an optimal boosting algorithm
that requires the minimal number of weak learners to achieve a certain accuracy.
Additionally, we propose an adaptive algorithm which is near optimal and enjoys
an excellent performance on real data due to its adaptive property.
1
Introduction
Boosting methods are a ensemble learning methods that aggregate several (not necessarily) weak
learners to build a stronger learner. When used to aggregate reasonably strong learners, boosting has
been shown to produce results competitive with other state-of-the-art methods (e.g., Korytkowski
et al. [1], Zhang and Wang [2]). Until recently theoretical development in this area has been focused
on batch binary settings where the learner can observe the entire training set at once, and the labels
are restricted to be binary (cf. Schapire and Freund [3]). In the past few years, progress has been
made to extend the theory and algorithms to more general settings.
Dealing with multiclass classification turned out to be more subtle than initially expected. Mukherjee
and Schapire [4] unify several different proposals made earlier in the literature and provide a general
framework for multiclass boosting. They state their weak learning conditions in terms of cost matrices
that have to satisfy certain restrictions: for example, labeling with the ground truth should have less
cost than labeling with some other labels. A weak learning condition, just like the binary condition,
states that the performance of a learner, now judged using a cost matrix, should be better than a
random guessing baseline. One particular condition they call the edge-over-random condition, proves
to be sufficient for boostability. The edge-over-random condition will also figure prominently in this
paper. They also consider a necessary and sufficient condition for boostability but it turns out to be
computationally intractable to be used in practice.
A recent trend in modern machine learning is to train learners in an online setting where the instances
come sequentially and the learner has to make predictions instantly. Oza [5] initially proposed an
online boosting algorithm that has accuracy comparable with the batch version, but it took several
years to design an algorithm with theoretical justification (Chen et al. [6]). Beygelzimer et al. [7]
achieved a breakthrough by proposing an optimal algorithm in online binary settings and an adaptive
algorithm that works quite well in practice. These theories in online binary boosting have led to
several extensions. For example, Chen et al. [8] combine one vs all method with binary boosting
algorithms to tackle online multiclass problems with bandit feedback, and Hu et al. [9] build a theory
of boosting in regression setting.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
In this paper, we combine the insights and techniques of Mukherjee and Schapire [4] and Beygelzimer
et al. [7] to provide a framework for online multiclass boosting. The cost matrix framework from the
former work is adopted to propose an online weak learning condition that defines how well a learner
can perform over a random guess (Definition 1). We show this condition is naturally derived from its
batch setting counterpart. From this weak learning condition, a boosting algorithm (Algorithm 1) is
proposed which is theoretically optimal in that it requires the minimal number of learners and sample
complexity to attain a specified level of accuracy. We also develop an adaptive algorithm (Algorithm
2) which allows learners to have variable strengths. This algorithm is theoretically less efficient than
the optimal one, but the experimental results show that it is quite comparable and sometimes even
better due to its adaptive property. Both algorithms not only possess theoretical proofs of mistake
bounds, but also demonstrate superior performance over preexisting methods.
2
Preliminaries
We first describe the basic setup for online boosting. While in the batch setting, an additional weak
learner is trained at every iteration, in the online setting, the algorithm starts with a fixed count
of N weak learners and a booster which manages the weak learners. There are k possible labels
[k] := {1, ? ? ? , k} and k is known to the learners. At each iteration t = 1, ? ? ? , T , an adversary picks
a labeled example (xt , yt ) ? X ? [k], where X is some domain, and reveals xt to the booster. Once
the booster observes the unlabeled data xt , it gathers the weak learners? predictions and makes a final
prediction. Throughout this paper, index i takes values from 1 to N ; t from 1 to T ; and l from 1 to k.
We utilize the cost matrix framework, first proposed by Mukherjee and Schapire [4], to develop
multiclass boosting algorithms. This is a key ingredient in the multiclass extension as it enables
different penalization for each pair of correct label and prediction, and we further develop this
framework to suit the online setting. The booster sequentially computes cost matrices {Cit ?
Rk?k | i = 1, ? ? ? , N }, sends (xt , Cit ) to the ith weak learner W Li , and gets its prediction lti ? [k].
Here the cost matrix Cit plays a role of loss function in that W Li tries to minimize the cumulative
P
cost t Cit [yt , lti ]. As the booster wants each learner to predict the correct label, it wants to set the
diagonal entries of Cit to be minimal among its row. At this stage, the true label yt is not revealed yet,
but the previous weak learners? predictions can affect the computation of the cost matrix for the next
learner. Given a matrix C, the (i, j)th entry will be denoted by C[i, j], and ith row vector by C[i].
Once all the learners make predictions, the booster makes the final prediction y?t by majority votes.
The booster can either take simple majority votes or weighted ones. In fact for the adaptive algorithm,
we will allow weighted votes so that the booster can assign more weights on well-performing learners.
The weight for W Li at iteration t will be denoted by ?ti . After observing the booster?s final decision,
the adversary reveals the true label yt , and the booster suffers 0-1 loss 1(?
yt 6= yt ). The booster also
shares the true label to the weak learners so that they can train on this data point.
Two main issues have to be resolved to design a good boosting algorithm. First, we need to design
the booster?s strategy for producing cost matrices. Second, we need to quantify weak learner?s
PT
ability to reduce the cumulative cost t=1 Cit [yt , lti ]. The first issue will be resolved by introducing
potential functions, which will be thoroughly discussed in Section 3.1. For the second issue, we
introduce our online weak learning condition, a generalization of the weak learning assumption in
Beygelzimer et al. [7], stating that for any adaptively given sequence of cost matrices, weak learners
can produce predictions whose cumulative cost is less than that incurred by random guessing. The
online weak learning condition will be discussed in the following section. For the analysis of the
adaptive algorithm, we use empirical edges instead of the online weak learning condition.
2.1
Online weak learning condition
In this section, we propose an online weak learning condition that states the weak learners are better
than a random guess. We first define a baseline condition that is better than a random guess. Let
?[k] denote a family of distributions over [k] and ul? ? ?[k] be a uniform distribution that puts ?
1??
1??
more weight on the label l. For example, u1? = ( 1??
k + ?, k , ? ? ? , k ). For a given sequence of
examples {(xt , yt ) | t = 1, ? ? ? , T }, U? ? RT ?k consists of rows uy?t . Then we restrict the booster?s
2
choice of cost matrices to
C1eor := {C ? Rk?k | ?l, r ? [k], C[l, l] = 0, C[l, r] ? 0, and ||C[l]||1 = 1}.
Note that diagonal entries are minimal among the row, and C1eor also has a normalization constraint.
A broader choice of cost matrices is allowed if one can assign importance weights on observations,
which is possible for various learners. Even if the learner does not take the importance weight as an
input, we can achieve a similar effect by sending to the learner an instance with probability that is
proportional to its weight. Interested readers can refer Beygelzimer et al. [7, Lemma 1]. From now
on, we will assume that our weak learners can take weight wt as an input.
We are ready to present our online weak learning condition. This condition is in fact naturally derived
from the batch setting counterpart that is well studied by Mukherjee and Schapire [4]. The link is
thoroughly discussed in Appendix A. For the scaling issue, we assume the weights wt lie in [0, 1].
Definition 1. (Online multiclass weak learning condition) For parameters ?, ? ? (0, 1), and
S > 0, a pair of online learner and an adversary is said to satisfy online weak learning condition
with parameters ?, ?, and S if for any sample length T , any adaptive sequence of labeled examples,
and for any adaptively chosen series of pairs of weight and cost matrix {(wt , Ct ) ? [0, 1] ? C1eor | t =
1, ? ? ? , T }, the learner can generate predictions y?t such that with probability at least 1 ? ?,
T
X
wt Ct [yt , y?t ] ? C ? U0? + S =
t=1
1??
||w||1 + S,
k
(1)
where C ? RT ?k consists of rows of wt Ct [yt ] and A ? B0 denotes the Frobenius inner product
Tr(AB0 ). w = (w1 , ? ? ? , wT ) and the last equality holds due to the normalized condition on C1eor . ?
is called an edge, and S an excess loss.
Remark. Notice that this condition is imposed on a pair of learner and adversary instead of solely
on a learner. This is because no learner can satisfy this condition if the adversary draws samples
in a completely adaptive manner. The probabilistic statement is necessary because many online
algorithms? predictions are not deterministic. The excess loss requirement is needed since an online
learner cannot produce meaningful predictions before observing a sufficient number of examples.
3
An optimal algorithm
In this section, we describe the booster?s optimal strategy for designing cost matrices. We first
introduce a general theory without specifying the loss, and later investigate the asymptotic behavior
of cumulative loss suffered by our algorithm under the specific 0-1 loss. We adopt the potential
function framework from Mukherjee and Schapire [4] and extend it to the online setting. Potential
functions help both in designing cost matrices and in proving the mistake bound of the algorithm.
3.1
A general online multiclass boost-by-majority (OnlineMBBM) algorithm
We will keep track of the weighted cumulative votes of the first i weak learners for the sample xt by
Pi
sit := j=1 ?tj elj , where ?ti is the weight of W Li , lti is its prediction and ej is the j th standard basis
t
vector. For the optimal algorithm, we assume that ?ti = 1, ?i, t. In other words, the booster makes
the final decision by simple majority votes. Given a cumulative vote s ? Rk , suppose we have a loss
function Lr (s) where r denotes the correct label. We call a loss function proper, if it is a decreasing
function of s[r] and an increasing function of other coordinates (we alert the reader that ?proper loss?
has at least one other meaning in the literature). From now on, we will assume that our loss function
is proper. A good example of proper loss is multiclass 0-1 loss:
Lr (s) := 1(max s[l] ? s[r]).
l6=r
(2)
The purpose of the potential function ?ri (s) is to estimate the booster?s loss when there remain i
learners until the final decision and the current cumulative vote is s. More precisely, we want potential
functions to satisfy the following conditions:
?r0 (s)
r
?i+1 (s)
= Lr (s),
= El?ur? ?ri (s + el ).
3
(3)
Algorithm 1 Online Multiclass Boost-by-Majority (OnlineMBBM)
1: for t = 1, ? ? ? , T do
2:
Receive example xt
3:
Set s0t = 0 ? Rk
4:
for i = 1, ? ? ? , N do
5:
Set the normalized cost matrix Dit according to (5) and pass it to W Li
6:
Get weak predictions lti = W Li (xt ) and update sit = sti?1 + elti
7:
end for
8:
Predict y?t := argmaxl sN
t [l] and receive true label yt
9:
for i = 1, ? ? ? , N do
Pk
10:
Set wi [t] = l=1 [?yNt?i (si?1
+ el ) ? ?yNt?i (sti?1 + eyt )]
t
11:
Pass training example with weight (xt , yt , wi [t]) to W Li
12:
end for
13: end for
Readers should note that ?ri (s) also inherits the proper property of the loss function, which can be
shown by induction. The condition (3) can be loosened by replacing both equalities by inequalities
???, but in practice we usually use equalities.
Now we describe the booster?s strategy for designing cost matrices. After observing xt , the booster
sequentially sets a cost matrix Cit for W Li , gets the weak learner?s prediction lti and uses this in the
computation of the next cost matrix Ci+1
t . Ultimately, booster wants to set
Cit [r, l] = ?rN ?i (si?1
+ el ).
t
(4)
C1eor ,
However, this cost matrix does not satisfy the condition of
and thus should be modified in order
to utilize the weak learning condition. First to make the cost for the true label equal to 0, we subtract
Cit [r, r] from every element of Cit [r]. Since the potential function is proper, our new cost matrix still
has non-negative elements after the subtraction. We then normalize the row so that each row has `1
norm equal to 1. In other words, we get new normalized cost matrix
Dit [r, l] =
?rN ?i (si?1
+ el ) ? ?rN ?i (si?1
+ er )
t
t
,
i
w [t]
(5)
Pk
where wi [t] := l=1 ?rN ?i (si?1
+ el ) ? ?rN ?i (si?1
+ er ) plays the role of weight. It is still possible
t
t
i
that a row vector Ct [r] is a zero vector so that normalization is impossible. In this case, we just leave
it as a zero vector. Our weak learning condition (1) still works with cost matrices some of whose row
vectors are zeros because however the learner predicts, it incurs no cost.
After defining cost matrices, the rest of the algorithm is straightforward except we have to estimate
||wi ||? to normalize the weight. This is necessary because the weak learning condition assumes
the weights lying in [0, 1]. We cannot compute the exact value of ||wi ||? until the last instance is
revealed, which is fine as we need this value only in proving the mistake bound. The estimate wi? for
||wi ||? requires to specify the loss, and we postpone the technical parts to Appendix B.2. Interested
readers may directly refer Lemma 10 before proceeding. Once the learners generate predictions after
observing cost matrices, the final decision is made by simple majority votes. After the true label
is revealed, the booster updates the weight and sends the labeled instance with weight to the weak
learners. The pseudocode for the entire algorithm is depicted in Algorithm 1. The algorithm is named
after Beygelzimer et al. [7, OnlineBBM], which is in fact OnlineMBBM with binary labels.
We present our first main result regarding the mistake bound of general OnlineMBBM. The proof
appears in Appendix B.1 where the main idea is adopted from Beygelzimer et al. [7, Lemma 3].
Theorem 2. (Cumulative loss bound for OnlineMBBM) Suppose weak learners and an adversary
satisfy the online weak learning condition (1) with parameters ?, ?, and S. For any T and N satisfying
? N1 , and any adaptive sequence of labeled examples generated by the adversary, the final loss
suffered by OnlineMBBM satisfies the following inequality with probability 1 ? N ?:
T
X
yt
L
(sN
t )
?
?1N (0)T
t=1
+S
N
X
i=1
4
wi? .
(6)
Here ?1N (0) plays a role of asymptotic error rate and the second term determines the sample complexity. We will investigate the behavior of those terms under the 0-1 loss in the following section.
3.2
Mistake bound under 0-1 loss and its optimality
From now on, we will specify the loss to be multiclass 0-1 loss defined in (2), which might be the
most relevant measure in multiclass problems. To present a specific mistake bound, two terms in
the RHS of (6) should be bounded. This requires an approximation of potentials, which is technical
and postponed to Appendix B.2. Lemma 9 and 10 provide the bounds for those terms. We also
mention another bound for the weight in the remark after Lemma 10 so that one can use whichever
tighter. Combining the above lemmas with Theorem 2 gives the following corollary. The additional
constraint on ? comes from Lemma 10.
Corollary 3. (0-1 loss bound of OnlineMBBM) Suppose weak learners and an adversary satisfy
the online weak learning condition (1) with parameters ?, ?, and S, where ? < 12 . For any T and
N satisfying ? N1 and any adaptive sequence of labeled examples generated by the adversary,
OnlineMBBM can generate predictions y?t that satisfy the following inequality with probability 1?N ?:
T
X
1(yt 6= y?t ) ? (k ? 1)e?
?2 N
2
?
? 5/2 N S).
T + O(k
(7)
t=1
Therefore in order to achieve error rate , it suffices to use N = ?( ?12 ln k ) weak learners, which
? k5/2 S).
gives an excess loss bound of ?(
?
5/2
? k S). If
Remark. Note that the above excess loss bound gives a sample complexity bound of ?(
?
we use alternative weight bound to get kN S as an upper bound for the second term in (6), we end up
?
? k2 S).
having O(kN
S). This will give an excess loss bound of ?(
?
We now provide lower bounds on the number of learners and sample complexity for arbitrary online
boosting algorithms to evaluate the optimality of OnlineMBBM under 0-1 loss. In particular, we
construct weak learners that satisfy the online weak learning condition (1) and have almost matching
asymptotic error rate and excess loss compared to those of OnlineMBBM as in (7). Indeed we
can prove that the number of learners and sample complexity of OnlineMBBM is optimal up to
logarithmic factors, ignoring the influence of the number of classes k. Our bounds are possibly
suboptimal up to polynomial factors in k, and the problem to fill the gap remains open. The detailed
proof and a discussion of the gap can be found in Appendix B.3. Our lower bound is a multiclass
version of Beygelzimer et al. [7, Theorem 3].
k ln( 1 )
Theorem 4. (Lower bounds for N and T ) For any ? ? (0, 14 ), ?, ? (0, 1), and S ? ? ? , there
exists an adversary with a family of learners satisfying the online weak learning condition (1) with
parameters ?, ?, and S, such that to achieve asymptotic error rate , an online boosting algorithm
k
requires at least ?( k21? 2 ln 1 ) learners and a sample complexity of ?( ?
S).
4
An adaptive algorithm
The online weak learning condition imposes minimal assumptions on the asymptotic accuracy of
learners, and obviously it leads to a solid theory of online boosting. However, it has two main practical
limitations. The first is the difficulty of estimating the edge ?. Given a learner and an adversary, it
is by no means a simple task to find the maximum edge that satisfies (1). The second issue is that
different learners may have different edges. Some learners may in fact be quite strong with significant
edges, while others are just slightly better than a random guess. In this case, OnlineMBBM has to
pick the minimum edge as it assumes common ? for all weak learners. It is obviously inefficient in
that the booster underestimates the strong learners? accuracy.
Our adaptive algorithm will discard the online weak learning condition to provide a more practical
method. Empirical edges ?1 , ? ? ? , ?N (see Section 4.2 for the definition) are measured for the weak
learners and are used to bound the number of mistakes made by the boosting algorithm.
5
4.1
Choice of loss function
Adaboost, proposed by Freund et al. [10], is arguably the most popular boosting algorithm in practice.
It aims to minimize the exponential loss, and has many variants which use some other surrogate
loss. The main reason of using a surrogate loss is ease of optimization; while 0-1 loss is not even
continuous, most surrogate losses are convex. We adopt the use of a surrogate loss for the same reason,
and throughout this section will discuss our choice of surrogate loss for the adaptive algorithm.
Exponential loss is a very strong candidate in that it provides a closed form for computing potential
functions, which are used to design cost matrices (cf. Mukherjee and Schapire [4, Theorem 13]).
One property of online setting, however, makes it unfavorable. Like OnlineMBBM, each data point
will have a different weight depending on weak learners? performance, and if the algorithm uses
exponential loss, this weight will be an exponential function of difference in weighted cumulative
votes. With this exponentially varying weights among samples, the algorithm might end up depending
on very small portion of observed samples. This is undesirable because it is easier for the adversary
to manipulate the sample sequence to perturb the learner.
To overcome exponentially varying weights, Beygelzimer et al. [7] use logistic loss in their adaptive
algorithm. Logistic loss is more desirable in that its derivative is bounded and thus weights will be
relatively smooth. For this reason, we will also use multiclass version of logistic loss:
X
Lr (s) =:
log(1 + exp(s[r] ? s[r])).
(8)
l6=r
We still need to compute potential functions from logistic loss in order to calculate cost matrices.
Unfortunately, Mukherjee and Schapire [4] use a unique property of exponential loss to get a closed
form for potential functions, which cannot be adopted to logistic loss. However, the optimal cost
matrix induced from exponential loss has a very close connection with the gradient of the loss (cf.
Mukherjee and Schapire [4, Lemma 22]). From this, we will design our cost matrices as following:
(
1
, if l 6= r
i?1
i?1
i
P t [r]?st [l])
(9)
Ct [r, l] := 1+exp(s
1
? j6=r 1+exp(si?1 [r]?si?1 [j]) , if l = r.
t
t
Cit [r]
Readers should note that the row vector
is simply the gradient of Lr (si?1
). Also note that this
t
matrix does not belong to C1eor , but it does guarantee that the correct prediction gets the minimal cost.
The choice of logistic loss over exponential loss is somewhat subjective. The undesirable property
of exponential loss does not necessarily mean that we cannot build an adaptive algorithm using this
loss. In fact, we can slightly modify Algorithm 2 to develop algorithms using different surrogates
(exponential loss and square hinge loss). However, their theoretical bounds are inferior to the one with
logistic loss. Interested readers can refer Appendix D, but it assumes understanding of Algorithm 2.
4.2
Adaboost.OLM
Our work is a generalization of Adaboost.OL by Beygelzimer et al. [7], from which the name
Adaboost.OLM comes with M standing for multiclass. We introduce a new concept of an expert.
From N weak learners, we can produce N experts where expert i makes its prediction by weighted
majority votes among the first i learners. Unlike OnlineMBBM, we allow
weights ?ti over
P varying
yt i
the learners. As we are working with logistic loss, we want to minimize t L (st ) for each i, where
the loss is given in (8). We want to alert the readers to note that even though the algorithm tries to
minimize the cumulative surrogate loss, its performance is still evaluated by 0-1 loss. The surrogate
loss only plays a role of a bridge that makes the algorithm adaptive.
We do not impose the online weak
learning condition on weak learners, but instead just measure the
P i
i
i
t Ct [yt ,lt ]
P
performance of W L by ?i :=
. This empirical edge will be used to bound the number of
i
t Ct [yt ,yt ]
mistakes made by Adaboost.OLM. By definition of cost matrix, we can check
Cit [yt , yt ] ? Cit [yt , l] ? ?Cit [yt , yt ], ?l ? [k],
from which we can prove ?1 ? ?i ? 1, ?i. If the online weak learning condition is met with edge ?,
then one can show that ?i ? ? with high probability when the sample size is sufficiently large.
6
Algorithm 2 Adaboost.OLM
1: Initialize: ?i, v1i = 1, ?1i = 0
2: for t = 1, ? ? ? , T do
3:
Receive example xt
4:
Set s0t = 0 ? Rk
5:
for i = 1, ? ? ? , N do
6:
Compute Cit according to (9) and pass it to W Li
7:
Set lti = W Li (xt ) and sit = si?1
+ ?ti elti
t
i
i
8:
Set y?t = argmaxl st [l], the prediction of expert i
9:
end for
10:
Randomly draw it with P(it = i) ? vti
11:
Predict y?t = y?tit and receive the true label yt
12:
for i = 1, ? ? ? , N do
0
i
13:
Set ?t+1
= ?(?ti ? ?t fti (?ti )) using (10) and ?t =
?
2 2?
(k?1) t
Ci [y ,y ]
t t
Set wi [t] = ? tk?1
and pass (xt , yt , wi [t]) to W Li
i
15:
Set vt+1
= vti ? exp(?1(yt 6= y?ti ))
16:
end for
17: end for
14:
Unlike the optimal algorithm, we cannot show the last expert that utilizes all the learners has the
best accuracy. However, we can show at least one expert has a good predicting power. Therefore
we will use classical Hedge algorithm (Littlestone and Warmuth [11] and Freund and Schapire [12])
to randomly choose an expert at each iteration with adaptive probability weight depending on each
expert?s prediction history.
Finally we need to address how to set the weight ?ti for each weak learner.
tries to
P As our algorithm
i
i
minimize the cumulative logistic loss, we want to set ?ti to minimize t Lyt (si?1
+
?
e
).
This
t
t lt
is again a classical topic in online learning, and we will use online gradient descent, proposed
by Zinkevich [13]. By letting, fti (?) := Lyt (si?1
+ ?elti ), we need an online algorithm ensuring
t
P i i
P i
i
f
(?
)
?
min
f
(?)
+
R
(T
)
where
F
is a feasible set to be specified later, and Ri (T )
??F
t
t t
t t
is a regret that is sublinear in T . To apply Zinkevich [13, Theorem 1], we need fti to be convex
and F to be compact. The first assumption is met by our choice of logistic loss, and for the second
assumption, we will set F = [?2, 2]. There is no harm to restrict the choice of ?ti by F because we
can always scale the weights without affecting the result of weighted majority votes.
By taking derivatives, we get
0
fti (?)
(
=
1
i?1 i
1+exp(si?1
P t [yt ]?st [lt ]??)
1
? j6=yt 1+exp(si?1 [j]+??s
i?1
[yt ])
t
t
, if lti 6= yt
, if lti = yt .
(10)
0
This provides |fti (?)| ? k ? 1. Now let ?(?) represent a projection onto ?
F : ?(?) :=
2 2?
i
i
i0
i
max{?2, min{2, ?}}. By setting ?t+1 = ?(?t ? ?t ft (?t )) where ?t = (k?1) t , we get
?
?
Ri (T ) ? 4 2(k ? 1) T . Readers should note that any learning rate of the form ?t = ?ct would
work, but our choice is optimized to ensure the minimal regret.
The pseudocode for Adaboost.OLM is presented in Algorithm 2. In fact, if we put k = 2, Adaboost.OLM has the same structure with Adaboost.OL. As in OnlineMBBM, the booster also needs
to pass the weight along with labeled instance. According to (9), it can be inferred that the weight is
proportional to ?Cit [yt , yt ].
4.3
Mistake bound and comparison to the optimal algorithm
Now we present our second main result that provides a mistake bound of Adaboost.OLM. The main
structure of the proof is adopted from Beygelzimer et al. [7, Theorem 4] but in a generalized cost
matrix framework. The proof appears in Appendix C.
7
Theorem 5. (Mistake bound of Adaboost.OLM) For any T and N , with probability 1 ? ?, the
number of mistakes made by Adaboost.OLM satisfies the following inequality:
T
X
t=1
2
? PkN
T + O(
),
N
2
2
i=1 ?i
i=1 ?i
8(k ? 1)
1(yt 6= y?t ) ? PN
? notation suppresses dependence on log 1 .
where O
?
Remark. Note that this theorem naturally implies Beygelzimer et al. [7, Theorem 4]. The difference
in coefficients is due to different scaling of ?i . In fact, their ?i ranges from [? 12 , 12 ].
Now that we have established a mistake bound, it is worthwhile to compare the bound with the
optimal boosting algorithm. Suppose the weak learners satisfy the weak learning
condition (1)
P i
i
t Ct [yt ,lt ]
P
??
with edge ?. For simplicity, we will ignore the excess loss S. As we have ?i =
i
t Ct [yt ,yt ]
8(k?1)
kN
?
with high probability, the mistake bound becomes 2 T + O( 2 ). In order to achieve error
? N
?
8(k?1)
? 2k2 4 ) sample size. Note that
learners
and
T
=
?(
2
?
?
? k2 )}. Adaboost.OLM is
? k5/2 ), ?(
?( ?12 ln k ) and T = min{?(
?
?
rate , Adaboost.OLM requires N ?
OnlineMBBM requires N =
obviously suboptimal, but due to its adaptive feature, its performance on real data is quite comparable
to that by OnlineMBBM.
5
Experiments
We compare the new algorithms to existing ones for online boosting on several UCI data sets, each
with k classes1 . Table 1 contains some highlights, with additional results and experimental details in
the Appendix E. Here we show both the average accuracy on the final 20% of each data set, as well as
the average run time for each algorithm. Best decision tree gives the performance of the best of 100
online decision trees fit using the VFDT algorithm in Domingos and Hulten [14], which were used as
the weak learners in all other algorithms, and Online Boosting is an algorithm taken from Oza [5].
Both provide a baseline for comparison with the new Adaboost.OLM and OnlineMBBM algorithms.
Best MBBM takes the best result from running the OnlineMBBM with five different values of the
edge parameter ?.
Despite being theoretically weaker, Adaboost.OLM often demonstrates similar accuracy and sometimes outperforms Best MBBM, which exemplifies the power of adaptivity in practice. This power
comes from the ability to use diverse learners efficiently, instead of being limited by the strength of
the weakest learner. OnlineMBBM suffers from high computational cost, as well as the difficulty of
choosing the correct value of ?, which in general is unknown, but when the correct value of ? is used
it peforms very well. Finally in all cases Adaboost.OLM and OnlineMBBM algorithms outperform
both the best tree and the preexisting Online Boosting algorithm, while also enjoying theoretical
accuracy bounds.
Table 1: Comparison of algorithm accuracy on final 20% of data set and run time in seconds. Best
accuracy on a data set reported in bold.
1
Data sets
k
Best decision tree
Online Boosting
Adaboost.OLM
Best MBBM
Balance
Mice
Cars
Mushroom
Nursery
ISOLET
Movement
3
8
4
2
4
26
5
0.768
0.608
0.924
0.999
0.953
0.515
0.915
0.772
0.399
0.914
1.000
0.941
0.149
0.870
0.754
0.561
0.930
1.000
0.966
0.521
0.962
0.821
42
0.695 2173
0.914
56
1.000
325
0.969 1510
0.635 64707
0.988 18676
8
105
39
241
526
470
1960
19
263
27
169
302
1497
3437
20
416
59
355
735
2422
5072
Codes are available at https://github.com/yhjung88/OnlineBoostingWithVFDT
8
Acknowledgments
We acknowledge the support of NSF under grants CAREER IIS-1452099 and CIF-1422157.
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6,294 | 6,694 | Matching on Balanced Nonlinear Representations for
Treatment Effects Estimation
Yun Fu
Northeastern University
Boston, MA
[email protected]
Sheng Li
Adobe Research
San Jose, CA
[email protected]
Abstract
Estimating treatment effects from observational data is challenging due to the
missing counterfactuals. Matching is an effective strategy to tackle this problem.
The widely used matching estimators such as nearest neighbor matching (NNM)
pair the treated units with the most similar control units in terms of covariates,
and then estimate treatment effects accordingly. However, the existing matching
estimators have poor performance when the distributions of control and treatment
groups are unbalanced. Moreover, theoretical analysis suggests that the bias of
causal effect estimation would increase with the dimension of covariates. In this
paper, we aim to address these problems by learning low-dimensional balanced and
nonlinear representations (BNR) for observational data. In particular, we convert
counterfactual prediction as a classification problem, develop a kernel learning
model with domain adaptation constraint, and design a novel matching estimator.
The dimension of covariates will be significantly reduced after projecting data
to a low-dimensional subspace. Experiments on several synthetic and real-world
datasets demonstrate the effectiveness of our approach.
1
Introduction
Causal questions exist in many areas, such as health care [24, 12], economics [14], political science [17], education [36], digital marketing [6, 43, 5, 15, 44], etc. In the field of health care, it is
critical to understand if a new medicine could cure a certain illness and perform better than the old
ones. In political science, it is of great importance to evaluate whether the government should fund a
job training program, by assessing if the program is the true factor that leads to the success of job
hunting. All of these causal questions can be addressed by the causal inference technique. Formally,
causal inference estimates the treatment effect on some units after interventions [33, 20]. In the
above example of heath care, the units could be patients, and the intervention would be taking new
medicines. Due to the wide applications of causal questions, effective causal inference techniques are
highly desired to address these problems.
Generally, the causal inference problems can be tackled by either experimental study or observational
study. Experimental study is popular in traditional causal inference problems, but it is time-consuming
and sometimes impractical. As an alternative strategy, observational study has attracted increasing
attention in the past decades, which extracts causal knowledge only from the observed data. Two
major paradigms for observational study have been developed in computer science and statistics,
including the causal graphical model [29] and the potential outcome framework [27, 33]. The former
builds directed acyclic graphs (DAG) from covariates, treatment and outcome, and uses probabilistic
inference to determine causal relationships; while the latter estimates counterfactuals for each treated
unit, and gives a precise definition of causal effect. The equivalence of two paradigms has been
discussed in [11]. In this paper, we mainly focus on the potential outcome framework.
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
A missing data problem needs to be dealt with in the potential outcome framework. As each unit is
either treated or not treated, it is impossible to observe its outcomes in both scenarios. In other words,
one has to predict the missing counterfactuals. A widely used solution to estimating counterfactuals
is matching. According to the (binary) treatment assignments, a set of units can be divided into a
treatment group and a control group. For each treated unit, matching methods select its counterpart in
the control group based on certain criteria, and treat the selected unit as a counterfactual. Then the
treatment effect can be estimated by comparing the outcomes of treated units and the corresponding
counterfactuals. Some popular matching estimators include nearest neighbor matching (NNM) [32],
propensity score matching [31], coarsened exact matching (CEM) [17], genetic matching [9], etc.
Existing matching methods have three major drawbacks. First, they either perform matching in the
original covariate space (e.g., NNM, CEM) or in the one-dimensional propensity score space (e.g.,
PSM). The potential of using intermediate representations has not been extensively studied before.
Second, existing methods work well for data with a moderate number of covariates, but may fail
for data with a large number of covariates, as theoretical analysis suggests that the bias of treatment
effect estimation would increase with the dimension of covariates [1]. Third, most matching methods
do not take into account whether the distributions of two groups are balanced or not. The matching
process would make no sense if the distributions of two groups have little overlap.
To address the above problems, we propose to learn balanced and nonlinear representations (BNR)
from observational data, and design a novel matching estimator named BNR-NNM. First, the
counterfactual prediction problem is converted to a multi-class classification problem, by categorizing
the outcomes to ordinal labels. Then, we propose a novel criterion named ordinal scatter discrepancy
(OSD) for supervised kernel learning on data with ordinal labels, and extract low-dimensional
nonlinear representations from covariates. Further, to achieve balanced distributions in the lowdimensional space, a maximum mean discrepancy (MMD) criterion [4] is incorporated to the model.
Finally, matching strategy is performed on the extracted balanced representations, in order to provide
a robust estimation of causal effect. In summary, the main contributions of our work include:
? We propose a novel matching estimator, BNR-NNM, which learns low-dimensional balanced
and nonlinear representations via kernel learning.
? We convert the counterfactual prediction problem into a multi-class classification problem,
and design an OSD criterion for nonlinear kernel learning with ordinal labels.
? We incorporate a domain adaptation constraint to feature learning by using the maximum
mean discrepancy criterion, which leads to balanced representations.
? We evaluate the proposed estimator on both synthetic datasets and real-world datasets, and
demonstrate its superiority over the state-of-the-art methods.
2
Background
Potential Outcome Framework. The potential outcome framework is proposed by Neyman and
Rubin [27, 33]. Considering binary treatments for a set of units, there are two possible outcomes for
each unit. Formally, for unit k, the outcome is defined as Yk (1) if it received treatment, and Yk (0) if
it did not. Then, the individual-level treatment effect is defined as ?k = Yk (1) ? Yk (0). Clearly, each
unit only belongs to one of the two groups, and therefore, we can only observe one of the two possible
outcomes. This is the well-known missing data problem in causal inference. In particular, if unit k
received treatment, Yk (1) is the observed outcome, and Yk (0) is missing data, i.e., counterfactual.
The potential outcome framework usually makes the following assumptions [19].
Assumption 1. Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for
any units do not vary with the treatments assigned to other units, and for each unit there are no
differences forms or versions of each treatment level, which lead to different potential outcomes.
|=
Assumption 2. Strongly Ignorable Treatment Assignment (SITA): Conditional on covariates xk ,
treatment Tk is independent of potential outcomes.
(Yk (1), Yk (0))
Tk |xk . (Unconfoundedness)
(1)
0 < Pr(Tk = 1|xk ) < 1. (Overlap)
These assumptions enable the modeling of treatment of one unit with respect to covariates, independent of outcomes and other units.
Matching Estimators. To address the aforementioned missing data problem, a simple yet effective
strategy has been developed, which is matching [32, 33, 14, 40]. The idea of matching is to estimate
2
the counterfactual for a treated unit by seeking its most similar counterpart in the control group.
Existing matching methods can be roughly divided into three categories: nearest neighbor matching
(NNM), weighting, and subclassification. We mainly focus on NNM in this paper.
Let XC ? Rd?NC and XT ? Rd?NT denote the covariates of a control group and a treatment group,
respectively, where d is the number of covariates, NC and NT are the group sizes. T is a binary
vector indicating if the units received treatments (i.e., Tk = 1) or not (i.e., Tk = 0). Y is an outcome
vector. For each treated unit k, NNM finds its nearest neighbor in the control group in terms of the
covariates. The outcome of the selected control unit is considered as an estimation of counterfactual.
Then, the average treatment effect on treated (ATT) is defined as:
1 X
AT T =
Yk (1) ? Y?k (0) ,
(2)
NT
k:Tk =1
where Y?k (0) is the counterfactual estimated from unit k?s nearest neighbor in the control group.
NNM can be implemented in various ways, such as using different distance metrics, or choosing
different number of neighbors. Euclidean distance and Mahalanobis distance are two widely-used
distance metrics for NNM. They work well when there are a few covariates with normal distributions [34]. Another important matching estimator is propensity score matching (PSM) [31]. PSM
estimates the propensity score (i..e., the probability of receiving treatment) for each unit via logistic
regression, and pairs the units from two groups with similar scores [35, 8, 30]. Most recently, a
covariate balancing propensity score (CBPS) method is developed to balance the distributions of two
groups by weighting the covariates, and has shown promising performance [18].
The key differences between the proposed BNR-NNM estimator and the traditional matching estimators are two-fold. First, BNR-NNM performs matching in an intermediate low-dimensional
subspace that could guarantee a low estimation bias, while the traditional estimators adopt either the
original covariate space or the one-dimensional space. Second, BNR-NNM explicitly considers the
balanced distributions across treatment and control groups, while the traditional estimators usually
fail to achieve such a property.
Machine Learning for Causal Inference. In recent years, researchers have been exploring the
relationships between causal inference and machine learning [39, 10, 38]. A number of predictive
models have been designed to estimate the causal effects, such as causal trees [3] and causal
forests [42]. Balancing the distributions of two groups is considered as a key issue in observational
study, which is closely related to covariate shift and in general domain adaptation [2]. Meanwhile,
causal inference has also been incorporated to improve the performance of domain adaptation [46, 45].
Most recently, the idea of representation learning is introduced to learn new features from covariates
through random projections [25], informative subspace learning [7], and deep neural networks [21,
37].
3
Learning Balanced and Nonlinear Representations (BNR)
In this section, we first define the notations that will be used throughout this paper. Then we introduce
how to convert the counterfactual prediction problem into a multi-class classification problem, and
justify the rationality of this strategy. We will also present the details of how to learn nonlinear and
balanced representations, and derive the closed-form solutions to the model.
Notations. Let X = [XC , XT ] ? Rd?N denote the covariates of all units, where XC ? Rd?NC is
the control group with NC units, and XT ? Rd?NT is the treatment group with NT units. N is the
total number of units, and d is the number of covariates for each unit. ? : x ? Rd ? ?(x) ? F is
a nonlinear mapping function from sample space R to an implicit feature space F. T ? RN ?1 is a
binary vector to indicate if the units received treatments or not. Y ? RN ?1 is an outcome vector. The
elements in Y could be either discrete or continuous values.
3.1
From Counterfactual Prediction to Multi-Class Classification
When estimating the treatment effects as shown in Eq.(2), we only have the observed outcome Yk (1),
but need to estimate the counterfactual Y?k (0). Ideally, we would train a model Y?k (0) = Fcf (xk ) that
can predict the counterfactual for any units, given the covariate vector xk . One strategy is to build a
predictive model (e.g., regression) that maps each unit xi to its output Yi , which has been extensively
3
studied before. Alternatively, we can convert the counterfactual prediction problem into a multi-class
classification problem.
Given a set of units X and the corresponding outcome vector Y , we aim to learn a predictive
model Fcf (xk ) that maps from the covariate space to the outcome space. In particular, we propose
to seek an intermediate representation space in which the units close to each other should have
very similar outcomes. The outcome vector Y usually contains continuous values. We categorize
outcomes in Y into multiple levels on the basis of the magnitude of outcome value, and consider
them as (pseudo) class labels. Clustering or kernel density estimation can be used for discretizing
Y . Finally, Y is converted to a (pseudo) class label vector Yc with c categories. For example,
Y = [0.3, 0.5, 1.1, 1.2, 2.4] could be categorized as Y3 = [1, 1, 2, 2, 3]. As a result, we could use Yc
and X to train a classifier.
Note that the Yc actually contains ordinal labels, as the discretized labels carry additional information.
In particular, the labels [1, 2, 3] are not totally independent. We actually assume that Class 1 should
be more close to Class 2 than Class 3, since the outcome values in Class 1 are closer to those in Class
2. We will make use of such ordinal label information when designing the classification model.
3.2
Learning Nonlinear Representations via Ordinal Scatter Discrepancy
To obtain effective representations from X, we propose to train a nonlinear classifier in a reproducing
kernel Hilbert space (RKHS). The reasons of employing the RKHS based nonlinear models are as
follows. First, compared to linear models, nonlinear models are usually more capable of dealing
with complicated data distributions. It is well known that the treatment and control groups might
have diverse distributions, and the nonlinear models would be able to tightly couple them in a shared
low-dimensional subspace. Second, the RKHS based nonlinear models usually have closed-form
solutions because of the kernel trick, which is beneficial for handling large-scale data.
Let ?(xi ) denote the mapped counterpart of xi in kernel space, and then ?(X) =
[?(x1 ), ?(x2 ), ? ? ? , ?(xN )]. In light of the maximum scatter difference criterion [26], we take
into account the ordinal label information, and propose a novel criterion named Ordinal Scatter Discrepancy (OSD) to achieve the desired data distribution after projecting ?(X) to a low-dimensional
subspace. In particular, OSD minimizes the within-class scatter, and meanwhile maximize the
noncontiguous-class scatter matrix. Let P denote a transformation matrix, OSD maps samples onto a
subspace by maximizing the differences of noncontiguous-class scatter and within-class scatter. We
perform OSD in kernel space to learn nonlinear representations, and have the following objective
function:
arg max F (P, ?(X), Yc ) = tr(P > (KI ? ?KW )P ),
P
(3)
s.t.
P > P = I,
where ? is a non-negative trade-off parameter, tr(?) is the trace operator for matrix, and I is an identity
matrix. The orthogonal constraint P > P = I is introduced to reduce the redundant information in
projection.
In Eq.(3), KI and KW are the noncontiguous-class scatter matrix and within-class scatter matrix in
kernel space, respectively. The detailed definitions are:
KI?
=
?
KW
c(c?1)
2
=
1
N
c
c
P
P
e(j?i) (mi ? mj )(mi ? mj )>
(4)
i=1 j=i+1
ni
c P
P
(?(xij ) ? m)(?(x
?
?i )>
ij ) ? m
(5)
i=1 j=1
where ?(xij ) = [k(x1 , xij ), k(x2 , xij ), ? ? ? , k(xN , xij )]> , mi is the mean vector of ?(xij ) that
belongs to the i-th class, m
? is the mean vector of all ?(xij ), and ni is the number of units in the
i-th class. k(xi , xj ) = h?(xi ), ?(xj )i is a kernel function, which is utilized to avoid calculating the
explicit form of function ? (i.e., the kernel trick).
Eq. (4) characterizes the scatter of a set of classes with (pseudo) ordinal labels. It measures the scatter
of every pair of classes. The factor e(j?i) is used to penalize the classes that are noncontiguous. The
intuition is that, for ordinal labels, we may expect the contiguous classes will be close to each other
after projection, while the noncontiguous classes should be pushed away. Therefore, we put larger
4
weights for the noncontiguous classes. For example, e(2?1) < e(3?1) , since Class 1 should be more
close to Class 2 than Class 3, as we explained in Section 3.1.
Eq. (5) measures the within-class scatter. We expect that the units having the same (pseudo) class
labels will be very close to each other in the feature space, and therefore they will have similar feature
representations after projection.
The differences between the proposed OSD criterion and other discriminative criteria (e.g., Fisher
criterion, maximum scatter difference criterion) are two-fold. (1) OSD criterion learns nonlinear
projection and feature representations in the RKHS space; (2) OSD explicitly makes use of the
ordinal label information that are usually ignored by existing criteria. Moreover, the maximum scatter
difference criterion is a special case of OSD.
3.3
Learning Balanced Representations via Maximum Mean Discrepancy
Balanced distributions of control and treatment groups, in terms of covariates, would greatly facilitate
the causal inference methods such as NNM. To this end, we adopt the idea of maximum mean
discrepancy (MMD) [4] when learning the transformation P , and finally obtain balanced nonlinear
representations. The MMD criterion has been successfully applied to some problems like domain
adaptation [28].
Assume that the control group XC and treatment group XT are random variable sets with distributions
P and Q, MMD implies the empirical estimation of the distance between P and Q. In particular,
MMD estimates the distance between nonlinear feature sets ?(XC ) and ?(XT ), which can be
formulated as:
n
nT
C
P
P
Dist(?(XC ), ?(XT )) = k N1C
?(XCi ) ? N1T
?(XT i )k2F ,
(6)
i=1
i=1
where F denotes a kernel space.
By utilizing the kernel trick, Dist(?(XC ), ?(XT )) in the original kernel space can be equivalently
converted to:
Dist(?(XC ), ?(XT )) = tr(KL),
(7)
KCC KCT
where K =
is a kernel matrix, KCC , KT T , and KT C are kernel matrices defined
KT C KT T
on control group, treatment group, and cross groups, respectively. L is a constant matrix. If
xi , xj ? XC , Lij = N12 ; if xi , xj ? XT , Lij = N12 ; otherwise, Lij = ? NC1NT .
C
T
As all the units are projected into a new space via projection P , we need to measure the MMD for
new representations ?(XC ) = P > ?(XC ) and ?(XT ) = P > ?(XT ), and rewrite Eq.(7) into the
following form after some derivations:
Dist(?(XC ), ?(XT )) = tr(P > KLKP ).
3.4
(8)
BNR Model and Solutions
The representation learning objectives described in Section 3.2 and Section 3.3 are actually performed
on the same data set with different partitions. For nonlinear representation learning, we merge the
control group and treatment group, assign a (pseudo) ordinal label for each unit, and then learn
discriminative nonlinear features accordingly. For balanced representation learning, we aim to
mitigate the distribution discrepancy between control group and treatment group. Two learning
objectives are motivated from different perspectives, and therefore they are complementary to each
other. By combing the objectives for nonlinear and balanced representations in Eq.(3) and Eq.(8), we
can extract effective representations for the purpose of treatment effect estimation.
The objective function of BNR is formulated as follows:
arg max F (P, ?(X), Yc ) ? ?Dist(?(XC ), ?(XT ))
P
s.t.
= tr(P > (KI ? ?KW )P ) ? ?tr(P > KLKP ),
P > P = I,
(9)
where ? is a trade-off parameter to balance the effects of two terms. A negative sign is added before
?Dist(?(XC ), ?(XT )) in order to adapt it into this maximization problem.
5
The problem Eq.(9) can be efficiently solved by using a closed-form solution described in Proposition
1. The proof is provided in the supplementary document due to space limit.
Proposition 1. The optimal solution of P in problem Eq.(9) is the eigenvectors of matrix (KI ?
?KW ? ?KLK), which correspond to the m leading eigenvalues.
4
BNR for Nearest Neighbor Matching
Leveraging on the balanced nonlinear representations extracted from observational data, we propose
a novel nearest neighbor matching estimator named BNR-NNM.
After obtaining the transformation P in kernel space, we could generate nonlinear and balanced
representations for control and treated units as: X?C = P > KC , X?T = P > KT , where KC and KT
are kernel matrices defined in control and treatment groups, respectively. Then we follow the basic
idea of nearest neighbor matching. On the new representations X?C and X?T , we calculate the distance
between each treated unit and control unit, and choose the one with the smallest distance. The
outcome of the selected control unit serves as the estimation of counterfactual. Finally, the average
treatment effect on treated (ATT) can be calculated, as defined in Eq.(2). The complete procedures of
BNR-NNM are summarized in Algorithm 1.
Algorithm 1. BNR-NNM
The estimated ATT is dependent on the transforInput: Treatment group XT ? Rd?Nt
mation matrix P . Although P is optimal for the
Control group XC ? Rd?Nc
representation learning model Eq.(9), it might not
Outcome vectors YT and YC
be optimal for the whole causal inference process,
Total sample size N
for three reasons. First, the model Eq.(9) contains
Kernel function k
two major hyperparameters, ? and ?. Different ?opParameters ?, ?, c
timal? transformations P would be obtained with
1: Convert outcomes to (pseudo) ordinal labels
different parameter settings. Second, the ground2: Construct KI and KW using Eqs.(4) and (5)
3: Construct kernel matrix K using Eq.(7)
truth label information required by supervised learn4: Learn the transformation P using Eq.(9)
ing are unknown. Recall that we categorize the
5: Construct kernel matrix KC and KT
outcome vector as pseudo labels, which introduces
6: Project KC and KT using P
considerable uncertainty. Third, the ground-truth
X?C = P > KC , X?T = P > KT .
information of causal effect is unknown in observa7: Perform NNM between X?C and X?T
tional studies with real-world data. Therefore, it is
8: Estimate the ATT A from Eq.(2)
impossible to use the faithful supervision informaOutput: Return A
tion of causal effect to guide the learning process.
These uncertainties from three perspectives might result in an unreliable estimation of ATT.
Thus, we present two strategies to tackle the above issue. (1) Median causal effect from multiple
estimations. Following the randomized NNM estimator [25], we implement multiple settings of
BNR-NNM with different parameters ?, ? and c, calculate multiple ATT values, and finally choose
the median value as the final estimation. In this way, a robust estimation of causal effect can be
obtained. (2) Model selection by cross-validation. Alternatively, the cross-validation strategy can be
employed to select proper values for ? and ?, by equally dividing the data and pseudo labels into k
subsets. Although the multiple runs in the above strategies would increase the computational cost,
our method is still efficient for three reasons. First, the dimension of covariates will be significantly
reduced, which enables a faster matching process. Second, owing to the closed-form solution to P
introduced in Proposition 1, the representation learning procedure is efficient. Third, these settings
are independent from each other, and therefore they can be executed in parallel.
5
Experiments and Analysis
Synthetic Dataset. Data Generation. We generate a synthetic dataset by following the protocols
described in [41, 25]. In particular, the sample size N is set to 1000, and the number of covariates d is
set to 100. The following basis functions are adopted in the data generation process: g1 (x) = x ? 0.5,
g2 (x) = (x ? 0.5)2 + 2, g3 (x) = x2 ? 1/3, g4 (x) = ?2 sin(2x), g5 (x) = e?x ? e?1 ? 1,
g6 (x) = e?x , g7 (x) = x2 , g8 (x) = x, g9 (x) = Ix>0 , and g10 (x) = cos(x). For each unit, the
covariates x1 , x2 , ? ? ? , xd are drawn independently from the standard normal distribution N (0, 1).
We only consider binary treatment in this paper, and define the treatment vector T as T |x = 1 if
P5
vector T , the
k=1 gk (xk ) > 0 and T |x = 0 otherwise. Given covariate vector x and the treatment
P5
outcome variables in Y are generated from the following model: Y |x, T ? N ( j=1 gj+5 (xj ) +
6
T, 1). It is obvious that Y contains continuous values. The first five covariates are correlated to the
treatments in T and the outcomes in Y , simulating a confounding effect, while the rest are noisy
components. By definition, the true causal effect (i.e., the ground truth of ATT) in this dataset is 1.
Baselines and Settings. We compare our matching estimator BNR-NNM with the following baseline
methods: Euclidean distance based NNM (Eud-NNM), Mahalanobis distance based NNM (MahNNM) [34], PSM [31], principal component analysis based NNM (PCA-NNM), locality preserving
projections based NNM (LPP-NNM), and randomized NNM (RNNM) [25].
Eud?NNM
PSM
Mah?NNM
PCA?NNM
LPP?NNM
RNNM
BNR?NNM (Ours)
0
10
Mean Square Error
PSM is a classical causal inference approach,
which estimates the propensity scores for each
control or treated unit using logistic regression,
and then perform matching on these scores. As
our approach learns new representations via
transformations, we also implement two matching estimators based on the popular subspace
learning methods PCA [22] and LPP [13]. The
nearest neighbor matching is performed on the
low-dimensional feature space learned by PCA
and LPP, respectively. RNNM is the state-ofthe-art matching estimator, especially for highdimensional data. It projects units to multiple
random subspaces, performs matching in each
of them, and finally selects the median value of
estimations. In RNNM, the number of random
projections is set to 20. The proposed BNRNNM and RNNM share a similar idea on projecting data to low-dimensional subspaces, but
they have different motivations and learn different data representations.
?1
10
?2
10
2
5
10
20
30
40
50
Dimension
60
70
80
90
100
Figure 1: MSE of different estimators on the synthetic dataset. Note that Eud-NNM and Mah-NNM
only involve matching in the original 100 dimensional data space.
The major parameters in BNR-NNM include ?, ?, and c. In the experiments, ? is empirically set to
1. ? is chosen from {10?3 , 10?1 , 1, 10, 103 }. The number of categories c is chosen from {2, 4, 6, 8}.
As described in Section 4, the median ATT of multiple estimations is used as the final result. We use
the Gaussian kernel function k(xi , xj ) = exp(?kxi ? xj k2 /2? 2 ), in which the bandwidth parameter
? is empirically set to 5. In the experiments we observe that our approach allows flexible settings
for these parameters, and intuitively selecting parameters from a wider range would lead to a robust
estimation of ATT.
Results and Discussions. To ensure a robust estimation of the performance of each matching
estimator, we repeat the data generation process 500 times, calculate the ATT for each estimator
in every replication, and compute the mean square error (MSE) with standard error (SD) for each
estimator over all of the replications. Eud-NNM and Mah-NNM perform matching in the original
covariate space, and PSM maps each unit to a single score. Thus we only have a single point
estimation for each of them. For PCA-NNM, LPP-NNM, RNNM and our method, we can choose the
dimension of feature space where the matching is conducted. Specifically, we increase the dimension
from 2 to 100, and calculate MSE and SD in each case. Figure 1 shows the MSE and SD (shown
as error bars) of each estimator when varying the dimensions. We observe from Figure 1 that the
proposed estimator BNR-NNM obtains lower MSE than all other methods in every case. The lowest
MSE is achieved when the dimension is 5. In addition, we have analyzed the sensitivity of parameter
settings. The detailed results are provided in the supplementary document.
IHDP Dataset with Simulated Outcomes. IHDP data [16] is an experimental dataset collected by
the Infant Health and Development Program. In particular, a randomized experiment was conducted,
where intensive high-quality care were provided to the low-birth-weight and premature infants. By
using the original data, an observation study can be conducted by removing a nonrandom subset of
the treatment group: all children with non-white mothers. After this preprocessing step, there are in
total 24 pretreatment covariates (excluding race) and 747 units, including 608 control units and 139
treatment units. The outcomes are simulated by using the pretreatment covariates and the treatment
assignment information, in order to hold the unconfoundedness assumption.
7
Due to the space limit, the outcome simulation procedures Table 1: Results on IHDP dataset.
are provided in the supplementary document. We repeat such
Method
?AT T
procedures for 200 times and generate 200 sets of simulated
Eu-NNM
0.18?0.06
outcomes, in order to conduct extensive evaluations. For each
Mah-NNM 0.31?0.12
set of simulated outcomes, we run our method and the basePSM
0.26?0.08
lines introduced above, and report the results in Table 1. We
PCA-NNM 0.19?0.11
use the error in average treatment effect on treated (ATT),
LPP-NNM
0.25?0.13
?AT T , as the evaluation metric. It is defined as the absolute
RNNM
0.16?0.07
[
difference between true ATT and estimated ATT (AT
T ), i.e.,
BNR-NNM 0.16?0.06
[
?AT T = |AT T ? AT
T |. Table 1 shows that the proposed
BNR-NNM estimator outperforms most baselines, which further validates the effectiveness of the
balanced and nonlinear representations.
LaLonde Dataset with Real Outcomes. The LaLonde dataset is a widely used benchmark for
observational studies [23]. It consists of a treatment group and a control group. The treatment group
contains 297 units from a randomized study of a job training program (the ?National Supported
Work Demonstration?), where an unbiased estimate of the average treatment effect is available. The
original LaLonde dataset contains 425 control units that are collected from the Current Population
Survey. Recently, Imai et al. augmented the data by including 2,490 units from the Panel Study
of Income Dynamics [18]. Thus, the sample size of control group is increased to 2,915. For each
sample, the covariates include age, education, race (black, white, or Hispanic), marriage status, high
school degree, earnings in 1974, and earnings in 1975. The outcome variable is earnings in 1978. In
this benchmark dataset, the unbiased estimation of ATT is $886 with a standard error of $448.
We compare our estimator with the baselines Table 2: Results on LaLonde dataset. BIAS (%) is
used in the previous experiments. In addi- the bias in percentage of the true effect.
tion, we also compare with a recently proMethod
ATT
SD
BIAS (%)
posed matching estimator, covariate balancing
Ground
Truth
886
488
N/A
propensity score (CBPS) [18] and a deep neural
Eu-NNM
-565.9 592.8
164%
network (DNN) method [37]. CBPS aims to
Mah-NNM
-67.9
526.1
108%
achieve balanced distributions between control
PSM
-947.6 567.9
201%
and treatment groups by adjusting the weights
PCA-NNM
-499.8
592.5
156%
for covariates. The DNN method utilizes a deep
LPP-NNM
-457.1 581.2
152%
neural network architecture for counterfactual
RNNM
-557.6 584.9
163%
regression, which is the state-of-the-art method
CBPS
423.3 1295.2
52%
on representation learning based counterfactual
DNN
742.0
N/A
16%
inference. For BNR-NNM, we use the same set783.6
546.3
12%
tings for ? and c as in the previous experiments. BNR-NNM
Table 2 shows the ground truth of ATT, and the estimations of different methods. We can observe
from Table 2 that CBPS and DNN obtain better results than other baselines, as both of them consider
the balanced property across treatment and control groups. Moreover, our BNR-NNM estimator
achieves the best result, due to the fully exploitation of balanced and nonlinear feature representations.
The evaluations on runtime behavior of each compared method are provided in the supplementary
document due to space limit.
6
Conclusions
In this paper, we propose a novel matching estimator based on balanced and nonlinear representations
for treatment effect estimation. Our method leverages on the predictive power of machine learning
models to estimate counterfactuals, and achieves balanced distributions in an intermediate feature
space. In particular, an ordinal scatter discrepancy criterion is designed to extract discriminative
features from observational data with ordinal pseudo labels, while a maximum mean discrepancy
criterion is incorporated to achieve balanced distributions. Extensive experimental results on three
synthetic and real-world datasets show that our approach provides more accurate estimation of causal
effects than the state-of-the-art matching estimators and representation learning methods. In future
work, we will extend the balanced representation learning model to other causal inference strategies
such as weighting and regression, and design estimators for multiple levels of treatments.
Acknowledgement. This research is supported in part by the NSF IIS award 1651902, ONR Young
Investigator Award N00014-14-1-0484, and U.S. Army Research Office Award W911NF-17-1-0367.
8
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6,295 | 6,695 | Learning Overcomplete HMMs
Vatsal Sharan
Stanford University
[email protected]
Sham Kakade
University of Washington
[email protected]
Percy Liang
Stanford University
[email protected]
Gregory Valiant
Stanford University
[email protected]
Abstract
We study the problem of learning overcomplete HMMs?those that have many
hidden states but a small output alphabet. Despite having significant practical
importance, such HMMs are poorly understood with no known positive or negative
results for efficient learning. In this paper, we present several new results?both
positive and negative?which help define the boundaries between the tractable and
intractable settings. Specifically, we show positive results for a large subclass of
HMMs whose transition matrices are sparse, well-conditioned, and have small
probability mass on short cycles. On the other hand, we show that learning is
impossible given only a polynomial number of samples for HMMs with a small
output alphabet and whose transition matrices are random regular graphs with large
degree. We also discuss these results in the context of learning HMMs which can
capture long-term dependencies.
1
Introduction
Hidden Markov Models (HMMs) are commonly used for data with natural sequential structure (e.g.,
speech, language, video). This paper focuses on overcomplete HMMs, where the number of output
symbols m is much smaller than the number of hidden states n. As an example, for an HMM that
outputs natural language documents one character at a time, the number of characters m is quite
small, but the number of hidden states n would need to be very large to encode the rich syntactic,
semantic, and discourse structure of the document.
Most algorithms for learning HMMs with provable guarantees assume the transition T ? Rn?n and
observation O ? Rm?n matrices are full rank [2, 3, 20] and hence do not apply to the overcomplete
regime. A notable exception is the recent work of Huang et al. [14] who studied this setting where
m n and showed that generic HMMs can be learned in polynomial time given exact moments
of the output process (which requires infinite data). Though understanding properties of generic
HMMs is an important first step, in reality, HMMs with a large number of hidden states typically have
structured, non-generic transition matrices?e.g., consider sparse transition matrices or transition
matrices of factorial HMMs [12]. Huang et al. [14] also assume access to exact moments, which
leaves open the question of when learning is possible with efficient sample complexity. Summarizing,
we are interested in the following questions:
1. What are the fundamental limitations for learning overcomplete HMMs?
2. What properties of HMMs make learning possible with polynomial samples?
3. Are there structured HMMs which can be learned in the overcomplete regime?
Our contributions. We make progress on all three questions in this work, sharpening our understanding of the boundary between tractable and intractable learning. We begin by stating a negative
result, which perhaps explains some of the difficulty of obtaining strong learning guarantees in the
overcomplete setting.
Theorem 1. The parameters of HMMs where i) the transition matrix encodes a random walk on a
regular graph on n nodes with degree polynomial in n, ii) the output alphabet m = polylog(n) and,
iii) the output distribution for each hidden state is chosen uniformly and independently at random,
cannot be learned (even approximately) using polynomially many samples over any window length
polynomial in n, with high probability over the choice of the observation matrix.
Theorem 1 is somewhat surprising, as parameters of HMMs with such transition matrices can be
easily learned in the non-overcomplete (m ? n) regime. This is because such transition matrices
are full-rank and their condition numbers are polynomial in n; hence spectral techniques such
as Anandkumar et al. [3] can be applied. Theorem 1 is also fundamentally of a different nature
as compared to lower bounds based on parity with noise reductions for HMMs [20], as ours is
information-theoretic.1 Also, it seems far more damning as the hard cases are seemingly innocuous
classes such as random walks on dense graphs. The lower bound also shows that analyzing generic or
random HMMs might not be the right framework to consider in the overcomplete regime as these
might not be learnable with polynomial samples even though they are identifiable. This further
motivates the need for understanding HMMs with structured transition matrices. We provide a proof
of Theorem 1 with more explicitly stated conditions in Appendix D.
For our positive results we focus on understanding properties of structured transition matrices
which make learning tractable. To disentangle additional complications due to the choice of the
observation matrix, we will assume that the observation matrix is drawn at random throughout the
paper. Long-standing open problems on learning aliased HMMs (HMMs where multiple hidden states
have identical output distributions) [7, 15, 23] hint that understanding learnability with respect to
properties of the observation matrix is a daunting task in itself, and is perhaps best studied separately
from understanding how properties of the transition matrix affect learning.
Our positive result on learnability (Theorem 2) depends on two natural graph-theoretic properties of
the transition matrix. We consider transition matrices which are i) sparse (hidden states have constant
degree) and ii) have small probability mass on cycles shorter than 10 logm n states?and show that
these HMMs can be learned efficiently using tensor decomposition and the method of moments,
given random observation matrices. The condition prohibiting short cycles might seem mysterious.
Intuitively, we need this condition to ensure that the Markov Chain visits a sufficient large portion of
the state space in a short interval of time, and in fact the condition stems from information-theoretic
considerations. We discuss these further in Sections 2.4 and 3.1. We also discuss how our results
relate to learning HMMs which capture long-term dependencies in their outputs, and introduce a new
notion of how well an HMM captures long-term dependencies. These are discussed in Section 5.
We also show new identifiability results for sparse HMMs. These results provide a finer picture of
identifiability than Huang et al. [14], as ours hold for sparse transition matrices which are not generic.
Technical contribution. To prove Theorem 2 we show that the Khatri-Rao product of dependent
random vectors is well-conditioned under certain conditions. Previously, Bhaskara et al. [6] showed
that the Khatri-Rao product of independent random vectors is well-conditioned to perform a smoothed
analysis of tensor decomposition, their techniques however do not extend to the dependent case. For
the dependent case, we show a similar result using a novel Markov chain coupling based argument
which relates the condition number to the best coupling of output distributions of two random walks
with disjoint starting distributions. The technique is outlined in Section 2.2.
Related work. Spectral methods for learning HMMs have been studied in Anandkumar et al.
[3], Bhaskara et al. [5], Allman et al. [1], Hsu et al. [13], but these results require m ? n. In Allman
et al. [1], the authors show that that HMMs are identifiable given moments of continuous observations
over a time interval of length N = 2? + 1 for some ? such that ? +m?1
? n. When m n
m?1
1/m
this requires ? = O(n
). Bhaskara et al. [5] give another bound on window size which requires
? = O(n/m). However, with a output alphabet of size m, specifying all moments in a N length
continuous time interval requires mN time and samples, and therefore all of these approaches lead to
exponential runtimes when m is constant with respect to n. Also relevant is the work by Anandkumar
et al. [4] on guarantees for learning certain latent variable models such as Gaussian mixtures in the
overcomplete setting through tensor decomposition. As mentioned earlier, the work closest to ours is
Huang et al. [14] who showed that generic HMMs are identifiable with ? = O(logm n), which gives
the first polynomial runtimes for the case when m is constant.
1
Parity with noise is information theoretically easy given observations over a window of length at least the
number of inputs to the parity. This is linear in the number of hidden states of the parity with noise HMM,
whereas Theorem 1 says that the sample complexity must be super polynomial for any polynomial sized window.
2
Outline. Section 2 introduces the notation and setup. It also provides examples and a high-level
overview of our proof approach. Section 3 states the learnability result, discusses our assumptions
and HMMs which satisfy these assumptions. Section 4 contains our identifiability results for sparse
HMMs. Section 5 discusses natural measures of long-term dependencies in HMMs. We conclude in
Section 6. Proof details are deferred to the Appendix.
2
Setup and preliminaries
In this section we first introduce the required notation, and then outline the method of moments
approach for parameter recovery. We also go over some examples to provide a better understanding
of the classes of HMMs we aim to learn, and give a high level proof strategy.
2.1
Notation and preliminaries
We will denote the output at time t by yt and the hidden state at time t by ht . Let the number of hidden
states be n and the number of observations be m. Assume that the output alphabet is {0, . . . , m ? 1}
without loss of generality. Let T be the transition matrix and O be the observation matrix of the
HMM, both of these are defined so that the columns add up to one. For any matrix A, we refer to
the ith column of A as Ai . T 0 is defined as the transition matrix of the time-reversed Markov chain,
but we do not assume reversibility and hence T may not equal T 0 . Let yij = yi , . . . , yj denote the
sequence of outputs from time i to time j. Let lij = li , . . . , lj refer to a string of length i + j ? 1
over the output alphabet, denoting a particular output sequence from time i to j. Define a bijective
mapping L which maps an output sequence l1? ? {0, . . . , m?1}? into an index L(l1? ) ? {1, . . . , m? }
and the associated inverse mapping L?1 .
Throughout the paper, we assume that the transition matrix T is ergodic, and hence has a stationary
distribution. We also assume that every hidden state has stationary probability at least 1/poly(n).
This is a necessary condition, as otherwise we might not even visit all states in poly(n) samples. We
also assume that the output process of the HMM is stationary. A stochastic process is stationary
if the distribution of any subset of random variables is invariant with respect to shifts in the time
? +T
?
?
?
?
index?that is, P[y??
= l??
] = P[y??
+T = l?? ] for any ?, T and string l?? . This is true if the
initial hidden state is chosen according to the stationary distribution.
Our results depend on the conditioning of the matrix T with respect to the `1 norm. We define
(1)
?min (T ) as the minimum `1 gain of the transition matrix T over all vectors x having unit `1 norm
(not just non-negative vectors x, for which the ratio would always be 1):
(1)
?min (T ) = minn
x?R
(1)
kT xk1
kxk1
(1)
?min (T ) is also a natural parameter to measure the long-term dependence of the HMM?if ?min (T )
is large then T preserves significant information about the distribution of hidden states at time 0 at a
future time t, for all initial distributions at time 0. We discuss this further in Section 5.
2.2
Method of moments for learning HMMs
Our algorithm for learning HMMs follows the method of moments based approach, outlined for
example in Anandkumar et al. [2] and Huang et al. [14]. In contrast to the more popular ExpectationMaximization (EM) approach which can suffer from slow convergence and local optima [21], the
method of moments approach ensures guaranteed recovery of the parameters under mild conditions.
More details about tensor decomposition and the method of moments approach to learning HMMs
can be found in Appendix A.
The method of moments approach to learning HMMs has two high-level steps. In the first step, we
write down a tensor of empirical moments of the data, such that the factors of the tensor correspond to
parameters of the underlying model. In the second step, we perform tensor decomposition to recover
the factors of the tensor?and then recover the parameters of the model from the factors. The key fact
that enables the second step is that tensors have a unique decomposition under mild conditions on
their factors, for example tensors have a unique decomposition if all the factors are full rank. The
uniqueness of tensor decomposition permits unique recovery of the parameters of the model.
?
We will learn the HMM using the moments of observation sequences y??
from time ?? to ? .
Since the output process is assumed to be stationary, the distribution of outputs is the same for
3
any contiguous time interval of the same length, and we use the interval ?? to ? in our setup for
convenience. We call the length of the observation sequences used for learning the window length
N = 2? + 1. Since the number of samples required to estimate moments over a window of length N
is mN , it is desirable to keep N small. Note that to ensure polynomial runtime and sample complexity
for the method of moments approach, the window length N must be O(logm n).
We will now define our moment tensor. Given moments over a window of length N = 2? + 1, we
?
?
can construct the third-order moment tensor M ? Rm ?m ?m using the mapping L from strings of
outputs to indices in the tensor:
?
?
M(L(l? ),L(l?? ),l0 ) = P[y??
= l??
].
1
?1
M is simply the tensor of the moments of the HMM over a window length N , and can be estimated
directly from data. We can write M as an outer product because of the Markov property:
M =A?B?C
?
where A ? Rm
state at time 0):
?n
, B ? Rm
?
?n
, C ? Rm?n are defined as follows (here h0 denotes the hidden
AL(l1? ),i = P[y1? = l1? | h0 = i]
??
??
BL(l?? ),i = P[y?1
= l?1
| h0 = i]
?1
Cl0 ,i = P[y0 = l, h0 = i]
T and O can be related in a simple manner to A, B and C. If we can decompose the tensor M into the
factors A, B and C, we can recover T and O from A, B and C. Kruskal?s condition [18] guarantees
that tensors have a unique decomposition whenever A and B are full rank and no two column of C
are the same. We refer the reader to Appendix A for more details, specifically Algorithm 1.
2.3
High-level proof strategy
As the transition and observation matrices can be recovered from the factors of the tensors, our goal
is to analyze the conditions under which the tensor decomposition step works provably. Note that the
factor matrix A is the likelihood of observing each sequence of observations conditioned on starting
at a given hidden state. We?ll refer to A as the likelihood matrix for this reason. B is the equivalent
matrix for the time-reversed Markov chain. If we show that A, B are full rank and no two columns of
C are the same, then the HMM can be learned provided the exact moments using the simultaneous
diagonalization algorithm, also known as Jennrich?s algorithm (see Algorithm 1). We show this
property for our identifiability results. For our learnability results, we show that the matrices A and
B are well-conditioned (have condition numbers polynomial in n), which implies learnability from
polynomial samples. This is the main technical contribution of the paper, and requires analyzing
the condition number of the Khatri-Rao product of dependent random vectors. Before sketching the
argument, we first introduce some notation. We can define A(t) as the likelihood matrix over t steps:
(t)
AL(lt ),i = P[y1t = l1t | h0 = i].
1
A(t) can be recursively written down as follows:
A(0) = OT, A(t) = (O A(t?1) )T
(1)
where A B, denotes the Khatri-Rao product of the matrices A and B. If A and B are two matrices
of size m1 ? r and m2 ? r then the Khatri-Rao product is a m1 m2 ? r matrix whose ith column is
the outer product Ai ? Bi flattened into a vector. Note that A(? ) is the same as A. We now sketch
our argument for showing that A(? ) is well-conditioned under appropriate conditions.
Coupling random walks to analyze the Khatri-Rao product. As mentioned in the introduction,
in this paper we are interested in the setting where the transition matrix is fixed but the observation
matrix is drawn at random. If we could draw fresh random matrices O at each time step of the
recursion in Eq. 1, then A would be well-conditioned by the smoothed analysis of the Khatri-Rao
product due to Bhaskara et al. [6]. However, our setting is significantly more difficult, as we do not
have access to fresh randomness at each time step, so the techniques of Bhaskara et al. [6] cannot be
applied here. As pointed out earlier, the condition number of A in this scenario depends crucially on
the transition matrix T , as A is not even full rank if T = I.
4
(a) Transition matrix is a cycle, or a
permutation on the hidden states.
(b) Transition matrix is a random walk on a
graph with small degree and no short cycles.
Figure 1: Examples of transition matrices which we can learn, refer to Section 2.4 and Section 3.2.
Instead, we analyze A by a coupling argument. To get some intuition for this, note that if A does not
have full rank, then there are two disjoint sets of columns of A whose linear combinations are equal,
and these combination weights can be used to setup the initial states of two random walks defined by
the transition matrix T which have the same output distribution for ? time steps. More generally, if
A is ill-conditioned then there are two random walks with disjoint starting states which have very
similar output distributions. We show that if two random walks have very similar output distributions
over ? time steps for a randomly chosen observation matrix O, then most of the probability mass in
(1)
these random walks can be coupled. On the other hand, if (?min (T ))? is sufficiently large, the total
variational distance between random walks starting at two different starting states must be at least
(1)
(?min (T ))? after ? time steps, and so there cannot be a good coupling, and A is well-conditioned.
We provide a sketch of the argument for a simple case in Appendix 1 before we prove Theorem 2.
2.4
Illustrative examples
We now provide a few simple examples which will illustrate some classes of HMMs we can and
cannot learn. We first provide an example of a class of simple HMMs which can be handled by our
results, but has non-generic transition matrices and hence does not fit into the framework of Huang
et al. [14]. Consider an HMM where the transition matrix is a permutation or cyclic shift on the
hidden states (see Fig. 1a). Our results imply that such HMMs are learnable in polynomial time from
polynomial samples if the output distributions of the hidden states are chosen at random. We will
try to provide some intuition about why an HMM with the transition matrix as in Fig. 1a should be
efficiently learnable. Let us consider the simple case when the the outputs are binary (so m = 2) and
each hidden state deterministically outputs a 0 or a 1, and is labeled by a 0 or a 1 accordingly. If
the labels are assigned at random, then with high probability the string of labels of any continuous
sequence of 2 log2 n hidden states in the cycle in Fig. 1a will be unique. This means that the output
distribution in a 2 log2 n time window is unique for every initial hidden state, and it can be shown
that this ensures that the moment tensor has a unique factorization. By showing that the output
distribution in a 2 log2 n time window is very different for different initial hidden states?in addition
to being unique?we can show that the factors of the moment tensor are well-conditioned, which
allows recovery with efficient sample complexity. As another slightly more complex example of
an HMM we can learn, Fig. 1b depicts an HMM whose transition matrix is a random walk on a
graph with small degree and no short cycles. Our learnability result can handle such HMMs having
structured transition matrices.
As an example of an HMM which cannot be learned in our framework, consider an HMM with
transition matrix T = I and binary observations (m = 2), see Fig. 2a. In this case, the probability of
an output sequence only depends on the total number of zeros or ones in the sequence. Therefore,
we only get t independent measurements from windows of length t, hence windows of length O(n)
instead of O(log2 n) are necessary for identifiability (also refer to Blischke [8] for more discussions
on this case). More generally, we prove in Proposition 1 that for small m a transition matrix composed
only of cycles of constant length (see Fig. 2b) requires the window length to be polynomial in n to
become identifiable.
Proposition 1. Consider an HMM on n hidden states and m observations with the transition
matrix
c
being a permutation composed of cycles of length c. Then windows of length O(n1/m ) are necessary
for the model to be identifiable, which is polynomial in n for constant c and m.
The root cause of the difficulty in learning HMMs having short cycles is that they do not visit a large
enough portion of the state space in O(logm n) steps, and hence moments over a O(logm n) time
5
(a) Transition matrix is the identity on
8 hidden states.
(b) Transition matrix is a union of 4 cycles,
each on 5 hidden states.
Figure 2: Examples of transition matrices which do not fit in our framework. Proposition 1 shows
that such HMMs where the transition matrix is composed of a union of cycles of constant length are
not even identifiable from short windows of length O(logm n)
window do not carry sufficient information for learning. Our results cannot handle such classes of
transition matrices, also see Section 3.1 for more discussion.
3
Learnability results for overcomplete HMMs
In this section, we state our learnability result, discuss the assumptions and provide examples of
HMMs which satisfy these assumptions. Our learnability results hold under the following conditions:
Assumptions: For fixed constants c1 , c2 , c3 > 1, the HMM satisfies the following properties for
some c > 0:
1. Transition matrix is well-conditioned: Both T and the transition matrix T 0 of the time
(1)
(1)
reversed Markov Chain are well-conditioned in the `1 -norm: ?min (T ), ?min (T 0 ) ? 1/mc/c1
2. Transition matrix does not have short cycles: For both T and T 0 , every state visits at least
10 logm n states in 15 logm n time except with probability ?1 ? 1/nc .
3. All hidden states have small ?degree?: There exists ?2 such that for every hidden state i, the
transition distributions Ti and Ti0 have cumulative mass at most ?2 on all but d states, with
d ? m1/c2 and ?2 ? 1/nc . Hence this is a soft ?degree? requirement.
4. Output distributions are random and have small support : There exists ?3 such that for every
hidden state i the output distribution Oi has cumulative mass at most ?3 on all but k outputs,
with k ? m1/c3 and ?3 ? 1/nc . Also, the output distribution Oi is drawn uniformly on
these k outputs.
The constants c1 , c2 , c3 are can be made explicit, for example, c1 = 20, c2 = 16 and c3 = 10 works.
Under these conditions, we show that HMMs can be learned using polynomially many samples:
Theorem 2. If an HMM satisfies the above conditions, then with high probability over the choice
of O, the parameters of the HMM are learnable to within additive error with observations over
windows of length 2? + 1, ? = 15 logm n, with the sample complexity poly(n, 1/).
Appendix C also states a corollary of Theorem 2 in terms of the minimum singular value ?min (T ) of
(1)
the matrix T , instead of ?min (T ). We discuss the conditions for Theorem 2 next, and subsequently
provide examples of HMMs which satisfy these conditions.
3.1
Discussion of the assumptions
1. Transition matrix is well-conditioned: Note that singular transition matrices might not even be
identifiable. Moreover, Mossel and Roch [20] showed that learning HMMs with singular transition
matrices is as hard as learning parity with noise, which is widely conjectured to be computationally
hard. Hence, it is necessary to exclude at least some classes of ill-conditioned transition matrices.
2. Transition matrix does not have short cycles: Due to Proposition 1, we know that a HMM might
not even be identifiable from short windows if it is composed of a union of short cycles, hence we
expect a similar condition for learning the HMM with polynomial samples; though there is a gap
between the upper and lower bounds in terms of the probability mass which is allowed on the short
cycles. We performed some simulations to understand how the length of cycles in the transition
matrix and the probability mass assigned to short cycles affects the condition number of the likelihood
matrix A; recall that the condition number of A determines the stability of the method of moments
6
Condition number of matrix A
Condition number of matrix A
100
80
60
40
Cycle length 2
Cycle length 4
Cycle length 8
20
0
0.1
0.2
0.3
Epsilon
0.4
0.5
(a) The conditioning becomes worse
when cycles are smaller or when more
probability mass is put on short cycles.
200
150
Degree 2
Degree 4
Degree 8
100
50
0
0.01
0.02
0.03
Epsilon
0.04
0.05
(b) The conditioning becomes worse as the
degree increases, and when more probabiltiy
mass is put on the dense part of T .
Figure 3: Experiments to study the effect of sparsity and short cycles on the learnability of HMMs.
The condition number of the likelihood matrix A determines the stability or sample complexity of the
method of moments approach. The condition numbers are averaged over 10 trials.
approach. We take the number of hidden states n = 128, and let P128 be a cycle on the n hidden
states (as in Fig. 1a). Let Pc be a union of short cycles of length c on the n states (refer to Fig. 2b
for an example). We take the transition matrix to be T = Pc + (1 ? )P128 for different values of
c and . Fig. 3a shows that the condition number of A becomes worse and hence learning requires
more samples if the cycles are shorter in length, and if more probability mass is assigned to the short
cycles, hinting that our conditions are perhaps not be too stringent.
3. All hidden states have a small degree: Condition 3 in Theorem 2 can be reinterpreted as saying
that the transition probabilities out of any hidden state must have mass at most 1/n1+c on any hidden
state except a set of d hidden states, for any c > 0. While this soft constraint is weaker than a hard
constraint on the degree, it natural to ask whether any sparsity is necessary to learn HMMs. As above,
we carry out simulations to understand how the degree affects the condition number of the likelihood
matrix A. We consider transition matrices on n = 128 hidden states which are a combination of a
dense part and a cycle. Define P128 to be a cycle as before. Define Gd as the adjacency matrix of a
directed regular graph with degree d. We take the transition matrix T = Gd + (1 ? d)P128 . Hence
the transition distribution of every hidden state has mass on a set of d neighbors, and the residual
probability mass is assigned to the permutation P128 . Fig. 3b shows that the condition number of
A becomes worse as the degree d becomes larger, and as more probability mass is assigned to
the dense part Gd of the transition matrix T , providing some weak evidence for the necessity of
Condition 3. Also, recall that Theorem 1 shows that HMMs where the transition matrix is a random
walk on an undirected regular graph with large degree (degree polynomial in n) cannot be learned
using polynomially many samples if m is a constant with
? respect to n. However, such graphs have
all eigenvalues except the first one to be less than O(1/ d), hence it is not clear if the hardness of
learning depends on the large degree itself or is only due to T being ill-conditioned. More concretely,
we pose the following open question:
Open question: Consider an HMM with a transition matrix T = (1 ? )P + U , where P is the
cyclic permutation on n hidden states (such as in Fig. 1a) and U is a random walk on a undirected,
regular graph with large degree (polynomial in n) and > 0 is a constant. Can this HMM be learned
using polynomial samples when m is small (constant) with respect to n? This example approximately
preserves ?min (T ) by the addition of the permutation, and hence the difficulty is only due to the
transition matrix having large degree.
4. Output distributions are random and have small support: As discussed in the introduction, if we
do not assume that the observation matrices are random, then even simple HMMs with a cycle or
permutation as the transition matrix might require long windows even to become identifiable, see Fig.
4. Hence some assumptions on the output distribution do seem necessary for learning the model from
short time windows, though our assumptions are probably not tight. For instance, the assumption that
the output distributions have a small support makes learning easier as it leads to the outputs being
more discriminative of the hidden states, but it is not clear that this is a necessary assumption. Ideally,
we would like to prove our learnability results under a smoothed model for O, where an adversary is
allowed to see the transition matrix T and pick any worst-case O, but random noise is then added to
7
the output distributions, which limits the power of the adversary. We believe our results should hold
under such a smoothed setting, but set this aside for future work.
Figure 4: Consider two HMMs with transition matrices being cycles on n = 16 states with binary
outputs, and outputs conditioned on the hidden states are deterministic. The states labeled as 0 always
emit a 0 and the states labeled as 1 always emit a 1. The two HMMs are not distinguishable from
windows of length less than 8. Hence with worst case O even simple HMMs like the cycle could
require long windows to even become identifiable.
3.2
Examples of transition matrices which satisfy our assumptions
We revisit the examples from Fig. 1a and Fig. 1b, showing that they satisfy our assumptions.
1. Transition matrices where the Markov Chain is a permutation: If the Markov chain is a permutation
with all cycles longer than 10 logm n then the transition matrix obeys all the conditions for Theorem
2. This is because all the singular values of a permutation are 1, the degree is 1 and all hidden states
visit 10 logm n different states in 15 logm n time steps.
2. Transition matrices which are random walks on graphs with small degree and large girth:
For directed graphs, Condition 2 can be equivalently stated as that the graph representation of the
transition matrix has a large girth (girth of a graph is defined as the length of its shortest cycle).
3. Transition matrices of factorial HMMs: Factorial HMMs [12] factor the latent state at any time into
D dimensions, each of which independently evolves according to a Markov process. For D = 2, this
is equivalent to saying that the hidden states are indexed by two labels (i, j) and if T1 and T2 represent
the transition matrices for the two dimensions, then P[(i1 , j1 ) ? (i2 , j2 )] = T1 (i2 , i1 )T2 (j2 , j1 ).
This naturally models settings where there are multiple latent concepts which evolve independently.
The following properties are easy to show:
1. If either of T1 or T2 visit N different states in 15 logm n time steps with probability (1 ? ?),
then T visits N different states in 15 logm n time steps with probability (1 ? ?).
2. ?min (T ) = ?min (T1 )?min (T2 )
3. If all hidden states in T1 and T2 have mass at most ? on all but d1 states and d2 states
respectively, then T has mass at most 2? on all but d1 d2 states.
Therefore, factorial HMMs are learnable with random O if the underlying processes obey conditions
similar to the assumptions for Theorem 2. If both T1 and T2 are well-conditioned and at least one of
them does not have short cycles, and either has small degree, then T is learnable with random O.
4
Identifiability of HMMs from short windows
As it is not obvious that some of the requirements for Theorem 2 are necessary, it is natural to attempt
to derive stronger results for just identifiability of HMMs having structured transition matrices. In this
section, we state our results for identifiability of HMMs from windows of size O(logm n). Huang
et al. [14] showed that all HMMs except those belonging to a measure zero set become identifiable
from windows of length 2? + 1 with ? = 8dlogm ne. However, the measure zero set itself might
possibly contain interesting classes of HMMs (see Fig. 1), for example sparse HMMs also belong to
a measure zero set. We refine the identifiability results in this section, and show that a natural sparsity
condition on the transition matrix guarantees identifiability from short windows. Given any transition
matrix T , we regard T as being supported by a set of indices S if the non-zero entries of T all lie in
S. We now state our result for identifiability of sparse HMMs.
Theorem 3. Let S be a set of indices which supports a permutation where all cycles have at least
2dlogm ne hidden states. Then the set T of all transition matrices with support S is identifiable from
windows of length 4dlogm ne + 1 for all observation matrices O except for a measure zero set of
transition matrices in T and observation matrices O.
8
We hypothesize that excluding a measure zero set of transition matrices in Theorem 3 should not be
necessary as long as the transition matrix is full rank, but are unable to show this. Note that our result
on identifiability is more flexible in allowing short cycles in transition matrices than Theorem 2, and
is closer to the lower bound on identifiability in Proposition 1.
We also strengthen the result of Huang et al. [14] for identifiability of generic HMMs. Huang
et al. [14] conjectured that windows of length 2dlogm ne + 1 are sufficient for generic HMMs to be
identifiable. The constant 2 is the information theoretic bound as an HMM on n hidden states and
m outputs has O(n2 + nm) independent parameters, and hence needs observations over a window
of size 2dlogm ne + 1 to be uniquely identifiable. Proposition 2 settles this conjecture, proving
the optimal window length requirement for generic HMMs to be identifiable. As the number of
possible outputs over a window of length t is mt , the size of the moment tensor in Section 2.2 is itself
exponential in the window length. Therefore even a factor of 2 improvement in the window length
requirement leads to a quadratic improvement in the sample and time complexity.
Proposition 2. The set of all HMMs is identifiable from observations over windows of length
2dlogm ne + 1 except for a measure zero set of transition matrices T and observation matrices O.
5
Discussion on long-term dependencies in HMMs
In this section, we discuss long-term dependencies in HMMs, and show how our results on overcomplete HMMs improve the understanding of how HMMs can capture long-term dependencies, both
(1)
with respect to the Markov chain and the outputs. Recall the definition of ?min (T ):
kT xk1
(1)
?min (T ) = minn
x?R
kxk1
(1)
We claim that if ?min (T ) is large, then the transition matrix preserves significant information about
the distribution of hidden states at time 0 at a future time t, for all initial distributions at time 0.
Consider any two distributions p0 and q0 at time 0. Let pt and qt be the distributions of the hidden
states at time t given that the distribution at time 0 is p0 and q0 respectively. Then the `1 distance
(1)
between pt and qt is kpt ? qt k1 ? (?min (T ))t kp0 ? q0 k1 , verifying our claim. It is interesting
to compare this notion with the mixing time of the transition matrix. Defining mixing time as
the time until the `1 distance between any two starting distributions is at most 1/2, it follows that
(1)
(1)
the mixing time ?mix ? 1/ log(1/?min (T )), therefore if ?min (T )) is large then the chain is slowly
(1)
mixing. However, the converse is not true??min (T ) might be small even if the chain never mixes,
for example if the graph is disconnected but the connected components mix very quickly. Therefore,
(1)
?min (T ) is possibly a better notion of the long-term dependence of the transition matrix, as it requires
that information is preserved about the past state ?in all directions?.
Another reasonable notion of the long-term dependence of the HMM is the long-term dependence in
the output process instead of in the hidden Markov chain, which is the utility of past observations
when making predictions about the distant future (given outputs y?? , . . . , y1 , y2 , . . . , yt , at time t
how far back do we need to remember about the past to make a good prediction about yt ?). This does
not depend in a simple way on the T and O matrices, but we do note that if the Markov chain is fast
mixing then the output process can certainly not have long-term dependencies. We also note that with
respect to long-term dependencies in the output process, the setting m n seems to be much more
interesting than when m is comparable to n. The reason is that in the small output alphabet setting
we only receive a small amount of information about the true hidden state at each step, and hence
longer windows are necessary to infer the hidden state and make a good prediction. We also refer the
reader to Kakade et al. [16] for related discussions on the memory of output processes of HMMs.
6
Conclusion and Future Work
The setting where the output alphabet m is much smaller than the number of hidden states n is
well-motivated in practice and seems to have several interesting theoretical questions about new
lower bounds and algorithms. Though some of our results are obtained in more restrictive conditions
than seems necessary, we hope the ideas and techniques pave the way for much sharper results in
this setting. Some open problems which we think might be particularly useful for improving our
understanding is relaxing the condition on the observation matrix being random to some structural
constraint on the observation matrix (such as on its Kruskal rank), and more thoroughly investigating
the requirement for the transition matrix being sparse and not having short cycles.
9
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10
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6,296 | 6,696 | GP CaKe: Effective brain connectivity with causal
kernels
Luca Ambrogioni
Radboud University
[email protected]
Max Hinne
Radboud University
[email protected]
Marcel A. J. van Gerven
Radboud University
[email protected]
Eric Maris
Radboud University
[email protected]
Abstract
A fundamental goal in network neuroscience is to understand how activity in one
brain region drives activity elsewhere, a process referred to as effective connectivity.
Here we propose to model this causal interaction using integro-differential equations and causal kernels that allow for a rich analysis of effective connectivity. The
approach combines the tractability and flexibility of autoregressive modeling with
the biophysical interpretability of dynamic causal modeling. The causal kernels are
learned nonparametrically using Gaussian process regression, yielding an efficient
framework for causal inference. We construct a novel class of causal covariance
functions that enforce the desired properties of the causal kernels, an approach
which we call GP CaKe. By construction, the model and its hyperparameters have
biophysical meaning and are therefore easily interpretable. We demonstrate the
efficacy of GP CaKe on a number of simulations and give an example of a realistic
application on magnetoencephalography (MEG) data.
1
Introduction
In recent years, substantial effort was dedicated to the study of the network properties of neural
systems, ranging from individual neurons to macroscopic brain areas. It has become commonplace to
describe the brain as a network that may be further understood by considering either its anatomical
(static) scaffolding, the functional dynamics that reside on top of that or the causal influence that
the network nodes exert on one another [1?3]. The latter is known as effective connectivity and has
inspired a surge of data analysis methods that can be used to estimate the information flow between
neural sources from their electrical or haemodynamic activity[2, 4]. In electrophysiology, the most
popular connectivity methods are variations on the autoregressive (AR) framework [5]. Specifically,
Granger causality (GC) and related methods, such as partial directed coherence and directed transfer
function, have been successfully applied to many kinds of neuroscientific data [6, 7]. These methods
can be either parametric or non-parametric, but are not based on a specific biophysical model [8, 9].
Consequently, the connectivity estimates obtained from these methods are only statistical in nature
and cannot be directly interpreted in terms of biophysical interactions [10]. This contrasts with the
framework of dynamic causal modeling (DCM), which allows for Bayesian inference (using Bayes
factors) with respect to biophysical models of interacting neuronal populations [11]. These models
are usually formulated in terms of either deterministic or stochastic differential equations, in which
the effective connectivity between neuronal populations depends on a series of scalar parameters
that specify the strength of the interactions and the conduction delays [12]. DCMs are usually
less flexible than AR models since they depend on an appropriate parametrization of the effective
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
connectivity kernel, which in turn depends on detailed prior biophysical knowledge or Bayesian
model comparison.
In this paper, we introduce a new method that is aimed to bridge the gap between biophysically
inspired models, such as DCM, and statistical models, such as AR, using the powerful tools of
Bayesian nonparametrics [13]. We model the interacting neuronal populations with a system of
stochastic integro-differential equations. In particular, the intrinsic dynamic of each population is
modeled using a linear differential operator while the effective connectivity between populations
is modeled using causal integral operators. The differential operators can account for a wide range
of dynamic behaviors, such as stochastic relaxation and stochastic oscillations. While this class of
models cannot account for non-linearities, it has the advantage of being analytically tractable. Using
the framework of Gaussian process (GP) regression, we can obtain the posterior distribution of the
effective connectivity kernel without specifying a predetermined parametric form. We call this new
effective connectivity method Gaussian process Causal Kernels (GP CaKe). The GP CaKe method
can be seen as a nonparametric extension of linear DCM for which the exact posterior distribution can
be obtained in closed-form without resorting to variational approximations. In this way, the method
combines the flexibility and statistical simplicity of AR modeling with the biophysical interpretability
of a linear DCM.
The paper is structured as follows. In Section 2 we describe the model for the activity of neuronal
populations and their driving interactions. In Section 3 we construct a Bayesian hierarchical model
that allows us to learn the causal interaction functions. Next, in Subsection 3.2, we show that these
causal kernels may be learned analytically using Gaussian process regression. Subsequently in
Section 4, we validate GP CaKe using a number of simulations and demonstrate its usefulness on
MEG data in Section 5. Finally, we discuss the wide array of possible extensions and applications of
the model in Section 6.
2
Neuronal dynamics
We model the activity of a neuronal population xj (t) using the stochastic differential equation
Dj xj (t) = Ij (t) + wj (t) ,
(1)
where Ij (t) is the total synaptic input coming from other neuronal populations and wj (t) is Gaussian
PP
dp
white noise with mean 0 and variance ? 2 . The differential operator Dj = ?0 + p=1 ?p dt
p specifies
the internal dynamic of the neuronal population. For example, oscillatory dynamic can be modeled
d
d2
2
using the damped harmonic operator DjH = dt
, where ?0 is the (undamped) peak
2 + ? dt + ?0
angular frequency and ? is the damping coefficient.
In Eq. 1, the term Ij (t) accounts for the effective connectivity between neuronal populations. Assuming that the interactions are linear and stationary over time, the most general form for Ij (t) is given
by a sum of convolutions:
N
X
Ij (t) =
ci?j ? xi (t) ,
(2)
i=1
where the function ci?j (t) is the causal kernel, modeling the effective connectivity from population i
to population j, and ? indicates the convolution operator. The causal kernel ci?j (t) gives a complete
characterization of the linear effective connectivity between the two neuronal populations, accounting
for the excitatory or inhibitory nature of the connection, the time delay, and the strength of the
interaction. Importantly, in order to preserve the causality of the system, we assume that ci?j (t) is
identically equal to zero for negative lags (t < 0).
Inserting Eq. 2 into Eq. 1, we obtain the following system of stochastic integro-differential equations:
Dj xj (t) =
N
X
ci?j ? xi (t) + wj (t),
j = 1...N ,
(3)
i=1
which fully characterizes the stochastic dynamic of a functional network consisting of N neuronal
populations.
2
3
The Bayesian model
We can frame the estimation of the effective connectivity between neuronal populations as a nonparametric Bayesian regression problem. In order to do this, we assign a GP prior distribution to the kernel
functions ci?j (t) for every presynaptic population i and postsynaptic population j. A stochastic function f (t) is said to follow a GP distribution when all its marginal distributions p(f (t1 ), . . . , f (tn )) are
distributed as a multivariate Gaussian [14]. Since these marginals are determined by their mean vector
and covariance matrix, the GP is fully specified by a mean and a covariance function, respectively
mf (t) = hf (t)i and Kf (t1 , t2 ) = h(f (t1 ) ? mf (t1 ))(f (t2 ) ? mf (t2 ))i. Using the results of the
previous subsection we can summarize the problem of Bayesian nonparametric effective connectivity
estimation in the following way:
ci?j (t) ? GP (0, K(t1 , t2 ))
wj (t) ? N (0, ? 2 )
Dj xj (t) =
N
X
(4)
(ci?j ? xi ) (t) + wj (t) ,
i=1
where expressions such as f (t) ? GP m(t), K(t1 , t2 ) mean that the stochastic process f (t) follows
a GP distribution with mean function m(t) and covariance function K(t1 , t2 ).
Our aim is to obtain the posterior distributions of the effective connectivity kernels given a set of
samples from all the neuronal processes. As a consequence of the time shift invariance, the system
of integro-differential equations becomes a system of decoupled linear algebraic equations in the
frequency domain. It is therefore convenient to rewrite the regression problem in the frequency
domain:
ci?j (?) ? CGP 0, K(?1 , ?2 )
wj (?) ? CN (0, ? 2 )
Pj (?)xj (?) =
N
X
(5)
xi (?)ci?j (?) + wj (?) ,
i=1
PP
where Pj (?) = p=0 ?p (?i?)p is a complex-valued polynomial since the application of a differential operator in the time domain is equivalent to multiplication with a polynomial in the frequency
domain. In the previous expression, CN (?, ?) denotes a circularly-symmetric complex normal
distribution with mean ? and variance ?, while CGP (m(t), K(?)) denotes a circularly-symmetric
complex valued GP with mean function m(?) and Hermitian covariance function K(?1 , ?2 ) [15].
Importantly, the complex valued Hermitian covariance function K(?1 , ?2 ) can be obtained from
K(t1 , t2 ) by taking the Fourier transform of both its arguments:
Z +? Z +?
(6)
K(?1 , ?2 ) =
e?i?1 t1 ?i?2 t2 K(t1 , t2 )dt1 dt2 .
??
3.1
??
Causal covariance functions
In order to be applicable for causal inference, the prior covariance function K(t1 , t2 ) must reflect
three basic assumptions about the connectivity kernel: I) temporal localization, II) causality and
III) smoothness. Since we perform the GP analysis in the frequency domain, we will work with
K(?1 , ?2 ), i.e. the double Fourier transform of the covariance function.
First, the connectivity kernel should be localized in time, as the range of plausible delays in axonal
communication between neuronal populations is bounded. In order to enforce this constraint, we
need a covariance function K(t1 , t2 ) that vanishes when either t1 or t2 becomes much larger than a
time constant ?. In the frequency domain, this temporal localization can be implemented by inducing
correlations between the Fourier coefficients of neighboring frequencies. In fact, local correlations in
the time domain are associated with a Fourier transform that vanishes for high values of ?. From
Fourier duality, this implies that local correlations in the frequency domain are associated with a
function that vanishes for high values of t. We model these spectral correlations using a squared
exponential covariance function:
KSE (?1 , ?2 ) = e??
(?2 ??1 )2
2
+its (?2 ??1 )
3
= e??
?2
2
+its ?
,
(7)
where ? = ?2 ? ?1 . Since we expect the connectivity to be highest after a minimal conduction delay
ts , we introduced a time shift factor its ? in the exponent that translates the peak of the variance from
0 to ts , which follows from the Fourier shift theorem. As this covariance function depends solely on
the difference between frequencies ?, it can be written (with a slight abuse of notation) as KSE (?).
Second, we want the connectivity kernel to be causal, meaning that information cannot propagate
back from the future. In order to enforce causality, we introduce a new family of covariance functions
that vanish when the lag t2 ? t1 is negative. In the frequency domain, a causal covariance function
can be obtained by adding an imaginary part to Eq. 7 that is equal to its Hilbert transform H [16].
Causal covariance functions are the Fourier dual of quadrature covariance functions, which define GP
distributions over the space of analytic functions, i.e. functions whose Fourier coefficients are zero
for all negative frequencies [15]. The causal covariance function is given by the following formula:
KC (?) = KSE (?) + iHKSE (?) .
(8)
Finally, as communication between neuronal populations is mediated by smooth biological processes
such as synaptic release of neurotransmitters and dendritic propagation of potentials, we want the
connectivity kernel to be a smooth function of the time lag. Smoothness in the time domain can be
imposed by discounting high frequencies. Here, we use the following discounting function:
f (?1 , ?2 ) = e??
2 +? 2
?1
2
2
.
(9)
This discounting function induces a process that is smooth (infinitely differentiable) and with time
scale equal to ? [14]. Our final covariance function is given by
K(?1 , ?2 ) = f (?1 , ?2 ) (KSE (?) + iHKSE (?)) .
(10)
Unfortunately, the temporal smoothing breaks the strict causality of the covariance function because
it introduces leakage from the positive lags to the negative lags. Nevertheless, the covariance function
closely approximates a causal covariance function when ? is not much bigger than ts .
3.2
Gaussian process regression
In order to explain how to obtain the posterior distribution of the causal kernel, we need to review some
basic results of nonparametric Bayesian regression and GP regression in particular. Nonparametric
Bayesian statistics deals with inference problems where the prior distribution has infinitely many
degrees of freedom [13]. We focus on the following nonparametric regression problem, where the aim
is to reconstruct a series of real-valued functions from a finite number of noisy mixed observations:
X
yt =
?i (t)fi (t) + wt ,
(11)
i
where yt is the t-th entry of the data vector y, fi (t) is an unknown latent function and wt is a
random variable that models the observation noise with diagonal covariance matrix D. The mixing
functions ?i (t) are assumed to be known and determine how the latent functions generate the data.
In nonparametric Bayesian regression, we specify prior probability distributions over the whole
(infinitely dimensional) space of functions fi (t). Specifically, in the GP regression framework this
distribution is chosen to be a zero-mean GP. In order to infer the value of the function f (t) at an
?
arbitrary set of target points T ? = {t?
1 , ..., tm }, we organize these values in the vector f with entries
?
fl = f (tl ). The posterior expected value of f , that we will denote as mfj |y , is given by
X
?1
mfj |y = Kf?j ?j
?i Kfi ?i + D
y,
(12)
i
where the covariance matrix Kf is defined by the entries [Kf ]uv = Kf (tu , tv ) and the crosscovariance matrix K?? is defined by the entries [Kf? ]uv = Kf (t?
u , tv ) [14]. The matrices ?i are
square and diagonal, with the entries [?i ]uu given by ?i (tu ).
It is easy to see that the problem defined by Eq. 5 has the exact same form as the generalized
regression problem given by Eq. 11, with ? as dependent variable. In particular, the weight functions
wj (?)
?2
?i (?) are given by Pxij(?)
(?) and the noise term Pj (?) has variance |Pj (?)|2 . Therefore, the expectation
of the posterior distributions p(ci?j (?)|{x1 (?h )}, . . . , {xN (?h )}) can be obtained in closed from
from Eq. 12.
4
4
Effective connectivity simulation study
We performed a simulation study to assess the performance of the GP CaKe approach in recovering
the connectivity kernel from a network of simulated sources. The neuronal time series xj (t) are
generated by discretizing a system of integro-differential equations, as expressed in Eq. 3. Time series
data was then generated for each of the sources using the Ornstein-Uhlenbeck process dynamic, i.e.
d
D(1) =
+?,
(13)
dt
where the positive parameter ? is the relaxation coefficient of the process. The bigger ? is, the faster
the process reverts to its mean (i.e. zero) after a perturbation. The discretization of this dynamic is
equivalent to a first order autoregressive process. As ground truth effective connectivity, we used
functions of the form
?
(14)
ci?j (? ) = ai?j ? e? s ,
where ? is a (non-negative) time lag, ai?j is the connectivity strength from i to j and s is the
connectivity time scale.
In order to recover the connectivity kernels ci?j (t) we first need to estimate the differential operator
D(1) . For simplicity, we estimated the parameters of the differential operator by maximizing the
univariate marginal likelihood of each individual source. This procedure requires that the variance
of the structured input from the other neuronal populations is smaller than the variance of the
unstructured white noise input so that the estimation of the intrinsic dynamic is not too much affected
by the coupling.
Since most commonly used effective connectivity measures (e.g. Granger causality, partial directed
coherence, directed transfer function) are obtained from fitted vector autoregression (VAR) coefficients, we use VAR as a comparison method. Since the least-squares solution for the VAR coefficients
is not regularized, we also compare with a ridge regularized VAR model, whose penalty term is
learned using cross-validation on separately generated training data. This comparison is particularly
natural since our connectivity kernel is the continuous-time equivalent of the lagged AR coefficients
between two time series.
4.1
Recovery of the effective connectivity kernels
We explore the effects of different parameter values to demonstrate the intuitiveness of the kernel
parameters. Whenever a parameter is not specifically adjusted, we use the following default values:
noise level ? = 0.05, temporal smoothing ? = 0.15 and temporal localization ? = ?. Furthermore,
we set ts = 0.05 throughout.
Figure 1 illustrates connectivity kernels recovered by GP CaKe. These kernels have a connection
strength of ai?j = 5.0 if i feeds into j and ai?j = 0 otherwise. This applies to both the two node
and the three node network. As these kernels show, our method recovers the desired shape as well
as the magnitude of the effective connectivity for both connected and disconnected edges. At the
same time, Fig. 1B demonstrates that the indirect pathway through two connections does not lead to a
non-zero estimated kernel. Note furthermore that the kernels become non-zero after the zero-lag mark
(indicated by the dashed lines), demonstrating that there is no significant anti-causal information
leakage.
The effects of the different kernel parameter settings are shown in Fig. 2A, where again the method is
estimating connectivity for a two node network with one active connection, with ai?j = 5.0. We
show the mean squared error (MSE) as well as the correlation between the ground truth effective
connectivity and the estimates obtained using our method. We do this for different values of the
temporal smoothing, the noise level and the temporal localization parameters. Figure 2B shows
the estimated kernels that correspond to these settings. As to be expected, underestimating the
temporal smoothness results in increased variance due to the lack of regularization. On the other hand,
overestimating the smoothness results in a highly biased estimate as well as anti-causal information
leakage. Overestimating the noise level does not induce anti-causal information leakage but leads to
substantial bias. Finally, overestimating the temporal localization leads to an underestimation of the
duration of the causal influence.
Figure 3 shows a quantitative comparison between GP CaKe and the (regularized and unregularized)
VAR model for the networks shown in Fig. 1A and Fig. 1B. The connection strength ai?j was
5
A
B
1
2
1
2
Ground truth
GP CaKe
3
0 lag
Figure 1: Example of estimated connectivity. A. The estimated connectivity kernels for two connections: one present (2 ? 1) and one absent (1 ? 2). B. A three-node network in which node 1 feeds
into node 2 and node 2 feeds into node 3. The disconnected edge from 1 to 3 is correctly estimated,
as the estimated kernel is approximately zero. For visual clarity, estimated connectivity kernels for
other absent connections (2 ? 1, 3 ? 2 and 3 ?1) are omitted in the second panel. The shaded
areas indicate the 95% posterior density interval over 200 trials.
varied to study its effect on the kernel estimation. It is clear that GP CaKe greatly outperforms both
VAR models and that ridge regularization is beneficial for the VAR approach. Note that, when the
connection strength is low, the MSE is actually smallest for the fully disconnected model. Conversely,
both GP CaKe and VAR always outperform the disconnected estimate with respect to the correlation
measure.
5
Brain connectivity
In this section we investigate the effective connectivity structure of a network of cortical sources. In
particular, we focus on sources characterized by alpha oscillations (8?12Hz), the dominant rhythm in
MEG recordings. The participant was asked to watch one-minute long video clips selected from an
American television series. During these blocks the participant was instructed to fixate on a cross
in the center of the screen. At the onset of each block a visually presented message instructed the
participant to pay attention to either the auditory or the visual stream. The experiment also included a
so-called ?resting state? condition in which the participant was instructed to fixate on a cross in the
center of a black screen. Brain activity was recorded using a 275 channels axial MEG system.
The GP CaKe method can be applied to a set of signals whose intrinsic dynamic can be characterized
by stochastic differential equations. Raw MEG measurements can be seen as a mixture of dynamical
signals, each characterized by a different intrinsic dynamic. Therefore, in order to apply the method
on MEG data, we need to isolate a set of dynamic components. We extracted a series of unmixed
neural sources by applying independent component analysis (ICA) on the sensor recordings. These
components were chosen to have a clear dipolar pattern, the signature of a localized cortical source.
These local sources have a dynamic that can be well approximated with a linear mixture of linear
stochastic differential equations [17]. We used the recently introduced temporal GP decomposition
in order to decompose the components? time series into a series of dynamic components [17]. In
particular, for each ICA source we independently extracted the alpha oscillation component, which
d2
d
2
we modeled with a damped harmonic oscillator: DjH = dt
2 + ? dt + ?0 . Note that the temporal GP
decomposition automatically estimates the parameters ? and ?0 through a non-linear least-squares
procedure [17].
We computed the effective connectivity between the sources that corresponded to occipital, parietal
and left- and right auditory cortices (see Fig. 4A) using GP CaKe with the following parameter
settings: temporal smoothing ? = 0.01, temporal shift ts = 0.004, temporal localization ? = 8?
and noise level ? = 0.05. To estimate the causal structure of the network, we performed a z-test
on the maximum values of the kernels for each of the three conditions. The results were corrected
6
Temporal smoothing
1.0
? = 0.10
? = 1.00
? = 10.00
1.0
3
2
0.2
0 lag
0.4
MSE
Correlation
5
4
0.0
1
10
1.0
Noise level
-1
?
0
10
1
10
0.0
0.8
Correlation
MSE
0.8
0
0.6
0.4
0.4
0.0
-2
10
-1
10
?
0
10
1
10
Temporal localization
1.0
0.0
1.0
2.0
0.0
1.0
2.0
0.0
1.0
? = 0.01
? = 0.10
? = 1.00
? = 10.00
?=?
? = 2?
? = 3?
? = 4?
2.0
1.0
0.2
0.2
2.0
MSE
0.6
1.0
GP CaKe
0.0
-2
10
0.0
0
0.30
0.8
Correlation
1.0
0.20
MSE
0.6
0.4
0.0
0.10
0.2
0.0
? = 0.01
6
0.6
Correlation
B
7
0.8
Ground truth
A
?
2?
?
3?
0.0
4?
1.0
2.0
Time lag (s)
0.0
1.0
Time lag (s)
2.0
0.0
1.0
Time lag (s)
2.0
0.0
1.0
Time lag (s)
2.0
Figure 2: The effect of the the temporal localization, smoothness and noise level parameters on a
present connection. A. The correlation and mean squared error between the ground truth connectivity
kernel and the estimation by GP CaKe. B. The shapes of the estimated kernels as determined by the
indicated parameter. Default values for the parameters that remain fixed are ? = 0.05, ? = 0.15 and
? = ?. The dashed line indicates the zero-lag moment at which point the causal effect deviates from
zero. The shaded areas indicate the 95% posterior density interval over 200 trials.
B
Two-node network
0.2
0.0
?0.2
1.0
5.0
10.0
Connection weight
100
10-1
10-2
108
0.2
101
Correlation
Mean squared error
Correlation
0.4
Three-node network
0.3
102
Mean squared error
A
0.1
0.0
?0.1
1.0
5.0
?0.2
10.0
Connection weight
GP CaKe
1.0
2.5
5.0
Connection weight
VAR, Ridge
VAR
106
104
102
100
10-2
1.0
2.5
5.0
Connection weight
Baseline
Figure 3: The performance of the recovery of the effective connectivity kernels in terms of the
correlation and mean squared error between the actual and the recovered kernel. Left column: results
for the two node graph shown in Fig. 1A. Right column: results for the three node graph shown in
Fig. 1B. The dashed line indicates the baseline that estimates all node pairs as disconnected.
for multiple comparisons using FDR correction with ? = 0.05. The resulting structure is shown
in Fig. 4A, with the corresponding causal kernels in Fig. 4B. The three conditions are clearly
distinguishable from their estimated connectivity structure. For example, during the auditory attention
condition, alpha band causal influence from parietal to occipital cortex is suppressed relative to the
other conditions. Furthermore, a number of connections (i.e. right to left auditory cortex, as well as
both auditory cortices to occipital cortex) are only present during the resting state.
7
B
R, V, A
0
?1
?2
?3
R.A.
audio
2
1
0
?1
?2
?3
1e?18
2
1
0
?1
?2
?3
video
2
1
0
?1
?2
?3
1e?18
rest
Par. cortex
A
R, V, A
,A
,V
,A
,V
R
R
V
R
R
,A
,V
R
L.A.
V
L. auditory cortex R. auditory cortex
1
1e?18
R
Parietal cortex
1e?18
2
onset
Par.
R, V
R. aud. cortex L. aud. cortex
Occ.
Occipital cortex
Occ. cortex
A
0.00
0.04
0.08
0.12
Time lag (s)
0.00
0.04
0.08
0.12
Time lag (s)
0.00
0.04
0.08
0.12
Time lag (s)
Time lag (s)
Figure 4: Effective connectivity using MEG for three conditions: I. resting state (R), II. attention to
video stream (V) and III. attention to audio stream (A). Shown are the connections between occipital
cortex, parietal cortex and left and right auditory cortices. A. The binary network for each of the
three conditions. B. The kernels for each of the connections. Note that the magnitude of the kernels
depends on the noise level ?, and as the true strength is unknown, this is in arbitrary units.
6
Discussion
We introduced a new effective connectivity method based on GP regression and integro-differential
dynamical systems, referred to as GP CaKe. GP CaKe can be seen as a nonparametric extension of
DCM [11] where the posterior distribution over the effective connectivity kernel can be obtained in
closed form. In order to regularize the estimation, we introduced a new family of causal covariance
functions that encode three basic assumptions about the effective connectivity kernel: (1) temporal
localization, (2) causality, and (3) temporal smoothness. The resulting estimated kernels reflect the
time-modulated causal influence that one region exerts on another. Using simulations, we showed
that GP CaKe produces effective connectivity estimates that are orders of magnitude more accurate
than those obtained using (regularized) multivariate autoregression. Furthermore, using MEG data,
we showed that GP CaKe is able to uncover interesting patterns of effective connectivity between
different brain regions, modulated by cognitive state.
The strategy for selecting the hyperparameters of the GP CaKe model depends on the specific study.
If they are hand-chosen they should be set in a conservative manner. For example, the temporal
localization should be longer than the highest biologically meaningful conduction delay. Analogously,
the smoothing parameter should be smaller than the time scale of the system of interest. In ideal
cases, such as for the analysis of the subthreshold postsynaptic response of the cellular membrane,
these values can be reasonably obtained from biophysical models. When prior knowledge is not
available, several off-the-shelf Bayesian hyperparameter selection or marginalization techniques can
be applied to GP CaKe directly since both the marginal likelihood and its gradient are available in
closed-form. In this paper, instead of proposing a particular hyper-parameter selection technique, we
decided to focus our exposition on the interpretability of the hyperparameters. In fact, biophysical
interpretability can help neuroscientists construct informed hyperprior distributions.
Despite its high performance, the current version of the GP CaKe method has some limitations.
First, the method can only be used on signals whose intrinsic dynamics are well approximated
by linear stochastic differential equations. Real-world neural recordings are often a mixture of
several independent dynamic components. In this case the signal needs to be preprocessed using
a dynamic decomposition technique [17]. The second limitation is that the intrinsic dynamics are
currently estimated from the univariate signals. This procedure can lead to biases when the neuronal
populations are strongly coupled. Therefore, future developments should focus on the integration of
dynamic decomposition with connectivity estimation within an overarching Bayesian model.
The model can be extended in several directions. First, the causal structure of the neural dynamical
system can be constrained using structural information in a hierarchical Bayesian model. Here,
structural connectivity may be provided as an a priori constraint, for example derived from diffusionweighted MRI [18], or learned from the functional data simultaneously [19]. This allows the model
to automatically remove connections that do not reflect a causal interaction, thereby regularizing
8
the estimation. Alternatively, the anatomical constraints on causal interactions may be integrated
into a spatiotemporal model of the brain cortex by using partial integro-differential neural field
equations [20] and spatiotemporal causal kernels. In addition, the nonparametric modeling of the
causal kernel can be integrated into a more complex and biophysically realistic model where the
differential equations are not assumed to be linear [12] or where the observed time series data are
filtered through a haemodynamic [21] or calcium impulse response function [22].
Finally, while our model explicitly refers to neuronal populations, we note that the applicability of
the GP CaKe framework is in no way limited to neuroscience and may also be relevant for fields such
as econometrics and computational biology.
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6,297 | 6,697 | Decoupling ?when to update? from ?how to update?
Eran Malach
School of Computer Science
The Hebrew University, Israel
[email protected]
Shai Shalev-Shwartz
School of Computer Science
The Hebrew University, Israel
[email protected]
Abstract
Deep learning requires data. A useful approach to obtain data is to be creative and
mine data from various sources, that were created for different purposes. Unfortunately, this approach often leads to noisy labels. In this paper, we propose a meta
algorithm for tackling the noisy labels problem. The key idea is to decouple ?when
to update? from ?how to update?. We demonstrate the effectiveness of our algorithm by mining data for gender classification by combining the Labeled Faces
in the Wild (LFW) face recognition dataset with a textual genderizing service,
which leads to a noisy dataset. While our approach is very simple to implement,
it leads to state-of-the-art results. We analyze some convergence properties of the
proposed algorithm.
1
Introduction
In recent years, deep learning achieves state-of-the-art results in various different tasks, however,
neural networks are mostly trained using supervised learning, where a massive amount of labeled
data is required. While collecting unlabeled data is relatively easy given the amount of data available
on the web, providing accurate labeling is usually an expensive task. In order to overcome this
problem, data science becomes an art of extracting labels out of thin air. Some popular approaches
to labeling are crowdsourcing, where the labeling is not done by experts, and mining available
meta-data, such as text that is linked to an image in a webpage. Unfortunately, this gives rise to a
problem of abundant noisy labels - labels may often be corrupted [19], which might deteriorate the
performance of neural-networks [12].
Let us start with an intuitive explanation as to why noisy labels are problematic. Common neural
network optimization algorithms start with a random guess of what the classifier should be, and
then iteratively update the classifier based on stochastically sampled examples from a given dataset,
optimizing a given loss function such as the hinge loss or the logistic loss. In this process, wrong
predictions lead to an update of the classifier that would hopefully result in better classification
performance. While at the beginning of the training process the predictions are likely to be wrong,
as the classifier improves it will fail on less and less examples, thus making fewer and fewer updates.
On the other hand, in the presence of noisy labels, as the classifier improves the effect of the noise
increases - the classifier may give correct predictions, but will still have to update due to wrong
labeling. Thus, in an advanced stage of the training process the majority of the updates may actually
be due to wrongly labeled examples, and therefore will not allow the classifier to further improve.
To tackle this problem, we propose to decouple the decision of ?when to update? from the decision
of ?how to update?. As mentioned before, in the presence of noisy labels, if we update only when
the classifier?s prediction differs from the available label, then at the end of the optimization process,
these few updates will probably be mainly due to noisy labels. We would therefore like a different
update criterion, that would let us decide whether it is worthy to update the classifier based on a
given example. We would like to preserve the behavior of performing many updates at the beginning of the training process but only a few updates when we approach convergence. To do so, we
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
suggest to train two predictors, and perform update steps only in case of disagreement between them.
This way, when the predictors get better, the ?area? of their disagreement gets smaller, and updates
are performed only on examples that lie in the disagreement area, therefore preserving the desired
behavior of the standard optimization process. On the other hand, since we do not perform an update based on disagreement with the label (which may be due to a problem in the label rather than
a problem in the predictor), this method keeps the effective amount of noisy labels seen throughout
the training process at a constant rate.
The idea of deciding ?when to update? based on a disagreement between classifiers is closely related
to approaches for active learning and selective sampling - a setup in which the learner does not have
unlimited access to labeled examples, but rather has to query for each instance?s label, provided at
a given cost (see for example [34]). Specifically, the well known query-by-committee algorithm
maintains a version space of hypotheses and at each iteration, decides whether to query the label of
a given instance by sampling two hypotheses uniformly at random from the version space [35, 14].
Naturally, maintaining the version space of deep networks seems to be intractable. Our algorithm
maintains only two deep networks. The difference between them stems from the random initialization. Therefore, unlike the original query-by-committee algorithm, that samples from the version
space at every iteration, we sample from the original hypotheses class only once (at the initialization), and from there on, we update these two hypotheses using the backpropagation rule, when
they disagree on the label. To the best of our knowledge, this algorithm was not proposed/analyzed
previously, not in the active learning literature and especially not as a method for dealing with noisy
labels.
To show that this method indeed improves the robustness of deep learning to noisy labels, we conduct an experiment that aims to study a real-world scenario of acquiring noisy labels for a given
dataset. We consider the task of gender classification based on images. We did not have a dedicated
dataset for this task. Instead, we relied on the Labeled Faces in the Wild (LFW) dataset, which contains images of different people along with their names, but with no information about their gender.
To find the gender for each image, we use an online service to match a gender to a given name (as is
suggested by [25]), a method which is naturally prone to noisy labels (due to unisex names). Applying our algorithm to an existing neural network architecture reduces the effect of the noisy labels,
achieving better results than similar available approaches, when tested on a clean subset of the data.
We also performed a controlled experiment, in which the base algorithm is the perceptron, and show
that using our approach leads to a noise resilient algorithm, which can handle an extremely high
label noise rates of up to 40%. The controlled experiments are detailed in Appendix B.
In order to provide theoretical guarantees for our meta algorithm, we need to tackle two questions:
1. does this algorithm converge? and if so, how quickly? and 2. does it converge to an optimum? We
give a positive answer to the first question, when the base algorithm is the perceptron and the noise
is label flip with a constant probability. Specifically, we prove that the expected number of iterations
required by the resulting algorithm equals (up to a constant factor) to that of the perceptron in the
noise-free setting. As for the second question, clearly, the convergence depends on the initialization
of the two predictors. For example, if we initialize the two predictors to be the same predictor, the
algorithm will not perform any updates. Furthermore, we derive lower bounds on the quality of the
solution even if we initialize the two predictors at random. In particular, we show that for some
distributions, the algorithm?s error will be bounded away from zero, even in the case of linearly
separable data. This raises the question of whether a better initialization procedure may be helpful.
Indeed, we show that for the same distribution mentioned above, even if we add random label noise,
if we initialize the predictors by performing few vanilla perceptron iterations, then the algorithm
performs much better. Despite this worst case pessimism, we show that empirically, when working
with natural data, the algorithm converges to a good solution. We leave a formal investigation of
distribution dependent upper bounds to future work.
2
Related Work
The effects of noisy labels was vastly studied in many different learning algorithms (see for example
the survey in [13]), and various solutions to this problem have been proposed, some of them with
theoretically provable bounds, including methods like statistical queries, boosting, bagging and more
[21, 26, 7, 8, 29, 31, 23, 27, 3]. Our focus in this paper is on the problem of noisy labels in the context
of deep learning. Recently, there have been several works aiming at improving the resilience of deep
2
learning to noisy labels. To the best of our knowledge, there are four main approaches. The first
changes the loss function. The second adds a layer that tries to mimic the noise behavior. The third
groups examples into buckets. The fourth tries to clean the data as a preprocessing step. Beyond
these approaches, there are methods that assume a small clean data set and another large, noisy, or
even unlabeled, data set [30, 6, 38, 1]. We now list some specific algorithms from these families.
[33] proposed to change the cross entropy loss function by adding a regularization term that takes
into account the current prediction of the network. This method is inspired by a technique called
minimum entropy regularization, detailed in [17, 16]. It was also found to be effective by [12],
which suggested a further improvement of this method by effectively increasing the weight of the
regularization term during the training procedure.
[28] suggested to use a probabilistic model that models the conditional probability of seeing a wrong
label, where the correct label is a latent variable of the model. While [28] assume that the probability
of label-flips between classes is known in advance, a follow-up work by [36] extends this method
to a case were these probabilities are unknown. An improved method, that takes into account the
fact that some instances might be more likely to have a wrong label, has been proposed recently
in [15]. In particular, they add another softmax layer to the network, that can use the output of
the last hidden layer of the network in order to predict the probability of the label being flipped.
Unfortunately, their method involves optimizing the biases of the additional softmax layer by first
training it on a simpler setup (without using the last hidden layer), which implies two-phase training
that further complicates the optimization process. It is worth noting that there are some other works
that suggest methods that are very similar to [36, 15], with a slightly different objective or training
method [5, 20], or otherwise suggest a complicated process which involves estimation of the classdependent noise probabilities [32]. Another method from the same family is the one described in
[37], who suggests to differentiate between ?confusing? noise, where some features of the example
make it hard to label, or otherwise a completely random label noise, where the mislabeling has no
clear reason.
[39] suggested to train the network to predict labels on a randomly selected group of images from
the same class, instead of classifying each image individually. In their method, a group of images
is fed as an input to the network, which merges their inner representation in a deeper level of the
network, along with an attention model added to each image, and producing a single prediction.
Therefore, noisy labels may appear in groups with correctly labeled examples, thus diminishing
their impact. The final setup is rather complicated, involving many hyper-parameters, rather than
providing a simple plug-and-play solution to make an existing architecture robust to noisy labels.
From the family of preprocessing methods, we mention [4, 10], that try to eliminate instances that are
suspected to be mislabeled. Our method shares a similar motivation of disregarding contaminated
instances, but without the cost of complicating the training process by a preprocessing phase.
In our experiment we test the performance of our method against methods that are as simple as
training a vanilla version of neural network. In particular, from the family of modified loss function
we chose the two variants of the regularized cross entropy loss suggested by [33] (soft and hard
bootsrapping). From the family of adding a layer that models the noise, we chose to compare to one
of the models suggested in [15] (which is very similar to the model proposed by [36]), because this
model does not require any assumptions or complication of the training process. We find that our
method outperformed all of these competing methods, while being extremely simple to implement.
Finally, as mentioned before, our ?when to update? rule is closely related to approaches for active
learning and selective sampling, and in particular to the query-by-committee algorithm. In [14] a
thorough analysis is provided for various base algorithms implementing the query-by-committee
update rule, and particularly they analyze the perceptron base algorithm under some strong distributional assumptions. In other works, an ensemble of neural networks is trained in an active learning
setup to improve the generalization of neural networks [11, 2, 22]. Our method could be seen as
a simplified member of ensemble methods. As mentioned before, our motivation is very different
than the active learning scenario, since our main goal is dealing with noisy labels, rather than trying
to reduce the number of label queries. To the best of our knowledge, the algorithm we propose was
not used or analyzed in the past for the purpose of dealing with noisy labels in deep learning.
3
3
METHOD
As mentioned before, to tackle the problem of noisy labels, we suggest to change the update rule
commonly used in deep learning optimization algorithms in order to decouple the decision of ?when
to update? from ?how to update?. In our approach, the decision of ?when to update? does not depend
on the label. Instead, it depends on a disagreement between two different networks. This method
could be generally thought of as a meta-algorithm that uses two base classifiers, performing updates
according to a base learning algorithm, but only on examples for which there is a disagreement
between the two classifiers.
To put this formally, let X be an instance space and Y be the label space, and assume we sample
? over X ? Y, with possibly noisy labels. We wish to train a classifier
examples from a distribution D
h, coming from a hypothesis class H. We rely on an update rule, U , that updates h based on its
current value as well as a mini-batch of b examples. The meta algorithm receives as input a pair of
two classifiers, h1 , h2 2 H, the update rule, U , and a mini batch size, b. A pseudo-code is given in
Algorithm 1.
Note that we do not specify how to initialize the two base classifiers, h1 , h2 . When using deep
learning as the base algorithm, the easiest approach is maybe to perform a random initialization.
Another approach is to first train the two classifiers while following the regular ?when to update?
rule (which is based on the label y), possibly training each classifier on a different subset of the data,
and switching to the suggested update rule only in an advanced stage of the training process. We
later show that the second approach is preferable.
At the end of the optimization process, we can simply return one of the trained classifiers. If a
small accurately labeled test data is available, we can choose to return the classifier with the better
accuracy on the clean test data.
Algorithm 1 Update by Disagreement
input:
an update rule U
batch size b
two initial predictors h1 , h2 2 H
for t = 1, 2, . . . , N do
?b
draw mini-batch (x1 , y1 ), . . . , (xb , yb ) ? D
let S = {(xi , yi ) : h1 (xi ) 6= h2 (xi )}
h1
U (h1 , S)
h2
U (h2 , S)
end for
4
Theoretical analysis
Since a convergence analysis for deep learning is beyond our reach even in the noise-free setting,
we focus on analyzing properties of our algorithm for linearly separable data, which is corrupted by
random label noise, and while using the perceptron as a base algorithm.
Let X = {x 2 Rd : kxk ? 1}, Y = {?1}, and let D be a probability distribution over X ? Y,
such that there exists w? for which D({(x, y) : yhw? , xi < 1}) = 0. The distribution we observe,
? is a noisy version of D. Specifically, to sample (x, y?) ? D
? one should sample (x, y) ? D
denoted D,
and output (x, y) with probability 1 ? and (x, y) with probability ?. Here, ? is in [0, 1/2).
Finally, let H be the class of linear classifiers, namely, H = {x 7! sign(hw, xi) : w 2 Rd }. We
use the perceptron?s update rule with mini-batch size of 1. That is, given the classifier wt 2 Rd , the
update on example (xt , yt ) 2 X ? Y is: wt+1 = U (wt , (xt , yt )) := wt + yt xt .
As mentioned in the introduction, to provide a full theoretical analysis of this algorithm, we need to
account for two questions:
1. does this algorithm converge? and if so, how quickly?
2. does it converge to an optimum?
4
Theorem 1 below provides a positive answer for the first question. It shows that the number of
updates of our algorithm is only larger by a constant factor (that depends on the initial vectors and
the amount of noise) relatively to the bound for the vanilla perceptron in the noise-less case.
Theorem 1 Suppose that the ?Update by Disagreement? algorithm is run on a sequence of random
? and with initial vectors w(1) , w(2) . Denote K = maxi kw(i) k. Let T be the
N examples from D,
0
0
0
number of updates performed by the ?Update by Disagreement? algorithm.
? 2
?
Then, E[T ] ? 3(1(4 K+1)
2?)2 kw k where the expectation is w.r.t. the randomness of sampling from D.
Proof It will be more convenient to rewrite the algorithm as follows. We perform N iterations,
(i)
(i)
where at iteration t we receive (xt , y?t ), and update wt+1 = wt + ?t y?t xt , where
(
(1)
(2)
1 if sign(hwt , xt i) 6= sign(hwt , xt i)
?t =
0 otherwise
Observe that we can write y?t = ?t yt , where (xt , yt ) ? D, and ?t is a random variables with
?
P[?t = 1] = 1 ? and P[?t = 1] = ?. We
Palso use the notation vt = yt hw , xt i and v?t = ?t vt .
Our goal is to upper bound T? := E[T ] = E[ t ?t ].
We start with showing that
E
"
N
X
?t v?t
t=1
#
(1
(1)
2?)T
Indeed, since ?t is independent of ?t and vt , we get that:
E[?t v?t ] = E[?t ?t vt ] = E[?t ] ? E[?t vt ] = (1
where in the last inequality we used the fact that vt
Summing over t we obtain that Equation 1 holds.
Next, we show that for i 2 {1, 2},
(i)
kwt k2
?
(i)
kw0 k2
(1)
+
N
X
2?) E[?t vt ]
1 with probability 1 and ?t is non-negative.
(2)
?t (2kw0
t=1
(2)
2?) E[?t ]
(1
(1)
(2)
w0 k + 1)
(1)
(2)
(1)
(2)
Indeed, since the update of wt+1 and wt+1 is identical, we have that kwt+1 wt+1 k = kw0
w0 k
(1)
(2)
for every t. Now, whenever ?t = 1 we have that either yt hwt 1 , xt i ? 0 or yt hwt 1 , xt i ? 0.
(1)
Assume w.l.o.g. that yt hwt 1 , xt i ? 0. Then,
(1)
Second,
(1)
1
kwt k2 = kwt
(2)
(1) 2
1k
+ yt xt k2 = kwt
(1)
1 , xt i
+ 2yt hwt
(1) 2
1k
+ kxt k2 ? kwt
(2)
(2) 2
(2)
2
2
1 + yt xt k = kwt 1 k + 2yt hwt 1 , xt i + kxt k
(2)
(2)
(1)
kwt 1 k2 + 2yt hwt 1 wt 1 , xt i + kxt k2
(2)
(2)
(1)
(2)
(2)
kwt 1 k2 + 2 kwt 1 wt 1 k + 1 = kwt 1 k2 + 2 kw0
+1
kwt k2 = kwt
?
?
(i)
(i) 2
1k
Therefore, the above two equations imply 8i 2 {1, 2}, kwt k2 ? kwt
Summing over t we obtain that Equation 2 holds.
(1)
w0 k + 1
(2)
+ 2 kw0
(1)
w0 k + 1.
Equipped with Equation 1 and Equation 2 we are ready to prove the theorem.
(i)
(2)
(1)
Denote K = maxi kw0 k and note that kw0
w0 k ? 2K. We prove the theorem by providing
(i)
upper and lower bounds on E[hwt , w? i]. Combining the update rule with Equation 1 we get:
"N
#
X
(i)
(i)
(i)
?
?
E[hwt , w i] = hw , w i + E
?t v?t
hw , w? i + (1 2?)T?
K kw? k + (1 2?)T?
0
0
t=1
To construct an upper bound, first note that Equation 2 implies that
(i)
(i)
(2)
E[kwt k2 ] ? kw0 k2 + (2kw0
(1)
w0 k + 1)T? ? K 2 + (4 K + 1) T?
5
Using the above and Jensen?s inequality, we get that
q
q
(i)
(i)
(i)
E[hwt , w? i] ? E[kwt k kw? k] ? kw? k E[kwt k2 ] ? kw? k K 2 + (4 K + 1)T?
Comparing the upper and lower bounds, we obtain that
q
K kw? k + (1 2?)T? ? kw? k K 2 + (4 K + 1)T?
p
p
p
Using a + b ? a + b, the above implies that
p
p
(1 2?)T? kw? k (4 K + 1) T? 2 K kw? k ? 0
p
p
Denote ? = kw? k (4 K + 1), then the above also implies that (1 2?)T? ? T? ? ? 0.
Denote = ?/(1 2?), using standard algebraic manipulations, the above implies that
T? ? + 2 + 1.5 ? 3 2 ,
where we used the fact that kw? k must be at least 1 for the separability assumption to hold, hence
1. This concludes our proof.
The above theorem tells us that our algorithm converges quickly. We next address the second question, regarding the quality of the point to which the algorithm converges. As mentioned in the
(1)
(2)
introduction, the convergence must depend on the initial predictors. Indeed, if w0 = w0 , then
(1)
the algorithm will not make any updates. The next question is what happens if we initialize w0
(2)
and w0 at random. The lemma below shows that this does not suffice to ensure convergence to the
optimum, even if the data is linearly separable without noise. The proof for this lemma is given in
Appendix A.
Lemma 1 Fix some 2 (0, 1) and let d be an integer greater than 40 log(1/ ). There exists a
distribution over Rd ? {?1}, which is separable by a weight vector w? for which kw? k2 = d, such
that running the ?Update by Disagreement? algorithm, with the perceptron as the underlying update
(1)
(2)
rule, and with every coordinate of w0 , w0 initialized according to any symmetric distribution over
R, will yield a solution whose error is at least 1/8, with probability of at least 1
.
Trying to circumvent the lower bound given in the above lemma, one may wonder what would
(1)
(2)
happen if we will initialize w0 , w0 differently. Intuitively, maybe noisy labels are not such a big
(1)
(2)
problem at the beginning of the learning process. Therefore, we can initialize w0 , w0 by running
the vanilla perceptron for several iterations, and only then switch to our algorithm. Trivially, for
the distribution we constructed in the proof of Lemma 1, this approach will work just because in
(1)
(2)
the noise-free setting, both w0 and w0 will converge to vectors that give the same predictions
?
as w . But, what would happen in the noisy setting, when we flip the label of every example with
probability of ?? The lemma below shows that the error of the resulting solution is likely to be order
of ?3 . Here again, the proof is given in Appendix A.
? over Rd ? {?1} such that to
Lemma 2 Consider a vector w? 2 {?1}d and the distribution D
sample a pair (x, y?) we first choose x uniformly at random from {e1 , . . . , ed }, set y = hw? , ei i, and
(1)
(2)
set y? = y with probability 1 ? and y? = y with probability ?. Let w0 , w0 be the result of
? for any number of iterations.
running the vanilla perceptron algorithm on random examples from D
Suppose that we run the ?Update by Disagreement? algorithm for an additional arbitrary number
of iterations. Then, the error of the solution is likely to be ?(?3 ).
To summarize, we see that without making additional assumptions on the data distribution, it is
impossible to prove convergence of our algorithm to a good solution. In the next section we show
that for natural data distributions, our algorithm converges to a very good solution.
5
EXPERIMENTS
We now demonstrate the merit of our suggested meta-algorithm using empirical evaluation. Our
main experiment is using our algorithm with deep networks in a real-world scenario of noisy labels.
6
In particular, we use a hypothesis class of deep networks and a Stochastic Gradient Descent with momentum as the basis update rule. The task is classifying face images according to gender. As training
data, we use the Labeled Faces in the Wild (LFW) dataset for which we had a labeling of the name
of the face, but we did not have gender labeling. To construct gender labels, we used an external
service that provides gender labels based on names. This process resulted in noisy labels. We show
that our method leads to state-of-the-art results on this task, compared to competing noise robustness
methods. We also performed controlled experiments to demonstrate our algorithm?s performance on
linear classification with varying levels of noise. These results are detailed in Appendix B.
5.1
Deep Learning
We have applied our algorithm with a Stochastic Gradient Descent (SGD) with momentum as the
base update rule on the task of labeling images of faces according to gender. The images were taken
from the Labeled Faces in the Wild (LFW) benchmark [18]. This benchmark consists of 13,233
images of 5,749 different people collected from the web, labeled with the name of the person in the
picture. Since the gender of each subject is not provided, we follow the method of [25] and use a
service that determines a person?s gender by their name (if it is recognized), along with a confidence
level. This method gives rise to ?natural? noisy labels due to ?unisex? names, and therefore allows
us to experiment with a real-world setup of dataset with noisy labels.
Name
Confidence
Kim
88%
Morgan
64%
Joan
82%
Leslie
88%
Correct
Mislabeled
Figure 1: Images from the dataset tagged as female
We have constructed train and test sets as follows. We first took all the individuals on which the
gender service gave 100% confidence. We divided this set at random into three subsets of equal
size, denoted N1 , N2 , N3 . We denote by N4 the individuals on which the confidence level is in
[90%, 100%), and by N5 the individuals on which the confidence level is in [0%, 90%). Needless to
say that all the sets N1 , . . . , N5 have zero intersection with each other.
We repeated each experiment three times, where in every time we used a different Ni as the test set,
for i 2 {1, 2, 3}. Suppose N1 is the test set, then for the training set we used two configurations:
1. A dataset consisting of all the images that belong to names in N2 , N3 , N4 , N5 , where unrecognized names were labeled as male (since the majority of subjects in LFW are males).
2. A dataset consisting of all the images that belong to names in N2 , N3 , N4 .
We use a network architecture suggested by [24], using an available tensorflow implementation1 .
It should be noted that we did not change any parameters of the network architecture or the optimization process, and use the default parameters in the implementation. Since the amount of male
and female subjects in the dataset is not balanced, we use an objective of maximizing the balanced
accuracy [9] - the average accuracy obtained on either class.
Training is done for 30,000 iterations on 128 examples mini-batch. In order to make the networks
disagreement meaningful, we initialize the two networks by training both of them normally (updating on all the examples) until iteration #5000, where we switch to training with the ?Update by
Disagreement? rule. Due to the fact that we are not updating on all examples, we decrease the weight
of batches with less than 10% of the original examples in the original batch to stabilize gradients. 2 .
1
2
https://github.com/dpressel/rude-carnie.
Code is available online on https://github.com/emalach/UpdateByDisagreement.
7
We inspect the balanced accuracy on our test data during the training process, comparing our method
to a vanilla neural network training, as well as to soft and hard bootstrapping described in [33] and
to the s-model described in [15], all of which are using the same network architecture. We use the
initialization parameters for [33, 15] that were suggested in the original papers. We show that while
in other methods, the accuracy effectively decreases during the training process due to overfitting the
noisy labels, in our method this effect is less substantial, allowing the network to keep improving.
We study two different scenarios, one in which a small clean test data is available for model selection,
and therefore we can choose the iteration with best test accuracy, and a more realistic scenario where
there is no clean test data at hand. For the first scenario, we observe the balanced accuracy of the best
available iteration. For the second scenario, we observe the balanced accuracy of the last iteration.
As can be seen in Figure 2 and the supplementary results listed in Table 1 in Appendix B, our
method outperforms the other methods in both situations. This is true for both datasets, although, as
expected, the improvement in performance is less substantial on the cleaner dataset.
The second best algorithm is the s-model described in [15]. Since our method can be applied to
any base algorithm, we also applied our method on top of the s-model. This yields even better
performance, especially when the data is less noisy, where we obtain a significant improvement.
Dataset #1 - more noise
Dataset #2 - less noise
Figure 2: Balanced accuracy of all methods on clean test data, trained on the two different datasets.
6
Discussion
We have described an extremely simple approach for supervised learning in the presence of noisy
labels. The basic idea is to decouple the ?when to update? rule from the ?how to update? rule. We
achieve this by maintaining two predictors, and update based on their disagreement. We have shown
that this simple approach leads to state-of-the-art results.
Our theoretical analysis shows that the approach leads to fast convergence rate when the underlying
update rule is the perceptron. We have also shown that proving that the method converges to an optimal solution must rely on distributional assumptions. There are several immediate open questions
that we leave to future work. First, suggesting distributional assumptions that are likely to hold in
practice and proving that the algorithm converges to an optimal solution under these assumptions.
Second, extending the convergence proof beyond linear predictors. While obtaining absolute convergence guarantees seems beyond reach at the moment, coming up with oracle based convergence
guarantees may be feasible.
Acknowledgements: This research is supported by the European Research Council (TheoryDL
project).
8
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11
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6,298 | 6,698 | Self-Normalizing Neural Networks
G?nter Klambauer
Thomas Unterthiner
Andreas Mayr
Sepp Hochreiter
LIT AI Lab & Institute of Bioinformatics,
Johannes Kepler University Linz
A-4040 Linz, Austria
{klambauer,unterthiner,mayr,hochreit}@bioinf.jku.at
Abstract
Deep Learning has revolutionized vision via convolutional neural networks (CNNs)
and natural language processing via recurrent neural networks (RNNs). However,
success stories of Deep Learning with standard feed-forward neural networks
(FNNs) are rare. FNNs that perform well are typically shallow and, therefore cannot
exploit many levels of abstract representations. We introduce self-normalizing
neural networks (SNNs) to enable high-level abstract representations. While
batch normalization requires explicit normalization, neuron activations of SNNs
automatically converge towards zero mean and unit variance. The activation
function of SNNs are ?scaled exponential linear units? (SELUs), which induce
self-normalizing properties. Using the Banach fixed-point theorem, we prove that
activations close to zero mean and unit variance that are propagated through many
network layers will converge towards zero mean and unit variance ? even under
the presence of noise and perturbations. This convergence property of SNNs allows
to (1) train deep networks with many layers, (2) employ strong regularization
schemes, and (3) to make learning highly robust. Furthermore, for activations
not close to unit variance, we prove an upper and lower bound on the variance,
thus, vanishing and exploding gradients are impossible. We compared SNNs on
(a) 121 tasks from the UCI machine learning repository, on (b) drug discovery
benchmarks, and on (c) astronomy tasks with standard FNNs, and other machine
learning methods such as random forests and support vector machines. For FNNs
we considered (i) ReLU networks without normalization, (ii) batch normalization,
(iii) layer normalization, (iv) weight normalization, (v) highway networks, and (vi)
residual networks. SNNs significantly outperformed all competing FNN methods
at 121 UCI tasks, outperformed all competing methods at the Tox21 dataset, and
set a new record at an astronomy data set. The winning SNN architectures are often
very deep.
1
Introduction
Deep Learning has set new records at different benchmarks and led to various commercial applications
[21, 26]. Recurrent neural networks (RNNs) [15] achieved new levels at speech and natural language
processing, for example at the TIMIT benchmark [10] or at language translation [29], and are already
employed in mobile devices [24]. RNNs have won handwriting recognition challenges (Chinese and
Arabic handwriting) [26, 11, 4] and Kaggle challenges, such as the ?Grasp-and Lift EEG? competition.
Their counterparts, convolutional neural networks (CNNs) [20] excel at vision and video tasks. CNNs
are on par with human dermatologists at the visual detection of skin cancer [8]. The visual processing
for self-driving cars is based on CNNs [16], as is the visual input to AlphaGo which has beaten one
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
BatchNorm Depth 8
BatchNorm Depth 16
BatchNorm Depth 32
SNN Depth 8
SNN Depth 16
SNN Depth 32
1
10
2
10
3
10
4
10
5
Training loss
10
0
250
500
750
1000
Iterations
1250
1500
1750
BatchNorm Depth 8
BatchNorm Depth 16
BatchNorm Depth 32
SNN Depth 8
SNN Depth 16
SNN Depth 32
100
10
1
10
2
10
3
10
4
10
5
Training loss
100
2000
0
250
500
750
1000
Iterations
1250
1500
1750
2000
Figure 1: The left panel and the right panel show the training error (y-axis) for feed-forward neural
networks (FNNs) with batch normalization (BatchNorm) and self-normalizing networks (SNN) across
update steps (x-axis) on the MNIST dataset the CIFAR10 dataset, respectively. We tested networks
with 8, 16, and 32 layers and learning rate 1e-5. FNNs with batch normalization exhibit high variance
due to perturbations. In contrast, SNNs do not suffer from high variance as they are more robust to
perturbations and learn faster.
of the best human GO players [27]. At vision challenges, CNNs are constantly winning, for example
at the large ImageNet competition [19, 13], but also almost all Kaggle vision challenges, such as the
?Diabetic Retinopathy? and the ?Right Whale? challenges [7, 12].
However, looking at Kaggle challenges that are not related to vision or sequential tasks, gradient
boosting, random forests, or support vector machines (SVMs) are winning most of the competitions.
Deep Learning is notably absent, and for the few cases where FNNs won, they are shallow. For
example, the HIGGS challenge, the Merck Molecular Activity challenge, and the Tox21 Data
challenge were all won by FNNs with at most four hidden layers. Surprisingly, it is hard to find
success stories with FNNs that have many hidden layers, though they would allow for different levels
of abstract representations of the input [2].
To robustly train very deep CNNs, batch normalization evolved into a standard to normalize neuron
activations to zero mean and unit variance [17]. Layer normalization [1] also ensures zero mean
and unit variance, while weight normalization [25] ensures zero mean and unit variance if in the
previous layer the activations have zero mean and unit variance. Natural neural networks [6] also aim
at normalizing the variance of activations by reparametrization of the weights. However, training with
normalization techniques is perturbed by stochastic gradient descent (SGD), stochastic regularization
(like dropout), and the estimation of the normalization parameters. Both RNNs and CNNs can
stabilize learning via weight sharing, therefore they are less prone to these perturbations. In contrast,
FNNs trained with normalization techniques suffer from these perturbations and have high variance in
the training error (see Figure 1). This high variance hinders learning and slows it down. Furthermore,
strong regularization, such as dropout, is not possible as it would further increase the variance
which in turn would lead to divergence of the learning process. We believe that this sensitivity to
perturbations is the reason that FNNs are less successful than RNNs and CNNs.
Self-normalizing neural networks (SNNs) are robust to perturbations and do not have high variance
in their training errors (see Figure 1). SNNs push neuron activations to zero mean and unit variance
thereby leading to the same effect as batch normalization, which enables to robustly learn many
layers. SNNs are based on scaled exponential linear units ?SELUs? which induce self-normalizing
properties like variance stabilization which in turn avoids exploding and vanishing gradients.
2
Self-normalizing Neural Networks (SNNs)
Normalization and SNNs. For a neural network with activation function f , we consider two
consecutive layers that are connected by a weight matrix W . Since the input to a neural network
is a random variable, the activations x in the lower layer, the network inputs z = W x, and the
activations y = f (z) in the higher layer are random variables as well. We assume that all activations
2
xi of the lower layer have mean ? := E(xi ) and variance ? := Var(xi ). An activation y in the higher
layer has mean ?
? := E(y) and variance ?? := Var(y). Here E(.) denotes the expectation and Var(.)
the variance of a random variable. A single activation y = f (z) has net input z = wT x. For n units
with activation xiP
, 1 6 i 6 n in the lower layer, we define n timesP
the mean of the weight vector
n
n
w ? Rn as ? := i=1 wi and n times the second moment as ? := i=1 wi2 .
We consider the mapping g that maps mean and variance of the activations from one layer to mean
and variance of the activations in the next layer g : (?, ?) 7? (?
?, ??). Normalization techniques
like batch, layer, or weight normalization ensure a mapping g that keeps (?, ?) and (?
?, ??) close to
predefined values, typically (0, 1).
Definition 1 (Self-normalizing neural net). A neural network is self-normalizing if it possesses a
mapping g : ? 7? ? for each activation y that maps mean and variance from one layer to the next
and has a stable and attracting fixed point depending on (?, ? ) in ?. Furthermore, the mean and
the variance remain in the domain ?, that is g(?) ? ?, where ? = {(?, ?) | ? ? [?min , ?max ], ? ?
[?min , ?max ]}. When iteratively applying the mapping g, each point within ? converges to this fixed
point.
Therefore, we consider activations of a neural network to be normalized, if both their mean and their
variance across samples are within predefined intervals. If mean and variance of x are already within
these intervals, then also mean and variance of y remain in these intervals, i.e., the normalization is
transitive across layers. Within these intervals, the mean and variance both converge to a fixed point
if the mapping g is applied iteratively.
Therefore, SNNs keep normalization of activations when propagating them through layers of the
network. The normalization effect is observed across layers of a network: in each layer the activations
are getting closer to the fixed point. The normalization effect can also observed be for two fixed
layers across learning steps: perturbations of lower layer activations or weights are damped in the
higher layer by drawing the activations towards the fixed point. If for all y in the higher layer, ? and
? of the corresponding weight vector are the same, then the fixed points are also the same. In this
case we have a unique fixed point for all activations y. Otherwise, in the more general case, ? and
? differ for different y but the mean activations are drawn into [?min , ?max ] and the variances are
drawn into [?min , ?max ].
Constructing Self-Normalizing Neural Networks. We aim at constructing self-normalizing neural networks by adjusting the properties of the function g. Only two design choices are available for
the function g: (1) the activation function and (2) the initialization of the weights.
For the activation function, we propose ?scaled exponential linear units? (SELUs) to render a FNN as
self-normalizing. The SELU activation function is given by
x
if x > 0
selu(x) = ?
.
(1)
?ex ? ? if x 6 0
SELUs allow to construct a mapping g with properties that lead to SNNs. SNNs cannot be derived
with (scaled) rectified linear units (ReLUs), sigmoid units, tanh units, and leaky ReLUs. The
activation function is required to have (1) negative and positive values for controlling the mean, (2)
saturation regions (derivatives approaching zero) to dampen the variance if it is too large in the lower
layer, (3) a slope larger than one to increase the variance if it is too small in the lower layer, (4) a
continuous curve. The latter ensures a fixed point, where variance damping is equalized by variance
increasing. We met these properties of the activation function by multiplying the exponential linear
unit (ELU) [5] with ? > 1 to ensure a slope larger than one for positive net inputs.
For the weight initialization, we propose ? = 0 and ? = 1 for all units in the higher layer. The
next paragraphs will show the advantages of this initialization. Of course, during learning these
assumptions on the weight vector will be violated. However, we can prove the self-normalizing
property even for weight vectors that are not normalized, therefore, the self-normalizing property can
be kept during learning and weight changes.
Deriving the Mean and Variance Mapping Function g. We assume that the xi are independent
from each other but share the same mean ? and variance ?. Of course, the independence assumptions
is not fulfilled in general. We will elaborate on the independence assumption below. The network
3
T
input
which we can infer the following moments E(z) =
Pn z in the higher layer is z = w x for P
n
i=1 wi xi ) = ? ? , where we used the independence
i=1 wi E(xi ) = ? ? and Var(z) = Var(
of the xi . The net input z is a weighted sum of independent, but not necessarily identically distributed
variables xi ,?
for which the central limit theorem
? (CLT) states that z approaches a normal distribution:
z ? N (??, ?? ) with density pN (z; ??, ?? ). According to the CLT, the larger n, the closer is z
to a normal distribution. For Deep Learning, broad layers with hundreds of neurons xi are common.
Therefore the assumption that z is normally distributed is met well for most currently used neural
networks (see Supplementary Figure S7). The function g maps the mean and variance of activations
in the lower layer to the mean ?
? = E(y) and variance ?? = Var(y) of the activations y in the next
layer:
Z ?
?
?
?
?
g:
7?
: ?
?(?, ?, ?, ? ) =
selu(z) pN (z; ??, ?? ) dz
(2)
?
??
??
Z ?
?
?2 .
??(?, ?, ?, ? ) =
selu(z)2 pN (z; ??, ?? ) dz ? ?
??
These integrals can be analytically computed and lead to following mappings of the moments:
1
??
?
? = ? (??) erf ? ?
+
(3)
2
2 ??
!
r
(??)2
?? + ??
??
2?
?
??+ ??
2 erfc
? ?
?? e 2(?? ) + ??
?e
? ? erfc ? ?
+
?
2 ??
2 ??
??
??
1
?? + ??
+ ?2 ?2e??+ 2 erfc ? ?
(4)
?? = ?2 (??)2 + ?? 2 ? erfc ? ?
2
2 ??
2 ??
!
r
(??)2
?
?? + 2??
??
2
2
? 2(?? )
2(??+?? )
+e
erfc ? ?
(??) ?? e
? (?
?)
+ erfc ? ?
+
?
2 ??
2 ??
Stable and Attracting Fixed Point (0, 1) for Normalized Weights. We assume a normalized
weight vector w with ? = 0 and ? = 1. Given a fixed point (?, ?), we can solve equations Eq. (3)
and Eq. (4) for ? and ?. We chose the fixed point (?, ?) = (0, 1), which is typical for activation
normalization. We obtain the fixed point equations ?
? = ? = 0 and ?? = ? = 1 that we solve for ?
and ? and obtain the solutions ?01 ? 1.6733 and ?01 ? 1.0507, where the subscript 01 indicates
that these are the parameters for fixed point (0, 1). The analytical expressions for ?01 and ?01 are
given in Supplementary Eq. (8). We are interested whether the fixed point (?, ?) = (0, 1) is stable
and attracting. If the Jacobian of g has a norm smaller than 1 at the fixed point, then g is a contraction
mapping and the fixed point is stable. The (2x2)-Jacobian J (?, ?) of g : (?, ?) 7? (?
?, ??) evaluated at
the fixed point (0, 1) with ?01 and ?01 is J (0, 1) = ((0.0, 0.088834), (0.0, 0.782648)). The spectral
norm of J (0, 1) (its largest singular value) is 0.7877 < 1. That means g is a contraction mapping
around the fixed point (0, 1) (the mapping is depicted in Figure 2). Therefore, (0, 1) is a stable
fixed point of the mapping g. The norm of the Jacobian also determines the convergence rate as a
consequence of the Banach fixed point theorem. The convergence rate around the fixed point (0,1) is
about 0.78. In general, the convergence rate depends on ?, ?, ?, ? and is between 0.78 and 1.
Stable and Attracting Fixed Points for Unnormalized Weights. A normalized weight vector w
cannot be ensured during learning. For SELU parameters ? = ?01 and ? = ?01 , we show in the next
theorem that if (?, ? ) is close to (0, 1), then g still has an attracting and stable fixed point that is close
to (0, 1). Thus, in the general case there still exists a stable fixed point which, however, depends
on (?, ? ). If we restrict (?, ?, ?, ? ) to certain intervals, then we can show that (?, ?) is mapped to
the respective intervals. Next we present the central theorem of this paper, from which follows that
SELU networks are self-normalizing under mild conditions on the weights.
Theorem 1 (Stable and Attracting Fixed Points). We assume ? = ?01 and ? = ?01 . We restrict the
range of the variables to the following intervals ? ? [?0.1, 0.1], ? ? [?0.1, 0.1], ? ? [0.8, 1.5], and
? ? [0.95, 1.1], that define the functions? domain ?. For ? = 0 and ? = 1, the mapping Eq. (2) has
the stable fixed point (?, ?) = (0, 1), whereas for other ? and ? the mapping Eq. (2) has a stable
and attracting fixed point depending on (?, ? ) in the (?, ?)-domain: ? ? [?0.03106, 0.06773] and
4
Figure 2: For ? = 0 and ? = 1, the mapping g of mean ? (x-axis) and variance ? (y-axis) to the
next layer?s mean ?
? and variance ?? is depicted. Arrows show in which direction (?, ?) is mapped by
g : (?, ?) 7? (?
?, ??). The fixed point of the mapping g is (0, 1).
? ? [0.80009, 1.48617]. All points within the (?, ?)-domain converge when iteratively applying the
mapping Eq. (2) to this fixed point.
Proof. We provide a proof sketch (see detailed proof in Supplementary Material). With the Banach
fixed point theorem we show that there exists a unique attracting and stable fixed point. To this end,
we have to prove that a) g is a contraction mapping and b) that the mapping stays in the domain, that
is, g(?) ? ?. The spectral norm of the Jacobian of g can be obtained via an explicit formula for the
largest singular value for a 2 ? 2 matrix. g is a contraction mapping if its spectral norm is smaller
than 1. We perform a computer-assisted proof to evaluate the largest singular value on a fine grid and
ensure the precision of the computer evaluation by an error propagation analysis of the implemented
algorithms on the according hardware. Singular values between grid points are upper bounded by the
mean value theorem. To this end, we bound the derivatives of the formula for the largest singular
value with respect to ?, ?, ?, ?. Then we apply the mean value theorem to pairs of points, where one
is on the grid and the other is off the grid. This shows that for all values of ?, ?, ?, ? in the domain ?,
the spectral norm of g is smaller than one. Therefore, g is a contraction mapping on the domain ?.
Finally, we show that the mapping g stays in the domain ? by deriving bounds on ?
? and ??. Hence,
the Banach fixed-point theorem holds and there exists a unique fixed point in ? that is attained.
Consequently, feed-forward neural networks with many units in each layer and with the SELU
activation function are self-normalizing (see definition 1), which readily follows from Theorem 1. To
give an intuition, the main property of SELUs is that they damp the variance for negative net inputs
and increase the variance for positive net inputs. The variance damping is stronger if net inputs are
further away from zero while the variance increase is stronger if net inputs are close to zero. Thus, for
large variance of the activations in the lower layer the damping effect is dominant and the variance
decreases in the higher layer. Vice versa, for small variance the variance increase is dominant and the
variance increases in the higher layer.
However, we cannot guarantee that mean and variance remain in the domain ?. Therefore, we next
treat the case where (?, ?) are outside ?. It is especially crucial to consider ? because this variable
has much stronger influence than ?. Mapping ? across layers to a high value corresponds to an
exploding gradient, since the Jacobian of the activation of high layers with respect to activations
in lower layers has large singular values. Analogously, mapping ? across layers to a low value
corresponds to an vanishing gradient. Bounding the mapping of ? from above and below would avoid
both exploding and vanishing gradients. Theorem 2 states that the variance of neuron activations of
SNNs is bounded from above, and therefore ensures that SNNs learn robustly and do not suffer from
exploding gradients.
Theorem 2 (Decreasing ?). For ? = ?01 , ? = ?01 and the domain ?+ : ?1 6 ? 6 1, ?0.1 6 ? 6
0.1, 3 6 ? 6 16, and 0.8 6 ? 6 1.25, we have for the mapping of the variance ??(?, ?, ?, ?, ?, ?)
given in Eq. (4): ??(?, ?, ?, ?, ?01 , ?01 ) < ?.
The proof can be found in Supplementary Material. Thus, when mapped across many layers, the
variance in the interval [3, 16] is mapped to a value below 3. Consequently, all fixed points (?, ?)
5
of the mapping g (Eq. (2)) have ? < 3. Analogously, Theorem 3 states that the variance of neuron
activations of SNNs is bounded from below, and therefore ensures that SNNs do not suffer from
vanishing gradients.
Theorem 3 (Increasing ?). We consider ? = ?01 , ? = ?01 and the domain ?? : ?0.1 6 ? 6 0.1,
and ?0.1 6 ? 6 0.1. For the domain 0.02 6 ? 6 0.16 and 0.8 6 ? 6 1.25 as well as for the
domain 0.02 6 ? 6 0.24 and 0.9 6 ? 6 1.25, the mapping of the variance ??(?, ?, ?, ?, ?, ?) given
in Eq. (4) increases: ??(?, ?, ?, ?, ?01 , ?01 ) > ?.
The proof can be found in the Supplementary Material. All fixed points (?, ?) of the mapping g
(Eq. (2)) ensure for 0.8 6 ? that ?? > 0.16 and for 0.9 6 ? that ?? > 0.24. Consequently, the variance
mapping Eq. (4) ensures a lower bound on the variance ?. Therefore SELU networks control the
variance of the activations and push it into an interval, whereafter the mean and variance move
toward the fixed point. Thus, SELU networks are steadily normalizing the variance and subsequently
normalizing the mean, too. In all experiments, we observed that self-normalizing neural networks
push the mean and variance of activations into the domain ? .
Since SNNsP
have a fixed point at zero mean and unit variance for normalized weights
Initialization.
Pn
n
? = i=1 wi = 0 and ? = i=1 wi2 = 1 (see above), we initialize SNNs such that these constraints
are fulfilled in expectation. We draw the weights from a Gaussian distribution with E(wi ) = 0 and
variance Var(wi ) = 1/n. Uniform and truncated Gaussian distributions with these moments led to
networks with similar behavior. The ?MSRA initialization? is similar since it uses zero mean and
variance 2/n to initialize the weights [14]. The additional factor 2 counters the effect of rectified
linear units.
New Dropout Technique. Standard dropout randomly sets an activation x to zero with probability
1 ? q for 0 < q 6 1. In order to preserve the mean, the activations are scaled by 1/q during
training. If x has mean E(x) = ? and variance Var(x) = ?, and the dropout variable d follows a
binomial distribution B(1, q), then the mean E(1/qdx) = ? is kept. Dropout fits well to rectified
linear units, since zero is in the low variance region and corresponds to the default value. For scaled
exponential linear units, the default and low variance value is limx??? selu(x) = ??? = ?0 .
Therefore, we propose ?alpha dropout?, that randomly sets inputs to ?0 . The new mean and new
variance is E(xd + ?0 (1 ? d)) = q? + (1 ? q)?0 , and Var(xd + ?0 (1 ? d)) = q((1 ? q)(?0 ?
?)2 + ?). We aim at keeping mean and variance to their original values after ?alpha dropout?, in
order to ensure the self-normalizing property even for ?alpha dropout?. The affine transformation
a(xd + ?0 (1 ? d)) + b allows to determine parameters a and b such that mean and variance are kept
to their values: E(a(x ? d + ?0 (1 ? d)) + b) = ? and Var(a(x ? d + ?0 (1 ? d)) + b) = ? . In
contrast to dropout, a and b will depend on ? and ?, however our SNNs converge to activations with
?1/2
zero mean and unit variance. With ? = 0 and ? = 1, we obtain a = q + ?02 q(1 ? q)
and
?1/2
02
0
b = ? q + ? q(1 ? q)
((1 ? q)? ). The parameters a and b only depend on the dropout rate
1 ? q and the most negative activation ?0 . Empirically, we found that dropout rates 1 ? q = 0.05 or
0.10 lead to models with good performance. ?Alpha-dropout? fits well to scaled exponential linear
units by randomly setting activations to the negative saturation value.
Applicability of the central limit theorem and independence assumption. In the derivative of
the mapping
(Eq. (2)), we used the central limit theorem (CLT) to approximate the network inputs
Pn
z = i=1 wi xi with a normal distribution. We justified normality because network inputs represent
a weighted sum of the inputs xi , where for Deep Learning n is typically large. The Berry-Esseen
theorem states that the convergence rate to normality is n?1/2 [18]. In the classical version of the CLT,
the random variables have to be independent and identically distributed, which typically does not
hold for neural networks. However, the Lyapunov CLT does not require the variable to be identically
distributed anymore. Furthermore, even under weak dependence, sums of random variables converge
in distribution to a Gaussian distribution [3].
Optimizers. Empirically, we found that SGD, momentum, Adadelta and Adamax worked well for
training SNNs, whereas for Adam we had to adjust the parameters (?2 = 0.99, = 0.01) to obtain
proficient networks.
6
3
Experiments
We compare SNNs to other deep networks at different benchmarks. Hyperparameters such as number
of layers (blocks), neurons per layer, learning rate, and dropout rate, are adjusted by grid-search
for each dataset on a separate validation set (see Supplementary Section S4). We compare the
following FNN methods: (1) ?MSRAinit?: FNNs without normalization and with ReLU activations
and ?Microsoft weight initialization? [14]. (2) ?BatchNorm?: FNNs with batch normalization [17].
(3) ?LayerNorm?: FNNs with layer normalization [1]. (4) ?WeightNorm?: FNNs with weight
normalization [25]. (5) ?Highway?: Highway networks [28]. (6) ?ResNet?: Residual networks
[13] adapted to FNNs using residual blocks with 2 or 3 layers with rectangular or diavolo shape.
(7) ?SNNs?: Self normalizing networks with SELUs with ? = ?01 and ? = ?01 and the proposed
dropout technique and initialization strategy.
121 UCI Machine Learning Repository datasets. The benchmark comprises 121 classification
datasets from the UCI Machine Learning repository [9] from diverse application areas, such as
physics, geology, or biology. The size of the datasets ranges between 10 and 130, 000 data points
and the number of features from 4 to 250. In abovementioned work [9], there were methodological
mistakes [30] which we avoided here. Each compared FNN method was optimized with respect to
its architecture and hyperparameters on a validation set that was then removed from the subsequent
analysis. The selected hyperparameters served to evaluate the methods in terms of accuracy on
the pre-defined test sets. The accuracies are reported in the Supplementary Table S8. We ranked
the methods by their accuracy for each prediction task and compared their average ranks. SNNs
significantly outperform all competing networks in pairwise comparisons (paired Wilcoxon test
across datasets) as reported in Table 1 (left panel).
Table 1: Left: Comparison of seven FNNs on 121 UCI tasks. We consider the average rank difference
to rank 4, which is the average rank of seven methods with random predictions. The first column gives
the method, the second the average rank difference, and the last the p-value of a paired Wilcoxon test
whether the difference to the best performing method is significant. SNNs significantly outperform
all other methods. Right: Comparison of 24 machine learning methods (ML) on the UCI datasets
with more than 1000 data points. The first column gives the method, the second the average rank
difference to rank 12.5, and the last the p-value of a paired Wilcoxon test whether the difference
to the best performing method is significant. Methods that were significantly worse than the best
method are marked with ?*?. SNNs outperform all competing methods.
Method
FNN method comparison
avg. rank diff. p-value
SNN
MSRAinit
LayerNorm
Highway
ResNet
WeightNorm
BatchNorm
-0.756
-0.240*
-0.198*
0.021*
0.273*
0.397*
0.504*
Method
ML method comparison
avg. rank diff. p-value
SNN
SVM
RandomForest
MSRAinit
LayerNorm
Highway
...
2.7e-02
1.5e-02
1.9e-03
5.4e-04
7.8e-07
3.5e-06
-6.7
-6.4
-5.9
-5.4*
-5.3
-4.6*
...
5.8e-01
2.1e-01
4.5e-03
7.1e-02
1.7e-03
...
We further included 17 machine learning methods representing diverse method groups [9] in the
comparison and the grouped the data sets into ?small? and ?large? data sets (for details see Supplementary Section S4.2). On 75 small datasets with less than 1000 data points, random forests and
SVMs outperform SNNs and other FNNs. On 46 larger datasets with at least 1000 data points, SNNs
show the highest performance followed by SVMs and random forests (see right panel of Table 1, for
complete results see Supplementary Tables S9 and S10). Overall, SNNs have outperformed state of
the art machine learning methods on UCI datasets with more than 1,000 data points.
Typically, hyperparameter selection chose SNN architectures that were much deeper than the selected
architectures of other FNNs, with an average depth of 10.8 layers, compared to average depths of 6.0
for BatchNorm, 3.8 WeightNorm, 7.0 LayerNorm, 5.9 Highway, and 7.1 for MSRAinit networks. For
ResNet, the average number of blocks was 6.35. SNNs with many more than 4 layers often provide
the best predictive accuracies across all neural networks.
7
Drug discovery: The Tox21 challenge dataset. The Tox21 challenge dataset comprises about
12,000 chemical compounds whose twelve toxic effects have to be predicted based on their chemical
structure. We used the validation sets of the challenge winners for hyperparameter selection (see
Supplementary Section S4) and the challenge test set for performance comparison. We repeated
the whole evaluation procedure 5 times to obtain error bars. The results in terms of average AUC
are given in Table 2. In 2015, the challenge organized by the US NIH was won by an ensemble
of shallow ReLU FNNs which achieved an AUC of 0.846 [23]. Besides FNNs, this ensemble
also contained random forests and SVMs. Single SNNs came close with an AUC of 0.845?0.003.
The best performing SNNs have 8 layers, compared to the runner-ups ReLU networks with layer
normalization with 2 and 3 layers. Also batchnorm and weightnorm networks, typically perform best
with shallow networks of 2 to 4 layers (Table 2). The deeper the networks, the larger the difference in
performance between SNNs and other methods (see columns 5?8 of Table 2). The best performing
method is an SNN with 8 layers.
Table 2: Comparison of FNNs at the Tox21 challenge dataset in terms of AUC. The rows represent
different methods and the columns different network depth and for ResNets the number of residual
blocks (6 and 32 blocks were omitted due to computational constraints). The deeper the networks,
the more prominent is the advantage of SNNs. The best networks are SNNs with 8 layers.
#layers / #blocks
method
2
3
4
6
8
16
32
SNN
Batchnorm
WeightNorm
LayerNorm
Highway
MSRAinit
ResNet
83.7 ? 0.3
80.0 ? 0.5
83.7 ? 0.8
84.3 ? 0.3
83.3 ? 0.9
82.7 ? 0.4
82.2 ? 1.1
84.4 ? 0.5
79.8 ? 1.6
82.9 ? 0.8
84.3 ? 0.5
83.0 ? 0.5
81.6 ? 0.9
80.0 ? 2.0
84.2 ? 0.4
77.2 ? 1.1
82.2 ? 0.9
84.0 ? 0.2
82.6 ? 0.9
81.1 ? 1.7
80.5 ? 1.2
83.9 ? 0.5
77.0 ? 1.7
82.5 ? 0.6
82.5 ? 0.8
82.4 ? 0.8
80.6 ? 0.6
81.2 ? 0.7
84.5 ? 0.2
75.0 ? 0.9
81.9 ? 1.2
80.9 ? 1.8
80.3 ? 1.4
80.9 ? 1.1
81.8 ? 0.6
83.5 ? 0.5
73.7 ? 2.0
78.1 ? 1.3
78.7 ? 2.3
80.3 ? 2.4
80.2 ? 1.1
81.2 ? 0.6
82.5 ? 0.7
76.0 ? 1.1
56.6 ? 2.6
78.8 ? 0.8
79.6 ? 0.8
80.4 ? 1.9
na
Astronomy: Prediction of pulsars in the HTRU2 dataset. Since a decade, machine learning
methods have been used to identify pulsars in radio wave signals [22]. Recently, the High Time
Resolution Universe Survey (HTRU2) dataset has been released with 1,639 real pulsars and 16,259
spurious signals. Currently, the highest AUC value of a 10-fold cross-validation is 0.976 which has
been achieved by Naive Bayes classifiers followed by decision tree C4.5 with 0.949 and SVMs with
0.929. We used eight features constructed by the PulsarFeatureLab as used previously [22]. We
assessed the performance of FNNs using 10-fold nested cross-validation, where the hyperparameters
were selected in the inner loop on a validation set (for details see Supplementary Section S4). Table 3
reports the results in terms of AUC. SNNs outperform all other methods and have pushed the
state-of-the-art to an AUC of 0.98.
Table 3: Comparison of FNNs and reference methods at HTRU2 in terms of AUC. The first, fourth
and seventh column give the method, the second, fifth and eight column the AUC averaged over 10
cross-validation folds, and the third and sixth column the p-value of a paired Wilcoxon test of the
AUCs against the best performing method across the 10 folds. FNNs achieve better results than Naive
Bayes (NB), C4.5, and SVM. SNNs exhibit the best performance and set a new record.
method
SNN
MSRAinit
WeightNorm
Highway
FNN methods
AUC
0.9803 ? 0.010
0.9791 ? 0.010
0.9786* ? 0.010
0.9766* ? 0.009
p-value
3.5e-01
2.4e-02
9.8e-03
method
LayerNorm
BatchNorm
ResNet
FNN methods
AUC
0.9762* ? 0.011
0.9760 ? 0.013
0.9753* ? 0.010
p-value
1.4e-02
6.5e-02
6.8e-03
ref. methods
method AUC
NB
C4.5
SVM
0.976
0.946
0.929
SNNs and convolutional neural networks. In initial experiments with CNNs, we found that SELU
activations work well at image classification tasks: On MNIST, SNN-CNNs (2x Conv, MaxPool, 2x
fully-connected, 30 Epochs) reach 99.2%?0.1 accuracy (ReLU: 99.2%?0.1) and on CIFAR10 (2x
Conv, MaxPool, 2x Conv, MaxPool, 2x fully-connected, 200 Epochs) SNN-CNNs reach 82.5?0.8%
8
accuracy (ReLU: 76.1?1.0%). This finding unsurprising since even standard ELUs without the
self-normalizing property have been shown to improve CNN training and accuracy[5].
4
Conclusion
To summarize, self-normalizing networks work well with the following configuration:
? SELU activation with parameters ? ? 1.0507 and ? ? 1.6733,
? inputs normalized to zero mean and unit variance,
? network weights initialized with variance 1/n, and
? regularization with ?alpha-dropout?.
We have introduced self-normalizing neural networks for which we have proved that neuron activations are pushed towards zero mean and unit variance when propagated through the network.
Additionally, for activations not close to unit variance, we have proved an upper and lower bound
on the variance mapping. Consequently, SNNs do not face vanishing and exploding gradient problems. Therefore, SNNs work well for architectures with many layers, allowed us to introduce a
novel regularization scheme and learn very robustly. On 121 UCI benchmark datasets, SNNs have
outperformed other FNNs with and without normalization techniques, such as batch, layer, and
weight normalization, or specialized architectures, such as Highway or Residual networks. SNNs
also yielded the best results on drug discovery and astronomy tasks. The best performing SNN
architectures are typically very deep in contrast to other FNNs.
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10
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6,299 | 6,699 | Learning to Pivot with Adversarial Networks
Gilles Louppe
New York University
[email protected]
Michael Kagan
SLAC National Accelerator Laboratory
[email protected]
Kyle Cranmer
New York University
[email protected]
Abstract
Several techniques for domain adaptation have been proposed to account for
differences in the distribution of the data used for training and testing. The majority
of this work focuses on a binary domain label. Similar problems occur in a scientific
context where there may be a continuous family of plausible data generation
processes associated to the presence of systematic uncertainties. Robust inference
is possible if it is based on a pivot ? a quantity whose distribution does not depend
on the unknown values of the nuisance parameters that parametrize this family
of data generation processes. In this work, we introduce and derive theoretical
results for a training procedure based on adversarial networks for enforcing the
pivotal property (or, equivalently, fairness with respect to continuous attributes) on
a predictive model. The method includes a hyperparameter to control the tradeoff between accuracy and robustness. We demonstrate the effectiveness of this
approach with a toy example and examples from particle physics.
1
Introduction
Machine learning techniques have been used to enhance a number of scientific disciplines, and they
have the potential to transform even more of the scientific process. One of the challenges of applying
machine learning to scientific problems is the need to incorporate systematic uncertainties, which
affect both the robustness of inference and the metrics used to evaluate a particular analysis strategy.
In this work, we focus on supervised learning techniques where systematic uncertainties can be
associated to a data generation process that is not uniquely specified. In other words, the lack of
systematic uncertainties corresponds to the (rare) case that the process that generates training data is
unique, fully specified, and an accurate representative of the real world data. By contrast, a common
situation when systematic uncertainty is present is when the training data are not representative
of the real data. Several techniques for domain adaptation have been developed to create models
that are more robust to this binary type of uncertainty. A more generic situation is that there are
several plausible data generation processes, specified as a family parametrized by continuous nuisance
parameters, as is typically found in scientific domains. In this broader context, statisticians have for
long been working on robust inference techniques based on the concept of a pivot ? a quantity whose
distribution is invariant with the nuisance parameters (see e.g., (Degroot and Schervish, 1975)).
Assuming a probability model p(X, Y, Z), where X are the data, Y are the target labels, and Z are the
nuisance parameters, we consider the problem of learning a predictive model f (X) for Y conditional
on the observed values of X that is robust to uncertainty in the unknown value of Z. We introduce a
flexible learning procedure based on adversarial networks (Goodfellow et al., 2014) for enforcing that
f (X) is a pivot with respect to Z. We derive theoretical results proving that the procedure converges
towards a model that is both optimal and statistically independent of the nuisance parameters (if
that model exists) or for which one can tune a trade-off between accuracy and robustness (e.g., as
driven by a higher level objective). In particular, and to the best of our knowledge, our contribution is
the first solution for imposing pivotal constraints on a predictive model, working regardless of the
31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Adversary r
Classifier f
Z
?1 (f (X; ?f ); ?r )
f (X; ?f )
...
X
...
?2 (f (X; ?f ); ?r )
P(?1 , ?2 , . . . )
...
p?r (Z|f (X; ?f ))
Lf (?f )
?f
?r
Lr (?f , ?r )
Figure 1: Architecture for the adversarial training of a binary classifier f against a nuisance parameters Z. The
adversary r models the distribution p(z|f (X; ?f ) = s) of the nuisance parameters as observed only through
the output f (X; ?f ) of the classifier. By maximizing the antagonistic objective Lr (?f , ?r ), the classifier f
forces p(z|f (X; ?f ) = s) towards the prior p(z), which happens when f (X; ?f ) is independent of the nuisance
parameter Z and therefore pivotal.
type of the nuisance parameter (discrete or continuous) or of its prior. Finally, we demonstrate the
effectiveness of the approach with a toy example and examples from particle physics.
2
Problem statement
We begin with a family of data generation processes p(X, Y, Z), where X ? X are the data, Y ? Y
are the target labels, and Z ? Z are the nuisance parameters that can be continuous or categorical. Let
us assume that prior to incorporating the effect of uncertainty in Z, our goal is to learn a regression
function f : X ? S with parameters ?f (e.g., a neural network-based probabilistic classifier) that
minimizes a loss Lf (?f ) (e.g., the cross-entropy). In classification, values s ? S = R|Y| correspond
to the classifier scores used for mapping hard predictions y ? Y, while S = Y for regression.
We augment our initial objective so that inference based on f (X; ?f ) will be robust to the value
z ? Z of the nuisance parameter Z ? which remains unknown at test time. A formal way of enforcing
robustness is to require that the distribution of f (X; ?f ) conditional on Z (and possibly Y ) be
invariant with the nuisance parameter Z. Thus, we wish to find a function f such that
p(f (X; ?f ) = s|z) = p(f (X; ?f ) = s|z 0 )
(1)
0
for all z, z ? Z and all values s ? S of f (X; ?f ). In words, we are looking for a predictive function
f which is a pivotal quantity with respect to the nuisance parameters. This implies that f (X; ?f ) and
Z are independent random variables.
As stated in Eqn. 1, the pivotal quantity criterion is imposed with respect to p(X|Z) where Y is
marginalized out. In some situations however (see e.g., Sec. 5.2), class conditional independence of
f (X; ?f ) on the nuisance Z is preferred, which can then be stated as requiring
p(f (X; ?f ) = s|z, y) = p(f (X; ?f ) = s|z 0 , y)
(2)
for one or several specified values y ? Y.
3
Method
Joint training of adversarial networks was first proposed by (Goodfellow et al., 2014) as a way to
build a generative model capable of producing samples from random noise z. More specifically, the
authors pit a generative model g : Rn ? Rp against an adversarial classifier d : Rp ? [0, 1] whose
antagonistic objective is to recognize real data X from generated data g(Z). Both models g and d are
trained simultaneously, in such a way that g learns to produce samples that are difficult to identify by
d, while d incrementally adapts to changes in g. At the equilibrium, g models a distribution whose
samples can be identified by d only by chance. That is, assuming enough capacity in d and g, the
distribution of g(Z) eventually converges towards the real distribution of X.
2
Algorithm 1 Adversarial training of a classifier f against an adversary r.
? ?
Inputs: training data {xi , yi , zi }N
i=1 ; Outputs: ?f , ?r .
1: for t = 1 to T do
2:
for k = 1 to K do
3:
Sample minibatch {xm , zm , sm = f (xm ; ?f )}M
m=1 of size M ;
4:
With ?f fixed, update r by ascending its stochastic gradient ??r E(?f , ?r ) :=
??r
M
X
log p?r (zm |sm );
m=1
5:
6:
7:
end for
Sample minibatch {xm , ym , zm , sm = f (xm ; ?f )}M
m=1 of size M ;
With ?r fixed, update f by descending its stochastic gradient ??f E(?f , ?r ) :=
??f
M
X
? log p?f (ym |xm ) + log p?r (zm |sm ) ,
m=1
where p?f (ym |xm ) denotes 1(ym = 0)(1 ? sm ) + 1(ym = 1)sm ;
8: end for
In this work, we repurpose adversarial networks as a means to constrain the predictive model f
in order to satisfy Eqn. 1. As illustrated in Fig. 1, we pit f against an adversarial model r :=
p?r (z|f (X; ?f ) = s) with parameters ?r and associated loss Lr (?f , ?r ). This model takes as input
realizations s of f (X; ?f ) and produces as output a function modeling the posterior probability
density p?r (z|f (X; ?f ) = s). Intuitively, if p(f (X; ?f ) = s|z) varies with z, then the corresponding
correlation can be captured by r. By contrast, if p(f (X; ?f ) = s|z) is invariant with z, as we require,
then r should perform poorly and be close to random guessing. Training f such that it additionally
minimizes the performance of r therefore acts as a regularization towards Eqn. 1.
If Z takes discrete values, then p?r can be represented as a probabilistic classifier R ? R|Z| whose
j th output (for j = 1, . . . , |Z|) is the estimated probability mass p?r (zj |f (X; ?f ) = s). Similarly, if
Z takes continuous values, then we can model the posterior probability density p(z|f (X; ?f ) = s)
with a sufficiently flexible parametric family of distributions P(?1 , ?2 , . . . ), where the parameters ?j
depend on f (X, ?f ) and ?r . The adversary r may take any form, i.e. it does not need to be a neural
network, as long as it exposes a differentiable function p?r (z|f (X; ?f ) = s) of sufficient capacity
to represent the true distribution. Fig. 1 illustrates a concrete example where p?r (z|f (X; ?f ) = s)
is a mixture of gaussians, as modeled with a mixture density network (Bishop, 1994)). The j th
output corresponds to the estimated value of the corresponding parameter ?j of that distribution (e.g.,
the mean, variance and mixing coefficients of its components). The estimated probability density
p?r (z|f (X; ?f ) = s) can then be evaluated for any z ? Z and any score s ? S.
As with generative adversarial networks, we propose to train f and r simultaneously, which we carry
out by considering the value function
E(?f , ?r ) = Lf (?f ) ? Lr (?f , ?r )
(3)
that we optimize by finding the minimax solution
??f , ??r = arg min max E(?f , ?r ).
(4)
?f
?r
Without loss of generality, the adversarial training procedure to obtain (??f , ??r ) is formally presented
in Algorithm 1 in the case of a binary classifier f : Rp ? [0, 1] modeling p(Y = 1|X). For reasons
further explained in Sec. 4, Lf and Lr are respectively set to the expected value of the negative
log-likelihood of Y |X under f and of Z|f (X; ?f ) under r:
Lf (?f ) = Ex?X Ey?Y |x [? log p?f (y|x)],
(5)
Lr (?f , ?r ) = Es?f (X;?f ) Ez?Z|s [? log p?r (z|s)].
(6)
The optimization algorithm consists in using stochastic gradient descent alternatively for solving
Eqn. 4. Finally, in the case of a class conditional pivot, the settings are the same, except that the
adversarial term Lr (?f , ?r ) is restricted to Y = y.
3
4
Theoretical results
In this section, we show that in the setting of Algorithm 1 where Lf and Lr are respectively set
to expected value of the negative log-likelihood of Y |X under f and of Z|f (X; ?f ) under r, the
minimax solution of Eqn. 4 corresponds to a classifier f which is a pivotal quantity.
In this setting, the nuisance parameter Z is considered as a random variable of prior p(Z), and
our goal is to find a function f (?; ?f ) such that f (X; ?f ) and Z are independent random variables.
Importantly, classification of Y with respect to X is considered in the context where Z is marginalized
out, which means that the classifier minimizing Lf is optimal with respect to Y |X, but not necessarily
with Y |X, Z. Results hold for a nuisance parameters Z taking either categorical or continuous values.
By abuse of notation, H(Z) denotes the differential entropy in this latter case. Finally, the proposition
below is derived in a non-parametric setting, by assuming that both f and r have enough capacity.
Proposition 1. If there exists a minimax solution (??f , ??r ) for Eqn. 4 such that E(??f , ??r ) =
H(Y |X) ? H(Z), then f (?; ??f ) is both an optimal classifier and a pivotal quantity.
Proof. For fixed ?f , the adversary r is optimal at
?
??r = arg max E(?f , ?r ) = arg min Lr (?f , ?r ),
?r
?r
(7)
in which case p ?? (z|f (X; ?f ) = s) = p(z|f (X; ?f ) = s) for all z and all s, and Lr reduces to the
?r
expected entropy Es?f (X;?f ) [H(Z|f (X; ?f ) = s)] of the conditional distribution of the nuisance
parameters. This expectation corresponds to the conditional entropy of the random variables Z and
f (X; ?f ) and can be written as H(Z|f (X; ?f )). Accordingly, the value function E can be restated
as a function depending on ?f only:
E 0 (?f ) = Lf (?f ) ? H(Z|f (X; ?f )).
(8)
In particular, we have the lower bound
H(Y |X) ? H(Z) ? Lf (?f ) ? H(Z|f (X; ?f ))
where the equality holds at ??f = arg min? E 0 (?f ) when:
(9)
f
? ??f minimizes the negative log-likelihood of Y |X under f , which happens when ??f are the
parameters of an optimal classifier. In this case, Lf reduces to its minimum value H(Y |X).
? ??f maximizes the conditional entropy H(Z|f (X; ?f )), since H(Z|f (X; ?)) ? H(Z) from
the properties of entropy. Note that this latter inequality holds for both the discrete and the
differential definitions of entropy.
By assumption, the lower bound is active, thus we have H(Z|f (X; ?f )) = H(Z) because of the
second condition, which happens exactly when Z and f (X; ?f ) are independent variables. In other
words, the optimal classifier f (?; ??f ) is also a pivotal quantity.
Proposition 1 suggests that if at each step of Algorithm 1 the adversary r is allowed to reach its
optimum given f (e.g., by setting K sufficiently high) and if f is updated to improve Lf (?f ) ?
H(Z|f (X; ?f )) with sufficiently small steps, then f should converge to a classifier that is both
optimal and pivotal, provided such a classifier exists. Therefore, the adversarial term Lr can be
regarded as a way to select among the class of all optimal classifiers a function f that is also pivotal.
Despite the former theoretical characterization of the minimax solution of Eqn. 4, let us note that
formal guarantees of convergence towards that solution by Algorithm 1 in the case where a finite
number K of steps is taken for r remains to be proven.
In practice, the assumption of existence of an optimal and pivotal classifier may not hold because the
nuisance parameter directly shapes the decision boundary. In this case, the lower bound
H(Y |X) ? H(Z) < Lf (?f ) ? H(Z|f (X; ?f ))
(10)
is strict: f can either be an optimal classifier or a pivotal quantity, but not both simultaneously. In
this situation, it is natural to rewrite the value function E as
E? (?f , ?r ) = Lf (?f ) ? ?Lr (?f , ?r ),
4
(11)
3.0
p(f(X)|Z = ? ?)
p(f(X)|Z = 0)
p(f(X)|Z = + ?)
3.5
3.0
p(f(X))
2.5
2.0
1.5
1.0
0.5
0.00.0
0.2
0.4
f(X)
0.6
1.0
0.9
0.8
2.0
Z= +?
0.7
1.5
0.6
1.0
Z=0
0.5
0.5
0.4
0.0
Z= ??
0.3
0.5
0.2
1.01.0 0.5 0.0 0.5 1.0 1.5 2.0 0.1
2.5
0.8
1.0
4.0
?0
?1 |Z = z
3.5
3.0
2.5
p(f(X))
4.0
3.0
p(f(X)|Z = ? ?)
p(f(X)|Z = 0)
p(f(X)|Z = + ?)
2.5
2.0
?0
?1 |Z = z
0.84
Z=?
1.5
0.72
0.60
2.0
1.0
1.5
0.5
1.0
0.0
0.5
0.5
0.00.0
1.01.0 0.5 0.0 0.5 1.0 1.5 2.0 0.12
0.2
0.4
f(X)
0.6
0.8
1.0
Z=0
Z= ??
0.48
0.36
0.24
Figure 2: Toy example. (Left) Conditional probability densities of the decision scores at Z = ??, 0, ? without
adversarial training. The resulting densities are dependent on the continuous parameter Z, indicating that f is
not pivotal. (Middle left) The associated decision surface, highlighting the fact that samples are easier to classify
for values of Z above ?, hence explaining the dependency. (Middle right) Conditional probability densities of
the decision scores at Z = ??, 0, ? when f is built with adversarial training. The resulting densities are now
almost identical to each other, indicating only a small dependency on Z. (Right) The associated decision surface,
illustrating how adversarial training bends the decision function vertically to erase the dependency on Z.
where ? ? 0 is a hyper-parameter controlling the trade-off between the performance of f and its
independence with respect to the nuisance parameter. Setting ? to a large value will preferably
enforces f to be pivotal while setting ? close to 0 will rather constraint f to be optimal. When the
lower bound is strict, let us note however that there may exist distinct but equally good solutions ?f , ?r
minimizing Eqn. 11. In this zero-sum game, an increase in accuracy would exactly be compensated
by a decrease in pivotality and vice-versa. How to best navigate this Pareto frontier to maximize a
higher-level objective remains a question open for future works.
Interestingly, let us finally emphasize that our results hold using only the (1D) output s of f (?; ?f ) as
input to the adversary. We could similarly enforce an intermediate representation of the data to be
pivotal, e.g. as in (Ganin and Lempitsky, 2014), but this is not necessary.
5
Experiments
In this section, we empirically demonstrate the effectiveness of the approach with a toy example
and examples from particle physics. Notably, there are no other other approaches to compare to in
the case of continuous nuisance parameters, as further explained in Sec. 6. In the case of binary
parameters, we do not expect results to be much different from previous works. The source code to
reproduce the experiments is available online 1 .
5.1
A toy example with a continous nuisance parameter
As a guiding toy example, let us consider the binary classification of 2D data drawn from multivariate
gaussians with equal priors, such that
1
?0.5
x ? N (0, 0),
when Y = 0,
(12)
?0.5
1
1 0
x|Z = z ? N (1, 1 + z),
when Y = 1.
(13)
0 1
The continuous nuisance parameter Z here represents our uncertainty about the location of the mean
of the second gaussian. Our goal is to build a classifier f (?; ?f ) for predicting Y given X, but such
that the probability distribution of f (X; ?f ) is invariant with respect to the nuisance parameter Z.
Assuming a gaussian prior z ? N (0, 1), we generate data {xi , yi , zi }N
i=1 , from which we train a
neural network f minimizing Lf (?f ) without considering its adversary r. The network architecture
comprises 2 dense hidden layers of 20 nodes respectively with tanh and ReLU activations, followed
by a dense output layer with a single node with a sigmoid activation. As shown in Fig. 2, the resulting
classifier is not pivotal, as the conditional probability densities of its decision scores f (X; ?f ) show
1
https://github.com/glouppe/paper-learning-to-pivot
5
7
6
67.5
68.0
68.5
69.0
69.5
70.0
70.5
? = 0|Z = 0
?=0
?=1
? = 10
? = 500
5
1.42
1.41
1.40
1.39
1.38
1.37
1.36
4
AMS
Lf
Lr
Lf ? ?Lr
8
0.70
0.65
0.60
0.55
0.50
0.45
3
2
1
0
0
50
100
150
1
0.0
200
T
Figure 3: Toy example. Training curves for Lf (?f ),
Lr (?f , ?r ) and Lf (?f ) ? ?Lr (?f , ?r ). Initialized
with a pre-trained classifier f , adversarial training
was performed for 200 iterations, mini-batches of
size M = 128, K = 500 and ? = 50.
0.2
0.4
0.6
threshold on f(X)
0.8
1.0
Figure 4: Physics example. Approximate median
significance as a function of the decision threshold
on the output of f . At ? = 10, trading accuracy
for independence to pileup results in a net benefit in
terms of statistical significance.
large discrepancies between values z of the nuisance parameters. While not shown here, a classifier
trained only from data generated at the nominal value Z = 0 would also not be pivotal.
Let us now consider the joint training of f against an adversary r implemented as a mixture density
network modeling Z|f (X; ?f ) as a mixture of five gaussians. The network architecture of r comprises
2 dense hidden layers of 20 nodes with ReLU activations, followed by an output layer of 15 nodes
corresponding to the means, standard deviations and mixture coefficients of the gaussians. Output
nodes for the mean values come with linear activations, output nodes for the standard deviations
with exponential activations to ensure positivity, while output nodes for the mixture coefficients
implement the softmax function to ensure positivity and normalization. When running Algorithm 1
as initialized with the classifier f obtained previously, adversarial training effectively reshapes the
decision function so it that becomes almost independent on the nuisance parameter, as shown in
Fig. 2. The conditional probability densities of the decision scores f (X; ?f ) are now very similar to
each other, indicating only a residual dependency on the nuisance, as theoretically expected. The
dynamics of adversarial training is illustrated in Fig. 3, where the losses Lf , Lr and Lf ? ?Lr are
evaluated after each iteration. In the first iterations, we observe that the global objective Lf ? ?Lr
is minimized by making the classifier less accurate, hence the corresponding increase of Lf , but
which results in a classifier that is more pivotal, hence the associated increase of Lr and the total
net benefit. As learning goes, minimizing E requires making predictions that are more accurate,
hence decreasing Lf , or that are even less dependent on Z, hence shaping p?r towards the prior p(Z).
Indeed, Lf eventually starts decreasing, while remaining bounded from below by min?f Lf (?f ) as
approximated by the dashed line in the first
? plot. Similarly, Lr tends towards the differential entropy
H(Z) of the prior (where H(Z) = log(? 2?e) = 1.419 in the case of a standard normal), as shown
by the dashed line in the second plot. Finally, let us note that the ideal situation of a classifier that
is both optimal and pivotal is unreachable for this problem, as shown in the third plot by the offset
between Lf ? ?Lr and the dashed line approximating H(Y |X) ? ?H(Z).
5.2
High energy physics examples
Binary Case Experiments at high energy colliders like the LHC (Evans and Bryant, 2008) are
searching for evidence of new particles beyond those described by the Standard Model (SM) of
particle physics. A wide array of theories predict the existence of new massive particles that would
decay to known particles in the SM such as the W boson. The W boson is unstable and can decay to
two quarks, each of which produce collimated sprays of particles known as jets. If the exotic particle is
heavy, then the W boson will be moving very fast, and relativistic effects will cause the two jets from
its decay to merge into a single ?W -jet?. These W -jets have a rich internal substructure. However,
jets are also produced ubiquitously at high energy colliders through more mundane processes in the
6
SM, which leads to a challenging classification problem that is beset with a number of sources of
systematic uncertainty. The classification challenge used here is common in jet substructure studies
(see e.g. (CMS Collaboration, 2014; ATLAS Collaboration, 2015, 2014)): we aim to distinguish
normal jets produced copiously at the LHC (Y = 0) and from W -jets (Y = 1) potentially coming
from an exotic process. We reuse the datasets used in (Baldi et al., 2016a).
Challenging in its own right, this classification problem is made all the more difficult by the presence
of pileup, or multiple proton-proton interactions occurring simultaneously with the primary interaction.
These pileup interactions produce additional particles that can contribute significant energies to jets
unrelated to the underlying discriminating information. The number of pileup interactions can vary
with the running conditions of the collider, and we want the classifier to be robust to these conditions.
Taking some liberty, we consider an extreme case with a categorical nuisance parameter, where
Z = 0 corresponds to events without pileup and Z = 1 corresponds to events with pileup, for which
there are an average of 50 independent pileup interactions overlaid.
We do not expect that we will be able to find a function f that simultaneously minimizes the
classification loss Lf and is pivotal. Thus, we need to optimize the hyper-parameter ? of Eqn. 11 with
respect to a higher-level objective. In this case, the natural higher-level context is a hypothesis test of
a null hypothesis with no Y = 1 events against an alternate hypothesis that is a mixture of Y = 0 and
Y = 1 events. In the absence of systematic uncertainties, optimizing Lf simultaneously optimizes
the power of a classical hypothesis test in the Neyman-Pearson sense. When we include systematic
uncertainties we need to balance the classification performance against the robustness to uncertainty
in Z. Since we are still performing a hypothesis test against the null, we only wish to impose the
pivotal property on Y = 0 events. To this end, we use as a higher level objective the Approximate
Median Significance (AMS), which is a natural generalization of the power of a hypothesis test when
systematic uncertainties are taken into account (see Eqn. 20 of Adam-Bourdarios et al. (2014)).
For several values of ?, we train a classifier using Algorithm 1 but consider the adversarial term
Lr conditioned on Y = 0 only, as outlined in Sec. 2. The architecture of f comprises 3 hidden
layers of 64 nodes respectively with tanh, ReLU and ReLU activations, and is terminated by a single
final output node with a sigmoid activation. The architecture of r is the same, but uses only ReLU
activations in its hidden nodes. As in the previous example, adversarial training is initialized with
f pre-trained. Experiments are performed on a subset of 150000 samples for training while AMS
is evaluated on an independent test set of 5000000 samples. Both training and testing samples are
weighted such that the null hypothesis corresponded to 1000 of Y = 0 events and the alternate
hypothesis included an additional 100 Y = 1 events prior to any thresholding on f . This allows us
to probe the efficacy of the method proposed here in a representative background-dominated high
energy physics environment. Results reported below are averages over 5 runs.
As Fig. 4 illustrates, without adversarial training (at ? = 0|Z = 0 when building a classifier at the
nominal value Z = 0 only, or at ? = 0 when building a classifier on data sampled from p(X, Y, Z)),
the AMS peaks at 7. By contrast, as the pivotal constraint is made stronger (for ? > 0) the AMS
peak moves higher, with a maximum value around 7.8 for ? = 10. Trading classification accuracy
for robustness to pileup thereby results in a net benefit in terms of the power of the hypothesis
test. Setting ? too high however (e.g. ? = 500) results in a decrease of the maximum AMS, by
focusing the capacity of f too strongly on independence with Z, at the expense of accuracy. In effect,
optimizing ? yields a principled and effective approach to control the trade-off between accuracy and
robustness that ultimately maximizes the power of the enveloping hypothesis test.
Continous Case Recently, an independent group has used our approach to learn jet classifiers that
are independent of the jet mass (Shimmin et al., 2017), which is a continuous attribute. The results of
their studies show that the adversarial training strategy works very well for real-world problems with
continuous attributes, thus enhancing the sensitivity of searches for new physics at the LHC.
6
Related work
Learning to pivot can be related to the problem of domain adaptation (Blitzer et al., 2006; Pan et al.,
2011; Gopalan et al., 2011; Gong et al., 2013; Baktashmotlagh et al., 2013; Ajakan et al., 2014;
Ganin and Lempitsky, 2014), where the goal is often stated as trying to learn a domain-invariant
representation of the data. Likewise, our method also relates to the problem of enforcing fairness
7
in classification (Kamishima et al., 2012; Zemel et al., 2013; Feldman et al., 2015; Edwards and
Storkey, 2015; Zafar et al., 2015; Louizos et al., 2015), which is stated as learning a classifier that is
independent of some chosen attribute such as gender, color or age. For both families of methods, the
problem can equivalently be stated as learning a classifier which is a pivotal quantity with respect
to either the domain or the selected feature. As an example, unsupervised domain adaptation with
labeled data from a source domain and unlabeled data from a target domain can be recast as learning
a predictive model f (i.e., trained to minimize Lf evaluated on labeled source data only) that is also a
pivot with respect to the domain Z (i.e., trained to maximize Lr evaluated on both source and target
data). In this context, (Ganin and Lempitsky, 2014; Edwards and Storkey, 2015) are certainly among
the closest to our work, in which domain invariance and fairness are enforced through an adversarial
minimax setup composed of a classifier and an adversarial discriminator. Following this line of work,
our method can be regarded as a unified generalization that also supports a continuously parametrized
family of domains or as enforcing fairness over continuous attributes.
Most related work is based on the strong and limiting assumption that Z is a binary random variable
(e.g., Z = 0 for the source domain, and Z = 1 for the target domain). In particular, (Pan et al., 2011;
Gong et al., 2013; Baktashmotlagh et al., 2013; Zemel et al., 2013; Ganin and Lempitsky, 2014;
Ajakan et al., 2014; Edwards and Storkey, 2015; Louizos et al., 2015) are all based on the minimization
of some form of divergence between the two distributions of f (X)|Z = 0 and f (X)|Z = 1. For this
reason, these works cannot directly be generalized to non-binary or continuous nuisance parameters,
both from a practical and theoretical point of view. Notably, Kamishima et al. (2012) enforces
fairness through a prejudice regularization term based on empirical estimates of p(f (X)|Z). While
this approach is in principle sufficient for handling non-binary nuisance parameters Z, it requires
accurate empirical estimates of p(f (X)|Z = z) for all values z, which quickly becomes impractical
as the cardinality of Z increases. By contrast, our approach models the conditional dependence
through an adversarial network, which allows for generalization without necessarily requiring an
exponentially growing number of training examples.
A common approach to account for systematic uncertainties in a scientific context (e.g. in high energy
physics) is to take as fixed a classifier f built from training data for a nominal value z0 of the nuisance
parameter, and then propagate uncertainty by estimating p(f (x)|z) with a parametrized calibration
procedure. Clearly, this classifier is however not optimal for z 6= z0 . To overcome this issue, the
classifier f is sometimes built instead on a mixture of training data generated from several plausible
values z0 , z1 , . . . of the nuisance parameter. While this certainly improves classification performance
with respect to the marginal model p(X, Y ), there is no reason to expect the resulting classifier to
be pivotal, as shown previously in Sec. 5.1. As an alternative, parametrized classifiers (Cranmer
et al., 2015; Baldi et al., 2016b) directly take (nuisance) parameters as additional input variables,
hence ultimately providing the most statistically powerful approach for incorporating the effect
of systematics on the underlying classification task. In practice, parametrized classifiers are also
computationally expensive to build and evaluate. In particular, calibrating their decision function,
i.e. approximating p(f (x, z)|y, z) as a continuous function of z, remains an open challenge. By
contrast, constraining f to be pivotal yields a classifier that can be directly used in a wider range of
applications, since the dependence on the nuisance parameter Z has already been eliminated.
7
Conclusions
In this work, we proposed a flexible learning procedure for building a predictive model that is
independent of continuous or categorical nuisance parameters by jointly training two neural networks
in an adversarial fashion. From a theoretical perspective, we motivated the proposed algorithm by
showing that the minimax value of its value function corresponds to a predictive model that is both
optimal and pivotal (if that models exists) or for which one can tune the trade-off between power and
robustness. From an empirical point of view, we confirmed the effectiveness of our method on a toy
example and a particle physics example.
In terms of applications, our solution can be used in any situation where the training data may not be
representative of the real data the predictive model will be applied to in practice. In the scientific
context, the presence of systematic uncertainty can be incorporated by considering a family of data
generation processes, and it would be worth revisiting those scientific problems that utilize machine
learning in light of this technique. The approach also extends to cases where independence of the
predictive model with respect to observed random variables is desired, as in fairness for classification.
8
Acknowledgements
We would like to thank the authors of (Baldi et al., 2016a) for sharing the data used in their studies.
KC and GL are both supported through NSF ACI-1450310, additionally KC is supported through
PHY-1505463 and PHY-1205376. MK is supported by the US Department of Energy (DOE) under
grant DE-AC02-76SF00515 and by the SLAC Panofsky Fellowship.
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ICML (3), 28:325?333.
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