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4,800 | 5,346 | Sequence to Sequence Learning
with Neural Networks
Ilya Sutskever
Google
[email protected]
Oriol Vinyals
Google
[email protected]
Quoc V. Le
Google
[email protected]
Abstract
Deep Neural Networks (DNNs) are powerful models that have achieved excellent performance on difficult learning tasks. Although DNNs work well whenever
large labeled training sets are available, they cannot be used to map sequences to
sequences. In this paper, we present a general end-to-end approach to sequence
learning that makes minimal assumptions on the sequence structure. Our method
uses a multilayered Long Short-Term Memory (LSTM) to map the input sequence
to a vector of a fixed dimensionality, and then another deep LSTM to decode the
target sequence from the vector. Our main result is that on an English to French
translation task from the WMT-14 dataset, the translations produced by the LSTM
achieve a BLEU score of 34.8 on the entire test set, where the LSTM?s BLEU
score was penalized on out-of-vocabulary words. Additionally, the LSTM did not
have difficulty on long sentences. For comparison, a phrase-based SMT system
achieves a BLEU score of 33.3 on the same dataset. When we used the LSTM
to rerank the 1000 hypotheses produced by the aforementioned SMT system, its
BLEU score increases to 36.5, which is close to the previous state of the art. The
LSTM also learned sensible phrase and sentence representations that are sensitive
to word order and are relatively invariant to the active and the passive voice. Finally, we found that reversing the order of the words in all source sentences (but
not target sentences) improved the LSTM?s performance markedly, because doing
so introduced many short term dependencies between the source and the target
sentence which made the optimization problem easier.
1
Introduction
Deep Neural Networks (DNNs) are extremely powerful machine learning models that achieve excellent performance on difficult problems such as speech recognition [13, 7] and visual object recognition [19, 6, 21, 20]. DNNs are powerful because they can perform arbitrary parallel computation
for a modest number of steps. A surprising example of the power of DNNs is their ability to sort
N N -bit numbers using only 2 hidden layers of quadratic size [27]. So, while neural networks are
related to conventional statistical models, they learn an intricate computation. Furthermore, large
DNNs can be trained with supervised backpropagation whenever the labeled training set has enough
information to specify the network?s parameters. Thus, if there exists a parameter setting of a large
DNN that achieves good results (for example, because humans can solve the task very rapidly),
supervised backpropagation will find these parameters and solve the problem.
Despite their flexibility and power, DNNs can only be applied to problems whose inputs and targets
can be sensibly encoded with vectors of fixed dimensionality. It is a significant limitation, since
many important problems are best expressed with sequences whose lengths are not known a-priori.
For example, speech recognition and machine translation are sequential problems. Likewise, question answering can also be seen as mapping a sequence of words representing the question to a
1
sequence of words representing the answer. It is therefore clear that a domain-independent method
that learns to map sequences to sequences would be useful.
Sequences pose a challenge for DNNs because they require that the dimensionality of the inputs and
outputs is known and fixed. In this paper, we show that a straightforward application of the Long
Short-Term Memory (LSTM) architecture [16] can solve general sequence to sequence problems.
The idea is to use one LSTM to read the input sequence, one timestep at a time, to obtain large fixeddimensional vector representation, and then to use another LSTM to extract the output sequence
from that vector (fig. 1). The second LSTM is essentially a recurrent neural network language model
[28, 23, 30] except that it is conditioned on the input sequence. The LSTM?s ability to successfully
learn on data with long range temporal dependencies makes it a natural choice for this application
due to the considerable time lag between the inputs and their corresponding outputs (fig. 1).
There have been a number of related attempts to address the general sequence to sequence learning
problem with neural networks. Our approach is closely related to Kalchbrenner and Blunsom [18]
who were the first to map the entire input sentence to vector, and is very similar to Cho et al. [5].
Graves [10] introduced a novel differentiable attention mechanism that allows neural networks to
focus on different parts of their input, and an elegant variant of this idea was successfully applied
to machine translation by Bahdanau et al. [2]. The Connectionist Sequence Classification is another
popular technique for mapping sequences to sequences with neural networks, although it assumes a
monotonic alignment between the inputs and the outputs [11].
Figure 1: Our model reads an input sentence ?ABC? and produces ?WXYZ? as the output sentence. The
model stops making predictions after outputting the end-of-sentence token. Note that the LSTM reads the
input sentence in reverse, because doing so introduces many short term dependencies in the data that make the
optimization problem much easier.
The main result of this work is the following. On the WMT?14 English to French translation task,
we obtained a BLEU score of 34.81 by directly extracting translations from an ensemble of 5 deep
LSTMs (with 380M parameters each) using a simple left-to-right beam-search decoder. This is
by far the best result achieved by direct translation with large neural networks. For comparison,
the BLEU score of a SMT baseline on this dataset is 33.30 [29]. The 34.81 BLEU score was
achieved by an LSTM with a vocabulary of 80k words, so the score was penalized whenever the
reference translation contained a word not covered by these 80k. This result shows that a relatively
unoptimized neural network architecture which has much room for improvement outperforms a
mature phrase-based SMT system.
Finally, we used the LSTM to rescore the publicly available 1000-best lists of the SMT baseline on
the same task [29]. By doing so, we obtained a BLEU score of 36.5, which improves the baseline
by 3.2 BLEU points and is close to the previous state-of-the-art (which is 37.0 [9]).
Surprisingly, the LSTM did not suffer on very long sentences, despite the recent experience of other
researchers with related architectures [26]. We were able to do well on long sentences because we
reversed the order of words in the source sentence but not the target sentences in the training and test
set. By doing so, we introduced many short term dependencies that made the optimization problem
much simpler (see sec. 2 and 3.3). As a result, SGD could learn LSTMs that had no trouble with
long sentences. The simple trick of reversing the words in the source sentence is one of the key
technical contributions of this work.
A useful property of the LSTM is that it learns to map an input sentence of variable length into
a fixed-dimensional vector representation. Given that translations tend to be paraphrases of the
source sentences, the translation objective encourages the LSTM to find sentence representations
that capture their meaning, as sentences with similar meanings are close to each other while different
2
sentences meanings will be far. A qualitative evaluation supports this claim, showing that our model
is aware of word order and is fairly invariant to the active and passive voice.
2
The model
The Recurrent Neural Network (RNN) [31, 28] is a natural generalization of feedforward neural
networks to sequences. Given a sequence of inputs (x1 , . . . , xT ), a standard RNN computes a
sequence of outputs (y1 , . . . , yT ) by iterating the following equation:
ht = sigm W hx xt + W hh ht?1
yt
=
W yh ht
The RNN can easily map sequences to sequences whenever the alignment between the inputs the
outputs is known ahead of time. However, it is not clear how to apply an RNN to problems whose
input and the output sequences have different lengths with complicated and non-monotonic relationships.
A simple strategy for general sequence learning is to map the input sequence to a fixed-sized vector
using one RNN, and then to map the vector to the target sequence with another RNN (this approach
has also been taken by Cho et al. [5]). While it could work in principle since the RNN is provided
with all the relevant information, it would be difficult to train the RNNs due to the resulting long
term dependencies [14, 4] (figure 1) [16, 15]. However, the Long Short-Term Memory (LSTM) [16]
is known to learn problems with long range temporal dependencies, so an LSTM may succeed in
this setting.
The goal of the LSTM is to estimate the conditional probability p(y1 , . . . , yT ? |x1 , . . . , xT ) where
(x1 , . . . , xT ) is an input sequence and y1 , . . . , yT ? is its corresponding output sequence whose length
T ? may differ from T . The LSTM computes this conditional probability by first obtaining the fixeddimensional representation v of the input sequence (x1 , . . . , xT ) given by the last hidden state of the
LSTM, and then computing the probability of y1 , . . . , yT ? with a standard LSTM-LM formulation
whose initial hidden state is set to the representation v of x1 , . . . , xT :
?
p(y1 , . . . , yT ? |x1 , . . . , xT ) =
T
Y
p(yt |v, y1 , . . . , yt?1 )
(1)
t=1
In this equation, each p(yt |v, y1 , . . . , yt?1 ) distribution is represented with a softmax over all the
words in the vocabulary. We use the LSTM formulation from Graves [10]. Note that we require that
each sentence ends with a special end-of-sentence symbol ?<EOS>?, which enables the model to
define a distribution over sequences of all possible lengths. The overall scheme is outlined in figure
1, where the shown LSTM computes the representation of ?A?, ?B?, ?C?, ?<EOS>? and then uses
this representation to compute the probability of ?W?, ?X?, ?Y?, ?Z?, ?<EOS>?.
Our actual models differ from the above description in three important ways. First, we used two
different LSTMs: one for the input sequence and another for the output sequence, because doing
so increases the number model parameters at negligible computational cost and makes it natural to
train the LSTM on multiple language pairs simultaneously [18]. Second, we found that deep LSTMs
significantly outperformed shallow LSTMs, so we chose an LSTM with four layers. Third, we found
it extremely valuable to reverse the order of the words of the input sentence. So for example, instead
of mapping the sentence a, b, c to the sentence ?, ?, ?, the LSTM is asked to map c, b, a to ?, ?, ?,
where ?, ?, ? is the translation of a, b, c. This way, a is in close proximity to ?, b is fairly close to
?, and so on, a fact that makes it easy for SGD to ?establish communication? between the input and
the output. We found this simple data transformation to greatly boost the performance of the LSTM.
3
Experiments
We applied our method to the WMT?14 English to French MT task in two ways. We used it to
directly translate the input sentence without using a reference SMT system and we it to rescore the
n-best lists of an SMT baseline. We report the accuracy of these translation methods, present sample
translations, and visualize the resulting sentence representation.
3
3.1
Dataset details
We used the WMT?14 English to French dataset. We trained our models on a subset of 12M sentences consisting of 348M French words and 304M English words, which is a clean ?selected?
subset from [29]. We chose this translation task and this specific training set subset because of the
public availability of a tokenized training and test set together with 1000-best lists from the baseline
SMT [29].
As typical neural language models rely on a vector representation for each word, we used a fixed
vocabulary for both languages. We used 160,000 of the most frequent words for the source language
and 80,000 of the most frequent words for the target language. Every out-of-vocabulary word was
replaced with a special ?UNK? token.
3.2
Decoding and Rescoring
The core of our experiments involved training a large deep LSTM on many sentence pairs. We
trained it by maximizing the log probability of a correct translation T given the source sentence S,
so the training objective is
X
1/|S|
log p(T |S)
(T,S)?S
where S is the training set. Once training is complete, we produce translations by finding the most
likely translation according to the LSTM:
T? = arg max p(T |S)
T
(2)
We search for the most likely translation using a simple left-to-right beam search decoder which
maintains a small number B of partial hypotheses, where a partial hypothesis is a prefix of some
translation. At each timestep we extend each partial hypothesis in the beam with every possible
word in the vocabulary. This greatly increases the number of the hypotheses so we discard all but
the B most likely hypotheses according to the model?s log probability. As soon as the ?<EOS>?
symbol is appended to a hypothesis, it is removed from the beam and is added to the set of complete
hypotheses. While this decoder is approximate, it is simple to implement. Interestingly, our system
performs well even with a beam size of 1, and a beam of size 2 provides most of the benefits of beam
search (Table 1).
We also used the LSTM to rescore the 1000-best lists produced by the baseline system [29]. To
rescore an n-best list, we computed the log probability of every hypothesis with our LSTM and took
an even average with their score and the LSTM?s score.
3.3
Reversing the Source Sentences
While the LSTM is capable of solving problems with long term dependencies, we discovered that
the LSTM learns much better when the source sentences are reversed (the target sentences are not
reversed). By doing so, the LSTM?s test perplexity dropped from 5.8 to 4.7, and the test BLEU
scores of its decoded translations increased from 25.9 to 30.6.
While we do not have a complete explanation to this phenomenon, we believe that it is caused by
the introduction of many short term dependencies to the dataset. Normally, when we concatenate a
source sentence with a target sentence, each word in the source sentence is far from its corresponding
word in the target sentence. As a result, the problem has a large ?minimal time lag? [17]. By
reversing the words in the source sentence, the average distance between corresponding words in
the source and target language is unchanged. However, the first few words in the source language
are now very close to the first few words in the target language, so the problem?s minimal time lag is
greatly reduced. Thus, backpropagation has an easier time ?establishing communication? between
the source sentence and the target sentence, which in turn results in substantially improved overall
performance.
Initially, we believed that reversing the input sentences would only lead to more confident predictions in the early parts of the target sentence and to less confident predictions in the later parts. However, LSTMs trained on reversed source sentences did much better on long sentences than LSTMs
4
trained on the raw source sentences (see sec. 3.7), which suggests that reversing the input sentences
results in LSTMs with better memory utilization.
3.4
Training details
We found that the LSTM models are fairly easy to train. We used deep LSTMs with 4 layers,
with 1000 cells at each layer and 1000 dimensional word embeddings, with an input vocabulary
of 160,000 and an output vocabulary of 80,000. We found deep LSTMs to significantly outperform
shallow LSTMs, where each additional layer reduced perplexity by nearly 10%, possibly due to their
much larger hidden state. We used a naive softmax over 80,000 words at each output. The resulting
LSTM has 380M parameters of which 64M are pure recurrent connections (32M for the ?encoder?
LSTM and 32M for the ?decoder? LSTM). The complete training details are given below:
? We initialized all of the LSTM?s parameters with the uniform distribution between -0.08
and 0.08
? We used stochastic gradient descent without momentum, with a fixed learning rate of 0.7.
After 5 epochs, we begun halving the learning rate every half epoch. We trained our models
for a total of 7.5 epochs.
? We used batches of 128 sequences for the gradient and divided it the size of the batch
(namely, 128).
? Although LSTMs tend to not suffer from the vanishing gradient problem, they can have
exploding gradients. Thus we enforced a hard constraint on the norm of the gradient [10,
25] by scaling it when its norm exceeded a threshold. For each training batch, we compute
s = kgk2 , where g is the gradient divided by 128. If s > 5, we set g = 5g
s .
? Different sentences have different lengths. Most sentences are short (e.g., length 20-30)
but some sentences are long (e.g., length > 100), so a minibatch of 128 randomly chosen
training sentences will have many short sentences and few long sentences, and as a result,
much of the computation in the minibatch is wasted. To address this problem, we made
sure that all sentences within a minibatch were roughly of the same length, which a 2x
speedup.
3.5
Parallelization
A C++ implementation of deep LSTM with the configuration from the previous section on a single GPU processes a speed of approximately 1,700 words per second. This was too slow for our
purposes, so we parallelized our model using an 8-GPU machine. Each layer of the LSTM was
executed on a different GPU and communicated its activations to the next GPU (or layer) as soon
as they were computed. Our models have 4 layers of LSTMs, each of which resides on a separate
GPU. The remaining 4 GPUs were used to parallelize the softmax, so each GPU was responsible
for multiplying by a 1000 ? 20000 matrix. The resulting implementation achieved a speed of 6,300
(both English and French) words per second with a minibatch size of 128. Training took about a ten
days with this implementation.
3.6
Experimental Results
We used the cased BLEU score [24] to evaluate the quality of our translations. We computed our
BLEU scores using multi-bleu.pl1 on the tokenized predictions and ground truth. This way
of evaluating the BELU score is consistent with [5] and [2], and reproduces the 33.3 score of [29].
However, if we evaluate the state of the art system of [9] (whose predictions can be downloaded
from statmt.org\matrix) in this manner, we get 37.0, which is greater than the 35.8 reported
by statmt.org\matrix.
The results are presented in tables 1 and 2. Our best results are obtained with an ensemble of
LSTMs that differ in their random initializations and in the random order of minibatches. While the
decoded translations of the LSTM ensemble do not beat the state of the art, it is the first time that
a pure neural translation system outperforms a phrase-based SMT baseline on a large MT task by
1
There several variants of the BLEU score, and each variant is defined with a perl script.
5
Method
Bahdanau et al. [2]
Baseline System [29]
Single forward LSTM, beam size 12
Single reversed LSTM, beam size 12
Ensemble of 5 reversed LSTMs, beam size 1
Ensemble of 2 reversed LSTMs, beam size 12
Ensemble of 5 reversed LSTMs, beam size 2
Ensemble of 5 reversed LSTMs, beam size 12
test BLEU score (ntst14)
28.45
33.30
26.17
30.59
33.00
33.27
34.50
34.81
Table 1: The performance of the LSTM on WMT?14 English to French test set (ntst14). Note that
an ensemble of 5 LSTMs with a beam of size 2 is cheaper than of a single LSTM with a beam of
size 12.
Method
Baseline System [29]
Cho et al. [5]
State of the art [9]
Rescoring the baseline 1000-best with a single forward LSTM
Rescoring the baseline 1000-best with a single reversed LSTM
Rescoring the baseline 1000-best with an ensemble of 5 reversed LSTMs
Oracle Rescoring of the Baseline 1000-best lists
test BLEU score (ntst14)
33.30
34.54
37.0
35.61
35.85
36.5
?45
Table 2: Methods that use neural networks together with an SMT system on the WMT?14 English
to French test set (ntst14).
a sizeable margin, despite its inability to handle out-of-vocabulary words. The LSTM is within 0.5
BLEU points of the previous state of the art by rescoring the 1000-best list of the baseline system.
3.7
Performance on long sentences
We were surprised to discover that the LSTM did well on long sentences, which is shown quantitatively in figure 3. Table 3 presents several examples of long sentences and their translations.
3.8
Model Analysis
15
I was given a card by her in the garden
4
Mary admires John
3
2
In the garden , she gave me a card
She gave me a card in the garden
10
Mary is in love with John
5
1
0
?1
0
Mary respects John
John admires Mary
?5
John is in love with Mary
?2
She was given a card by me in the garden
In the garden , I gave her a card
?3
?10
?4
?5
?6
?8
?15
John respects Mary
?6
?4
?2
0
2
4
6
8
?20
?15
10
I gave her a card in the garden
?10
?5
0
5
10
15
20
Figure 2: The figure shows a 2-dimensional PCA projection of the LSTM hidden states that are obtained
after processing the phrases in the figures. The phrases are clustered by meaning, which in these examples is
primarily a function of word order, which would be difficult to capture with a bag-of-words model. Notice that
both clusters have similar internal structure.
One of the attractive features of our model is its ability to turn a sequence of words into a vector
of fixed dimensionality. Figure 2 visualizes some of the learned representations. The figure clearly
shows that the representations are sensitive to the order of words, while being fairly insensitive to the
6
Type
Our model
Sentence
Ulrich UNK , membre du conseil d? administration du constructeur automobile Audi ,
affirme qu? il s? agit d? une pratique courante depuis des ann?ees pour que les t?el?ephones
portables puissent e? tre collect?es avant les r?eunions du conseil d? administration afin qu? ils
ne soient pas utilis?es comme appareils d? e? coute a` distance .
Ulrich Hackenberg , membre du conseil d? administration du constructeur automobile Audi ,
d?eclare que la collecte des t?el?ephones portables avant les r?eunions du conseil , afin qu? ils
ne puissent pas e? tre utilis?es comme appareils d? e? coute a` distance , est une pratique courante
depuis des ann?ees .
? Les t?el?ephones cellulaires , qui sont vraiment une question , non seulement parce qu? ils
pourraient potentiellement causer des interf?erences avec les appareils de navigation , mais
nous savons , selon la FCC , qu? ils pourraient interf?erer avec les tours de t?el?ephone cellulaire
lorsqu? ils sont dans l? air ? , dit UNK .
? Les t?el?ephones portables sont v?eritablement un probl`eme , non seulement parce qu? ils
pourraient e? ventuellement cr?eer des interf?erences avec les instruments de navigation , mais
parce que nous savons , d? apr`es la FCC , qu? ils pourraient perturber les antennes-relais de
t?el?ephonie mobile s? ils sont utilis?es a` bord ? , a d?eclar?e Rosenker .
Avec la cr?emation , il y a un ? sentiment de violence contre le corps d? un e? tre cher ? ,
qui sera ? r?eduit a` une pile de cendres ? en tr`es peu de temps au lieu d? un processus de
d?ecomposition ? qui accompagnera les e? tapes du deuil ? .
Il y a , avec la cr?emation , ? une violence faite au corps aim?e ? ,
qui va e? tre ? r?eduit a` un tas de cendres ? en tr`es peu de temps , et non apr`es un processus de
d?ecomposition , qui ? accompagnerait les phases du deuil ? .
Truth
Our model
Truth
Our model
Truth
Table 3: A few examples of long translations produced by the LSTM alongside the ground truth
translations. The reader can verify that the translations are sensible using Google translate.
LSTM (34.8)
baseline (33.3)
40
40
35
BLEU score
BLEU score
35
30
25
20
LSTM (34.8)
baseline (33.3)
30
25
478
12
17
22
28
35
test sentences sorted by their length
20
0
79
500
1000
1500
2000
2500
3000
test sentences sorted by average word frequency rank
3500
Figure 3: The left plot shows the performance of our system as a function of sentence length, where the
x-axis corresponds to the test sentences sorted by their length and is marked by the actual sequence lengths.
There is no degradation on sentences with less than 35 words, there is only a minor degradation on the longest
sentences. The right plot shows the LSTM?s performance on sentences with progressively more rare words,
where the x-axis corresponds to the test sentences sorted by their ?average word frequency rank?.
replacement of an active voice with a passive voice. The two-dimensional projections are obtained
using PCA.
4
Related work
There is a large body of work on applications of neural networks to machine translation. So far,
the simplest and most effective way of applying an RNN-Language Model (RNNLM) [23] or a
7
Feedforward Neural Network Language Model (NNLM) [3] to an MT task is by rescoring the nbest lists of a strong MT baseline [22], which reliably improves translation quality.
More recently, researchers have begun to look into ways of including information about the source
language into the NNLM. Examples of this work include Auli et al. [1], who combine an NNLM
with a topic model of the input sentence, which improves rescoring performance. Devlin et al. [8]
followed a similar approach, but they incorporated their NNLM into the decoder of an MT system
and used the decoder?s alignment information to provide the NNLM with the most useful words in
the input sentence. Their approach was highly successful and it achieved large improvements over
their baseline.
Our work is closely related to Kalchbrenner and Blunsom [18], who were the first to map the input
sentence into a vector and then back to a sentence, although they map sentences to vectors using
convolutional neural networks, which lose the ordering of the words. Similarly to this work, Cho et
al. [5] used an LSTM-like RNN architecture to map sentences into vectors and back, although their
primary focus was on integrating their neural network into an SMT system. Bahdanau et al. [2] also
attempted direct translations with a neural network that used an attention mechanism to overcome
the poor performance on long sentences experienced by Cho et al. [5] and achieved encouraging
results. Likewise, Pouget-Abadie et al. [26] attempted to address the memory problem of Cho et
al. [5] by translating pieces of the source sentence in way that produces smooth translations, which
is similar to a phrase-based approach. We suspect that they could achieve similar improvements by
simply training their networks on reversed source sentences.
End-to-end training is also the focus of Hermann et al. [12], whose model represents the inputs and
outputs by feedforward networks, and map them to similar points in space. However, their approach
cannot generate translations directly: to get a translation, they need to do a look up for closest vector
in the pre-computed database of sentences, or to rescore a sentence.
5
Conclusion
In this work, we showed that a large deep LSTM with a limited vocabulary can outperform a standard SMT-based system whose vocabulary is unlimited on a large-scale MT task. The success of
our simple LSTM-based approach on MT suggests that it should do well on many other sequence
learning problems, provided they have enough training data.
We were surprised by the extent of the improvement obtained by reversing the words in the source
sentences. We conclude that it is important to find a problem encoding that has the greatest number
of short term dependencies, as they make the learning problem much simpler. In particular, while
we were unable to train a standard RNN on the non-reversed translation problem (shown in fig. 1),
we believe that a standard RNN should be easily trainable when the source sentences are reversed
(although we did not verify it experimentally).
We were also surprised by the ability of the LSTM to correctly translate very long sentences. We
were initially convinced that the LSTM would fail on long sentences due to its limited memory,
and other researchers reported poor performance on long sentences with a model similar to ours
[5, 2, 26]. And yet, LSTMs trained on the reversed dataset had little difficulty translating long
sentences.
Most importantly, we demonstrated that a simple, straightforward and a relatively unoptimized approach can outperform a mature SMT system, so further work will likely lead to even greater translation accuracies. These results suggest that our approach will likely do well on other challenging
sequence to sequence problems.
6
Acknowledgments
We thank Samy Bengio, Jeff Dean, Matthieu Devin, Geoffrey Hinton, Nal Kalchbrenner, Thang Luong, Wolfgang Macherey, Rajat Monga, Vincent Vanhoucke, Peng Xu, Wojciech Zaremba, and the Google Brain team
for useful comments and discussions.
8
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9
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phenomenon:1 |
4,801 | 5,347 | How transferable are features in deep neural
networks?
Jason Yosinski,1 Jeff Clune,2 Yoshua Bengio,3 and Hod Lipson4
1
Dept. Computer Science, Cornell University
2
Dept. Computer Science, University of Wyoming
3
Dept. Computer Science & Operations Research, University of Montreal
4
Dept. Mechanical & Aerospace Engineering, Cornell University
Abstract
Many deep neural networks trained on natural images exhibit a curious phenomenon in common: on the first layer they learn features similar to Gabor filters
and color blobs. Such first-layer features appear not to be specific to a particular
dataset or task, but general in that they are applicable to many datasets and tasks.
Features must eventually transition from general to specific by the last layer of
the network, but this transition has not been studied extensively. In this paper we
experimentally quantify the generality versus specificity of neurons in each layer
of a deep convolutional neural network and report a few surprising results. Transferability is negatively affected by two distinct issues: (1) the specialization of
higher layer neurons to their original task at the expense of performance on the
target task, which was expected, and (2) optimization difficulties related to splitting networks between co-adapted neurons, which was not expected. In an example network trained on ImageNet, we demonstrate that either of these two issues
may dominate, depending on whether features are transferred from the bottom,
middle, or top of the network. We also document that the transferability of features decreases as the distance between the base task and target task increases, but
that transferring features even from distant tasks can be better than using random
features. A final surprising result is that initializing a network with transferred
features from almost any number of layers can produce a boost to generalization
that lingers even after fine-tuning to the target dataset.
1
Introduction
Modern deep neural networks exhibit a curious phenomenon: when trained on images, they all tend
to learn first-layer features that resemble either Gabor filters or color blobs. The appearance of these
filters is so common that obtaining anything else on a natural image dataset causes suspicion of
poorly chosen hyperparameters or a software bug. This phenomenon occurs not only for different
datasets, but even with very different training objectives, including supervised image classification
(Krizhevsky et al., 2012), unsupervised density learning (Lee et al., 2009), and unsupervised learning of sparse representations (Le et al., 2011).
Because finding these standard features on the first layer seems to occur regardless of the exact cost
function and natural image dataset, we call these first-layer features general. On the other hand, we
know that the features computed by the last layer of a trained network must depend greatly on the
chosen dataset and task. For example, in a network with an N-dimensional softmax output layer that
has been successfully trained toward a supervised classification objective, each output unit will be
specific to a particular class. We thus call the last-layer features specific. These are intuitive notions
of general and specific for which we will provide more rigorous definitions below. If first-layer
1
features are general and last-layer features are specific, then there must be a transition from general
to specific somewhere in the network. This observation raises a few questions:
? Can we quantify the degree to which a particular layer is general or specific?
? Does the transition occur suddenly at a single layer, or is it spread out over several layers?
? Where does this transition take place: near the first, middle, or last layer of the network?
We are interested in the answers to these questions because, to the extent that features within a
network are general, we will be able to use them for transfer learning (Caruana, 1995; Bengio
et al., 2011; Bengio, 2011). In transfer learning, we first train a base network on a base dataset and
task, and then we repurpose the learned features, or transfer them, to a second target network to
be trained on a target dataset and task. This process will tend to work if the features are general,
meaning suitable to both base and target tasks, instead of specific to the base task.
When the target dataset is significantly smaller than the base dataset, transfer learning can be a
powerful tool to enable training a large target network without overfitting; Recent studies have
taken advantage of this fact to obtain state-of-the-art results when transferring from higher layers
(Donahue et al., 2013a; Zeiler and Fergus, 2013; Sermanet et al., 2014), collectively suggesting that
these layers of neural networks do indeed compute features that are fairly general. These results
further emphasize the importance of studying the exact nature and extent of this generality.
The usual transfer learning approach is to train a base network and then copy its first n layers to
the first n layers of a target network. The remaining layers of the target network are then randomly
initialized and trained toward the target task. One can choose to backpropagate the errors from
the new task into the base (copied) features to fine-tune them to the new task, or the transferred
feature layers can be left frozen, meaning that they do not change during training on the new task.
The choice of whether or not to fine-tune the first n layers of the target network depends on the
size of the target dataset and the number of parameters in the first n layers. If the target dataset is
small and the number of parameters is large, fine-tuning may result in overfitting, so the features
are often left frozen. On the other hand, if the target dataset is large or the number of parameters is
small, so that overfitting is not a problem, then the base features can be fine-tuned to the new task
to improve performance. Of course, if the target dataset is very large, there would be little need to
transfer because the lower level filters could just be learned from scratch on the target dataset. We
compare results from each of these two techniques ? fine-tuned features or frozen features ? in
the following sections.
In this paper we make several contributions:
1. We define a way to quantify the degree to which a particular layer is general or specific, namely,
how well features at that layer transfer from one task to another (Section 2). We then train pairs
of convolutional neural networks on the ImageNet dataset and characterize the layer-by-layer
transition from general to specific (Section 4), which yields the following four results.
2. We experimentally show two separate issues that cause performance degradation when using transferred features without fine-tuning: (i) the specificity of the features themselves, and
(ii) optimization difficulties due to splitting the base network between co-adapted neurons on
neighboring layers. We show how each of these two effects can dominate at different layers of
the network. (Section 4.1)
3. We quantify how the performance benefits of transferring features decreases the more dissimilar
the base task and target task are. (Section 4.2)
4. On the relatively large ImageNet dataset, we find lower performance than has been previously
reported for smaller datasets (Jarrett et al., 2009) when using features computed from random
lower-layer weights vs. trained weights. We compare random weights to transferred weights?
both frozen and fine-tuned?and find the transferred weights perform better. (Section 4.3)
5. Finally, we find that initializing a network with transferred features from almost any number
of layers can produce a boost to generalization performance after fine-tuning to a new dataset.
This is particularly surprising because the effect of having seen the first dataset persists even
after extensive fine-tuning. (Section 4.1)
2
2
Generality vs. Specificity Measured as Transfer Performance
We have noted the curious tendency of Gabor filters and color blobs to show up in the first layer of
neural networks trained on natural images. In this study, we define the degree of generality of a set
of features learned on task A as the extent to which the features can be used for another task B. It
is important to note that this definition depends on the similarity between A and B. We create pairs
of classification tasks A and B by constructing pairs of non-overlapping subsets of the ImageNet
dataset.1 These subsets can be chosen to be similar to or different from each other.
To create tasks A and B, we randomly split the 1000 ImageNet classes into two groups each containing 500 classes and approximately half of the data, or about 645,000 examples each. We train
one eight-layer convolutional network on A and another on B. These networks, which we call baseA
and baseB, are shown in the top two rows of Figure 1. We then choose a layer n from {1, 2, . . . , 7}
and train several new networks. In the following explanation and in Figure 1, we use layer n = 3 as
the example layer chosen. First, we define and train the following two networks:
? A selffer network B3B: the first 3 layers are copied from baseB and frozen. The five higher
layers (4?8) are initialized randomly and trained on dataset B. This network is a control for the
next transfer network. (Figure 1, row 3)
? A transfer network A3B: the first 3 layers are copied from baseA and frozen. The five higher
layers (4?8) are initialized randomly and trained toward dataset B. Intuitively, here we copy
the first 3 layers from a network trained on dataset A and then learn higher layer features on top
of them to classify a new target dataset B. If A3B performs as well as baseB, there is evidence
that the third-layer features are general, at least with respect to B. If performance suffers, there
is evidence that the third-layer features are specific to A. (Figure 1, row 4)
We repeated this process for all n in {1, 2, . . . , 7}2 and in both directions (i.e. AnB and BnA). In
the above two networks, the transferred layers are frozen. We also create versions of the above two
networks where the transferred layers are fine-tuned:
? A selffer network B3B+ : just like B3B, but where all layers learn.
? A transfer network A3B+ : just like A3B, but where all layers learn.
To create base and target datasets that are similar to each other, we randomly assign half of the 1000
ImageNet classes to A and half to B. ImageNet contains clusters of similar classes, particularly dogs
and cats, like these 13 classes from the biological family Felidae: {tabby cat, tiger cat, Persian cat,
Siamese cat, Egyptian cat, mountain lion, lynx, leopard, snow leopard, jaguar, lion, tiger, cheetah}.
On average, A and B will each contain approximately 6 or 7 of these felid classes, meaning that
base networks trained on each dataset will have features at all levels that help classify some types
of felids. When generalizing to the other dataset, we would expect that the new high-level felid
detectors trained on top of old low-level felid detectors would work well. Thus A and B are similar
when created by randomly assigning classes to each, and we expect that transferred features will
perform better than when A and B are less similar.
Fortunately, in ImageNet we are also provided with a hierarchy of parent classes. This information
allowed us to create a special split of the dataset into two halves that are as semantically different
from each other as possible: with dataset A containing only man-made entities and B containing
natural entities. The split is not quite even, with 551 classes in the man-made group and 449 in the
natural group. Further details of this split and the classes in each half are given in the supplementary
material. In Section 4.2 we will show that features transfer more poorly (i.e. they are more specific)
when the datasets are less similar.
1
The ImageNet dataset, as released in the Large Scale Visual Recognition Challenge 2012 (ILSVRC2012)
(Deng et al., 2009) contains 1,281,167 labeled training images and 50,000 test images, with each image labeled
with one of 1000 classes.
2
Note that n = 8 doesn?t make sense in either case: B8B is just baseB, and A8B would not work because
it is never trained on B.
3
WA1
WA2
WA3
WA4
WA5
WA6
WA7
WA8
input
A
WB1
WB2
WB3
WB4
WB5
input
B
WB1
WB2
WB3
or
or
or
WA1
WA2
WA3
or
or
or
WB6
WB7
labels
A
baseA
labels
B
baseB
WB8
B3B
and
B3B+
A3B
and
A3B+
Figure 1: Overview of the experimental treatments and controls. Top two rows: The base networks
are trained using standard supervised backprop on only half of the ImageNet dataset (first row: A
half, second row: B half). The labeled rectangles (e.g. WA1 ) represent the weight vector learned for
that layer, with the color indicating which dataset the layer was originally trained on. The vertical,
ellipsoidal bars between weight vectors represent the activations of the network at each layer. Third
row: In the selffer network control, the first n weight layers of the network (in this example, n = 3)
are copied from a base network (e.g. one trained on dataset B), the upper 8 ? n layers are randomly
initialized, and then the entire network is trained on that same dataset (in this example, dataset B).
The first n layers are either locked during training (?frozen? selffer treatment B3B) or allowed to
learn (?fine-tuned? selffer treatment B3B+ ). This treatment reveals the occurrence of fragile coadaptation, when neurons on neighboring layers co-adapt during training in such a way that cannot
be rediscovered when one layer is frozen. Fourth row: The transfer network experimental treatment
is the same as the selffer treatment, except that the first n layers are copied from a network trained
on one dataset (e.g. A) and then the entire network is trained on the other dataset (e.g. B). This
treatment tests the extent to which the features on layer n are general or specific.
3
Experimental Setup
Since Krizhevsky et al. (2012) won the ImageNet 2012 competition, there has been much interest
and work toward tweaking hyperparameters of large convolutional models. However, in this study
we aim not to maximize absolute performance, but rather to study transfer results on a well-known
architecture. We use the reference implementation provided by Caffe (Jia et al., 2014) so that our
results will be comparable, extensible, and useful to a large number of researchers. Further details of
the training setup (learning rates, etc.) are given in the supplementary material, and code and parameter files to reproduce these experiments are available at http://yosinski.com/transfer.
4
Results and Discussion
We performed three sets of experiments. The main experiment has random A/B splits and is discussed in Section 4.1. Section 4.2 presents an experiment with the man-made/natural split. Section 4.3 describes an experiment with random weights.
4
0.66
Top-1 accuracy (higher is better)
0.64
0.62
0.60
0.58
baseB
selffer BnB
selffer BnB +
transfer AnB
transfer AnB +
0.56
0.54
0.52
0
1
2
3
4
5
6
7
5: Transfer + fine-tuning improves generalization
Top-1 accuracy (higher is better)
0.64
3: Fine-tuning recovers co-adapted interactions
0.62
2: Performance drops
due to fragile
co-adaptation
4: Performance
drops due to
representation
specificity
0.60
0.58
0.56
0.54
0
1
2
3
4
5
Layer n at which network is chopped and retrained
6
7
Figure 2: The results from this paper?s main experiment. Top: Each marker in the figure represents
the average accuracy over the validation set for a trained network. The white circles above n =
0 represent the accuracy of baseB. There are eight points, because we tested on four separate
random A/B splits. Each dark blue dot represents a BnB network. Light blue points represent
BnB+ networks, or fine-tuned versions of BnB. Dark red diamonds are AnB networks, and light
red diamonds are the fine-tuned AnB+ versions. Points are shifted slightly left or right for visual
clarity. Bottom: Lines connecting the means of each treatment. Numbered descriptions above each
line refer to which interpretation from Section 4.1 applies.
4.1
Similar Datasets: Random A/B splits
The results of all A/B transfer learning experiments on randomly split (i.e. similar) datasets are
shown3 in Figure 2. The results yield many different conclusions. In each of the following interpretations, we compare the performance to the base case (white circles and dotted line in Figure 2).
3
AnA networks and BnB networks are statistically equivalent, because in both cases a network is trained
on 500 random classes. To simplify notation we label these BnB networks. Similarly, we have aggregated the
statistically identical BnA and AnB networks and just call them AnB.
5
1. The white baseB circles show that a network trained to classify a random subset of 500 classes
attains a top-1 accuracy of 0.625, or 37.5% error. This error is lower than the 42.5% top-1 error
attained on the 1000-class network. While error might have been higher because the network is
trained on only half of the data, which could lead to more overfitting, the net result is that error is
lower because there are only 500 classes, so there are only half as many ways to make mistakes.
2. The dark blue BnB points show a curious behavior. As expected, performance at layer one is
the same as the baseB points. That is, if we learn eight layers of features, save the first layer of
learned Gabor features and color blobs, reinitialize the whole network, and retrain it toward the
same task, it does just as well. This result also holds true for layer 2. However, layers 3, 4, 5,
and 6, particularly 4 and 5, exhibit worse performance. This performance drop is evidence that
the original network contained fragile co-adapted features on successive layers, that is, features
that interact with each other in a complex or fragile way such that this co-adaptation could not be
relearned by the upper layers alone. Gradient descent was able to find a good solution the first
time, but this was only possible because the layers were jointly trained. By layer 6 performance
is nearly back to the base level, as is layer 7. As we get closer and closer to the final, 500-way
softmax output layer 8, there is less to relearn, and apparently relearning these one or two layers
is simple enough for gradient descent to find a good solution. Alternately, we may say that
there is less co-adaptation of features between layers 6 & 7 and between 7 & 8 than between
previous layers. To our knowledge it has not been previously observed in the literature that such
optimization difficulties may be worse in the middle of a network than near the bottom or top.
3. The light blue BnB+ points show that when the copied, lower-layer features also learn on the
target dataset (which here is the same as the base dataset), performance is similar to the base
case. Such fine-tuning thus prevents the performance drop observed in the BnB networks.
4. The dark red AnB diamonds show the effect we set out to measure in the first place: the transferability of features from one network to another at each layer. Layers one and two transfer almost
perfectly from A to B, giving evidence that, at least for these two tasks, not only are the first-layer
Gabor and color blob features general, but the second layer features are general as well. Layer
three shows a slight drop, and layers 4-7 show a more significant drop in performance. Thanks
to the BnB points, we can tell that this drop is from a combination of two separate effects: the
drop from lost co-adaptation and the drop from features that are less and less general. On layers
3, 4, and 5, the first effect dominates, whereas on layers 6 and 7 the first effect diminishes and
the specificity of representation dominates the drop in performance.
Although examples of successful feature transfer have been reported elsewhere in the literature
(Girshick et al., 2013; Donahue et al., 2013b), to our knowledge these results have been limited
to noticing that transfer from a given layer is much better than the alternative of training strictly
on the target task, i.e. noticing that the AnB points at some layer are much better than training
all layers from scratch. We believe this is the first time that (1) the extent to which transfer is
successful has been carefully quantified layer by layer, and (2) that these two separate effects
have been decoupled, showing that each effect dominates in part of the regime.
5. The light red AnB+ diamonds show a particularly surprising effect: that transferring features
and then fine-tuning them results in networks that generalize better than those trained directly on
the target dataset. Previously, the reason one might want to transfer learned features is to enable
training without overfitting on small target datasets, but this new result suggests that transferring
features will boost generalization performance even if the target dataset is large. Note that this
effect should not be attributed to the longer total training time (450k base iterations + 450k finetuned iterations for AnB+ vs. 450k for baseB), because the BnB+ networks are also trained
for the same longer length of time and do not exhibit this same performance improvement.
Thus, a plausible explanation is that even after 450k iterations of fine-tuning (beginning with
completely random top layers), the effects of having seen the base dataset still linger, boosting
generalization performance. It is surprising that this effect lingers through so much retraining.
This generalization improvement seems not to depend much on how much of the first network
we keep to initialize the second network: keeping anywhere from one to seven layers produces
improved performance, with slightly better performance as we keep more layers. The average
boost across layers 1 to 7 is 1.6% over the base case, and the average if we keep at least five
layers is 2.1%.4 The degree of performance boost is shown in Table 1.
4
We aggregate performance over several layers because each point is computationally expensive to obtain
(9.5 days on a GPU), so at the time of publication we have few data points per layer. The aggregation is
6
Table 1: Performance boost of AnB+ over controls, averaged over different ranges of layers.
layers
aggregated
1-7
3-7
5-7
4.2
mean boost
over
baseB
1.6%
1.8%
2.1%
mean boost
over
selffer BnB+
1.4%
1.4%
1.7%
Dissimilar Datasets: Splitting Man-made and Natural Classes Into Separate Datasets
As mentioned previously, the effectiveness of feature transfer is expected to decline as the base and
target tasks become less similar. We test this hypothesis by comparing transfer performance on
similar datasets (the random A/B splits discussed above) to that on dissimilar datasets, created by
assigning man-made object classes to A and natural object classes to B. This man-made/natural split
creates datasets as dissimilar as possible within the ImageNet dataset.
The upper-left subplot of Figure 3 shows the accuracy of a baseA and baseB network (white circles)
and BnA and AnB networks (orange hexagons). Lines join common target tasks. The upper of the
two lines contains those networks trained toward the target task containing natural categories (baseB
and AnB). These networks perform better than those trained toward the man-made categories, which
may be due to having only 449 classes instead of 551, or simply being an easier task, or both.
4.3
Random Weights
We also compare to random, untrained weights because Jarrett et al. (2009) showed ? quite strikingly ? that the combination of random convolutional filters, rectification, pooling, and local normalization can work almost as well as learned features. They reported this result on relatively small
networks of two or three learned layers and on the smaller Caltech-101 dataset (Fei-Fei et al., 2004).
It is natural to ask whether or not the nearly optimal performance of random filters they report carries
over to a deeper network trained on a larger dataset.
The upper-right subplot of Figure 3 shows the accuracy obtained when using random filters for the
first n layers for various choices of n. Performance falls off quickly in layers 1 and 2, and then
drops to near-chance levels for layers 3+, which suggests that getting random weights to work in
convolutional neural networks may not be as straightforward as it was for the smaller network size
and smaller dataset used by Jarrett et al. (2009). However, the comparison is not straightforward.
Whereas our networks have max pooling and local normalization on layers 1 and 2, just as Jarrett
et al. (2009) did, we use a different nonlinearity (relu(x) instead of abs(tanh(x))), different layer
sizes and number of layers, as well as other differences. Additionally, their experiment only considered two layers of random weights. The hyperparameter and architectural choices of our network
collectively provide one new datapoint, but it may well be possible to tweak layer sizes and random
initialization details to enable much better performance for random weights.5
The bottom subplot of Figure 3 shows the results of the experiments of the previous two sections
after subtracting the performance of their individual base cases. These normalized performances
are plotted across the number of layers n that are either random or were trained on a different,
base dataset. This comparison makes two things apparent. First, the transferability gap when using
frozen features grows more quickly as n increases for dissimilar tasks (hexagons) than similar tasks
(diamonds), with a drop by the final layer for similar tasks of only 8% vs. 25% for dissimilar tasks.
Second, transferring even from a distant task is better than using random filters. One possible reason
this latter result may differ from Jarrett et al. (2009) is because their fully-trained (non-random)
networks were overfitting more on the smaller Caltech-101 dataset than ours on the larger ImageNet
informative, however, because the performance at each layer is based on different random draws of the upper
layer initialization weights. Thus, the fact that layers 5, 6, and 7 result in almost identical performance across
random draws suggests that multiple runs at a given layer would result in similar performance.
5
For example, the training loss of the network with three random layers failed to converge, producing only
chance-level validation performance. Much better convergence may be possible with different hyperparameters.
7
Man-made/Natural split
Top-1 accuracy
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.1
0.3
0
Relative top-1 accuracy (higher is better)
Random, untrained filters
0.6
1
2
3
4
5
6
0.0
7
0
1
2
3
4
5
6
7
0.00
?0.05
?0.10
?0.15
reference
mean AnB, random splits
mean AnB, m/n split
random features
?0.20
?0.25
?0.30
0
1
2
3
4
5
Layer n at which network is chopped and retrained
6
7
Figure 3: Performance degradation vs. layer. Top left: Degradation when transferring between dissimilar tasks (from man-made classes of ImageNet to natural classes or vice versa). The upper line
connects networks trained to the ?natural? target task, and the lower line connects those trained toward the ?man-made? target task. Top right: Performance when the first n layers consist of random,
untrained weights. Bottom: The top two plots compared to the random A/B split from Section 4.1
(red diamonds), all normalized by subtracting their base level performance.
dataset, making their random filters perform better by comparison. In the supplementary material,
we provide an extra experiment indicating the extent to which our networks are overfit.
5
Conclusions
We have demonstrated a method for quantifying the transferability of features from each layer of
a neural network, which reveals their generality or specificity. We showed how transferability is
negatively affected by two distinct issues: optimization difficulties related to splitting networks in
the middle of fragilely co-adapted layers and the specialization of higher layer features to the original
task at the expense of performance on the target task. We observed that either of these two issues
may dominate, depending on whether features are transferred from the bottom, middle, or top of
the network. We also quantified how the transferability gap grows as the distance between tasks
increases, particularly when transferring higher layers, but found that even features transferred from
distant tasks are better than random weights. Finally, we found that initializing with transferred
features can improve generalization performance even after substantial fine-tuning on a new task,
which could be a generally useful technique for improving deep neural network performance.
Acknowledgments
The authors would like to thank Kyunghyun Cho and Thomas Fuchs for helpful discussions, Joost
Huizinga, Anh Nguyen, and Roby Velez for editing, as well as funding from the NASA Space
Technology Research Fellowship (JY), DARPA project W911NF-12-1-0449, NSERC, Ubisoft, and
CIFAR (YB is a CIFAR Fellow).
8
References
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Bengio, Y., Bastien, F., Bergeron, A., Boulanger-Lewandowski, N., Breuel, T., Chherawala, Y., Cisse, M., C?ot?e,
M., Erhan, D., Eustache, J., Glorot, X., Muller, X., Pannetier Lebeuf, S., Pascanu, R., Rifai, S., Savard, F.,
and Sicard, G. (2011). Deep learners benefit more from out-of-distribution examples. In JMLR W&CP:
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Caruana, R. (1995). Learning many related tasks at the same time with backpropagation. pages 657?664,
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Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. (2009). ImageNet: A Large-Scale Hierarchical
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Krizhevsky, A., Sutskever, I., and Hinton, G. (2012). ImageNet classification with deep convolutional neural
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Le, Q. V., Karpenko, A., Ngiam, J., and Ng, A. Y. (2011). ICA with reconstruction cost for efficient overcomplete feature learning. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger, editors,
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9
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4,802 | 5,348 | Convolutional Kernel Networks
Julien Mairal, Piotr Koniusz, Zaid Harchaoui, and Cordelia Schmid
Inria?
[email protected]
Abstract
An important goal in visual recognition is to devise image representations that are
invariant to particular transformations. In this paper, we address this goal with a
new type of convolutional neural network (CNN) whose invariance is encoded by
a reproducing kernel. Unlike traditional approaches where neural networks are
learned either to represent data or for solving a classification task, our network
learns to approximate the kernel feature map on training data.
Such an approach enjoys several benefits over classical ones. First, by teaching CNNs to be invariant, we obtain simple network architectures that achieve a
similar accuracy to more complex ones, while being easy to train and robust to
overfitting. Second, we bridge a gap between the neural network literature and
kernels, which are natural tools to model invariance. We evaluate our methodology on visual recognition tasks where CNNs have proven to perform well, e.g.,
digit recognition with the MNIST dataset, and the more challenging CIFAR-10
and STL-10 datasets, where our accuracy is competitive with the state of the art.
1
Introduction
We have recently seen a revival of attention given to convolutional neural networks (CNNs) [22]
due to their high performance for large-scale visual recognition tasks [15, 21, 30]. The architecture
of CNNs is relatively simple and consists of successive layers organized in a hierarchical fashion;
each layer involves convolutions with learned filters followed by a pointwise non-linearity and a
downsampling operation called ?feature pooling?. The resulting image representation has been empirically observed to be invariant to image perturbations and to encode complex visual patterns [33],
which are useful properties for visual recognition. Training CNNs remains however difficult since
high-capacity networks may involve billions of parameters to learn, which requires both high computational power, e.g., GPUs, and appropriate regularization techniques [18, 21, 30].
The exact nature of invariance that CNNs exhibit is also not precisely understood. Only recently, the
invariance of related architectures has been characterized; this is the case for the wavelet scattering
transform [8] or the hierarchical models of [7]. Our work revisits convolutional neural networks,
but we adopt a significantly different approach than the traditional one. Indeed, we use kernels [26],
which are natural tools to model invariance [14]. Inspired by the hierarchical kernel descriptors
of [2], we propose a reproducing kernel that produces multi-layer image representations.
Our main contribution is an approximation scheme called convolutional kernel network (CKN) to
make the kernel approach computationally feasible. Our approach is a new type of unsupervised
convolutional neural network that is trained to approximate the kernel map. Interestingly, our network uses non-linear functions that resemble rectified linear units [1, 30], even though they were not
handcrafted and naturally emerge from an approximation scheme of the Gaussian kernel map.
By bridging a gap between kernel methods and neural networks, we believe that we are opening
a fruitful research direction for the future. Our network is learned without supervision since the
?
LEAR team, Inria Grenoble, Laboratoire Jean Kuntzmann, CNRS, Univ. Grenoble Alpes, France.
1
label information is only used subsequently in a support vector machine (SVM). Yet, we achieve
competitive results on several datasets such as MNIST [22], CIFAR-10 [20] and STL-10 [13] with
simple architectures, few parameters to learn, and no data augmentation. Open-source code for
learning our convolutional kernel networks is available on the first author?s webpage.
1.1
Related Work
There have been several attempts to build kernel-based methods that mimic deep neural networks;
we only review here the ones that are most related to our approach.
Arc-cosine kernels. Kernels for building deep large-margin classifiers have been introduced
in [10]. The multilayer arc-cosine kernel is built by successive kernel compositions, and each layer
relies on an integral representation. Similarly, our kernels rely on an integral representation, and
enjoy a multilayer construction. However, in contrast to arc-cosine kernels: (i) we build our sequence of kernels by convolutions, using local information over spatial neighborhoods (as opposed
to compositions, using global information); (ii) we propose a new training procedure for learning a
compact representation of the kernel in a data-dependent manner.
Multilayer derived kernels. Kernels with invariance properties for visual recognition have been
proposed in [7]. Such kernels are built with a parameterized ?neural response? function, which consists in computing the maximal response of a base kernel over a local neighborhood. Multiple layers
are then built by iteratively renormalizing the response kernels and pooling using neural response
functions. Learning is performed by plugging the obtained kernel in an SVM. In contrast to [7], we
propagate information up, from lower to upper layers, by using sequences of convolutions. Furthermore, we propose a simple and effective data-dependent way to learn a compact representation of
our kernels and show that we obtain near state-of-the-art performance on several benchmarks.
Hierarchical kernel descriptors. The kernels proposed in [2, 3] produce multilayer image representations for visual recognition tasks. We discuss in details these kernels in the next section: our
paper generalizes them and establishes a strong link with convolutional neural networks.
2
Convolutional Multilayer Kernels
The convolutional multilayer kernel is a generalization of the hierarchical kernel descriptors introduced in computer vision [2, 3]. The kernel produces a sequence of image representations that are
built on top of each other in a multilayer fashion. Each layer can be interpreted as a non-linear transformation of the previous one with additional spatial invariance. We call these layers image feature
maps1 , and formally define them as follows:
Definition 1. An image feature map ? is a function ? : ? ? H, where ? is a (usually discrete)
subset of [0, 1]d representing normalized ?coordinates? in the image and H is a Hilbert space.
For all practical examples in this paper, ? is a two-dimensional grid and corresponds to different
locations in a two-dimensional image. In other words, ? is a set of pixel coordinates. Given z
in ?, the point ?(z) represents some characteristics of the image at location z, or in a neighborhood
of z. For instance, a color image of size m ? n with three channels, red, green, and blue, may be
represented by an initial feature map ?0 : ?0 ? H0 , where ?0 is an m ? n regular grid, H0 is the
Euclidean space R3 , and ?0 provides the color pixel values. With the multilayer scheme, non-trivial
feature maps will be obtained subsequently, which will encode more complex image characteristics.
With this terminology in hand, we now introduce the convolutional kernel, first, for a single layer.
Definition 2 (Convolutional Kernel with Single Layer). Let us consider two images represented
by two image feature maps, respectively ? and ?? : ? ? H, where ? is a set of pixel locations,
and H is a Hilbert space. The one-layer convolutional kernel between ? and ?? is defined as
2
XX
2
? 1 z?z? k ? 12 k?(z)?
?
?
?? (z? )k
2 e 2?
H,
K(?, ?? ) :=
k?(z)kH k?? (z? )kH e 2?2 k
(1)
z?? z? ??
1
In the kernel literature, ?feature map? denotes the mapping between data points and their representation in
a reproducing kernel Hilbert space (RKHS) [26]. Here, feature maps refer to spatial maps representing local
image characteristics at everly location, as usual in the neural network literature [22].
2
where ? and ? are smoothing parameters of Gaussian kernels, and ?(z)
?
:= (1/ k?(z)kH ) ?(z)
? ?
if ?(z) 6= 0 and ?(z)
?
= 0 otherwise. Similarly, ?? (z ) is a normalized version of ?? (z? ).2
It is easy to show that the kernel K is positive definite (see Appendix A). It consists of a sum of
pairwise comparisons between the image features ?(z) and ?? (z? ) computed at all spatial locations z
and z? in ?. To be significant in the sum, a comparison needs the corresponding z and z? to be
close in ?, and the normalized features ?(z)
?
and ??? (z? ) to be close in the feature space H. The
parameters ? and ? respectively control these two definitions of ?closeness?. Indeed, when ? is
large, the kernel K is invariant to the positions z and z? but when ? is small, only features placed
at the same location z = z? are compared to each other. Therefore, the role of ? is to control how
much the kernel is locally shift-invariant. Next, we will show how to go beyond one single layer,
but before that, we present concrete examples of simple input feature maps ?0 : ?0 ? H0 .
Gradient map. Assume that H0 = R2 and that ?0 (z) provides the two-dimensional gradient of the
image at pixel z, which is often computed with first-order differences along each dimension. Then,
the quantity k?0 (z)kH0 is the gradient intensity, and ??0 (z) is its orientation, which can be characterized by a particular angle?that is, there exists ? in [0; 2?] such that ??0 (z) = [cos(?), sin(?)]. The
resulting kernel K is exactly the kernel descriptor introduced in [2, 3] for natural image patches.
Patch map. In that setting, ?0 associates to a location z an image patch of size m ? m centered
at z. Then, the space H0 is simply Rm?m , and ??0 (z) is a contrast-normalized version of the patch,
which is a useful transformation for visual recognition according to classical findings in computer
vision [19]. When the image is encoded with three color channels, patches are of size m ? m ? 3.
We now define the multilayer convolutional kernel, generalizing some ideas of [2].
Definition 3 (Multilayer Convolutional Kernel). Let us consider a set ?k?1 ? [0, 1]d and a Hilbert
space Hk?1 . We build a new set ?k and a new Hilbert space Hk as follows:
(i) choose a patch shape Pk defined as a bounded symmetric subset of [?1, 1]d , and a set of coordinates ?k such that for all location zk in ?k , the patch {zk } + Pk is a subset of ?k?1 ;3 In other
words, each coordinate zk in ?k corresponds to a valid patch in ?k?1 centered at zk .
(ii) define the convolutional kernel Kk on the ?patch? feature maps Pk ? Hk?1 , by replacing
in (1): ? by Pk , H by Hk?1 , and ?, ? by appropriate smoothing parameters ?k , ?k . We denote
by Hk the Hilbert space for which the positive definite kernel Kk is reproducing.
An image represented by a feature map ?k?1 : ?k?1 ? Hk?1 at layer k?1 is now encoded in the k-th
layer as ?k : ?k ? Hk , where for all zk in ?k , ?k (zk ) is the representation in Hk of the patch
feature map z 7? ?k?1 (zk + z) for z in Pk .
Concretely, the kernel Kk between two patches of ?k?1 and ??k?1 at respective locations zk and z?k is
2
2
X X
? 12 kz?z? k ? 12 k?
?k?1 (zk +z)??
??k?1 (z?k +z? )k
2
k?k?1 (zk + z)k k??k?1 (z?k + z? )k e 2?k
, (2)
e 2?k
z?Pk z? ?Pk
where k.k is the Hilbertian norm of Hk?1 . In Figure 1(a), we illustrate the interactions between the
sets of coordinates ?k , patches Pk , and feature spaces Hk across layers. For two-dimensional grids,
a typical patch shape is a square, for example P := {?1/n, 0, 1/n} ? {?1/n, 0, 1/n} for a 3 ? 3
patch in an image of size n ? n. Information encoded in the k-th layer differs from the (k?1)-th one
in two aspects: first, each point ?k (zk ) in layer k contains information about several points from
the (k?1)-th layer and can possibly represent larger patterns; second, the new feature map is more
locally shift-invariant than the previous one due to the term involving the parameter ?k in (2).
The multilayer convolutional kernel slightly differs from the hierarchical kernel descriptors of [2]
but exploits similar ideas. Bo et al. [2] define indeed several ad hoc kernels for representing local
information in images, such as gradient, color, or shape. These kernels are close to the one defined
in (1) but with a few variations. Some of them do not use normalized features ?(z),
?
and these kernels
use different weighting strategies for the summands of (1) that are specialized to the image modality,
e.g., color, or gradient, whereas we use the same weight k?(z)kH k?? (z? )kH for all kernels. The
generic formulation (1) that we propose may be useful per se, but our main contribution comes in
the next section, where we use the kernel as a new tool for learning convolutional neural networks.
R
P
When ? is not discrete, the notation
in (1) should be replaced by the Lebesgue integral in the paper.
3
For two sets A and B, the Minkowski sum A + B is defined as {a + b : a ? A, b ? B}.
2
3
?2 (z2 ) ? H2
?k (z)
?2
??k
{z2 } + P2
?1
?k?1
?1 (z1 ) ? H1
Gaussian filtering
+ downsampling
= pooling
?k (zk?1 )
pk
convolution
+ non-linearity
?
{zk?1 }+Pk?1
?0 (z0 ) ? H0
{z1 } + P1
?0
??k?1
?k?1 (z)
?k?1 (zk?1 )
(patch extraction)
(b) Zoom between layer k?1 and k of the CKN.
(a) Hierarchy of image feature maps.
Figure 1: Left: concrete representation of the successive layers for the multilayer convolutional
kernel. Right: one layer of the convolutional neural network that approximates the kernel.
3
Training Invariant Convolutional Kernel Networks
Generic schemes have been proposed for approximating a non-linear kernel with a linear one, such
as the Nystr?om method and its variants [5, 31], or random sampling techniques in the Fourier domain for shift-invariant kernels [24]. In the context of convolutional multilayer kernels, such an
approximation is critical because computing the full kernel matrix on a database of images is computationally infeasible, even for a moderate number of images (? 10 000) and moderate number of
layers. For this reason, Bo et al. [2] use the Nystr?om method for their hierarchical kernel descriptors.
In this section, we show that when the coordinate sets ?k are two-dimensional regular grids, a
natural approximation for the multilayer convolutional kernel consists of a sequence of spatial convolutions with learned filters, pointwise non-linearities, and pooling operations, as illustrated in
Figure 1(b). More precisely, our scheme approximates the kernel map of K defined in (1) at layer k
by finite-dimensional spatial maps ?k : ??k ? Rpk , where ??k is a set of coordinates related to ?k ,
and pk is a positive integer controlling the quality of the approximation. Consider indeed two images
represented at layer k by image feature maps ?k and ??k , respectively. Then,
(A) the corresponding maps ?k and ?k? are learned such that K(?k?1 , ??k?1 ) ? h?k , ?k? i, where h., .i
?
is the Euclidean inner-product acting as if ?k and ?k? were vectors in R|?k |pk ;
(B) the set ??k is linked to ?k by the relation ??k = ?k + Pk? where Pk? is a patch shape, and
?
the quantities ?k (zk ) in Hk admit finite-dimensional approximations ?k (zk ) in R|Pk |pk ; as
illustrated in Figure 1(b), ?k (zk ) is a patch from ?k centered at location zk with shape Pk? ;
(C) an activation map ?k : ?k?1 7? Rpk is computed from ?k?1 by convolution with pk filters
followed by a non-linearity. The subsequent map ?k is obtained from ?k by a pooling operation.
We call this approximation scheme a convolutional kernel network (CKN). In comparison to CNNs,
our approach enjoys similar benefits such as efficient prediction at test time, and involves the same
set of hyper-parameters: number of layers, numbers of filters pk at layer k, shape Pk? of the filters,
sizes of the feature maps. The other parameters ?k , ?k can be automatically chosen, as discussed
later. Training a CKN can be argued to be as simple as training a CNN in an unsupervised manner [25] since we will show that the main difference is in the cost function that is optimized.
3.1
Fast Approximation of the Gaussian Kernel
A key component of our formulation is the Gaussian kernel. We start by approximating it by a linear
operation with learned filters followed by a pointwise non-linearity. Our starting point is the next
lemma, which can be obtained after a simple calculation.
4
Lemma 1 (Linear expansion of the Gaussian Kernel). For all x and x? in Rm , and ? > 0,
m2 Z
2
?
2
1
1
2
? 2?12 kx?x? k22
=
(3)
e? ?2 kx?wk2 e? ?2 kx ?wk2 dw.
e
2
??
w?Rm
?
2
2
The lemma gives us a mapping of any x in Rm to the function w 7? Ce?(1/? )kx?wk2 in L2 (Rm ),
where the kernel is linear, and C is the constant in front of the integral. To obtain a finite-dimensional
representation, we need to approximate the integral with a weighted finite sum, which is a classical
problem arising in statistics (see [29] and chapter 8 of [6]). Then, we consider two different cases.
Small dimension, m ? 2. When the data lives in a compact set of Rm , the integral in (3) can be
approximated by uniform sampling over a large enough
set. We choose such a strategy for two types
2
2
? 2?12 kz?z? k
? 1
?(z)?
?
?
?? (z? )k
2 ; (ii) the terms e ( 2? 2 )k
H
of kernels from Eq. (1): (i) the spatial kernels e
when ? is the ?gradient map? presented in Section 2. In the latter case, H = R2 and ?(z)
?
is the
gradient orientation. We typically sample a few orientations as explained in Section 4.
Higher dimensions. To prevent the curse of dimensionality, we learn to approximate the kernel on
training data, which is intrinsically low-dimensional. We optimize importance weights ? = [?l ]pl=1
in Rp+ and sampling points W = [wl ]pl=1 in Rm?p on n training pairs (xi , yi )i=1,...,n in Rm ? Rm :
X
p
n
2
X
1
? ?12 kxi ?wl k22 ? ?12 kyi ?wl k22
? 2?12 kxi ?yi k22
.
(4)
?
e
?
e
e
min
l
m?p
n i=1
??Rp
+ ,W?R
l=1
Interestingly, we may already draw some links with neural networks. When applied to unit-norm
vectors xi and yi , problem (4) produces sampling points wl whose norm is close to one. After
2
2
?
learning, a new unit-norm point x in Rm is mapped to the vector [ ?l e?(1/? )kx?wl k2 ]pl=1 in Rp ,
which may be written as [f (wl? x)]pl=1 , assuming that the norm of wl is always one, where f is the
2
function u 7? e(2/? )(u?1) for u = wl? x in [?1, 1]. Therefore, the finite-dimensional representation
of x only involves a linear operation followed by a non-linearity, as in typical neural networks. In
Figure 2, we show that the shape of f resembles the ?rectified linear unit? function [30].
2
f (u)
-1
0
f (u) = e(2/? )(u?1)
f (u) = max(u, 0)
1
u
Figure 2: In dotted red, we plot the ?rectified linear unit? function u 7? max(u, 0). In blue, we plot
non-linear functions of our network for typical values of ? that we use in our experiments.
3.2
Approximating the Multilayer Convolutional Kernel
We have now all the tools in hand to build our convolutional kernel network. We start by making assumptions on the input data, and then present the learning scheme and its approximation principles.
The zeroth layer. We assume that the input data is a finite-dimensional map ?0 : ??0 ? Rp0 , and
that ?0 : ?0 ? H0 ?extracts? patches from ?0 . Formally, there exists a patch shape P0? such that
?
??0 = ?0 + P0? , H0 = Rp0 |P0 | , and for all z0 in ?0 , ?0 (z0 ) is a patch of ?0 centered at z0 . Then,
property (B) described at the beginning of Section 3 is satisfied for k = 0 by choosing ?0 = ?0 .
The examples of input feature maps given earlier satisfy this finite-dimensional assumption: for the
gradient map, ?0 is the gradient of the image along each direction, with p0 = 2, P0? = {0} is a 1?1
patch, ?0 = ??0 , and ?0 = ?0 ; for the patch map, ?0 is the input image, say with p0 = 3 for RGB data.
The convolutional kernel network. The zeroth layer being characterized, we present in Algorithms 1 and 2 the subsequent layers and how to learn their parameters in a feedforward manner. It
is interesting to note that the input parameters of the algorithm are exactly the same as a CNN?that
is, number of layers and filters, sizes of the patches and feature maps (obtained here via the subsampling factor). Ultimately, CNNs and CKNs only differ in the cost function that is optimized for
learning the filters and in the choice of non-linearities. As we show next, there exists a link between
the parameters of a CKN and those of a convolutional multilayer kernel.
5
Algorithm 1 Convolutional kernel network - learning the parameters of the k-th layer.
1
2
input ?k?1
, ?k?1
, . . . : ??k?1 ? Rpk?1 (sequence of (k?1)-th maps obtained from training images);
?
Pk?1 (patch shape); pk (number of filters); n (number of training pairs);
?
1
2
1: extract at random n pairs (xi , yi ) of patches with shape Pk?1
from the maps ?k?1
, ?k?1
, . . .;
2: if not provided by the user, set ?k to the 0.1 quantile of the data (kxi ? yi k2 )n
;
i=1
?
3: unsupervised learning: optimize (4) to obtain the filters Wk in R|Pk?1 |pk?1 ?pk and ? k in Rpk ;
output Wk , ? k , and ?k (smoothing parameter);
Algorithm 2 Convolutional kernel network - computing the k-th map form the (k?1)-th one.
?
input ?k?1 : ??k?1 ? Rpk?1 (input map); Pk?1
(patch shape); ?k ? 1 (subsampling factor); pk (numk
k
ber of filters); ?k (smoothing parameter); Wk = [wkl ]pl=1
and ? k = [?kl ]pl=1
(layer parameters);
1: convolution and non-linearity: define the activation map ?k : ?k?1 ? Rpk as
?k?1 (z)?wkl k2 pk
? 12 k?
?
?
2
,
(5)
?k : z 7? k?k?1 (z)k2
?kl e k
l=1
?
where ?k?1 (z) is a vector representing a patch from ?k?1 centered at z with shape Pk?1
, and the
?
vector ?k?1 (z) is an ?2 -normalized version of ?k?1 (z). This operation can be interpreted as a
spatial convolution of the map ?k?1 with the filters wkl followed by pointwise non-linearities;
2: set ?k to be ?k times the spacing between two pixels in ?k?1 ;
3: feature pooling: ??k is obtained by subsampling ?k?1 by a factor ?k and we define a new map
?k : ??k ? Rpk obtained from ?k by linear pooling with Gaussian weights:
?k : z 7?
X ? 12 ku?zk22
p
e ?k
2/?
?k (u).
(6)
u??k?1
output ?k : ??k ? Rpk (new map);
Approximation principles. We proceed recursively to show that the kernel approximation property (A) is satisfied; we assume that (B) holds at layer k?1, and then, we show that (A) and (B) also
hold at layer k. This is sufficient for our purpose since we have previously assumed (B) for the zeroth layer. Given two images feature maps ?k?1 and ??k?1 , we start by approximating K(?k?1 , ??k?1 )
by replacing ?k?1 (z) and ??k?1 (z? ) by their finite-dimensional approximations provided by (B):
2
X
?k?1 (z)??
?? (z? )k2
? 12 kz?z? k ? 12 k?
k?1
?
2
2
e 2?k
. (7)
k?k?1 (z)k2 k?k?1
(z? )k2 e 2?k
K(?k?1 , ??k?1 ) ?
z,z? ??k?1
Then, we use the finite-dimensional approximation of the Gaussian kernel involving ?k and
2
X
? 12 kz?z? k
2
?k (z)? ?k? (z? )e 2?k
,
K(?k?1 , ??k?1 ) ?
(8)
z,z? ??k?1
where ?k is defined in (5) and ?k? is defined similarly by replacing ?? by ??? . Finally, we approximate
the remaining Gaussian kernel by uniform sampling on ??k , following Section 3.1. After exchanging
sums and grouping appropriate terms together, we obtain the new approximation
? X
X
2
? 12 kz? ?uk ?
? 12 kz?uk22
2 X
?
?
?
?
2
k
k
K(?k?1 , ?k?1 ) ?
e
?k (z)
?k (z ) ,
(9)
e
?
?
?
u??k
z ??k?1
z??k?1
where the constant 2/? comes from the multiplication of the constant 2/(??k2 ) from (3) and the
weight ?k2 of uniform sampling orresponding to the square of the distance between two pixels of ??k .4
As a result, the right-hand side is exactly h?k , ?k? i, where ?k is defined in (6), giving us property (A).
It remains to show that property (B) also holds, specifically that the quantity (2) can be approximated
by the Euclidean inner-product h?k (zk ), ?k? (z?k )i with the patches ?k (zk ) and ?k? (z?k ) of shape Pk? ;
we assume for that purpose that Pk? is a subsampled version of the patch shape Pk by a factor ?k .
4
The choice of ?k in Algorithm 2 is driven by signal processing principles. The feature pooling step can
indeed be interpreted as a downsampling operation that reduces the resolution of the map from ?k?1 to ?k by
using a Gaussian anti-aliasing filter, whose role is to reduce frequencies above the Nyquist limit.
6
We remark that the kernel (2) is the same as (1) applied to layer k?1 by replacing ?k?1 by {zk }+Pk .
By doing the same substitution in (9), we immediately obtain an approximation of (2). Then, all
Gaussian terms are negligible forPall u and
each other?say
Pz that are far fromP
P when ku?zk2 ? 2?k .
Thus, we may replace the sums u??? z,z? ?{zk }+Pk by u?{zk }+P ? z,z? ??k?1 , which has the
k
k
same set of ?non-negligible? terms. This yields exactly the approximation h?k (zk ), ?k? (z?k )i.
Optimization. Regarding problem (4), stochastic gradient descent (SGD) may be used since a
potentially infinite amount of training data is available. However, we have preferred to use L-BFGSB [9] on 300 000 pairs of randomly selected training data points, and initialize W with the K-means
algorithm. L-BFGS-B is a parameter-free state-of-the-art batch method, which is not as fast as SGD
but much easier to use. We always run the L-BFGS-B algorithm for 4 000 iterations, which seems
to ensure convergence to a stationary point. Our goal is to demonstrate the preliminary performance
of a new type of convolutional network, and we leave as future work any speed improvement.
4
Experiments
We now present experiments that were performed using Matlab and an L-BFGS-B solver [9] interfaced by Stephen Becker. Each image is represented by the last map ? k of the CKN, which is used
in a linear SVM implemented in the software package LibLinear [16]. These representations are
centered, rescaled to have unit ?2 -norm on average, and the regularization parameter of the SVM is
always selected on a validation set or by 5-fold cross-validation in the range 2i , i = ?15 . . . , 15.
The patches Pk? are typically small; we tried the sizes m ? m with m = 3, 4, 5 for the first
layer, and m = 2, 3 for the upper ones. The number of filters pk in our experiments is in the
set {50, 100, 200, 400, 800}. The downsampling factor ?k is always chosen to be 2 between two consecutive layers, whereas the last layer is downsampled to produce final maps ?k of a small size?say,
2
?
?
5?5 or 4?4. For the gradient map ?0 , we approximate the Gaussian kernel e(1/?1 )k?0 (z)??0 (z )kH0
by uniformly sampling p1 = 12 orientations, setting ?1 = 2?/p1 . Finally, we also use a small off?
set ? to prevent numerical instabilities in the normalization steps ?(z)
= ?(z)/ max(k?(z)k2 , ?).
4.1
Discovering the Structure of Natural Image Patches
Unsupervised learning was first used for discovering the underlying structure of natural image
patches by Olshausen and Field [23]. Without making any a priori assumption about the data except a parsimony principle, the method is able to produce small prototypes that resemble Gabor
wavelets?that is, spatially localized oriented basis functions. The results were found impressive by
the scientific community and their work received substantial attention. It is also known that such
results can also be achieved with CNNs [25]. We show in this section that this is also the case for
convolutional kernel networks, even though they are not explicitly trained to reconstruct data.
Following [23], we randomly select a database of 300 000 whitened natural image patches of
size 12 ? 12 and learn p = 256 filters W using the formulation (4). We initialize W with Gaussian
random noise without performing the K-means step, in order to ensure that the output we obtain is
not an artifact of the initialization. In Figure 3, we display the filters associated to the top-128 largest
weights ?l . Among the 256 filters, 197 exhibit interpretable Gabor-like structures and the rest was
less interpretable. To the best of our knowledge, this is the first time that the explicit kernel map of
the Gaussian kernel for whitened natural image patches is shown to be related to Gabor wavelets.
4.2
Digit Classification on MNIST
The MNIST dataset [22] consists of 60 000 images of handwritten digits for training and 10 000
for testing. We use two types of initial maps in our networks: the ?patch map?, denoted by CNKPM and the ?gradient map?, denoted by CNK-GM. We follow the evaluation methodology of [25]
Figure 3: Filters obtained by the first layer of the convolutional kernel network on natural images.
7
Tr.
size
300
1K
2K
5K
10K
20K
40K
60K
CNN
[25]
7.18
3.21
2.53
1.52
0.85
0.76
0.65
0.53
Scat-1
[8]
4.7
2.3
1.3
1.03
0.88
0.79
0.74
0.70
Scat-2
[8]
5.6
2.6
1.8
1.4
1
0.58
0.53
0.4
CKN-GM1
(12/50)
4.39
2.60
1.85
1.41
1.17
0.89
0.68
0.58
CKN-GM2
(12/400)
4.24
2.05
1.51
1.21
0.88
0.60
0.51
0.39
CKN-PM1
(200)
5.98
3.23
1.97
1.41
1.18
0.83
0.64
0.63
CKN-PM2
(50/200)
4.15
2.76
2.28
1.56
1.10
0.77
0.58
0.53
[32]
[18]
[19]
0.47
NA
NA
NA
NA
NA
NA
NA
0.45
0.53
Table 1: Test error in % for various approaches on the MNIST dataset without data augmentation.
The numbers in parentheses represent the size p1 and p2 of the feature maps at each layer.
for comparison when varying the training set size. We select the regularization parameter of the
SVM by 5-fold cross validation when the training size is smaller than 20 000, or otherwise, we
keep 10 0000 examples from the training set for validation. We report in Table 1 the results obtained
for four simple architectures. CKN-GM1 is the simplest one: its second layer uses 3 ? 3 patches and
only p2 = 50 filters, resulting in a network with 5 400 parameters. Yet, it achieves an outstanding
performance of 0.58% error on the full dataset. The best performing, CKN-GM2, is similar to
CKN-GM1 but uses p2 = 400 filters. When working with raw patches, two layers (CKN-PM2)
gives better results than one layer. More details about the network architectures are provided in the
supplementary material. In general, our method achieves a state-of-the-art accuracy for this task
since lower error rates have only been reported by using data augmentation [11].
4.3
Visual Recognition on CIFAR-10 and STL-10
We now move to the more challenging datasets CIFAR-10 [20] and STL-10 [13]. We select the
best architectures on a validation set of 10 000 examples from the training set for CIFAR-10, and
by 5-fold cross-validation on STL-10. We report in Table 2 results for CKN-GM, defined in the
previous section, without exploiting color information, and CKN-PM when working on raw RGB
patches whose mean color is subtracted. The best selected models have always two layers, with 800
filters for the top layer. Since CKN-PM and CKN-GM exploit a different information, we also report
a combination of such two models, CKN-CO, by concatenating normalized image representations
together. The standard deviations for STL-10 was always below 0.7%. Our approach appears to
be competitive with the state of the art, especially on STL-10 where only one method does better
than ours, despite the fact that our models only use 2 layers and require learning few parameters.
Note that better results than those reported in Table 2 have been obtained in the literature by using
either data augmentation (around 90% on CIFAR-10 for [18, 30]), or external data (around 70% on
STL-10 for [28]). We are planning to investigate similar data manipulations in the future.
Method
CIFAR-10
STL-10
[12]
82.0
60.1
[27]
82.2
58.7
[18]
88.32
NA
[13]
79.6
51.5
[4]
NA
64.5
[17]
83.96
62.3
[32]
84.87
NA
CKN-GM
74.84
60.04
CKN-PM
78.30
60.25
CKN-CO
82.18
62.32
Table 2: Classification accuracy in % on CIFAR-10 and STL-10 without data augmentation.
5
Conclusion
In this paper, we have proposed a new methodology for combining kernels and convolutional neural
networks. We show that mixing the ideas of these two concepts is fruitful, since we achieve near
state-of-the-art performance on several datasets such as MNIST, CIFAR-10, and STL10, with simple
architectures and no data augmentation. Some challenges regarding our work are left open for the
future. The first one is the use of supervision to better approximate the kernel for the prediction task.
The second consists in leveraging the kernel interpretation of our convolutional neural networks to
better understand the theoretical properties of the feature spaces that these networks produce.
Acknowledgments
This work was partially supported by grants from ANR (project MACARON ANR-14-CE23-000301), MSR-Inria joint centre, European Research Council (project ALLEGRO), CNRS-Mastodons
program (project GARGANTUA), and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025).
8
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4,803 | 5,349 | Learning Deep Features for Scene Recognition
using Places Database
Bolei Zhou1 , Agata Lapedriza1,3 , Jianxiong Xiao2 , Antonio Torralba1 , and Aude Oliva1
1
Massachusetts Institute of Technology
2
Princeton University
3
Universitat Oberta de Catalunya
Abstract
Scene recognition is one of the hallmark tasks of computer vision, allowing definition of a context for object recognition. Whereas the tremendous recent progress
in object recognition tasks is due to the availability of large datasets like ImageNet
and the rise of Convolutional Neural Networks (CNNs) for learning high-level features, performance at scene recognition has not attained the same level of success.
This may be because current deep features trained from ImageNet are not competitive enough for such tasks. Here, we introduce a new scene-centric database called
Places with over 7 million labeled pictures of scenes. We propose new methods
to compare the density and diversity of image datasets and show that Places is as
dense as other scene datasets and has more diversity. Using CNN, we learn deep
features for scene recognition tasks, and establish new state-of-the-art results on
several scene-centric datasets. A visualization of the CNN layers? responses allows us to show differences in the internal representations of object-centric and
scene-centric networks.
1
Introduction
Understanding the world in a single glance is one of the most accomplished feats of the human brain:
it takes only a few tens of milliseconds to recognize the category of an object or environment, emphasizing an important role of feedforward processing in visual recognition. One of the mechanisms
subtending efficient human visual recognition is our capacity to learn and remember a diverse set of
places and exemplars [11]; by sampling the world several times per second, our neural architecture
constantly registers new inputs even for a very short time, reaching an exposure to millions of natural images within just a year. How much would an artificial system have to learn before reaching
the scene recognition abilities of a human being?
Besides the exposure to a dense and rich variety of natural images, one important property of the
primate brain is its hierarchical organization in layers of increasing processing complexity, an architecture that has inspired Convolutional Neural Networks or CNNs [2, 14]. These architectures
together with recent large databases (e.g., ImageNet [3]) have obtained astonishing performance on
object classification tasks [12, 5, 20]. However, the baseline performance reached by these networks
on scene classification tasks is within the range of performance based on hand-designed features
and sophisticated classifiers [24, 21, 4]. Here, we show that one of the reasons for this discrepancy
is that the higher-level features learned by object-centric versus scene-centric CNNs are different:
iconic images of objects do not contain the richness and diversity of visual information that pictures
of scenes and environments provide for learning to recognize them.
Here we introduce Places, a scene-centric image dataset 60 times larger than the SUN database [24].
With this database and a standard CNN architecture, we establish new baselines of accuracies on
1
various scene datasets (Scene15 [17, 13], MIT Indoor67 [19], SUN database [24], and SUN Attribute
Database [18]), significantly outperforming the results obtained by the deep features from the same
network architecture trained with ImageNet1 .
The paper is organized as follows: in Section 2 we introduce the Places database and describe the
collection procedure. In Section 3 we compare Places with the other two large image datasets:
SUN [24] and ImageNet [3]. We perform experiments on Amazon Mechanical Turk (AMT) to compare these 3 datasets in terms of density and diversity. In Section 4 we show new scene classification
performance when training deep features from millions of labeled scene images. Finally, we visualize the units? responses at different layers of the CNNs, demonstrating that an object-centric network
(using ImageNet [12]) and a scene-centric network (using Places) learn different features.
2
Places Database
The first benchmark for scene classification was the Scene15 database [13] based on [17]. This
dataset contains only 15 scene categories with a few hundred images per class, where current classifiers are saturating this dataset nearing human performance at 95%. The MIT Indoor67 database
[19] has 67 categories on indoor places. The SUN database [24] was introduced to provide a wide
coverage of scene categories. It is composed of 397 categories containing more than 100 images per
category.
Despite those efforts, all these scene-centric datasets are small in comparison with current object
datasets such as ImageNet (note that ImageNet also contains scene categories but in a very small
proportion as is shown in Fig. 2). Complementary to ImageNet (mostly object-centric), we present
here a scene-centric database, that we term the Places database. As now, Places contain more than 7
million images from 476 place categories, making it the largest image database of scenes and places
so far and the first scene-centric database competitive enough to train algorithms that require huge
amounts of data, such as CNNs.
2.1
Building the Places Database
Since the SUN database [24] has a rich scene taxonomy, the Places database has inherited the same
list of scene categories. To generate the query of image URL, 696 common adjectives (messy, spare,
sunny, desolate, etc), manually selected from a list of popular adjectives in English, are combined
with each scene category name and are sent to three image search engines (Google Images, Bing
Images, and Flickr). Adding adjectives to the queries allows us to download a larger number of
images than what is available in ImageNet and to increase the diversity of visual appearances. We
then remove duplicated URLs and download the raw images with unique URLs. To date, more than
40 million images have been downloaded. Only color images of 200?200 pixels or larger are kept.
PCA-based duplicate removal is conducted within each scene category in the Places database and
across the same scene category in the SUN database, which ensures that Places and the SUN do not
contain the same images, allowing us to combine the two datasets.
The images that survive this initial selection are sent to Amazon Mechanical Turk for two rounds
of individual image annotation. For a given category name, its definition as in [24], is shown at
the top of a screen, with a question like is this a living room scene? A single image at a time
is shown centered in a large window, and workers are asked to press a Yes or No key. For the
first round of labeling, the default answer is set to No, requiring the worker to actively pick up the
positive images. The positive images resulting from the first round annotation are further sent for a
second round annotation, in which the default answer is set to Yes (to pick up the remaining negative
images). In each HIT(one assignment for each worker), 750 downloaded images are included for
annotation, and an additional 30 positive samples and 30 negative samples with ground truth from
the SUN database are also randomly injected as control. Valid HITs kept for further analyses require
an accuracy of 90% or higher on these control images. After the two rounds of annotation, and as this
paper is published, 7,076,580 images from 476 scene categories are included in the Places database.
Fig. 1 shows image samples obtained with some of the adjectives used in the queries.
1
The database and pre-trained networks are available at http://places.csail.mit.edu
2
spare bedroom
wooded kitchen
teenage bedroom
messy kitchen
romantic bedroom
darkest forest path
stylish kitchen
rocky coast
wintering forest path
misty coast
greener forest path
sunny coast
Figure 1: Image samples from the scene categories grouped by their queried adjectives.
100000
Places
ImageNet
SUN
10000
100
bridge
cemetery
tower
train railway
canyon
pond
fountain
castle
lighthouse
valley
harbor
skyscraper
aquarium
palace
arch
highway
bedroom
creek
botanical garden
restaurant
kitchen
ocean
railroad track
river
baseball field
rainforest
stadium baseball
art gallery
office building
golf course
mansion
staircase
windmill
coast
stadium football
parking lot
basilica
building facade
lobby
abbey
vegetable garden
volcano
amusement park
shed
herb garden
alley
pasture
marsh
raft
dock
playground
mountain
hotel room
sea cliff
courtyard
badlands
office
boardwalk
desert sand
patio
living room
runway
plaza
sky
motel
underwater coral reef
driveway
dining room
train station platform
hospital
viaduct
forest path
construction site
campsite
mausoleum
music studio
mountain snowy
basement
cottage garden
boat deck
coffee shop
pagoda
shower
classroom
ballroom
corn field
parlor
yard
hot spring
kitchenette
art studio
butte
orchard
gas station
forest road
corridor
closet
fire station
dam
ski slope
field wild
ski resort
iceberg
fairway
phone booth
swamp
airport terminal
auditorium
wheat field
wind farm
bookstore
fire escape
supermarket
bar
water tower
rice paddy
cockpit
home office
crosswalk
bakery shop
bayou
veranda
slum
formal garden
chalet
ruin
attic
track outdoor
clothing store
tree farm
residential neighborhood
courthouse
restaurant patio
engine room
market outdoor
excavation
inn outdoor
trench
schoolhouse
conference room
pavilion
aqueduct
temple east asia
conference center
hospital room
rock arch
racecourse
shopfront
topiary garden
field cultivated
church outdoor
pulpit
museum indoor
dinette home
ice cream parlor
gift shop
boxing ring
laundromat
nursery
martial arts gym
swimming pool outdoor
food court
cathedral outdoor
reception
temple south asia
amphitheater
medina
pantry
galley
apartment building outdoor
watering hole
islet
banquet hall
crevasse
jail cell
candy store
kindergarden classroom
dorm room
bowling alley
ice skating rink outdoor
garbage dump
assembly line
picnic area
locker room
monastery outdoor
game room
kasbah
hotel outdoor
bus interior
doorway outdoor
television studio
butchers shop
waiting room
bamboo forest
restaurant kitchen
subway station platform
desert vegetation
beauty salon
rope bridge
stage indoor
snowfield
cafeteria
shoe shop
sandbar
igloo
1000
Figure 2: Comparison of the number of images per scene category in three databases.
We made 2 subsets of Places that will be used across the paper as benchmarks. The first one is Places
205, with the 205 categories with more than 5000 images. Fig. 2 compares the number of images in
Places 205 with ImageNet and SUN. Note that ImageNet only has 128 of the 205 categories, while
SUN contains all of them (we will call this set SUN 205, and it has, at least, 50 images per category).
The second subset of Places used in this paper is Places 88. It contains the 88 common categories
with ImageNet such that there are at least 1000 images in ImageNet. We call the corresponding
subsets SUN 88 and ImageNet 88.
3
Comparing Scene-centric Databases
Despite the importance of benchmarks and training datasets in computer vision, comparing datasets
is still an open problem. Even datasets covering the same visual classes have notable differences
providing different generalization performance when used to train a classifier [23]. Beyond the
number of images and categories, there are aspects that are important but difficult to quantify, like
the variability in camera poses, in decoration styles or in the objects that appear in the scene.
Although the quality of a database will be task dependent, it is reasonable to assume that a good
database should be dense (with a high degree of data concentration), and diverse (it should include
a high variability of appearances and viewpoints). Both quantities, density and diversity, are hard to
estimate in image sets, as they assume some notion of similarity between images which, in general,
is not well defined. Two images of scenes can be considered similar if they contain similar objects,
and the objects are in similar spatial configurations and pose, and have similar decoration styles.
However, this notion is loose and subjective so it is hard to answer the question are these two images
similar? For this reason, we define relative measures for comparing datasets in terms of density and
diversity that only require ranking similarities. In this section we will compare the densities and
diversities of SUN, ImageNet and Places using these relative measures.
3
3.1
Relative Density and Diversity
Density is a measure of data concentration. We assume that, in an image set, high density is equivalent to the fact that images have, in general, similar neighbors. Given two databases A and B, relative
density aims to measure which one of the two sets has the most similar nearest neighbors. Let a1
be a random image from set A and b1 from set B and let us take their respective nearest neighbors
in each set, a2 from A and b2 from B. If A is denser than B, then it would be more likely that a1
and a2 are closer to each other than b1 and b2 . From this idea we define the relative density as
DenB (A) = p (d(a1 , a2 ) < d(b1 , b2 )), where d(a1 , a2 ) is a distance measure between two images
(small distance implies high similarity). With this definition of relative density we have that A is
denser than B if, and only if, DenB (A) > DenA (B). This definition can be extended to an arbitrary
number of datasets, A1 , ..., AN :
DenA2 ,...,AN (A1 ) = p(d(a11 , a12 ) < min d(ai1 , ai2 ))
i=2:N
(1)
where ai1 ? Ai are randomly selected and ai2 ? Ai are near neighbors of their respective ai1 .
The quality of a dataset can not be measured just by its density. Imagine, for instance, a dataset
composed of 100,000 images all taken within the same bedroom. This dataset would have a very
high density but a very low diversity as all the images would look very similar. An ideal dataset,
expected to generalize well, should have high diversity as well.
There are several measures of diversity, most of them frequently used in biology to characterize the
richness of an ecosystem (see [9] for a review). In this section, we will use a measure inspired by
Simpson index of diversity [22]. Simpson index measures the probability that two random individuals from an ecosystem belong to the same species. It is a measure of how well distributed are the
individuals across different species in an ecosystem and it is related to the entropy of the distribution. Extending this measure for evaluating the diversity of images within a category is non-trivial if
there are no annotations of sub-categories. For this reason, we propose to measure relative diversity
of image datasets A and B based on this idea: if set A is more diverse than set B, then two random
images from set B are more likely to be visually similar than two random samples from A. Then,
the diversity of A with respect to B can be defined as DivB (A) = 1 ? p(d(a1 , a2 ) < d(b1 , b2 )),
where a1 , a2 ? A and b1 , b2 ? B are randomly selected. With this definition of relative diversity we
have that A is more diverse than B if, and only if, DivB (A) > DivA (B). For an arbitrary number of
datasets, A1 , ..., AN :
DivA2 ,...,AN (A1 ) = 1 ? p(d(a11 , a12 ) < min d(ai1 , ai2 ))
i=2:N
(2)
where ai1 , ai2 ? Ai are randomly selected.
3.2
Experimental Results
We measured the relative densities and diversities between SUN, ImageNet and Places using AMT.
Both measures used the same experimental interface: workers were presented with different pairs
of images and they had to select the pair that contained the most similar images. We observed that
different annotators are consistent in deciding whether a pair of images is more similar than another
pair of images.
In these experiments, the only difference when estimating density and diversity is how the pairs are
generated. For the diversity experiment, the pairs are randomly sampled from each database. Each
trial is composed of 4 pairs from each database, giving a total of 12 pairs to chose from. We used
4 pairs per database to increase the chances of finding a similar pair and avoiding users having to
skip trials. AMT workers had to select the most similar pair on each trial. We ran 40 trials per
category and two observers per trial, for the 88 categories in common between ImageNet, SUN and
Places databases. Fig. 3a shows some examples of pairs from one of the density experiments.The
pair selected by AMT workers as being more similar is highlighted.
For the density experiments, we selected pairs that were more likely to be visually similar. This
would require first finding the true nearest neighbor of each image, which would be experimentally
costly. Instead we used visual similarity as measured by using the Euclidean distance between
the Gist descriptor [17] of two images. Each pair of images was composed from one randomly
selected image and its 5-th nearest neighbor using Gist (we ignored the first 4 neighbors to avoid
4
ImageNet Places
ImageNet Places
1
Places
0.9
0.8
ImageNet
Diversity
0.7
0.6
0.5
SUN
0.4
0.3
SUN
SUN
0.2
0.1
a)
b)
c)
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Density
1
Figure 3: a) Examples of pairs for the diversity experiment. b) Examples of pairs for the density
experiment. c) Scatter plot of relative diversity vs. relative density per each category and dataset.
Test on SUN 88
70
70
Test on ImageNet Scene 88
Test on Places 88
55
50
50
40
30
Train on Places 88 [69.5]
Train on SUN 88 [63.3]
Train on ImageNet 88 [62.8]
20
10 0
10
a)
10
1
10
2
10
3
Number of training samples per category
Classification accuracy
60
Classification accuracy
Classification accuracy
60
50
40
30
Train on ImageNet 88 [65.6]
Train on Places 88 [60.3]
Train on SUN 88 [49.2]
20
10
10 0
10
4
b)
10
1
10
2
10
3
Number of training samples per category
45
40
35
30
25
Train on Places 88 [54.2]
Train on ImageNet 88 [44.6]
Train on SUN 88 [37.0]
20
15
10
4
10 0
10
c)
10
1
10
2
10
3
10
4
Number of training samples per category
Figure 4: Cross dataset generalization of training on the 88 common scenes between Places, SUN
and ImageNet then testing on the 88 common scenes from: a) SUN, b) ImageNet and c) Places
database.
near duplicates, which would give a wrong sense of high density). In this case we also show 12
pairs of images at each trial, but run 25 trials per category instead of 40 to avoid duplicate queries.
Fig. 3b shows some examples of pairs per one of the density experiments and also the selected pair
is highlighted. Notice that in the density experiment (where we computed neighbors) the pairs look,
in general, more similar than in the diversity experiment.
Fig. 3c shows a scatter plot of relative diversity vs. relative density for all the 88 categories and the
three databases. The point of crossing between the two black lines indicates the point where all the
results should fall if all the datasets were identical in terms of diversity and density. The figure also
shows the average of the density and diversity over all categories for each dataset.
In terms of density, the three datasets are, on average, very similar. However, there is a larger
variation in terms of diversity, showing Places to be the most diverse of the three datasets. The
average relative diversity on each dataset is 0.83 for Places, 0.67 for ImageNet and 0.50 for SUN.
In the experiment, users selected pairs from the SUN database to be the closest to each other 50%
of the time, while the pairs from the Places database were judged to be the most similar only on
17% of the trials. The categories with the largest variation in diversity across the three datasets are
playground, veranda and waiting room.
3.3
Cross Dataset Generalization
As discussed in [23], training and testing across different datasets generally results in a drop of
performance due to the dataset bias problem. In this case, the bias between datasets is due, among
other factors, to the differences in the density and diversity between the three datasets. Fig. 4 shows
the classification results obtained from the training and testing on different permutations of the 3
datasets. For these results we use the features extracted from a pre-trained ImageNet-CNN and a
linear SVM. In all three cases training and testing on the same dataset provides the best performance
for a fixed number of training examples. As the Places database is very large, it achieves the best
performance on two of the test sets when all the training data is used. In the next section we will
show that a CNN network trained using the Places database achieves a significant improvement over
scene-centered benchmarks in comparison with a network trained using ImageNet.
5
Table 1: Classification accuracy on the test set of Places 205 and the test set of SUN 205.
Places 205 SUN 205
Places-CNN
50.0%
66.2%
ImageNet CNN feature+SVM
40.8%
49.6%
4
Training Neural Network for Scene Recognition and Deep Features
Deep convolutional neural networks have obtained impressive classification performance on the
ImageNet benchmark [12]. For the training of Places-CNN, we randomly select 2,448,873 images
from 205 categories of Places (referred to as Places 205) as the train set, with minimum 5,000 and
maximum 15,000 images per category. The validation set contains 100 images per category and the
test set contains 200 images per category (a total of 41,000 images). Places-CNN is trained using
the Caffe package on a GPU NVIDIA Tesla K40. It took about 6 days to finish 300,000 iterations of
training. The network architecture of Places-CNN is the same as the one used in the Caffe reference
network [10]. The Caffe reference network, which is trained on 1.2 million images of ImageNet
(ILSVRC 2012), has approximately the same architecture as the network proposed by [12]. We call
the Caffe reference network as ImageNet-CNN in the following comparison experiments.
4.1
Visualization of the Deep Features
Through the visualization of the responses of the units for various levels of network layers, we
can have a better understanding of the differences between the ImageNet-CNN and Places-CNN
given that they share the same architecture. Fig.5 visualizes the learned representation of the units
at the Conv 1, Pool 2, Pool 5, and FC 7 layers of the two networks. Whereas Conv 1 units can
be directly visualized (they capture the oriented edges and opponent colors from both networks),
we use the mean image method to visualize the units of the higher layers: we first combine the
test set of ImageNet LSVRC2012 (100,000 images) and SUN397 (108,754 images) as the input
for both networks; then we sort all these images based on the activation response of each unit at
each layer; finally we average the top 100 images with the largest responses for each unit as a kind
of receptive field (RF) visualization of each unit. To compare the units from the two networks,
Fig. 5 displays mean images sorted by their first principal component. Despite the simplicity of
the method, the units in both networks exhibit many differences starting from Pool 2. From Pool
2 to Pool 5 and FC 7, gradually the units in ImageNet-CNN have RFs that look like object-blobs,
while units in Places-CNN have more RFs that look like landscapes with more spatial structures.
These learned unit structures are closely relevant to the differences of the training data. In future
work, it will be fascinating to relate the similarity and differences of the RF at different layers of
the object-centric network and scene-centric network with the known object-centered and scenecentered neural cortical pathways identified in the human brain (for a review, [16]). In the next
section we will show that these two networks (only differing in the training sets) yield very different
performances on a variety of recognition benchmarks.
4.2
Results on Places 205 and SUN 205
After the Places-CNN is trained, we use the final layer output (Soft-max) of the network to classify
images in the test set of Places 205 and SUN 205. The classification result is listed in Table 1. As
a baseline comparison, we show the results of a linear SVM trained on ImageNet-CNN features
of 5000 images per category in Places 205 and 50 images per category in SUN 205 respectively.
Places-CNN performs much better. We further compute the performance of the Places-CNN in the
terms of the top-5 error rate (one test sample is counted as misclassified if the ground-truth label is
not among the top 5 predicted labels of the model). The top-5 error rate for the test set of the Places
205 is 18.9%, while the top-5 error rate for the test set of SUN 205 is 8.1%.
4.3
Generic Deep Features for Visual Recognition
We use the responses from the trained CNN as generic features for visual recognition tasks. Responses from the higher-level layers of CNN have proven to be effective generic features with stateof-the-art performance on various image datasets [5, 20]. Thus we evaluate performance of the
6
Pool 2
Pool 5
FC 7
Pla c es -CNN
ImageNet-CNN
Conv 1
Figure 5: Visualization of the units? receptive fields at different layers for the ImageNet-CNN and
Places-CNN. Conv 1 units contains 96 filters. The Pool 2 feature map is 13?13?256; The Pool 5
feature map is 6?6?256; The FC 7 feature map is 4096?1. Subset of units at each layer are shown.
Table 2: Classification accuracy/precision on scene-centric databases and object-centric databases
for the Places-CNN feature and ImageNet-CNN feature. The classifier in all the experiments is a
linear SVM with the same parameters for the two features.
SUN397
MIT Indoor67
Scene15
SUN Attribute
Places-CNN feature
54.32?0.14
68.24
90.19?0.34
91.29
ImageNet-CNN feature 42.61?0.16
56.79
84.23?0.37
89.85
Caltech101
Caltech256
Action40
Event8
Places-CNN feature
65.18?0.88
45.59?0.31
42.86?0.25
94.12?0.99
ImageNet-CNN feature 87.22?0.92
67.23?0.27
54.92?0.33
94.42?0.76
deep features from the Places-CNN on the following scene and object benchmarks: SUN397 [24],
MIT Indoor67 [19], Scene15 [13], SUN Attribute [18], Caltech101 [7], Caltech256 [8], Stanford
Action40 [25], and UIUC Event8 [15]. All the experiments follow the standards in those papers 2 .
As a comparison, we evaluate the deep feature?s performance from the ImageNet-CNN on those
same benchmarks. Places-CNN and ImageNet-CNN have exactly the same network architecture,
but they are trained on scene-centric data and object-centric data respectively. We use the deep
features from the response of the Fully Connected Layer (FC) 7 of the CNNs, which is the final fully
connected layer before producing the class predictions. There is only a minor difference between
the feature of FC 7 and the feature of FC 6 layer [5]. The deep feature for each image is a 4096dimensional vector.
Table 2 summarizes the classification accuracy on various datasets for the ImageNet-CNN feature
and the Places-CNN feature. Fig.6 plots the classification accuracy for different visual features
on SUN397 database and SUN Attribute dataset. The classifier is a linear SVM with the same
default parameters for the two deep features (C=1) [6]. The Places-CNN feature shows impressive
performance on scene classification benchmarks, outperforming the current state-of-the-art methods
for SUN397 (47.20% [21]) and for MIT Indoor67 (66.87% [4]). On the other hand, the ImageNetCNN feature shows better performance on object-related databases. Importantly, our comparison
2
Detailed experimental setups are included in the supplementary materials.
7
Benchmark on SUN397 Dataset
Benchmark on SUN Attribute Dataset
70
Combined kernel [37.5]
HoG2x2 [26.3]
0.9
DenseSIFT [23.5]
60
Texton [21.6]
Gist [16.3]
0.85
LBP [14.7]
50
ImageNet?CNN [42.6]
Average Precision
Classification accuracy
Places?CNN [54.3]
40
30
0.8
0.75
0.7
Places?CNN [0.912]
ImageNet?CNN [0.898]
20
Combined kernel [0.879]
0.65
HoG2x2 [0.848]
Self?similarity [0.820]
10
Geometric Color Hist [0.783]
0.6
Gist [0.799]
0
1
5
10
20
0.55
1/1
50
Number of training samples per category
5/5
20/20
50/50
150/150
Number of training samples per attribute (positive/negative)
Figure 6: Classification accuracy on the SUN397 Dataset and average precision on the SUN Attribute Dataset with increasing size of training samples for the ImageNet-CNN feature and the
Places-CNN feature. Results of other hand-designed features/kernels are fetched from [24] and
[18] respectively.
Table 3: Classification accuracy/precision on various databases for Hybrid-CNN feature. The numbers in bold indicate the results outperform the ImageNet-CNN feature or Places-CNN feature.
SUN397
53.86?0.21
MIT Indoor67
70.80
Scene15
91.59?0.48
SUN Attribute
91.56
Caltech101
84.79?0.66
Caltech256
65.06?0.25
Action40
55.28?0.64
Event8
94.22?0.78
shows that Places-CNN and ImageNet-CNN have complementary strengths on scene-centric tasks
and object-centric tasks, as expected from the benchmark datasets used to train these networks.
Furthermore, we follow the same experimental setting of train and test split in [1] to fine tune
Places-CNN on SUN397: the fine-tuned Places-CNN achieves the accuracy of 56.2%, compared to
the accuracy of 52.2% achieved by the fine-tuned ImageNet-CNN in [1]. Note that the final output
of the fine-tuned CNN is directly used to predict scene category.
Additionally, we train a Hybrid-CNN, by combining the training set of Places-CNN and training set
of ImageNet-CNN. We remove the overlapping scene categories from the training set of ImageNet,
and then the training set of Hybrid-CNN has 3.5 million images from 1183 categories. HybridCNN is trained over 700,000 iterations, under the same network architecture of Places-CNN and
ImageNet-CNN. The accuracy on the validation set is 52.3%. We evaluate the deep feature (FC
7) from Hybrid-CNN on benchmarks shown in Table 3. Combining the two datasets yields an
additional increase in performance for a few benchmarks.
5
Conclusion
Deep convolutional neural networks are designed to benefit and learn from massive amounts of data.
We introduce a new benchmark with millions of labeled images, the Places database, designed to
represent places and scenes found in the real world. We introduce a novel measure of density and
diversity, and show the usefulness of these quantitative measures for estimating dataset biases and
comparing different datasets. We demonstrate that object-centric and scene-centric neural networks
differ in their internal representations, by introducing a simple visualization of the receptive fields
of CNN units. Finally, we provide the state-of-the-art performance using our deep features on all
the current scene benchmarks.
Acknowledgement. Thanks to Aditya Khosla for valuable discussions. This work is supported by the National
Science Foundation under Grant No. 1016862 to A.O, ONR MURI N000141010933 to A.T, as well as MIT
Big Data Initiative at CSAIL, Google and Xerox Awards, a hardware donation from NVIDIA Corporation, to
A.O and A.T., Intel and Google awards to J.X, and grant TIN2012-38187-C03-02 to A.L. This work is also supported by the Intelligence Advanced Research Projects Activity (IARPA) via Air Force Research Laboratory,
contract FA8650-12-C-7211 to A.T. The U.S. Government is authorized to reproduce and distribute reprints for
Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: 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 IARPA, AFRL, or the U.S. Government.
8
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4,804 | 535 | Unsupervised learning
of distributions on binary vectors
using two layer networks
David Haussler
Computer and Information Sciences
University of California Santa Cruz
Santa Cruz , CA 95064
Yoav Freund?
Computer and Information Sciences
University of California Santa Cruz
Santa Cruz, CA 95064
Abstract
We study a particular type of Boltzmann machine with a bipartite graph structure called a harmonium. Our interest is in using such a machine to model a probability distribution on binary input
vectors . We analyze the class of probability distributions that can be modeled by such machines.
showing that for each n ~ 1 this class includes arbitrarily good appwximations to any distribution
on the set of all n-vectors of binary inputs. We then present two learning algorithms for these
machines .. The first learning algorithm is the standard gradient ascent heuristic for computing
maximum likelihood estimates for the parameters (i.e. weights and thresholds) of the modeL Here
we give a closed form for this gradient that is significantly easier to compute than the corresponding
gradient for the general Boltzmann machine . The second learning algorithm is a greedy method
that creates the hidden units and computes their weights one at a time. This method is a variant
of the standard method for projection pursuit density estimation . We give experimental results for
these learning methods on synthetic data and natural data from the domain of handwritten digits.
1
Introduction
=
Let us suppose that each example in our in put data is a binary vector i
{x I, ... , x n } E {? l}n. and that
each such example is generated independently at random according some unknown distribution on {?l}n.
This situation arises. for instance. when each example consists of (possibly noisy) measurements of n different
binary attributes of a randomly selected object . In such a situation, unsupervised learning can be usefully
defined as using the input data to find a good model of the unknown distribution on {? l}n and thereby
learning the structure in the data.
The process of learning an unknown distribution from examples is usually called denszty estimation or
parameter estimation in statistics, depending on the nature of the class of distributions used as models.
Connectionist models of this type include Bayes networks (14). mixture models [3.13], and Markov random
fields [14,8]. Network models based on the notion of energy minimization such as Hopfield nets [9] and
Boltzmann machines [1] can also be used as models of probability distributions .
? yoavGcis.ucsc.edu
912
Unsupervised learning of distributions on binary vectors using 2-layer networks
The models defined by Hopfield networks are a special case of the more general Markov random field models
in which the local interactions are restricted to symmetric pairwise interactions between components of
the input. Boltzmann machines also use only pairwise interactions, but in addition they include hidden
units, which correspond to unobserved variables. These unobserved variables interact with the observed
variables represented by components of the input vector. The overall distribution on the set of possible
input vectors is defined as the marginal distribution induced on the components of the input vector by the
Markov random field over all variables, both observed and hidden . While the Hopfield network is relatively
well understood, it is limited in the types of distributions that it can model. On the other hand, Boltzmann
machines are universal in the sense that they are powerful enough to model any distribution (to any degree
of approximation), but the mathematical analysis of their capabilities is often intractable. Moreover, the
standard learning algorithm for the Boltzmann machine, a gradient ascent heuristic to compute the maximum
likelihood estimates for the weights and thresholds, requires repeated stochastic approximation, which results
in unacceptably slow learning. I In this work we attempt to narrow the gap between Hopfield networks and
Boltzmann machines by finding a model that will be powerful enough to be universal, 2 yet simple enough
to be analyzable and computationally efficient. 3 We have found such a model in a minor variant of the
special type of Boltzmann machine defined by Smolensky in his harmony theory [16][Ch.6J. This special type
of Boltzmann machine is defined by a network with a simple bipartite graph structure, which he called a
harmonium.
The harmonium consists of two types of units: input units, each of which holds one component of the input
vector, and hidden units, representing hidden variables. There is a weighted connection between each input
unit and each hidden unit, and no connections between input units or between hidden units (see Figure (1)) .
The presence of the hidden units induces dependencies, or correlations, between the variables modeled by
input units . To illustrate the kind of model that results, consider the distribution of people that visit a
specific coffee shop on Sunday. Let each of the n input variables represent the presence (+ 1) or absence (-1)
of a particular person that Sunday. These random variables are clearly not independent, e.g. if Fred's wife
and daughter are there, it is more likely that Fred is there , if you see three members of the golf club, you
expect to see other members of the golf club, if Bill is there you are unlikely to see Brenda there, etc. This
situation can be modeled by a harmonium model in which each hidden variable represents the presence or
absence of a social group . The weights connecting a hidden unit and an ipput unit measure the tendency of
the corresponding person to be associated with the corresponding group . In this coffee shop situation, several
social groups may be present at one time , exerting a combined influence on the distribution of customers.
This can be mo'deled easily with the harmonium , but is difficult to model using Bayes networks or mixture
models . <4
2
The Model
Let us begin by formalizing the harmonium model. To model a distribution on {?I}" we will use n input
units and some number m ~ 0 of hidden units. These units are connected in a bipartite graph as illustrated
in Figure (I) .
The random variables represented by the input units each take values in {+ I , -I}, while the hidden variables,
represented by the hidden units, take values in to, I} . The state of the machine is defined by the values
of these random variables. Define i
(XI," " xn) E {?l}n to be the state of the input units , and h
(hi , ... , hm ) E {O,l}m to be the state of the hidden units .
=
=
The connection weights between the input ~nits and the ith hidden unit are denoted 5 by w(') E R n and the
bias of the ith hidden unit is denoted by 9(') E R. The parameter vector ~
{(w(l),O(l?, . .. ,(w(m),o(m?))
=
lOne possible solution to this is tbe mean-field approximation [15], discussed furtber in section 2 below.
'In (4) we show tbat any distribution over (?1)" can be approximated to within any desired accuracy by a
harmonium model using 2" bidden units.
lSee also otber work relating Bayes nets and Boltzmann machines [12,1] .
t Noisy-OR gates have been introduced in the framework of Bayes Networks to allow for such combinations.
However, using this in networks with hidden units has not been studied, to the best of our knowledge.
~In (16)[Ch .6J, binary connection weights are used . Here we use real-valued weights .
913
914
Freund and Haussler
Hidden Units
Input
m=3
Units
2:1
2:2
2:3
2:4
2:5
Figure 1: The bipartite graph of the harmonium
defines the entire network, and thus also the probability model induced by the network. For a given
energy of a. state configuration of hidden and input units is defined to be
,p, the
m
E(i, hl~)
=-
L(w(i) . i
+ 8(i?)h i
(1)
i=!
and the probability of a configuration is
Pr(i,hl?l)
1
- l.) where Z =
= -Ze-E(Z,h
~-
L.,e-;.E(Z,h l .).
z,;;
Summing over h, it is easy to show that in the general case the probability distribution over possible state
vectors on the input units is given by
This product form is particular to the harmonium structure, and does not hold for general Boltzmann
machines. Product form distribution models have been used for density estimation in Projection Pursuit
[10,6,5] . We shall look further into this relationship in section 5.
3
Discussion of the model
The right hand side of Equation (2) has a simple intuitive interpretation . The ith factor in the product
corresponds to the hidden variable h. and is an increasing function of the dot product between i and the
weight vector of the ith hidden unit. Hence an input vector i will tend to have large probability when it is
in the direction of one of the weight vectors WCi) (i .e. when wei) . i is large). and small probability otherwise.
This is the way that the hidden variables can be seen to exert their" influence"; each corresponds to a.
preferred or "prototypical" direction in space .
The next to the last formula. in Equation (2) shows that the harmonium model can be written as a mixture
of 2m distributions of the form
~ exp (f)W(i) .i + 8('?)h.) ,
Z(h)
i=!
Unsupervised learning of distributions on binary vectors using 2-layer networks
where ii E to, l}m and Z(Ii) is the appropriate normalization factor. It is easily verified that each of these
distributions is in fact a product of n Bernoulli distributions on {+l, -l}, one for each input variable Xj.
Hence the harmonium model can be interpreted as a kind of mixture model. However, the number of
components in the mixture represented by a harmonium is exponential in the number of hidden units.
It is interesting to compare the class of harmonium models to the standard class of models defined by a
mixture of products of Bernoulli distributions. The same bipartite graph described in Figure (1) can be
used to define a standard mixture model. Assign each of the m hidden units a weight vector <.i;) and a
probability Pi such that I:~l Pi
1. To generate an example, choose one of the hidden units according to
te.;J(?) ?I. where Zi
the distribution defined by the Pi'S, and then choose the vector i according to P;(i)
is the appropriate normalization factor so that LIE{?I}" P;(i) 1. We thus get the distribution
=
=
=
m
P(i)
= L Pi e
i=1
W(') I
(3)
Z;
This form for presenting the standard mixture model emphasizes the similarity between this model and the
harmonium model. A vector i will have large probability if the dot product <.ii) ?x is large for some 1 :s i :s m
(so long as Pi is not too small). However, unlike the standard mixture model, the harmonium model allows
more than one hidden variable to be +1 for any generated example. This means that several hidden influences
can combine in the generation of a single example, because several hidden variables can be +1 at the same
time. To see why this is useful, consider the coffee shop example given in the introduction . At any moment
of time it is reasonable to find severa/social groups of people sitting in the shop . The harmonium model will
have a natural representation for this situation, while in order for the standard mixture model to describe
it accurately, a hidden variable has to be assigned to each combination of social groups that is likely to be
found in the shop at the same time. In such cases the harmonium model is exponentially more succinct than
the standard mixture model.
4
Learning by gradient ascent on the log-likelihood
We now suppose that we are given a sample consisting of a set 5 of vectors in {? l}n drawn independently
at random froro some unknown distribution . Our goal is use the sample 5 to find a good model for this
unknown distribution using a harmonium with m hidden units, if possible. The method we investigate here
is the method of maximum likelihood estimation using gradient ascent . The goal of learning is thus reduced
to finding the set of parameters for the harmonium that maximize the (log of the) probability of the set
of examples S. In fact, this gives the standard learning algorithm for general Boltzmann machines. For
a general Boltzmann machine this would require stochastic estimation of the parameters. As stochastic
estimation is very time-consuming, the result is that learning is very slow . In this section we show that
stochastic estimation need not be used for the harmonium model.
= {;{
I), ;(2), ... ,?(N)}, given a particular setting
From (2), the log likelihood of a sample of input vectors 5
{(w(l), 0(1? ?. ..? (w(m) , Oem?~} of the parameters of the model is:
?J
=
.
.
10g-hkehhood(?J)
=Lin Pr(i!?J) = Lm ( L In(l + e'"
-(.)
IES
.=1
(.)
H'
)) -
N In Z .
(4)
IES
Taking the gradient of the log-likelihood results in the following formula for the jth component of wei)
{}
~i) log-likelihood(?) = L x, 1 + e-(W~') 1+9(.1)
wJ
-
IES
N
L
Pr(il?J)x , 1 + e-(W!.IH,(,I)
(5)
IE!:!}"
A similar formula holds for the derivative of the bias term.
The purpose of the clamped and unclamped phases in the Boltzmann machine learning algorithm is to
approximate these two terms. In general, this requires stochastic methods. However , here the clamped term
is easy to calculate, it requires summing a logistic type function over all training examples. The same term
915
916
Freund and Haussler
is obtained by making the mean field approximation for the clamped phase in the general algorithm [15],
which is exact in this case. It is more difficult to compute the sleep phase term, as it is an explicit sum over
the entire input space, and within each term of this sum there is an implicit sum over the entire space of
configurations of hidden units in the factor Pr(i!,p) . However, again taking advantage of the special structure
of the harmonium, We can reduce this sleep phase gradient term to a sum only over the configurations of the
hidden units, yielding for each component of w(i)
8(i)log-likelibood(?l)
8w j
= L: Zj 1 + e-(W~')'I+I('?
les
where
Pr(hl?l)
=
-
N
L
Pr(hl?l)h i tanh(E hkWy?
he{O,I}"
(6)
k=1
exp(L~1 hi9(i? 0;=1 cosh(L~l hiW}i?
.
E.ii'e{o,I}" exp(E~1 h;9(i? OJ: 1 cosh(L~1 h;wJ'})]
Direct computation of (6) is fast for small m in contrast to the case for general Boltzmann machines (we
have performed experiments with m $ 10). However, for large m it is not possible to compute all 2m
terms. There is a way to avoid this exponential explosion if we can assume that a small number of terms
dominate the sums. If, for instance, we assume that the probability that more than k hidden units are
acti ve (+ I) at the same time is negligibly small we can get a good approximation by computing only O( mk)
terms . Alternately, if we are not sure which states of the hidden units have non-negligible probability, we
can dynamically search, as part of the learning process, for the significant terms in the sum . This way we
get an algorithm that is always accurate, and is efficient when the number of significant terms is small. In
the extreme case where we assume that only one hidden unit is active at a time (i.e. k = 1), the harmonium
model essentially reduces to the standard mixture model as discussed is section 3. For larger k, this type of
assumption provides a middle ground between the generality of the harmonium model and the simplicity of
the mixture model.
5
Projection Pursuit methods
A statistical method that has a close relationship with the harmonium model is the Projection Pursuit (PP)
technique [10,6 i5). The use of projection pursuit in the context of neural networks has been studied by
several researchers (e.g. [11]). Most of the work is in exploratory projection pursuit and projection pursuit
regreSSIOn. In this paper we are interested in projection pursuit dellslty estimation. Here PP avoids the
exponential blowup of the standard gradient ascent technique, and also has that advantage that the number
m of hidden units is estimated from the sample as well, rather than being specified in advance.
Projection pursuit density estimation [6] is based on several types of analysis, using the central limit theorem,
that lead to the following general conclusion. If i E R" is a random vector for which the different coordinates
are Independent, and w E R" is a vector from the n dimellsiollal ullit sphere, then the distribution of the
projectIon w? i is close to gaussian for most w. Thus searching for those directions wfor which the projection
of a sample is most non-gaussian is a way for detecting dependencies between the coordinates in high
dimensional distributions . Several "projection-indices" have been studied in the literature for measuring the
"non-gaussianity" of projection, each enhancing different properties of the projected distribution . In order
to find more than one projection direction, several methods of "structure elimination" have been devised .
These methods transform the sample in such a way that the the direction in which non-gaussianity has been
detected appears to be gaussian, thus enabling the algorithm to detect non-gaussian projections that would
otherwise be obscured. The search for a description of the distribution of a sample in terms of its projections
can be formalized in the context of maximal likelihood density estimation [6] . In order to create a formal
relation between the harmonium model and projection pursuit, we define a variant of the model that defines
a density over R" instead of a distribution over {?l}". Based on this form we devise a projection index and
a structure removal method that are the basis of the following learning algorithm (described fully in [4])
? Initialization
Set So to be the input sample.
Set Po to be the initial distribution (Gaussian).
Unsupervised learning of distributions on binary vectors using 2-layer networks
? Iteration
Repeat the following steps for i
1,2 . . . until no single-variable harmonium model has a significantly
higher likelihood than the Gaussian distribution with respect to Si'
1. Perform an estimate-maximize (EM) [2) search on the log-likelihood of a single hidden variable
model on the sample Si-I . Denote by 8i and wei) the parameters found by the search, and create
a new hidden unit with associated binary r. v. hi with these weights and bias.
2. Transform Si-l into Si using the following structure removal procedure.
For each example; E S'_1 compute the probability that the hidden variable h; found in the last
step is 1 on this input:
P(h;
1)
(1 + e-<I.+W(') .I))-I
=
= =
Flip a coin that has probability of "head" equal to P(h;
add; - WCi) to S; else add; to Si.
3. Set Pie;) to be Pi_l(i)Z,l (1
6
= 1).
If the coin turns out "head" then
+ el,+W(').I).
Experimental work
We have carried out several experiments to test the performance of unsupervised learning using the harmonium model. These are not, at this stage, extensive experimental comparisons, but they do provide initial
insights into the issues regarding our learning algorithms and the use of the harmonium model for learning
real world tasks .
The first set of experiments studies two methods for learning the harmonium model. The first is the gradient
ascent method, and the second is the projection pursuit method . The experiments in this set were performed
on synthetically generated data. The input consisted of binary vectors of 64 bits that represent 8 x 8 binary
images. The images are synthesized using a harmonium model with 10 hidden units whose weights were set
as in Figure (2,a) . The ultimate goal of the learning algorithms was to retrieve the model that generated
the data . To measure the quality of the models generated by the algorithms we use three different measures.
The likelihood.of the model, 6 the fraction of correct predictions the model makes when used to predict the
value of a single input bit given all the other bits, and the performance of the model when used to reconstruct
the inpu t from the most probable state of the hidden units. 7 All experiments use a test set and a train set,
each containing 1000 examples. The gradient ascent method used a standard momentum term, and typically
needed about 1000 epochs to stabilize. In the projection pursuit algorithm, 4 iterations of EM per hidden
unit proved sufficient to find a stable solution . The results are summarized in the following table and in
Figure (2) .
gradient ascent for 1000 epochs
proJectIOn pursuit
ProJection pursuit followed by
gradient ascent for 100 epochs
ProJection pursuit followed by
gradient ascent for 1000 epochs
true model
likelihood
train
test
0.399 0.425
0.799 0.802
single bit predictlOn
train
test
0.098
0.100
0.119
0.114
Input reconstructIOn
train
test
0.311
0.338
0.475
0.480
0.411
0.430
0.091
0.089
0.315
0.334
0.377
0.372
0.405
0.404
0.071
0.062
0.082
0.071
0.261
0.252
0.287
0.283
Looking at the table and Figure (2), and taking into account execution times, it appears that gradient
ascent is slow but eventually finds much of the underlying structure in the distribution, although several
of the hidden units (see units 1,2,6,7, counting from the left, in Figure (2,a? have no obvious relation to
the true model, In contrast, PP is fast and finds all of the features of the true model albeit sometimes
aWe present the negation of the log-likelihood, scaled so that the uniform distribution will have likelihood 1.0
1More precisely, for each input unit I we compute the probability p. that it has value +1. Then (or example
(XI, . . . ,1' .. ), we measure - L:~.I log,(1/2 + %,(p, - 1/2? .
917
918
Freund and Haussler
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Figure 2: The weight vectors of the models in the synthetic data experiments. Each matrix represents the
64 weights of one hidden unit. The square above the matrix represents the units bias. positive weights are
displayed as full squares and negative weights as empty squares, the area of the square is proportional to
the absolute value of the weight. (a) The weights in the model found by gradient ascent alone. (b) The
weights in the model found by projection pursuit alone. (c) The weights in the model used for generating
the data. (d) The weights in the model found by projection pursuit followed by gradient ascent. For this
last model we also show the histograms of the projection of the examples on the directions defined by those
weight vectors; the bimodality expected from projection pursuit analysis is evident .
in combinations, However, the error measurements show that something is still missing from the models
found by our implementation of PP. Following PP by a gradient ascent phase seems to give the best of both
algoflthms, finding a good approximation after only 140 epochs (40 PP + 100 gradient) and recovering the
true model almost exactly after 1040 epochs.
In the second set of experiments we compare the performance of the harmonium model to that of the mixture
model. The comparison uses real world data extracted from the NIST handwritten data base 8, Examples
are 16 x 16 binary images (see Figure (3)). We use 60 hidden units to model the distribution in both of the
models . Because of the large number of hidden units we cannot use gradient ascent learning and instead
use projection pursuit. For the same reason it was not possible to compute the likelihood of the harmonium
model and only the other two measures of error were used . Each test was run several times to get accuracy
bounds on the measurements. The results are summarized in the following table
Mixture model
HarmOnium model
smgle bIt predictIon
train
test
0.185 ? 0.005 0.258 ? 0.005
0.20 ? 0.01
0.21 ? om
anput reconstructIon
test
train
0.518 ? 0.002 0.715 ? 0.002
0.66 ? 0.03
0.63 ? 0.05
In Figure (4) we show some typical weight vectors found for the mixture model and for the harmonium
model, it is clear that while the mixture model finds weIghts that are some kind of average prototypes of
complete digits, the harmonium model finds weights that correspond to local features such as lines and
contrasts. There is a small but definite improvement in the errors of the harmonium model with respect to
the errors of the mixture model. As the experiments on synthetic data have shown that PP does not reach
INIST Special Database 1, HWDB RelI-l.l, May 1990.
Unsupervised learning of distributions on binary vectors using 2-layer networks
Figure 3: A few examples from the handwritten digits sample.
..............
.:~:I~~~;~L!i:
ll:~;~~! .l~.
!::::~ !:?i:::::
......?....
..
'
~~:
,~:::
::',
' . ' ? ? ? ? ? ? ? ? I"
?
.. "
. ..
???? 0
?? ??
'"
:
...:.?????????
,' ..
......
?.t ?
~::::::.::r::.
.:lIIl"!i:::::;
::::=,II~ : ::: ;:
~. :"~''';:::~::
'0'
eo. ? ?
....
?? 1. so ? ? ?
? ???? ? ??
?????? II . ???
~'~~~~~~'
.'
.. ~
............
.II... ..,
........
:,ii~~:!:~i
........
-, ....
;.II'?~ .~~:a::
III::;:~,;I~:
!!i:H'
.......:i:::;:
..
~
Figure 4: Typical weight vectors found by the mixture model (left) and the harmonium model (right)
optimal solutions by itself we expect the advantage of the harmonium model over the mixture model will
increase further by using improved learning methods. Of course, the harmonium model is a very general
distribution model and is not specifically tuned to the domain of handwritten digit images, thus it cannot be
compared to models specifically developed to capture structures in this domain. However, the experimental
results supports our claim that the harmonium model is a simple and tractable mathematical model for
describing distributions in which several correlation patterns combine to generate each individual example .
References
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(12) R. M. Neal. Leuning stochastic feedforwlLrd networks. Technical report, Deputment of Computer Science,
UniverSity of Toronto, Nov. 1990.
(13) S. NowlCLn . Ma.ximum likelihood competitive learning. In D. Touretsky, editor, Advance!
Proceumg Sy!teml, volume 2, pages 514-582. Morgan Kau(mlLnn, 1990 .
In
Neurolinformation
[14J J. Peul. Probabi/i!hc Retuoning in Intelligent Sy~tem!. Morgan KlLufmann, 1988.
(15] C. Peterson and J. R. Anderson. A mean field theory learnillg algorithm (or neural networks. Complex SIiItem! ,
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Cognition . Volume 1,' FoundatIon!. MIT Press, Cambridge, Mass., 1986.
In
the Mlcro!tructure of
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4,805 | 5,350 | Learning to Discover
Efficient Mathematical Identities
Wojciech Zaremba
Dept. of Computer Science
Courant Institute
New York Unviersity
Karol Kurach
Google Zurich &
Dept. of Computer Science
University of Warsaw
Rob Fergus
Dept. of Computer Science
Courant Institute
New York Unviersity
Abstract
In this paper we explore how machine learning techniques can be applied to the
discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a grammar of math
operators, we build trees that combine them in different ways, looking for compositions that are analytically equivalent to a target expression but of lower computational complexity. However, as the space of trees grows exponentially with the
complexity of the target expression, brute force search is impractical for all but
the simplest of expressions. Consequently, we introduce two novel learning approaches that are able to learn from simpler expressions to guide the tree search.
The first of these is a simple n-gram model, the other being a recursive neuralnetwork. We show how these approaches enable us to derive complex identities,
beyond reach of brute-force search, or human derivation.
1
Introduction
Machine learning approaches have proven highly effective for statistical pattern recognition problems, such as those encountered in speech or vision. However, their use in symbolic settings has
been limited. In this paper, we explore how learning can be applied to the discovery of mathematical
identities. Specifically, we propose methods for finding computationally efficient versions of a given
target expression. That is, finding a new expression which computes an identical result to the target,
but has a lower complexity (in time and/or space).
We introduce a framework based on attribute grammars [14] that allows symbolic expressions to be
expressed as a sequence of grammar rules. Brute-force enumeration of all valid rule combinations
allows us to discover efficient versions of the target, including those too intricate to be discovered by
human manipulation. But for complex target expressions this strategy quickly becomes intractable,
due to the exponential number of combinations that must be explored. In practice, a random search
within the grammar tree is used to avoid memory problems, but the chance of finding a matching
solution becomes vanishingly small for complex targets.
To overcome this limitation, we use machine learning to produce a search strategy for the grammar
trees that selectively explores branches likely (under the model) to yield a solution. The training
data for the model comes from solutions discovered for simpler target expressions. We investigate
several different learning approaches. The first group are n-gram models, which learn pairs, triples
etc. of expressions that were part of previously discovered solutions, thus hopefully might be part
of the solution for the current target. We also train a recursive neural network (RNN) that operates
within the grammar trees. This model is first pretrained to learn a continuous representation for
symbolic expressions. Then, using this representation we learn to predict the next grammar rule to
add to the current expression to yield an efficient version of the target.
Through the use of learning, we are able to dramatically widen the complexity and
scope of expressions that can be handled in our framework. We show examples of (i) O n3 target expressions
which can be computed in O n2 time (e.g. see Examples 1 & 2), and (ii) cases where naive eval1
uation of the target would require exponential time, but can be computed in O n2 or O n3 time.
The majority of these examples are too complex to be found manually or by exhaustive search and,
as far as we are aware, are previously undiscovered. All code and evaluation data can be found at
https://github.com/kkurach/math_learning.
In summary our contributions are:
? A novel grammar framework for finding efficient versions of symbolic expressions.
? Showing how machine learning techniques can be integrated into this framework, and
demonstrating how training models on simpler expressions can help which the discovery
of more complex ones.
? A novel application of a recursive neural-network to learn a continuous representation of
mathematical structures, making the symbolic domain accessible to many other learning
approaches.
? The discovery of many new mathematical identities which offer a significant reduction in
computational complexity for certain expressions.
n?m
Example 1: Assume we are given matrices A
, BP? Rm?p
. We wish to compute the
P? R
n Pm Pp
target expression: sum(sum(A*B)), i.e. : n,p AB = i=1 j=1 k=1 Ai,j Bj,k which
naively takes O(nmp) time. Our framework is able to discover an efficient version of the
formula, that computes the same result in O(n(m + p)) time: sum((sum(A, 1) * B)?, 1).
Our framework builds grammar trees that explore valid compositions of expressions from the
grammar, using a search strategy. In this example, the naive strategy of randomly choosing
permissible rules suffices and we can find another tree which matches the target expression in
reasonable time. Below, we show trees for (i) the original expression and (ii) the efficient
formula which avoids the use of a matrix-matrix multiply operation, hence is efficient to
compute.
???????
Example 2: Consider the target expression: sum(sum((A*B)k )), where k = 6. For an
expression of this degree, there are 9785 possible grammar trees and the naive strategy used in
Example 1 breaks down. We therefore learn a search strategy, training a model on successful
trees from simpler expressions, such as those for k = 2, 3, 4, 5. Our learning approaches capture
the common structure within the solutions, evident below, so can find an efficient O (nm)
expression for this target:
k = 2: sum((((((sum(A, 1)) * B) * A) * B)?), 1)
k = 3: sum((((((((sum(A, 1)) * B) * A) * B) * A) * B)?), 1)
k = 4: sum((((((((((sum(A, 1)) * B) * A) * B) * A) * B) * A) * B)?), 1)
k = 5: sum((((((((((((sum(A, 1)) * B) * A) * B) * A) * B) * A) * B) * A) * B)?), 1)
k = 6: sum(((((((((((((sum(A, 1) * B) * A) * B) *A) * B) * A) * B)* A) * B) * A) * B)?), 1)
1.1
Related work
The problem addressed in this paper overlaps with the areas of theorem proving [5, 9, 11], program
induction [18, 28] and probabilistic programming [12, 20]. These domains involve the challenging
issues of undecidability, the halting problem, and a massive space of potential computation. However, we limit our domain to computation of polynomials with fixed degree k, where undecidability
and the halting problem are not present, and the space of computation is manageable (i.e. it grows
exponentially, but not super-exponentially). Symbolic computation engines, such as Maple [6] and
Mathematica [27] are capable of simplifying expressions by collecting terms but do not explicitly
seek versions of lower complexity. Furthermore, these systems are rule based and do not use learning approaches, the major focus of this paper. In general, there has been very little exploration of
statistical machine learning techniques in these fields, one of the few attempts being the recent work
of Bridge et al. [4] who use learning to select between different heuristics for 1st order reasoning. In
contrast, our approach does not use hand-designed heuristics, instead learning them automatically
from the results of simpler expressions.
2
Rule
Input
Matrix-matrix multiply
Matrix-element multiply
Matrix-vector multiply
Matrix transpose
Column sum
Row sum
Column repeat
Row repeat
Element repeat
X
X
X
X
X
X
X
X
X
?
?
?
?
?
?
?
?
?
Output
Rn?m , Y ? Rm?p
Rn?m , Y ? Rn?m
Rn?m , Y ? Rm?1
Rn?m
Rn?m
Rn?m
Rn?1
R1?m
R1?1
Z
Z
Z
Z
Z
Z
Z
Z
Z
?
?
?
?
?
?
?
?
?
Rn?p
Rn?m
Rn?n
Rm?n
Rn?1
R1?m
Rn?m
Rn?m
Rn?m
Computation
Complexity
Z
Z
Z
Z
Z
Z
Z
Z
Z
O (nmp)
O (nm)
O (nm)
O (nm)
O (nm)
O (nm)
O (nm)
O (nm)
O (nm)
=
=
=
=
=
=
=
=
=
X * Y
X .* Y
X * Y
XT
sum(X,1)
sum(X,2)
repmat(X,1,m)
repmat(X,n,1)
repmat(X,n,m)
Table 1: The grammar G used in our experiments.
The attribute grammar, originally developed in 1968 by Knuth [14] in context of compiler construction, has been successfully used as a tool for design and formal specification. In our work, we
apply attribute grammars to a search and optimization problem. This has previously been explored
in a range of domains: from well-known algorithmic problems like knapsack packing [19], through
bioinformatics [26] to music [10]. However, we are not aware of any previous work related to discovering mathematical formulas using grammars, and learning in such framework. The closest work
to ours can be found in [7] which involves searching over the space of algorithms and the grammar
attributes also represent computational complexity.
Classical techniques in natural language processing make extensive use of grammars, for example
to parse sentences and translate between languages. In this paper, we borrow techniques from NLP
and apply them to symbolic computation. In particular, we make use of an n-gram model over
mathematical operations, inspired by n-gram language models. Recursive neural networks have
also been recently used in NLP, for example by Luong et al. [15] and Socher et al. [22, 23], as well
as generic knowledge representation Bottou [2]. In particular, Socher et al. [23], apply them to parse
trees for sentiment analysis. By contrast, we apply them to trees of symbolic expressions. Our work
also has similarities to Bowman [3] who shows that a recursive network can learn simple logical
predicates.
Our demonstration of continuous embeddings for symbolic expressions has parallels with the embeddings used in NLP for words and sentence structure, for example, Collobert & Weston [8], Mnih
& Hinton [17], Turian et al. [25] and Mikolov et al. [16].
2
Problem Statement
Problem Definition: We are given a symbolic target expression T that combines a set of variables V
to produce an output O, i.e. O = T(V). We seek an alternate expression S, such that S(V) = T(V),
but has lower computational complexity, i.e. O (S) < O (T).
In this paper we consider the restricted setting where: (i) T is a homogeneous polynomial of degree
k ? , (ii) V contains a single matrix or vector A and (iii) O is a scalar. While these assumptions may
seem quite restrictive, they still permit a rich family of expressions for our algorithm to explore.
For example, by combining multiple polynomial terms, an efficient Taylor series approximation
can be found for expressions involving trigonometric or exponential operators. Regarding (ii), our
framework can easily handle multiple variables, e.g. Figure 1, which shows expressions using two
matrices, A and B. However, the rest of the paper considers targets based on a single variable. In
Section 8, we discuss these restrictions further.
Notation: We adopt Matlab-style syntax for expressions.
3
Attribute Grammar
We first define an attribute grammar G, which contains a set of mathematical operations, each with
an associated complexity (the attribute). Since T contains exclusively polynomials, we use the
grammar rules listed in Table 1.
Using these rules we can develop trees that combine rules to form expressions involving V, which
for the purposes of this paper is a single matrix A. Since we know T involves expressions of degree
?
I.e. It only contains terms of degree k. E.g. ab + a2 + ac is a homogeneous polynomial of degree 2, but
a2 + b is not homogeneous (b is of degree 1, but a2 is of degree 2).
3
k, each tree must use A exactly k times. Furthermore, since the output is a scalar, each tree must
also compute a scalar quantity. These two constraints limit the depth of each tree. For some targets
T whose complexity is only O (() n3 ), we remove the matrix-matrix multiply rule, thus ensuring
that if any solution is found its complexity is at most O (() n2 ) (see Section 7.2 for more details).
Examples of trees are shown in Fig. 1. The search strategy for determining which rules to combine
is addressed in Section 6.
4
Representation of Symbolic Expressions
We need an efficient way to check if the expression produced by a given tree, or combination of trees
(see Section 5), matches T. The conventional approach would be to perform this check symbolically,
but this is too slow for our purposes and is not amenable to integration with learning methods. We
therefore explore two alternate approaches.
4.1 Numerical Representation
In this representation, each expression is represented by its evaluation of a randomly drawn set of
N points, where N is large (typically 1000). More precisely, for each variable in V, N different
copies are made, each populated with randomly drawn elements. The target expression evaluates
each of these copies, producing a scalar value for each, so yielding a vector t of length N which
uniquely characterizes T. Formally, tn = T(Vn ). We call this numerical vector t the descriptor
of the symbolic expression T. The size of the descriptor N , must be sufficiently large to ensure
that different expressions are not mapped to the same descriptor. Furthermore, when the descriptors
are used in the linear system of Eqn. 5 below, N must also be greater than the number of linear
equations. Any expression S formed by the grammar can be used to evaluate each Vn to produce
another N -length descriptor vector s, which can then be compared to t. If the two match, then
S(V) = T(V).
In practice, using floating point values can result in numerical issues that prevent t and s matching,
even if the two expressions are equivalent. We therefore use an integer-based descriptor in the form
of Zp ? , where p is a large prime number. This prevents both rounding issues as well as numerical
overflow.
4.2 Learned Representation
We now consider how to learn a continuous representation for symbolic expressions, that is learn a
projection ? which maps expressions S to l-dimensional vectors: ?(S) ? Rl . We use a recursive
neural network (RNN) to do this, in a similar fashion to Socher et al. [23] for natural language
and Bowman et al. [3] for logical expressions. This potentially allows many symbolic tasks to be
performed by machine learning techniques, in the same way that the word-vectors (e.g.[8] and [16])
enable many NLP tasks to be posed a learning problems.
We first create a dataset of symbolic expressions, spanning the space of all valid expressions up to
degree k. We then group them into clusters of equivalent expressions (using the numerical representation to check for equality),
and give each
cluster a discrete label 1 . . . C. For example, A, (AT )T
P P
P P
might have label 1, and i j Ai,j , j i Ai,j might have label 2 and so on. For k = 6, the
dataset consists of C = 1687 classes, examples of which are show in Fig. 1. Each class is split
80/20 into train/test sets.
We then train a recursive neural network (RNN) to classify a grammar tree into one of the C clusters.
Instead of representing each grammar rule by its underlying arithmetic, we parameterize it by a
weight matrix or tensor (for operations with one or two inputs, respectively) and use this to learn
the concept of each operation, as part of the network. A vector a ? Rl , where l = 30? is used
to represent each input variable. Working along the grammar tree, each operation in S evolves this
vector via matrix/tensor multiplications (preserving its length) until the entire expression is parsed,
resulting in a single vector ?(S) of length l, which is passed to the classifier to determine the class
of the expression, and hence which other expressions it is equivalent to.
Fig. 2 shows this procedure for two different expressions. Consider the first expression S = (A. ?
A)0 ? sum(A, 2). The first operation here is .?, which is implemented in the RNN by taking the
?
Integers modulo p
This was selected by cross-validation to control the capacity of the RNN, since it directly controls the
number of parameters in the model.
?
4
two (identical) vectors a and applies a weight tensor W3 (of size l ? l ? l, so that the output is
also size l), followed by a rectified-linear non-linearity. The output of this stage is this max((W3 ?
a) ? a, 0). This vector is presented to the next operation, a matrix transpose, whose output is thus
max(W2 ? max((W3 ? a) ? a, 0), 0). Applying the remaining operations produces a final output:
?(S) = max((W4 ? max(W2 ? max((W3 ? a) ? a, 0), 0)) ? max(W1 ? a, 0)). This is presented to a
C-way softmax classifier to predict the class of the expression. The weights W are trained using a
cross-entropy loss and backpropagation.
(((sum((sum((A * (A?)), 1)), 2)) * ((A * (((sum((A?), 1)) * A)?))?)) * A)
(sum(((sum((A * (A?)), 2)) * ((sum((A?), 1)) * (A * ((A?) * A)))), 1))
(((sum(A, 1)) * (((sum(A, 2)) * (sum(A, 1)))?)) * (A * ((A?) * A)))
((((sum((sum((A * (A?)), 1)), 2)) * ((sum((A?), 1)) * (A * ((A?) * A))))?)?)
((sum(A, 1)) * (((A?) * (A * ((A?) * ((sum(A, 2)) * (sum(A, 1))))))?))
((sum((sum((A * (A?)), 1)), 2)) * ((sum((A?), 1)) * (A * ((A?) * A))))
(((sum((sum((A * (A?)), 1)), 2)) * ((sum((A?), 1)) * A)) * ((A?) * A))
((A?) * ((sum(A, 2)) * ((sum((A?), 1)) * (A * (((sum((A?), 1)) * A)?)))))
(sum(((A?) * ((sum(A, 2)) * ((sum((A?), 1)) * (A * ((A?) * A))))), 2))
((((sum(A, 2)) * ((sum((A?), 1)) * A))?) * (A * (((sum((A?), 1)) * A)?)))
(((sum((A?), 1)) * (A * ((A?) * ((sum(A, 2)) * ((sum((A?), 1)) * A)))))?)
((((sum((A?), 1)) * A)?) * ((sum((A?), 1)) * (A * (((sum((A?), 1)) * A)?))))
(((A * ((A?) * ((sum(A, 2)) * ((sum((A?), 1)) * A))))?) * (sum(A, 2)))
(((A?) * ((sum(A, 2)) * ((sum((A?), 1)) * A))) * (sum(((A?) * A), 2)))
(b) Class B
(a) Class A
Figure 1: Samples from two classes of degree k = 6 in our dataset of expressions, used to learn
a continuous representation of symbolic expressions via an RNN. Each line represents a different
expression, but those in the same class are equivalent to one another.
(a) (A. ? A)0 ? sum(A, 2)
(b) (A0 . ? A0 ) ? sum(A, 2)
.
.
Figure 2: Our RNN applied to two expressions. The matrix A is represented by a fixed random
vector a (of length l = 30). Each operation in the expression applies a different matrix (for single
input operations) or tensor (for dual inputs, e.g. matrix-element multiplication) to this vector. After
each operation, a rectified-linear non-linearity is applied. The weight matrices/tensors for each
operation are shared across different expressions. The final vector is passed to a softmax classifier
(not shown) to predict which class they belong to. In this example, both expressions are equivalent,
thus should be mapped to the same class.
When training the RNN, there are several important details that are crucial to obtaining high classification accuracy:
? The weights should be initialized to the identity, plus a small amount of Gaussian noise
added to all elements. The identity allows information to flow the full length of the network,
up to the classifier regardless of its depth [21]. Without this, the RNN overfits badly,
producing test accuracies of ? 1%.
? Rectified linear units work much better in this setting than tanh activation functions.
? We learn using a curriculum [1], starting with the simplest expressions of low degree and
slowly increasing k.
? The weight matrix in the softmax classifier has much larger (?100) learning rate than the
rest of the layers. This encourages the representation to stay still even when targets are
replaced, for example, as we move to harder examples.
? As well as updating the weights of the RNN, we also update the initial value of a (i.e we
backpropagate to the input also).
When the RNN-based representation is employed for identity discovery (see Section 6.3), the vector
?(S) is used directly (i.e. the C-way softmax used in training is removed from the network).
5
Linear Combinations of Trees
For simple targets, an expression that matches the target may be contained within a single grammar
tree. But more complex expressions typically require a linear combination of expressions from
different trees.
5
To handle this, we can use the integer-based descriptors for each tree in a linear system and solve
for a match to the target descriptor (if one exists). Given a set of M trees, each with its own integer
descriptor vector f , we form an M by N linear system of equations and solve it:
F w = t mod Zp
where F = [f1 , . . . , fM ] holds the tree representations, w is the weighting on each of the trees
and t is the target representation. The system is solved using Gaussian elimination, where addition
and multiplication is performed modulo p. The number of solutions can vary: (a) there can be no
solution, which means that no linear combination of the current set of trees can match the target
expression. If all possible trees have been enumerated, then this implies the target expression is
outside the scope of the grammar. (b) There can be one or more solutions, meaning that some
combination of the current set of trees yields a match to the target expression.
6
Search Strategy
So far, we have proposed a grammar which defines the computations that are permitted (like a
programming language grammar), but it gives no guidance as to how explore the space of possible
expressions. Neither do the representations we introduced help ? they simply allow us to determine
if an expression matches or not. We now describe how to efficiently explore the space by learning
which paths are likely to yield a match.
Our framework uses two components: a scheduler, and a strategy. The scheduler is fixed, and traverses space of expressions according to recommendations given by the selected strategy (e.g. ?Random? or ?n-gram? or ?RNN?). The strategy assesses which of the possible grammar rules is likely
to lead to a solution, given the current expression. Starting with the variables V (in our case a single
element A, or more generally, the elements A, B etc.), at each step the scheduler receives scores
for each rule from the strategy and picks the one with the highest score. This continues until the
expression reaches degree k and the tree is complete. We then run the linear solver to see if a linear
combination of the existing set of trees matches the target. If not, the scheduler starts again with
a new tree, initialized with the set of variables V. The n-gram and RNN strategies are learned in
an
fashion, starting with simple target expressions (i.e. those of low degree k, such as
P incremental
T
become training examples used to improve the
ij AA ). Once solutions to these are found, they
P
strategy, needed for tackling harder targets (e.g. ij AAT A).
6.1 Random Strategy
The random strategy involves no learning, thus assigns equal scores to all valid grammar rules,
hence the scheduler randomly picks which expression to try at each step. For simple targets, this
strategy may succeed as the scheduler may stumble upon a match to the target within a reasonable
time-frame. But for complex target expressions of high degree k, the search space is huge and the
approach fails.
6.2 n-gram
In this strategy, we simply count how often subtrees of depth n occur in solutions to previously
solved targets. As the number of different subtrees of depth n is large, the counts become very
sparse as n grows. Due to this, we use a weighted linear combination of the score from all depths
up to n. We found an effective weighting to be 10k , where k is the depth of the tree.
6.3 Recursive Neural Network
Section 4.2 showed how to use an RNN to learn a continuous representation of grammar trees. Recall
that the RNN ? maps expressions to continuous vectors: ?(S) ? Rl . To build a search strategy from
this, we train a softmax layer on top of the RNN to predict which rule should be applied to the current
expression (or expressions, since some rules have two inputs), so that we match the target.
Formally, we have two current branches b1 and b2 (each corresponding to an expression) and wish
to predict the root operation r that joins them (e.g. .?) from among the valid grammar rules (|r|
in total). We first use the previously trained RNN to compute ?(b1 ) and ?(b2 ). These are then
presented to a |r|-way softmax layer (whose weight matrix U is of size 2l ? |r|). If only one branch
exists, then b2 is set to a fixed random vector. The training data for U comes from trees that give
efficient solutions to targets of lower degree k (i.e. simpler targets). Training of the softmax layer
is performed by stochastic gradient descent. We use dropout [13] as the network has a tendency to
overfit and repeat exactly the same expressions for the next value of k. Thus, instead of training on
exactly ?(b1 ) and ?(b2 ), we drop activations as we propagate toward the top of the tree (the same
6
fraction for each depth), which encourages the RNN to capture more local structures. At test time,
the probabilities from the softmax become the scores used by the scheduler.
7
Experiments
We first show results relating to the learned representation for symbolic expressions (Section 4.2).
Then we demonstrate our framework discovering efficient identities. For brevity, the identities discovered are listed in the supplementary material [29].
7.1 Expression Classification using Learned Representation
Table 2 shows the accuracy of the RNN model on expressions of varying degree, ranging from k = 3
to k = 6. The difficulty of the task can be appreciated by looking at the examples in Fig. 1. The low
error rate of ? 5%, despite the use of a simple softmax classifier, demonstrates the effectiveness of
our learned representation.
Test accuracy
Number of classes
Number of expressions
Degree k = 3
100% ? 0%
12
126
Degree k = 4
96.9% ? 1.5%
125
1520
Degree k = 5
94.7% ? 1.0%
970
13038
Degree k = 6
95.3% ? 0.7%
1687
24210
Table 2: Accuracy of predictions using our learned symbolic representation (averaged over 10 different initializations). As the degree increases tasks becomes more challenging, because number of
classes grows, and computation trees become deeper. However our dataset grows larger too (training
uses 80% of examples).
7.2
Efficient Identity Discovery
In our experiments we consider 5 different families of expressions, chosen to fall within the scope
of our grammar rules:
P
P
1. ( AAT )k : A is an Rn?n matrix. The k-th term is i,j (AAT )bk/2c for even k
P
P
P
and i,j (AAT )bk/2c A , for odd k. E.g. for k = 2 : i,j AAT ; for k = 3 : i,j AAT A;
P
for k = 4 : i,j AAT AAT etc. Naive evaluation is O kn3 .
P
n?n
2. P
( (A. ? A)AT )k : A is an RP
matrix and let B = A. ? A. The k-th
Pterm is
T bk/2c
T
bk/2c
(BA
)
for
even
k
and
(BA
B)
,
for
odd
k.
E.g.
for
k
=
2
:
i,j
i,j
i,j (A.?
P
P
T
T
T
A)A ; for k = 3 : i,j (A. ? A)A (A. ? A); for k = 4 : i,j (A. ? A)A (A. ? A)AT etc.
Naive evaluation is O kn3 .
P
3. Symk : Elementary symmetric polynomials. A is a vector in Rn?1 . For k = 1 : i Ai , for
P
P
k = 2 : i<j Ai Aj , for k = 3 : i<j<k Ai Aj Ak , etc. Naive evaluation is O nk .
n?1
4. (RBM-1)
. v is a binary n-vector. The k-th term is:
k : A is a vector in R
P
T
k
(v
A)
.
Naive
evaluation
is O (2n ).
v?{0,1}n
5. (RBM-2)k : Taylor series terms for the partition function of
Pan RBM. A is a matrix in
Rn?n . v and h are a binary n-vectors. The k-th term is v?{0,1}n ,h?{0,1}n (v T Ah)k .
Naive evaluation is O 22n .
Note that (i) for all families, the expressions yield a scalar output; (ii) the families are ordered in
rough order of ?difficulty?; (iii) we are not aware of any
Pprevious exploration
P of these expressions,
except for Symk , which is well studied [24]. For the ( AAT )k and ( (A. ? A)AT )k families
we remove the matrix-multiply rule from the grammar, thus ensuring
that if any solution
is found
it will be efficient since the remaining rules are at most O kn2 , rather than O kn3 . The other
families use the full grammar, given in Table
1. However, the limited set of rules means that if any
solution is found, it can at most be O n3 , rather than exponential in n, as the naive evaluations
would be. For each family, we apply our framework, using the three different search strategies
introduced in Section 6. For each run we impose a fixed cut-off time of 10 minutes? beyond which
we terminate the search. At each value of k, we repeat the experiments 10 times with different
random initializations and count the number of runs that find an efficient solution. Any non-zero
count is deemed a success, since each identity only needs to be discovered once. However, in Fig. 3,
we show the fraction of successful runs, which gives a sense of how quickly the identity was found.
?
Running on a 3Ghz 16-core Intel Xeon. Changing the cut-off has little effect on the plots, since the search
space grows exponentially fast.
7
We start with k = 2 and increase up to k = 15, using the solutions from previous values of k as
training data for the current degree. The search space quickly grows with k, as shown in Table 3.
Fig. 3 shows results for four of the families. We use n-grams for n = 1 . . . 5, as well as the RNN with
two different dropout rates (0.125 and 0.3). The learning approaches generally do much better than
the random strategy for large values of k, with the 3-gram, 4-gram and 5-gram models outperforming
the RNN.
For the first two families, the 3-gram model reliably finds solutions. These solutions involve repetition of a local patterns (e.g. Example 2), which can easily be captured with n-gram models. However, patterns that don?t have a simple repetitive structure are much more difficult to generalize. The
(RBM-2)k family is the most challenging, involving a double exponential sum, and the solutions
have highly complex trees (see supplementary material [29]). In this case, none of our approaches
performed better than the random strategy and no solutions were discovered for k > 5. However,
the k = 5 solution was found by the RNN consistently faster than the random strategy (100 ? 12 vs
438 ? 77 secs).
(
( AA T ) )
k
( A. * A ) A T ) )
(
1
k
Sym k
1
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
0.6
0.6
RNN0.3
RNN0.13
0.4
1?gram
2?gram
3?gram
4?gram
5?gram
Random
0.3
0.2
0.1
0
2
3
4
5
0.5
RNN0.3
RNN0.13
0.4
1?gram
2?gram
3?gram
4?gram
5?gram
Random
0.3
0.2
0.1
6
7
8
k
9
10 11 12 13 14 15
0
2
3
4
5
0.5
RNN0.3
RNN0.13
0.4
1?gram
2?gram
3?gram
4?gram
5?gram
Random
0.3
0.2
0.1
6
7
8
k
9
10 11 12 13 14 15
0
2
p(Success)
0.9
p(Success)
0.9
0.5
3
4
5
( RBM-1)
1
0.9
p(Success)
p(Success)
1
0.9
0.6
0.5
RNN0.3
RNN0.13
0.4
1?gram
2?gram
3?gram
4?gram
5?gram
Random
0.3
0.2
0.1
6
7
8
k
9
10 11 12 13 14 15
k
0
2
3
4
5
6
7
8
k
9
10 11 12 13 14 15
Figure 3: Evaluation on four different families of expressions. As the degree k increases, we
see that the random strategy consistently fails but the learning approaches can still find solutions
(i.e. p(Success) is non-zero). Best viewed in electronic form.
# Terms ? O n2
3
# Terms ? O n
k=2
39
41
k=3
171
187
k=4
687
790
k=5
2628
3197
k=6
9785
10k+
k = 7 and higher
Out of memory
Table 3: The number of possible expressions for different degrees k.
8
Discussion
We have introduced a framework based on a grammar of symbolic operations for discovering mathematical identities. Through the novel application of learning methods, we have shown how the
exploration of the search space can be learned from previously successful solutions to simpler expressions. This allows us to discover complex expressions that random or brute-force strategies
cannot find (the identities are given in the supplementary material [29]).
Some of the families considered in this paper are close to expressions often encountered in machine
learning. For example, dropout involves an exponential sum over binary masks, which is related to
the RBM-1 family. Also, the partition function of an RBM can be approximated by the RBM-2
family. Hence the identities we have discovered could potentially be used to give a closed-form
version of dropout, or compute the RBM partition function efficiently (i.e. in polynomial time).
Additionally, the automatic nature of our system naturally lends itself to integration with compilers,
or other optimization tools, where it could replace computations with efficient versions thereof.
Our framework could potentially be applied to more general settings, to discover novel formulae in
broader areas of mathematics. To realize this, additional grammar rules, e.g. involving recursion or
trigonometric functions would be needed. However, this would require a more complex scheduler
to determine when to terminate a given grammar tree. Also, it is surprising that a recursive neural
network can generate an effective continuous representation for symbolic expressions. This could
have broad applicability in allowing machine learning tools to be applied to symbolic computation.
The problem addressed in this paper involves discrete search within a combinatorially large space
? a core problem with AI. Our successful use of machine learning to guide the search gives hope
that similar techniques might be effective in other AI tasks where combinatorial explosions are
encountered.
Acknowledgements
The authors would like to thank Facebook and Microsoft Research for their support.
8
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9
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4,806 | 5,351 | Searching for Higgs Boson Decay Modes
with Deep Learning
Pierre Baldi
Department of Computer Science
University of California, Irvine
Irvine, CA 92617
[email protected]
Peter Sadowski
Department of Computer Science
University of California, Irvine
Irvine, CA 92617
[email protected]
Daniel Whiteson
Department of Physics and Astronomy
University of California, Irvine
Irvine, CA 92617 Address
[email protected]
Abstract
Particle colliders enable us to probe the fundamental nature of matter by observing exotic particles produced by high-energy collisions. Because the experimental measurements from these collisions are necessarily incomplete and imprecise,
machine learning algorithms play a major role in the analysis of experimental
data. The high-energy physics community typically relies on standardized machine learning software packages for this analysis, and devotes substantial effort
towards improving statistical power by hand-crafting high-level features derived
from the raw collider measurements. In this paper, we train artificial neural networks to detect the decay of the Higgs boson to tau leptons on a dataset of 82 million simulated collision events. We demonstrate that deep neural network architectures are particularly well-suited for this task with the ability to automatically
discover high-level features from the data and increase discovery significance.
1
Introduction
The Higgs boson was observed for the first time in 2011-2012 and will be the central object of
study when the Large Hadron Collider (LHC) comes back online to collect new data in 2015. The
observation of the Higgs boson in ZZ, ??, and W W decay modes at the LHC confirms its role
in electroweak symmetry-breaking [1, 2]. However, to establish that it also provides the interaction
which gives mass to the fundamental fermions, it must be demonstrated that the Higgs boson couples
to fermions through direct decay modes. Of the available modes, the most promising is the decay
to a pair of tau leptons (? ), which balances a modest branching ratio with manageable backgrounds.
From the measurements collected in 2011-2012, the LHC collaborations report data consistent with
h ? ? ? decays, but without statistical power to cross the 5? threshold, the standard for claims of
discovery in high-energy physics.
Machine learning plays a major role in processing data at particle colliders. This occurs at two levels:
the online filtering of streaming detector measurements, and the offline analysis of data once it has
been recorded [3], which is the focus of this work. Machine learning classifiers learn to distinguish
between different types of collision events by training on simulated data from sophisticated MonteCarlo programs. Single-hidden-layer, shallow neural networks are one of the primary techniques
used for this analysis, and standardized implementations are included in the prevalent multi-variate
1
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g
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+
#+
? "!
q
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#"?
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Z
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q?
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#+
Figure 1: Diagrams for the signal gg ? h ? ? ? ? `? ??`+ ?? and the dominant background
q q? ? Z ? ? ? ? `? ??`+ ??.
analysis software tools used by physicists. Efforts to increase statistical power tend to focus on
developing new features for use with the existing machine learning classifiers ? these high-level
features are non-linear functions of the low-level measurements, derived using knowledge of the
underlying physical processes.
However, the abundance of labeled simulation training data and the complex underlying structure
make this an ideal application for large, deep neural networks. In this work, we show how deep
neural networks can simplify and improve the analysis of high-energy physics data by automatically
learning high-level features from the data. We begin by describing the nature of the data and explaining the difference between the low-level and high-level features used by physicists. Then we
demonstrate that deep neural networks increase the statistical power of this analysis even without
the help of manually-derived high-level features.
2
Data
Collisions of protons at the LHC annhiliate the proton constituents, quarks and gluons. In a small
fraction of collisions, a new heavy state of matter is formed, such as a Higgs or Z boson. Such states
are very unstable and decay rapidly and successively into lighter particles until stable particles are
produced. In the case of Higgs boson production, the process is:
gg ? H ? ? + ? ?
(1)
followed by the subsequent decay of the ? leptons into lighter leptons (e and ?) and pairs of neutrinos
(?), see Fig. 1.
The point of collision is surrounded by concentric layers of detectors that measure the momentum
and direction of the final stable particles. The intermediate states are not observable, such that two
different processes with the same set of final stable particles can be difficult to distinguish. For
example, Figure 1 shows how the process q q? ? Z ? ? + ? ? yields the identical list of particles as
a process that produces the Higgs boson.
The primary approach to distinguish between two processes with identical final state particles is
via the momentum and direction of the particles, which contain information about the identity of
the intermediate state. With perfect measurement resolution and complete information of final state
2
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Figure 2: Low-level input features from basic kinematic quantities in `` + p6 T events for simulated
signal (black) and background (red) benchmark events. Shown are the distributions of transverse
momenta (pT ) of each observed particle as well as the imbalance of momentum in the final state.
Momentum angular information for each observed particle is also available to the network, but is
not shown, as the one-dimensional projections have little information.
particles B and C, we could calculate the invariant mass of the short-lived intermediate state A in
the process A ? B + C, via:
m2A = m2B+C = (EB + EC )2 ? |(pB + pC )|2
(2)
However, finite measurement resolution and escaping neutrinos (which are invisible to the detectors)
make it impossible to calculate the intermediate state mass precisely. Instead, the momentum and
direction of the final state particles are studied. This is done using simulated collisions from sophisticated Monte Carlo programs [4, 5, 6] that have been carefully tuned to provide highly faithful
descriptions of the collider data. Machine learning classifiers are trained on the simulated data to
recognize small differences in these processes, then the trained classifiers are used to analyze the
experimental data.
2.1
Low-level features
There are ten low-level features that comprise the essential measurements provided by the detectors:
? The three-dimensional momenta, p, of the charged leptons;
? The imbalance of momentum (6pT ) in the final state transverse to the beam direction, due to
unobserved or mismeasured particles;
? The number and momenta of particle ?jets? due to radiation of gluons or quarks.
Distributions of these features are given in Fig. 2.
2.2
High-level features
There is a vigorous effort in the physics community to construct non-linear combinations of these
low-level features that improve discrimination between Higgs-boson production and Z-boson production. High-level features that have been considered include:
? Axial missing momentum, p6 T ? p`+ `? ;
3
? Scalar sum of the observed momenta, |p`+ | + |p`? | + |6pT | +
P
i
|pjet |;
i
? Relative missing momentum, p6 T if ??(p, p6 T ) ? ?/2, and p6 T ? sin(??(p, p6 T ) if
??(p, p6 T ) < ?/2, where p is the momentum of any charged lepton or jet;
? Difference in lepton azimuthal angles, ??(`+ , `? );
? Difference in lepton polar angles, ??(`+ , `? );
p
? Angular distance between leptons, ?R = (??)2 + (??)2 ;
? Invariant mass of the two leptons, m`+ `? ;
? Missing mass, mMMC [7];
? Sphericity and transverse sphericity;
? Invariant mass of all visible objects (leptons and jets).
Distributions of these features are given in Fig. 3.
3
3.1
Methods
Current approach
Standard machine learning techniques in high-energy physics include methods such as boosted decision trees and single-layer feed-forward neural networks. The TMVA package [8] contains a
standardized implementation of these techniques that is widely-used by physicists. However, we
have found that our own hyperparameter-optimized, single-layer neural networks perform better
than the TMVA implementations. Therefore, we use our own hyperparameter-optimized shallow
neural networks trained on fast graphics processors as a benchmark for comparison.
3.2
Deep learning
Deep neural networks can automatically learn a complex hierarchy of non-linear features from data.
Training deep networks often requires additional computation and a careful selection of hyperparameters, but these difficulties have diminished substantially with the advent of inexpensive graphics processing hardware. We demonstrate here that deep neural networks provide a practical tool
for learning deep feature hierarchies and improving classifier accuracy while reducing the need for
physicists to carefully derive new features by hand. Many exploratory experiments were carried
with different architectures, training protocols, and hyperparameter optimization strategies. Some
of these experiments are still ongoing and, for conciseness, we report only the main results obtained
so far.
3.3
Hyperparameter optimization
Hyperparameters were optimized separately for shallow and deep neural networks. Shallow network
hyperparameters were chosen from combinations of the parameters listed in Table 1, while deep
network hyperparameters were chosen from combinations of those listed in Table 2. These were
selected based on classification performance (cross-entropy error) on the validation set, using the
full set of available features: 10 low-level features plus 15 high-level features. The best architectures
were the largest ones: a deep network with 300 hidden units in each of five hidden layers and an
initial learning rate of 0.03, and a shallow network with 15000 hidden units and an initial learning
rate of 0.01. These neural networks have approximately the same number of tunable parameters,
with 369,301 parameters in the deep network and 405,001 parameters in the shallow network.
Table 1: Hyperparameter options for shallow networks.
Hyperparameter
Options
Hidden units
100, 300, 1000, 15000
Initial learning rate 0.03, 0.01, 0.003, 0.001
4
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Figure 3: Distribution of high-level input features from invariant mass calculations in `?jjb?b events
for simulated signal (black) and background (red) events.
3.4
Training details
The problem is a basic classification task with two classes. The data set is balanced and contains
82 million examples. A validation set of 1 million examples was randomly set aside for tuning the
hyperparameters. Different cross validation strategies were used with little influence on the results
reported since these are obtained in a regime far away from overfitting.
5
Table 2: Hyperparameter options for deep networks.
Hyperparameter
Options
Number of layers
3,4,5,6
Hidden units per layer 100, 300
Initial learning rate
0.03, 0.01, 0.003
The following neural network hyperparameters were predetermined without optimization. The tanh
activation function was used for all hidden units, while the the logistic function was used for the
output. Weights were initialized from a normal distribution with zero mean and standard deviation
0.1 in the first layer, 0.001 in the output layer, and ?1k for all other hidden layers, where k was the
number of units in the previous layer. Gradient computations were made on mini-batches of size
100. A momentum term increased linearly over the first 25 epochs from 0.5 to 0.99, then remained
constant. The learning rate decayed by a factor of 1.0000002 every batch update until it reached a
minimum of 10?6 . All networks were trained for 50 epochs.
Computations were performed using machines with 16 Intel Xeon cores, an NVIDIA Tesla C2070
graphics processor, and 64 GB memory. Training was performed using the Theano and Pylearn2
software libraries [9, 10].
4
Results
The performance of each neural network architecture in terms of the Area Under the signal-rejection
Curve (AUC) is shown in Table 3. As expected, the shallow neural networks (one hidden layer)
perform better with the high-level features than the low-level features alone; the high-level features
were specifically designed to discriminate between the two hypotheses. However, this difference
disappears in deep neural networks, and in fact performance is better with the 10 low-level features
than with the 15 high-level features alone. This, along with the fact that the complete set of features
always performs best, suggests that there is information in the low-level measurements that is not
captured by the high-level features, and that the deep networks are exploiting this information.
Table 3: Comparison of performance for neural network architectures: shallow neural networks
(NN), and deep neural networks (DN) with different numbers of hidden units and layers. Each network architecture was trained on three sets of input features: low-level features, high-level features,
and the complete set of features. The table displays the test set AUC and the expected significance
of a discovery (in units of Gaussian ?) for 100 signal events and 5000 background events with a 5%
relative uncertainty.
AUC
Technique
Low-level High-level Complete
NN 300
0.788
0.792
0.798
NN 1000
0.788
0.792
0.798
NN 15000
0.788
0.792
0.798
DN 3-layer 0.796
0.794
0.801
DN 4-layer 0.797
0.797
0.802
DN 5-layer 0.798
0.798
0.803
DN 6-layer 0.799
0.797
0.803
Discovery significance
Technique
Low-level High-level Complete
NN 15000
1.7?
2.0?
2.0?
DN 6-layer 2.1?
2.2?
2.2?
The best networks are trained with the complete set of features, which provides both the raw measurements and the physicist?s domain knowledge. Figure 4 plots the empirical distribution of predictions (neural network output) for the test samples from each class, and shows how both the shallow
and deep networks trained on the complete feature set are more confident about their correct predictions.
6
NN lo-level
NN hi-level
NN lo+hi-level
0.0
0.2
0.4
0.6
Prediction
0.8
1.0
0.8
1.0
DN lo-level
DN hi-level
DN lo+hi-level
0.0
0.2
0.4
0.6
Prediction
Figure 4: Empirical distribution of predictions for signal events (solid) and background events
(dashed) from the test set.
Figure 5 shows how the AUC translates into discovery significance [11]. On this metric too, the sixlayer deep network trained on the low-level features outperforms the best shallow network (15000
hidden units) trained with the best feature set.
5
Discussion
While deep learning has led to significant advances in computer vision, speech, and natural language
processing, it is clearly useful for a wide range of applications, including a host of applications in the
natural sciences. The problems in high-energy physics are particularly suitable for deep learning,
having large data sets with complex underlying structure. Our results show that deep neural networks
provide a powerful and practical approach to analyzing particle collider data, and that the high-level
features learned from the data by deep neural networks increase the statistical power more than
the common high-level features handcrafted by the physicists. While the improvements may seem
small, they are very significant, especially when considering the billion-dollar cost of accelerator
experiments.
These preliminary experiments demonstrate the advantages of deep neural networks, but we have
not yet pushed the limits of what deep learning can do for this application. The deep architectures
in this work have less than 500,000 parameters and have not even begun to overfit the training data.
7
Shallow networks
Deep networks
all inputs
human-assisted
0.5
raw inputs
1.0
all inputs
1.5
human-assisted
2.0
raw inputs
Discovery significance (?)
2.5
0.0
Figure 5: Comparison of discovery significance for the traditional learning method (left) and the
deep learning method (right) using the low-level features, the high-level features, and the complete
set of features.
Experiments with larger architectures, including ensembles, with a variety of shapes and neuron
types, are currently in progress.
Since the high-level features are derived from the low-level features, it is interesting to note that one
could train a regression neural network to learn this relationship. Such a network would then be able
to predict the physicist-derived features from the low-level measurements. Some of these high-level
features may be more difficult to compute than others, requiring neural networks of a particular size
and depth, and it would be interesting to analyze the complexity of the high-level features in this
way. We are in the process of training such regression networks which could then be incorporated
into a larger prediction architecture, either by freezing their weights, or by allowing them to learn
further. In combination, these deep learning approaches should yield a system ready to sift through
the new Large Hadron Collider data in 2015.
References
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8
[10] Goodfellow, I. J., Warde-Farley, D., et al. Pylearn2: a machine learning research library. arXiv
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4,807 | 5,352 | Semi-supervised Learning with
Deep Generative Models
?
Diederik P. Kingma? , Danilo J. Rezende? , Shakir Mohamed? , Max Welling?
Machine Learning Group, Univ. of Amsterdam, {D.P.Kingma, M.Welling}@uva.nl
?
Google Deepmind, {danilor, shakir}@google.com
Abstract
The ever-increasing size of modern data sets combined with the difficulty of obtaining label information has made semi-supervised learning one of the problems
of significant practical importance in modern data analysis. We revisit the approach to semi-supervised learning with generative models and develop new models that allow for effective generalisation from small labelled data sets to large
unlabelled ones. Generative approaches have thus far been either inflexible, inefficient or non-scalable. We show that deep generative models and approximate
Bayesian inference exploiting recent advances in variational methods can be used
to provide significant improvements, making generative approaches highly competitive for semi-supervised learning.
1
Introduction
Semi-supervised learning considers the problem of classification when only a small subset of the
observations have corresponding class labels. Such problems are of immense practical interest in a
wide range of applications, including image search (Fergus et al., 2009), genomics (Shi and Zhang,
2011), natural language parsing (Liang, 2005), and speech analysis (Liu and Kirchhoff, 2013), where
unlabelled data is abundant, but obtaining class labels is expensive or impossible to obtain for the
entire data set. The question that is then asked is: how can properties of the data be used to improve
decision boundaries and to allow for classification that is more accurate than that based on classifiers
constructed using the labelled data alone. In this paper we answer this question by developing
probabilistic models for inductive and transductive semi-supervised learning by utilising an explicit
model of the data density, building upon recent advances in deep generative models and scalable
variational inference (Kingma and Welling, 2014; Rezende et al., 2014).
Amongst existing approaches, the simplest algorithm for semi-supervised learning is based on a
self-training scheme (Rosenberg et al., 2005) where the the model is bootstrapped with additional
labelled data obtained from its own highly confident predictions; this process being repeated until
some termination condition is reached. These methods are heuristic and prone to error since they
can reinforce poor predictions. Transductive SVMs (TSVM) (Joachims, 1999) extend SVMs with
the aim of max-margin classification while ensuring that there are as few unlabelled observations
near the margin as possible. These approaches have difficulty extending to large amounts of unlabelled data, and efficient optimisation in this setting is still an open problem. Graph-based methods
are amongst the most popular and aim to construct a graph connecting similar observations; label
information propagates through the graph from labelled to unlabelled nodes by finding the minimum
energy (MAP) configuration (Blum et al., 2004; Zhu et al., 2003). Graph-based approaches are sensitive to the graph structure and require eigen-analysis of the graph Laplacian, which limits the scale
to which these methods can be applied ? though efficient spectral methods are now available (Fergus et al., 2009). Neural network-based approaches combine unsupervised and supervised learning
For an updated version of this paper, please see http://arxiv.org/abs/1406.5298
1
by training feed-forward classifiers with an additional penalty from an auto-encoder or other unsupervised embedding of the data (Ranzato and Szummer, 2008; Weston et al., 2012). The Manifold
Tangent Classifier (MTC) (Rifai et al., 2011) trains contrastive auto-encoders (CAEs) to learn the
manifold on which the data lies, followed by an instance of TangentProp to train a classifier that is
approximately invariant to local perturbations along the manifold. The idea of manifold learning
using graph-based methods has most recently been combined with kernel (SVM) methods in the Atlas RBF model (Pitelis et al., 2014) and provides amongst most competitive performance currently
available.
In this paper, we instead, choose to exploit the power of generative models, which recognise the
semi-supervised learning problem as a specialised missing data imputation task for the classification problem. Existing generative approaches based on models such as Gaussian mixture or hidden
Markov models (Zhu, 2006), have not been very successful due to the need for a large number
of mixtures components or states to perform well. More recent solutions have used non-parametric
density models, either based on trees (Kemp et al., 2003) or Gaussian processes (Adams and Ghahramani, 2009), but scalability and accurate inference for these approaches is still lacking. Variational
approximations for semi-supervised clustering have also been explored previously (Li et al., 2009;
Wang et al., 2009).
Thus, while a small set of generative approaches have been previously explored, a generalised and
scalable probabilistic approach for semi-supervised learning is still lacking. It is this gap that we
address through the following contributions:
? We describe a new framework for semi-supervised learning with generative models, employing rich parametric density estimators formed by the fusion of probabilistic modelling
and deep neural networks.
? We show for the first time how variational inference can be brought to bear upon the problem of semi-supervised classification. In particular, we develop a stochastic variational
inference algorithm that allows for joint optimisation of both model and variational parameters, and that is scalable to large datasets.
? We demonstrate the performance of our approach on a number of data sets providing stateof-the-art results on benchmark problems.
? We show qualitatively generative semi-supervised models learn to separate the data classes
(content types) from the intra-class variabilities (styles), allowing in a very straightforward
fashion to simulate analogies of images on a variety of datasets.
2
Deep Generative Models for Semi-supervised Learning
We are faced with data that appear as pairs (X, Y) = {(x1 , y1 ), . . . , (xN , yN )}, with the i-th observation xi ? RD and the corresponding class label yi ? {1, . . . , L}. Observations will have
corresponding latent variables, which we denote by zi . We will omit the index i whenever it is clear
that we are referring to terms associated with a single data point. In semi-supervised classification,
only a subset of the observations have corresponding class labels; we refer to the empirical distribution over the labelled and unlabelled subsets as pel (x, y) and peu (x), respectively. We now develop
models for semi-supervised learning that exploit generative descriptions of the data to improve upon
the classification performance that would be obtained using the labelled data alone.
Latent-feature discriminative model (M1): A commonly used approach is to construct a model
that provides an embedding or feature representation of the data. Using these features, a separate
classifier is thereafter trained. The embeddings allow for a clustering of related observations in a
latent feature space that allows for accurate classification, even with a limited number of labels.
Instead of a linear embedding, or features obtained from a regular auto-encoder, we construct a
deep generative model of the data that is able to provide a more robust set of latent features. The
generative model we use is:
p(z) = N (z|0, I);
p? (x|z) = f (x; z, ?),
(1)
where f (x; z, ?) is a suitable likelihood function (e.g., a Gaussian or Bernoulli distribution) whose
probabilities are formed by a non-linear transformation, with parameters ?, of a set of latent variables z. This non-linear transformation is essential to allow for higher moments of the data to be
captured by the density model, and we choose these non-linear functions to be deep neural networks.
2
Approximate samples from the posterior distribution over the latent variables p(z|x) are used as features to train a classifier that predicts class labels y, such as a (transductive) SVM or multinomial
regression. Using this approach, we can now perform classification in a lower dimensional space
since we typically use latent variables whose dimensionality is much less than that of the observations. These low dimensional embeddings should now also be more easily separable since we
make use of independent latent Gaussian posteriors whose parameters are formed by a sequence of
non-linear transformations of the data. This simple approach results in improved performance for
SVMs, and we demonstrate this in section 4.
Generative semi-supervised model (M2): We propose a probabilistic model that describes the data
as being generated by a latent class variable y in addition to a continuous latent variable z. The data
is explained by the generative process:
p(y) = Cat(y|?);
p(z) = N (z|0, I);
p? (x|y, z) = f (x; y, z, ?),
(2)
where Cat(y|?) is the multinomial distribution, the class labels y are treated as latent variables if
no class label is available and z are additional latent variables. These latent variables are marginally
independent and allow us, in case of digit generation for example, to separate the class specification from the writing style of the digit. As before, f (x; y, z, ?) is a suitable likelihood function,
e.g., a Bernoulli or Gaussian distribution, parameterised by a non-linear transformation of the latent
variables. In our experiments, we choose deep neural networks as this non-linear function. Since
most labels y are unobserved, we integrate over the class of any unlabelled data during the inference process, thus performing classification as inference. Predictions for any missing labels are
obtained from the inferred posterior distribution p? (y|x). This model can also be seen as a hybrid
continuous-discrete mixture model where the different mixture components share parameters.
Stacked generative semi-supervised model (M1+M2): We can combine these two approaches by
first learning a new latent representation z1 using the generative model from M1, and subsequently
learning a generative semi-supervised model M2, using embeddings from z1 instead of the raw data
x. The result is a deep generative model with two layers of stochastic variables: p? (x, y, z1 , z2 ) =
p(y)p(z2 )p? (z1 |y, z2 )p? (x|z1 ), where the priors p(y) and p(z2 ) equal those of y and z above, and
both p? (z1 |y, z2 ) and p? (x|z1 ) are parameterised as deep neural networks.
3
Scalable Variational Inference
3.1 Lower Bound Objective
In all our models, computation of the exact posterior distribution is intractable due to the nonlinear,
non-conjugate dependencies between the random variables. To allow for tractable and scalable
inference and parameter learning, we exploit recent advances in variational inference (Kingma and
Welling, 2014; Rezende et al., 2014). For all the models described, we introduce a fixed-form
distribution q? (z|x) with parameters ? that approximates the true posterior distribution p(z|x). We
then follow the variational principle to derive a lower bound on the marginal likelihood of the model
? this bound forms our objective function and ensures that our approximate posterior is as close as
possible to the true posterior.
We construct the approximate posterior distribution q? (?) as an inference or recognition model,
which has become a popular approach for efficient variational inference (Dayan, 2000; Kingma and
Welling, 2014; Rezende et al., 2014; Stuhlm?uller et al., 2013). Using an inference network, we avoid
the need to compute per data point variational parameters, but can instead compute a set of global
variational parameters ?. This allows us to amortise the cost of inference by generalising between
the posterior estimates for all latent variables through the parameters of the inference network, and
allows for fast inference at both training and testing time (unlike with VEM, in which we repeat
the generalized E-step optimisation for every test data point). An inference network is introduced
for all latent variables, and we parameterise them as deep neural networks whose outputs form the
parameters of the distribution q? (?). For the latent-feature discriminative model (M1), we use a
Gaussian inference network q? (z|x) for the latent variable z. For the generative semi-supervised
model (M2), we introduce an inference model for each of the latent variables z and y, which we we
assume has a factorised form q? (z, y|x) = q? (z|x)q? (y|x), specified as Gaussian and multinomial
distributions respectively.
M1: q? (z|x) = N (z|?? (x), diag(? 2? (x))),
(3)
M2: q? (z|y, x) = N (z|?? (y, x), diag(? 2? (x))); q? (y|x) = Cat(y|? ? (x)),
3
(4)
where ? ? (x) is a vector of standard deviations, ? ? (x) is a probability vector, and the functions
?? (x), ? ? (x) and ? ? (x) are represented as MLPs.
3.1.1
Latent Feature Discriminative Model Objective
For this model, the variational bound J (x) on the marginal likelihood for a single data point is:
log p? (x) ? Eq? (z|x) [log p? (x|z)] ? KL[q? (z|x)kp? (z)] = ?J (x),
(5)
The inference network q? (z|x) (3) is used during training of the model using both the labelled and
unlabelled data sets. This approximate posterior is then used as a feature extractor for the labelled
data set, and the features used for training the classifier.
3.1.2
Generative Semi-supervised Model Objective
For this model, we have two cases to consider. In the first case, the label corresponding to a data
point is observed and the variational bound is a simple extension of equation (5):
log p? (x, y) ? Eq? (z|x,y) [log p? (x|y, z) + log p? (y) + log p(z) ? log q? (z|x, y)] = ?L(x, y), (6)
For the case where the label is missing, it is treated as a latent variable over which we perform
posterior inference and the resulting bound for handling data points with an unobserved label y is:
log p? (x) ? Eq? (y,z|x) [log p? (x|y, z) + log p? (y) + log p(z) ? log q? (y, z|x)]
X
=
q? (y|x)(?L(x, y)) + H(q? (y|x)) = ?U(x).
y
The bound on the marginal likelihood for the entire dataset is now:
X
X
J =
L(x, y) +
U(x)
(x,y)?e
pl
x?e
pu
(7)
(8)
The distribution q? (y|x) (4) for the missing labels has the form a discriminative classifier, and
we can use this knowledge to construct the best classifier possible as our inference model. This
distribution is also used at test time for predictions of any unseen data.
In the objective function (8), the label predictive distribution q? (y|x) contributes only to the second
term relating to the unlabelled data, which is an undesirable property if we wish to use this distribution as a classifier. Ideally, all model and variational parameters should learn in all cases. To remedy
this, we add a classification loss to (8), such that the distribution q? (y|x) also learns from labelled
data. The extended objective function is:
J ? = J + ? ? Epel (x,y) [? log q? (y|x)] ,
(9)
where the hyper-parameter ? controls the relative weight between generative and purely discriminative learning. We use ? = 0.1 ? N in all experiments. While we have obtained this objective function
by motivating the need for all model components to learn at all times, the objective 9 can also be
derived directly using the variational principle by instead performing inference over the parameters
? of the categorical distribution, using a symmetric Dirichlet prior over these parameterss.
3.2
Optimisation
The bounds in equations (5) and (9) provide a unified objective function for optimisation of both
the parameters ? and ? of the generative and inference models, respectively. This optimisation can
be done jointly, without resort to the variational EM algorithm, by using deterministic reparameterisations of the expectations in the objective function, combined with Monte Carlo approximation ?
referred to in previous work as stochastic gradient variational Bayes (SGVB) (Kingma and Welling,
2014) or as stochastic backpropagation (Rezende et al., 2014). We describe the core strategy for the
latent-feature discriminative model M1, since the same computations are used for the generative
semi-supervised model.
When the prior p(z) is a spherical Gaussian distribution p(z) = N (z|0, I) and the variational distribution q? (z|x) is also a Gaussian distribution as in (3), the KL term in equation (5) can be computed
4
Algorithm 1 Learning in model M1
while generativeTraining() do
D ? getRandomMiniBatch()
zi ? qP
? (zi |xi ) ?xi ? D
J ? n J (xi )
?J
(g? , g? ) ? ( ?J
?? , ?? )
(?, ?) ? (?, ?) + ?(g? , g? )
end while
while discriminativeTraining() do
D ? getLabeledRandomMiniBatch()
zi ? q? (zi |xi ) ?{xi , yi } ? D
trainClassifier({zi , yi } )
end while
Algorithm 2 Learning in model M2
while training() do
D ? getRandomMiniBatch()
yi ? q? (yi |xi ) ?{xi , yi } ?
/O
zi ? q? (zi |yi , xi )
J ? ? eq. (9)
?
?L?
(g? , g? ) ? ( ?L
?? , ?? )
(?, ?) ? (?, ?) + ?(g? , g? )
end while
analytically and the log-likelihood term can be rewritten, using the location-scale transformation for
the Gaussian distribution, as:
Eq? (z|x) [log p? (x|z)] = EN (|0,I) log p? (x|?? (x) + ? ? (x) ) ,
(10)
where indicates the element-wise product. While the expectation (10) still cannot be solved
analytically, its gradients with respect to the generative parameters ? and variational parameters ?
can be efficiently computed as expectations of simple gradients:
?{?,?} Eq? (z|x) [log p? (x|z)] = EN (|0,I) ?{?,?} log p? (x|?? (x) + ? ? (x) ) .
(11)
The gradients of the loss (9) for model M2 can be computed by a direct application of the chain rule
and by noting that the conditional bound L(xn , y) contains the same type of terms as the loss (9).
The gradients of the latter can then be efficiently estimated using (11) .
During optimization we use the estimated gradients in conjunction with standard stochastic gradientbased optimization methods such as SGD, RMSprop or AdaGrad (Duchi et al., 2010). This results
in parameter updates of the form: (? t+1 , ?t+1 ) ? (? t , ?t ) + ?t (gt? , gt? ), where ? is a diagonal
preconditioning matrix that adaptively scales the gradients for faster minimization. The training procedure for models M1 and M2 are summarised in algorithms 1 and 2, respectively. Our experimental
results were obtained using AdaGrad.
3.3
Computational Complexity
The overall algorithmic complexity of a single joint update of the parameters (?, ?) for M1 using the
estimator (11) is CM1 = M SCMLP where M is the minibatch size used , S is the number of samples
of the random variate , and CMLP is the cost of an evaluation of the MLPs in the conditional
distributions p? (x|z) and q? (z|x). The cost CMLP is of the form O(KD2 ) where K is the total
number of layers and D is the average dimension of the layers of the MLPs in the model. Training
M1 also requires training a supervised classifier, whose algorithmic complexity, if it is a neural net,
it will have a complexity of the form CMLP .
The algorithmic complexity for M2 is of the form CM2 = LCM1 , where L is the number of labels
and CM1 is the cost of evaluating the gradients of each conditional bound Jy (x), which is the same
as for M1. The stacked generative semi-supervised model has an algorithmic complexity of the
form CM1 + CM2 . But with the advantage that the cost CM2 is calculated in a low-dimensional space
(formed by the latent variables of the model M1 that provides the embeddings).
These complexities make this approach extremely appealing, since they are no more expensive than
alternative approaches based on auto-encoder or neural models, which have the lowest computational complexity amongst existing competitive approaches. In addition, our models are fully probabilistic, allowing for a wide range of inferential queries, which is not possible with many alternative
approaches for semi-supervised learning.
5
Table 1: Benchmark results of semi-supervised classification on MNIST with few labels.
N
100
600
1000
3000
4
NN
25.81
11.44
10.7
6.04
CNN
22.98
7.68
6.45
3.35
TSVM
16.81
6.16
5.38
3.45
CAE
13.47
6.3
4.77
3.22
MTC
12.03
5.13
3.64
2.57
AtlasRBF
8.10 (? 0.95)
?
3.68 (? 0.12)
?
M1+TSVM
11.82 (? 0.25)
5.72 (? 0.049)
4.24 (? 0.07)
3.49 (? 0.04)
M2
11.97 (? 1.71)
4.94 (? 0.13)
3.60 (? 0.56)
3.92 (? 0.63)
M1+M2
3.33 (? 0.14)
2.59 (? 0.05)
2.40 (? 0.02)
2.18 (? 0.04)
Experimental Results
Open source code, with which the most important results and figures can be reproduced, is available at http://github.com/dpkingma/nips14-ssl. For the latest experimental results,
please see http://arxiv.org/abs/1406.5298.
4.1
Benchmark Classification
We test performance on the standard MNIST digit classification benchmark. The data set for semisupervised learning is created by splitting the 50,000 training points between a labelled and unlabelled set, and varying the size of the labelled from 100 to 3000. We ensure that all classes are
balanced when doing this, i.e. each class has the same number of labelled points. We create a number of data sets using randomised sampling to confidence bounds for the mean performance under
repeated draws of data sets.
For model M1 we used a 50-dimensional latent variable z. The MLPs that form part of the generative
and inference models were constructed with two hidden layers, each with 600 hidden units, using
softplus log(1+ex ) activation functions. On top, a transductive SVM (TSVM) was learned on values
of z inferred with q? (z|x). For model M2 we also used 50-dimensional z. In each experiment, the
MLPs were constructed with one hidden layer, each with 500 hidden units and softplus activation
functions. In case of SVHN and NORB, we found it helpful to pre-process the data with PCA.
This makes the model one level deeper, and still optimizes a lower bound on the likelihood of the
unprocessed data.
Table 1 shows classification results. We compare to a broad range of existing solutions in semisupervised learning, in particular to classification using nearest neighbours (NN), support vector
machines on the labelled set (SVM), the transductive SVM (TSVM), and contractive auto-encoders
(CAE). Some of the best results currently are obtained by the manifold tangent classifier (MTC)
(Rifai et al., 2011) and the AtlasRBF method (Pitelis et al., 2014). Unlike the other models in this
comparison, our models are fully probabilistic but have a cost in the same order as these alternatives.
Results: The latent-feature discriminative model (M1) performs better than other models based
on simple embeddings of the data, demonstrating the effectiveness of the latent space in providing
robust features that allow for easier classification. By combining these features with a classification
mechanism directly in the same model, as in the conditional generative model (M2), we are able to
get similar results without a separate TSVM classifier.
However, by far the best results were obtained using the stack of models M1 and M2. This combined model provides accurate test-set predictions across all conditions, and easily outperforms the
previously best methods. We also tested this deep generative model for supervised learning with
all available labels, and obtain a test-set performance of 0.96%, which is among the best published
results for this permutation-invariant MNIST classification task.
4.2
Conditional Generation
The conditional generative model can be used to explore the underlying structure of the data, which
we demonstrate through two forms of analogical reasoning. Firstly, we demonstrate style and content separation by fixing the class label y, and then varying the latent variables z over a range of
values. Figure 1 shows three MNIST classes in which, using a trained model with two latent variables, and the 2D latent variable varied over a range from -5 to 5. In all cases, we see that nearby
regions of latent space correspond to similar writing styles, independent of the class; the left region
represents upright writing styles, while the right-side represents slanted styles.
As a second approach, we use a test image and pass it through the inference network to infer a
value of the latent variables corresponding to that image. We then fix the latent variables z to this
6
(a) Handwriting styles for MNIST obtained by fixing the class label and varying the 2D latent variable z
(b) MNIST analogies
(c) SVHN analogies
Figure 1: (a) Visualisation of handwriting styles learned by the model with 2D z-space. (b,c)
Analogical reasoning with generative semi-supervised models using a high-dimensional z-space.
The leftmost columns show images from the test set. The other columns show analogical fantasies
of x by the generative model, where the latent variable z of each row is set to the value inferred from
the test-set image on the left by the inference network. Each column corresponds to a class label y.
Table 2: Semi-supervised classification on
the SVHN dataset with 1000 labels.
KNN
77.93
(? 0.08)
TSVM
66.55
(? 0.10)
M1+KNN
65.63
(? 0.15)
M1+TSVM
54.33
(? 0.11)
Table 3: Semi-supervised classification on
the NORB dataset with 1000 labels.
M1+M2
36.02
(? 0.10)
KNN
78.71
(? 0.02)
TSVM
26.00
(? 0.06)
M1+KNN
65.39
(? 0.09)
M1+TSVM
18.79
(? 0.05)
value, vary the class label y, and simulate images from the generative model corresponding to that
combination of z and y. This again demonstrate the disentanglement of style from class. Figure 1
shows these analogical fantasies for the MNIST and SVHN datasets (Netzer et al., 2011). The
SVHN data set is a far more complex data set than MNIST, but the model is able to fix the style of
house number and vary the digit that appears in that style well. These generations represent the best
current performance in simulation from generative models on these data sets.
The model used in this way also provides an alternative model to the stochastic feed-forward networks (SFNN) described by Tang and Salakhutdinov (2013). The performance of our model significantly improves on SFNN, since instead of an inefficient Monte Carlo EM algorithm relying on
importance sampling, we are able to perform efficient joint inference that is easy to scale.
4.3
Image Classification
We demonstrate the performance of image classification on the SVHN, and NORB image data sets.
Since no comparative results in the semi-supervised setting exists, we perform nearest-neighbour
and TSVM classification with RBF kernels and compare performance on features generated by
our latent-feature discriminative model to the original features. The results are presented in tables 2
and 3, and we again demonstrate the effectiveness of our approach for semi-supervised classification.
7
4.4
Optimization details
The parameters were initialized by sampling randomly from N (0, 0.0012 I), except for the bias parameters which were initialized as 0. The objectives were optimized using minibatch gradient ascent
until convergence, using a variant of RMSProp with momentum and initialization bias correction, a
constant learning rate of 0.0003, first moment decay (momentum) of 0.1, and second moment decay
of 0.001. For MNIST experiments, minibatches for training were generated by treating normalised
pixel intensities of the images as Bernoulli probabilities and sampling binary images from this distribution. In the M2 model, a weight decay was used corresponding to a prior of (?, ?) ? N (0, I).
5
Discussion and Conclusion
The approximate inference methods introduced here can be easily extended to the model?s parameters, harnessing the full power of variational learning. Such an extension also provides a principled
ground for performing model selection. Efficient model selection is particularly important when the
amount of available data is not large, such as in semi-supervised learning.
For image classification tasks, one area of interest is to combine such methods with convolutional
neural networks that form the gold-standard for current supervised classification methods. Since all
the components of our model are parametrised by neural networks we can readily exploit convolutional or more general locally-connected architectures ? and forms a promising avenue for future
exploration.
A limitation of the models we have presented is that they scale linearly in the number of classes
in the data sets. Having to re-evaluate the generative likelihood for each class during training is an
expensive operation. Potential reduction of the number of evaluations could be achieved by using a
truncation of the posterior mass. For instance we could combine our method with the truncation algorithm suggested by Pal et al. (2005), or by using mechanisms such as error-correcting output codes
(Dietterich and Bakiri, 1995). The extension of our model to multi-label classification problems that
is essential for image-tagging is also possible, but requires similar approximations to reduce the
number of likelihood-evaluations per class.
We have developed new models for semi-supervised learning that allow us to improve the quality of
prediction by exploiting information in the data density using generative models. We have developed
an efficient variational optimisation algorithm for approximate Bayesian inference in these models
and demonstrated that they are amongst the most competitive models currently available for semisupervised learning. We hope that these results stimulate the development of even more powerful
semi-supervised classification methods based on generative models, of which there remains much
scope.
Acknowledgements. We are grateful for feedback from the reviewers. We would also like to
thank the SURFFoundation for the use of the Dutch national e-infrastructure for a significant part of
the experiments.
8
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9
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4,808 | 5,353 | Two-Stream Convolutional Networks
for Action Recognition in Videos
Karen Simonyan
Andrew Zisserman
Visual Geometry Group, University of Oxford
{karen,az}@robots.ox.ac.uk
Abstract
We investigate architectures of discriminatively trained deep Convolutional Networks (ConvNets) for action recognition in video. The challenge is to capture
the complementary information on appearance from still frames and motion between frames. We also aim to generalise the best performing hand-crafted features
within a data-driven learning framework.
Our contribution is three-fold. First, we propose a two-stream ConvNet architecture which incorporates spatial and temporal networks. Second, we demonstrate
that a ConvNet trained on multi-frame dense optical flow is able to achieve very
good performance in spite of limited training data. Finally, we show that multitask learning, applied to two different action classification datasets, can be used to
increase the amount of training data and improve the performance on both. Our
architecture is trained and evaluated on the standard video actions benchmarks of
UCF-101 and HMDB-51, where it is competitive with the state of the art. It also
exceeds by a large margin previous attempts to use deep nets for video classification.
1
Introduction
Recognition of human actions in videos is a challenging task which has received a significant amount
of attention in the research community [11, 14, 17, 26]. Compared to still image classification, the
temporal component of videos provides an additional (and important) clue for recognition, as a
number of actions can be reliably recognised based on the motion information. Additionally, video
provides natural data augmentation (jittering) for single image (video frame) classification.
In this work, we aim at extending deep Convolutional Networks (ConvNets) [19], a state-of-theart still image representation [15], to action recognition in video data. This task has recently been
addressed in [14] by using stacked video frames as input to the network, but the results were significantly worse than those of the best hand-crafted shallow representations [20, 26]. We investigate
a different architecture based on two separate recognition streams (spatial and temporal), which
are then combined by late fusion. The spatial stream performs action recognition from still video
frames, whilst the temporal stream is trained to recognise action from motion in the form of dense
optical flow. Both streams are implemented as ConvNets. Decoupling the spatial and temporal nets
also allows us to exploit the availability of large amounts of annotated image data by pre-training
the spatial net on the ImageNet challenge dataset [1]. Our proposed architecture is related to the
two-streams hypothesis [9], according to which the human visual cortex contains two pathways: the
ventral stream (which performs object recognition) and the dorsal stream (which recognises motion);
though we do not investigate this connection any further here.
The rest of the paper is organised as follows. In Sect. 1.1 we review the related work on action
recognition using both shallow and deep architectures. In Sect. 2 we introduce the two-stream
architecture and specify the Spatial ConvNet. Sect. 3 introduces the Temporal ConvNet and in
particular how it generalizes the previous architectures reviewed in Sect. 1.1. A mult-task learning
framework is developed in Sect. 4 in order to allow effortless combination of training data over
multiple datasets. Implementation details are given in Sect. 5, and the performance is evaluated
in Sect. 6 and compared to the state of the art. Our experiments on two challenging datasets (UCF101 [24] and HMDB-51 [16]) show that the two recognition streams are complementary, and our
1
deep architecture significantly outperforms that of [14] and is competitive with the state of the art
shallow representations [20, 21, 26] in spite of being trained on relatively small datasets.
1.1
Related work
Video recognition research has been largely driven by the advances in image recognition methods,
which were often adapted and extended to deal with video data. A large family of video action
recognition methods is based on shallow high-dimensional encodings of local spatio-temporal features. For instance, the algorithm of [17] consists in detecting sparse spatio-temporal interest points,
which are then described using local spatio-temporal features: Histogram of Oriented Gradients
(HOG) [7] and Histogram of Optical Flow (HOF). The features are then encoded into the Bag Of
Features (BoF) representation, which is pooled over several spatio-temporal grids (similarly to spatial pyramid pooling) and combined with an SVM classifier. In a later work [28], it was shown that
dense sampling of local features outperforms sparse interest points.
Instead of computing local video features over spatio-temporal cuboids, state-of-the-art shallow
video representations [20, 21, 26] make use of dense point trajectories. The approach, first introduced in [29], consists in adjusting local descriptor support regions, so that they follow dense
trajectories, computed using optical flow. The best performance in the trajectory-based pipeline
was achieved by the Motion Boundary Histogram (MBH) [8], which is a gradient-based feature,
separately computed on the horizontal and vertical components of optical flow. A combination of
several features was shown to further boost the accuracy. Recent improvements of trajectory-based
hand-crafted representations include compensation of global (camera) motion [10, 16, 26], and the
use of the Fisher vector encoding [22] (in [26]) or its deeper variant [23] (in [21]).
There has also been a number of attempts to develop a deep architecture for video recognition. In
the majority of these works, the input to the network is a stack of consecutive video frames, so the
model is expected to implicitly learn spatio-temporal motion-dependent features in the first layers,
which can be a difficult task. In [11], an HMAX architecture for video recognition was proposed
with pre-defined spatio-temporal filters in the first layer. Later, it was combined [16] with a spatial
HMAX model, thus forming spatial (ventral-like) and temporal (dorsal-like) recognition streams.
Unlike our work, however, the streams were implemented as hand-crafted and rather shallow (3layer) HMAX models. In [4, 18, 25], a convolutional RBM and ISA were used for unsupervised
learning of spatio-temporal features, which were then plugged into a discriminative model for action
classification. Discriminative end-to-end learning of video ConvNets has been addressed in [12]
and, more recently, in [14], who compared several ConvNet architectures for action recognition.
Training was carried out on a very large Sports-1M dataset, comprising 1.1M YouTube videos of
sports activities. Interestingly, [14] found that a network, operating on individual video frames,
performs similarly to the networks, whose input is a stack of frames. This might indicate that
the learnt spatio-temporal features do not capture the motion well. The learnt representation, finetuned on the UCF-101 dataset, turned out to be 20% less accurate than hand-crafted state-of-the-art
trajectory-based representation [20, 27].
Our temporal stream ConvNet operates on multiple-frame dense optical flow, which is typically
computed in an energy minimisation framework by solving for a displacement field (typically at
multiple image scales). We used a popular method of [2], which formulates the energy based on
constancy assumptions for intensity and its gradient, as well as smoothness of the displacement field.
Recently, [30] proposed an image patch matching scheme, which is reminiscent of deep ConvNets,
but does not incorporate learning.
2
Two-stream architecture for video recognition
Video can naturally be decomposed into spatial and temporal components. The spatial part, in the
form of individual frame appearance, carries information about scenes and objects depicted in the
video. The temporal part, in the form of motion across the frames, conveys the movement of the
observer (the camera) and the objects. We devise our video recognition architecture accordingly,
dividing it into two streams, as shown in Fig. 1. Each stream is implemented using a deep ConvNet,
softmax scores of which are combined by late fusion. We consider two fusion methods: averaging
and training a multi-class linear SVM [6] on stacked L2 -normalised softmax scores as features.
Spatial stream ConvNet operates on individual video frames, effectively performing action recognition from still images. The static appearance by itself is a useful clue, since some actions are
2
Spatial stream ConvNet
single frame
conv1
conv2
conv3
conv4
conv5
full6
full7
7x7x96
stride 2
norm.
pool 2x2
5x5x256
stride 2
norm.
pool 2x2
3x3x512
stride 1
3x3x512
stride 1
3x3x512
stride 1
pool 2x2
4096
dropout
2048
dropout
softmax
class
score
fusion
Temporal stream ConvNet
input
video
multi-frame
optical flow
conv1
conv2
conv3
conv4
conv5
full6
full7
7x7x96
stride 2
norm.
pool 2x2
5x5x256
stride 2
pool 2x2
3x3x512
stride 1
3x3x512
stride 1
3x3x512
stride 1
pool 2x2
4096
dropout
2048
dropout
softmax
Figure 1: Two-stream architecture for video classification.
strongly associated with particular objects. In fact, as will be shown in Sect. 6, action classification
from still frames (the spatial recognition stream) is fairly competitive on its own. Since a spatial
ConvNet is essentially an image classification architecture, we can build upon the recent advances
in large-scale image recognition methods [15], and pre-train the network on a large image classification dataset, such as the ImageNet challenge dataset. The details are presented in Sect. 5. Next, we
describe the temporal stream ConvNet, which exploits motion and significantly improves accuracy.
3
Optical flow ConvNets
In this section, we describe a ConvNet model, which forms the temporal recognition stream of our
architecture (Sect. 2). Unlike the ConvNet models, reviewed in Sect. 1.1, the input to our model is
formed by stacking optical flow displacement fields between several consecutive frames. Such input
explicitly describes the motion between video frames, which makes the recognition easier, as the
network does not need to estimate motion implicitly. We consider several variations of the optical
flow-based input, which we describe below.
(a)
(b)
(c)
(d)
(e)
Figure 2: Optical flow. (a),(b): a pair of consecutive video frames with the area around a moving hand outlined with a cyan rectangle. (c): a close-up of dense optical flow in the outlined area;
(d): horizontal component dx of the displacement vector field (higher intensity corresponds to positive values, lower intensity to negative values). (e): vertical component dy . Note how (d) and (e)
highlight the moving hand and bow. The input to a ConvNet contains multiple flows (Sect. 3.1).
3.1
ConvNet input configurations
Optical flow stacking. A dense optical flow can be seen as a set of displacement vector fields dt
between the pairs of consecutive frames t and t + 1. By dt (u, v) we denote the displacement vector
at the point (u, v) in frame t, which moves the point to the corresponding point in the following
frame t + 1. The horizontal and vertical components of the vector field, dxt and dyt , can be seen
as image channels (shown in Fig. 2), well suited to recognition using a convolutional network. To
represent the motion across a sequence of frames, we stack the flow channels dx,y
of L consecutive
t
frames to form a total of 2L input channels. More formally, let w and h be the width and height
of a video; a ConvNet input volume I? ? Rw?h?2L for an arbitrary frame ? is then constructed as
follows:
I? (u, v, 2k ? 1) = dx?+k?1 (u, v),
I? (u, v, 2k) =
dy? +k?1 (u, v),
(1)
u = [1; w], v = [1; h], k = [1; L].
For an arbitrary point (u, v), the channels I? (u, v, c), c = [1; 2L] encode the motion at that point
over a sequence of L frames (as illustrated in Fig. 3-left).
Trajectory stacking. An alternative motion representation, inspired by the trajectory-based descriptors [29], replaces the optical flow, sampled at the same locations across several frames, with
3
the flow, sampled along the motion trajectories. In this case, the input volume I? , corresponding to
a frame ? , takes the following form:
I? (u, v, 2k ? 1) = dx?+k?1 (pk ),
I? (u, v, 2k) = dy? +k?1 (pk ),
(2)
u = [1; w], v = [1; h], k = [1; L].
where pk is the k-th point along the trajectory, which starts at the location (u, v) in the frame ? and
is defined by the following recurrence relation:
p1 = (u, v);
pk = pk?1 + d? +k?2 (pk?1 ), k > 1.
Compared to the input volume representation (1), where the channels I? (u, v, c) store the displacement vectors at the locations (u, v), the input volume (2) stores the vectors sampled at the locations
pk along the trajectory (as illustrated in Fig. 3-right).
input volume channels
at point
input volume channels
at point
Figure 3: ConvNet input derivation from the multi-frame optical flow. Left: optical flow stacking (1) samples the displacement vectors d at the same location in multiple frames. Right: trajectory
stacking (2) samples the vectors along the trajectory. The frames and the corresponding displacement vectors are shown with the same colour.
Bi-directional optical flow. Optical flow representations (1) and (2) deal with the forward optical
flow, i.e. the displacement field dt of the frame t specifies the location of its pixels in the following
frame t + 1. It is natural to consider an extension to a bi-directional optical flow, which can be
obtained by computing an additional set of displacement fields in the opposite direction. We then
construct an input volume I? by stacking L/2 forward flows between frames ? and ? +L/2 and L/2
backward flows between frames ? ? L/2 and ? . The input I? thus has the same number of channels
(2L) as before. The flows can be represented using either of the two methods (1) and (2).
Mean flow subtraction. It is generally beneficial to perform zero-centering of the network input,
as it allows the model to better exploit the rectification non-linearities. In our case, the displacement
vector field components can take on both positive and negative values, and are naturally centered in
the sense that across a large variety of motions, the movement in one direction is as probable as the
movement in the opposite one. However, given a pair of frames, the optical flow between them can
be dominated by a particular displacement, e.g. caused by the camera movement. The importance
of camera motion compensation has been previously highlighted in [10, 26], where a global motion
component was estimated and subtracted from the dense flow. In our case, we consider a simpler
approach: from each displacement field d we subtract its mean vector.
Architecture. Above we have described different ways of combining multiple optical flow displacement fields into a single volume I? ? Rw?h?2L . Considering that a ConvNet requires a fixed-size
input, we sample a 224 ? 224 ? 2L sub-volume from I? and pass it to the net as input. The hidden layers configuration remains largely the same as that used in the spatial net, and is illustrated
in Fig. 1. Testing is similar to the spatial ConvNet, and is described in detail in Sect. 5.
3.2
Relation of the temporal ConvNet architecture to previous representations
In this section, we put our temporal ConvNet architecture in the context of prior art, drawing connections to the video representations, reviewed in Sect. 1.1. Methods based on feature encodings [17, 29] typically combine several spatio-temporal local features. Such features are computed
from the optical flow and are thus generalised by our temporal ConvNet. Indeed, the HOF and MBH
local descriptors are based on the histograms of orientations of optical flow or its gradient, which
can be obtained from the displacement field input (1) using a single convolutional layer (containing
4
orientation-sensitive filters), followed by the rectification and pooling layers. The kinematic features
of [10] (divergence, curl and shear) are also computed from the optical flow gradient, and, again, can
be captured by our convolutional model. Finally, the trajectory feature [29] is computed by stacking
the displacement vectors along the trajectory, which corresponds to the trajectory stacking (2). In the
supplementary material we visualise the convolutional filters, learnt in the first layer of the temporal
network. This provides further evidence that our representation generalises hand-crafted features.
As far as the deep networks are concerned, a two-stream video classification architecture of [16]
contains two HMAX models which are hand-crafted and less deep than our discriminatively trained
ConvNets, which can be seen as a learnable generalisation of HMAX. The convolutional models
of [12, 14] do not decouple spatial and temporal recognition streams, and rely on the motionsensitive convolutional filters, learnt from the data. In our case, motion is explicitly represented
using the optical flow displacement field, computed based on the assumptions of constancy of the
intensity and smoothness of the flow. Incorporating such assumptions into a ConvNet framework
might be able to boost the performance of end-to-end ConvNet-based methods, and is an interesting
direction for future research.
4
Multi-task learning
Unlike the spatial stream ConvNet, which can be pre-trained on a large still image classification
dataset (such as ImageNet), the temporal ConvNet needs to be trained on video data ? and the
available datasets for video action classification are still rather small. In our experiments (Sect. 6),
training is performed on the UCF-101 and HMDB-51 datasets, which have only: 9.5K and 3.7K
videos respectively. To decrease over-fitting, one could consider combining the two datasets into
one; this, however, is not straightforward due to the intersection between the sets of classes. One
option (which we evaluate later) is to only add the images from the classes, which do not appear in
the original dataset. This, however, requires manual search for such classes and limits the amount
of additional training data.
A more principled way of combining several datasets is based on multi-task learning [5]. Its aim
is to learn a (video) representation, which is applicable not only to the task in question (such as
HMDB-51 classification), but also to other tasks (e.g. UCF-101 classification). Additional tasks act
as a regulariser, and allow for the exploitation of additional training data. In our case, a ConvNet
architecture is modified so that it has two softmax classification layers on top of the last fullyconnected layer: one softmax layer computes HMDB-51 classification scores, the other one ? the
UCF-101 scores. Each of the layers is equipped with its own loss function, which operates only on
the videos, coming from the respective dataset. The overall training loss is computed as the sum of
the individual tasks? losses, and the network weight derivatives can be found by back-propagation.
5
Implementation details
ConvNets configuration. The layer configuration of our spatial and temporal ConvNets is schematically shown in Fig. 1. It corresponds to CNN-M-2048 architecture of [3] and is similar to the
network of [31]. All hidden weight layers use the rectification (ReLU) activation function; maxpooling is performed over 3 ? 3 spatial windows with stride 2; local response normalisation uses the
same settings as [15]. The only difference between spatial and temporal ConvNet configurations is
that we removed the second normalisation layer from the latter to reduce memory consumption.
Training. The training procedure can be seen as an adaptation of that of [15] to video frames, and
is generally the same for both spatial and temporal nets. The network weights are learnt using the
mini-batch stochastic gradient descent with momentum (set to 0.9). At each iteration, a mini-batch
of 256 samples is constructed by sampling 256 training videos (uniformly across the classes), from
each of which a single frame is randomly selected. In spatial net training, a 224 ? 224 sub-image is
randomly cropped from the selected frame; it then undergoes random horizontal flipping and RGB
jittering. The videos are rescaled beforehand, so that the smallest side of the frame equals 256. We
note that unlike [15], the sub-image is sampled from the whole frame, not just its 256 ? 256 center.
In the temporal net training, we compute an optical flow volume I for the selected training frame as
described in Sect. 3. From that volume, a fixed-size 224 ? 224 ? 2L input is randomly cropped and
flipped. The learning rate is initially set to 10?2 , and then decreased according to a fixed schedule,
which is kept the same for all training sets. Namely, when training a ConvNet from scratch, the rate
is changed to 10?3 after 50K iterations, then to 10?4 after 70K iterations, and training is stopped
5
after 80K iterations. In the fine-tuning scenario, the rate is changed to 10?3 after 14K iterations, and
training stopped after 20K iterations.
Testing. At test time, given a video, we sample a fixed number of frames (25 in our experiments)
with equal temporal spacing between them. From each of the frames we then obtain 10 ConvNet
inputs [15] by cropping and flipping four corners and the center of the frame. The class scores for the
whole video are then obtained by averaging the scores across the sampled frames and crops therein.
Pre-training on ImageNet ILSVRC-2012. When pre-training the spatial ConvNet, we use the
same training and test data augmentation as described above (cropping, flipping, RGB jittering).
This yields 13.5% top-5 error on ILSVRC-2012 validation set, which compares favourably to 16.0%
reported in [31] for a similar network. We believe that the main reason for the improvement is
sampling of ConvNet inputs from the whole image, rather than just its center.
Multi-GPU training. Our implementation is derived from the publicly available Caffe toolbox [13],
but contains a number of significant modifications, including parallel training on multiple GPUs
installed in a single system. We exploit the data parallelism, and split each SGD batch across several
GPUs. Training a single temporal ConvNet takes 1 day on a system with 4 NVIDIA Titan cards,
which constitutes a 3.2 times speed-up over single-GPU training.
Optical flow is computed using the off-the-shelf GPU implementation of [2] from the OpenCV
toolbox. In spite of the fast computation time (0.06s for a pair of frames), it would still introduce
a bottleneck if done on-the-fly, so we pre-computed the flow before training. To avoid storing
the displacement fields as floats, the horizontal and vertical components of the flow were linearly
rescaled to a [0, 255] range and compressed using JPEG (after decompression, the flow is rescaled
back to its original range). This reduced the flow size for the UCF-101 dataset from 1.5TB to 27GB.
6
Evaluation
Datasets and evaluation protocol. The evaluation is performed on UCF-101 [24] and
HMDB-51 [16] action recognition benchmarks, which are among the largest available annotated
video datasets1 . UCF-101 contains 13K videos (180 frames/video on average), annotated into 101
action classes; HMDB-51 includes 6.8K videos of 51 actions. The evaluation protocol is the same
for both datasets: the organisers provide three splits into training and test data, and the performance
is measured by the mean classification accuracy across the splits. Each UCF-101 split contains 9.5K
training videos; an HMDB-51 split contains 3.7K training videos. We begin by comparing different
architectures on the first split of the UCF-101 dataset. For comparison with the state of the art, we
follow the standard evaluation protocol and report the average accuracy over three splits on both
UCF-101 and HMDB-51.
Spatial ConvNets. First, we measure the performance of the spatial stream ConvNet. Three scenarios are considered: (i) training from scratch on UCF-101, (ii) pre-training on ILSVRC-2012
followed by fine-tuning on UCF-101, (iii) keeping the pre-trained network fixed and only training
the last (classification) layer. For each of the settings, we experiment with setting the dropout regularisation ratio to 0.5 or to 0.9. From the results, presented in Table 1a, it is clear that training the
ConvNet solely on the UCF-101 dataset leads to over-fitting (even with high dropout), and is inferior
to pre-training on a large ILSVRC-2012 dataset. Interestingly, fine-tuning the whole network gives
only marginal improvement over training the last layer only. In the latter setting, higher dropout
over-regularises learning and leads to worse accuracy. In the following experiments we opted for
training the last layer on top of a pre-trained ConvNet.
Temporal ConvNets. Having evaluated spatial ConvNet variants, we now turn to the temporal
ConvNet architectures, and assess the effect of the input configurations, described in Sect. 3.1. In
particular, we measure the effect of: using multiple (L = {5, 10}) stacked optical flows; trajectory
stacking; mean displacement subtraction; using the bi-directional optical flow. The architectures
are trained on the UCF-101 dataset from scratch, so we used an aggressive dropout ratio of 0.9 to
help improve generalisation. The results are shown in Table 1b. First, we can conclude that stacking
multiple (L > 1) displacement fields in the input is highly beneficial, as it provides the network with
long-term motion information, which is more discriminative than the flow between a pair of frames
1
Very recently, [14] released the Sports-1M dataset of 1.1M automatically annotated YouTube sports videos.
Processing the dataset of such scale is very challenging, and we plan to address it in future work.
6
Table 1: Individual ConvNets accuracy on UCF-101 (split 1).
(a) Spatial ConvNet.
Dropout ratio
Training setting
0.5
0.9
From scratch
42.5% 52.3%
Pre-trained + fine-tuning 70.8% 72.8%
Pre-trained + last layer
72.7% 59.9%
(b) Temporal ConvNet.
Mean subtraction
off
on
Single-frame optical flow (L = 1)
73.9%
Optical flow stacking (1) (L = 5)
80.4%
Optical flow stacking (1) (L = 10)
79.9% 81.0%
Trajectory stacking (2)(L = 10)
79.6% 80.2%
Optical flow stacking (1)(L = 10), bi-dir.
81.2%
Input configuration
(L = 1 setting). Increasing the number of input flows from 5 to 10 leads to a smaller improvement,
so we kept L fixed to 10 in the following experiments. Second, we find that mean subtraction is
helpful, as it reduces the effect of global motion between the frames. We use it in the following
experiments as default. The difference between different stacking techniques is marginal; it turns
out that optical flow stacking performs better than trajectory stacking, and using the bi-directional
optical flow is only slightly better than a uni-directional forward flow. Finally, we note that temporal
ConvNets significantly outperform the spatial ConvNets (Table 1a), which confirms the importance
of motion information for action recognition.
We also implemented the ?slow fusion? architecture of [14], which amounts to applying a ConvNet
to a stack of RGB frames (11 frames in our case). When trained from scratch on UCF-101, it
achieved 56.4% accuracy, which is better than a single-frame architecture trained from scratch
(52.3%), but is still far off the network trained from scratch on optical flow. This shows that while
multi-frame information is important, it is also important to present it to a ConvNet in an appropriate
manner.
Multi-task learning of temporal ConvNets. Training temporal ConvNets on UCF-101 is challenging due to the small size of the training set. An even bigger challenge is to train the ConvNet on
HMDB-51, where each training split is 2.6 times smaller than that of UCF-101. Here we evaluate
different options for increasing the effective training set size of HMDB-51: (i) fine-tuning a temporal
network pre-trained on UCF-101; (ii) adding 78 classes from UCF-101, which are manually selected
so that there is no intersection between these classes and the native HMDB-51 classes; (iii) using the
multi-task formulation (Sect. 4) to learn a video representation, shared between the UCF-101 and
HMDB-51 classification tasks. The results are reported in Table 2. As expected, it is beneficial to
Table 2: Temporal ConvNet accuracy on HMDB-51 (split 1 with additional training data).
Training setting
Training on HMDB-51 without additional data
Fine-tuning a ConvNet, pre-trained on UCF-101
Training on HMDB-51 with classes added from UCF-101
Multi-task learning on HMDB-51 and UCF-101
Accuracy
46.6%
49.0%
52.8%
55.4%
utilise full (all splits combined) UCF-101 data for training (either explicitly by borrowing images, or
implicitly by pre-training). Multi-task learning performs the best, as it allows the training procedure
to exploit all available training data.
We have also experimented with multi-task learning on the UCF-101 dataset, by training a network
to classify both the full HMDB-51 data (all splits combined) and the UCF-101 data (a single split).
On the first split of UCF-101, the accuracy was measured to be 81.5%, which improves on 81.0%
achieved using the same settings, but without the additional HMDB classification task (Table 1b).
Two-stream ConvNets. Here we evaluate the complete two-stream model, which combines the
two recognition streams. One way of combining the networks would be to train a joint stack of
fully-connected layers on top of full6 or full7 layers of the two nets. This, however, was not feasible
in our case due to over-fitting. We therefore fused the softmax scores using either averaging or
a linear SVM. From Table 3 we conclude that: (i) temporal and spatial recognition streams are
complementary, as their fusion significantly improves on both (6% over temporal and 14% over
spatial nets); (ii) SVM-based fusion of softmax scores outperforms fusion by averaging; (iii) using
bi-directional flow is not beneficial in the case of ConvNet fusion; (iv) temporal ConvNet, trained
using multi-task learning, performs the best both alone and when fused with a spatial net.
Comparison with the state of the art. We conclude the experimental evaluation with the comparison against the state of the art on three splits of UCF-101 and HMDB-51. For that we used a
7
Table 3: Two-stream ConvNet accuracy on UCF-101 (split 1).
Spatial ConvNet
Pre-trained + last layer
Pre-trained + last layer
Pre-trained + last layer
Pre-trained + last layer
Temporal ConvNet
bi-directional
uni-directional
uni-directional, multi-task
uni-directional, multi-task
Fusion Method
averaging
averaging
averaging
SVM
Accuracy
85.6%
85.9%
86.2%
87.0%
spatial net, pre-trained on ILSVRC, with the last layer trained on UCF or HMDB. The temporal
net was trained on UCF and HMDB using multi-task learning, and the input was computed using
uni-directional optical flow stacking with mean subtraction. The softmax scores of the two nets were
combined using averaging or SVM. As can be seen from Table 4, both our spatial and temporal nets
alone outperform the deep architectures of [14, 16] by a large margin. The combination of the two
nets further improves the results (in line with the single-split experiments above), and is comparable
to the very recent state-of-the-art hand-crafted models [20, 21, 26].
Table 4: Mean accuracy (over three splits) on UCF-101 and HMDB-51.
Method
Improved dense trajectories (IDT) [26, 27]
IDT with higher-dimensional encodings [20]
IDT with stacked Fisher encoding [21] (based on Deep Fisher Net [23])
Spatio-temporal HMAX network [11, 16]
?Slow fusion? spatio-temporal ConvNet [14]
Spatial stream ConvNet
Temporal stream ConvNet
Two-stream model (fusion by averaging)
Two-stream model (fusion by SVM)
7
UCF-101
85.9%
87.9%
65.4%
73.0%
83.7%
86.9%
88.0%
HMDB-51
57.2%
61.1%
66.8%
22.8%
40.5%
54.6%
58.0%
59.4%
Conclusions and directions for improvement
We proposed a deep video classification model with competitive performance, which incorporates
separate spatial and temporal recognition streams based on ConvNets. Currently it appears that
training a temporal ConvNet on optical flow (as here) is significantly better than training on raw
stacked frames [14]. The latter is probably too challenging, and might require architectural changes
(for example, a combination with the deep matching approach of [30]). Despite using optical flow
as input, our temporal model does not require significant hand-crafting, since the flow is computed
using a method based on the generic assumptions of constancy and smoothness.
As we have shown, extra training data is beneficial for our temporal ConvNet, so we are planning to
train it on large video datasets, such as the recently released collection of [14]. This, however, poses
a significant challenge on its own due to the gigantic amount of training data (multiple TBs).
There still remain some essential ingredients of the state-of-the-art shallow representation [26],
which are missed in our current architecture. The most prominent one is local feature pooling
over spatio-temporal tubes, centered at the trajectories. Even though the input (2) captures the optical flow along the trajectories, the spatial pooling in our network does not take the trajectories into
account. Another potential area of improvement is explicit handling of camera motion, which in our
case is compensated by mean displacement subtraction.
Acknowledgements
This work was supported by ERC grant VisRec no. 228180. We gratefully acknowledge the support
of NVIDIA Corporation with the donation of the GPUs used for this research.
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4,809 | 5,354 | Rounding-based Moves for Metric Labeling
M. Pawan Kumar
Ecole Centrale Paris & INRIA Saclay
[email protected]
Abstract
Metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given metric distance function over the
label set. Popular methods for solving metric labeling include (i) move-making
algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear
programming (LP) relaxation based approach. In order to convert the fractional
solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class
of parallel rounding procedures, and design move-making algorithms that closely
mimic them. We prove that the multiplicative bound of a move-making algorithm
exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results
for move-making algorithms as special cases.
1
Introduction
A Markov random field (MRF) is a graph whose vertices are random variables, and whose edges
specify a neighborhood over the random variables. Each random variable can be assigned a value
from a set of labels, resulting in a labeling of the MRF. The putative labelings of an MRF are
quantitatively distinguished from each other by an energy function, which is the sum of potential
functions that depend on the cliques of the graph. An important optimization problem associate with
the MRF framework is energy minimization, that is, finding a labeling with the minimum energy.
Metric labeling is a special case of energy minimization, which models several useful low-level
vision tasks [3, 4, 18]. It is characterized by a finite, discrete label set and a metric distance function
over the labels. The energy function in metric labeling consists of arbitrary unary potentials and
pairwise potentials that are proportional to the distance between the labels assigned to them. The
problem is known to be NP-hard [20]. Two popular approaches for metric labeling are: (i) movemaking algorithms [4, 8, 14, 15, 21], which iteratively improve the labeling by solving a minimum
st-cut problem; and (ii) linear programming (LP) relaxation [5, 13, 17, 22], which is obtained by
dropping the integral constraints in the corresponding integer programming formulation. Movemaking algorithms are very efficient due to the availability of fast minimum st-cut solvers [2] and
are very popular in the computer vision community. In contrast, the LP relaxation is significantly
slower, despite the development of specialized solvers [7, 9, 11, 12, 16, 19, 22, 23, 24, 25]. However,
when used in conjunction with randomized rounding algorithms, the LP relaxation provides the best
known polynomial-time theoretical guarantees for metric labeling [1, 5, 10].
At first sight, the difference between move-making algorithms and the LP relaxation appears to be
the standard accuracy vs. speed trade-off. However, for some special cases of distance functions,
it has been shown that appropriately designed move-making algorithms can match the theoretical
guarantees of the LP relaxation [14, 15, 20]. In this paper, we extend this result for a large class
of randomized rounding procedures, which we call parallel rounding. In particular we prove that
for any arbitrary (semi-)metric distance function, there exist move-making algorithms that match
the theoretical guarantees provided by parallel rounding. The proofs, the various corollaries of our
1
theorems (which cover all previously known guarantees) and our experimental results are deferred
to the accompanying technical report.
2
Preliminaries
Metric Labeling. The problem of metric labeling is defined over an undirected graph G =
(X, E). The vertices X = {X1 , X2 , ? ? ? , Xn } are random variables, and the edges E specify a
neighborhood relationship over the random variables. Each random variable can be assigned a value
from the label set L = {l1 , l2 , ? ? ? , lh }. We assume that we are also provided with a metric distance
function d : L ? L ? R+ over the labels.
We refer to an assignment of values to all the random variables as a labeling. In other words, a
labeling is a vector x ? Ln , which specifies the label xa assigned to each random variable Xa . The
hn different labelings are quantitatively distinguished from each other by an energy function Q(x),
which is defined as follows:
X
X
Q(x) =
wab d(xa , xb ).
?a (xa ) +
Xa ?X
(Xa ,Xb )?E
Here, the unary potentials ?a (?) are arbitrary, and the edge weights wab are non-negative. Metric
labeling requires us to find a labeling with the minimum energy. It is known to be NP-hard.
Multiplicative Bound. As metric labeling plays a central role in low-level vision, several approximate algorithms have been proposed in the literature. A common theoretical measure of accuracy
for an approximate algorithm is the multiplicative bound. In this work, we are interested in the
multiplicative bound of an algorithm with respect to a distance function. Formally, given a distance
function d, the multiplicative bound of an algorithm is said to be B if the following condition is
satisfied for all possible values of unary potentials ?a (?) and non-negative edge weights wab :
X
X
X
X
wab d(x?a , x?b ). (1)
wab d(?
xa , x
?b ) ?
?a (x?a ) + B
?a (?
xa ) +
Xa ?X
Xa ?X
(Xa ,Xb )?E
(Xa ,Xb )?E
? is the labeling estimated by the algorithm for the given values of unary potentials and edge
Here, x
weights, and x? is an optimal labeling. Multiplicative bounds are greater than or equal to 1, and are
invariant to reparameterizations of the unary potentials. A multiplicative bound B is said to be tight
if the above inequality holds as an equality for some value of unary potentials and edge weights.
Linear Programming Relaxation. An overcomplete representation of a labeling can be specified
using the following variables: (i) unary variables ya (i) ? {0, 1} for all Xa ? X and li ? L such
that ya (i) = 1 if and only if Xa is assigned the label li ; and (ii) pairwise variables yab (i, j) ? {0, 1}
for all (Xa , Xb ) ? E and li , lj ? L such that yab (i, j) = 1 if and only if Xa and Xb are assigned
labels li and lj respectively. This allows us to formulate metric labeling as follows:
X X
X
X
min
?a (li )ya (i) +
wab d(li , lj )yab (i, j),
y
s.t.
Xa ?X li ?L
(Xa ,Xb )?E li ,lj ?L
X
ya (i) = 1, ?Xa ? X,
X
yab (i, j) = ya (i), ?(Xa , Xb ) ? E, li ? L,
X
yab (i, j) = yb (j), ?(Xa , Xb ) ? E, lj ? L,
li ?L
lj ?L
li ?L
ya (i) ? {0, 1}, yab (i, j) ? {0, 1}, ?Xa ? X, (Xa , Xb ) ? E, li , lj ? L.
By relaxing the final set of constraints such that the optimization variables can take any value between 0 and 1 inclusive, we obtain a linear program (LP). The computational complexity of solving
the LP relaxation is polynomial in the size of the problem.
Rounding Procedure. In order to prove theoretical guarantees of the LP relaxation, it is common
to use a rounding procedure that can covert a feasible fractional solution y of the LP relaxation to
? of the integer linear program. Several rounding procedures have been
a feasible integer solution y
2
proposed in the literature. In this work, we focus on the randomized parallel rounding procedures
proposed in [5, 10]. These procedures have the property that, given a fractional solution y, the
probability of assigning a label li ? L to a random variable Xa ? X is equal to ya (i), that is,
Pr(?
ya (i) = 1) = ya (i).
(2)
We will describe the various rounding procedures in detail in sections 3-5. For now, we would like
to note that our reason for focusing on the parallel rounding of [5, 10] is that they provide the best
known polynomial-time theoretical guarantees for metric labeling. Specifically, we are interested in
their approximation factor, which is defined next.
Approximation Factor. Given a distance function d, the approximation factor for a rounding procedure is said to be F if the following condition is satisfied for all feasible fractional solutions y:
?
?
X
X
d(li , lj )yab (i, j).
(3)
d(li , lj )?
ya (i)?
yb (j)? ? F
E?
li ,lj ?L
li ,lj ?L
? refers to the integer solution, and the expectation is taken with respect to the randomized
Here, y
rounding procedure applied to the feasible solution y.
Given a rounding procedure with an approximation factor of F , an optimal fractional solution y? of
? that satisfies the following condition:
the LP relaxation can be rounded to a labeling y
?
?
X
X
X X
wab d(li , lj )?
ya (i)?
yb (j)?
?a (li )?
ya (i) +
E?
Xa ?X li ?L
?
X X
Xa ?X li ?L
?a (li )ya? (i)
(Xa ,Xb )?E li ,lj ?L
+F
X
X
?
wab d(li , lj )yab
(i, j).
(Xa ,Xb )?E li ,lj ?L
The above inequality follows directly from properties (2) and (3). Similar to multiplicative bounds,
approximation factors are always greater than or equal to 1, and are invariant to reparameterizations
of the unary potentials. An approximation factor F is said to be tight if the above inequality holds
as an equality for some value of unary potentials and edge weights.
Submodular Energy Function. We will use the following important fact throughout this paper.
Given an energy function defined using arbitrary unary potentials, non-negative edge weights and a
submodular distance function, an optimal labeling can be computed in polynomial time by solving
an equivalent minimum st-cut problem [6]. Recall that a submodular distance function d? over a
label set L = {l1 , l2 , ? ? ? , lh } satisfies the following properties: (i) d? (li , lj ) ? 0 for all li , lj ? L,
and d? (li , lj ) = 0 if and only if i = j; and (ii) d? (li , lj ) + d? (li+1 , lj+1 ) ? d? (li , lj+1 ) + d? (li+1 , lj )
for all li , lj ? L\{lh } (where \ refers to set difference).
3
Complete Rounding and Complete Move
We start with a simple rounding scheme, which we call complete rounding. While complete rounding is not very accurate, it would help illustrate the flavor of our results. We will subsequently
consider its generalizations, which have been useful in obtaining the best-known approximation
factors for various special cases of metric labeling.
The complete rounding procedure consists of a single stage where we use the set of all unary variables to obtain a labeling (as opposed to other rounding procedures discussed subsequently). Algorithm 1 describes its main steps. Intuitively, it treats the value of the unary variable ya (i) as the
probability of assigning the label li to the random variable Xa . It obtains a labeling by sampling
from all the distributions ya = [ya (i), ?li ? L] simultaneously using the same random number.
It can be shown that using a different random number to sample the distributions ya and yb of
two neighboring random variables (Xa , Xb ) ? E results in an infinite approximation factor. For
example, let y a (i) = y b (i) = 1/h for all li ? L, where h is the number of labels. The pairwise
variables yab that minimize the energy function are y ab (i, i) = 1/h and y ab (i, j) = 0 when i 6= j.
For the above feasible solution of the LP relaxation, the RHS of inequality (3) is 0 for any finite F ,
while the LHS of inequality (3) is strictly greater than 0 if h > 1. However, we will shortly show that
using the same random number r for all random variables provides a finite approximation factor.
3
Algorithm 1 The complete rounding procedure.
input A feasible solution y of the LP relaxation.
1: Pick a real number r uniformly from [0, 1].
2: for all Xa ? X do
P
3:
Define Ya (0) = 0 and Ya (i) = ij=1 ya (j) for all li ? L.
4:
Assign the label li ? L to the random variable Xa if Ya (i ? 1) < r ? Ya (i).
5: end for
We now turn our attention to designing a move-making algorithm whose multiplicative bound
matches the approximation factor of the complete rounding procedure. To this end, we modify
the range expansion algorithm proposed in [15] for truncated convex pairwise potentials to a general
(semi-)metric distance function. Our method, which we refer to as the complete move-making algorithm, considers all putative labels of all random variables, and provides an approximate solution
in a single iteration. Algorithm 2 describes its two main steps. First, it computes a submodular
overestimation of the given distance function by solving the following optimization problem:
d=
argmin t
(4)
d?
s.t.
d? (li , lj ) ? td(li , lj ), ?li , lj ? L,
d? (li , lj ) ? d(li , lj ), ?li , lj ? L,
d? (li , lj ) + d? (li+1 , lj+1 ) ? d? (li , lj+1 ) + d? (li+1 , lj ), ?li , lj ? L\{lh }.
The above problem minimizes the maximum ratio of the estimated distance to the original distance
over all pairs of labels, that is, maxi6=j d? (li , lj )/d(li , lj ). We will refer to the optimal value of
problem (4) as the submodular distortion of the distance function d. Second, it replaces the original
distance function by the submodular overestimation and computes an approximate solution to the
original metric labeling problem by solving a single minimum st-cut problem. Note that, unlike
the range expansion algorithm [15] that uses the readily available submodular overestimation of
a truncated convex distance (namely, the corresponding convex distance function), our approach
estimates the submodular overestimation via the LP (4). Since the LP (4) can be solved for any
arbitrary distance function, it makes complete move-making more generally applicable.
Algorithm 2 The complete move-making algorithm.
input Unary potentials ?a (?), edge weights wab , distance function d.
1: Compute a submodular overestimation of d by solving problem (4).
2: Using the approach of [6], solve the following problem via an equivalent minimum st-cut problem:
X
X
? = argmin
wab d(xa , xb ).
x
?a (xa ) +
x?Ln
Xa ?X
(Xa ,Xb )?E
The following theorem establishes the theoretical guarantees of the complete move-making algorithm and the complete rounding procedure.
Theorem 1. The tight multiplicative bound of the complete move-making algorithm is equal to the
submodular distortion of the distance function. Furthermore, the tight approximation factor of the
complete rounding procedure is also equal to the submodular distortion of the distance function.
In terms of computational complexities, complete move-making is significantly faster than solving
the LP relaxation. Specifically, given an MRF with n random variables and m edges, and a label
set with h labels, the LP relaxation requires at least O(m3 h3 log(m2 h3 )) time, since it consists
of O(mh2 ) optimization variables and O(mh) constraints. In contrast, complete move-making
requires O(nmh3 log(m)) time, since the graph constructed using the method of [6] consists of
O(nh) nodes and O(mh2 ) arcs. Note that complete move-making also requires us to solve the
linear program (4). However, since problem (4) is independent of the unary potentials and the edge
weights, it only needs to be solved once beforehand in order to compute the approximate solution
for any metric labeling problem defined using the distance function d.
4
4
Interval Rounding and Interval Moves
Theorem 1 implies that the approximation factor of the complete rounding procedure is very large
for distance functions that are highly non-submodular. For example, consider the truncated linear
distance function defined as follows over a label set L = {l1 , l2 , ? ? ? , lh }:
d(li , lj ) = min{|i ? j|, M }.
Here, M is a user specified parameter that determines the maximum distance. The tightest submodular overestimation of the above distance function is the linear distance function, that is,
d(li , lj ) = |i ? j|. This implies that the submodular distortion of the truncated linear metric is
(h ? 1)/M , and therefore, the approximation factor for the complete rounding procedure is also
(h ? 1)/M . In order to avoid this large approximation factor, Chekuri et al. [5] proposed an interval
rounding procedure, which captures the intuition that it is beneficial to assign similar labels to as
many random variables as possible.
Algorithm 3 provides a description of interval rounding. The rounding procedure chooses an interval
of at most q consecutive labels (step 2). It generates a random number r (step 3), and uses it to
attempt to assign labels to previously unlabeled random variables from the selected interval (steps
4-7). It can be shown that the overall procedure converges in a polynomial number of iterations with
a probability of 1 [5]. Note that if we fix q = h and z = 1, interval rounding becomes equivalent
to complete rounding. However, the analyses in [5, 10] shows that other values of q provide better
approximation factors for various special cases.
Algorithm 3 The interval rounding procedure.
input A feasible solution y of the LP relaxation.
1: repeat
2:
Pick an integer z uniformly from [?q + 2, h]. Define an interval of labels I = {ls , ? ? ? , le },
where s = max{z, 1} is the start index and e = min{z + q ? 1, h} is the end index.
3:
Pick a real number r uniformly from [0, 1].
4:
for all Unlabeled random variables
a do
PX
s+i?1
5:
Define Ya (0) = 0 and Ya (i) = j=s ya (j) for all i ? {1, ? ? ? , e ? s + 1}.
6:
Assign the label ls+i?1 ? I to the Xa if Ya (i ? 1) < r ? Ya (i).
7:
end for
8: until All random variables have been assigned a label.
Our goal is to design a move-making algorithm whose multiplicative bound matches the approximation factor of interval rounding for any choice of q. To this end, we propose the interval move-making
algorithm that generalizes the range expansion algorithm [15], originally proposed for truncated convex distances, to arbitrary distance functions. Algorithm 4 provides its main steps. The central idea
? by allowing each random variable Xa to either retain
of the method is to improve a given labeling x
its current label x
?a or to choose a new label from an interval of consecutive labels. In more detail, let
I = {ls , ? ? ? , le } ? L be an interval of labels of length at most q (step 4). For
S the sake of simplicity,
let us assume that x
?a ?
/ I for any random variable Xa . We define Ia = I {?
xa } (step 5). For each
pair of neighboring random variables (Xa , Xb ) ? E, we compute a submodular distance function
dx?a ,?xb : Ia ? Ib ? R+ by solving the following linear program (step 6):
dx?a ,?xb =
s.t.
argmin t
(5)
d?
d? (li , lj ) ? td(li , lj ), ?li ? Ia , lj ? Ib ,
d? (li , lj ) ? d(li , lj ), ?li ? Ia , lj ? Ib ,
d? (li , lj ) + d? (li+1 , lj+1 ) ? d? (li , lj+1 ) + d? (li+1 , lj ), ?li , lj ? I\{le },
d? (li , le ) + d? (li+1 , x
?b ) ? d? (li , x
?b ) + d? (li+1 , le ), ?li ? I\{le },
?
?
?
d (le , lj ) + d (?
xa , lj+1 ) ? d (le , lj+1 ) + d? (?
xa , lj ), ?lj ? I\{le },
?
?
?
d (le , le ) + d(?
xa , x
?b ) ? d (le , x
?b ) + d (?
xa , le ).
Similar to problem (4), the above problem minimizes the maximum ratio of the estimated distance
to the original distance. However, instead of introducing constraints for all pairs of labels, it is only
5
considers pairs of labels li and lj where li ? Ia and lj ? Ib . Furthermore, it does not modify the
distance between the current labels x
?a and x
?b (as can be seen in the last constraint of problem (5)).
Given the submodular distance functions dx?a ,?xb , we can compute a new labeling x by solving the
following optimization problem via minimum st-cut using the method of [6] (step 7):
X
X
wab dx?a ,?xb (xa , xb )
x=
argmin
?a (xa ) +
x
s.t.
Xa ?X
(Xa ,Xb )?E
xa ? Ia , ?Xa ? X.
(6)
? , then we update our
If the energy of the new labeling x is less than that of the current labeling x
labeling to x (steps 8-10). Otherwise, we retain the current estimate of the labeling and consider
another interval. The algorithm converges when the energy does not decrease for any interval of
length at most q. Note that, once again, the main difference between interval move-making and the
range expansion algorithm is the use of an appropriate optimization problem, namely the LP (5), to
obtain a submodular overestimation of the given distance function. This allows us to use interval
move-making for the general metric labeling problem, instead of focusing on only truncated convex
models.
Algorithm 4 The interval move-making algorithm.
input Unary potentials ?a (?), edge weights wab , distance function d, initial labeling x0 .
? = x0 .
1: Set current labeling to initial labeling, that is, x
2: repeat
3:
for all z ? [?q + 2, h] do
4:
Define an interval of labels I = {ls , ? ? ? , le }, where s = max{z, 1} is the start index and
e = min{z + qS? 1, h} is the end index.
5:
Define Ia = I {?
xa } for all random variables Xa ? X.
6:
Obtain submodular overestimates dx?a ,?xb for each pair of neighboring random variables
(Xa , Xb ) ? E by solving problem (5).
7:
Obtain a new labeling x by solving problem (6).
? then
8:
if Energy of x is less than energy of x
? = x.
9:
Update x
10:
end if
11:
end for
12: until Energy cannot be decreased further.
The following theorem establishes the theoretical guarantees of the interval move-making algorithm
and the interval rounding procedure.
Theorem 2. The tight multiplicative bound of the interval move-making algorithm is equal to the
tight approximation factor of the interval rounding procedure.
An interval move-making algorithm that uses an interval length of q runs for at most O(h/q) iterations. This follows from a simple modification of the result by Gupta and Tardos [8] (specifically,
theorem 3.7). Hence, the total time complexity of interval move-making is O(nmhq 2 log(m)),
since each iteration solves a minimum st-cut problem of a graph with O(nq) nodes and O(mq 2 )
arcs. In other words, interval move-making is at most as computationally complex as complete
move-making, which in turn is significantly less complex than solving the LP relaxation. Note that
problem (5), which is required for interval move-making, is independent of the unary potentials
and the edge weights. Hence, it only needs to be solved once beforehand for all pairs of labels
(?
xa , x
?b ) ? L ? L in order to obtain a solution for any metric labeling problem defined using the
distance function d.
5
Hierarchical Rounding and Hierarchical Moves
We now consider the most general form of parallel rounding that has been proposed in the literature,
namely the hierarchical rounding procedure [10]. The rounding relies on a hierarchical clustering
of the labels. Formally, we denote a hierarchical clustering of m levels for the label set L by C =
{C(i), i = 1, ? ? ? , m}. At each level i, the clustering C(i) = {C(i, j) ? L, j = 1, ? ? ? , hi } is
6
mutually exclusive and collectively exhaustive, that is,
[
C(i, j) = L, C(i, j) ? C(i, j ? ) = ?, ?j 6= j ? .
j
Furthermore, for each cluster C(i, j) at the level i > 2, there exists a unique cluster C(i ? 1, j ? ) in
the level i ? 1 such that C(i, j) ? C(i ? 1, j ? ). We call the cluster C(i ? 1, j ? ) the parent of the
cluster C(i, j) and define p(i, j) = j ? . Similarly, we call C(i, j) a child of C(i ? 1, j ? ). Without
loss of generality, we assume that there exists a single cluster at level 1 that contains all the labels,
and that each cluster at level m contains a single label.
Algorithm 5 The hierarchical rounding procedure.
input A feasible solution y of the LP relaxation.
1: Define fa1 = 1 for all Xa ? X.
2: for all i ? {2, ? ? ? , m} do
3:
for all Xa ? X do
4:
Define zai (j) for all j ? {1, ? ? ? , hi } as follows:
P
k,lk ?C(i,j) ya (k)
zai (j) =
0
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
if p(i, j) = fai?1 ,
otherwise.
Define yai (j) for all j ? {1, ? ? ? , hi } as follows:
z i (j)
yai (j) = Phi a
i ?
j ? =1 za (j )
end for
Using a rounding procedure (complete or interval) on yi = [yai (j), ?Xa ? X, j ?
? i.
{1, ? ? ? , hi }], obtain an integer solution y
for all Xa ? X do
Let ka ? {1, ? ? ? , hi } such that y?i (ka ) = 1. Define fai = ka .
end for
end for
for all Xa ? X do
Let lk be the unique label present in the cluster C(m, fam ). Assign lk to Xa .
end for
Algorithm 5 describes the hierarchical rounding procedure. Given a clustering C, it proceeds in a
top-down fashion through the hierarchy while assigning each random variable to a cluster in the
current level. Let fai be the index of the cluster assigned to the random variable Xa in the level
i. In the first step, the rounding procedure assigns all the random variables to the unique cluster
C(1, 1) (step 1). At each step i, it assigns each random variable to a unique cluster in the level i
i
by computing a conditional probability distribution as follows. The conditional
Pprobability ya (j)
of assigning the random variable Xa to the cluster C(i, j) is proportional to lk ?C(i,j) ya (k) if
p(i, j) = fai?1 (steps 3-6). The conditional probability yai (j) = 0 if p(i, j) 6= fai?1 , that is, a
random variable cannot be assigned to a cluster C(i, j) if it wasn?t assigned to its parent in the
previous step. Using a rounding procedure (complete or interval) for yi , we obtain an assignment
of random variables to the clusters at level i (step 7). Once such an assignment is obtained, the
values fai are computed for all random variables Xa (steps 8-10). At the end of step m, hierarchical
rounding would have assigned each random variable to a unique cluster in the level m. Since each
cluster at level m consists of a single label, this provides us with a labeling of the MRF (steps 12-14).
Our goal is to design a move-making algorithm whose multiplicative bound matches the approximation factor of the hierarchical rounding procedure for any choice of hierarchical clustering C. To
this end, we propose the hierarchical move-making algorithm, which extends the hierarchical graph
cuts approach for hierarchically well-separated tree (HST) metrics proposed in [14]. Algorithm 6
provides its main steps. In contrast to hierarchical rounding, the move-making algorithm traverses
the hierarchy in a bottom-up fashion while computing a labeling for each cluster in the current level.
Let xi,j be the labeling corresponding to the cluster C(i, j). At the first step, when considering the
level m of the clustering, all the random variables are assigned the same label. Specifically, xm,j
a
7
Algorithm 6 The hierarchical move-making algorithm.
input Unary potentials ?a (?), edge weights wab , distance function d.
1: for all j ? {1, ? ? ? , h} do
2:
Let lk be the unique label is the cluster C(m, j). Define xm,j
= lk for all Xa ? X.
a
3: end for
4: for all i ? {2, ? ? ? , m} do
5:
for all j ? {1, ? ? ? , hm?i+1 } do
?
6:
Define Lm?i+1,j
= {xm?i+2,j
, p(m ? i + 2, j ? ) = j, j ? ? {1, ? ? ? , hm?i+2 }}.
a
a
7:
Using a move-making algorithm (complete or interval), compute the labeling xm?i+1,j
under the constraint xm?i+1,j
? Lm?i+1,j
.
a
a
8:
end for
9: end for
10: The final solution is x1,1 .
is equal to the unique label contained in the cluster C(m, j) (steps 1-3). At step i, it computes the
labeling xm?i+1,j for each cluster C(m ? i + 1, j) by using the labelings computed in the previous
step. Specifically, it restricts the label assigned to a random variable Xa in the labeling xm?i+1,j
to the subset of labels that were assigned to it by the labelings corresponding to the children of
C(m ? i + 1, j) (step 6). Under this restriction, the labeling xm?i+1,j is computed by approximately minimizing the energy using a move-making algorithm (step 7). Implicit in our description
is the assumption that that we will use a move-making algorithm (complete or interval) in step 7 of
Algorithm 6 whose multiplicative bound matches the approximation factor of the rounding procedure (complete or interval) used in step 7 of Algorithm 5. Note that, unlike the hierarchical graph
cuts approach [14], the hierarchical move-making algorithm can be used for any arbitrary clustering
and not just the one specified by an HST metric.
The following theorem establishes the theoretical guarantees of the hierarchical move-making algorithm and the hierarchical rounding procedure.
Theorem 3. The tight multiplicative bound of the hierarchical move-making algorithm is equal to
the tight approximation factor of the hierarchical rounding procedure.
Note that hierarchical move-making solves a series of problems defined on a smaller label set. Since
the complexity of complete and interval move-making is superlinear in the number of labels, it can
be verified that the hierarchical move-making algorithm is at most as computationally complex as
the complete move-making algorithm (corresponding to the case when the clustering consists of
only one cluster that contains all the labels). Hence, hierarchical move-making is significantly faster
than solving the LP relaxation.
6
Discussion
For any general distance function that can be used to specify the (semi-)metric labeling problem, we
proved that the approximation factor of a large family of parallel rounding procedures is matched by
the multiplicative bound of move-making algorithms. This generalizes previously known results on
the guarantees of move-making algorithms in two ways: (i) in contrast to previous results [14, 15, 20]
that focused on special cases of distance functions, our results are applicable to arbitrary semi-metric
distance functions; and (ii) the guarantees provided by our theorems are tight. Our experiments
(described in the technical report) confirm that the rounding-based move-making algorithms provide
similar accuracy to the LP relaxation, while being significantly faster due to the use of efficient
minimum st-cut solvers.
Several natural questions arise. What is the exact characterization of the rounding procedures for
which it is possible to design matching move-making algorithms? Can we design rounding-based
move-making algorithms for other combinatorial optimization problems? Answering these questions will not only expand our theoretical understanding, but also result in the development of efficient and accurate algorithms.
Acknowledgements. This work is funded by the European Research Council under the European Community?s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement number 259112.
8
References
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9
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4,810 | 5,355 | Stochastic Gradient Descent, Weighted Sampling, and
the Randomized Kaczmarz algorithm
Nathan Srebro
Toyota Technological Institute at Chicago
and Dept. of Computer Science, Technion
[email protected]
Deanna Needell
Department of Mathematical Sciences
Claremont McKenna College
Claremont CA 91711
[email protected]
Rachel Ward
Department of Mathematics
Univ. of Texas, Austin
[email protected]
Abstract
We improve a recent guarantee of Bach and Moulines on the linear convergence
of SGD for smooth and strongly convex objectives, reducing a quadratic dependence on the strong convexity to a linear dependence. Furthermore, we show how
reweighting the sampling distribution (i.e. importance sampling) is necessary in
order to further improve convergence, and obtain a linear dependence on average
smoothness, dominating previous results, and more broadly discus how importance sampling for SGD can improve convergence also in other scenarios. Our
results are based on a connection between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature
studying each of the two methods.
1
Introduction
This paper concerns two algorithms which until now have remained somewhat disjoint in the literature: the randomized Kaczmarz algorithm for solving linear systems and the stochastic gradient
descent (SGD) method for optimizing a convex objective using unbiased gradient estimates. The
connection enables us to make contributions by borrowing from each body of literature to the other.
In particular, it helps us highlight the role of weighted sampling for SGD and obtain a tighter guarantee on the linear convergence regime of SGD.
Our starting point is a recent analysis on convergence of the SGD iterates. Considering a stochastic
objective F (x) = Ei [fi (x)], classical analyses of SGD show a polynomial rate on the suboptimality
of the objective value F (xk ) ? F (x? ). Bach and Moulines [1] showed that if F (x) is ?-strongly
convex, fi (x) are Li -smooth (i.e. their gradients are Li -Lipschitz), and x? is a minimizer of (almost)
all fi (x) (i.e. Pi (?fi (x? ) = 0) = 1), then Ekxk ? x? k goes to zero exponentially, rather then
polynomially, in k. That is, reaching a desired accuracy of Ekxk ? x? k2 ? ? requires a number of
steps that scales only logarithmically in 1/?. Bach and Moulines?s bound on the required number of
iterations further depends on the average squared conditioning number E[(Li /?)2 ].
In a seemingly independent line of research, the Kaczmarz method was proposed as an iterative
method for solving overdetermined systems of linear equations [7]. The simplicity of the method
makes it popular in applications ranging from computer tomography to digital signal processing [5,
1
9, 6]. Recently, Strohmer and Vershynin [19] proposed a variant of the Kaczmarz method which
selects rows with probability proportional to their squared norm, and showed that using this selection
strategy, a desired accuracy of ? can be reached in the noiseless setting in a number of steps that
scales with log(1/?) and only linearly in the condition number. As we discuss in Section 5, the
randomized Kaczmarz algorithm is in fact a special case of stochastic gradient descent.
Inspired by the above analysis, we prove improved convergence results for generic SGD, as well as
for SGD with gradient estimates chosen based on a weighted sampling distribution, highlighting the
role of importance sampling in SGD:
We first show that without perturbing the sampling distribution, we can obtain a linear dependence
on the uniform conditioning (sup Li /?), but it is not possible to obtain a linear dependence on
the average conditioning E[Li ]/?. This is a quadratic improvement over [1] in regimes where the
components have similar Lipschitz constants (Theorem 2.1 in Section 2).
We then show that with weighted sampling we can obtain a linear dependence on the average conditioning E[Li ]/?, dominating the quadratic dependence of [1] (Corollary 3.1 in Section 3).
In Section 4, we show how also for smooth but not-strongly-convex objectives, importance sampling
can improve a dependence on a uniform bound over smoothness, (sup Li ), to a dependence on the
average smoothness E[Li ]?such an improvement is not possible without importance sampling.
For non-smooth objectives, we show that importance sampling can eliminate a dependence on the
variance in the Lipschitz constants of the components.
Finally, in Section 5, we turn to the Kaczmarz algorithm, and show we can improve known guarantees in this context as well.
2
SGD for Strongly Convex Smooth Optimization
We consider the problem of minimizing a strongly convex function of the form F (x) = Ei?D fi (x)
where fi : H ? R are smooth functionals over H = Rd endowed with the standard Euclidean
norm k?k2 , or over a Hilbert space H with the norm k?k2 . Here i is drawn from some source
distribution D over an arbitrary probability space. Throughout this manuscript, unless explicitly
specified otherwise, expectations will be with respect to indices drawn from the source distribution
D. We denote the unique minimum x? = arg min F (x) and denote by ? 2 the ?residual? quantity at
the minimum, ? 2 = Ek?fi (x? )k22 .
Assumptions Our bounds will be based on the following assumptions and quantities: First, F has
strong convexity parameter ?; that is, hx ? y, ?F (x) ? ?F (y)i ? ?kx ? yk22 for all vectors x
and y. Second, each fi is continuously differentiable and the gradient function ?fi has Lipschitz
constant Li ; that is, k?fi (x) ? ?fi (y)k2 ? Li kx ? yk2 for all vectors x and y. We denote sup L
the supremum of the support of Li , i.e. the smallest L such that Li ? L a.s., and similarly denote
inf L the infimum. We denote the average Lipschitz constant as L = ELi .
An unbiased gradient estimate for F (x) can be obtained by drawing i ? D and using ?fi (x) as the
estimate. The SGD updates with (fixed) step size ? based on these gradient estimates are given by:
xk+1 ? xk ? ??fik (xk )
(2.1)
x? k22
where {ik } are drawn i.i.d. from D. We are interested in the distance kxk ?
from the unique minimum, and denote the initial distance by ?0 = kx0 ? x? k22 .
of the iterates
Bach and Moulines [1, Theorem 1] considered this setting1 and established that
EL2
?2
i
k = 2 log(?0 /?)
+ 2
(2.2)
2
?
? ?
SGD iterations of the form (2.1), with an appropriate step-size, are sufficient to ensure
Ekxk ? x? k22 ? ?, where the expectation is over the random sampling. As long as ? 2 = 0, i.e. the
1
Bach and Moulines?s results are somewhat more general. Their Lipschitz requirement is a bit weaker and
more complicated, but in terms of Li yields (2.2). They also study the use of polynomial decaying step-sizes,
but these do not lead to improved runtime if the target accuracy is known ahead of time.
2
same minimizer x? minimizes all components fi (x) (though of course it need not be a unique minimizer of any of them); this yields linear convergence to x? , with a graceful degradation as ? 2 > 0.
However, in the linear convergence regime, the number of required iterations scales with the expected squared conditioning EL2i /?2 . In this paper, we reduce this quadratic dependence to a linear
dependence. We begin with a guarantee ensuring linear dependence on sup L/?:
Theorem 2.1 Let each fi be convex where ?fi has Lipschitz constant Li , with Li ? sup L a.s.,
and let F (x) = Efi (x) be ?-strongly convex. Set ? 2 = Ek?fi (x? )k22 , where x? = argminx F (x).
Suppose that ? ? 1/?. Then the SGD iterates given by (2.1) satisfy:
h
ik
?? 2
.
Ekxk ? x? k22 ? 1 ? 2??(1 ? ? sup L)
kx0 ? x? k22 +
(2.3)
? 1 ? ? sup L
That is, for any desired ?, using a step-size of
?=
??
2?? sup L + 2? 2
sup L
?2
ensures that after k = 2 log(?0 /?)
+ 2
?
? ?
(2.4)
SGD iterations, Ekxk ? x? k22 ? ?, where ?0 = kx0 ? x? k22 and where both expectations are with
respect to the sampling of {ik }.
Proof sketch: The crux of the improvement over [1] is a tighter recursive equation. Instead of:
kxk+1 ? x? k22 ? 1 ? 2?? + 2? 2 L2ik kxk ? x? k22 + 2? 2 ? 2 ,
we use the co-coercivity Lemma (Lemma A.1 in the supplemental material) to obtain:
kxk+1 ? x? k22 ? 1 ? 2?? + 2? 2 ?Lik kxk ? x? k22 + 2? 2 ? 2 .
The significant difference is that one of the factors of Lik , an upper bound on the second derivative
(where ik is the random index selected in the kth iteration) in the third term inside the parenthesis,
is replaced by ?, a lower bound on the second derivative of F . A complete proof can be found in the
supplemental material.
Comparison to [1] Our bound (2.4) improves a quadratic dependence on ?2 to a linear dependence and replaces the dependence on the average squared smoothness EL2i with a linear dependence
on the smoothness bound sup L. When all Lipschitz constants Li are of similar magnitude, this is a
quadratic improvement in the number of required iterations. However, when different components
fi have widely different scaling, i.e. Li are highly variable, the supremum might be significantly
larger then the average square conditioning.
Tightness Considering the above, one might hope to obtain a linear dependence on the average
smoothness L. However, as the following example shows, this is not possible. Consider a uniform
source distribution over N + 1 quadratics, with the first quadratic f1 being N (x[1] ? b)2 and all
others being x[2]2 , and b = ?1. Any method must examine f1 in order to recover x to within
error less then one, but by uniformly sampling indices i, this takes N iterations in expectation.
2
?1)
?1)
We can calculate sup L = L1 = 2N , L = 2(2N
, EL2i = 4(N +N
, and ? = 1. Both
N
N
2
2
sup L/? = ELi /? = O(N ) scale correctly with the expected number of iterations, while error
reduction in O(L/?) = O(1) iterations is not possible for this example.
We therefore see that the choice between EL2i and sup L is unavoidable. In the next Section, we
will show how we can obtain a linear dependence on the average smoothness L, using importance
sampling, i.e. by sampling from a modified distribution.
3
Importance Sampling
For a weight function w(i) which assigns a non-negative weight w(i) ? 0 to each index i, the
weighted distribution D(w) is defined as the distribution such that
PD(w) (I) ? Ei ?D [1I (i)w(i)] ,
3
where I is an event (subset of indices) and 1I (?) its indicator function. For a discrete distribution
D with probability mass function p(i) this corresponds to weighting the probabilities to obtain a
new probability mass function, which we write as p(w) (i) ? w(i)p(i). Similarly, for a continuous
distribution, this corresponds to multiplying the density by w(i) and renormalizing. Importance
sampling has appeared in both the Kaczmarz method [19] and in coordinate-descent methods [14,
15], where the weights are proportional to some power of the Lipschitz constants (of the gradient
coordinates). Here we analyze this type of sampling in the context of SGD.
One way to construct D(w) is through rejection sampling: sample i ? D, and accept with probability
w(i)/W , for some W ? supi w(i). Otherwise, reject and continue to re-sample until a suggestion
i is accepted. The accepted samples are then distributed according to D(w) .
We use E(w) [?] = Ei?D(w) [?] to denote expectation where indices are sampled from the weighted
distribution D(w) . An important property of such an expectation is that for any quantity X(i):
h
i
1
E(w) w(i)
X(i) = E [w(i)] ? E [X(i)] ,
(3.1)
where recall that the expectationsh on the r.h.s.
i are with respect to i ? D. In particular, when
1
E[w(i)] = 1, we have that E(w) w(i)
X(i) = EX(i). In fact, we will consider only weights
s.t. E[w(i)] = 1, and refer to such weights as normalized.
Reweighted SGD For any normalized weight function w(i), we can write:
(w)
fi
(x) =
1
fi (x) and
w(i)
(w)
F (x) = E(w) [fi
(x)].
(3.2)
This is an equivalent, and equally valid, stochastic representation of the objective F (x), and we can
just as well base SGD on this representation. In this case, at each iteration we sample i ? D(w)
(w)
1
and then use ?fi (x) = w(i)
?fi (x) as an unbiased gradient estimate. SGD iterates based on the
representation (3.2), which we will refer to as w-weighted SGD, are then given by
xk+1 ? xk ?
?
?fik (xk )
w(ik )
(3.3)
where {ik } are drawn i.i.d. from D(w) .
The important observation here is that all SGD guarantees are equally valid for the w-weighted
updates (3.3)?the objective is the same objective F (x), the sub-optimality is the same, and the
minimizer x? is the same. We do need, however, to calculate the relevant quantities controlling SGD
(w)
convergence with respect to the modified components fi and the weighted distribution D(w) .
Strongly Convex Smooth Optimization using Weighted SGD We now return to the analysis of
strongly convex smooth optimization and investigate how re-weighting can yield a better guarantee.
(w)
(w)
(w)
1
Li .
The Lipschitz constant Li of each component fi is now scaled, and we have Li = w(i)
The supremum is then given by:
(w)
sup L(w) = sup Li
= sup
i
i
Li
.
w(i)
(3.4)
It is easy to verify that (3.4) is minimized by the weights
w(i) =
Li
,
L
so that
sup L(w) = sup
i
Li
= L.
Li /L
(3.5)
Before applying Theorem 2.1, we must also calculate:
(w)
2
?(w)
= E(w) [k?fi
(x? )k22 ] = E[
1
L
L 2
k?fi (x? )k22 ] = E[ k?fi (x? )k22 ] ?
? .
w(i)
Li
inf L
4
(3.6)
Now, applying Theorem 2.1 to the w-weighted SGD iterates (3.3) with weights (3.5), we have that,
with an appropriate stepsize,
2
sup L
L
?(w)
?2
L
(w)
k = 2 log(?0 /?)
= 2 log(?0 /?)
(3.7)
+ 2
+
? 2
?
? ?
?
inf L ? ?
iterations are sufficient for E(w) kxk ? x? k22 ? ?, where x? , ? and ?0 are exactly as in Theorem 2.1.
If ? 2 = 0, i.e. we are in the ?realizable? situation, with true linear convergence, then we also have
2
?(w)
= 0. In this case, we already obtain the desired guarantee: linear convergence with a linear
dependence on the average conditioning L/?, strictly improving over the best known results [1].
However, when ? 2 > 0 we get a dissatisfying scaling of the second term, by a factor of L/inf L.
Fortunately, we can easily overcome this factor. To do so, consider sampling from a distribution
which is a mixture of the original source distribution and its re-weighting:
1 1 Li
w(i) = + ? .
(3.8)
2 2 L
We refer to this as partially biased sampling. Instead of an even mixture as in (3.9), we could also
use a mixture with any other constant proportion, i.e. w(i) = ? + (1 ? ?)Li /L for 0 < ? < 1.
Using these weights, we have
1
1
2
sup L(w) = sup 1 1 Li Li ? 2L and ?(w)
= E[ 1 1 Li k?fi (x? )k22 ] ? 2? 2 . (3.9)
i 2 + 2 ?
2 + 2 ? L
L
Corollary 3.1 Let each fi be convex where ?fi has Lipschitz constant Li and let F (x) =
Ei?D [fi (x)], where F (x) is ?-strongly convex. Set ? 2 = Ek?fi (x? )k22 , where x? =
argminx F (x). For any desired ?, using a stepsize of
L
?2
??
ensures
that
after
k
=
4
log(?
/?)
+
(3.10)
?=
0
?
?2 ?
4(??L + ? 2 )
iterations of w-weighted SGD (3.3) with weights specified by (3.8), E(w) kxk ? x? k22 ? ?, where
?0 = kx0 ? x? k22 and L = ELi .
This result follows by substituting (3.9) into Theorem 2.1. We now obtain the desired linear scaling
on L/?, without introducing any additional factor to the residual term, except for a constant factor.
We thus obtain a result which dominates Bach and Moulines (up to a factor of 2) and substantially
improves upon it (with a linear rather than quadratic dependence on the conditioning). Such ?partially biased weights? are not only an analysis trick, but might indeed improve actual performance
over either no weighting or the ?fully biased? weights (3.5), as demonstrated in Figure 1.
Implementing Importance Sampling In settings where linear systems need to be solved repeatedly, or when the Lipschitz constants are easily computed from the data, it is straightforward to
sample by the weighted distribution. However, when we only have sampling access to the source
distribution D (or the implied distribution over gradient estimates), importance sampling might be
difficult. In light of the above results, one could use rejection sampling to simulate sampling from
D(w) . For the weights (3.5), this can be done by accepting samples with probability proportional
to Li / sup L. The overall probability of accepting a sample is then L/ sup L, introducing an additional factor of sup L/L. This yields a sample complexity with a linear dependence on sup L, as
in Theorem 2.1, but a reduction in the number of actual gradient calculations and updates. In even
less favorable situations, if Lipschitz constants cannot be bounded for individual components, even
importance sampling might not be possible.
4
Importance Sampling for SGD in Other Scenarios
In the previous Section, we considered SGD for smooth and strongly convex objectives, and were
particularly interested in the regime where the residual ? 2 is low, and the linear convergence term is
dominant. Weighted SGD is useful also in other scenarios, and we now briefly survey them, as well
as relate them to our main scenario of interest.
5
Error || xk ? x* ||2
Error || xk ? x* ||2
?=0
? = 0.2
?=1
1
0
10
?1
10
0
10
10
10
0
1000
2000
Iteration k
3000
0
10
?1
?1
10
?=0
? = 0.2
?=1
1
10
Error || xk ? x* ||2
?=0
? = 0.2
?=1
1
10
0
1000
2000 3000
Iteration k
4000
5000
0
1000
2000 3000
Iteration k
4000
5000
Figure 1: Performance of SGD with weights w(i) = ? + (1 ? ?) LLi on synthetic overdetermined least
squares problems of the form (5.1) (? = 1 is unweighted, ? = 0 is fully weighted). Left: ai are standard
spherical Gaussian, bi = hai , x0 i + N (0, 0.12 ). Center: ai is spherical Gaussian with variance i, bi =
hai , x0 i + N (0, 202 ). Right: ai is spherical Gaussian with variance i, bi = hai , x0 i + N (0, 0.12 ). In all
cases, matrix A with rows ai is 1000 ? 100 and the corresponding least squares problem is strongly convex;
the stepsize was chosen as in (3.10).
2
2
?=0
?=.4
?=1
0
10
?2
10
Error F(xk) ? F(x*)
k
0
10
Error F(xk) ? F(x*)
10
?=0
?=.4
?=1
*
Error F(x ) ? F(x )
10
?=0
? = .4
?=1
1
10
0
0
2000
4000
Iteration k
6000
8000
0
5000
10000
Iteration k
15000
10
0
5000
10000
Iteration k
15000
Figure 2: Performance of SGD with weights w(i) = ? + (1 ? ?) LLi on synthetic underdetermined least
squares problems of the form (5.1) (? = 1 is unweighted, ? = 0 is fully weighted). We consider 3 cases. Left:
ai are standard spherical Gaussian, bi = hai , x0 i+N (0, 0.12 ). Center: ai is spherical Gaussian with variance
i, bi = hai , x0 i + N (0, 202 ). Right: ai is spherical Gaussian with variance i, bi = hai , x0 i + N (0, 0.12 ).
In all cases, matrix A with rows ai is 50 ? 100 and so the corresponding least squares problem is not strongly
convex; the step-size was chosen as in (3.10).
Smooth, Not Strongly Convex When each component fi is convex, non-negative, and has an
Li -Lipschitz gradient, but the objective F (x) is not necessarily strongly convex, then after
(sup L)kx? k22 F (x? ) + ?
k=O
?
(4.1)
?
?
iterations of SGD with an appropriately chosen step-size we will have F (xk ) ? F (x? ) + ?, where
xk is an appropriate averaging of the k iterates [18]. The relevant quantity here determining the iteration complexity is again sup L. Furthermore, the dependence on the supremum is unavoidable and
cannot be replaced with the average Lipschitz constant L [3, 18]: if we sample gradients according
to the source distribution D, we must have a linear dependence on sup L.
The only quantity in the bound (4.1) that changes with a re-weighting is sup L?all other quantities
(kx? k22 , F (x? ), and the sub-optimality ?) are invariant to re-weightings. We can therefore replace
the dependence on sup L with a dependence on sup L(w) by using a weighted SGD as in (3.3). As we
already calculated, the optimal weights are given by (3.5), and using them we have sup L(w) = L.
In this case, there is no need for partially biased sampling, and we obtain that
Lkx? k22 F (x? ) + ?
k=O
?
(4.2)
?
?
iterations of weighed SGD updates (3.3) using the weights (3.5) suffice. Empirical evidence suggests
that this is not a theoretical artifact; full weighted sampling indeed exhibits better convergence rates
compared to partially biased sampling in the non-strongly convex setting (see Figure 2), in contrast
6
to the strongly convex regime (see Figure 1). We again see that using importance sampling allows us
to reduce the dependence on sup L, which is unavoidable without biased sampling, to a dependence
on L. An interesting question for further consideration is to what extent importance sampling can
also help with stochastic optimization procedures such as SAG [8] and SDCA [17] which achieve
faster convergence on finite data sets. Indeed, weighted sampling was shown empirically to achieve
faster convergence rates for SAG [16], but theoretical guarantees remain open.
Non-Smooth Objectives We now turn to non-smooth objectives, where the components fi might
not be smooth, but each component is Gi -Lipschitz. Roughly speaking, Gi is a bound on the first
derivative (the subgradients) of fi , while Li is a bound on the second derivatives of fi . Here,
the performance of SGD (actually stochastic subgradient decent) depends on the second moment
G2 = E[G2i ] [12]. The precise iteration complexity depends on whether the objective is strongly
convex or whether x? is bounded, but in either case depends linearly on G2 .
Using weighted SGD, we get linear dependence on
2
2
i
h
Gi
Gi
(w)
G2(w) = E(w) (Gi )2 = E(w)
=
E
w(i)2
w(i)
(w)
(4.3)
(w)
where Gi = Gi /w(i) is the Lipschitz constant of the scaled fi . This is minimized by the
2
weights w(i) = Gi /G, where G = EGi , yielding G2(w) = G . Using importance sampling, we
2
therefore reduce the dependence on G2 to a dependence on G . Its helpful to recall that G2 =
2
G + Var[Gi ]. What we save is thus exactly the variance of the Lipschitz constants Gi .
Parallel work we recently became aware of [22] shows a similar improvement for a non-smooth
composite objective. Rather than relying on a specialized analysis as in [22], here we show this
follows from SGD analysis applied to different gradient estimates.
Non-Realizable Regime Returning to the smooth and strongly convex setting of Sections 2 and 3,
let us consider more carefully the residual term ? 2 = Ek?fi (x? )k22 . This quantity depends on the
weighting, and in Section 3, we avoided increasing it, introducing partial biasing for this purpose.
However, if this is the dominant term, we might want to choose weights to minimize this term. The
optimal weights here would be proportional to k?fi (x? )k2 , which is not known in general.
An alternative approach is to bound k?fi (x? )k2 ? Gi and so ? 2 ? G2 . Taking this bound, we are
back to the same quantity as in the non-smooth case, and the optimal weights are proportional to Gi .
Note that this differs from using weights proportional to Li , which optimize the linear-convergence
term as studied in Section 3.
To understand how weighting according to Gi and Li are different, consider a generalized linear
objective fi (x) = ?i (hzi , xi), where ?i is a scalar function with bounded |?0i | , |?00i |. We have
that Gi ? kzi k2 while Li ? kzi k22 . Weighting according to (3.5), versus weighting with w(i) =
Gi /G, thus corresponds to weighting according to kzi k22 versus kzi k2 , and are rather different. E.g.,
weighting by Li ? kzi k22 yields G2(w) = G2 : the same sub-optimal dependence as if no weighting
at all were used. A good solution could be to weight by a mixture of Gi and Li , as in the partial
weighting scheme of Section 3.
5
The least squares case and the Randomized Kaczmarz Method
A special case of interest is the least squares problem, where
n
F (x) =
1
1X
(hai , xi ? bi )2 = kAx ? bk22
2 i=1
2
(5.1)
with b ? Cn , A an n ? d matrix with rows ai , and x? = argminx 21 kAx ? bk22 is the least-squares
solution. We can also write (5.1) as a stochastic objective, where the source distribution D is uniform
over {1, 2, . . . , n} and fi = n2 (hai , xi ? bi )2 . In this setting, ? 2 = kAx? ? bk22 is the residual error
7
at the least squares solution x? , which can also be interpreted as noise variance in a linear regression
model.
The randomized Kaczmarz method introduced for solving the least squares problem (5.1) in the
case where A is an overdetermined full-rank matrix, begins with an arbitrary estimate x0 , and in the
kth iteration selects a row i at random from the matrix A and iterates by:
xk+1 = xk + c ?
bi ? hai , xk i
ai ,
kai k22
(5.2)
where c = 1 in the standard method. This is almost an SGD update with step-size ? = c/n, except
for the scaling by kai k22 .
Strohmer and Vershynin [19] provided the first non-asymptotic convergence rates, showing that
drawing rows proportionally to kai k22 leads to provable exponential convergence in expectation [19].
With such a weighting, (5.2) is exactly weighted SGD, as in (3.3), with the fully biased weights
(3.5).
The reduction of the quadratic dependence on the conditioning to a linear dependence in Theorem
2.1, and the use of biased sampling, was inspired by the analysis of [19]. Indeed, applying Theorem
2.1 to the weighted SGD iterates with weights as in (3.5) and a stepsize of ? = 1 yields precisely the
guarantee of [19]. Furthermore, understanding the randomized Kaczmarz method as SGD, allows
us to obtain the following improvements:
Partially Biased Sampling. Using partially biased sampling weights (3.8) yields a better dependence on the residual over the fully biased sampling weights (3.5) considered by [19].
Using Step-sizes. The randomized Kaczmarz method with weighted sampling exhibits exponential
convergence, but only to within a radius, or convergence horizon, of the least-squares solution [19,
10]. This is because a step-size of ? = 1 is used, and so the second term in (2.3) does not vanish.
It has been shown [21, 2, 20, 4, 11] that changing the step size can allow for convergence inside of
this convergence horizon, but only asymptotically. Our results allow for finite-iteration guarantees
with arbitrary step-sizes and can be immediately applied to this setting.
Uniform Row Selection. Strohmer and Vershynin?s variant of the randomized Kaczmarz method
calls for weighted row sampling, and thus requires pre-computing all the row norms. Although
certainly possible in some applications, in other cases this might be better avoided. Understanding
the randomized Kaczmarz as SGD allows us to apply Theorem 2.1 also with uniform weights (i.e. to
the unbiased SGD), and obtain a randomized Kaczmarz using uniform sampling, which converges
to the least-squares solution and enjoys finite-iteration guarantees.
6
Conclusion
We consider this paper as making three main contributions. First, we improve the dependence on
the conditioning for smooth and strongly convex SGD from quadratic to linear. Second, we investigate SGD and importance sampling and show how it can yield improvements not possible without
reweighting. Lastly, we make connections between SGD and the randomized Kaczmarz method.
This connection along with our new results show that the choice in step-size of the Kaczmarz method
offers a tradeoff between convergence rate and horizon and also allows for a convergence bound
when the rows are sampled uniformly.
For simplicity, we only considered SGD with fixed step-size ?, which is appropriate when the target
accuracy in known in advance. Our analysis can be adapted also to decaying step-sizes.
Our discussion of importance sampling is limited to a static reweighting of the sampling distribution.
A more sophisticated approach would be to update the sampling distribution dynamically as the
method progresses, and as we gain more information about the relative importance of components
(e.g. about k?fi (x? )k). Such dynamic sampling is sometimes attempted heuristically, and obtaining
a rigorous framework for this would be desirable.
8
References
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9
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4,811 | 5,356 | An Accelerated Proximal Coordinate Gradient Method
Qihang Lin
University of Iowa
Iowa City, IA, USA
[email protected]
Zhaosong Lu
Simon Fraser University
Burnaby, BC, Canada
[email protected]
Lin Xiao
Microsoft Research
Redmond, WA, USA
[email protected]
Abstract
We develop an accelerated randomized proximal coordinate gradient (APCG)
method, for solving a broad class of composite convex optimization problems.
In particular, our method achieves faster linear convergence rates for minimizing
strongly convex functions than existing randomized proximal coordinate gradient methods. We show how to apply the APCG method to solve the dual of the
regularized empirical risk minimization (ERM) problem, and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned
ERM problems, our method obtains improved convergence rates than the state-ofthe-art stochastic dual coordinate ascent (SDCA) method.
1
Introduction
Coordinate descent methods have received extensive attention in recent years due to their potential
for solving large-scale optimization problems arising from machine learning and other applications.
In this paper, we develop an accelerated proximal coordinate gradient (APCG) method for solving
convex optimization problems with the following form:
def
F (x) = f (x) + ?(x) ,
(1)
minimize
x?RN
where f is differentiable on dom (?), and ? has a block separable structure. More specifically,
?(x) =
n
X
?i (xi ),
(2)
i=1
where each xi denotes a sub-vector of x with cardinality Ni , and each ?i : RNi ? R ? {+?}
is a closed convex function. We assume the collection {xi : i = 1, . . . , n} form a partition of
the components of x ? RN . In addition to the capability of modeling nonsmooth regularization
terms such as ?(x) = ?kxk1 , this model also includes optimization problems with block separable
constraints. More precisely, each block constraint xi ? Ci , where Ci is a closed convex set, can be
modeled by an indicator function defined as ?i (xi ) = 0 if xi ? Ci and ? otherwise.
At each iteration, coordinate descent methods choose one block of coordinates xi to sufficiently
reduce the objective value while keeping other blocks fixed. One common and simple approach
for choosing such a block is the cyclic scheme. The global and local convergence properties of the
cyclic coordinate descent method have been studied in, for example, [21, 11, 16, 2, 5]. Recently,
randomized strategies for choosing the block to update became more popular. In addition to its theoretical benefits [13, 14, 19], numerous experiments have demonstrated that randomized coordinate
descent methods are very powerful for solving large-scale machine learning problems [3, 6, 18, 19].
Inspired by the success of accelerated full gradient methods (e.g., [12, 1, 22]), several recent work
applied Nesterov?s acceleration schemes to speed up randomized coordinate descent methods. In
particular, Nesterov [13] developed an accelerated randomized coordinate gradient method for minimizing unconstrained smooth convex functions, which corresponds to the case of ?(x) ? 0 in (1).
1
Lu and Xiao [10] gave a sharper convergence analysis of Nesterov?s method, and Lee and Sidford [8] developed extensions with weighted random sampling schemes. More recently, Fercoq
and Richt?arik [4] proposed an APPROX (Accelerated, Parallel and PROXimal) coordinate descent
method for solving the more general problem (1) and obtained accelerated sublinear convergence
rates, but their method cannot exploit the strong convexity to obtain accelerated linear rates.
In this paper, we develop a general APCG method that achieves accelerated linear convergence
rates when the objective function is strongly convex. Without the strong convexity assumption, our
method recovers the APPROX method [4]. Moreover, we show how to apply the APCG method to
solve the dual of the regularized empirical risk minimization (ERM) problem, and devise efficient
implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems,
our method obtains faster convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method [19], and the improved iteration complexity matches the accelerated SDCA
method [20]. We present numerical experiments to illustrate the advantage of our method.
1.1
Notations and assumptions
For any partition of x ? RN into {xi ? RNi : i = 1, . . . , n}, there is an N ? N permutation
matrix U partitioned as U = [U1 ? ? ? Un ], where Ui ? RN ?Ni , such that
n
X
Ui xi ,
and xi = UiT x, i = 1, . . . , n.
x=
i=1
For any x ? RN , the partial gradient of f with respect to xi is defined as
?i f (x) = UiT ?f (x), i = 1, . . . , n.
We associate each subspace RNi , for i = 1, . . . , n, with the standard Euclidean norm, denoted
by k ? k. We make the following assumptions which are standard in the literature on coordinate
descent methods (e.g., [13, 14]).
Assumption 1. The gradient of function f is block-wise Lipschitz continuous with constants Li , i.e.,
k?i f (x + Ui hi ) ? ?i f (x)k ? Li khi k, ? hi ? RNi , i = 1, . . . , n, x ? RN .
For convenience, we define the following norm in the whole space RN :
X
1/2
n
kxkL =
Li kxi k2
, ? x ? RN .
(3)
i=1
Assumption 2. There exists ? ? 0 such that for all y ? RN and x ? dom (?),
?
f (y) ? f (x) + h?f (x), y ? xi + ky ? xk2L .
2
The convexity parameter of f with respect to the norm k ? kL is the largest ? such that the above
inequality holds. Every convex function satisfies Assumption 2 with ? = 0. If ? > 0, the function f
is called strongly convex.
We note that an immediate consequence of Assumption 1 is
Li
f (x + Ui hi ) ? f (x) + h?i f (x), hi i + khi k2 , ? hi ? RNi ,
2
This together with Assumption 2 implies ? ? 1.
2
i = 1, . . . , n,
x ? RN . (4)
The APCG method
In this section we describe the general APCG method, and several variants that are more suitable
for implementation under different assumptions. These algorithms extend Nesterov?s accelerated
gradient methods [12, Section 2.2] to the composite and coordinate descent setting.
We first explain the notations used in our algorithms. The algorithms proceed in iterations, with k
being the iteration counter. Lower case letters x, y, z represent vectors in the full space RN , and
x(k) , y (k) and z (k) are their values at the kth iteration. Each block coordinate is indicated with a
(k)
subscript, for example, xi represents the value of the ith block of the vector x(k) . The Greek letters
?, ?, ? are scalars, and ?k , ?k and ?k represent their values at iteration k.
2
Algorithm 1: the APCG method
(0)
Input: x ? dom (?) and convexity parameter ? ? 0.
Initialize: set z (0) = x(0) and choose 0 < ?0 ? [?, 1].
Iterate: repeat for k = 0, 1, 2, . . .
1. Compute ?k ? (0, n1 ] from the equation
n2 ?k2 = (1 ? ?k ) ?k + ?k ?,
and set
?k ?
.
?k+1
?k ?k z (k) + ?k+1 x(k) .
?k+1 = (1 ? ?k )?k + ?k ?,
2. Compute y (k) as
y (k) =
1
?k ?k + ?k+1
(5)
?k =
(6)
(7)
3. Choose ik ? {1, . . . , n} uniformly at random and compute
o
n n?
2
k
z (k+1) = arg min
x?(1??k )z (k) ??k y (k)
L +h?ik f (y (k) ), xik i+?ik (xik ) .
2
x?RN
4. Set
x(k+1) = y (k) + n?k (z (k+1) ? z (k) ) +
? (k)
(z ? y (k) ).
n
(8)
The general APCG method is given as Algorithm 1. At each iteration k, it chooses a random
coordinate ik ? {1, . . . , n} and generates y (k) , x(k+1) and z (k+1) . One can observe that x(k+1) and
z (k+1) depend on the realization of the random variable
?k = {i0 , i1 , . . . , ik },
while y (k) is independent of ik and only depends on ?k?1 . To better understand this method, we
make the following observations. For convenience, we define
n n?
o
2
k
z?(k+1) = arg min
x ? (1 ? ?k )z (k) ? ?k y (k)
L + h?f (y (k) ), x ? y (k) i + ?(x) , (9)
2
x?RN
which is a full-dimensional update version of Step 3. One can observe that z (k+1) is updated as
( (k+1)
z?i
if i = ik ,
(k+1)
zi
=
(10)
(k)
(k)
(1 ? ?k )zi + ?k yi
if i 6= ik .
Notice that from (5), (6), (7) and (8) we have
x(k+1) = y (k) + n?k z (k+1) ? (1 ? ?k )z (k) ? ?k y (k) ,
which together with (10) yields
?
? y (k) + n? z (k+1) ? z (k) +
k
(k+1)
i
i
i
xi
=
? y (k)
i
?
n
(k)
zi
(k)
? yi
(k+1)
That is, in Step 4, we only need to update the block coordinates xik
if i = ik ,
if i 6= ik .
(11)
(k)
and set the rest to be yi .
We now state a theorem concerning the expected rate of convergence of the APCG method, whose
proof can be found in the full report [9].
Theorem 1. Suppose Assumptions 1 and 2 hold. Let F ? be the optimal value of problem (1), and
{x(k) } be the sequence generated by the APCG method. Then, for any k ? 0, there holds:
(
2 )
? k
?
?0
2n
(k)
?
1?
E?k?1 [F (x )] ? F ? min
F (x(0) ) ? F ? + R02 ,
,
?
n
2n + k ?0
2
where
def
?
R0 = min
kx(0) ? x? kL ,
?
?
x ?X
and X is the set of optimal solutions of problem (1).
3
(12)
Our result in Theorem 1 improves upon the convergence rates of the proximal coordinate gradient
methods in [14, 10], which have convergence rates on the order of
n
o
k
n
O min 1 ? n? , n+k
.
For n = 1, our result matches exactly that of the accelerated full gradient method in [12, Section 2.2].
2.1
Two special cases
Here we give two simplified versions of the APCG method, for the special cases of ? = 0 and
? > 0, respectively. Algorithm 2 shows the simplified version for ? = 0, which can be applied to
problems without strong convexity, or if the convexity parameter ? is unknown.
Algorithm 2: APCG with ? = 0
(0)
Input: x ? dom (?).
Initialize: set z (0) = x(0) and choose ?0 ? (0, n1 ].
Iterate: repeat for k = 0, 1, 2, . . .
1. Compute y (k) = (1 ? ?k )x(k) + ?k z (k) .
2. Choose ik ? {1, . . . , n}nuniformly at random and compute
o
n?k Lik
(k+1)
x ? z (k)
2 + h?i f (y (k) ), x ? y (k) i + ?i (x) .
zik
= arg minx?RN
k
k
ik
ik
2
(k+1)
and set zi
(k)
= zi
for all i 6= ik .
3. Set x(k+1) = y (k) + n?k (z (k+1) ? z (k) ).
p
?k4 + 4?k2 ? ?k2 .
4. Compute ?k+1 = 12
According to Theorem 1, Algorithm 2 has an accelerated sublinear convergence rate, that is
2
1
2n
F (x(0) ) ? F ? + R02 .
E?k?1 [F (x(k) )] ? F ? ?
2n + kn?0
2
With the choice of ?0 = 1/n, Algorithm 2 reduces to the APPROX method [4] with single block
update at each iteration (i.e., ? = 1 in their Algorithm 1).
For the strongly convex case with ? > ?
0, we can initialize Algorithm 1 with the parameter ?0 = ?,
which implies ?k = ? and ?k = ?k = ?/n for all k ? 0. This results in Algorithm 3.
Algorithm 3: APCG with ?0 = ? > 0
Input: x
(0)
? dom (?) and convexity parameter ? > 0.
Initialize: set z (0) = x(0) and and ? =
Iterate: repeat for k = 0, 1, 2, . . .
1. Compute y (k) =
?
?
n .
x(k) +?z (k)
.
1+?
2. Choose ik ? {1, . . . , n} uniformly at random and compute
o
n
2
(k)
(k)
??y (k)
L +h?ik f (y (k) ), xik ?yik i+?ik (xik ) .
z (k+1) = arg min n?
2 x?(1??)z
x?RN
3. Set x(k+1) = y (k) + n?(z (k+1) ? z (k) ) + n?2 (z (k) ? y (k) ).
As a direct corollary of Theorem 1, Algorithm 3 enjoys an accelerated linear convergence rate:
? k
?
?
F (x(0) ) ? F ? + R02 .
E?k?1 [F (x(k) )] ? F ? ? 1 ?
n
2
To the best of our knowledge, this is the first time such an accelerated rate is obtained for solving
the general problem (1) (with strong convexity) using coordinate descent type of methods.
4
2.2
Efficient implementation
The APCG methods we presented so far all need to perform full-dimensional vector operations at
each iteration. For example, y (k) is updated as a convex combination of x(k) and z (k) , and this
can be very costly since in general they can be dense vectors. Moreover, for the strongly convex case (Algorithms 1 and 3), all blocks of z (k+1) need to be updated at each iteration, although
only the ik -th block needs to compute the partial gradient and perform a proximal mapping. These
full-dimensional vector updates cost O(N ) operations per iteration and may cause the overall computational cost of APCG to be even higher than the full gradient methods (see discussions in [13]).
In order to avoid full-dimensional vector operations, Lee and Sidford [8] proposed a change of
variables scheme for accelerated coordinated descent methods for unconstrained smooth minimization. Fercoq and Richt?arik [4] devised a similar scheme for efficient implementation in the ? = 0
case for composite minimization. Here we show that such a scheme can also be developed for the
case of ? > 0 in the composite optimization setting. For simplicity, we only present an equivalent
implementation of the simplified APCG method described in Algorithm 3.
Algorithm 4: Efficient implementation of APCG with ?0 = ? > 0
Input: x
(0)
? dom (?) and convexity parameter ? > 0.
?
?
(0)
Initialize: set ? = n and ? = 1??
= 0 and v (0) = x(0) .
1+? , and initialize u
Iterate: repeat for k = 0, 1, 2, . . .
1. Choose ik ? {1, . . . , n} uniformly at random and compute
o
n
n?Lik
(k)
(k)
(k)
?ik = arg min
k?k2 + h?ik f (?k+1 u(k) +v (k) ), ?i + ?ik (??k+1 uik +vik +?) .
2
??R
Ni
k
2. Let u(k+1) = u(k) and v (k+1) = v (k) , and update
(k+1)
uik
(k)
= uik ?
1?n? (k)
? ,
2?k+1 ik
(k+1)
vik
(k)
= vik +
1+n? (k)
2 ?ik .
(13)
Output: x(k+1) = ?k+1 u(k+1) + v (k+1)
The following Proposition is proved in the full report [9].
Proposition 1. The iterates of Algorithm 3 and Algorithm 4 satisfy the following relationships:
x(k) = ?k u(k) + v (k) ,
y (k) = ?k+1 u(k) + v (k) ,
z (k) = ??k u(k) + v (k) .
(14)
We note that in Algorithm 4, only a single block coordinate of the vectors u(k) and v (k) are updated
at each iteration, which cost O(Ni ). However, computing the partial gradient ?ik f (?k+1 u(k) +v (k) )
may still cost O(N ) in general. In the next section, we show how to further exploit structure in many
ERM problems to completely avoid full-dimensional vector operations.
3
Application to regularized empirical risk minimization (ERM)
Let A1 , . . . , An be vectors in Rd , ?1 , . . . , ?n be a sequence of convex functions defined on R, and g
be a convex function on Rd . Regularized ERM aims to solve the following problem:
n
minimize P (w),
w?Rd
with
1X
P (w) =
?i (ATi w) + ?g(w),
n i=1
where ? > 0 is a regularization parameter. For example, given a label bi ? {?1} for each vector Ai ,
for i = 1, . . . , n, we obtain the linear SVM problem by setting ?i (z) = max{0, 1?bi z} and g(w) =
(1/2)kwk22 . Regularized logistic regression is obtained by setting ?i (z) = log(1+exp(?bi z)). This
formulation also includes regression problems. For example, ridge regression is obtained by setting
(1/2)?i (z) = (z ? bi )2 and g(w) = (1/2)kwk22 , and we get Lasso if g(w) = kwk1 .
5
Let ??i be the convex conjugate of ?i , that is, ??i (u) = maxz?R (zu ? ?i (z)). The dual of the
regularized ERM problem is (see, e.g., [19])
n
1X
1
?
?
maximize
D(x),
with
D(x)
=
Ax
,
??
(?x
)
?
?g
i
i
x?Rn
n i=1
?n
def
where A = [A1 , . . . , An ]. This is equivalent to minimize F (x) = ?D(x), that is,
n
X
1
def 1
?
?
F
(x)
=
minimize
Ax
.
?
(?x
)
+
?g
i
x?Rn
n i=1 i
?n
1
Ax and
The structure of F (x) above matches the formulation in (1) and (2) with f (x) = ?g ? ?n
?i (xi ) = n1 ??i (?xi ), and we can apply the APCG method to minimize F (x). In order to exploit
the fast linear convergence rate, we make the following assumption.
Assumption 3. Each function ?i is 1/? smooth, and the function g has unit convexity parameter 1.
Here we slightly abuse the notation by overloading ?, which also appeared in Algorithm 1. But
in this section it solely represents the (inverse) smoothness parameter of ?i . Assumption 3 implies
that each ??i has strong convexity parameter ? (with respect to the local Euclidean norm) and g ?
is differentiable and ?g ? has Lipschitz constant 1. In the following, we split the function F (x) =
f (x) + ?(x) by relocating the strong convexity term as follows:
n
1
?
1 X ?
?
?
f (x) = ?g
(15)
Ax +
kxk2 ,
?(x) =
? (?xi ) ? kxi k2 .
?n
2n
n i=1
2
As a result, the function f is strongly convex and each ?i is still convex. Now we can apply the
APCG method to minimize F (x) = ?D(x), and obtain the following guarantee.
Theorem 2. Suppose Assumption 3 holds and kAi k ? R for all i = 1, . . . , n. In order to obtain an
expected dual optimality gap E[D? ? D(x(k) )] ? ? by using the APCG method, it suffices to have
q
2
k ? n + nR
log(C/?).
(16)
??
where D? = maxx?Rn D(x) and the constant C = D? ? D(x(0) ) + (?/(2n))kx(0) ? x? k2 .
2
2
?
R +??n
ik
Proof. The function f (x) in (15) has coordinate Lipschitz constants Li = kA
?n2 + n ?
?n2
?
and convexity parameter n with respect to the unweighted Euclidean norm. The strong convexity
parameter of f (x) with respect to the norm k ? kL defined in(3) is
. 2
+??n
= R2??n
? = n? R ?n
2
+??n .
?
? k
?
?
According to Theorem 1, we have E[D? ?D(x(0) )] ? 1 ? n
C ? exp ? n k C. Therefore
it suffices to have the number of iterations k to be larger than
q
q
q
R2 +??n
nR2
2 + nR2 log(C/?) ?
?n log(C/?) = n
log(C/?).
log(C/?)
=
n
+
n
?
??n
??
??
This finishes the proof.
Several state-of-the-art algorithms for ERM, including SDCA [19], SAG [15, 17] and SVRG [7, 23]
obtain the iteration complexity
2
O n+ R
log(1/?)
.
(17)
??
We note that our result in (16) can be much better for ill-conditioned problems, i.e., when the condi2
tion number R
?? is larger than n. This is also confirmed by our numerical experiments in Section 4.
The complexity bound in (17) for the aforementioned work is for minimizing the primal objective
P (x) or the duality gap P (x) ? D(x), but our result in Theorem 2 is in terms of the dual optimality.
In the full report [9], we show that the same guarantee on accelerated primal-dual convergence can be
obtained by our method with an extra primal gradient step, without affecting the overall complexity.
The experiments in Section 4 illustrate superior performance of our algorithm on reducing the primal
objective value, even without performing the extra step.
6
We note that Shalev-Shwartz and Zhang
an accelerated SDCA method
[20]qrecently
developed
n
which achieves the same complexity O n + ?? log(1/?) as our method. Their method calls
the SDCA method in a full-dimensional accelerated gradient method in an inner-outer iteration procedure. In contrast, our APCG method is a straightforward single loop coordinate gradient method.
3.1
Implementation details
Here we show how to exploit the structure of the regularized ERM problem to efficiently compute
the coordinate gradient ?ik f (y (k) ), and totally avoid full-dimensional updates in Algorithm 4. We
focus on the special case g(w) = 21 kwk2 and show how to compute ?ik f (y (k) ). According to (15),
?ik f (y (k) ) =
1
T
(k)
)
?n2 Ai (Ay
(k)
+ n? yik .
Since we do not form y (k) in Algorithm 4, we update Ay (k) by storing and updating two vectors
in Rd : p(k) = Au(k) and q (k) = Av (k) . The resulting method is detailed in Algorithm 5.
Algorithm 5: APCG for solving dual ERM
Input: x
(0)
? dom (?) and convexity parameter ? > 0.
?
?
(0)
Initialize: set ? = n and ? = 1??
= 0, v (0) = x(0) , p(0) = 0 and q (0) = Ax(0) .
1+? , and let u
Iterate: repeat for k = 0, 1, 2, . . .
1. Choose ik ? {1, . . . , n} uniformly at random, compute the coordinate gradient
(k)
(k)
(k)
?ik = ?n1 2 ?k+1 ATik p(k) + ATik q (k) + n? ?k+1 uik + vik .
2. Compute coordinate increment
o
n
?(kAik k2 +??n)
(k)
(k)
(k)
(k)
k?k2 + h?ik , ?i + n1 ??ik (?k+1 uik ? vik ? ?) .
?ik = arg min
2?n
3. Let u
??R
(k+1)
Ni
k
= u(k) and v (k+1) = v (k) , and update
(k+1)
uik
(k)
= uik ?
p(k+1) = p(k) ?
1?n? (k)
? ,
2?k+1 ik
(k+1)
vik
(k)
1?n?
A ? ,
2?k+1 ik ik
Output: approximate primal and dual solutions
1
w(k+1) = ?n
?k+2 p(k+1) + q (k+1) ,
(k)
= vik +
q (k+1) = q (k) +
1+n? (k)
2 ?ik ,
(k)
1+n?
2 Aik ?ik .
(18)
x(k+1) = ?k+1 u(k+1) + v (k+1) .
Each iteration of Algorithm 5 only involves the two inner products ATik p(k) , ATik q (k) in computing
(k)
?ik and the two vector additions in (18). They all cost O(d) rather than O(n). When the Ai ?s are
sparse (the case of most large-scale problems), these operations can be carried out very efficiently.
Basically, each iteration of Algorithm 5 only costs twice as much as that of SDCA [6, 19].
4
Experiments
In our experiments, we solve ERM problems with smoothed hinge loss for binary classification.
That is, we pre-multiply each feature vector Ai by its label bi ? {?1} and use the loss function
?
if a ? 1,
? 0
?
1
?
a
?
if a ? 1 ? ?,
?(a) =
2
? 1 (1 ? a)
2
otherwise.
2?
The conjugate function of ? is ?? (b) = b + ?2 b2 if b ? [?1, 0] and ? otherwise. Therefore we have
?xi
?
1 ?
if xi ? [0, 1]
n
? (?xi ) ? kxi k2 =
?i (xi ) =
?
otherwise.
n
2
The dataset used in our experiments are summarized in Table 1.
7
?
?5
10
rcv1
100
100
10?3
10?3
10?3
10?6
10?6
10?6
AFG
SDCA
APCG
10?12
10?15
0
20
40
60
80
10?12
100
10?15
0
20
40
60
80
10?12
100
10?15
0
100
10?3
10?3
10?3
10?6
10?6
10?6
20
40
60
80
100
10?9
0
20
40
60
80
AFG
SDCA
APCG
10?9
100
100
10?9
0
100
100
100
10?1
10?2
10?3
10?4
10?5
10?6
0
10?1
10?2
10?3
10?4
10?5
10?6
0
10?1
10?2
10?3
10?4
10?5
10?6
0
20
40
60
80
100
100
AFG
SDCA
APCG
10?1
10?8
AFG
SDCA
APCG
10?9
100
10?9
0
10?7
news20
100
10?9
10?6
covertype
10?2
20
40
60
80
100
100
100
10?1
10?1
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
10?2
10?2
10?3
10?3
10?3
10?4
0
20
40
60
80
100
10?4
0
10?4
20
40
60
80
100
10?5
0
Figure 1: Comparing the APCG method with SDCA and the accelerated full gradient method (AFG)
with adaptive line search. In each plot, the vertical axis is the primal objective gap P (w(k) )?P ? , and
the horizontal axis is the number of passes through the entire dataset. The three columns correspond
to the three datasets, and each row corresponds to a particular value of the regularization parameter ?.
In our experiments, we compare the APCG method with SDCA and the accelerated full gradient
method (AFG) [12] with an additional line search procedure to improve efficiency. When the regularization parameter ? is not too small (around 10?4 ), then APCG performs similarly as SDCA as
predicted by our complexity results, and they both outperform AFG by a substantial margin.
Figure 1 shows the results in the ill-conditioned setting, with ? varying form 10?5 to 10?8 . Here
we see that APCG has superior performance in reducing the primal objective value compared with
SDCA and AFG, even though our theory only gives complexity for solving the dual ERM problem.
AFG eventually catches up for cases with very large condition number (see the plots for ? = 10?8 ).
datasets
rcv1
covtype
news20
number of samples n
20,242
581,012
19,996
number of features d
47,236
54
1,355,191
sparsity
0.16%
22%
0.04%
Table 1: Characteristics of three binary classification datasets (available from the LIBSVM web
page: http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets).
8
References
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9
| 5356 |@word msr:2 version:3 norm:6 hsieh:2 tr:2 reduction:2 cyclic:3 series:1 bc:1 ati:1 existing:1 ka:1 com:1 comparing:1 luo:2 numerical:2 partition:2 plot:2 update:9 zik:1 ith:1 iterates:1 zhang:5 mathematical:2 direct:1 ik:40 introductory:1 news20:2 expected:2 inspired:1 cardinality:1 totally:1 moreover:2 notation:3 developed:4 guarantee:2 every:1 concave:1 sag:1 exactly:1 k2:11 unit:1 local:2 consequence:1 subscript:1 solely:1 abuse:1 inria:1 twice:1 au:1 studied:1 bi:5 block:20 procedure:2 sdca:15 empirical:4 maxx:1 composite:5 pre:1 get:1 cannot:1 uiowa:1 convenience:2 risk:4 www:1 equivalent:2 maxz:1 demonstrated:1 straightforward:1 attention:1 convex:20 simplicity:1 roux:2 coordinate:42 increment:1 updated:4 suppose:2 aik:1 programming:2 associate:1 updating:1 kxk1:1 csie:1 wang:1 richt:4 counter:1 substantial:1 convexity:15 complexity:10 ui:4 nesterov:6 dom:7 depend:1 solving:9 predictive:1 upon:1 efficiency:2 completely:1 afg:9 fast:2 describe:1 choosing:2 shalev:4 whose:1 kai:1 solve:4 larger:2 otherwise:4 advantage:1 differentiable:3 sequence:2 product:1 loop:1 realization:1 ky:1 r02:3 convergence:19 illustrate:2 develop:3 ac:1 montreal:1 received:1 strong:7 predicted:1 involves:1 implies:3 greek:1 stochastic:9 libsvmtools:1 suffices:2 ntu:1 proposition:2 extension:1 hold:4 sufficiently:1 around:1 exp:2 mapping:1 achieves:3 label:2 largest:1 city:1 weighted:1 minimization:11 arik:4 aim:1 rather:1 avoid:5 shrinkage:1 varying:1 corollary:1 ax:5 focus:1 contrast:1 i0:1 burnaby:1 entire:1 tak:1 france:1 i1:1 arg:6 dual:14 ill:4 overall:2 denoted:1 aforementioned:1 classification:2 art:3 special:3 initialize:7 sampling:1 represents:2 broad:1 progressive:1 icml:3 nonsmooth:1 report:6 saha:1 beck:2 keerthi:1 microsoft:4 n1:5 huge:1 multiply:1 zhaosong:2 primal:7 partial:3 condi2:1 euclidean:3 theoretical:1 column:1 modeling:1 teboulle:1 sidford:3 cost:6 nr2:2 johnson:1 too:1 kn:1 proximal:12 kxi:3 chooses:1 st:1 international:3 randomized:8 siam:4 lee:3 together:2 choose:8 li:5 potential:1 b2:1 summarized:1 includes:2 coordinated:1 satisfy:1 depends:1 tion:1 closed:2 capability:1 parallel:2 simon:1 minimize:6 ni:5 became:1 variance:2 characteristic:1 efficiently:2 yield:1 ofthe:1 correspond:1 basically:1 lu:4 confirmed:1 explain:1 proof:3 recovers:1 proved:1 dataset:2 popular:1 knowledge:1 improves:1 manuscript:3 higher:1 improved:2 formulation:2 though:1 strongly:6 horizontal:1 web:1 logistic:1 indicated:1 hal:1 usa:2 regularization:4 hong:1 ay:2 ridge:1 performs:1 cp:1 wise:1 recently:2 common:1 superior:2 extend:1 kluwer:1 kwk2:1 sundararajan:1 ai:4 smoothness:1 approx:3 unconstrained:2 rd:4 similarly:1 recent:2 inequality:1 binary:2 success:1 kwk1:1 yi:3 devise:2 additional:1 r0:1 maximize:1 lik:2 full:18 reduces:1 smooth:3 technical:3 match:3 faster:3 bach:2 lin:7 devised:1 concerning:1 fraser:1 a1:2 variant:1 regression:3 basic:1 arxiv:6 iteration:19 represent:2 addition:3 affecting:1 extra:2 rest:1 ascent:4 pass:1 kwk22:2 call:1 split:1 iterate:5 finish:1 gave:1 zi:5 lasso:1 reduce:1 inner:2 vik:7 accelerating:1 proceed:1 cause:1 yik:2 tewari:2 detailed:1 http:1 outperform:1 qihang:2 notice:1 relocating:1 arising:1 per:1 threshold:1 k4:1 libsvm:1 imaging:1 year:1 sum:1 inverse:2 letter:2 powerful:1 sfu:1 def:4 hi:5 bound:1 covertype:1 constraint:2 precisely:1 generates:1 u1:1 speed:1 fercoq:3 min:8 optimality:2 performing:1 separable:2 rcv1:2 according:3 combination:1 conjugate:2 slightly:1 partitioned:1 tw:1 erm:13 equation:1 eventually:1 cjlin:1 available:1 operation:7 apply:4 observe:2 schmidt:2 tetruashvili:1 denotes:1 hinge:1 exploit:4 objective:6 strategy:1 costly:1 nr:1 gradient:27 kth:1 subspace:1 minx:1 outer:1 nondifferentiable:1 tseng:3 modeled:1 relationship:1 minimizing:5 sharper:1 xik:5 atik:4 implementation:8 unknown:1 perform:2 av:1 observation:1 vertical:1 datasets:4 finite:2 descent:23 immediate:1 rn:18 smoothed:1 canada:2 unpublished:1 paris:1 kl:3 extensive:1 redmond:1 appeared:1 sparsity:1 max:1 including:1 ia:1 suitable:1 regularized:11 indicator:1 scheme:6 improve:1 numerous:1 axis:2 carried:1 catch:1 literature:1 l2:1 asymptotic:1 loss:6 lecture:1 permutation:1 sublinear:2 iowa:2 rni:5 xiao:6 storing:1 row:1 course:1 repeat:5 keeping:1 svrg:1 enjoys:1 understand:1 sparse:1 benefit:1 apcg:31 uit:2 unweighted:1 collection:1 adaptive:1 simplified:3 far:1 approximate:1 obtains:2 global:1 xi:21 shwartz:4 un:1 continuous:1 search:2 iterative:1 table:2 ca:1 dense:1 whole:1 n2:4 razaviyayn:1 uik:7 sub:1 khi:2 exponential:1 kxk2:1 jmlr:1 theorem:8 zu:1 r2:2 svm:2 covtype:1 exists:1 overloading:1 ci:3 conditioned:4 kx:2 margin:1 gap:3 boston:1 scalar:1 chang:2 corresponds:2 satisfies:1 acceleration:1 lipschitz:3 change:1 specifically:1 uniformly:4 reducing:2 called:1 duality:1 accepted:1 support:1 accelerated:26 |
4,812 | 5,357 | Inference by Learning: Speeding-up Graphical
Model Optimization via a Coarse-to-Fine Cascade of
Pruning Classifiers
Bruno Conejo?
GPS Division, California Institute of Technology, Pasadena, CA, USA
Universite Paris-Est, Ecole des Ponts ParisTech, Marne-la-Vallee, France
[email protected]
Nikos Komodakis
Universite Paris-Est, Ecole des Ponts ParisTech, Marne-la-Vallee, France
[email protected]
Sebastien Leprince & Jean Philippe Avouac
GPS Division, California Institute of Technology, Pasadena, CA, USA
[email protected] [email protected]
Abstract
We propose a general and versatile framework that significantly speeds-up graphical model optimization while maintaining an excellent solution accuracy. The
proposed approach, refereed as Inference by Learning or in short as IbyL, relies
on a multi-scale pruning scheme that progressively reduces the solution space by
use of a coarse-to-fine cascade of learnt classifiers. We thoroughly experiment
with classic computer vision related MRF problems, where our novel framework
constantly yields a significant time speed-up (with respect to the most efficient
inference methods) and obtains a more accurate solution than directly optimizing
the MRF. We make our code available on-line [4].
1
Introduction
Graphical models in computer vision Optimization of undirected graphical models such as
Markov Random Fields, MRF, or Conditional Random Fields, CRF, is of fundamental importance in
computer vision. Currently, a wide spectrum of problems including stereo matching [25, 13], optical flow estimation [27, 16], image segmentation [23, 14], image completion and denoising [10], or,
object recognition [8, 2] rely on finding the mode of the distribution associated to the random field,
i.e., the Maximum A Posteriori (MAP) solution. The MAP estimation, often referred as the labeling
problem, is posed as an energy minimization task. While this task is NP-Hard, strong optimum solutions or even the optimal solutions can be obtained [3]. Over the past 20 years, tremendous progress
has been made in term of computational cost, and, many different techniques have been developed
such as move making approaches [3, 19, 22, 21, 28], and message passing methods [9, 32, 18, 20].
A review of their effectiveness has been published in [31, 12]. Nevertheless, the ever increasing
dimensionality of the problems and the need for larger solution space greatly challenge these tech?
This work was supported by USGS through the Measurements of surface ruptures produced by continental
earthquakes from optical imagery and LiDAR project (USGS Award G13AP00037), the Terrestrial Hazard
Observation and Reporting Center of Caltech, and the Moore foundation through the Advanced Earth Surface
Observation Project (AESOP Grant 2808).
1
niques as even the best ones have a highly super-linear computational cost and memory requirement
relatively to the dimensionality of the problem.
Our goal in this work is to develop a general MRF optimization framework that can provide a
significant speed-up for such methods while maintaining the accuracy of the estimated solutions.
Our strategy for accomplishing this goal will be to gradually reduce (by a significant amount) the
size of the discrete state space via exploiting the fact that an optimal labeling is typically far from
being random. Indeed, most MRF optimization problems favor solutions that are piecewise smooth.
In fact, this spatial structure of the MAP solution has already been exploited in prior work to reduce
the dimensionality of the solution space.
Related work A first set of methods of this type, referred here for short as the super-pixel approach
[30], defines a grouping heuristic to merge many random variables together in super-pixels. The
grouping heuristic can be energy-aware if it is based on the energy to minimize as in [15], or, energyagnostic otherwise as in [7, 30]. All random variables belonging to the same super-pixel are forced
to take the same label. This restricts the solution space and results in an optimization speed-up as
a smaller number of variables needs to be optimized. The super-pixel approach has been applied
with segmentation, stereo and object recognition [15]. However, if the grouping heuristic merges
variables that should have a different label in the MAP solution, only an approximate labeling is
computed. In practice, defining general yet efficient grouping heuristics is difficult. This represents
the key limitation of super-pixel approaches.
One way to overcome this limitation is to mimic the multi-scale scheme used in continuous optimization by building a coarse to fine representation of the graphical model. Similarly to the superpixel approach, such a multi-scale method, relies again on a grouping of variables for building the
required coarse to fine representation [17, 24, 26]. However, contrary to the super-pixel approach,
if the grouping merges variables that should have a different label in the MAP solution, there always exists a scale at which these variables are not grouped. This property thus ensures that the
MAP solution can still be recovered. Nevertheless, in order to manage a significant speed-up of
the optimization, the multi-scale approach also needs to progressively reduce the number of labels
per random variable (i.e., the solution space). Typically, this is achieved by use, for instance, of a
heuristic that keeps only a small fixed number of labels around the optimal label of each node found
at the current scale, while pruning all other labels, which are therefore not considered thereafter [5].
This strategy, however, may not be optimal or even valid for all types of problems. Furthermore,
such a pruning heuristic is totally inappropriate (and can thus lead to errors) for nodes located along
discontinuity boundaries of an optimal solution, where such boundaries are always expected to exist
in practice. An alternative strategy followed by some other methods relies on selecting a subset of
the MRF nodes at each scale (based on some criterion) and then fixing their labels according to the
optimal solution estimated at the current scale (essentially, such methods contract the entire label
set of a node to a single label). However, such a fixing strategy may be too aggressive and can also
easily lead to eliminating good labels.
Proposed approach Our method simultaneously makes use of the following two strategies for
speeding-up the MRF optimization process:
(i) it solves the problem through a multi-scale approach that gradually refines the MAP estimation based on a coarse-to-fine representation of the graphical model,
(ii) and, at the same time, it progressively reduces the label space of each variable by cleverly
utilizing the information computed during the above coarse-to-fine process.
To achieve that, we propose to significantly revisit the way that the pruning of the solution space
takes place. More specifically:
(i) we make use of and incorporate into the above process a fine-grained pruning scheme that
allows an arbitrary subset of labels to be discarded, where this subset can be different for
each node,
(ii) additionally, and most importantly, instead of trying to manually come up with some criteria
for deciding what labels to prune or keep, we introduce the idea of relying entirely on
a sequence of trained classifiers for taking such decisions, where different classifiers per
scale are used.
2
We name such an approach Inference by Learning, and show that it is particularly efficient and effective in reducing the label space while omitting very few correct labels. Furthermore, we demonstrate
that the training of these classifiers can be done based on features that are not application specific
but depend solely on the energy function, which thus makes our approach generic and applicable
to any MRF problem. The end result of this process is to obtain both an important speed-up and
a significant decrease in memory consumption as the solution space is progressively reduced. Furthermore, as each scale refines the MAP estimation, a further speed-up is obtained as a result of a
warm-start initialization that can be used when transitioning between different scales.
Before proceeding, it is worth also noting that there exists a body of prior work [29] that focuses on
fixing the labels of a subset of nodes of the graphical model by searching for a partial labeling with
the so-called persistency property (which means that this labeling is provably guaranteed to be part
of an optimal solution). However, finding such a set of persistent variables is typically very time
consuming. Furthermore, in many cases only a limited number of these variables can be detected.
As a result, the focus of these works is entirely different from ours, since the main motivation in our
case is how to obtain a significant speed-up for the optimization.
Hereafter, we assume without loss of generality that the graphical model is a discrete pairwise
CRF/MRF. However, one can straightforwardly apply our approach to higher order models.
Outline of the paper We briefly review the optimization problem related to a discrete pairwise
MRF and introduce the necessary notations in section 2. We describe our general multi-scale pruning
framework in section 3. We explain how classifiers are trained in section 4. Experimental results
and their analysis are presented in 5. Finally, we conclude the paper in section 6.
2
Notation and preliminaries
To represent a discrete MRF model M, we use the following notation
M = V, E, L, {?i }i?V , {?ij }(i,j)?E .
(1)
Here V and E represent respectively the nodes and edges of a graph, and L represents a discrete
label set. Furthermore, for every i ? V and (i, j) ? E, the functions ?i : L ? R and ?ij : L2 ? R
represent
and pairwise costs (that are also known connectively as MRF potentials
respectively unary
? = {?i }i?V , {?ij }(i,j)?E ). A solution x = (xi )i?V of this model consists of one variable per
vertex i, taking values in the label set L, and the total cost (energy) E(x|M) of such a solution is
E(x|M) =
X
X
?i (xi ) +
i?V
?ij (xi , xj ) .
(i,j)?E
The goal of MAP estimation is to find a solution that has minimum energy, i.e., computes
xMAP = arg min E(x|M) .
x?L|V|
The above minimization takes place over the full solution space of model M, which is L|V| . Here
we will also make use of a pruned solution space S(M, A), which is defined based on a binary
function A : V ? L ? {0, 1} (referred to as the pruning matrix hereafter) that specifies the status
(active or pruned) of a label for a given vertex, i.e.,
1
if label l is active at vertex i
A(i, l) =
(2)
0
if label l is pruned at vertex i
During optimization, active labels are retained while pruned labels are discarded. Based on a given
A, the corresponding pruned solution space of model M is defined as
n
o
S(M, A) = x ? L|V| | (?i), A(i, xi ) = 1 .
3
Multiscale Inference by Learning
In this section we describe the overall structure of our MAP estimation framework, beginning by
explaining how to construct the coarse-to-fine representation of the input graphical model.
3
3.1
Model coarsening
Given a model M (defined as in (1)), we wish to create a ?coarser? version of this model M0 =
V 0 , E 0 , L, {?0i }i?V 0 , {?0ij }(i,j)?E 0 . Intuitively, we want to partition the nodes of M into groups,
and treat each group as a single node of the coarser model M0 (the implicit assumption is that nodes
of M that are grouped together are assigned the same label). To that end, we will make use of a
grouping function g : V ? N . The nodes and edges of the coarser model are then defined as follows
V 0 = {i0 | ?i ? V, i0 = g(i)} ,
0
0
0
(3)
0
0
0
0
E = {(i , j ) | ?(i, j) ? E, i = g(i), j = g(j), i 6= j } .
(4)
Furthermore, the unary and pairwise potentials of M0 are given by
(?i0 ? V 0 ),
Figure 1: Black circles:
V, Black lines: E, Red
squares: V 0 , Blue lines:
E 0.
(?(i0 , j 0 ) ? E 0 ),
?0i0 (l)
?0i0 j 0 (l0 , l1 ) =
P
= P i?V|i0 =g(i)
+ (i,j)?E|i0 =g(i)=g(j)
X
(i,j)?E|i0 =g(i),j 0 =g(j)
?i (l)
, (5)
?ij (l, l)
?ij (l0 , l1 ) . (6)
With a slight abuse of notation, we will hereafter use g(M) to denote the coarser model resulting
from M when using the grouping function g, i.e., we define g(M) = M0 . Also, given a solution x0
of M0 , we can ?upsample? it into a solution x of M by setting xi = x0g(i) for each i ? V. We will
use the following notation in this case: g ?1 (x0 ) = x. We provide a toy example in supplementary
materials.
3.2
Coarse-to-fine optimization and label pruning
To estimate the MAP of an input model M, we first construct a series of N +1 progressively coarser
models (M(s) )0?s?N by use of a sequence of N grouping functions (g (s) )0?s<N , where
M(0) = M and (?s), M(s+1) = g (s) (M(s) ) .
This provides a multiscale (coarse-to-fine) representation of the original model., where the elements
of the resulting models are denoted as follows:
(s)
(s)
M(s) = V (s) , E (s) , L, {?i }i?V (s) , {?ij }(i,j)?E (s)
In our framework, MAP estimation proceeds from the coarsest to the finest scale (i.e., from model
M(N ) to M(0) ). During this process, a pruning matrix A(s) is computed at each scale s, which is
used for defining a restricted solution space S(M(s) , A(s) ). The elements of the matrix A(N ) at the
coarsest scale are all set equal to 1 (i.e., no label pruning is used in this case), whereas in all other
scales A(s) is computed by use of a trained classifier f (s) .
More specifically, at any given scale s, the following steps take place:
i. We approximately minimize (via any existing MRF optimization method) the energy of the
model M(s) over the restricted solution space S(M(s) , A(s) ), i.e., we compute
x(s) ? arg minx?S(M(s) ,A(s) ) E(x|M(s) ) .
ii. Given the estimated solution x(s) , a feature map z (s) : V (s) ? L ? RK is computed at
the current scale, and a trained classifier f (s) : RK ? {0, 1} uses this feature map z (s) to
construct the pruning matrix A(s?1) for the next scale as follows
(?i ? V (s?1) , ?l ? L), A(s?1) (i, l) = f (s) (z (s) (g (s?1) (i), l)) .
iii. Solution x(s) is ?upsampled? into x(s?1) = [g (s?1) ]?1 (x(s) ) and used as the initialization for the optimization at the next scale s ? 1. Note that, due to (5) and (6), it holds
E(x(s?1) |M(s?1) ) = E(x(s) |M(s) ). Therefore, this initialization ensures that energy will
continually decrease if the same is true for the optimization applied per scale. Furthermore,
it can allow for a warm-starting strategy when transitioning between scales.
The pseudocode of the resulting algorithm appears in Algo. 1.
4
Algorithm 1: Inference by learning framework
Data: Model M, grouping functions (g (s) )0?s<N , classifiers (f (s) )0<s?N
Result: x(0)
Compute the coarse to fine sequence of MRFs:
M(0) ? M
for s = [0 . . . N ? 1] do
M(s+1) ? g (s) (M(s) )
Optimize the coarse to fine sequence of MRFs over pruned solution spaces:
(?i ? V (N ) , ?l ? L), A(N ) (i, l) ? 1
Initialize x(N )
for s = [N...0] do
Update x(s) by iterative minimization: x(s) ? arg minx?S(M(s) ,A(s) ) E(x|M(s) )
if s 6= 0 then
Compute feature map z (s)
Update pruning matrix for next finer scale: A(s?1) (i, l) = f (s) (z (s) (g (s?1) (i), l))
Upsample x(s) for initializing solution x(s?1) at next scale: x(s?1) ? [g (s?1) ]?1 (x(s) )
4
Features and classifier for label pruning
For each scale s, we explain how the set of features comprising the feature map z (s) is computed
and how we train (off-line) the classifier f (s) . This is a crucial step for our approach. Indeed, if the
classifier wrongly prunes labels that belong to the MAP solution, then, only an approximate labeling
might be found at the finest scale. Moreover, keeping too many active labels will result in a poor
speed-up for MAP estimation.
4.1
Features
The feature map z (s) : V (s) ? L ? RK is formed by stacking K individual real-valued features
defined on V (s) ? L. We propose to compute features that are not application specific but depend
solely on the energy function and the current solution x(s) . This makes our approach generic and
applicable to any MRF problem. However, as we establish a general framework, specific application
features can be straightforwardly added in future work.
Presence of strong discontinuity This binary feature, PSD(s) , accounts for the existence of dis(s)
(s)
continuity in solution x(s) when a strong link (i.e., ?ij (xi , xj ) > ?) exists between neighbors.
Its definition follows for any vertex i ? V (s) and any label l ? L :
(s)
(s)
1
?(i, j) ? E (s) | ?ij (xi , xj ) > ?
PSD(s) (i, l) =
(7)
0
otherwise
Local energy variation This feature represents the local variation of the energy around the current
solution x(s) . It accounts for both the unary and pairwise terms associated to a vertex and a label.
As in [11], we remove the local energy of the current solution as it leads to a higher discriminative
power. The local energy variation feature, LEV(s) , is defined for any i ? V (s) and l ? L as follows:
(s)
LEV
(s)
(i, l) =
(s)
(s)
NV (i)
(s)
(s)
?i (l) ? ?i (xi )
(s)
(s)
(s)
(s)
X
?ij (l, xj ) ? ?ij (xi , xj )
j:(i,j)?E (s)
NE (i)
+
(s)
(s)
(8)
(s)
with NV (i) = card{i0 ? V (s?1) : g (s?1) (i0 ) = i} and NE (i) = card{(i0 , j 0 ) ? E (s?1) :
g (s?1) (i0 ) = i, g (s?1) (j 0 ) = j}.
Unary ?coarsening? This feature, UC(s) , aims to estimate an approximation of the coarsening
induced in the MRF unary terms when going from model M(s?1) to model M(s) , i.e., as a result of
5
applying the grouping function g (s?1) . It is defined for any i ? V (s) and l ? L as follows
(s?1)
UC(s) (i, l) =
|?i0
X
i0 ?V (s?1) |g (s?1) (i0 )=i
(s)
(l) ?
?i (l)
(s)
NV (i)
(s)
|
(9)
NV (i)
Feature normalization The features are by design insensitive to any additive term applied on all
the unary and pairwise terms. However, we still need to apply a normalization to the LEV(s) and
UC(s) features to make them insensitive to any positive global scaling factor applied on both the
unary and pairwise terms (such scaling variations are commonly used in computer vision). Hence,
we simply divide group of features, LEV(s) and UC(s) by their respective mean value.
4.2
Classifier
To train the classifiers, we are given as input a set of MRF instances (all of the same class, e.g.,
stereo-matching) along with the ground truth MAP solutions. We extract a subset of MRFs for offline learning and a subset for on-line testing. For each MRF instance in the training set, we apply
the algorithm 1 without any pruning (i.e., A(s) ? 1) and, at each scale, we keep track of the features
(s)
z (s) and also compute the binary function XMAP : V (s) ? L ? {0, 1} defined as follows:
1, if l is the ground truth label for node i
(0)
(?i ? V, ?l ? L), XMAP (i, l) =
0, otherwise
_
(s)
(s?1)
(?s > 0)(?i ? V (s) , ?l ? L), XMAP (i, l) =
XMAP (i0 , l) ,
i0 ?V (s?1) :g (s) (i0 )=i
(s)
W
where denotes the binary OR operator. The values 0 and 1 in XMAP define respectively the two
classes c0 and c1 when training the classifier f (s) , where c0 means that the label can be pruned and
c1 that the label should not be pruned.
To treat separately the nodes that are on the border of a strong discontinuity, we split the feature map
(s)
(s)
(s)
(s)
z (s) into two groups z0 and z1 , where z0 contains only features where PSD(s) = 0 and z1
contains only features where PSD(s) = 1 (strong discontinuity). For each group, we train a standard
linear C-SVM classifier with l2 -norm regularization (regularization parameter was set to C = 10).
The linear classifiers give good enough accuracy during training while also being fast to evaluate at
test time
During training (and for each group), we also introduce weights to balance the different number of
elements in each class (c0 is much larger than c1 ), and to also strongly penalize misclassification in
c1 (as such misclassification can have a more drastic impact on the accuracy of MAP estimation). To
card(c0 )
accomplish that, we set the weight for class c0 to 1, and the weight for class c1 to ? card(c
, where
1)
card(?) counts the number of training samples in each class. Parameter ? is a positive scalar (common to both groups) used for tuning the penalization of misclassification in c1 (it will be referred
to as the pruning aggressiveness factor hereafter as it affects the amount of labels that get pruned).
During on-line testing, depending on the value of the PSD feature, f (s) applies the linear classifier
(s)
(s)
learned on group z0 if PSD(s) = 0, or the linear classifier learned on group z1 if PSD(s) = 1.
5
Experimental results
We evaluate our framework on pairwise MRFs from stereo-matching, image restoration, and, optical
flow estimation problems. The corresponding MRF graphs consist of regular 4-connected grids in
this case. At each scale, the grouping function merges together vertices of 2 ? 2 subgrids. We leave
more advanced grouping functions [15] for future work. As MRF optimization subroutine, we use
the Fast-PD algorithm [21]. We make our code available on-line [4].
Experimental setup For the stereo matching problem, we estimate the disparity map from images
IR and IL where each label encodes a potential disparity d (discretized at quarter of a pixel precision), with MRF potentials ?p (d) = ||IL (yp , xp )?IR (yp , xp ?d)||1 and ?pq (d0 , d1 ) = wpq |d0 ?d1 |,
with the weight wpq varying based on the image gradient (parameters are adjusted for each sequence). We train the classifier on the well-known Tsukuba stereo-pair (61 labels), and use all other
6
(a) Speed-up
(b) Active label ratio
(c) Energy ratio
(d) Label agreement
Figure 2: Performance of our Inference by Learning framework: (Top row) stereo matching, (Middle row)
optical flow, (Bottom row) image restoration. For stereo matching, the Average Middlebury curve represents
the average value of the statistic for the entire Middlebury dataset [6] (2001, 2003, 2005 and 2006) (37 stereopairs).
stereo-pairs of [6] (2001, 2003, 2005 and 2006) for testing. For image restoration, we estimate the
pixel intensity of a noisy and incomplete image I with MRF potentials ?p (l) = ||I(yp , xp ) ? l||22
and ?(l0 , l1 ) = 25 min(||l0 ? l1 ||22 , 200). We train the classifier on the Penguin image stereo-pair
(256 labels), and use House (256 labels) for testing (dataset [31]). For the optical flow estimation,
we estimate a subpixel-accurate 2D displacement field between two frames by extending the stereo
matching formulation to 2D. Using the dataset of [1], we train the classifier on Army (1116 labels),
and test on RubberWhale (625 labels) and Dimetrodon (483 labels). For all experiments, we use 5
scales and set in (7) ? = 5w
?pq with w
?pq being the mean value of edge weights.
Evaluations We evaluate three optimization strategies: the direct optimization (i.e., optimizing
the full MRF at the finest scale), the multi-scale optimization (? = 0, i.e., our framework without
any pruning), and our Inference by Learning optimization, where we experiment with different error
ratios ? that range between 0.001 and 1.
We assess the performance by computing the energy ratio, i.e., the ratio between the current energy
and the energy computed by the direct optimization, the best label agreement, i.e., the proportion
of labels that coincides with the labels of the lowest computed energy, the speed-up factor, i.e., the
ratio of computation time between the direct optimization and the current optimization strategy, and,
the active label ratio, i.e., the percentage of active labels at the finest scale.
Results and discussion For all problems, we present in Fig. 2 the performance of our Inference
by Learning approach for all tested aggressiveness factors and show in Fig. 3 estimated results for
? = 0.01. We present additional results in the supplementary material.
For every problem and aggressiveness factors until ? = 0.1, our pruning-based optimization obtains
a lower energy (column (c) of Fig. 2) in less computation time, achieving a speed-up factor (column
(a) of Fig. 2) close to 5 for Stereo-matching, above 10 for Optical-flow and up to 3 for image
restoration. (note that these speed-up factors are with respect to an algorithm, FastPD, that was the
most efficient one in recent comparisons [12]). The percentage of active labels (Fig. 2 column (b))
strongly correlates with the speed-up factor. The best labeling agreement (Fig. 2 column (d)) is
never worse than 97% (except for the image restoration problems because of the in-painted area)
7
Tsukuba
Venus
Teddy
Army
Dimet.
House
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: Results of our Inference by Learning framework for ? = 0.1. Each row is a different MRF problem.
(a) original image, (b) ground truth, (c) solution of the pruning framework, (d,e,f) percentage of active labels
per vertex for scale 0, 1 and 2 (black 0%, white 100%).
and is always above 99% for ? 6 0.1. As expected, less pruning happens near label discontinuities
as illustrated in column (d,e,f) of Fig. 3 justifying the use of a dedicated linear classifier. Moreover,
large homogeneously labeled regions are pruned earlier in the coarse to fine scale.
6
Conclusion and future work
Our Inference by Learning approach consistently speeds-up the graphical model optimization by
a significant amount while maintaining an excellent accuracy of the labeling estimation. On most
experiments, it even obtains a lower energy than direct optimization.
In future work, we plan to experiment with problems that require general pairwise potentials where
message-passing techniques can be more effective than graph-cut based methods but are at the same
time much slower. Our framework is guaranteed to provide an even more dramatic speedup in this
case since the computational complexity of message-passing methods is quadratic with respect to
the number of labels while being linear for graph-cut based methods used in our experiments. We
also intend to explore the use of application specific features, learn the grouping functions used in
the coarse-to-fine scheme, jointly train the cascade of classifiers, and apply our framework to high
order graphical models.
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4,813 | 5,358 | Probabilistic low-rank matrix completion on finite
alphabets
?
Eric
Moulines
Institut Mines-T?el?ecom
T?el?ecom ParisTech
CNRS LTCI
Olga Klopp
CREST et MODAL?X
Universit?e Paris Ouest
Jean Lafond
Institut Mines-T?el?ecom
T?el?ecom ParisTech
CNRS LTCI
[email protected]
[email protected]
[email protected]
Joseph Salmon
Institut Mines-T?el?ecom
T?el?ecom ParisTech
CNRS LTCI
[email protected]
Abstract
The task of reconstructing a matrix given a sample of observed entries is known
as the matrix completion problem. It arises in a wide range of problems, including recommender systems, collaborative filtering, dimensionality reduction,
image processing, quantum physics or multi-class classification to name a few.
Most works have focused on recovering an unknown real-valued low-rank matrix from randomly sub-sampling its entries. Here, we investigate the case where
the observations take a finite number of values, corresponding for examples to
ratings in recommender systems or labels in multi-class classification. We also
consider a general sampling scheme (not necessarily uniform) over the matrix
entries. The performance of a nuclear-norm penalized estimator is analyzed theoretically. More precisely, we derive bounds for the Kullback-Leibler divergence
between the true and estimated distributions. In practice, we have also proposed
an efficient algorithm based on lifted coordinate gradient descent in order to tackle
potentially high dimensional settings.
1
Introduction
Matrix completion has attracted a lot of contributions over the past decade. It consists in recovering
the entries of a potentially high dimensional matrix, based on their random and partial observations.
In the classical noisy matrix completion problem, the entries are assumed to be real valued and observed in presence of additive (homoscedastic) noise. In this paper, it is assumed that the entries take
values in a finite alphabet that can model categorical data. Such a problem arises in analysis of voting patterns, recovery of incomplete survey data (typical survey responses are true/false, yes/no or
do not know, agree/disagree/indifferent), quantum state tomography [13] (binary outcomes), recommender systems [18, 2] (for instance in common movie rating datasets, e.g., MovieLens or Neflix,
ratings range from 1 to 5) among many others. It is customary in this framework that rows represent
individuals while columns represent items e.g., movies, survey responses, etc. Of course, the observations are typically incomplete, in the sense that a significant proportion of the entries are missing.
Then, a crucial question to be answered is whether it is possible to predict the missing entries from
these partial observations.
1
Since the problem of matrix completion is ill-posed in general, it is necessary to impose a lowdimensional structure on the matrix, one particularly popular example being a low rank constraint.
The classical noisy matrix completion problem (real valued observations and additive noise), can be
solved provided that the unknown matrix is low rank, either exactly or approximately; see [7, 15, 17,
20, 5, 16] and the references therein. Most commonly used methods amount to solve a least square
program under a rank constraint or a convex relaxation of a rank constraint provided by the nuclear
(or trace norm) [10].
The problem of probabilistic low rank matrix completion over a finite alphabet has received much
less attention; see [22, 8, 6] among others. To the best of our knowledge, only the binary case
(also referred to as the 1-bit matrix completion problem) has been covered in depth. In [8], the
authors proposed to model the entries as Bernoulli random variables whose success rate depend
upon the matrix to be recovered through a convex link function (logistic and probit functions being
natural examples). The estimated matrix is then obtained as a solution of a maximization of the
log-likelihood of the observations under an explicit low-rank constraint. Moreover, the sampling
model proposed in [8] assumes that the entries are sampled uniformly at random. Unfortunately,
this condition is not totally realistic in recommender system applications: in such a context some
users are more active than others and some popular items are rated more frequently. Theoretically,
an important issue is that the method from [8] requires the knowledge of an upper bound on the
nuclear norm or on the rank of the unknown matrix.
Variations on the 1-bit matrix completion was further considered in [6] where a max-norm (though
the name is similar, this is different from the sup-norm) constrained minimization is considered. The
method of [6] allows more general non-uniform samplings but still requires an upper bound on the
max-norm of the unknown matrix.
In the present paper we consider a penalized maximum log-likelihood method, in which the loglikelihood of the observations is penalized by the nuclear norm (i.e., we focus on the Lagrangian
version rather than on the constrained one). We first establish an upper bound of the KullbackLeibler divergence between the true and the estimated distribution under general sampling distributions; see Section 2 for details. One should note that our method only requires the knowledge of
an upper bound on the maximum absolute value of the probabilities, and improves upon previous
results found in the literature.
Last but not least, we propose an efficient implementation of our statistical procedure, which is
adapted from the lifted coordinate descent algorithm recently introduced in [9, 14]. Unlike other
methods, this iterative algorithm is designed to solve the convex optimization and not (possibly nonconvex) approximated formulation as in [21]. It also has the benefit that it does not need to perform
full/partial SVD (Singular Value Decomposition) at every iteration; see Section 3 for details.
Notation
Define m1 ? m2 := min(m1 , m2 ) and m1 ? m2 := max(m1 , m2 ). We equip the set of m1 ? m2
matrices with real entries (denoted Rm1 ?m2 ) with the scalar product hX|X 0 i := tr(X > X 0 ). For a
given matrix X ? Rm1 ?m2 we write kXk? := maxi,j |Xi,j | and, for q ? 1, we denote its Schatten
q-norm by
!1/q
mX
1 ?m2
q
kXk?,q :=
?i (X)
,
i=1
where ?i (X) are the singular values of X ordered in decreasing order (see [1] for more details on
such norms). The operator norm of X is given by kXk?,? := ?1 (X). Consider two vectors of
j
p ? 1 matrices (X j )p?1
and (X 0j )p?1
j=1 such that for any (k, l) ? [m1 ] ? [m2 ] we have Xk,l ? 0,
Pp?1 j=1j
Pp?1 0j
0j
Xk,l ? 0, 1 ? j=1 Xk,l ? 0 and 1 ? j=1 Xk,l ? 0. Their square Hellinger distance is
d2H (X, X 0 ) :=
1
m1 m2
?
v
?v
?2 ?
u
2 u
p?1 q
p?1
p?1
q
X ?X
X
X
u
u
j
0j
j
0j ? ?
Xk,l
+?t1 ?
Xk,l
Xk,l
? Xk,l
? t1 ?
?
?
k?[m1 ] j=1
l?[m2 ]
j=1
2
j=1
and their Kullback-Leibler divergence is
?
Pp?1 j ?
j
p?1
p?1
X
X X
1
?
X
1
j=1 Xk,l
k,l
j
j
?
KL (X, X 0 ) :=
Xk,l
) log
Xk,l
log 0j + (1 ?
Pp?1 0j ? .
m1 m2
Xk,l
1 ? j=1 Xk,l
j=1
k?[m1 ] j=1
l?[m2 ]
Given an integer p > 1, a function f : Rp?1 ? Rp?1 is called a p-link function if for any x ? Rp?1
Pp?1
it satisfies f j (x) ? 0 for j ? [p ? 1] and 1 ? j=1 f j (x) ? 0. For any collection of p ? 1 matrices
j
j
j p?1
(X j )p?1
j=1 , f (X) denotes the vector of matrices (f (X) )j=1 such that f (X)k,l = f (Xk,l ) for any
(k, l) ? [m1 ] ? [m2 ] and j ? [p ? 1].
2
Main results
Let p denote the cardinality of our finite alphabet, that is the number of classes of the logistic model
(e.g., ratings have p possible values or surveys p possible answers). For a vector of p ? 1 matrices
m1 ?m2
and an index ? ? [m1 ] ? [m2 ], we denote by X? the vector (X?j )p?1
X = (X j )p?1
j=1 of R
j=1 .
We consider an i.i.d. sequence (?i )1?i?n over [m1 ] ? [m2 ], with a probability distribution function
? that controls the way the matrix entries are revealed. It is customary to consider the simple
uniform sampling distribution over the set [m1 ] ? [m2 ], though more general sampling schemes
could be considered as well. We observe n independent random elements (Yi )1?i?n ? [p]n . The
observations (Y1 , . . . , Yn ) are assumed to be independent and to follow a multinomial distribution
with success probabilities given by
? ?1 , . . . , X
? ?p?1 ) j ? [p ? 1] and
P(Yi = j) = f (X
i
i
j
P(Yi = p) = 1 ?
p?1
X
P(Yi = j)
j=1
?
? j p?1
where {f j }p?1
j=1 is a p-link function and X = (X )j=1 is the vector of true (unknown) parameters
? i instead of X
? ? . Let us denote by ?Y
we aim at recovering. For ease of notation, we often write X
i
the (normalized) negative log-likelihood of the observations:
?
?
??
p?1
p?1
n
X
1 X ?X
1{Yi =j} log f j (Xi ) + 1{Yi =p} log ?1 ?
f j (Xi )?? ,
(1)
?Y (X) = ?
n i=1 j=1
j=1
For any ? > 0 our proposed estimator is the following:
?=
X
arg min
X?(Rm1 ?m2 )p?1
maxj?[p?1] kX j k? ??
??Y (X) ,
where ??Y (X) = ?Y (X) + ?
p?1
X
kX j k?,1 ,
(2)
j=1
with ? > 0 being a regularization parameter controlling the rank of the estimator. In the rest of the
paper we assume that the negative log-likelihood ?Y is convex (this is the case for the multinomial
logit function, see for instance [3]).
? in the binomial setting
In this section we present two results controlling the estimation error of X
(i.e., when p = 2). Before doing so, let us introduce some additional notation and assumptions. The
?
score function (defined as the gradient of the negative log-likelihood) taken at the true parameter X,
? := ? ?Y (X).
? We also need the following constants depending on the link function
is denoted by ?
f and ? > 0:
M? = sup 2| log(f (x))| ,
|x|??
|f 0 (x)|
|f 0 (x)|
L? = max sup
, sup
|x|?? f (x) |x|?? 1 ? f (x)
K? = inf
|x|??
f 0 (x)2
.
8f (x)(1 ? f (x))
3
!
,
In our framework, we allow for a general distribution for observing the coefficients. However, we
need to control deviations of the sampling mechanism from the uniform distribution and therefore
we consider the following assumptions.
H1. There exists a constant ? ? 1 such that for all indexes (k, l) ? [m1 ] ? [m2 ]
min(?k,l ) ? 1/(?m1 m2 ) .
k,l
with ?k,l := ?(?1 = (k, l)).
Pm2
Pm1
?k,l ) for any l ? [m2 ] (resp. k ? [m1 ]) the
?k,l (resp. Rk := l=1
Let us define Cl := k=1
probability of sampling a coefficient in column l (resp. in row k).
H2. There exists a constant ? ? 1 such that
max(Rk , Cl ) ? ?/(m1 ? m2 ) ,
k,l
Assumption H1 ensures that each coefficient has a non-zero probability of being sampled whereas
H2 requires that no column nor row is sampled with too high probability (see also [11, 16] for more
details on this condition).
We define the sequence of matrices (Ei )ni=1 associated to the revealed coefficient (?i )ni=1 by
0 m2
1
Ei := eki (e0li )> where (ki , li ) = ?i and with (ek )m
k=1 (resp. (el )l=1 ) being the canonical bam2
m1
sis of R (resp. R ). Furthermore, if (?i )1?i?n is a Rademacher sequence independent from
(?i )ni=1 and (Yi )1?i?n we define
n
1X
?R :=
?i Ei .
n i=1
We can now state our first result. For completeness, the proofs can be found in the supplementary
material.
? ?,? and kXk
? ? ? ?. Then, with probability at least
Theorem 1. Assume H1 holds, ? ? 2k?k
1 ? 2/d the Kullback-Leibler divergence between the true and estimated distribution is bounded by
!
p
log(d)
?2
2
? 2
2
?
?
?
KL f (X), f (X) ? 8 max
m1 m2 rank(X) ? + c L? (Ek?R k?,? ) , ?eM?
,
K?
n
where c? is a universal constant.
? ?,? is stochastic and that its expectation Ek?R k?,? is unknown. However, thanks to
Note that k?k
Assumption H2 these quantities can be controlled.
To ease notation let us also define m := m1 ? m2 , M := m1 ? m2 and d := m1 + m2 .
? ? ? ?. Assume in addition that n ?
Theorem 2. Assume H 1 and H 2p
hold and that kXk
2m log(d)/(9?). Taking ? = 6L? 2? log(d)/(mn), then with probability at least 1 ? 3/d the
folllowing holds
!
p
? ? Xk
? 2
? log(d)
kX
??2 L2? M rank(X)
log(d)
?,2
?
?
? KL f (X), f (X) ? max c?
, 8?eM?
,
K?
m1 m2
K?
n
n
where c? is a universal constant.
Remark. Let us compare the rate of convergence of Theorem 2 with those obtained in previous
? is estimated by minimizing the negative
works on 1-bit matrix completion. In [8], the parameter X
?
log-likelihood under the constraints kXk? ? ? and kXk?,1 ? ? rm1 m2 for some r > 0. Under
? ? r, they could prove that
the assumption that rank(X)
r
? ? Xk
? 2
kX
rd
?,2
? C?
,
m1 m2
n
where C? is a constant depending on ? (see [8, Theorem 1]). This rate of convergence is slower
than the rate of convergence given by Theorem 2. [6] studied a max-norm constrained maximum
likelihood estimate and obtained a rate of convergence similar to [8].
4
3
Numerical Experiments
Implementation For numerical experiments, data were simulated according to a multinomial
logit distribution. In this setting, an observation Yk,l associated to row k and column l is distributed
p?1
1
as P(Yk,l = j) = f j (Xk,l
, . . . , Xk,l
) where
?
??1
p?1
X
f j (x1 , . . . , xp?1 ) = exp(xj ) ?1 +
exp(xj )? , for j ? [p ? 1] .
(3)
j=1
With this choice, ?Y is convex and problem (2) can be solved using convex optimization algorithms. Moreover, following the advice of [8] we considered the unconstrained version of problem
(2) (i.e., with no constraint on kXk? ), which reduces significantly the computation burden and has
no significant impact on the solution in practice. To solve this problem, we have extended to the
multinomial case the coordinate gradient descent algorithm introduced by [9]. This type of algorithm has the advantage, say over the Soft-Impute [19] or the SVT [4] algorithm, that it does not
require the computation of a full SVD at each step of the main loop of an iterative (proximal) algorithm (bare in mind that the proximal operator associated to the nuclear norm is the soft-thresholding
operator of the singular values). The proposed version only computes the largest singular vectors
and singular values. This potentially decreases the computation by a factor close to the value of the
upper bound on the rank commonly used (see the aforementioned paper for more details).
Let us present the algorithm. Any vector of p ? 1 matrices X = (X j )p?1
j=1 is identified as an element
of the tensor product space Rm1 ?m2 ? Rp?1 and denoted by:
X=
p?1
X
X j ? ej ,
(4)
j=1
p?1
where again (ej )p?1
and ? stands for the tensor product. The set of
j=1 is the canonical basis on R
normalized rank-one matrices is denoted by
M := M ? Rm1 ?m2 |M = uv > | kuk = kvk = 1, u ? Rm1 , v ? Rm2 .
Define ? the linear space of real-valued functions on M with finite support, i.e., ?(M
P ) = 0 except
for a finite number of M ? M. This space is equipped with the `1 -norm k?k1 = M ?M |?(M )|.
Define by ?+ the positive orthant, i.e., the cone of functions ? ? ? such that ?(M ) ? 0 for all
p?1
M ? M. Any tensor X can be associated with a vector ? = (?1 , . . . , ?p?1 ) ? ?+
, i.e.,
X=
p?1 X
X
?j (M )M ? ej .
(5)
j=1 M ?M
Such representations are not unique, and among them, the one associated to the SVD plays a key
role, as we will see below. For a given X represented by (4) and for any j ? {1, . . . , p ? 1}, denote
j
j
by {?kj }nk=1 the (non-zero) singular values of the matrix X j and {ujk ,vkj }nk=1 the associated singular
vectors. Then, X may be expressed as
j
X=
p?1 X
n
X
?kj ujk (vkj )> ? ej .
(6)
j=1 k=1
Defining ?j the function ?j (M ) = ?kj if M = ujk (vkj )> , k ? [nj ] and ?j (M ) = 0 otherwise, one
obtains a representation of the type given in Eq. (5).
Conversely, for any ? = (?1 , . . . , ?p?1 ) ? ?p?1 , define the map
W : ? ? W? :=
p?1
X
W?j ? ej
with W?j :=
X
?j (M )M
M ?M
j=1
and the auxiliary objective function
? ?Y (?) = ?
?
p?1 X
X
j=1 M ?M
5
?j (M ) + ?Y (W? ) .
(7)
The map ? 7? W? is a continuous linear map from (?p?1 , k ? k1 ) to Rm1 ?m2 ? Rp?1 , where
Pp?1 P
p?1
k?k1 = j=1 M ?M |?j (M )|. In addition, for all ? ? ?+
p?1
X
kW?j k?,1 ? k?k1 ,
j=1
Pp?1
j
j=1 kW? k?,1
and one obtains k?k1 =
when ? is the representation associated to the SVD decomposition. An important consequence, outlined in [9, Proposition 3.1], is that the minimization of (7)
is actually equivalent to the minimization of (2); see [9, Theorem 3.2].
The proposed coordinate gradient descent algorithm updates at each step the nonnegative finite support function ?. For ? ? ? we denote by supp(?) the support of ? and for M ? M, by ?M ? ? the
Dirac function on M satisfying ?M (M ) = 1 and ?M (M 0 ) = 0 if M 0 6= M . In our experiments we
have set to zero the initial ?0 .
Algorithm 1: Multinomial lifted coordinate gradient descent
Data: Observations: Y , tuning parameter ?
initial parameter: ?0 ? ?p?1
+ ; tolerance: ; maximum number of iterations: K
Result: ? ? ?p?1
+
Initialization: ? ? ?0 , k ? 0
while k ? K do
for j = 0 to p ? 1 do
Compute top singular vectors pair of (?? ?Y (W? ))j : uj , vj
Let g = ? + minj=1,...,p?1 h? ?Y | uj (v j )> i
if g ? ?/2 then
? ? ? + (b0 ?u0 (v0 )> , . . . , bp?1 ?up?1 (vp?1 )> )
(?0 , . . . , ?p?1 ) =
arg min
?
Y
(b0 ,...,bp?1 )?Rp?1
+
? ? ? + (?0 ?u0 (v0 )> , . . . , ?p?1 ?up?1 (vp?1 )> )
k ?k+1
else
Let gmax = maxj?[p?1] maxuj (vj )> ?supp(?j ) |? + h? ?Y | uj (v j )> i|
if gmax ? then
break
else
? ?Y (?0 )
??
arg min
?
? 0 ??p?1
,supp(? 0j )?supp(? j ),j?[p?1]
+
k ?k+1
A major interest of Algorithm 1 is that it requires to store the value of the parameter entries only
for the indexes which are actually observed. Since in practice the number of observations is much
smaller than the total number of coefficients m1 m2 , this algorithm is both memory and computationally efficient. Moreover, using an SVD algorithm such as Arnoldi iterations to compute the top
singular values and vector pairs (see [12, Section 10.5] for instance) allows us to take full advantage
of gradient sparse structure. Algorithm 1 was implemented in C and Table 1 gives a rough idea
of the execution time for the case of two classes on a 3.07Ghz w3550 Xeon CPU (RAM 1.66 Go,
Cache 8Mo).
Simulated experiments To evaluate our procedure we have performed simulations for matrices with p = 2 or 5. For each class matrix X j we sampled uniformly five unitary vector pairs
(ujk , vkj )5k=1 . We have then generated matrices of rank equals to 5, such that
5
X
?
X j = ? m1 m2
?k ujk (vkj )> ,
k=1
?
with (?1 , . . . , ?5 ) = (2, 1, 0.5, 0.25, 0.1) and ? is a scaling factor. The m1 m2 factor, guarantees
that E[kX j k? ] does not depend on the sizes of the problem m1 and m2 .
6
Parameter Size
Observations
Execution Time (s.)
103 ? 103
105
4.5
3 ? 103 ? 3 ? 103
105
52
104 ? 104
107
730
Table 1: Execution time of the proposed algorithm for the binary case.
We then sampled the entries uniformly and the observations according to a logit distribution given
by Eq. (3). We have then considered and compared the two following estimators both computed
using Algorithm 1:
? the logit version of our method (with the link function given by Eq. (3))
? N ), that consists in using the Gaussian
? the Gaussian completion method (denoted by X
log-likelihood instead of the multinomial in (2), i.e., using a classical squared Frobenius
norm (the implementation being adapted mutatis mutandis). Moreover an estimation of the
standard deviation is obtained by the classical analysis of the residue.
Contrary to the logit version, the Gaussian matrix completion does not directly recover the probabilities of observing a rating. However, we can estimate this probability by the following quantity:
?
if j = 1 ,
?
?0
?N
j?0.5?X
N
?
k,l
P(Xk,l = j) = FN (0,1) (pj+1 ) ? FN (0,1) (pj ) with pj =
if 0 < j < p
?
?
?
?
1
if j = p ,
where FN (0,1) is the cdf of a zero-mean standard Gaussian random variable.
As we see on Figure 1, the logistic estimator outperforms the Gaussian for both cases p = 2 and
p = 5 in terms of the Kullback-Leibler divergence. This was expected because the Gaussian model
allows uniquely symmetric distributions with the same variance for all the ratings, which is not the
case for logistic distributions. The choice of the ? parameter has been set for both methods by
performing 5-fold cross-validation on a geometric grid of size 0.8 log(n).
Table 2 and Table 3 summarize the results obtained for a 900 ? 1350 matrix respectively for p = 2
and p = 5. For both the binomial case p = 2 and the multinomial case p = 5, the logistic model
slightly outperforms the Gaussian model. This is partly due to the fact that in the multinomial case,
some ratings can have a multi-modal distribution. In such a case, the Gaussian model is unable
to predict these ratings, because its distribution is necessarily centered around a single value and
is not flexible enough. For instance consider the case of a rating distribution with high probability
of seeing 1 or 5, low probability of getting 2, 3 and 4, where we observed both 1?s and 5?s. The
estimator based on a Gaussian model will tend to center its distribution around 2.5 and therefore
misses the bimodal shape of the distribution.
Observations
Gaussian prediction error
Logistic prediction error
10 ? 103
0.49
0.42
50 ? 103
0.34
0.30
100 ? 103
0.29
0.27
500 ? 103
0.26
0.24
Table 2: Prediction errors for a binomial (2 classes) underlying model, for a 900 ? 1350 matrix.
Observations
Gaussian prediction error
Logistic prediction error
10 ? 103
0.78
0.75
50 ? 103
0.76
0.54
100 ? 103
0.73
0.47
500 ? 103
0.69
0.43
Table 3: Prediction Error for a multinomial (5 classes) distribution against a 900 ? 1350 matrix.
Real dataset We have also run the same estimators on the MovieLens 100k dataset. In the case
of real data we cannot calculate the Kullback-Leibler divergence since no ground truth is available.
Therefore, to compare the prediction errors, we randomly selected 20% of the entries as a test set,
and the remaining entries were split between a training set (80%) and a validation set (20%).
7
Normalized KL divergence for logistic (plain), Gaussian (dashed)
Normalized KL divergence for logistic (plain), Gaussian (dashed)
0.20
0.20
size: 100x150
size: 300x450
size: 900x1350
0.15
Mean KL divergence
Mean KL divergence
size: 100x150
size: 300x450
size: 900x1350
0.10
0.10
0.05
0.05
0.00
0.15
100000
200000
300000
400000
0.00
500000
100000
Number of observations
200000
300000
400000
500000
Number of observations
Figure 1: Kullback-Leibler divergence between the estimated and the true model for different
matrices sizes and sampling fraction, normalized by number of classes. Right figure: binomial
and Gaussian models ; left figure: multinomial with five classes and Gaussian model. Results are
averaged over five samples.
For this dataset, ratings range from 1 to 5. To consider the benefit of a binomial model, we have
tested each rating against the others (e.g., ratings 5 are set to 0 and all others are set to 1). Interestingly we see that the Gaussian prediction error is significantly better when choosing labels ?1, 1
instead of labels 0, 1. This is another motivation for not using the Gaussian version: the sensibility to
the alphabet choice seems to be crucial for the Gaussian version, whereas the binomial/multinomial
ones are insensitive to it. These results are summarized in table 4.
Rating
Gaussian prediction error (labels ?1 and 1)
Gaussian prediction error (labels 0 and 1)
Logistic prediction error
1
0.06
0.12
0.06
2
0.12
0.20
0.11
3
0.28
0.39
0.27
4
0.35
0.46
0.34
5
0.19
0.30
0.20
Table 4: Binomial prediction error when performing one versus the others procedure on the MovieLens 100k dataset.
4
Conclusion and future work
We have proposed a new nuclear norm penalized maximum log-likelihood estimator and have provided strong theoretical guarantees on its estimation accuracy in the binary case. Compared to
previous works on 1-bit matrix completion, our method has some important advantages. First, it
works under quite mild assumptions on the sampling distribution. Second, it requires only an upper bound on the maximal absolute value of the unknown matrix. Finally, the rates of convergence
given by Theorem 2 are faster than the rates of convergence obtained in [8] and [6]. In future work,
we could consider the extension to more general data fitting terms, and to possibly generalize the
results to tensor formulations, or to penalize directly the nuclear norm of the matrix probabilities
themselves.
Acknowledgments
Jean Lafond is grateful for fundings from the Direction G?en?erale de l?Armement (DGA) and to the
labex LMH through the grant no ANR-11-LABX-0056-LMH in the framework of the ?Programme
des Investissements d?Avenir?. Joseph Salmon acknowledges Chair Machine Learning for Big Data
for partial financial support. The authors would also like to thank Alexandre Gramfort for helpful
discussions.
8
References
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9
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4,814 | 5,359 | Controlling privacy in recommender systems
Tommi Jaakkola
CSAIL, MIT
[email protected]
Yu Xin
CSAIL, MIT
[email protected]
Abstract
Recommender systems involve an inherent trade-off between accuracy of recommendations and the extent to which users are willing to release information about
their preferences. In this paper, we explore a two-tiered notion of privacy where
there is a small set of ?public? users who are willing to share their preferences
openly, and a large set of ?private? users who require privacy guarantees. We
show theoretically and demonstrate empirically that a moderate number of public
users with no access to private user information already suffices for reasonable
accuracy. Moreover, we introduce a new privacy concept for gleaning relational
information from private users while maintaining a first order deniability. We
demonstrate gains from controlled access to private user preferences.
1
Introduction
Recommender systems exploit fragmented information available from each user. In a realistic system there?s also considerable ?churn?, i.e., users/items entering or leaving the system. The core
problem of transferring the collective experience of many users to an individual user can be understood in terms of matrix completion ([13, 14]). Given a sparsely populated matrix of preferences,
where rows and columns of the matrix correspond to users and items, respectively, the goal is to
predict values for the missing entries.
Matrix completion problems can be solved as convex regularization problems, using trace norm
as a convex surrogate to rank. A number of algorithms are available for solving large-scale tracenorm regularization problems. Such algorithms typically operate by iteratively building the matrix
from rank-1 components (e.g., [7, 17]). Under reasonable assumptions (e.g., boundedness, noise,
restricted strong convexity), the resulting empirical estimators have been shown to converge to the
underlying matrix with high probability ([12, 8, 2]). Consistency guarantees have mostly involved
matrices of fixed dimension, i.e., generalization to new users is not considered. In this paper, we
reformulate the regularization problem in a manner that depends only on the item (as opposed to
user) features, and characterize the error for out-of-sample users.
The completion accuracy depends directly on the amount of information that each user is willing to share with the system ([1]). It may be possible in some cases to side-step this statistical
trade-off by building Peer-to-Peer networks with homomorphic encryption that is computationally
challenging([3, 11]). We aim to address the statistical question directly.
The statistical trade-off between accuracy and privacy further depends on the notion of privacy we
adopt. A commonly used privacy concept is Differential Privacy (DP) ([6]), first introduced to
protect information leaked from database queries. In a recommender context, users may agree to a
trusted party to hold and aggregate their data, and perform computations on their behalf. Privacy
guarantees are then sought for any results published beyond the trusted party (including back to
the users). In this setting, differential privacy can be achieved through obfuscation (adding noise)
without a significant loss of accuracy ([10]).
1
In contrast to [10], we view the system as an untrusted entity, and assume that users wish to guard
their own data. We depart from differential privacy and separate computations that can be done
locally (privately) by individual users and computations that must be performed by the system (e.g.,
aggregation). For example, in terms of low rank matrices, only the item features have to be solved by
the system. The corresponding user features can be obtained locally by the users and subsequently
used for ranking.
From this perspective, we divide the set of users into two pools, the set of public users who openly
share their preferences, and the larger set of private users who require explicit privacy guarantees.
We show theoretically and demonstrate empirically that a moderate number of public users suffice
for accurate estimation of item features. The remaining private users can make use of these item
features without any release of information. Moreover, we propose a new 2nd order privacy concept
which uses limited (2nd order) information from the private users as well, and illustrate how recommendations can be further improved while maintaining marginal deniability of private information.
2
Problem formulation and summary of results
Recommender setup without privacy Consider a recommendation problem with n users and
? ? Rn?m . If only a few
m items. The underlying complete rating matrix to be recovered is X
? can be assumed to have low rank. As such, it is also
latent factors affect user preferences, X
recoverable from a small number of observed entries. We assume that entries are observed with
noise. Specifically,
?i,j + i,j , (i, j) ? ?
Yi,j = X
(1)
where ? denotes the set of observed entries. Noise is assumed to be i.i.d and follows a zeromean sub-Gaussian distribution with parameter kk?2 = ?. Following [16], we refer to the noise
distribution as Sub(? 2 ).
To bias our estimated rating matrix X to have low rank, we use convex relaxation of rank P
in the form
of trace norm. The trace-norm is the sum of singular values of the matrix or kXk? = i ?i (X).
The basic estimation problem, without any privacy considerations, is then given by
1 X
?
min
(Yi,j ? Xi,j )2 + ?
kXk?
(2)
m?n
N
mn
X?R
(i,j)??
where ?
? is a regularization parameter and N = |?| is the total number of observed ratings. The
factor mn ensures that the regularization does not grow with either dimension.
The above formulation requires the server to explicitly
? obtain predictions for each user, i.e., solve
for X. We can instead write X = U V T and ? = (1/ mn)V V T , and solve for ? only. If the server
then communicates the resulting low rank ? (or just V ) to each user, the users can reconstruct the
relevant part of U locally, and reproduce X as it pertains to them. To this end, let ?i = {j : (i, j) ?
?} be the set of observed entries for user i, and let Yi,?i be a column vector of user i?s ratings. Then
we can show that Eq.(2) is equivalent to solving
min+
??S
n
X
T
Yi,?
(?0 I + ??i ,?i )Yi,?i +
i
?
nm k?k?
(3)
i=1
?
?
where S + is the set of positive semi-definite m ? m matrices and ?0 = ?N/ nm. By solving ?,
c
we can predict ratings for unobserved items (index set ?i for user i) by
? i,?c = ??c ,? (?0 I + ?? ,? )?1 Yi,?
X
i
i
i
i
i
i
(4)
Note that we have yet to address any privacy concerns. The solution to Eq.(3) still requires access
to full ratings Yi,?i for each user.
Recommender setup with privacy Our privacy setup assumes an untrusted server. Any user
interested in guarding their data must therefore keep and process their data locally, releasing information to the server only in a controlled manner. We will initially divide users into two broad
2
categories, public and private. Public users are willing to share all their data with the server while
private users are unwilling to share any. This strict division is removed later when we permit private
users to release, in a controlled manner, limited information pertaining to their ratings (2nd order
information) so as to improve recommendations.
Any data made available to the server enables the server to model the collective experience of users,
for example, to solve Eq.(3). We will initially consider the setting where Eq.(3) is solved on the
basis of public users only. We use an EM type algorithm for training. In the E-step, the current ?
is sent to public users to complete their rating vectors and send back to the server. In the M-step,
? (or V? ) can be subsequently
? is then updated based on these full rating vectors. The resulting ?
shared with the private users, enabling the private users (their devices) to locally rank candidate
? is then improved by asking
items without any release of private information. The estimation of ?
private users to share 2nd order relational information about their ratings without any release of
marginal selections/ratings.
Note that we do not consider privacy beyond ratings. In other words, we omit any subsequent release
of information due to users exploring items recommended to them.
Summary of contributions We outline here our major contributions towards characterizing the
role of public users and the additional controlled release of information from private users.
p
?T X/
? ?nm can be estimated in a consistent, accurate manner on the basis
1) We show that ?
?= X
? ??
of public users alone. In particular, we express the error k?
?kF as a function of the total number
of observations. Moreover, if the underlying public user ratings can be thought of as i.i.d. samples,
we also bound k?
? ? ?? kF in terms of the number of public users. Here ?? is the true limiting
estimate. See section 3.1 for details.
? i,?c for private users relates to the accuracy of
2) We show how the accuracy of predicted ratings X
i
? (primarily from public users). Since the ratings for user i may not be related to the
estimating ?
? lies in, we can only characterize the accuracy when sufficient overlap exists. We
subspace that ?
? i,?c ? X
?i,?c k depends on this overlap, accuracy of ?,
? and
quantify this overlap, and show how kX
i
i
the observation noise. See section 3.2 for details.
3) Having established the accuracy of predictions based on public users alone, we go one step further
and introduce a new privacy mechanism and algorithms for gleaning additional relational (2nd order)
information from private users. This 2nd order information is readily used by the server to estimate
? The privacy concept constructively maintains first order (marginal) deniability for private users.
?.
We demonstrate empirically the gains from the additional 2nd order information. See section 4.
3
3.1
Analysis
?
Statistical Consistency of ?
?T U
? be a solution to Eq.(2). We can write X
? = U
? V? T , where U
? = I?m with 0/1 diagonal.
Let X
p
1
T
?
?
?
?
? To this end,
Since ? = ?
X X we can first analyze errors in X and then relate them to ?.
mn
we follow the restricted strong convexity (RSC) analysis[12]. However, their result depends on
the inverse of the minimum number of ratings of all users and items. In practice (see below), the
number of ratings decays exponentially across sorted users, making such a result loose. We provide
a modified analysis that depends only on the total number of observations N .
?i,? belongs to a fixed r dimensional
Throughout the analysis, we assume that each row vector X
?i,j | ?
subspace. We also assume that both noiseless and noisy entries are bounded,
i.e. |Yi,j |, |X
P
2
?, ?(i, j). For brevity, we use kY ? Xk? to denote the empirical loss (i,j)?? (Yi,j ? Xi,j )2 . The
restricted strong convexity property (RSC) assumes that there exists a constant ? > 0 such that
? ?
? 2F ? 1 kX
? ? Xk
? 2?
kX ? Xk
mn
N
3
(5)
? ?X
? in a certain subset. RSC provides the step from approximating observations to apfor X
proximating the full underlying matrix. It is satisfied with high probability provided that N =
(m + n) log(m + n)).
?=P
?S Q
?T , and let row(X) and col(X) denote the row and column spaces of
Assume the SVD of X
X. We define the following two sets,
A(P, Q)
B(P, Q)
?, col(X) ? Q}
?
:= {X, row(X) ? P
?? , col(X) ? Q
?? }
:= {X, row(X) ? P
(6)
Let ?A (X) and ?B (X) be the projection of X onto sets A and B, respectively, and ?A = I ? ?A ,
? ?X
? be the difference between the estimated and the underlying rating
?B = I ? ?B . Let ? = X
matrices. Our first lemma demonstrates that ? lies primarily in a restricted subspace and the second
one guarantees that the noise remains bounded.
Lemma 3.1. Assume i,j for (i, j) ? ? are i.i.d. sub-gaussian with ? = ki,j k?1 . Then with
2 2?
e
probability 1 ? N 4ch
, k?B (?)k? ? k?B (?)k? + 2c ?N ? mn log2 N . Here h > 0 is an absolute
constant associated with the sub-gaussian noise.
pn
2 2?
2
mn log N
N p mn
? N , then c ?
If ? = ?0 c? log
= c? log
N?
?0
N = b log N
N where we leave the deN
pendence on n explicit. Let D(b, n, N ) denote the set of difference matrices that satisfy lemma 3.1
above. By combining the lemma and the RSC property, we obtain the following theorem.
Theorem 3.2. Assume RSC for the set D(b, n, N ) with parameter ? > 0 where b =
? N,
?0 c? log
N
then we have
?=
where h, c > 0 are constants.
? 1 k?kF
mn
?
2c?( ?1?
?
+
2r log
?N
? ) N
?
c? m
?0 .
Let
e
with probability at least 1? N 4ch
The bound in the theorem consists of two terms, pertaining to the noise and the regularization. In
contrast to [12], the terms only relate to the total number of observations N .
? First, we map the accuracy of X
? to that of ?
? using a
We now turn our focus on the accuracy of ?.
perturbation bound for polar decomposition (see [9]).
?
? ? Xk
? F ? ?, then k?
? ??
Lemma 3.3. If ? 1 kX
?kF ? 2?
mn
? As a
This completes our analysis in terms of recovering ?
? for a fixed size underlying matrix X.
final step, we turn to the question of how the estimation error changes as the number of users or n
?T ?
?i be the underlying rating vector for user i and define ?n = 1 Pn X
grows. Let X
i=1 i Xi . Then
mn
1
1
n
?
n
?
?
?
? = (? ) 2 . If ? is the limit of ? , then ? = (? ) 2 . We bound the distance between ?
? and ?? .
?i are i.i.d samples from a distribution with support only in a subspace
Theorem 3.4. Assume X
?i k ? ??m. Let ?1 and ?r be the smallest and largest
of dimension r and bounded norm kX
?
eigenvalues of ? . Then, for large enough n, with probability at least 1 ? nr2 ,
s
?
?r log n
log n
?
k?
? ? ? kF ? 2 r?
+ o(
)
(7)
?1 n
n
Combining the two theorems and using triangle inequality, we obtain a high probability bound on
? ? ?? kF . Assuming the number of ratings for each user is larger than ?m, then N > ?nm and
k?
?
the bound grows in the rate of ?(log n/ n) with ? being a constant that depends on ?. For large
enough ?, the required n to achieve a certain error bound is small. Therefore a few public users with
large number of ratings could be enough to obtain a good estimate of ?? .
3.2
Prediction accuracy
? i,?c for all users as defined in
We are finally ready to characterize the error in the predicted ratings X
i
? ? ?? k obtained on the basis of our results
Eq.(4). For brevity, we use ? to represent the bound on k?
above. We also use x? and x?c as shorthands for Xi,?i and Xi,?ci with the idea that x? typically
refers to a new private user.
4
The key issue for us here is that the partial rating vector x? may be of limited use. For example,
if the number of observed ratings is less than rank r, then we would be unable to identify a rating
vector in the r dimensional subspace even without noise. We seek to control this in our analysis by
assuming that the observations have enough signal to be useful. Let SVD of ?? be Q? S ? (Q? )T ,
and ?1 be its minimum eigenvalue. We constrain the index set of observations ? such that it belongs
to the set
m
2
2
? T
D(?) = ? ? {1, . . . , m}, s.t.kxkF ? ? kx? kF , ?x ? row((Q ) )
|?|
The parameter ? depends on how the low dimensional sub-space is aligned with the coordinate axes.
We are only interested in characterizing prediction errors for observations that are in D(?). This is
quite different from the RSC property. Our main result is then
Theorem 3.5. Suppose k? ? ?? kF ? ? ?1 , ? ? D(?). For any ?
x ? row((Q? )T ), our
observation x? = ?
x? + ? where ? ? Sub(? 2 ) is the noise vector. The predicted ratings over
the remaining entries p
are given by x
??c = ??c ,? (?0 I + ??,? )?1 x? . Then, with probability at least
1 ? exp(?c2 min(c41 , |?|c21 )),
r
1
?
m
k?
xk F
2c1 ?|?| 4
0
c
c
?
?
kx? ? ?
x? kF ? 2 ? + ?( ?
+ 1)(
+
)
|?|
?1
?0
where c1 , c2 > 0 are constants.
?
All the proofs are provided in the supplementary material. The term proportional
to k?
xkF / ?1 is
?
1
due to the estimation error of ?? , while the term proportional to 2c1 ?|?| 4 / ?0 comes from the
noise in the observed ratings.
4
Controlled privacy for private users
Our theoretical results already demonstrate that a relatively small number of public users with many
ratings suffices for a reasonable performance guarantee for both public and private users. Empirical
results (next section) support this claim. However, since public users enjoy no privacy guarantees,
we would like to limit the required number of such users by requesting private users to contribute in
a limited manner while maintaining specific notions of privacy.
Definition 4.1. : Privacy preserving mechanism. Let M : Rm?1 ? Rm?1 be a random mechanism that takes a rating vector r as input and outputs M(r) of the same dimension with j th element
M(r)j . We say that M(r) is element-wise privacy preserving if Pr(M(r)j = z) = p(z) for
j = 1, ..., m, and some fixed distribution p.
For example, a privacy preserving mechanism M(r) is element-wise private if its coordinates follow the same marginal distribution such as uniform. Note that such a mechanism can still release
information about how different ratings interact (co-vary) which is necessary for estimation.
Discrete values. Assume that each element in r and M(r) belongs to a discrete set S with |S| = K.
A natural privacy constraint is to insist that the marginal distribution of M(r)j is uniform, i.e.,
Pr(M(r)j = z) = 1/K, for z ? S. A trivial mechanism that satisfies the privacy constraint is to
select each value uniformly at random from S. In this case, the returned rating vector contributes
nothing to the server model. Our goal is to design a mechanism that preserves useful 2nd order
information.
We assume that a small number of public user profiles are available, from which we can learn
an initial model parameterized by (?, V ), where ? is the item mean vector and V is a low rank
component of ?. The server provides each private user the pair (?, V ) and asks, once, for a response
M(r). Here r is the user?s full rating vector, completed (privately) with the help of the server model
(?, V ).
The mechanism M(r) is assumed to be element-wise privacy preserving, thus releasing nothing
about a single element in isolation. What information should it carry? If each user i provided their
Pn
1
1
full rating vector ri , the server could estimate ? according to ?nm
( i=1 (ri ??)(ri ??)T ) 2 . Thus,
5
if M(r) preserves the second order statistics to the extent possible, the server could still obtain an
accurate estimate of ?.
Consider a particular user and their completed rating vector r. Let P(x) = Pr(M(r) = x). We
select distribution P(x) by solving the following optimization problem geared towards preserving
interactions between the ratings under the uniform marginal constraint.
min
P
Ex?P k(x ? ?)(x ? ?)T ? (r ? ?)(r ? ?)T k2F
s.t.
P(xi = s) = 1/K, ?i, ?s ? S.
(8)
where K = |S|. The exact solution is difficult to obtain because the number of distinct assignments
of x is K m . Instead, we consider an approximate solution. Let x1 , ..., xK ? Rm?1 be K different
vectors such that, for each i, {x1i , x2i , ..., xK
i } forms a permutation of S. If we choose x with
Pr(x = xj ) = 1/K, then the marginal distribution of each element is uniform therefore maintaining
element-wise privacy. It remains to optimize the set x1 , ..., xK to capture interactions between
ratings.
We use a greedy coordinate descent algorithm to optimize x1 , ..., xK . For each coordinate i, we
randomly select a pair xp and xq , and switch xpi and xqi if the objective function in (8) is reduced.
The process is repeated a few times before we move on to the next coordinate. The algorithm can
be implemented efficiently because each operation deals only with a single coordinate.
Finally, according to the mechanism, the private user selects one of xj , j = 1, . . . , K, uniformly
at random and sends the discrete vector back to the server. Since the resulting rating vectors from
private users are noisy, the server decreases their weight relative to the information from public users
as part of the overall M-step for estimating ?.
Continuous values. Assuming the rating values are continuous and unbounded, we require instead
that the returned rating vectors follow the marginal distributions with a given mean and variance.
Specifically, E[M(r)i ] = 0 and Var[M(r)i ] = v where v is a constant that remains to be determined. Note that, again, any specific element of the returned vector will not, in isolation, carry any
information specific to the element.
As before, we search for the distribution P so as to minimize the L2 error of the second order
statistics under marginal constraints. For simplicity, denote r0 = r ? ? where r is the true completed
rating vector, and ui = M(r)i . The objective is given by
min
P,v
Eu?P kuuT ? r0 r0T k2F
s.t.
E[ui ] = 0, Var[ui ] = v, ?i.
(9)
Note that the formulation does not directly constrain that P has identical marginals, only that the
means and variances agree. However, the optimal solution does, as shown next.
Pm
Theorem 4.2. Let zi = sign(ri0 ) and h = ( i=1 |ri0 |)/m. The minimizing distribution of (9) is
given by Pr(u = zh) = Pr(u = ?zh) = 1/2.
We leave the proof in the supplementary material. A few remarks are in order. The mechanism in this
case is a two component mixture distribution, placing a probability mass on sign(r0 )h (vectorized)
and ?sign(r0 )h with equal probability. As a result, the server, knowing the algorithm that private
users follow, can reconstruct sign(r0 ) up to an overall randomly chosen sign. Note also that the
value of h is computed from user?s private rating vector and therefore releases some additional
information about r0 = r ? ? albeit weakly. To remove this information altogether, we could use
the same h for all users and estimate it based on public users.
The privacy constraints will clearly have a negative impact on the prediction accuracy in comparison
to having direct access to all the ratings. However, the goal is to improve accuracy beyond the public
users alone by obtaining limited information from private users. While improvements are possible,
the limited information surfaces in several ways. First, since private users provide no first order
information, the estimation of mean rating values cannot be improved beyond public users. Second,
the sampling method we use to enforce element-wise privacy adds noise to the aggregate second
order information from which V is constructed. Finally, the server can run the M-step with respect to
the private users? information only once, whereas the original EM algorithm could entertain different
completions for user ratings iteratively. Nevertheless, as illustrated in the next section, the algorithm
can still achieve a good accuracy, improving with each additional private user.
6
5
Experiments
We perform experiments on the Movielens 10M dataset which contains 10 million ratings from
69878 users on 10677 movies. The test set contains 10 ratings for each user. We begin by demonstrating that indeed a few public users suffice for making accurate recommendations. Figure 1 left
shows the test performance of both weighted (see [12]) and unweighted (uniform) trace norm regularization as we add users. Here users with most ratings are added first.
0.96
1.5
Uniform
Weighted
0.95
Most ratings
Random
1.4
0.94
1.3
0.92
Test RMSE
Test RMSE
0.93
0.91
0.9
1.2
1.1
0.89
1
0.88
0.9
0.87
0.86
0
0.2
0.4
0.6
Percentage of Users
0.8
1
0.8
200
400
600
800 1000 1200 1400 1600 1800 2000
Number of ratings (k)
Figure 1: Left: Test RMSE as a function of the percentage of public users; Right: Performance
changes with different rating numbers.
With only 1% of public users added, the test RMSE of unweighted trace norm regularization is
0.876 which is already close to the optimal prediction error. Note that the loss of weighted trace
norm regularization actually starts to go up when the number of users increases. The reason is that
the weighted trace norm penalizes less for users with few ratings. As a result, the resulting low
dimensional subspace used for prediction is influenced more by users with few ratings.
The statistical convergence bound in section 3.1 involves both terms that decrease as a function of
the number of ratings N and the number of public users n. The two factors are usually coupled. It
is interesting to see how they impact performance individually. Given a number of total ratings, we
compare two different methods of selecting public users. In the first method, users with most ratings
are selected first, whereas the second method selects users uniformly at random. As a result, if we
equalize the total number of ratings from each method, the second method selects a lot more users.
Figure 1 Right shows that the second method achieves better performance. An interpretation, based
on the theory, is that the right side of error bound (7) decreases as the number of users increases.
We also show how performance improves based on controlled access to private user preferences.
First, we take the top 100 users with the most ratings as the public users, and learn the initial
prediction model from their ratings. To highlight possible performance gains, private users with
more ratings are selected first. The results remain close if we select private users uniformly.
The rating values are from 0.5 to 5 with totally 10 different discrete values. Following the privacy
mechanism for discrete values, each private user generates ten different candidate vectors and returns
one of them uniformly at random. In the M-step, the weight for each private user is set to 1/2
compared to 1 for public users. During training, after processing w = 20 private users, we update
parameters (?, V ), re-complete the rating vectors of public users, making predictions for next batch
of private users more accurate. The privacy mechanism for continuous values is also tested under
the same setup. We denote the two privacy mechanism as PMD and PMC, respectively.
We compare five different scenarios. First, we use a standard DP method that adds Laplace noise to
the completed rating vector. Let the DP parameter be , because the maximum difference between
rating values is 4.5, the noise follows Lap(0, 4.5/). As before, we give a smaller weight to the
noisy rating vectors and this is determined by cross validation. Second, [5] proposed a notion of
?local privacy? in which differential privacy is guaranteed for each user separately. An optimal
strategy for d-dimensional multinomial distribution in this case reduces effective samples from n to
n2 /d where is the DP parameter. In our case the dimension corresponds to the number of items
7
0.92
0.915
0.91
Test RMSE
0.905
0.9
0.895
0.89
0.885
0.88
0.875
0.87
0
PMC
PMD
Lap eps=1
Lap eps=5
SSLP eps=5
Exact 2nd order
Full EM
50
100
150
200
250
Number of ??private?? users
300
350
400
Figure 2: Test RMSE as a function of private user numbers. PMC: the privacy mechanism for
continuous values; PMD: the privacy mechanism for discrete values; Lap eps=1: DP with Laplace
noise, = 1; Lap eps=5: same as before except = 5; SSLP eps=5: sampling strategy described
in [4] with DP parameter = 5; Exact 2nd order: with exact second order statistics from private
users (not a valid privacy mechanism); Full EM: EM without any privacy protection.
making estimation challenging under DP constraints. We also compare to this method and denote it
as SSLP (sampling strategy for local privacy).
In addition, to understand how our approximation to second order statistics affects the performance,
we also compare to the case that r0 a is given to the server directly where a = {?1, 1} with probability 1/2. In this way, the server can obtain the exact second order statistics using r0 r0T . Note that
this is not a valid privacy preserving mechanism. Finally, we compare to the case that the algorithm
can access private user rating vectors multiple times and update the parameters iteratively. Again,
this is not a valid mechanism but illustrates how much could be gained.
Figure 2 shows the performance as a function of the number of private users. The standard Laplace
noise method performs reasonably well when = 5, however the corresponding privacy guarantee
is very weak. SSLP improves the accuracy mildly.
In contrast, with the privacy mechanism we defined in section 4 the test RMSE decreases significantly as more private users are added. If we use the exact second order information without the
sampling method, the final test RMSE would be reduced by 0.07 compared to PMD. Lastly, full
EM without privacy protection reduces the test RMSE by another 0.08. These performance gaps are
expected because there is an inherent trade-off between accuracy and privacy.
6
Conclusion
Our contributions in this paper are three-fold. First, we provide explicit guarantees for estimating
item features in matrix completion problems. Second, we show how the resulting estimates, if shared
with new users, can be used to predict their ratings depending on the degree of overlap between
their private ratings and the relevant item subspace. The empirical results demonstrate that only a
small number of public users with large number of ratings suffices for a good performance. Third,
we introduce a new privacy mechanism for releasing 2nd order information needed for estimating
item features while maintaining 1st order deniability. The experiments show that this mechanism
indeed performs well in comparison to other mechanisms. We believe that allowing different levels
of privacy is an exciting research topic. An extension of our work would be applying the privacy
mechanism to the learning of graphical models in which 2nd or higher order information plays an
important role.
7
Acknowledgement
The work was partially supported by Google Research Award and funding from Qualcomm Inc.
8
References
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9
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4,815 | 536 | Tangent Prop - A formalism for specifying
selected invariances in an adaptive network
Patrice Simard
AT&T Bell Laboratories
101 Crawford Corner Rd
Holmdel, NJ 07733
Yann Le Cun
AT&T Bell Laboratories
101 Crawford Corner Rd
Holmdel, NJ 07733
Bernard Victorri
Universite de Caen
Caen 14032 Cedex
France
John Denker
AT&T Bell Laboratories
101 Crawford Corner Rd
Holmdel, NJ 07733
Abstract
In many machine learning applications, one has access, not only to training
data, but also to some high-level a priori knowledge about the desired behavior of the system. For example, it is known in advance that the output
of a character recognizer should be invariant with respect to small spatial distortions of the input images (translations, rotations, scale changes,
etcetera).
We have implemented a scheme that allows a network to learn the derivative of its outputs with respect to distortion operators of our choosing.
This not only reduces the learning time and the amount of training data,
but also provides a powerful language for specifying what generalizations
we wish the network to perform.
1
INTRODUCTION
In machine learning, one very often knows more about the function to be learned
than just the training data. An interesting case is when certain directional derivatives of the desired function are known at certain points. For example, an image
895
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Simard, Victorri, Le Cun, and Denker
Figure 1: Top: Small rotations of an original digital image of the digit "3" (center).
Middle: Representation of the effect of the rotation in the input vector space space
(assuming there are only 3 pixels). Bottom: Images obtained by moving along the
tangent to the transformation curve for the same original digital image (middle).
recognition system might need to be invariant with respect to small distortions of
the input image such as translations, rotations, scalings, etc.; a speech recognition
system n.ight need to be invariant to time distortions or pitch shifts. In other
words, the derivative of the system's output should be equal to zero when the input
is transformed in certain ways.
Given a large amount of training data and unlimited training time, the system
could learn these invariances from the data alone, but this is often infeasible. The
limitation on data can be overcome by training the system with additional data
obtained by distorting (translating, rotating, etc.) the original patterns (Baird,
1990). The top of Fig. 1 shows artificial data generated by rotating a digital image of
the digit "3" (with the original in the center). This procedure, called the "distortion
model" , has two drawbacks. First, the user must choose the magnitude of distortion
and how many instances should be generated. Second, and more importantly, the
distorted data is highly correlated with the original data. This makes traditional
learning algorithms such as back propagation very inefficient. The distorted data
carries only a very small incremental amount of information, since the distorted
patterns are not very different from the original ones. It may not be possible to
adjust the learning system so that learning the invariances proceeds at a reasonable
rate while learning the original points is non-divergent.
The key idea in this paper is that it is possible to directly learn the effect on
the output of distorting the input, independently from learning the undistorted
Tangent Prop-A formalism for specifying selected invariances in an adaptive network
F(x)
F(x)
x1
x2
x3
x4
x
x1
x2
x3
x4
x
Figure 2: Learning a given function (solid line) from a limited set of example (Xl
to X4). The fitted curves are shown in dotted line. Top: The only constraint is that
the fitted curve goes through the examples. Bottom: The fitted curves not only
goes through each examples but also its derivatives evaluated at the examples agree
with the derivatives of the given function.
patterns. When a pattern P is transformed (e.g. rotated) with a transformation
s that depends on one parameter a (e.g. the angle of the rotation), the set of all
the transformed patterns S(P) = {sea, P) Va} is a one dimensional curve in the
vector space of the inputs (see Fig. 1). In certain cases, such as rotations of digital
images, this curve must be made continuous using smoothing techniques, as will be
shown below. When the set of transformations is parameterized by n parameters
ai (rotation, translation, scaling, etc.), S(P) is a manifold of at most n dimensions.
The patterns in S(P) that are obtained through small transformations of P, i.e.
the part of S( P) that is close to P, can be approximated by a plane tangent to
the manifold S(P) at point P. Small transformations of P can be obtained by
adding to P a linear combination of vectors that span the tangent plane (tangent
vectors). The images at the bottom of Fig. 1 were obtained by that procedure.
More importantly, the tangent vectors can be used to specify high order constraints
on the function to be learned, as explained below.
To illustrate the method, consider the problem of learning a single-valued function
F from a limited set of examples. Fig. 2 (left) represents a simple case where the
desired function F (solid line) is to be approximated by a function G (dotted line)
from four examples {(Xi, F(Xi))}i=1,2,3,4. As exemplified in the picture, the fitted
function G largely disagrees with the desired function F between the examples. If
the functions F and G are assumed to be differentiable (which is generally the case),
the approximation G can be greatly improved by requiring that G's derivatives
evaluated at the points {xd are equal to the derivatives of F at the same points
(Fig. 2 right). This result can be extended to multidimensional inputs. In this case,
we can impose the equality of the derivatives of F and G in certain directions, not
necessarily in all directions of the input space.
Such constraints find immediate use in traditional learning problems. It is often the
case that a priori knowledge is available on how the desired function varies with
897
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Simard, Victorri, Le Cun, and Denker
pattern P
rotated by ex
pattern P
tangent
vector
--
Figure 3: How to compute a tangent vector for a given transformation (in this case
a rotation).
respect to some transformations of the input. It is straightforward to derive the
corresponding constraint on the directional derivatives of the fitted function G in
the directions of the transformations (previously named tangent vectors). Typical
examples can be found in pattern recognition where the desired classification function is known to be invariant with respect to some transformation of the input such
as translation, rotation, scaling, etc., in other words, the directional derivatives of
the classification function in the directions of these transformations is zero.
2
IMPLEMENTATION
The implementation can be divided into two parts. The first part consists in computing the tangent vectors. This part is independent from the learning algorithm
used subsequently. The second part consists in modifying the learning algorithm
(for instance backprop) to incorporate the information about the tangent vectors.
Part I: Let x be an input pattern and s be a transformation operator acting
on the input space and depending on a parameter a. If s is a rotation operator
for instance, then s( a, x) denotes the input x rotated by the angle a. We will
require that the transformation operator s be differentiable with respect to a and
x, and that s(O, x) = x. The tangent vector is by definition 8s(a, x)/8a. It can be
approximated by a finite difference, as shown in Fig. 3. In the figure, the input space
is a 16 by 16 pixel image and the patterns are images of handwritten digits. The
transformations considered are rotations of the digit images. The tangent vector
is obtained in two steps. First the image is rotated by an infinitesimal amount a.
This is done by computing the rotated coordinates of each pixel and interpolating
the gray level values at the new coordinates. This operation can be advantageously
combined with some smoothing using a convolution. A convolution with a Gaussian
provides an efficient interpolation scheme in O(nm) multiply-adds, where nand m
are the (gaussian) kernel and image sizes respectively. The next step is to subtract
(pixel by pixel) the rotated image from the original image and to divide the result
Tangent Prop-A formalism for specifying selected invariances in an adaptive network
by the scalar 0 (see Fig. 3). If Ie types of transformations are considered, there
will be Ie different tangent vectors per pattern. For most algorithms, these do not
require any storage space since they can be generated as needed from the original
pattern at negligible cost.
Part IT: Tangent prop is an extension of the backpropagation algorithm, allowing
it to learn directional derivatives. Other algorithms such as radial basis functions
can be extended in a similar fashion.
To implement our idea, we will modify the usual weight-update rule:
~w =
oE
-7] ow
is replaced with
~w
=
0
-7] ow (E
+ J.tEr)
(1)
where 7] is the learning rate, E the usual objective function, Er an additional objective function (a regularizer) that measures the discrepancy between the actual and
desired directional derivatives in the directions of some selected transformations,
and J.t is a weighting coefficient.
=
Let x be an input pattern, y G(x) be the input-output function of the network.
The regularizer Er is of the form
Er(x)
:e
etrainingset
where Er(x) is
(2)
Here, Ki(x) is the desired directional derivative of G in the direction induced by
transformation Si applied to pattern x. The second term in the norm symbol is the
actual directional derivative, which can be rewritten as
=G'{x). OSi(O, x)
0=0
00
0=0
where G'(x) is the Jacobian of G for pattern x, and OSi(O, x)Joo is the tangent
vector associated to transformation Si as described in Part I. Multiplying the tangent
vector by the Jacobian involves one forward propagation through a "linearized"
version of the network. In the special case where local invariance with respect to
the Si'S is desired, Ki(x) is simply set to o.
Composition of transformations: The theory of Lie groups (Gilmore, 1974)
ensures that compositions of local (small) transformations Si correspond to linear
combinations of the corresponding tangent vectors (the local transformations Si
have a structure of Lie algebra). Consequently, if Er{x) = 0 is verified, the network
derivative in the direction of a linear combination of the tangent vectors is equal
to the same linear combination of the desired derivatives. In other words if the
network is successfully trained to be locally invariant with respect to, say, horizontal
translation and vertical translations, it will be invariant with respect to compositions
thereof.
We have derived and implemented an efficient algorithm, "tangent prop" , for performing the weight update (Eq. 1). It is analogous to ordinary backpropagation,
899
900
Simard, Victorri, Le Cun, and Denker
W l+ 1
W'+l
Iti
Iti
e: l
,
b'.-l
j3J-1
,
x?'-I
Network
e;-I
Jacobian nework
Figure 4: forward propagated variables (a, x, a, e), and backward propagated variables (b, y, p, t/J) in the regular network (roman symbols) and the Jacobian (linearized) network (greek symbols)
but in addition to propagating neuron activations, it also propagates the tangent
vectors. The equations can be easily derived from Fig. 4.
Forward propagation:
a~
?
=~
wL x '.-l
L...J I, ,
x~
= u(aD
(3)
i
Tangent forward propagation:
,_
ai -
~
,
~'-1
L...J wW"i
i
e! = u'(a~)a~
(4)
Tangent gradient backpropagation:
~ w'+1.I.l+1
(3i1 -- L...J
Iti ?lit
(5)
It
Gradient backpropagation:
~ w1+ 1yl+1
bi' -- L...J
Iti It
(6)
It
Weight update:
8[E(W, Up) + I'Er (W, Up, Tp)] _ 1-1 , + ~'-l.I.'
- Xi Yi I'\oi ?Ii
8w??'
I,
(7)
Tangent Prop--A formalism for specifying selected invariances in an adaptive network
60
50
%Erroron
the test set
20
10
160
320
Training set size
Figure 5: Generalization performance curve as a function of the training set size for
the tangent prop and the backprop algorithms
The regularization parameter jJ is tremendously important, because it determines
the tradeoff between minimizing the usual objective function and minimizing the
directional derivative error.
3
RESULTS
Two experiments illustrate the advantages of tangent prop. The first experiment
is a classification task, using a small (linearly separable) set of 480 binarized handwritten digit . The training sets consist of 10, 20, 40, 80, 160 or 320 patterns, and
the training set contains the remaining 160 patterns. The patterns are smoothed
using a gaussian kernel with standard deviation of one half pixel. For each of the
training set patterns, the tangent vectors for horizontal and vertical translation
are computed. The network has two hidden layers with locally connected shared
weights, and one output layer with 10 units (5194 connections, 1060 free parameters) (Le Cun, 1989). The generalization performance as a function of the training
set size for traditional backprop and tangent prop are compared in Fig. 5. We have
conducted additional experiments in which we implemented not only translations
but also rotations, expansions and hyperbolic deformations. This set of 6 generators is a basis for all linear transformations of coordinates for two dimensional
images. It is straightforward to implement other generators including gray-Ievelshifting, "smooth" segmentation, local continuous coordinate transformations and
independent image segment transformations.
The next experiment is designed to show that in applications where data is highly
901
902
Simard, Victorri, Le Cun, and Denker
A-. NMSE VI.
Av"ge NMSE VI 1ge
0.15
.15
0.1
.1
o
o
oL-~~==~~=;~==+=~~~
1000 2000 3000 4000 5000 6000 7000 8000 0000 10000
0
.....
1000 2000 3000 4000 5000 6000 7000 8000 0000 10000
.....
15
"
-
o
15
"
-
0
-0.5
-.5
-1
-1
+--_+_-_--+_-_+_-_-_
-1
-0.5
0
0.5
1.5
-1 .5
-1 .5
Distortion model
+--_+_-_--+--_+_-_-__t
.5
1.5
- .5
o
-1.5
-1.5
-1
Tangent prop
Figure 6: Comparison of the distortion model (left column) and tangent prop (right
column). The top row gives the learning curves (error versus number of sweeps
through the training set). The bottom row gives the final input-output function of
the network; the dashed line is the result for unadorned back prop.
Tangent Prop-A formalism for specifying selected invariances in an adaptive network
correlated, tangent prop yields a large speed advantage. Since the distortion model
implies adding lots of highly correlated data, the advantage of tangent prop over
the distortion model becomes clear.
The task is to approximate a function that has plateaus at three locations. We want
to enforce local invariance near each of the training points (Fig. 6, bottom). The
network has one input unit, 20 hidden units and one output unit. Two strategies are
possible: either generate a small set of training point covering each of the plateaus
(open squares on Fig. 6 bottom), or generate one training point for each plateau
(closed squares), and enforce local invariance around them (by setting the desired
derivative to 0). The training set of the former method is used as a measure the
performance for both methods. All parameters were adjusted for approximately
optimal performance in all cases. The learning curves for both models are shown in
Fig. 6 (top). Each sweep through the training set for tangent prop is a little faster
since it requires only 6 forward propagations, while it requires 9 in the distortion
model. As can be seen, stable performance is achieved after 1300 sweeps for the
tangent prop, versus 8000 for the distortion model. The overall speedup is therefore
about 10.
Tangent prop in this example can take advantage of a very large regularization term.
The distortion model is at a disadvantage because the only parameter that effectively controls the amount of regularization is the magnitude of the distortions, and
this cannot be increased to large values because the right answer is only invariant
under small distortions.
4
CONCLUSIONS
When a priori information about invariances exists, this information must be made
available to the adaptive system. There are several ways of doing this, including the
distortion model and tangent prop. The latter may be much more efficient in some
applications, and it permits separate control of the emphasis and learning rate for
the invariances, relative to the original training data points. Training a system to
have zero derivatives in some directions is a powerful tool to express invariances to
transformations of our choosing. Tests of this procedure on large-scale applications
(handwritten zipcode recognition) are in progress.
References
Baird, H. S. (1990). Document Image Defect Models. In IAPR 1990 Workshop on
Sytactic and Structural Pattern Recognition, pages 38-46, Murray Hill, NJ.
Gilmore, R. (1974). Lie Groups, Lie Algebras and some of their Applications. Wiley,
New York.
Le Cun, Y. (1989) . Generalization and Network Design Strategies. In Pfeifer, R.,
Schreter, Z., Fogelman, F., and Steels, L., editors, Connectionism in Perspective, Zurich, Switzerland. Elsevier. an extended version was published as a
technical report of the University of Toronto.
903
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4,816 | 5,360 | Content-based recommendations
with Poisson factorization
Laurent Charlin
Department of Computer Science
Columbia University
New York, NY 10027
[email protected]
Prem Gopalan
Department of Computer Science
Princeton University
Princeton, NJ 08540
[email protected]
David M. Blei
Departments of Statistics & Computer Science
Columbia University
New York, NY 10027
[email protected]
Abstract
We develop collaborative topic Poisson factorization (CTPF), a generative model
of articles and reader preferences. CTPF can be used to build recommender systems by learning from reader histories and content to recommend personalized
articles of interest. In detail, CTPF models both reader behavior and article texts
with Poisson distributions, connecting the latent topics that represent the texts
with the latent preferences that represent the readers. This provides better recommendations than competing methods and gives an interpretable latent space for
understanding patterns of readership. Further, we exploit stochastic variational
inference to model massive real-world datasets. For example, we can fit CPTF
to the full arXiv usage dataset, which contains over 43 million ratings and 42
million word counts, within a day. We demonstrate empirically that our model
outperforms several baselines, including the previous state-of-the art approach.
1
Introduction
In this paper we develop a probabilistic model of articles and reader behavior data. Our model is
called collaborative topic Poisson factorization (CTPF). It identifies the latent topics that underlie the articles, represents readers in terms of their preferences for those topics, and captures how
documents about one topic might be interesting to the enthusiasts of another.
As a recommendation system, CTPF performs well in the face of massive, sparse, and long-tailed
data. Such data is typical because most readers read or rate only a few articles, while a few readers
may read thousands of articles. Further, CTPF provides a natural mechanism to solve the ?cold start?
problem, the problem of recommending previously unread articles to existing readers. Finally, CTPF
provides a new exploratory window into the structure of the collection. It organizes the articles
according to their topics and identifies important articles both in terms of those important to their
topic and those that have transcended disciplinary boundaries.
We illustrate the model with an example. Consider the classic paper ?Maximum likelihood from
incomplete data via the EM algorithm? [5]. This paper, published in the Journal of the Royal Statistical Society (B) in 1977, introduced the expectation-maximization (EM) algorithm. The EM
algorithm is a general method for finding maximum likelihood estimates in models with hidden
random variables. As many readers will know, EM has had an enormous impact on many fields,
1
including computer vision, natural language processing, and machine learning. This original paper
has been cited over 37,000 times.
Figure 1 illustrates the CTPF representation of the EM paper. (This model was fit to the shared
libraries of scientists on the Mendeley website; the number of readers is 80,000 and the number of
articles is 261,000.) In the figure, the horizontal axes contains topics, latent themes that pervade
the collection [2]. Consider the black bars in the left figure. These represent the topics that the
EM paper is about. (These were inferred from the abstract of the paper.) Specifically, it is about
probabilistic modeling and statistical algorithms. Now consider the red bars on the right, which are
summed with the black bars. These represent the preferences of the readers who have the EM paper
in their libraries. CTPF has uncovered the interdisciplinary impact of the EM paper. It is popular
with readers interested in many fields outside of those the paper discusses, including computer vision
and statistical network analysis.
The CTPF representation has advantages. For forming recommendations, it naturally interpolates
between using the text of the article (the black bars) and the inferred representation from user behavior data (the red bars). On one extreme, it recommends rarely or never read articles based mainly on
their text; this is the cold start problem. On the other extreme, it recommends widely-read articles
based mainly on their readership. In this setting, it can make good inferences about the red bars.
Further, in contrast to traditional matrix factorization algorithms, we combine the space of preferences and articles via interpretable topics. CTPF thus offers reasons for making recommendations,
readable descriptions of reader preferences, and an interpretable organization of the collection. For
example, CTPF can recognize the EM paper is among the most important statistics papers that has
had an interdisciplinary impact.
In more detail, CTPF draws on ideas from two existing models: collaborative topic regression [20]
and Poisson factorization [9]. Poisson factorization is a form of probabilistic matrix factorization [17] that replaces the usual Gaussian likelihood and real-valued representations with a Poisson
likelihood and non-negative representations. Compared to Gaussian factorization, Poisson factorization enjoys more efficient inference and better handling of sparse data. However, PF is a basic
recommendation model. It cannot handle the cold start problem or easily give topic-based representations of readers and articles.
Collaborative topic regression is a model of text and reader data that is based on the same intuitions
as we described above. (Wang and Blei [20] also use the EM paper as an example.) However, in its
implementation, collaborative topic regression is a non-conjugate model that is complex to fit, difficult to work with on sparse data, and difficult to scale without stochastic optimization. Further, it is
based on a Gaussian likelihood of reader behavior. Collaborative Poisson factorization, because it is
based on Poisson and gamma variables, enjoys an easier-to-implement and more efficient inference
algorithm and a better fit to sparse real-world data. As we show below, it scales more easily and
provides significantly better recommendations than collaborative topic regression.
2
The collaborative topic Poisson factorization model
In this section we describe the collaborative topic Poisson factorization model (CTPF), and discuss
its statistical properties. We are given data about users (readers) and documents (articles), where
each user has read or placed in his library a set of documents. The rating rud equals one if user u
consulted document d, can be greater than zero if the user rated the document and is zero otherwise.
Most of the values of the matrix y are typically zero, due to sparsity of user behavior data.
Background: Poisson factorization. CTPF builds on Poisson matrix factorization [9]. In collaborative filtering, Poisson factorization (PF) is a probabilistic model of users and items. It associates
each user with a latent vector of preferences, each item with a latent vector of attributes, and constrains both sets of vectors to be sparse and non-negative. Each cell of the observed matrix is
assumed drawn from a Poisson distribution, whose rate is a linear combination of the corresponding
user and item attributes. Poisson factorization has also been used as a topic model [3], and developed
as an alternative text model to latent Dirichlet allocation (LDA). In both applications Poisson factorization has been shown to outperform competing methods [3, 9]. PF is also more easily applicable
to real-life preference datasets than the popular Gaussian matrix factorization [9].
2
40
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!
30
30
20
algorithm, efficient,
optimal, clustering,
optimization, show
probability, prior,
bayesian, likelihood,
inference, maximum
20
network, connected,
modules, nodes,
links, topology
!
!
10
image, object,
matching, tracking"
motion,segmentation
10
!
!
!
!
!
!!!
!
!
!
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!!!
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Topic
Topic
Figure 1: We visualized the inferred topic intensities ? (the black bars) and the topic offsets (the
red bars) of an article in the Mendeley [13] dataset. The plots are for the statistics article titled
?Maximum likelihood from incomplete data via the EM algorithm?. The black bars represent the
topics that the EM paper is about. These include probabilistic modeling and statistical algorithms.
The red bars represent the preferences of the readers who have the EM paper in their libraries. It
is popular with readers interested in many fields outside of those the paper discusses, including
computer vision and statistical network analysis.
Collaborative topic Poisson factorization. CTPF is a latent variable model of user ratings and
document content. CTPF uses Poisson factorization to model both types of data. Rather than modeling them as independent factorization problems, we connect the two latent factorizations using a
correction term [20] which we?ll describe below.
Suppose we have data containing D documents and U users. CTPF assumes a collection of K
unormalized topics ?1:K . Each topic ?k is a collection of word intensities on a vocabulary of size
V . Each component ?vk of the unnormalized topics is drawn from a Gamma distribution. Given
the topics, CTPF assumes that a document d is generated with a vector of K latent topic intensities
?d , and represents users with a vector of K latent topic preferences ?u . Additionally, the model
associates each document with K latent topic offsets d that capture the document?s deviation from
the topic intensities. These deviations occur when the content of a document is insufficient to explain
its ratings. For example, these variables can capture that a machine learning article is interesting to
a biologist, because other biologists read it.
We now define a generative process for the observed word counts in documents and observed user
ratings of documents under CTPF:
1. Document model:
(a) Draw topics ?vk ? Gamma(a, b)
(b) Draw document topic intensities ?dk ? Gamma(c, d)
(c) Draw word count wdv ? Poisson(?dT ?v ).
2. Recommendation model:
(a) Draw user preferences ?uk ? Gamma(e, f )
(b) Draw document topic offsets dk ? Gamma(g, h)
(c) Draw rud ? Poisson(?uT (?d + d )).
CTPF specifies that the conditional probability that a user u rated document d with rating rud is
drawn from a Poisson distribution with rate parameter ?uT (?d + d ). The form of the factorization
couples the user preferences for the document topic intensities ?d and the document topic offsets d .
This allows the user preferences to be interpreted as affinity to latent topics.
CTPF has two main advantages over previous work (e.g., [20]), both of which contribute to its
superior empirical performance (see Section 5). First, CTPF is a conditionally conjugate model
when augmented with auxiliary variables. This allows CTPF to conveniently use standard variational
inference with closed-form updates (see Section 3). Second, CTPF is built on Poisson factorization;
it can take advantage of the natural sparsity of user consumption of documents and can analyze
massive real-world data. This follows from the likelihood of the observed data under the model [9].
3
We analyze user preferences and document content with CTPF via its posterior distribution over
latent variables p(?1:K , ?1:D , 1:D , ?1:U |w, r). By estimating this distribution over the latent structure, we can characterize user preferences and document readership in many useful ways. Figure 1
gives an example.
Recommending old and new documents. Once the posterior is fit, we use CTPF to recommend
in-matrix documents and out-matrix or cold-start documents to users. We define in-matrix documents as those that have been rated by at least one user in the recommendation system. All other
documents are new to the system. A cold-start recommendation of a new document is based entirely
on its content. For predicting both in-matrix and out-matrix documents, we rank each user?s unread
documents by their posterior expected Poisson parameters,
scoreud = E[?uT (?d + d )|w, r].
(1)
The intuition behind the CTPF posterior is that when there is no reader data, we depend on the
topics to make recommendations. When there is both reader data and article content, this gives
information about the topic offsets. We emphasize that under CTPF the in-matrix recommendations
and cold-start recommendations are not disjoint tasks. There is a continuum between these tasks.
For example, the model can provide better predictions for articles with few ratings by leveraging its
latent topic intensities ?d .
3
Approximate posterior inference
Given a set of observed document ratings r and their word counts w, our goal is to infer the topics
?1:K , the user preferences ?1:U , the document topic intensities ?1:D , the document topic offsets 1:D .
With estimates of these quantities, we can recommend in-matrix and out-matrix documents to users.
Computing the exact posterior distribution p(?1:K , ?1:D , 1:D , ?1:U |w, r) is intractable; we use variational inference [15]. We first develop a coordinate ascent algorithm?a batch algorithm that iterates over only the non-zero document-word counts and the non-zero user-document ratings. We then
present a more scalable stochastic variational inference algorithm.
In variational inference we first define a parameterized family of distributions over the hidden variables. We then fit the parameters to find a distribution that minimizes the KL divergence to the
posterior. The model is conditionally conjugate if the complete conditional of each latent variable
is in the exponential family and is in the same family as its prior. (The complete conditional is the
conditional distribution of a latent variable given the observations and the other latent variables in
the model [8].) For the class of conditionally conjugate models, we can perform this optimization
with a coordinate ascent algorithm and closed form updates.
Auxiliary variables. To facilitate inference, we first augment CTPF with auxiliary variables. Following Ref. [6] and Ref. [9],
P we add K latent variables zdv,k ? Poisson(?dk ?vk ), which are
integers such that wdv =
k zdv,k . Similarly, for each observed rating rud , we add K latent
a
b
variables yud,k
? Poisson(?uk ?dk ) and K latent variables yud,k
? Poisson(?uk dk ) such that
P a
b
rud = k yud,k + yud,k . A sum of independent Poisson random variables is itself a Poisson with
rate equal to the sum of the rates. Thus, these new latent variables preserve the marginal distribution
of the observations, wdv and rud . Further, when the observed counts are 0, these auxiliary variables
are not random. Consequently, our inference procedure need only consider the auxiliary variables
for non-zero observations.
CTPF with the auxiliary variables is conditionally conjugate; its complete conditionals are shown
in Table 1. The complete conditionals of the Gamma variables ?vk , ?dk , dk , and ?uk are Gamma
distributions with shape and rate parameters as shown in Table 1. For the auxiliary Poisson variables,
observe that zdv is a K-dimensional latent vector of Poisson counts, which when conditioned on
their observed sum wdv , is distributed as a multinomial [14, 4]. A similar reasoning underlies the
conditional for yud which is a 2K-dimensional latent vector of Poisson counts. With our complete
conditionals in place, we now derive the coordinate ascent algorithm for the expanded set of latent
variables.
4
Latent Variable
Type
?dk
?vk
?uk
dk
zdv
Gamma
Gamma
Gamma
Gamma
Mult
yud
Mult
Complete conditional
P
P a
P
P
c + v zdv,k + u yud,k
, d + v ?vk + u ?uk
P
P
a + d zdv,k , b + d ?dk
P a
P b
P
e + d yud,k
+ d yud,k
, f + d (?dk + dk )
P b
P
g + u yud,k , h + u ?uk
log
( ?dk + log ?vk
log ?uk + log ?dk if k < K,
log ?uk + log dk if K ? k < 2K
Variational parameters
shp ?rte
??dk
, ?dk
shp ?rte
?
?vk
, ?vk
shp
rte
??uk
, ??uk
shp rte
?dk , ?dk
?dv
?ud
Table 1: CTPF: latent variables, complete conditionals and variational parameters.
Variational family. We define the mean-field variational family q(?, ?, ?, , z, y) over the latent
variables where we consider these variables to be independent and each governed by its own distribution,
Y
Y
Y
Y
Y
q(?, ?, , ?, z, y) =
q(?vk )
q(?dk )q(dk )
q(?uk )
q(yud,k )
q(zdv,k ).
(2)
v,k
d,k
u,k
ud,k
dv,k
The variational factors for topic components ?vk , topic intensities ?dk , user preferences ?uk are
all Gamma distributions?the same as their conditional distributions?with freely set shape and
rate variational parameters. For example, the variational distribution for the topic intensities ?dk
shp ?rte
is Gamma(?dk ; ??dk
, ?dk ). We denote shape with the superscript ?shp? and rate with the superscript
?rte?. The variational factor for zdv is a multinomial Mult(wdv , ?dv ) where the variational parameter
a
b
?dv is a point on the K-simplex. The variational factor for yud = (yud
, yud
) is also a multinomial
Mult(rud , ?ud ) but here ?ud is a point in the 2K-simplex.
Optimal coordinate updates. In coordinate ascent we iteratively optimize each variational parameter while holding the others fixed. Under the conditionally conjugate augmented CTPF, we
can optimize each coordinate in closed form by setting the variational parameter equal to the expected natural parameter (under q) of the complete conditional. For a given random variable, this
expected conditional parameter is the expectation of a function of the other random variables and
observations. (For details, see [9, 10]). We now describe two of these updates; the other updates
are similarly derived.
The update for the variational shape and rate parameters of topic intensities ?dk is
??dk
= hc +
P
v
wdv ?dv,k +
P
u rud ?ud,k , d
+
shp
??vk
v ??rte
vk
P
+
shp
??uk
rte i.
u ??uk
P
(3)
The Gamma update in Equation 3 derives from the expected natural parameter (under q) of the
complete conditional for ?dk in Table 1. In the shape parameter for topic intensities for document d,
a
we use that Eq [zdv,k ] = wdv ?dv,k for the word indexed by v and Eq [yud,k
] = rud ?ud,k for the user
indexed by u. In the rate parameter, we use that the expectation of a Gamma variable is the shape
divided by the rate.
The update for the multinomial ?dv is
?dv
?
shp
shp
rte
rte
exp{?(??dk
) ? log ??dk
+ ?(??vk
) ? log ??vk
},
(4)
where ?(?) is the digamma function (the first derivative of the log ? function). This update comes
shp
rte
from the expectation of the log of a Gamma variable, for example, Eq [log ?dk ] = ?(??dk
) ? log ??dk
.
Coordinate ascent algorithm. The CTPF coordinate ascent algorithm is illustrated in Figure 2.
Similar to the algorithm of [9], our algorithm is efficient on sparse matrices. In steps 1 and 2, we
need to only update variational multinomials for the non-zero word counts wdv and the non-zero
ratings rud . In step 3, the sums over the expected zdv,k and the expected yud,k need only to consider
non-zero observations. This efficiency comes from the likelihood of the full matrix depending only
on the non-zero observations [9].
5
Initialize the topics ?1:K and topic intensities ?1:D using LDA [2] as described in Section 3.
Repeat until convergence:
1. For each word count wdv > 0, set ?dv to the expected conditional parameter of zdv .
2. For each rating rud > 0, set ?ud to the expected conditional parameter of yud .
3. For each document d and each k, update the block of variational topic intensities ??dk to
their expected conditional parameters using Equation 3. Perform similar block updates
for ??vk , ??uk and ?dk , in sequence.
Figure 2: The CTPF coordinate ascent algorithm. The expected conditional parameters of the latent
variables are computed from Table 1.
Stochastic algorithm. The CTPF coordinate ascent algorithm is efficient: it only iterates over the
non-zero observations in the observed matrices. The algorithm computes approximate posteriors for
datasets with ten million observations within hours (see Section 5). To fit to larger datasets, within
hours, we develop an algorithm that subsamples a document and estimates variational parameters
using stochastic variational inference [10]. The stochastic algorithm is also useful in settings where
new items continually arrive in a stream. The CTPF SVI algorithm is described in the Appendix.
Computational efficiency. The SVI algorithm is more efficient than the batch algorithm. The
batch algorithm has a per-iteration computational complexity of O((W + R)K) where R and W
are the total number of non-zero observations in the document-user and document-word matrices,
respectively. For the SVI algorithm, this is O((wd + rd )K) where rd is the number of users rating
the sampled document d and wd is the number of unique words in it. (We assume that a single
document is sampled in each iteration.) In Figure 2, the sums involving the multinomial parameters
can be tracked for efficient memory usage. The bound on memory usage is O((D + V + U )K).
Hyperparameters, initialization and stopping criteria: Following [9], we fix each Gamma shape
and rate hyperparameter at 0.3. We initialize the variational parameters for ?uk and dk to the prior
on the corresponding latent variables and add small uniform noise. We initialize ??vk and ??dk using
estimates of their normalized counterparts from LDA [2] fitted to the document-word matrix w. For
the SVI algorithm described in the Appendix, we set learning rate parameters ?0 = 1024, ? = 0.5
and use a mini-batch size of 1024. In both algorithms, we declare convergence when the change in
expected predictive likelihood is less than 0.001%.
4
Related work
Several research efforts propose joint models of item covariates and user activity. Singh and Gordon [19] present a framework for simultaneously factorizing related matrices, using generalized link
functions and coupled latent spaces. Hong et al. [11] propose Co-factorization machines for modeling user activity on twitter with tweet features, including content. They study several design choices
for sharing latent spaces. While CTPF is roughly an instance of these frameworks, we focus on the
task of recommending articles to readers.
Agarwal and Chen [1] propose fLDA, a latent factor model which combines document features
through their empirical LDA [2] topic intensities and other covariates, to predict user preferences.
The coupling of matrix decomposition and topic modeling through shared latent variables is also
considered in [18, 22]. Like fLDA, both papers tie latent spaces without corrective terms. Wang
and Blei [20] have shown the importance of using corrective terms through the collaborative topic
regression (CTR) model which uses a latent topic offset to adjust a document?s topic proportions.
CTR has been shown to outperform a variant of fLDA [20]. Our proposed model CTPF uses the
CTR approach to sharing latent spaces.
CTR [20] combines topic modeling using LDA [2] with Gaussian matrix factorization for one-class
collaborative filtering [12]. Like CTPF, the underlying MF algorithm has a per-iteration complexity that is linear in the number of non-zero observations. Unlike CTPF, CTR is not conditionally
6
Mean precision
CTPF (Section 2)
Decoupled PF (Section 5)
mendeley.in
Content Only
Ratings Only [9]
mendeley.out
Collaborative Topic Regression [20]
arxiv.in
arxiv.out
0.4%
0.4%
1.0%
2.0%
1.5%
1.0%
0.5%
0.3%
0.2%
0.5%
0.1%
10 30 50 70
100
10 30 50 70
mendeley.in
4%
2%
50
70
100
100
10 30 50 70
100
Collaborative Topic Regression [20]
arxiv.in
arxiv.out
4%
5%
4%
3%
2%
1%
30
0.1%
10 30 50 70
mendeley.out
6%
10
0.2%
Decoupled PF (Section
5)
Content
Only
Ratings Only [9]
Number
of recommendations
CTPF (Section 2)
Mean recall
100
0.3%
2.0%
1.5%
1.0%
0.5%
3%
2%
1%
0%
10
30
50
70
100
10
30
50
70
100
10
30
50
70
100
Number of recommendations
Figure 3: The CTPF coordinate ascent algorithm outperforms CTR and other competing algorithms on both
in-matrix and out-matrix predictions. Each panel shows the in-matrix or out-matrix recommendation task on
the Mendeley data set or the 1-year arXiv data set. Note that the Ratings-only model cannot make out-matrix
predictions. The mean precision and mean recall are computed from a random sample of 10,000 users.
conjugate, and the inference algorithm depends on numerical optimization of topic intensities. Further, CTR requires setting confidence parameters that govern uncertainty around a class of observed
ratings. As we show in Section 5, CTPF scales more easily and provides significantly better recommendations than CTR.
.
5
Empirical results
We use the predictive approach to evaluating model fitness [7], comparing the predictive accuracy
of the CTPF coordinate ascent algorithm in Figure 2 to collaborative topic regression (CTR) [21].
We also compare to variants of CTPF to demonstrate that coupling the latent spaces using corrective
terms is essential for good predictive performance, and that CTPF predicts significantly better than
its variants and CTR. Finally, we explore large real-world data sets revealing the interaction patterns
between readers and articles.
Data sets. We study the CTPF algorithm of Figure 2 on two data sets. The Mendeley data set [13]
of scientific articles is a binary matrix of 80,000 users and 260,000 articles with 5 million observations. Each cell corresponds to the presence or absence of an article in a scientist?s online library.
The arXiv data set is a matrix of 120,297 users and 825,707 articles, with 43 million observations.
Each observation indicates whether or not a user has consulted an article (or its abstract). This data
was collected from the access logs of registered users on the http://arXiv.org paper repository. The
articles and the usage data spans a timeline of 10 years (2003-2012). In our experiments on predictive performance, we use a subset of the data set, with 64,978 users 636,622 papers and 7.6 million
clicks, which spans one year of usage data (2012). We treat the user clicks as implicit feedback and
specifically as binary data. For each article in the above data sets, we remove stop words and use
tf-idf to choose the top 10,000 distinct words (14,000 for arXiv) as the vocabulary. We implemented
the batch and stochastic algorithms for CTPF in 4500 lines of C++ code.1
Competing methods. We study the predictive performance of the following models. With the
exception of the Poisson factorization [9], which does not model content, the topics and topic intensities (or proportions) in all CTPF models are initialized using LDA [2], and fit using batch
variational inference. We set K = 100 in all of our experiments.
? CTPF: CTPF is our proposed model (Section 2) with latent user preferences tied to a single
vector ?u , and interpreted as affinity to latent topics ?.
1
Our source code is available from: https://github.com/premgopalan/collabtm
7
Topic: "Statistical Inference Algorithms"
Topic: ?Information Retrieval?
A) Articles about the topic; readers in the field
On the ergodicity properties of adaptive MCMC algorithms
Particle filtering within adaptive Metropolis Hastings sampling
An Adaptive Sequential Monte Carlo Sampler
A) Articles about the topic; readers in the field
The anatomy of a large-scale hypertextual Web search engine
Authoritative sources in a hyperlinked environment
A translation approach to portable ontology specifications
B) Articles outside the topic; readers in the field
A comparative review of dimension reduction methods in ABC
Computational methods for Bayesian model choice
The Proof of Innocence
B) Articles outside the topic; readers in the field
How to choose a good scientific problem.
Practical Guide to Support Vector Classification
Maximum likelihood from incomplete data via the EM?
C) Articles about this field; readers outside the field
Introduction to Monte Carlo Methods
An introduction to Monte Carlo simulation of statistical...
The No-U-Turn Sampler: Adaptively setting path lengths...
C) Articles about this field; readers outside the field
Data clustering: a review
Defrosting the digital library: bibliographic tools?
Top 10 algorithms in data mining
Figure 4: The top articles by the expected weight ?dk from a component discovered by our stochastic variational inference in the arXiv data set (Left) and Mendeley (Right). Using the expected topic proportions ?dk
and the expected topic offsets dk , we identified subclasses of articles: A) corresponds to the top articles by
topic proportions in the field of ?Statistical inference algorithms? for arXiv and ?Ontologies and applications?
for Mendeley; B) corresponds to the top articles with low topic proportions in this field, but a large ?dk + dk ,
demonstrating the outside interests of readers of that field (e.g., very popular papers often appear such as ?The
Proof of Innocence? which describes a rigorous way to ?fight your traffic tickets?). C) corresponds to the top
articles with high topic proportions in this field but that also draw significant interest from outside readers.
? Decoupled Poisson Factorization: This model is similar to CTPF but decouples the user
latent preferences into distinct components pu and qu , each of dimension K. We have,
wdv ? Poisson(?dT ?v ); rud ? Poisson(pTu ?d + quT d ).
(5)
The user preference parameters for content and ratings can vary freely. The qu are independent of topics and offer greater modeling flexibility, but they are less interpretable than
the ?u in CTPF. Decoupling the factorizations has been proposed by Porteous et al. [16].
? Content Only: We use the CTPF model without the document topic offsets d . This resembles the idea developed in [1] but using Poisson generating distributions.
? Ratings Only [9]: We use Poisson factorization to the observed ratings. This model can
only make in-matrix predictions.
? CTR [20]: A full optimization of this model does not scale to the size of our data sets
despite running for several days. Accordingly, we fix the topics and document topic proportions to their LDA values. This procedure is shown to perform almost as well as jointly
optimizing the full model in [20]. We follow the authors? experimental settings. Specifically, for hyperparameter selection we started with the values of hyperparameters suggested
by the authors and explored various values of the learning rate as well as the variance of
the prior over the correction factor (?v in [20]). Training convergence was assessed using
the model?s complete log-likelihood on the training observations. (CTR does not use a
validation set.)
Evaluation. Prior to training models, we randomly select 20% of ratings and 1% of documents in
each data set to be used as a held-out test set. Additionally, we set aside 1% of the training ratings as
a validation set (20% for arXiv) and use it to determine convergence. We used the CTPF settings described in Section 3 across both data sets. During testing, we generate the top M recommendations
for each user as those items with the highest predictive score under each method. Figure 3 shows the
mean precision and mean recall at varying number of recommendations for each method and data
set. We see that CTPF outperforms CTR and the Ratings-only model on all data sets. CTPF outperforms the Decoupled PF model and the Content-only model on all data sets except on cold-start
predictions on the arXiv data set, where it performs equally well. The Decoupled PF model lacks
CTPF?s interpretable latent space. The Content-only model performs poorly on most tasks; it lacks a
corrective term on topics to account for user ratings. In Figure 4, we explored the Mendeley and the
arXiv data sets using CTPF. We fit the Mendeley data set using the coordinate ascent algorithm, and
the full arXiv data set using the stochastic algorithm from Section 3. Using the expected document
topic intensities ?dk and the expected document topic offsets dk , we identified interpretable topics
and subclasses of articles that reveal the interaction patterns between readers and articles.
8
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4,817 | 5,361 | Minimax-optimal Inference from Partial Rankings
Bruce Hajek
UIUC
[email protected]
Sewoong Oh
UIUC
[email protected]
Jiaming Xu
UIUC
[email protected]
Abstract
This paper studies the problem of rank aggregation under the Plackett-Luce model.
The goal is to infer a global ranking and related scores of the items, based on partial rankings provided by multiple users over multiple subsets of items. A question
of particular interest is how to optimally assign items to users for ranking and how
many item assignments are needed to achieve a target estimation error. Without
any assumptions on how the items are assigned to users, we derive an oracle lower
bound and the Cram?er-Rao lower bound of the estimation error. We prove an upper bound on the estimation error achieved by the maximum likelihood estimator,
and show that both the upper bound and the Cram?er-Rao lower bound inversely depend on the spectral gap of the Laplacian of an appropriately defined comparison
graph. Since random comparison graphs are known to have large spectral gaps,
this suggests the use of random assignments when we have the control. Precisely,
the matching oracle lower bound and the upper bound on the estimation error imply that the maximum likelihood estimator together with a random assignment is
minimax-optimal up to a logarithmic factor. We further analyze a popular rankbreaking scheme that decompose partial rankings into pairwise comparisons. We
show that even if one applies the mismatched maximum likelihood estimator that
assumes independence (on pairwise comparisons that are now dependent due to
rank-breaking), minimax optimal performance is still achieved up to a logarithmic
factor.
1
Introduction
Given a set of individual preferences from multiple decision makers or judges, we address the problem of computing a consensus ranking that best represents the preference of the population collectively. This problem, known as rank aggregation, has received much attention across various
disciplines including statistics, psychology, sociology, and computer science, and has found numerous applications including elections, sports, information retrieval, transportation, and marketing
[1, 2, 3, 4]. While consistency of various rank aggregation algorithms has been studied when a
growing number of sampled partial preferences is observed over a fixed number of items [5, 6],
little is known in the high-dimensional setting where the number of items and number of observed
partial rankings scale simultaneously, which arises in many modern datasets. Inference becomes
even more challenging when each individual provides limited information. For example, in the well
known Netflix challenge dataset, 480,189 users submitted ratings on 17,770 movies, but on average
a user rated only 209 movies. To pursue a rigorous study in the high-dimensional setting, we assume
that users provide partial rankings over subsets of items generated according to the popular PlackettLuce (PL) model [7] from some hidden preference vector over all the items and are interested in
estimating the preference vector (see Definition 1).
Intuitively, inference becomes harder when few users are available, or each user is assigned few
items to rank, meaning fewer observations. The first goal of this paper is to quantify the number of
item assignments needed to achieve a target estimation error. Secondly, in many practical scenarios
such as crowdsourcing, the systems have the control over the item assignment. For such systems, a
1
natural question of interest is how to optimally assign the items for a given budget on the total number of item assignments. Thirdly, a common approach in practice to deal with partial rankings is to
break them into pairwise comparisons and apply the state-of-the-art rank aggregation methods specialized for pairwise comparisons [8, 9]. It is of both theoretical and practical interest to understand
how much the performance degrades when rank breaking schemes are used.
Notation. For any set S, let |S| denote its cardinality. Let sn1 = {s1 , . . . , sn } denote a set with
n elements. For any positive integer N , let [N ] = {1, . . . , N }. We use standard big O notations,
e.g., for any sequences {an } and {bn }, an = ?(bn ) if there is an absolute constant C > 0 such that
1/C ? an /bn ? C. For a partial ranking ? over S, i.e., ? is a mapping from [|S|] to S, let ? ?1
denote the inverse mapping. All logarithms are natural unless the base is explicitly specified. We
say a sequence of events {An } holds with high probability if P[An ] ? 1 ? c1 n?c2 for two positive
constants c1 , c2 .
1.1
Problem setup
We describe our model in the context of recommender systems, but it is applicable to other systems
with partial rankings. Consider a recommender system with m users indexed by [m] and n items
indexed by [n]. For each item i ? [n], there is a hidden parameter ?i? measuring the underlying
preference. Each user j, independent of everyone else, randomly generates a partial ranking ?j
over a subset of items Sj ? [n] according to the PL model with the underlying preference vector
?? = (?1? , . . . , ?n? ).
Definition 1 (PL model). A partial ranking ? : [|S|] ? S is generated from {?i? , i ? S} under
the PL model in two steps: (1) independently assign each item i ? S an unobserved value Xi ,
?
exponentially distributed with mean e??i ; (2) select ? so that X?(1) ? X?(2) ? ? ? ? ? X?(|S|) .
The PL model can be equivalently described in the following sequential manner.
To generate a
P
?
?i?0
partial ranking ?, first select ?(1) in S randomly from the distribution e?i /
e
; secondly,
i0 ?S
P
?i?0
?i?
select ?(2) in S \ {?(1)} with the probability distribution e /
; continue the
i0 ?S\{?(1)} e
process in the same fashion until all the items in S are assigned. The PL model is a special case of
the following class of models.
Definition 2 (Thurstone model, or random utility model (RUM) ). A partial ranking ? : [|S|] ? S
is generated from {?i? , i ? S} under the Thurstone model for a given CDF F in two steps: (1)
independently assign each item i ? S an unobserved utility Ui , with CDF F (c ? ?i? ); (2) select ? so
that U?(1) ? U?(2) ? ? ? ? ? U?(|S|) .
To recover the PL model from the Thurstone model, take F to be the CDF for the standard Gumbel
?c
distribution: F (c) = e?(e ) . Equivalently, take F to be the CDF of ? log(X) such that X has the
exponential distribution with mean one. For this choice of F, the utility Ui having CDF F (c ? ?i? ),
?
is equivalent to Ui = ? log(Xi ) such that Xi is exponentially distributed with mean e??i . The
corresponding partial permutation ? is such that X?(1) ? X?(2) ? ? ? ? ? X?(|S|) , or equivalently,
U?(1) ? U?(2) ? ? ? ? ? U?(|S|) . (Note the opposite ordering of X?s and U ?s.)
Given the observation of all partial rankings {?j }j?[m] over the subsets {Sj }j?[m] of items, the
task is to infer the underlying preference vector ?? . For the PL model, and more generally for the
Thurstone model, we see that ?? and ?? + a1 for any a ? R are statistically indistinguishable,
where 1 is an all-ones vector. Indeed, under our model, the preference vector ?? is the equivalence
class [?? ] = {? : P
?a ? R, ? = ?? + a1}. To get a unique representation of the equivalence
n
class, we assumeP i=1 ?i? = 0. Then the space of all possible preference vectors is given by
n
? = {? ? Rn : i=1 ?i = 0}. Moreover, if ?i? ? ?i?0 becomes arbitrarily large for all i0 6= i, then
with high probability item i is ranked higher than any other item i0 and there is no way to estimate
?i to any accuracy. Therefore, we further put the constraint that ?? ? [?b, b]n for some b ? R
and define ?b = ? ? [?b, b]n . The parameter b characterizes the dynamic range of the underlying
preference. In this paper, we assume b is a fixed constant. As observed in [10], if b were scaled with
n, then it would be easy to rank items with high preference versus items with low preference and
one can focus on ranking items with close preference.
2
We denote the number of items
Pm assigned to user j by kj := |Sj | and the average number of assigned
1
items per use by k = m j=1 kj ; parameter k may scale with n in this paper. We consider two
scenarios for generating the subsets {Sj }m
j=1 : the random item assignment case where the Sj ?s are
chosen independently and uniformly at random from all possible subsets of [n] with sizes given by
the kj ?s, and the deterministic item assignment case where the Sj ?s are chosen deterministically.
Our main results depend on the structure of a weighted undirected graph G defined as follows.
Definition 3 (Comparison graph G). Each item i ? [n] corresponds to a vertex i ? [n]. For any pair
of vertices i, i0 , there is a P
weighted edge between them if there exists a user who ranks both items i
and i0 ; the weight equals j:i,i0 ?Sj kj1?1 .
P
Let A denote the weighted adjacency matrix of G. Let di = j Aij , so di is the number of users
who rank item i, and without loss of generality assume d1 ? d2 ? ? ? ? ? dn . Let D denote the n ? n
diagonal matrix formed by {di , i ? [n]} and define the graph Laplacian L as L = D ? A. Observe
that L is positive semi-definite and the smallest eigenvalue of L is zero with the corresponding
eigenvector given by the normalized all-one vector. Let 0 = ?1 ? ?2 ? ? ? ? ? ?n denote the
eigenvalues of L in ascending order.
Summary
Pn 1 of main results. Theorem 1 gives a lower bound for the estimation error that scales as
i=2 di . The lower bound is derived based on a genie-argument and holds for both the PL model
andPthe more general Thurstone model. Theorem 2 shows that the Cram?er-Rao lower bound scales
n
as i=2 ?1i . Theorem 3 gives an upper bound for the squared error of the maximum likelihood (ML)
log n
?
estimator that scales as (?mk
2 . Under the full rank breaking scheme that decomposes a k-way
2 ? ?n )
n
k
comparison into 2 pairwise comparisons, Theorem 4 gives an upper bound that scales as mk?log
.
2
2
If the comparison graph is an expander graph, i.e., ?2 ? ?n and mk = ?(n
log
n),
our
lower
and
P
P
upper bounds match up to a log n factor. This follows from the fact that i ?i = i di = mk,
and for expanders mk = ?(n?2 ). Since the Erd?os-R?enyi random graph is an expander graph with
high probability for average degree larger than log n, when the system is allowed to choose the
item assignment, we propose a random assignment scheme under which the items for each user are
chosen independently and uniformly at random. It follows from Theorem 1 that mk = ?(n) is
necessary for any item assignment scheme to reliably infer the underlying preference vector, while
our upper bounds imply that mk = ?(n log n) is sufficient with the random assignment scheme and
can be achieved by either the ML estimator or the full rank breaking or the independence-preserving
breaking that decompose a k-way comparison into bk/2c non-intersecting pairwise comparisons,
proving that rank breaking schemes are also nearly optimal.
1.2
Related Work
There is a vast literature on rank aggregation, and here we can only hope to cover a fraction of them
we see most relevant. In this paper, we study a statistical learning approach, assuming the observed
ranking data is generated from a probabilistic model. Various probabilistic models on permutations
have been studied in the ranking literature (see, e.g., [11, 12]). A nonparametric approach to modeling distributions over rankings using sparse representations has been studied in [13]. Most of the
parametric models fall into one of the following three categories: noisy comparison model, distance
based model, and random utility model. The noisy comparison model assumes that there is an underlying true ranking over n items, and each user independently gives a pairwise comparison which
agrees with the true ranking with probability p > 1/2. It is shown in [14] that O(n log n) pairwise
comparisons, when chosen adaptively, are sufficient for accurately estimating the true ranking.
The Mallows model is a distance-based model, which randomly generates a full ranking ? over n
?
items from some underlying true ranking ? ? with probability proportional to e??d(?,? ) , where ? is
a fixed spread parameter and d(?, ?) can be any permutation distance such as the Kemeny distance.
It is shown in [14] that the true ranking ? ? can be estimated accurately given O(log n) independent
full rankings generated under the Mallows model with the Kemeny distance.
In this paper, we study a special case of random utility models (RUMs) known as the Plackett-Luce
(PL) model. It is shown in [7] that the likelihood function under the PL model is concave and the
ML estimator can be efficiently found using a minorization-maximization (MM) algorithm which is
3
a variation of the general EM algorithm. We give an upper bound on the error achieved by such
an ML estimator, and prove that this is matched by a lower bound. The lower bound is derived by
comparing to an oracle estimator which observes the random utilities of RUM directly. The BradleyTerry (BT) model is the special case of the PL model where we only observe pairwise comparisons.
For the BT model, [10] proposes RankCentrality algorithm based on the stationary distribution of a
random walk over a suitably defined comparison graph and shows ?(npoly(log n)) randomly chosen
pairwise comparisons are sufficient to accurately estimate the underlying parameters; one corollary
of our result is a matching performance guarantee for the ML estimator under the BT model. More
recently, [15] analyzed various algorithms including RankCentrality and the ML estimator under a
general, not necessarily uniform, sampling scheme.
In a PL model with priors, MAP inference becomes computationally challenging. Instead, an efficient message-passing algorithm is proposed in [16] to approximate the MAP estimate. For a more
general family of random utility models, Soufiani et al. in [17, 18] give a sufficient condition under
which the likelihood function is concave, and propose a Monte-Carlo EM algorithm to compute the
ML estimator for general RUMs. More recently in [8, 9], the generalized method of moments together with the rank-breaking is applied to estimate the parameters of the PL model and the random
utility model when the data consists of full rankings.
2
Main results
In this section, we present our theoretical findings and numerical experiments.
2.1
Oracle lower bound
In this section, we derive an oracle lower bound for any estimator of ?? . The lower bound is constructed by considering an oracle who reveals all the hidden scores in the PL model as side information and holds for the general Thurstone models.
Theorem 1. Suppose ?1m are generated from the Thurstone model for some CDF F. For any estib
mator ?,
n
X
1
1
(n ? 1)2
1
? 2
b
inf sup E[||? ? ? ||2 ] ?
,
?
2
2
di
mk
2I(?) + b2 (d2?
2I(?) + b2 (d2?
?b ? ? ??b
1 +d2 ) i=2
1 +d2 )
R (?0 (x))2
where ? is the probability density function of F , i.e., ? = F 0 and I(?) =
?(x) dx; the second
inequality follows from the Jensen?s inequality. For the PL model, which is a special case of the
Thurstone models with F being the standard Gumbel distribution, I(?) = 1.
Pn
Theorem 1 shows that the oracle lower bound scales as i=2 d1i . We remark that the summation
begins with 1/d2 . This makes some sense, in view of the fact that the parameters ?i? need to sum
to zero. For example, if d1 is a moderate value and all the other di ?s are very large, then with the
hidden scores as side information, we may be able to accurately estimate ?i? for i 6= 1 and therefore
accurately estimate ?1? . The oracle lower bound also depends on the dynamic range b and is tight for
b = 0, because a trivial estimator that always outputs the all-zero vector achieves the lower bound.
Comparison to previous work Theorem 1 implies that mk = ?(n) is necessary for any item
assignment scheme to reliably infer ?? , i.e., ensuring E[||?b? ?? ||22 ] = o(n). It provides the first converse result on inferring the parameter vector under the general Thurstone models to our knowledge.
For the Bradley-Terry model, which is a special case of the PL model where all the partial rankings
reduce to the pairwise comparisons, i.e., k = 2, it is shown in [10] that m = ?(n) is necessary
for the random item assignment scheme to achieve the reliable inference based on the informationtheoretic argument. In contrast, our converse result is derived based on the Bayesian Cram?e-Rao
lower bound [19], applies to the general models with any item assignment, and is considerably
tighter if di ?s are of different orders.
2.2
Cram?er-Rao lower bound
In this section, we derive the Cram?er-Rao lower bound for any unbiased estimator of ?? .
4
Theorem 2. Let kmax = maxj?[m] kj and U denote the set of all unbiased estimators of ?? , i.e.,
b ? = ?] = ?, ?? ? ?b . If b > 0, then
?b ? U if and only if E[?|?
!?1 n
!?1
kX
kX
max
max
X 1
1
1
1
1
(n ? 1)2
inf sup E[k?b ? ?? k22 ] ? 1 ?
? 1?
,
b
kmax
`
?
kmax
`
mk
? ? ??b
??U
i=2 i
`=1
`=1
where the second inequality follows from the Jensen?s inequality.
Pn
The Cram?er-Rao lower bound scales as i=2 ?1i . When G is disconnected, i.e., all the items can be
partitioned into two groups such that no user ever compares an item in one group with an item in
the other group, ?2 = 0 and the Cram?er-Rao lower bound is infinity, which is valid (and of course
tight) because there is no basis for gauging any item in one connected component with respect to any
item in the other connected component and the accurate inference is impossible for any estimator.
Although the Cram?er-Rao lower bound only holds for any unbiased estimator, we suspect that a
lower bound with the same scaling holds for any estimator, but we do not have a proof.
2.3
ML upper bound
In this section, we study the ML estimator based on the partial rankings. The ML estimator of ?? is
defined as ?bML ? arg max???b L(?), where L(?) is the log likelihood function given by
L(?) =
log P? [?1m ]
j ?1
m kX
X
??j (`) ? log exp(??j (`) ) + ? ? ? + exp(??j (kj ) ) .
=
(1)
j=1 `=1
As observed in [7], L(?) is concave in ? and thus the ML estimator can be efficiently computed
either via the gradient descent method or the EM type algorithms.
The following theorem gives an upper bound on the error rates inversely dependent on ?2 . Intuitively, by the well-known Cheeger?s inequality, if the spectral gap ?2 becomes larger, then there are
more edges across any bi-partition of G, meaning more pairwise comparisons are available between
any bi-partition of movies, and therefore ?? can be estimated more accurately.
Theorem 3. Assume ?n ? C log n for a sufficiently large constant C in the case with k > 2. Then
with high probability,
(
?
4(1 + e2b )2 ??1
2? m log n If k = 2,
?
b
k?ML ? ? k2 ?
8e4b 2mk
? log n
If k > 2.
? ?16e2b ? log n
2
n
We compare
the above upper bound with the Cram?er-RaoP
lower bound given by Theorem 2. Notice
Pn
n
1
that i=1 ?i = mk and ?1 = 0. Therefore, mk
?
2
i=2 ?i and the upper bound is always
?2
larger than the Cram?er-Rao lower bound. When the comparison graph G is an expander and mk =
?(n log n), by the well-known Cheeger?s inequality, ?2 ? ?n = ?(log n) , the upper bound is only
larger than the Cram?er-Rao lower bound by a logarithmic factor. In particular, with the random item
assignment scheme, we show that ?2 , ?n ? mk
n if mk ? C log n and as a corollary of Theorem 3,
?
b
mk = ?(n log n) is sufficient to ensure k?ML ??? k2 = o( n), proving the random item assignment
scheme with the ML estimation is minimax-optimal up to a log n factor.
Corollary 1. Suppose S1m are chosen independently and uniformly at random among all possible
subsets of [n]. Then there exists a positive constant C > 0 such that if m ? Cn log n when k = 2
and mk ? Ce2b log n when k > 2, then with high probability
?
q
? 4(1 + e2b )2 n2 log n ,
if k = 2,
m
q
k?bML ? ?? k2 ?
2
log n
?
32e4b 2n mk
,
if k > 2.
Comparison to previous work Theorem 3 provides the first finite-sample error rates for inferring
the parameter vector under the PL model to our knowledge. For the Bradley-Terry model, which
is a special case of the PL model with k = 2, [10] derived the similar performance guarantee by
analyzing the rank centrality algorithm and the ML estimator. More recently, [15] extended the
results to the non-uniform sampling scheme of item pairs, but the performance guarantees obtained
when specialized
to the uniform sampling scheme require at least m = ?(n4 log n) to ensure k?b ?
?
?
? k2 = o( n), while our results only require m = ?(n log n).
5
2.4
Rank breaking upper bound
In this section, we study two rank-breaking schemes which decompose partial rankings into pairwise
comparisons.
Definition 4. Given a partial ranking ? over the subset S ? [n] of size k, the independencepreserving breaking scheme (IB) breaks ? into bk/2c non-intersecting pairwise comparisons of form
bk/2c
{it , i0t , yt }t=1 such that {is , i0s } ? {it , i0t } = ? for any s 6= t and yt = 1 if ? ?1 (it ) < ? ?1 (i0t ) and
bk/2c
0 otherwise. The random IB chooses {it , i0t }t=1 uniformly at random among all possibilities.
If ? is generated under the PL model, then the IB breaks ? into independent pairwise comparisons
generated under the PL model. Hence, we can first break partial rankings ?1m into independent pairwise comparisons using the random IB and then apply the ML estimator on the generated pairwise
comparisons with the constraint that ? ? ?b , denoted by ?bIB . Under the random assignment scheme,
?
as a corollary of Theorem 3, mk = ?(n log n) is sufficient to ensure k?bIB ? ?? k2 = o( n), proving
the random item assignment scheme with the random IB is minimax-optimal up to a log n factor in
view of the oracle lower bound in Theorem 1.
Corollary 2. Suppose S1m are chosen independently and uniformly at random among all possible
subsets of [n] with size k. There exists a positive constant C > 0 such that if mk ? Cn log n, then
with high probability,
r
2n2 log n
?
2b 2
b
k?IB ? ? k2 ? 4(1 + e )
.
mk
Definition 5. Given a partial ranking ? over the subset S ? [n] of size k, the full breaking scheme
(k2)
such that yt = 1 if
(FB) breaks ? into all k2 possible pairwise comparisons of form {it , i0t , yt }t=1
? ?1 (it ) < ? ?1 (i0t ) and 0 otherwise.
If ? is generated under the PL model, then the FB breaks ? into pairwise comparisons which are not
independently generated under the PL model. We pretend the pairwise comparisons induced from
the full breaking are all independent and maximize the weighted log likelihood function given by
m
X
X
1
L(?) =
?i I{??1 (i)<??1 (i0 )} + ?i0 I{??1 (i)>??1 (i0 )} ? log e?i + e?i0
j
j
j
j
2(kj ? 1) 0
j=1
i,i ?Sj
(2)
with the constraint that ? ? ?b . Let ?bFB denote the maximizer. Notice that we put the weight kj1?1
to adjust the contributions of the pairwise comparisons generated from the partial rankings over
subsets with different sizes.
?
Theorem 4. With high probability, k?bFB ? ?? k2 ? 2(1 + e2b )2 mk?2log n . Furthermore, suppose S1m
are chosen independently and uniformly at random among all possible subsets of [n]. There exists
a positive constant
C > 0 such that if mk ? Cn log n, then with high probability, k?bFB ? ?? k2 ?
q
4(1 + e2b )2
n2 log n
mk .
Theorem 4 shows that the error rates of ?bFB inversely depend on ?2 . When the comparison graph G
is an expander, i.e., ?2 ? ?n , the upper bound is only larger than the Cram?er-Rao lower bound by a
logarithmic factor. The similar observation holds for the ML estimator as shown in Theorem 3. With
the random item assignment scheme, Theorem 4 imply that the FB only need mk = ?(n log n)
to achieve the reliable inference, which is optimal up to a log n factor in view of the oracle lower
bound in Theorem 1.
Comparison to previous work The rank breaking schemes considered in [8, 9] breaks the full
rankings according to rank positions while our schemes break the partial rankings according to the
item indices. The results in [8, 9] establish the consistency of the generalized method of moments
under the rank breaking schemes when the data consists of full rankings. In contrast, Corollary 2 and
Theorem 4 apply to the more general setting with partial rankings and provide the finite-sample error
rates, proving the optimality of the random IB and FB with the random item assignment scheme.
6
2.5
Numerical experiments
Suppose there are n = 1024 items and ?? is uniformly distributed over [?b, b]. We first generate
d full rankings over 1024 items according to the PL model with parameter ?? . Then for each fixed
k ? {512, 256, . . . , 2}, we break every full ranking ? into n/k partial rankings over subsets of
n/k
size k as follows: Let {Sj }j=1 denote a partition of [n] generated uniformly at random such that
n/k
Sj ? Sj 0 = ? for j 6= j 0 and |Sj | = k for all j; generate {?j }j=1 such that ?j is the partial ranking
over set Sj consistent with ?. In this way, in total we get m = dn/k k-way comparisons which
are all independently generated from the PL model. We apply the minorization-maximization (MM)
algorithm proposed in [7] to compute the ML estimator ?bML based on the k-way comparisons and
the estimator ?bFB based on the pairwise comparisons induced by the
error is
FB. The estimation
mk b
? 2
measured by the rescaled mean square error (MSE) defined by log2 n2 k? ? ? k2 .
We run the simulation with b = 2 and d = 16, 64. The results are depicted in Fig. 1. We also plot
?1
Pk
as per Theorem 2. The oracle lower
the Cram?er-Rao (CR) limit given by log2 1 ? k1 l=1 1l
bound in Theorem 1 implies that the rescaled MSE is at least 0. We can see that the rescaled MSE of
the ML estimator ?bML is close to the CR limit and approaches the oracle lower bound as k becomes
large, suggesting the ML estimator is minimax-optimal. Furthermore, the rescaled MSE of ?bFB under
FB is approximately twice larger than the CR limit, suggesting that the FB is minimax-optimal up
to a constant factor.
3
FB (d=16)
FB (d=64)
d=16
d=64
CR Limit
Rescaled MSE
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
log2(k)
Figure 1: The error rate based on nd/k k-way comparisons with and without full breaking.
Finally, we point out that when d = 16 and log2 (k) = 1, the MSE returned by the MM algorithm
is infinity. Such singularity occurs for the following reason. Suppose we consider a directed comparison graph with nodes corresponding to items such that for each (i, j), there is a directed edge
(i ? j) if item i is ever ranked higher than j. If the graph is not strongly connected, i.e., if there
exists a partition of the items into two groups A and B such that items in A are always ranked higher
than items in B, then if all {?i : i ? A} are increased by a positive constant a, and all {?i : i ? B}
are decreased by another positive constant a0 such that all {?i , i ? [n]} still sum up to zero, the log
likelihood (1) must increase; thus, the log likelihood has no maximizer over the parameter space
?, and the MSE returned by the MM algorithm will diverge. Theoretically, if b is a constant and
d exceeds the order of log n, the directed comparison graph will be strongly connected with high
probability and so such singularity does not occur in our numerical experiments when d ? 64. In
practice we can deal with this singularity issue in three ways: 1) find the strongly connected components and then run MM in each component to come up with an estimator of ?? restricted to each
component; 2) introduce a proper prior on the parameters and use Bayesian inference to come up
with an estimator (see [16]); 3) add to the log likelihood objective function a regularization term
based on k?k2 and solve the regularized ML using the gradient descent algorithms (see [10]).
7
3
Proofs
We sketch the proof of our two upper bounds given by Theorem 3 and Theorem 4. The proofs of
other results can be found in the supplementary file. We introduce some additional notations used
in the proof. For a vector x, let kxk2 denote the usual l2 norm. Let 1 denote the all-one vector
and 0 denote the all-zero vector with the appropriate dimension. Let S n denote the set of n ? n
symmetric matrices with real-valued entries. For X ? S n , let ?1P
(X) ? ?2 (X) ? ? ? ? ? ?n (X)
n
denote its eigenvalues sorted in increasing order. Let Tr(X) = i=1 ?i (X) denote its trace and
kXk = max{??1 (X), ?n (X)} denote its spectral norm. For two matrices X, Y ? S n , we write
X ? Y if Y ?X is positive semi-definite, i.e., ?1 (Y ?X) ? 0. Recall that L(?) is the log likelihood
function. Let ?L(?) denote its gradient and H(?) ? S n denote its Hessian matrix.
3.1
Proof of Theorem 3
The main idea of the proof is inspired from the proof of [10, Theorem 4]. We first introduce several
key auxiliary results used in the proof. Observe that E?? [?L(?? )] = 0. The following lemma upper
bounds the deviation of ?L(?? ) from its mean.
Lemma 1. With probability at least 1 ?
2e2
n ,
?
k?L(? )k2 ?
p
2mk log n
(3)
Observed that ?H(?) is positive semi-definite with the smallest eigenvalue equal to zero. The
following lemma lower bounds its second smallest eigenvalue.
Lemma 2. Fix any ? ? ?b . Then
(
e2b
?2 If k = 2,
(1+e2b )2
(4)
?2 (?H(?)) ?
?
1
2b
?2 ? 16e
?n log n If k > 2,
4e4b
where the inequality holds with probability at least 1 ? n?1 in the case with k > 2.
Proof of Theorem 3. Define ? = ?bML ? ?? . It follows from the definition that ? is orthogonal to
the all-one vector. By the definition of the ML estimator, L(?bML ) ? L(?? ) and thus
L(??ML ) ? L(?? ) ? h?L(?? ), ?i ? ?h?L(?? ), ?i ? ?k?L(?? )k2 k?k2 ,
(5)
where the last inequality holds due to the Cauchy-Schwartz inequality. By the Taylor expansion,
there exists a ? = a?bML + (1 ? a)?? for some a ? [0, 1] such that
1
1
L(??ML ) ? L(?? ) ? h?L(?? ), ?i = ?> H(?)? ? ? ?2 (?H(?))k?k22 ,
(6)
2
2
where the last inequality holds because the Hessian matrix ?H(?) is positive semi-definite with
H(?)1 = 0 and ?> 1 = 0. Combining (5) and (6),
k?k2 ? 2k?L(?? )k2 /?2 (?H(?)).
(7)
Note that ? ? ?b by definition. The theorem follows by Lemma 1 and Lemma 2.
3.2
Proof of Theorem 4
It follows from the definition of L(?) given by (2) that
X
X
1
exp(?i? )
?
,
?i L(? ) =
I{??1 (i)<??1 (i0 )} ?
j
j
kj ? 1 0
exp(?i? ) + exp(?i?0 )
0
j:i?Sj
(8)
i ?Sj :i 6=i
which is a sum of di independent ?
random variables with mean zero and bounded by 1. By Ho?
effding?s inequality,
|?
L(?
)|
?
di log n with probability at least 1 ? 2n?2 . By union bound,
i
?
?
k?L(? )k2 ? mk log n with probability at least 1 ? 2n?1 . The Hessian matrix is given by
m
X
X
1
exp(?i + ?i0 )
H(?) = ?
(ei ? ei0 )(ei ? ei0 )>
2.
2(kj ? 1) 0
[exp(?i ) + exp(?i0 )]
j=1
i,i ?Sj
exp(?i +?i0 )
If |?i | ? b, ?i ? [n], [exp(?
2 ?
i )+exp(?i0 )]
and the theorem follows from (7).
e2b
.
(1+e2b )2
8
It follows that ?H(?) ?
e2b
L
(1+e2b )2
for ? ? ?b
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9
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4,818 | 5,362 | Efficient Optimization for Average Precision SVM
Pritish Mohapatra
IIIT Hyderabad
[email protected]
C.V. Jawahar
IIIT Hyderabad
[email protected]
M. Pawan Kumar
Ecole Centrale Paris & INRIA Saclay
[email protected]
Abstract
The accuracy of information retrieval systems is often measured using average
precision (AP). Given a set of positive (relevant) and negative (non-relevant) samples, the parameters of a retrieval system can be estimated using the AP - SVM
framework, which minimizes a regularized convex upper bound on the empirical AP loss. However, the high computational complexity of loss-augmented inference, which is required for learning an AP - SVM, prohibits its use with large
training datasets. To alleviate this deficiency, we propose three complementary
approaches. The first approach guarantees an asymptotic decrease in the computational complexity of loss-augmented inference by exploiting the problem structure. The second approach takes advantage of the fact that we do not require a
full ranking during loss-augmented inference. This helps us to avoid the expensive step of sorting the negative samples according to their individual scores. The
third approach approximates the AP loss over all samples by the AP loss over difficult samples (for example, those that are incorrectly classified by a binary SVM),
while ensuring the correct classification of the remaining samples. Using the PAS CAL VOC action classification and object detection datasets, we show that our
approaches provide significant speed-ups during training without degrading the
test accuracy of AP - SVM.
1
Introduction
Information retrieval systems require us to rank a set of samples according to their relevance to a
query. The parameters of a retrieval system can be estimated by minimizing the prediction risk on a
training dataset, which consists of positive and negative samples. Here, positive samples are those
that are relevant to a query, and negative samples are those that are not relevant to the query. Several
risk minimization frameworks have been proposed in the literature, including structured support
vector machines (SSVM) [15, 16], neural networks [14], decision forests [11] and boosting [13]. In
this work, we focus on SSVMs for clarity while noting the methods we develop are also applicable
to other learning frameworks.
The SSVM framework provides a linear prediction rule to obtain a structured output for a structured
input. Specifically, the score of a putative output is the dot product of the parameters of an SSVM
with the joint feature vector of the input and the output. The prediction requires us to maximize
the score over all possible outputs for an input. During training, the parameters of an SSVM are
estimated by minimizing a regularized convex upper bound on a user-specified loss function. The
loss function measures the prediction risk, and should be chosen according to the evaluation criterion
for the system. While in theory the SSVM framework can be employed in conjunction with any loss
function, in practice its feasibility depends on the computational efficiency of the corresponding lossaugmented inference. In other words, given the current estimate of the parameters, it is important to
be able to efficiently maximize the sum of the score and the loss function over all possible outputs.
1
A common measure of accuracy for information retrieval is average precision (AP), which is used
in several standard challenges such as the PASCAL VOC object detection, image classification and
action classification tasks [7], and the TREC Web Track corpora. The popularity of AP inspired Yue
et al. [19] to propose the AP - SVM framework, which is a special case of SSVM. The input of AP SVM is a set of samples, the output is a ranking and the loss function is one minus the AP of the
ranking. In order to learn the parameters of an AP - SVM, Yue et al. [19] developed an optimal greedy
algorithm for loss-augmented inference. Their algorithm consists of two stages. First, it sorts the
positive samples P and the negative samples N separately in descending order of their individual
scores. The individual score of a sample is equal to the dot product of the parameters with the feature
vector of the sample. Second, starting from the negative sample with the highest score, it iteratively
finds the optimal interleaving rank for each of the |N | negative samples. The interleaving rank for a
negative sample is the index of the highest ranked positive sample ranked below it. which requires
at most O(|P|) time per iteration. The overall algorithm is described in detail in the next section.
Note that, typically |N | |P|, that is, the negative samples significantly outnumber the positive
samples.
While the AP - SVM has been successfully applied for ranking using high-order information in mid to
large size datasets [5], many methods continue to use the simpler binary SVM framework for large
datasets. Unlike AP - SVM, a binary SVM optimizes the surrogate 0-1 loss. Its main advantage is
the efficiency of the corresponding loss-augmented inference algorithm, which has a complexity of
O(|P| + |N |). However, this gain in training efficiency often comes at the cost of a loss in testing
accuracy, which is especially significant when training with weakly supervised datasets [1].
In order to facilitate the use of AP - SVM, we present three complementary approaches to speed-up
its learning. Our first approach exploits an interesting structure in the problem corresponding to the
computation of the rank of the j-th negative sample. Specifically, we show that when j > |P|, the
rank of the j-th negative sample is obtained by maximizing a discrete unimodal function. Here, a
discrete function defined over points {1, ? ? ? , p} is said to be unimodal if it is non-decreasing from
{1, ? ? ? , k} and non-increasing from {k, ? ? ? , p} for some k ? {1, ? ? ? , p}. Since the mode of a
discrete unimodal function can be computed efficiently using binary search, it reduces the computational complexity of computing the rank of the j-th negative sample from O(|P|) to O(log(|P|)). To
the best of our knowledge, ours is the first work to improve the speed of loss-augmented inference
for AP - SVM by taking advantage of the special structure of the problem. Unlike [2] which proposes
an efficient method for a similar framework of structured output ranking, our method optimizes the
APloss.
Our second approach relies on the fact that in many cases we do not need to explicitly compute the
optimal interleaving rank for all the negative samples. Specifically, we only need to compute the
interleaving rank for the set of negative samples that would have an interleaving rank of less than
|P| + 1. We identify this set using a binary search over the list of negative samples. While training,
after the initial few training iterations the size of this set rapidly reduces, allowing us to significantly
reduce the training time in practice.
Our third approach uses the intuition that the 0-1 loss and the AP loss differ only when some of the
samples are difficult to classify (that is, some positive samples that can be confused as negatives and
vice versa). In other words, when the 0-1 loss over the training dataset is 0, then the AP loss is also
0. Thus, instead of optimizing the AP loss over all the samples, we adopt a two-stage approximate
strategy. In the first stage, we identify a subset of difficult samples (specifically, those that are
incorrectly classified by a binary SVM). In the second stage, we optimize the AP loss over the subset
of difficult samples, while ensuring the correct classification of the remaining easy samples. Using
the PASCAL VOC action classification and object detection datasets, we empirically demonstrate that
each of our approaches greatly reduces the training time of AP - SVM while not decreasing the testing
accuracy.
2
The AP-SVM Framework
We provide a brief overview of the AP - SVM framework, highlighting only those aspects that are
necessary for the understanding of this paper. For a detailed description, we refer the reader to [19].
Input and Output. The input of an AP - SVM is a set of n samples, which we denote by X =
{xi , i = 1, ? ? ? , n}. Each sample can either belong to the positive class (that is, the sample is
2
relevant) or the negative class (that is, the sample is not relevant). The indices for the positive and
negative samples are denoted by P and N respectively. In other words, if i ? P and j ? N then xi
belongs to positive class and xj belongs to the negative class.
The desired output is a ranking matrix R of size n ? n, such that (i) Rij = 1 if xi is ranked higher
than xj ; (ii) Rij = ?1 if xi is ranked lower than xj ; and (iii) Rij = 0 if xi and xj are assigned
the same rank. During training, the ground-truth ranking matrix R? is defined as: (i) R?ij = 1 and
R?ji = ?1 for all i ? P and j ? N ; (ii) R?ii0 = 0 and R?jj 0 = 0 for all i, i0 ? P and j, j 0 ? N .
Joint Feature Vector. For a sample xi , let ?(xi ) denote its feature vector. The joint feature vector
of the input X and an output R is specified as
1 XX
?(X, R) =
Rij (?(xi ) ? ?(xj )).
(1)
|P||N |
i?P j?N
In other words, the joint feature vector is the scaled sum of the difference between the features of all
pairs of samples, where one sample is positive and the other is negative.
Parameters and Prediction. The parameter vector of AP - SVM is denoted by w, and is of the same
size as the joint feature vector. Given the parameters w, the ranking of an input X is predicted by
maximizing the score, that is,
R = argmax w> ?(X, R).
(2)
R
Yue et al. [19] showed that the above optimization can be performed efficiently by sorting the samples xk in descending order of their individual scores, that is, sk = w> ?(xk ).
Parameter Estimation. Given the input X and the ground-truth ranking matrix R? , we estimate
the AP - SVM parameters by optimizing a regularized upper bound on the empirical AP loss. The
AP loss of an output R is defined as 1 ? AP(R? , R), where AP(?, ?) corresponds to the AP of the
ranking R with respect to the true ranking R? . Specifically, the parameters are obtained by solving
the following convex optimization problem:
min
w
s.t.
1
||w||2 + C?,
2
w> ?(X, R? ) ? w> ?(X, R) ? ?(R? , R) ? ?, ?R
(3)
The computational complexity of solving the above problem depends on the complexity of the corresponding loss-augmented inference, that is,
? = argmax w> ?(X, R) + ?(R? , R).
R
(4)
R
For a given set of parameters w, the above problem requires us to find the most violated ranking,
that is, the ranking that maximizes the sum of the score and the AP loss. To be more precise, what
? and the AP loss ?(R? , R)
? corresponding to the most
we require is the joint feature vector ?(X, R)
violated ranking. Yue et al. [19] provided an optimal greedy algorithm for problem (4), which is
summarized in Algorithm 1. It consists of two stages. First, it sorts the positive and the negative
samples separately in descending order of their scores (steps 1-2). This takes O(|P| log(|P|) +
|N | log(|N |)) time. Second, starting with the highest scoring negative sample, it iteratively finds
the interleaving rank of each negative sample xj . This involves maximizing the quantity ?j (i),
defined in equation (5), over all i ? {1, ? ? ? , |P|} (steps 3-7), which takes O(|P||N |) time.
3
Efficient Optimization for AP-SVM
In this section, we propose three methods to speed up the training procedure of AP - SVM. The first
two methods are exact. Specifically, they reduce the time taken to perform loss-augmented inference
while ensuring the computation of the same most violated ranking as Algorithm 1. The third method
provides a framework for a sensible trade-off between training efficiency and test accuracy.
3.1 Efficient Search for Loss-Augmented Inference
In order to find the most violated ranking, the greedy algorithm of Yue et al. [19] iteratively computes the optimal interleaving rank optj ? {1, ? ? ? , |P| + 1} for each negative sample xj (step 5
3
Algorithm 1 The optimal greedy algorithm for loss-augmented inference for training AP - SVM.
input Training samples X containing positive samples P and negative samples N , parameters w.
p
1: Sort the positive samples in descending order of the scores si = w> ?(xi ), i ? {1, . . . , |P|}.
>
2: Sort the negative samples in descending order of the scores sn
j = w ?(xj ), j ? {1, . . . , |N |}.
3: Set j = 1.
4: repeat
5:
Compute the interleaving rank optj = argmaxi?{1,??? ,|P|} ?j (i), where
|P|
X
2(spk ? snj )
1
j
j?1
?
?
.
?j (i) =
|P| j + k j + k ? 1
|P||N |
(5)
k=i
The j-th negative sample is ranked between the (optj ? 1)-th and the optj -th positive sample.
6:
Set j ? j + 1.
7: until j > |N |.
of Algorithm 1). The interleaving rank optj specifies that the negative sample xj must be ranked
between the (optj ? 1)-th and the optj -th positive sample. The computation of the optimal interleaving rank for a particular negative sample requires us to maximize the discrete function ?j (i) over
the domain i ? {1, ? ? ? , |P|}. Yue et al. [19] use a simple linear algorithm for this step, which takes
O(|P|) time. In contrast, we propose a more efficient algorithm to maximize ?j (?), which exploits
the special structure of this discrete function.
Before we describe our efficient algorithm in detail, we require the definition of a unimodal function.
A discrete function f : {1, ? ? ? , p} ? R is said to be unimodal if and only if there exists a k ?
{1, ? ? ? , p} such that
f (i) ? f (i + 1), ?i ? {1, ? ? ? , k ? 1},
f (i ? 1) ? f (i), ?i ? {k + 1, ? ? ? , p}.
(6)
In other words, a unimodal discrete function is monotonically non-decreasing in the interval [1, k]
and monotonically non-increasing in the interval [k, p]. The maximization of a unimodal discrete
function over its domain {1, ? ? ? , p} simply requires us to find the index k that satisfies the above
properties. The maximization can be performed efficiently, in O(log(p)) time, using binary search.
We are now ready to state the main result that allows us to compute the optimal interleaving rank of
a negative sample efficiently.
Proposition 1. The discrete function ?j (i), defined in equation (5), is unimodal in the domain
{1, ? ? ? , p}, where p = min{|P|, j}.
The proof of the above proposition is provided in Appendix A (supplementary material).
Algorithm 2 Efficient search for the optimal interleaving rank of a negative sample.
input {?j (i), i = 1, ? ? ? , |P|}.
1: p = min{|P|, j}.
2: Compute an interleaving rank i1 as
ii = argmax ?j (i).
(7)
i?{1,??? ,p}
3: Compute an interleaving rank i2 as
i2 =
argmax
?j (i).
(8)
i?{p+1,??? ,|P|}
4: Compute the optimal interleaving rank optj as
optj =
i1
i2
if ?j (i1 ) ? ?j (i2 ),
otherwise.
4
(9)
Using the above proposition, the discrete function ?j (i) can be optimized over the domain
{1, ? ? ? , |P|} efficiently as described in Algorithm 2. Briefly, our efficient search algorithm finds
an interleaving ranking i1 over the domain {1, ? ? ? , p}, where p is set to min{|P|, j} in order to
ensure that the function ?j (?) is unimodal (step 2 of Algorithm 2). Since i1 can be computed using binary search, the computational complexity of this step is O(log(p)). Furthermore, we find an
interleaving ranking i2 over the domain {p + 1, ? ? ? , |P|} (step 3 of Algorithm 2). Since i2 needs
to be computed using linear search, the computational complexity of this step is O(|P| ? p) when
p < |P| and 0 otherwise. The optimal interleaving ranking optj of the negative sample xj can then
be computed by comparing the values of ?j (i1 ) and ?j (i2 ) (step 4 of Algorithm 2).
Note that, in a typical training dataset, the negative samples significantly outnumber the positive
samples, that is, |N | |P|. For all the negative samples xj where j ? |P|, p will be equal to |P|.
Hence, the maximization of ?j (?) can be performed efficiently over the entire domain {1, ? ? ? , |P|}
using binary search in O(log(|P|)) as opposed to the O(|P|) time suggested in [19].
3.2
Selective Ranking for Loss-Augmented Inference
While the efficient search algorithm described in the previous subsection allows us to find the optimal interleaving rank for a particular negative sample, the overall loss-augmented inference would
still remain computationally inefficient when the number of negative samples is large (as is typically the case). This is due to the following two reasons. First, loss-augmented inference spends a
considerable amount of time sorting the negative samples according to their individual scores (step
2 of Algorithm 1). Second, if we were to apply our efficient search algorithm to every negative
sample, the total computational complexity of the second stage of loss-augmented inference (step
3-7 of Algorithm 1) will still be O(|P|2 + (|N | ? |P|) log(|P|)).
In order to overcome the above computational issues, we exploit two key properties of lossaugmented inference in AP - SVM. First, if a negative sample xj has the optimal interleaving rank
optj = |P| + 1, then all the negative samples that have lower score than xj would also have the
same optimal interleaving rank (that is, optk = optj = |P| + 1 for all k > j). This property follows
directly from the analysis of Yue et al. [19] who showed that, for k < j, optk ? optj and for any
negative sample xj , optj ? [1, |P| + 1]. We refer the reader to [19] for a detailed proof. Second, we
? but the
note that the desired output of loss-augmented inference is not the most violated ranking R,
? and the AP loss AP(R? , R).
? From the definition of the joint feature
joint feature vector ?(X, R)
vector and the AP loss, it follows that they do not depend on the relative ranking of the negative
samples that share the same optimal interleaving rank. Specifically, both the joint feature vector
and the AP loss only depend on the number of negatives that are ranked higher and lower than each
positive sample.
The above two observations suggest the following alternate strategy to Algorithm 1. Instead of explicitly computing the optimal interleaving rank for each negative
sample (which can be computationally expensive), we
compute it only for negative samples that are expected
to have optimal interleaving rank less than |P| + 1. Algorithm 3 outlines the procedure we propose in detail. We
first find the score s? such that every negative sample xj
with score snj < s? has optj = |P| + 1. We do a binary
search over the list of scores of negative samples to find s?
(step 4 of algorithm 3). We do not need to sort the scores
of all the negative samples, as we use the quick select algorithm to find the k-th highest score wherever required. Figure 1: A row corresponds to the interleaving ranks of the negative samIf the output of the loss-augmented inference is such that
ples after a training iteration. Here,
a large number of negative samples have optimal interthere are 4703 negative samples, and
leaving rank as |P| + 1, then this alternate strategy would
131 training iterations. The interleavresult in a significant speed-up during training. In our
ing ranks are represented using a heat
experiments, we found that in later iterations of the opmap where the deepest red represents
timization, this is indeed the case in practice. Figure 1
interleaving rank of |P| + 1. (The figshows how the number of negative samples with optimal
ure is best viewed in colour.)
interleaving rank equal to |P| + 1, rapidly increases after
5
Algorithm 3 The selective ranking algorithm for loss-augmented inference in AP - SVM.
input S x , S x? , |P|, |N |
1: Sort the positive samples in descending order of their scores S x .
2: Do binarysearch over S x? to find s?.
3: Set Nl = j ? N |sn
?
j <s
4: Sort Nl in descending order of the scores.
5: for all j ? Nl do
6:
Compute optj using Algorithm 2.
7: end for
8: Set Nr = N ? Nl .
9: for all j ? Nr do
10:
Set optj = |P| + 1.
11: end for
output optj , ?j ? N
a few training iterations for a typical experiment. A large number of negative samples have optimal
interleaving rank equal to |P| + 1, while the negative samples that have other values of optimal
interleaving rank decrease considerably.
It would be worth taking note that here, even though we take advantage of the fact that a long
sequence of negative samples at the end of the list take the same optimal interleaving rank, such
sequences also occur at other locations throughout the list. This can be leveraged for further speedup by computing the interleaving rank for only the boundary samples of such sequences and setting
all the intermediate samples to the same interleaving rank as the boundary samples. We can use a
method similar to the one presented in this section to search for such sequences by using the quick
select algorithm to compute the interleaving rank for any particular negative sample on the list.
3.3
Efficient Approximation of AP-SVM
The previous two subsections provide exact algorithms for loss-augmented inference that reduce
the time require for training an AP - SVM. However, despite these improvements, AP - SVM might
be slower to learn compared to simpler frameworks such as the binary SVM, which optimizes the
surrogate 0-1 loss. The disadvantage of using the binary SVM is that, in general, the 0-1 loss is a
poor approximation for the AP loss. However, the quality of the approximation is not uniformly
poor for all samples, but depends heavily on their separability. Specifically, when the 0-1 loss of a
set of samples is 0 (that is, they are linearly separable by a binary SVM), their AP loss is also 0. This
observation inspires us to approximate the AP loss over the entire set of training samples using the
AP loss over the subset of difficult samples. In this work, we define the subset of difficult samples
as those that are incorrectly classified by a simple binary SVM.
Formally, given the complete input X and the ground-truth ranking matrix R? , we represent individual samples as xi and their class as yi . In other words, yi = 1 if i ? P and yi = ?1 if i ? N .
In order to approximate the AP - SVM, we adopt a two stage strategy. In the first stage, we learn
a binary SVM by minimizing the regularized convex upper bound on the 0-1 loss over the entire
training set. Since the loss-augmented inference for 0-1 loss is very fast, the parameters w0 of the
binary SVM can be estimated efficiently. We use the binary SVM to define the set of easy samples as
Xe = {xi , yi w0> ?i (x) ? 1}. In other words, a positive sample is easy if it is assigned a score that
is greater than 1 by the binary SVM. Similarly, a negative sample is easy if it is assigned a score that
is less than -1 by the binary SVM. The remaining difficult samples are denoted by Xd = X ? Xe
and the corresponding ground-truth ranking matrix by R?d . In the second stage, we approximate the
AP loss over the entire set of samples X by the AP loss over the difficult samples Xd while ensuring
that the samples Xe are correctly classified. In order to accomplish this, we solve the following
optimization problem:
min
w
s.t.
1
||w||2 + C?
2
w> ?(Xd , R?d ) ? w> ?(Xd , Rd ) ? ?(R?d , Rd ) ? ?, ?Rd ,
yi w> ?(xi ) > 1, ?xi ? Xe .
6
(10)
In practice, we can choose to retain only the top k% of Xe ranked in descending order of their score
and push the remaining samples into the difficult set Xd . This gives the AP - SVM more flexibility to
update the parameters at the cost of some additional computation.
4
Experiments
We demonstrate the efficacy of our methods, described in the previous section, on the challenging
problems of action classification and object detection.
4.1 Action Classification
Dataset. We use the PASCAL VOC 2011 [7] action classification dataset for our experiments. This
dataset consists of 4846 images, which include 10 different action classes. The dataset is divided
into two parts: 3347 ?trainval? person bounding boxes and 3363 ?test? person bounding boxes. We
use the ?trainval? bounding boxes for training since their ground-truth action classes are known.
We evaluate the accuracy of the different instances of SSVM on the ?test? bounding boxes using the
PASCAL evaluation server.
Features. We use the standard poselet [12] activation features to define the sample feature for
each person bounding box. The feature vector consists of 2400 action poselet activations and 4
object detection scores. We refer the reader to [12] for details regarding the feature vector.
Methods. We present results on five different methods. First, the standard binary SVM, which
optimizes the 0-1 loss. Second, the standard AP - SVM, which uses the inefficient loss-augmented
inference described in Algorithm 1. Third, AP - SVM - SEARCH, which uses efficient search to compute the optimal interleaving rank for each negative sample using Algorithm 2. Fourth, AP - SVM SELECT, which uses the selective ranking strategy outlined in Algorithm 3. Fifth, AP - SVM - APPX ,
which employs the approximate AP - SVM framework described in subsection 3.3. Note that, AP SVM , AP - SVM - SEARCH and AP - SVM - SELECT are guaranteed to provide the same set of parameters
since both efficient search and selective ranking are exact methods. The hyperparameters of all five
methods are fixed using 5-fold cross-validation on the ?trainval? set.
Results. Table 1 shows the AP for the rankings obtained by the five methods for ?test? set. Note that
AP - SVM (and therefore, AP - SVM - SEARCH and AP - SVM - SELECT) consistently outperforms binary
SVM by optimizing a more appropriate loss function during training. The approximate AP - SVM APPX provides comparable results to the exact AP - SVM formulations by optimizing the AP loss over
difficult samples, while ensuring the correct classification of easy samples. The time required to
compute the most violated rankings for each of the five methods in shown in Table 2. Note that
all three methods described in this paper result in substantial improvement in training time. The
overall time required for loss-augmented inference is reduced by a factor of 5 ? 10 compared to the
original AP - SVM approach. It can also be observed that though each loss-augmented inference step
for binary SVM is significantly more efficient than for AP - SVM (Table 3), in some cases we observe
that we required more cutting plane iterations for binary SVM to converge. As a result, in some cases
training binary SVM is slower than training AP - SVM with our proposed speed-ups.
Object class
Jumping
Phoning
Playing instrument
Reading
Riding bike
Running
Taking photo
Using computer
Walking
Riding horse
Binary SVM
52.580
32.090
35.210
27.410
72.240
73.090
21.880
30.620
54.400
79.820
AP - SVM
55.230
32.630
41.180
26.600
81.060
76.850
25.980
32.050
57.090
83.290
AP - SVM - APPX
k=25%
54.660
31.380
40.510
27.100
80.660
75.720
25.360
32.460
57.380
83.650
k=50%
55.640
30.660
38.650
25.530
79.950
74.670
23.680
32.810
57.430
83.560
k=75%
54.570
29.610
37.260
24.980
78.660
72.550
22.860
32.840
55.790
82.390
Table 1: Test AP for the different action classes of PASCAL VOC 2011 action dataset. For AP - SVM APPX , we report test results for 3 different values of k, which is the percentage of samples that are
included in the easy set among all the samples that the binary SVM had classified with margin > 1.
7
Binary SVM
0.1068
AP - SVM
AP - SVM - SEARCH
AP - SVM - SELECT
AP - SVM - APPX ( K =50)
ALL
0.5660
0.0671
0.0404
0.2341
0.0251
Table 2: Computation time (in seconds) for computing the most violated ranking when using the
different methods. The reported time is averaged over the training for all the action classes.
Binary SVM
0.637
AP - SVM
AP - SVM - SEARCH
AP - SVM - SELECT
AP - SVM - APPX ( K =50)
ALL
13.192
1.565
0.942
8.217
0.689
Table 3: Computation time (in milli-seconds) for computing the most violated ranking per iteration
when using the different methods. The reported time is averaged over all training iterations and
over all the action classes.
4.2 Object Detection
Dataset. We use the PASCAL VOC 2007 [6] object detection dataset, which consists of a total of
9963 images. The dataset is divided into a ?trainval? set of 5011 images and a ?test? set of 4952
images. All the images are labelled to indicate the presence or absence of the instances of 20
different object categories. In addition, we are also provided with tight bounding boxes around the
object instances, which we ignore during training and testing. Instead, we treat the location of the
objects as a latent variable. In order to reduce the latent variable space, we use the selective-search
algorithm [17] in its fast mode, which generates an average of 2000 candidate windows per image.
Features. For each of the candidate windows, we use a feature representation that is extracted
from a trained Convolutional Neural Network (CNN). Specifically, we pass the image as input to the
CNN and use the activation vector of the penultimate layer of the CNN as the feature vector. Inspired
by the work of Girshick et al. [9], we use the CNN that is trained on the ImageNet dataset [4], by
rescaling each candidate window to a fixed size of 224 ? 224. The length of the resulting feature
vector is 4096.
Methods. We train latent AP - SVMs [1] as object detectors for 20 object categories. In our experiments, we determine the value of the hyperparameters using 5-fold cross-validation. During testing,
we evaluate each candidate window generated by selective search, and use non-maxima suppression
to prune highly overlapping detections.
Results. This experiment places high computational demands due to the size of the dataset (5011
?trainval? images), as well as the size of the latent space (2000 candidate windows per image). We
compare the computational efficiency of the loss-augmented inference algorithm proposed in [19]
and the exact methods proposed by us. The total time taken for loss-augmented inference during
training, averaged over the all the 20 classes, is 0.3302 sec for our exact methods (SEARCH+SELECT)
which is significantly better than the 6.237 sec taken by the algorithm used in [19].
5
Discussion
We proposed three complementary approaches to improve the efficiency of learning AP - SVM. The
first two approaches exploit the problem structure to speed-up the computation of the most violated
ranking using exact loss-augmented inference. The third approach provides an accurate approximation of AP - SVM, which facilitates the trade-off of test accuracy and training time.
As mentioned in the introduction, our approaches can also be used in conjunction with other learning
frameworks, such as the popular deep convolutional neural networks. A combination of methods
proposed in this paper and the speed-ups proposed in [10] may prove to be effective in such a
framework. The efficacy of optimizing AP efficiently using other frameworks needs to be empirically
evaluated. Another computational bottleneck of all SSVM frameworks is the computation of the joint
feature vector. An interesting direction of future research would be to combine our approaches with
those of sparse feature coding [3, 8, 18] to improve the speed to AP - SVM learning further.
6
Acknowledgement
This work is partially funded by the European Research Council under the European Community?s
Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement number 259112. Pritish is
supported by the TCS Research Scholar Program.
8
References
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9
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4,819 | 5,363 | Ranking via Robust Binary Classification
Hyokun Yun
Amazon
Seattle, WA 98109
[email protected]
Parameswaran Raman, S. V. N. Vishwanathan
Department of Computer Science
University of California
Santa Cruz, CA 95064
{params,vishy}@ucsc.edu
Abstract
We propose RoBiRank, a ranking algorithm that is motivated by observing a close
connection between evaluation metrics for learning to rank and loss functions
for robust classification. It shows competitive performance on standard benchmark datasets against a number of other representative algorithms in the literature.
We also discuss extensions of RoBiRank to large scale problems where explicit
feature vectors and scores are not given. We show that RoBiRank can be efficiently parallelized across a large number of machines; for a task that requires
386, 133 ? 49, 824, 519 pairwise interactions between items to be ranked, RoBiRank finds solutions that are of dramatically higher quality than that can be found
by a state-of-the-art competitor algorithm, given the same amount of wall-clock
time for computation.
1
Introduction
Learning to rank is a problem of ordering a set of items according to their relevances to a given
context [8]. While a number of approaches have been proposed in the literature, in this paper we
provide a new perspective by showing a close connection between ranking and a seemingly unrelated
topic in machine learning, namely, robust binary classification.
In robust classification [13], we are asked to learn a classifier in the presence of outliers. Standard
models for classification such as Support Vector Machines (SVMs) and logistic regression do not
perform well in this setting, since the convexity of their loss functions does not let them give up
their performance on any of the data points [16]; for a classification model to be robust to outliers,
it has to be capable of sacrificing its performance on some of the data points. We observe that
this requirement is very similar to what standard metrics for ranking try to evaluate. Discounted
Cumulative Gain (DCG) [17] and its normalized version NDCG, popular metrics for learning to
rank, strongly emphasize the performance of a ranking algorithm at the top of the list; therefore, a
good ranking algorithm in terms of these metrics has to be able to give up its performance at the
bottom of the list if that can improve its performance at the top.
In fact, we will show that DCG and NDCG can indeed be written as a natural generalization of robust
loss functions for binary classification. Based on this observation we formulate RoBiRank, a novel
model for ranking, which maximizes the lower bound of (N)DCG. Although the non-convexity
seems unavoidable for the bound to be tight [9], our bound is based on the class of robust loss
functions that are found to be empirically easier to optimize [10]. Indeed, our experimental results
suggest that RoBiRank reliably converges to a solution that is competitive as compared to other
representative algorithms even though its objective function is non-convex.
While standard deterministic optimization algorithms such as L-BFGS [19] can be used to estimate
parameters of RoBiRank, to apply the model to large-scale datasets a more efficient parameter estimation algorithm is necessary. This is of particular interest in the context of latent collaborative
1
retrieval [24]; unlike standard ranking task, here the number of items to rank is very large and explicit feature vectors and scores are not given.
Therefore, we develop an efficient parallel stochastic optimization algorithm for this problem. It has
two very attractive characteristics: First, the time complexity of each stochastic update is independent of the size of the dataset. Also, when the algorithm is distributed across multiple number of
machines, no interaction between machines is required during most part of the execution; therefore,
the algorithm enjoys near linear scaling. This is a significant advantage over serial algorithms, since
it is very easy to deploy a large number of machines nowadays thanks to the popularity of cloud
computing services, e.g. Amazon Web Services.
We apply our algorithm to latent collaborative retrieval task on Million Song Dataset [3] which consists of 1,129,318 users, 386,133 songs, and 49,824,519 records; for this task, a ranking algorithm
has to optimize an objective function that consists of 386, 133 ? 49, 824, 519 number of pairwise
interactions. With the same amount of wall-clock time given to each algorithm, RoBiRank leverages
parallel computing to outperform the state-of-the-art with a 100% lift on the evaluation metric.
2
Robust Binary Classification
Suppose we are given training data which consists of n data points (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ),
where each xi ? Rd is a d-dimensional feature vector and yi ? {?1, +1} is a label associated
with it. A linear model attempts to learn a d-dimensional parameter ?, and for a given feature
vector x it predicts label +1 if hx, ?i ? 0 and ?1 otherwise. Here h?, ?i denotes the Euclidean dot
product between
Pn two vectors. The quality of ? can be measured by the number of mistakes it makes:
L(?) := i=1 I(yi ? hxi , ?i < 0). The indicator function I(? < 0) is called the 0-1 loss function,
because it has a value of 1 if the decision rule makes a mistake, and 0 otherwise. Unfortunately,
since the 0-1 loss is a discrete function its minimization is difficult [11]. The most popular solution
to this problem in machine learning is to upper bound the 0-1 loss by an easy to optimize function
[2]. For example, logistic regression uses the logistic loss function ?0 (t) := log2 (1 + 2?t ), to come
up with a continuous and convex objective function
L(?) :=
n
X
i=1
?0 (yi ? hxi , ?i),
(1)
which upper bounds L(?). It is clear that for each i, ?0 (yi ? hxi , ?i) is a convex function in ?;
therefore, L(?), a sum of convex functions, is also a convex function which is relatively easier to
optimize [6]. Support Vector Machines (SVMs) on the other hand can be recovered by using the
hinge loss to upper bound the 0-1 loss.
However, convex upper bounds such as L(?) are known to be sensitive to outliers [16]. The basic
intuition here is that when yi ? hxi , ?i is a very large negative number for some data point i, ?(yi ?
hxi , ?i) is also very large, and therefore the optimal solution of (1) will try to decrease the loss on
such outliers at the expense of its performance on ?normal? data points.
In order to construct robust loss functions, consider the following two transformation functions:
?1 (t) := log2 (t + 1), ?2 (t) := 1 ?
1
,
log2 (t + 2)
(2)
which, in turn, can be used to define the following loss functions:
?1 (t) := ?1 (?0 (t)), ?2 (t) := ?2 (?0 (t)).
(3)
One can see that ?1 (t) ? ? as t ? ??, but at a much slower rate than ?0 (t) does; its derivative
?10 (t) ? 0 as t ? ??. Therefore, ?1 (?) does not grow as rapidly as ?0 (t) on hard-to-classify data
points. Such loss functions are called Type-I robust loss functions by Ding [10], who also showed
that they enjoy statistical robustness properties. ?2 (t) behaves even better: ?2 (t) converges to a
constant as t ? ??, and therefore ?gives up? on hard to classify data points. Such loss functions
are called Type-II loss functions, and they also enjoy statistical robustness properties [10].
In terms of computation, of course, ?1 (?) and ?2 (?) are not convex, and therefore the objective
function based on such loss functions is more difficult to optimize. However, it has been observed
2
in Ding [10] that models based on optimization of Type-I functions are often empirically much
more successful than those which optimize Type-II functions. Furthermore, the solutions of Type-I
optimization are more stable to the choice of parameter initialization. Intuitively, this is because
Type-II functions asymptote to a constant, reducing the gradient to almost zero in a large fraction
of the parameter space; therefore, it is difficult for a gradient-based algorithm to determine which
direction to pursue. See Ding [10] for more details.
3
Ranking Model via Robust Binary Classification
Let X = {x1 , x2 , . . . , xn } be a set of contexts, and Y = {y1 , y2 , . . . , ym } be a set of items to
be ranked. For example, in movie recommender systems X is the set of users and Y is the set of
movies. In some problem settings, only a subset of Y is relevant to a given context x ? X ; e.g.
in document retrieval systems, only a subset of documents is relevant to a query. Therefore, we
define Yx ? Y to be a set of items relevant to context x. Observed data can be described by a set
W := {Wxy }x?X ,y?Yx where Wxy is a real-valued score given to item y in context x.
We adopt a standard problem setting used in the literature of learning to rank. For each context x
and an item y ? Yx , we aim to learn a scoring function f (x, y) : X ? Yx ? R that induces a
ranking on the item set Yx ; the higher the score, the more important the associated item is in the
given context. To learn such a function, we first extract joint features of x and y, which will be
denoted by ?(x, y). Then, we parametrize f (?, ?) using a parameter ?, which yields the linear model
f? (x, y) := h?(x, y), ?i, where, as before, h?, ?i denotes the Euclidean dot product between two
vectors. ? induces a ranking on the set of items Yx ; we define rank? (x, y) to be the rank of item y
in a given context x induced by ?. Observe that rank? (x, y) can also be written as a sum of 0-1 loss
functions (see e.g. Usunier et al. [23]):
X
rank? (x, y) =
I (f? (x, y) ? f? (x, y 0 ) < 0) .
(4)
y 0 ?Yx ,y 0 6=y
3.1
Basic Model
If an item y is very relevant in context x, a good parameter ? should position y at the top of the list;
in other words, rank? (x, y) has to be small, which motivates the following objective function [7]:
X
X
L(?) :=
cx
v(Wxy ) ? rank? (x, y),
(5)
x?X
y?Yx
where cx is an weighting factor for each context x, and v(?) : R+ ? R+ quantifies the relevance
level of y on x. Note that {cx } and v(Wxy ) can be chosen to reflect the metric the model is going to
be evaluated on (this will be discussed in Section 3.2). Note that (5) can be rewritten using (4) as a
sum of indicator functions. Following the strategy in Section 2, one can form an upper bound of (5)
by bounding each 0-1 loss function by a logistic loss function:
X X
X
L(?) :=
cx
v (Wxy ) ?
?0 (f? (x, y) ? f? (x, y 0 )) .
(6)
x?X
y?Yx
y 0 ?Yx ,y 0 6=y
Just like (1), (6) is convex in ? and hence easy to minimize.
3.2
DCG
Although (6) enjoys convexity, it may not be a good objective function for ranking. This is because
in most applications of learning to rank, it is more important to do well at the top of the list than at
the bottom, as users typically pay attention only to the top few items. Therefore, it is desirable to
give up performance on the lower part of the list in order to gain quality at the top. This intuition is
similar to that of robust classification in Section 2; a stronger connection will be shown below.
Discounted Cumulative Gain (DCG) [17] is one of the most popular metrics for ranking. For each
context x ? X , it is defined as:
X
v (Wxy )
DCG(?) := cx
,
(7)
log2 (rank? (x, y) + 2)
y?Yx
3
where v(t) = 2t ? 1 and cx = 1. Since 1/ log(t + 2) decreases quickly and then asymptotes to
a constant as t increases, this metric emphasizes the quality of the ranking at the top of the list.
Normalized DCG (NDCG) simply normalizes the metric to bound it between 0 and 1 by calculating
the maximum achievable DCG value mx and dividing by it [17].
3.3
RoBiRank
Now we formulate RoBiRank, which optimizes the lower bound of metrics for ranking in form (7).
Observe that max? DCG(?) can be rewritten as
X
X
1
min
cx
v (Wxy ) ? 1 ?
.
(8)
?
log2 (rank? (x, y) + 2)
x?X
y?Yx
Using (4) and the definition of the transformation function ?2 (?) in (2), we can rewrite the objective
function in (8) as:
?
?
X
X
X
L2 (?) :=
cx
v (Wxy ) ? ?2 ?
I (f? (x, y) ? f? (x, y 0 ) < 0)? .
(9)
x?X
y?Yx
y 0 ?Yx ,y 0 6=y
Since ?2 (?) is a monotonically increasing function, we can bound (9) with a continuous function by
bounding each indicator function using the logistic loss:
?
?
X
X
X
L2 (?) :=
cx
v (Wxy ) ? ?2 ?
?0 (f? (x, y) ? f? (x, y 0 ))? .
(10)
x?X
y?Yx
y 0 ?Yx ,y 0 6=y
This is reminiscent of the basic model in (6); as we applied the transformation ?2 (?) on the logistic
loss ?0 (?) to construct the robust loss ?2 (?) in (3), we are again applying the same transformation
on (6) to construct a loss function that respects the DCG metric used in ranking. In fact, (10) can be
seen as a generalization of robust binary classification by applying the transformation on a group of
logistic losses instead of a single loss. In both robust classification and ranking, the transformation
?2 (?) enables models to give up on part of the problem to achieve better overall performance.
As we discussed in Section 2, however, transformation of logistic loss using ?2 (?) results in Type-II
loss function, which is very difficult to optimize. Hence, instead of ?2 (?) we use an alternative transformation ?1 (?), which generates Type-I loss function, to define the objective function of RoBiRank:
?
?
X
X
X
L1 (?) :=
cx
v (Wxy ) ? ?1 ?
?0 (f? (x, y) ? f? (x, y 0 ))? .
(11)
x?X
y?Yx
y 0 ?Yx ,y 0 6=y
Since ?1 (t) ? ?2 (t) for every t > 0, we have L1 (?) ? L2 (?) ? L2 (?) for every ?. Note
that L1 (?) is continuous and twice differentiable. Therefore, standard gradient-based optimization
techniques can be applied to minimize it. As is standard, a regularizer on ? can be added to avoid
overfitting; for simplicity, we use the `2 -norm in our experiments.
3.4
Standard Learning to Rank Experiments
We conducted experiments to check the performance of RoBiRank (11) in a standard learning to
rank setting, with a small number of labels to rank. We pitch RoBiRank against the following
algorithms: RankSVM [15], the ranking algorithm of Le and Smola [14] (called LSRank in the sequel), InfNormPush [22], IRPush [1], and 8 standard ranking algorithms implemented in RankLib1
namely MART, RankNet, RankBoost, AdaRank, CoordAscent, LambdaMART, ListNet and RandomForests. We use three sources of datasets: LETOR 3.0 [8] , LETOR 4.02 and YAHOO LTRC
[20], which are standard benchmarks for ranking (see Table 2 for summary statistics). Each dataset
consists of five folds; we consider the first fold, and use the training, validation, and test splits provided. We train with different values of regularization parameter, and select one with the best NDCG
1
2
http://sourceforge.net/p/lemur/wiki/RankLib
http://research.microsoft.com/en-us/um/beijing/projects/letor/letor4dataset.aspx
4
on the validation dataset. The performance of the model with this parameter on the test dataset is
reported. We used implementation of the L-BFGS algorithm provided by the Toolkit for Advanced
Optimization (TAO)3 for estimating the parameter of RoBiRank. For the other algorithms, we either
implemented them using our framework or used the implementations provided by the authors.
TD 2004
1
TD 2004
1
RoBiRank
RankSVM
LSRank
InfNormPush
IRPush
0.8
NDCG@k
NDCG@k
0.8
0.6
0.4
RoBiRank
MART
RankNet
RankBoost
AdaRank
CoordAscent
LambdaMART
ListNet
RandomForests
0.6
5
10
15
0.4
20
k
5
10
15
20
k
Figure 1: Comparison of RoBiRank with a number of competing algorithms.
We use values of NDCG at different levels of truncation as our evaluation metric [17]; see Figure 1.
RoBiRank outperforms its competitors on most of the datasets; due to space constraints, we only
present plots for the TD 2004 dataset here and other plots can be found in Appendix B. The performance of RankSVM seems insensitive to the level of truncation for NDCG. On the other hand,
RoBiRank, which uses non-convex loss function to concentrate its performance at the top of the
ranked list, performs much better especially at low truncation levels. It is also interesting to note
that the NDCG@k curve of LSRank is similar to that of RoBiRank, but RoBiRank consistently outperforms at each level. RoBiRank dominates Inf-Push and IR-Push at all levels. When compared to
standard algorithms, Figure 1 (right), again RoBiRank outperforms especially at the top of the list.
Overall, RoBiRank outperforms IRPush and InfNormPush on all datasets except TD 2003 and
OHSUMED where IRPush seems to fare better at the top of the list. Compared to the 8 standard
algorithms, again RobiRank either outperforms or performs comparably to the best algorithm except
on two datasets (TD 2003 and HP 2003), where MART and Random Forests overtake RobiRank at
few values of NDCG. We present a summary of the NDCG values obtained by each algorithm in
Table 2 in the appendix. On the MSLR30K dataset, some of the additional algorithms like InfNormPush and IRPush did not complete within the time period available; indicated by dashes in the table.
4
Latent Collaborative Retrieval
For each context x and an item y ? Y, the standard problem setting of learning to rank requires
training data to contain feature vector ?(x, y) and score Wxy assigned on the x, y pair. When the
number of contexts |X | or the number of items |Y| is large, it might be difficult to define ?(x, y)
and measure Wxy for all x, y pairs. Therefore, in most learning to rank problems we define the set
of relevant items Yx ? Y to be much smaller than Y for each context x, and then collect data only
for Yx . Nonetheless, this may not be realistic in all situations; in a movie recommender system, for
example, for each user every movie is somewhat relevant.
On the other hand, implicit user feedback data is much more abundant. For example, a lot of users
on Netflix would simply watch movie streams on the system but do not leave an explicit rating. By
the action of watching a movie, however, they implicitly express their preference. Such data consist
only of positive feedback, unlike traditional learning to rank datasets which have score Wxy between
each context-item pair x, y. Again, we may not be able to extract feature vectors for each x, y pair.
In such a situation, we can attempt to learn the score function f (x, y) without a feature vector ?(x, y)
by embedding each context and item in an Euclidean latent space; specifically, we redefine the score
function to be: f (x, y) := hUx , Vy i, where Ux ? Rd is the embedding of the context x and Vy ? Rd
3
http://www.mcs.anl.gov/research/projects/tao/index.html
5
is that of the item y. Then, we can learn these embeddings by a ranking model. This approach was
introduced in Weston et al. [24], and was called latent collaborative retrieval.
Now we specialize RoBiRank model for this task. Let us define ? to be the set of context-item pairs
(x, y) which was observed in the dataset. Let v(Wxy ) = 1 if (x, y) ? ?, and 0 otherwise; this is a
natural choice since the score information is not available. For simplicity, we set cx = 1 for every
x. Now RoBiRank (11) specializes to:
?
?
X
X
L1 (U, V ) =
?1 ?
?0 (f (Ux , Vy ) ? f (Ux , Vy0 ))? .
(12)
y 0 6=y
(x,y)??
Note that now the summation inside the parenthesis of (12) is over all items Y instead of a smaller
set Yx , therefore we omit specifying the range of y 0 from now on. To avoid overfitting, a regularizer
is added to (12); for simplicity we use the Frobenius norm of U and V in our experiments.
4.1
Stochastic Optimization
When the size of the data |?| or the number of items |Y| is large, however, methods that require
exact evaluation of the function value and its gradient will become very slow since the evaluation
takes O (|?| ? |Y|) computation. In this case, stochastic optimization methods are desirable [4];
in this subsection, we will develop a stochastic gradient descent algorithm whose complexity is
independent of |?| and |Y|.
For simplicity, let ? be a concatenation of all parameters {Ux }x?X , {Vy }y?Y . The gradient
?? L1 (U, V ) of (12) is
?
?
X
X
?? ? 1 ?
?0 (f (Ux , Vy ) ? f (Ux , Vy0 ))? .
(x,y)??
y 0 6=y
Finding an unbiased estimator of the gradient whose computation is independent of |?| is not difficult; if we sample a pair (x, y) uniformly from ?, then it is easy to see that the following estimator
?
?
X
|?| ? ?? ?1 ?
?0 (f (Ux , Vy ) ? f (Ux , Vy0 ))?
(13)
y 0 6=y
is unbiased. This still involves a summation over Y, however, so it requires O(|Y|) calculation.
Since ?1 (?) is a nonlinear function it seems unlikely that an unbiased stochastic gradient which
randomizes over Y can be found; nonetheless, to achieve convergence guarantees of the stochastic
gradient descent algorithm, unbiasedness of the estimator is necessary [18].
We attack this problem by linearizing the objective function by parameter expansion. Note the
following property of ?1 (?) [5]:
?1 (t) = log2 (t + 1) ? ? log2 ? +
? ? (t + 1) ? 1
.
log 2
(14)
1
This holds for any ? > 0, and the bound is tight when ? = t+1
. Now introducing an auxiliary
parameter ?xy for each (x, y) ? ? and applying this bound, we obtain an upper bound of (12) as
P
0
?xy
X
y 0 6=y ?0 (f (Ux , Vy ) ? f (Ux , Vy )) + 1 ? 1
L(U, V, ?) :=
? log2 ?xy +
. (15)
log 2
(x,y)??
Now we propose an iterative algorithm in which, each iteration consists of (U, V )-step and ?-step;
in the (U, V )-step we minimize (15) in (U, V ) and in the ?-step we minimize in ?. Pseudo-code can
be found in Algorithm 1 in Appendix C.
(U, V )-step The partial derivative
of (15) in terms of U and V can be calculated as:
P
P
0
?U,V L(U, V, ?) := log1 2 (x,y)?? ?xy
y 0 6=y ?U,V ?0 (f (Ux , Vy ) ? f (Ux , Vy )) . Now it is easy
to see that the following stochastic procedure unbiasedly estimates the above gradient:
6
RoBiRank 4
RoBiRank 16
RoBiRank 32
RoBiRank 1
0.1
0
0
0.5
1
1.5
2
2.5
3
0.2
Mean Precision@10
0.2
0.2
Weston et al. (2012)
RoBiRank 1
RoBiRank 4
RoBiRank 16
RoBiRank 32
0.3
Mean Precision@1
Mean Precision@1
0.3
0.1
0
Weston et al. (2012)
RoBiRank 1
RoBiRank 4
RoBiRank 16
RoBiRank 32
0.15
0.1
5 ? 10?2
0
0.2
number of machines ? seconds elapsed ?106
0.4
0.6
seconds elapsed
0.8
0
1
?105
0
0.2
0.4
0.6
seconds elapsed
0.8
1
?105
Figure 2: Left: Scaling of RoBiRank on Million Song Dataset. Center, Right: Comparison of
RoBiRank and Weston et al. [24] when the same amount of wall-clock computation time is given.
? Sample (x, y) uniformly from ?
? Sample y 0 uniformly from Y \ {y}
? Estimate the gradient by
|?| ? (|Y| ? 1) ? ?xy
? ?U,V ?0 (f (Ux , Vy ) ? f (Ux , Vy0 )).
log 2
(16)
Therefore a stochastic gradient descent algorithm based on (16) will converge to a local minimum
of the objective function (15) with probability one [21]. Note that the time complexity of calculating
(16) is independent of |?| and |Y|. Also, it is a function of only Ux and Vy ; the gradient is zero in
terms of other variables.
?-step When U and V are fixed, minimization of ?xy variable is independent of each other and
a simple analytic solution exists: ?xy = P 0 ?0 (f (Ux ,V1y )?f (Ux ,V 0 ))+1 . This of course requires
y
y 6=y
O(|Y|) work. In principle, we can avoid summation over Y by taking stochastic gradient in terms of
?xy as we did for U and V . However, since the exact solution is simple to compute and also because
most of the computation time is spent on (U, V )-step, we found this update rule to be efficient.
Parallelization The linearization trick in (15) not only enables us to construct an efficient stochastic gradient algorithm, but also makes possible to efficiently parallelize the algorithm across multiple
number of machines. Due to lack of space, details are relegated to Appendix D.
4.2
Experiments
In this subsection we use the Million Song Dataset (MSD) [3], which consists of 1,129,318 users
(|X |), 386,133 songs (|Y|), and 49,824,519 records (|?|) of a user x playing a song y in the training
dataset. The objective is to predict the songs from the test dataset that a user is going to listen to4 .
Since explicit ratings are not given, NDCG is not applicable for this task; we use precision at 1 and
10 [17] as our evaluation metric. In our first experiment we study the scaling behavior of RoBiRank
as a function of number of machines. RoBiRank p denotes the parallel version of RoBiRank which
is distributed across p machines. In Figure 2 (left) we plot mean Precision@1 as a function of the
number of machines ? the number of seconds elapsed; this is a proxy for CPU time. If an algorithm
linearly scales across multiple processors, then all lines in the figure should overlap with each other.
As can be seen RoBiRank exhibits near ideal speed up when going from 4 to 32 machines5 .
In our next experiment we compare RoBiRank with a state of the art algorithm from Weston et al.
[24], which optimizes a similar objective function (17). We compare how fast the quality of the
solution improves as a function of wall clock time. Since the authors of Weston et al. [24] do not
make available their code, we implemented their algorithm within our framework using the same
data structures and libraries used by our method. Furthermore, for a fair comparison, we used the
same initialization for U and V and performed an identical grid-search over the step size parameter.
4
the original data also provides the number of times a song was played by a user, but we ignored this in our
experiment.
5
The graph for RoBiRank 1 is hard to see because it was run for only 100,000 CPU-seconds.
7
It can be seen from Figure 2 (center, right) that on a single machine the algorithm of Weston et al.
[24] is very competitive and outperforms RoBiRank. The reason for this might be the introduction
of the additional ? variables in RoBiRank, which slows down convergence. However, RoBiRank
training can be distributed across processors, while it is not clear how to parallelize the algorithm
of Weston et al. [24]. Consequently, RoBiRank 32 which uses 32 machines for its computation can
produce a significantly better model within the same wall clock time window.
5
Related Work
In terms of modeling, viewing ranking problems as generalization of binary classification problems
is not a new idea; for example, RankSVM defines the objective function as a sum of hinge losses,
similarly to our basic model (6) in Section 3.1. However, it does not directly optimize the ranking
metric such as NDCG; the objective function and the metric are not immediately related to each
other. In this respect, our approach is closer to that of Le and Smola [14] which constructs a convex
upper bound on the ranking metric and Chapelle et al. [9] which improves the bound by introducing
non-convexity. The objective function of Chapelle et al. [9] is also motivated by ramp loss, which
is used for robust classification; nonetheless, to our knowledge the direct connection between the
ranking metrics in form (7) (DCG, NDCG) and the robust loss (3) is our novel contribution. Also,
our objective function is designed to specifically bound the ranking metric, while Chapelle et al. [9]
proposes a general recipe to improve existing convex bounds.
Stochastic optimization of the objective function for latent collaborative retrieval has been also explored in Weston et al. [24]. They attempt to minimize
?
?
X
X
? ?1 +
I(f (Ux , Vy ) ? f (Ux , Vy0 ) < 0)? ,
(17)
(x,y)??
y 0 6=y
Pt
where ?(t) = k=1 k1 . This is similar to our objective function (15); ?(?) and ?2 (?) are asymptotically equivalent. However, we argue that our formulation (15) has two major advantages. First, it is
a continuous and differentiable function, therefore gradient-based algorithms such as L-BFGS and
stochastic gradient descent have convergence guarantees. On the other hand, the objective function
of Weston et al. [24] is not even continuous, since their formulation is based on a function ?(?)
that is defined for only natural numbers. Also, through the linearization trick in (15) we are able
to obtain an unbiased stochastic gradient, which is necessary for the convergence guarantee, and to
parallelize the algorithm across multiple machines as discussed in Appendix D. It is unclear how
these techniques can be adapted for the objective function of Weston et al. [24].
6
Conclusion
In this paper, we developed RoBiRank, a novel model on ranking, based on insights and techniques
from robust binary classification. Then, we proposed a scalable and parallelizable stochastic optimization algorithm that can be applied to latent collaborative retrieval task which large-scale data
without feature vectors and explicit scores have to take care of. Experimental results on both learning
to rank datasets and latent collaborative retrieval dataset suggest the advantage of our approach.
As a final note, the experiments in Section 4.2 are arguably unfair towards WSABIE. For instance,
one could envisage using clever engineering tricks to derive a parallel variant of WSABIE (e.g.,
by averaging gradients from various machines), which might outperform RoBiRank on the MSD
dataset. While performance on a specific dataset might be better, we would lose global convergence
guarantees. Therefore, rather than obsess over the performance of a specific algorithm on a specific
dataset, via this paper we hope to draw the attention of the community to the need for developing
principled parallel algorithms for this important problem.
Acknowledgments We thank anonymous reviewers for their constructive comments, and Texas
Advanced Computing Center for infrastructure and support for experiments. This material is partially based upon work supported by the National Science Foundation under grant No. IIS-1117705.
8
References
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9
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4,820 | 5,364 | Tight Bounds for Influence in Diffusion Networks and
Application to Bond Percolation and Epidemiology
R?emi Lemonnier1,2
Kevin Scaman1
Nicolas Vayatis1
1
2
CMLA ? ENS Cachan, CNRS, France, 1000mercis, Paris, France
{lemonnier, scaman, vayatis}@cmla.ens-cachan.fr
Abstract
In this paper, we derive theoretical bounds for the long-term influence of a node
in an Independent Cascade Model (ICM). We relate these bounds to the spectral
radius of a particular matrix and show that the behavior is sub-critical when this
spectral radius is lower than 1. More specifically,
? we point out that, in general
networks, the sub-critical regime behaves in O( n) where n is the size of the
network, and that this upper bound is met for star-shaped networks. We apply our
results to epidemiology and percolation on arbitrary networks, and derive a bound
for the critical value beyond which a giant connected component arises. Finally,
we show empirically the tightness of our bounds for a large family of networks.
1
Introduction
The emergence of social graphs of the World Wide Web has had a considerable effect on propagation of ideas or information. For advertisers, these new diffusion networks have become a favored
vector for viral marketing operations, that consist of advertisements that people are likely to share
by themselves with their social circle, thus creating a propagation dynamics somewhat similar to
the spreading of a virus in epidemiology ([1]). Of particular interest is the problem of influence
maximization, which consists of selecting the top-k nodes of the network to infect at time t = 0
in order to maximize in expectation the final number of infected nodes at the end of the epidemic.
This problem was first formulated by Domingues and Richardson in [2] and later expressed in [3]
as an NP-hard discrete optimization problem under the Independent Cascade (IC) framework, a
widely-used probabilistic model for information propagation.
From an algorithmic point of view, influence maximization has been fairly well studied. Assuming
the transmission probability of all edges are known, Kempe, Kleinberg and Tardos ([3]) derived
a greedy algorithm based on Monte-Carlo simulations that was shown to approximate the optimal
solution up to a factor 1 ? 1e , building on classical results of optimization theory. Since then, various
techniques were proposed in order to significantly improve the scalability of this algorithm ([4, 5, 6,
7]), and also to provide an estimate of the transmission probabilities from real data ([8, 9]). Recently,
a series of papers ([10, 11, 12]) introduced continuous-time diffusion networks in which infection
spreads during a time period T at varying rates across the different edges. While these models
provide a more accurate representation of real-world networks for finite T , they are equivalent to the
IC model when T ? ?. In this paper, will focus on such long-term behavior of the contagion.
From a theoretical point of view, little is known about the influence maximization problem under the
IC model framework. The most celebrated result established by Newman ([13]) proves the equivalence between bond percolation and the Susceptible-Infected-Removed (SIR) model in epidemiology
([14]) that can be identified to a special case of IC model where transmission probability are equal
amongst all infectious edges.
In this paper, we propose new bounds on the influence of any set of nodes. Moreover, we prove the
existence of an epidemic threshold for a key quantity defined by the spectral radius of a given hazard
1
matrix.
? Under this threshold, the influence of any given set of nodes in a network of size n will be
O( n), while the influence of a randomly chosen set of nodes will be O(1). We provide empirical
evidence that these bounds are sharp for a family of graphs and sets of initial influencers and can
therefore be used as what is to our knowledge the first closed-form formulas for influence estimation.
We show that these results generalize bounds obtained on the SIR model by Draief, Ganesh and
Massouli?e ([15]) and are closely related to recent results on percolation on finite inhomogeneous
random graphs ([16]).
The rest of the paper is organized as follows. In Sec. 2, we recall the definition of Information
Cascades Model and introduce useful notations. In Sec. 3, we derive theoretical bounds for the
influence. In Sec. 4, we show that our results also apply to the fields of percolation and epidemiology
and generalize existing results in these fields. In Sec. 5, we illustrate our results by applying them
on simple networks and retrieving well-known results. In Sec. 6, we perform experiments in order
to show that our bounds are sharp for a family of graphs and sets of initial nodes.
2
2.1
Information Cascades Model
Influence in random networks and infection dynamics
Let G = (V, E) be a directed network of n nodes and A ? V be a set of n0 nodes that are initially
contagious (e.g. aware of a piece of information, infected by a disease or adopting a product). In
the sequel, we will refer to A as the influencers. The behavior of the cascade is modeled using a
probabilistic framework. The influencer nodes spread the contagion through the network by means
of transmission through the edges of the network. More specifically, each contagious node can infect
its neighbors with a certain probability. The influence of A, denoted as ?(A), is the expected number
of nodes reached by the contagion originating from A, i.e.
X
?(A) =
P(v is infected by the contagion |A).
(1)
v?V
We consider three infection dynamics that we will show in the next section to be equivalent regarding
the total number of infected nodes at the end of the epidemic.
Discrete-Time Information Cascades [DT IC(P)] At time t = 0, only the influencers are infected. Given a matrix P = (pij )ij ? [0, 1]n?n , each node i that receives the contagion at time t
may transmit it at time t + 1 along its outgoing edge (i, j) ? E with probability pij . Node i cannot
make any attempt to infect its neighbors in subsequent rounds. The process terminates when no
more infections are possible.
Continuous-Time Information Cascades [CT IC(F, T )] At time t = 0, only the influencers
are infected. Given a matrix F = (fij )ij of non-negative integrable functions, each node i that
receives the contagion at time t may transmit it at time s > t along its outgoing edge (i, j) ? E
with stochastic rate of occurrence fij (s ? t). The process terminates at a given deterministic time
T > 0. This model is much richer than Discrete-time IC, but we will focus here on its behavior
when T = ?.
Random Networks [RN (P)] Given a matrix P = (pij )ij ? [0, 1]n?n , each edge (i, j) ? E is
removed independently of the others with probability 1 ? pij . A node i ? V is said to be infected if
i is linked to at least one element of A in the spanning subgraph G 0 = (V, E 0 ) where E 0 ? E is the
set of non-removed edges.
For any v ? V, we will designate by influence of v the influence of the set containing only v,
i.e. ?({v}). We will show in Section 4.2 that, if P is symmetric and G undirected, these three
infection processes are equivalent to bond percolation and the influence of a node v is also equal
to the expected size of the connected component containing v in G 0 . This will make our results
applicable to percolation in arbitrary networks. Following the percolation literature, we will denote
as sub-critical a cascade whose influence is not proportional to the size of the network n.
2
2.2
The hazard matrix
In order to linearize the influence problem and derive upper bounds, we introduce the concept of
hazard matrix, which describes the behavior of the information cascade. As we will see in the
following, in the case of Continuous-time Information Cascades, this matrix gives, for each edge of
the network, the integral of the instantaneous rate of transmission (known as hazard function). The
spectral radius of this matrix will play a key role in the influence of the cascade.
Definition. For a given graph G = (V, E) and edge transmission probabilities pij , let H be the
n ? n matrix, denoted as the hazard matrix, whose coefficients are
? ln(1 ? pij ) if (i, j) ? E
Hij =
.
(2)
0
otherwise
Next lemma shows the equivalence between the three definitions of the previous section.
Lemma 1. For a given graph G = (V, E), set of influencers A, and transmission probabilities matrix P, the distribution of the set of infected nodes is equal under
R ?the infection dynamics
DT IC(P), CT IC(F, ?) and RN (P), provided that for any (i, j) ? E, 0 fij (t)dt = Hij .
Definition. For a given set of influencers A ? V, we will denote as H(A) the hazard matrix except
for zeros along the columns whose indices are in A:
H(A)ij = 1{j ?A}
Hij .
/
(3)
We recall that for any square matrix M , its spectral radius ?(M ) is defined by ?(M ) = maxi (|?i |)
where ?1 , ..., ?n are the (possibly repeated) eigenvalues of matrix M . We will also use that, when
>
>
M is a real square matrix with positive entries, ?( M +M
) = supX XX >MXX .
2
Remark. When the pij are small, the hazard matrix is very close to the transmission matrix P.
This implies that, for low pij values, the spectral radius of H will be very close to that of P. More
specifically, a simple calculation holds
?(P) ? ?(H) ?
? ln(1 ? kPk? )
?(P),
kPk?
(4)
for x ? 1? implies that the
where kPk? = maxi,j pij . The relatively slow increase of ? ln(1?x)
x
behavior of ?(P) and ?(H) will be of the same order of magnitude even for high (but lower than 1)
values of kPk? .
3
Upper bounds for the influence of a set of nodes
Given A ? V the set of influencer nodes and |A| = n0 < n, we derive here two upper bounds
for the influence of A. The first bound (Proposition 1) applies to any set of influencers A such
that |A| = n0 . Intuitively, this result correspond to a best-case scenario (or a worst-case scenario,
depending on the viewpoint), since we can target any set of nodes so as to maximize the resulting
contagion.
>
Proposition 1. Define ?c (A) = ?( H(A)+H(A)
). Then, for any A such that |A| = n0 < n, denoting
2
by ?(A) the expected number of nodes reached by the cascade starting from A:
?(A) ? n0 + ?1 (n ? n0 ),
where ?1 is the smallest solution in [0, 1] of the following equation:
?c (A)n0
?1 ? 1 + exp ??c (A)?1 ?
= 0.
?1 (n ? n0 )
Corollary 1. Under the same assumptions:
3
(5)
(6)
s
? if ?c (A) < 1,
? if ?c (A) ? 1,
?(A) ? n0 +
?c (A) p
n0 (n ? n0 ),
1 ? ?c (A)
2?c (A)
!
.
?(A) ? n ? (n ? n0 ) exp ??c (A) ? p
4n/n0 ? 3 ? 1
?
In particular, when ?c (A) < 1, ?(A) = O( n) and the regime is sub-critical.
The second result (Proposition 2) applies in the case where A is drawn from a uniform distribution
over the ensemble of sets of n0 nodes chosen amongst n (denoted as Pn0 (V)). This result corresponds to the average-case scenario in a setting where the initial influencer nodes are not known and
drawn independently of the transmissions over each edge.
>
Proposition 2. Define ?c = ?( H+H
). Assume the set of influencers A is drawn from a uniform
2
distribution over Pn0 (V). Then, denoting by ?uniform the expected number of nodes reached by the
cascade starting from A:
?uniform ? n0 + ?2 (n ? n0 ),
where ?2 is the unique solution in [0, 1] of the following equation:
?c n 0
?2 ? 1 + exp ??c ?2 ?
= 0.
n ? n0
(7)
(8)
Corollary 2. Under the same assumptions:
? if ?c < 1,
? if ?c ? 1,
n0
,
1 ? ?c
?c
?uniform ? n ? (n ? n0 ) exp ?
.
1 ? nn0
?uniform ?
In particular, when ?c < 1, ?uniform = O(1) and the regime is sub-critical.
?
The difference in the sub-critical regime between O( n) and O(1) for the worst and average case
influence is an important feature of our results, and is verified in our experiments (see Sec. 6). Intuitively, when the network is inhomogeneous and contains highly central nodes (e.g. scale-free networks), there will be a significant difference between specifically targeting the most central nodes
and random targeting (which will most probably target a peripheral node).
4
Application to epidemiology and percolation
Building on the celebrated equivalences between the fields of percolation, epidemiology and influence maximization, we show that our results generalize existing results in these fields.
4.1
Susceptible-Infected-Removed (SIR) model in epidemiology
We show here that Proposition 1 further improves results on the SIR model in epidemiology. This
widely used model was introduced by Kermac and McKendrick ([14]) in order to model the propagation of a disease in a given population. In this setting, nodes represent individuals, that can be
in one of three possible states, susceptible (S), infected (I) or removed (R). At t = 0, a subset A of
n0 nodes is infected and the epidemic spreads according to the following evolution. Each infected
node transmits the infection along its outgoing edge (i, j) ? E at stochastic rate of occurrence ? and
is removed from the graph at stochastic rate of occurrence ?. The process ends for a given T > 0.
It is straightforward that, if the removed events are not observed, this infection process is equivalent
to CT IC(F, T ) where for any (i, j) ? E,fij (t) = ? exp(??t). The hazard matrix H is therefore
equal to ?? A where A = 1{(i,j)?E} ij is the adjacency matrix of the underlying network. Note
4
that, by Lemma 1, our results can be used in order to model the total number of infected nodes in a
setting where infection and recovery rates of a given node exhibit a non-exponential behavior. For
instance, incubation periods for different individuals generally follow a log-normal distribution [17],
which indicates that continuous-time IC with a log-normal rate of removal might be well-suited to
model some kind of infections.
It was recently shown by Draief, Ganesh and Massouli?e ([15]) that, in the case of undirected networks, and if ??(A) < ?,
?
nn0
?(A) ?
.
(9)
?
1 ? ? ?(A)
?
This result shows, that, when ?(H) = ?? ?(A) < 1, the influence of set of nodes A is O( n).
We show in the next lemma that this result is a direct consequence of Corollary 1: the condition
?c (A) < 1 is weaker than ?(H) < 1 and, under these conditions, the bound of Corollary 1 is tighter.
Lemma 2. For any symmetric adjacency matrix A, initial set of influencers A such that |A| = n0 <
?
n, ? > 0 and ? < ?(A)
, we have simultaneously ?c (A) ? ?? ?(A) and
s
?
nn0
?c (A) p
,
(10)
n0 (n ? n0 ) ?
n0 +
?
1 ? ?c (A)
1 ? ? ?(A)
where the condition ? <
?
?(A)
imposes that the regime is sub-critical.
Moreover, these new bounds capture with more accuracy the behavior of the influence in extreme
cases. In the limit ? ? 0, the difference between the two?
bounds is significant, because Proposition
1 yields ?(A) ? n0 whereas (9) only ensures ?(A) ? nn0 . When n = n0 , Proposition 1 also
0
ensures that ?(A) = n0 whereas (9) yields ?(A) ? 1? ?n?(A)
. Secondly, Proposition 1 gives also
?
bounds in the case ??(A) ? ?. Finally, Proposition 1 applies to more general cases that the classical
homogeneous SIR model, and allows infection and recovery rates to vary across individuals.
4.2
Bond percolation
Given a finite undirected graph G = (V, E), bond percolation theory describes the behavior of
connected clusters of the spanning subgraph of G obtained by retaining a subset E 0 ? E of edges
of G according to a given distribution.When these removals occur independently along each edge
with same probability 1 ? p, this process is called homogeneous percolation and is fairly well known
(see e.g [18]). The inhomogeneous case, where the independent edge removal probabilities 1 ? pij
vary across the edges, is more intricate and has been the subject of recent studies. In particular,
results on critical probabilities and size of the giant component have been obtained by Bollobas,
Janson and Riordan in [16]. However, these bounds hold for a particular class of asymptotic graphs
(inhomogeneous random graphs) when n ? ?. In the next lemma, we show that our results can be
used in order to obtain bounds that hold in expectation for any fixed graph.
Lemma 3. Let P = (pij )ij ? [0, 1]n?n be a symmetric matrix. Let G 0 = (V, E 0 ) be the undirected
subgraph of G such that each edge {i, j} ? E is removed independently with probability 1 ? pij . Let
Gd = (V, Ed ) be the directed graph such that (i, j) ? Ed ?? {i, j} ? E. Then, for any v ? V,
the expected size of the connected component containing v in G 0 is equal to the influence of v in Gd
under the infection process DT IC(P).
We now derive an upper bound for C1 (G 0 ), the size of the largest connected component of the
spanning subgraph G 0 = (V, E 0 ). In the following, we will denote by E[C1 (G 0 )] the expected value
of this random variable, given P = (pij )ij .
Proposition 3. Let G = (V, E) be an undirected network where each edge {i, j} ? E has an independent probability 1 ? pij of being removed. The expected size of the largest connected component
of the resulting subgraph G 0 is upper bounded by:
?
(11)
E[C1 (G 0 )] ? n ?3 ,
where ?3 is the unique solution in [0, 1] of the following equation:
n?1
n
?3 ? 1 +
exp ?
?(H)?3 = 0.
n
n?1
5
(12)
Moreover, the resulting network has a probability of being connected upper bounded by:
P(G 0 is connected) ? ?3 .
(13)
In the case ?(H) < 1, we can further simplify our bounds in the same way than for Propositions 1
and 2.
q
n
Corollary 3. In the case ?(H) < 1, E[C1 (G 0 )] ? 1??(H)
.
Whereas our results hold for any n ? N, classical results in percolation theory study the asymptotic
behavior of sequences of graphs when n ? ?. In order to further compare our results, we therefore
consider sequences of spanning subgraphs (G 0 n )n ?N , obtained by removing each edge of graphs
of n nodes (Gn )n ?N with probability 1 ? pnij . A previous result ([16], Corollary 3.2 of section
5) states that, for particular sequences known as inhomogeneous random graphs and under a given
sub-criticality condition, C1 (G 0 n ) = o(n) asymptotically almost surely (a.a.s.), i.e with probability
going to 1 as n ? ?. Using Proposition 3, we get for our part the following result:
Corollary 4. Assume the sequence Hn = ? ln(1 ? pnij ) ij
is such that
n ?N
n
lim sup ?(H ) < 1.
(14)
n??
Then, for any > 0, we have asymptotically almost surely when n ? ?,
C1 (Gn0 ) = o(n1/2+ ).
(15)
This result is to our knowledge the first to bound the expected size of the largest connected component in general arbitrary networks.
5
Application to particular networks
In order to illustrate our theoretical results, we now apply our bounds to three specific networks and
compare them to existing results, showing that our bounds are always of the same order than these
specific results. We consider three particular networks: 1) star-shaped networks, 2) Erd?os-R?enyi
networks and 3) random graphs with an expected degree distribution. In order to simplify these
problems and exploit existing theorems, we will consider in this section that pij = p is fixed for
each edge {i, j} ? E. Infection dynamics thus only depend on p, the set of influencers A, and the
structure of the underlying network.
5.1
Star-shaped networks
For a star shaped network centered around a given node v1 , and A = {v1 }, the exact influence is
computable and writes ?({v1 }) = 1 + p(n ? 1). As H(A)ij = ? ln(1 ? p)1{i=1,j6=1} , the spectral
radius is given by
H(A) + H(A)>
? ln(1 ? p) ?
?
=
n ? 1.
(16)
2
2
Therefore, Proposition 1 states that ?({v1 }) ? 1 + (n ? 1)?1 where ?1 is the solution of equation
?
1
ln(1 ? p)
1 ? ?1 = exp
?1 n ? 1 + ?
.
(17)
2
?1 n ? 1
1
1
It is worth mentionning that, when p = ?n?1
, ?1 = ?n?1
is solution of (17) and therefore the
?
bound is ?({v1 }) ? 1 + n ? 1 which is tight. Note that, in the case of star-shaped networks, the
influence does not present a critical behavior and is always linear with respect to the total number of
nodes n.
5.2
Erd?os-R?enyi networks
For Erd?os-R?enyi networks G(n, p) (i.e. an undirected network with n nodes where each couple of
nodes (i, j) ? V 2 belongs to E independently of the others with probability p), the exact influence
6
of a set of nodes is not known. However, percolation theory characterizes the limit behavior of the
giant connected component when n ? ?. In the simplest case of Erd?os-R?enyi networks G(n, nc )
the following result holds:
Lemma 4. (taken from [16]) For a given sequence of Erd?os-R?enyi networks G(n, nc ), we have:
? if c < 1, C1 (G(n, nc )) ?
3
(1?c)2
log(n) a.a.s.
? if c > 1, C1 (G(n, nc )) = (1 + o(1))?n a.a.s. where ? ? 1 + exp(??c) = 0.
As previously stated, our results hold for any given graph, and not only asymptotically. However,
we get an asymptotic behavior consistent with the aforementioned result. Indeed, using notations of
n
section 4.2, Hij
= ? ln(1 ? nc )1{i6=j} and ?(Hn ) = ?(n ? 1) ln(1 ? nc ). Using Proposition 3, and
noting that ?3 = (1 + o(1))?, we get that, for any > 0:
? if c < 1, C1 (G(n, nc )) = o(n1/2+ ) a.a.s.
? if c > 1, C1 (G(n, nc )) ? (1 + o(1))?n1+ a.a.s., where ? ? 1 + exp(??c) = 0.
5.3
Random graphs with given expected degree distribution
In this section, we apply our bounds to random graphs whose expected degree distribution is fixed
(see e.g [19], section 13.2.2). More specifically, let w = (wi )i?{1,...,n} be the expected degree of
each node of the network. For a fixed w, let G(w) be a random graph whose edges are selected
independently and randomly with probability
1{i6=j} wi wj
.
(18)
qij = P
k wk
For these graphs, results on the volume of connected components (i.e the expected sum of degrees
of the nodes in these components) were derived in [20] but our work gives to our knowledge the first
result on the size of the giant component. Note that Erd?os-R?enyi G(n, p) networks are a special case
of (18) where wi = np for any i ? V.
In order to further compare our results, we note that these graphs are also very similar to the widely
used configuration model where node degrees are fixed to a sequence w, the main difference being
that the occupation probabilities pij are in this P
case not independent
anymore. For configuration
P
models, a giant component exists if and only if i wi2 > 2 i wi ([21, 22]). In theP
case of graphs
P
with given expected degree distribution, we retrieve the key role played by the ratio i wi2 / i wi
>
) < 1 where
in our criterion of non-existence of the giant component given by ?( H+H
2
P
w2
H + H>
?
? ?((qij )ij ) ? Pi i .
(19)
2
i wi
The left-hand approximation is particularly good when the qij are small. This is for instance the case
?
as soon as there exists ? < 1 such that, for any i ? V, wi = o(n
P ). The right-hand side
P is based
P on
2
the fact that the spectral radius of the matrix (qij + 1{i=j} wi / k wk )ij is given by i wi2 / i wi .
6
Experimental results
In this section, we show that the bounds given in Sec. 3 are tight (i.e. very close to empirical results in
particular graphs), and are good approximations of the influence on a large set of random networks.
Fig. 1a compares experimental simulations of the influence to the bound derived in proposition 1.
The considered networks have n = 1000 nodes and are of 6 types (see e.g [19] for further details on
these different networks): 1) Erd?os-R?enyi networks, 2) Preferential attachment networks, 3) Smallworld networks, 4) Geometric random networks ([23]), 5) 2D regular grids and 6) totally connected
networks with fixed weight b ? [0, 1] except for the ingoing and outgoing edges of the influencer
node A = {v1 } having weight a ? [0, 1]. Except for totally connected networks, edge probabilities
are set to the same value p for each edge (this parameter was used to tune the spectral radius ?c (A)).
All points of the plots are averages over 100 simulations. The results show that the bound in proposition 1 is tight (see totally connected networks in Fig. 1a) and close to the real influence for a large
7
class of random
? networks. In particular, the tightness of the bound around ?c (A) = 1 validates the
behavior in n of the worst-case influence in the sub-critical regime. Similarly, Fig. 1b compares
1000
900
900
800
800
700
700
uniform
600
influence (?
influence (?(A))
)
1000
500
400
totally connected
erdos renyi
preferential attachment
small World
geometric random
2D grid
upper bound
300
200
100
0
0
2
4
6
8
spectral radius of the Hazard matrix (?c(A))
(a) Fixed set of influencers
600
500
400
totally connected
erdos renyi
preferential attachment
small World
geometric random
2D grid
upper bound
300
200
100
0
10
0
2
4
6
8
spectral radius of the Hazard matrix (?c)
10
(b) Uniformly distributed set of influencers
Figure 1: Empirical influence on random networks of various types. The solid lines are the upper
bounds in propositions 1 (for Fig. 1a) and 2 (for Fig. 1b).
experimental simulations of the influence to the bound derived in proposition 2 in the case of random
initial influencers. While this bound is not as tight as the previous one, the behavior of the bound
agrees with experimental simulations, and proves a relatively good approximation of the influence
under a random set of initial influencers. It is worth mentioning that the bound is tight for the subcritical regime and shows that corollary 2 is a good approximation of ?uniform when ?c < 1. In
order to verify the criticality of ?c (A) = 1, we compared the behavior of ?(A)?w.r.t the size of the
network n. When ?c (A) < 1 (see Fig. 2a in which ?c (A) = 0.5), ?(A) = O( n), and the bound
is tight. On the contrary, when ?c (A) > 1 (see Fig. 2b in which ?c (A) = 1.5), ?(A) = O(n), and
?(A) is linear w.r.t. n for most random networks.
totally connected
erdos renyi
preferential attachment
small World
geometric random
2D grid
upper bound
influence (?(A))
25
20
400
15
300
200
10
100
5
0
totally connected
erdos renyi
preferential attachment
small World
geometric random
2D grid
upper bound
500
influence (?(A))
30
0
200
400
600
size of the network (n)
800
1000
(a) Sub-critical regime: ?c (A) = 0.5
0
0
200
400
600
size of the network (n)
800
1000
(b) Super-critical regime: ?c (A) = 1.5
Figure 2: Influence w.r.t. the size of the network in the sub-critical and super-critical regime. The
solid line is the upper bound in proposition 1. Note the square-root versus linear behavior.
7
Conclusion
In this paper, we derived the first upper bounds for the influence of a given set of nodes in any
finite graph under the Independent Cascade Model (ICM) framework, and relate them to the spectral
radius of a given hazard matrix. We show that these bounds can also be used to generalize previous
results in the fields of epidemiology and percolation. Finally, we provide empirical evidence that
these bounds are close to the best possible for general graphs.
Acknowledgments
This research is part of the SODATECH project funded by the French Government within the program of ?Investments for the Future ? Big Data?.
8
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lemma:8 total:3 called:1 experimental:4 people:1 mark:2 arises:1 outgoing:4 |
4,821 | 5,365 | Shaping Social Activity by Incentivizing Users
Mehrdad Farajtabar?
Nan Du?
Manuel Gomez-Rodriguez?
?
Isabel Valera
Hongyuan Zha?
Le Song?
?
?
Georgia Institute of Technology
MPI for Software Systems
Univ. Carlos III in Madrid?
{mehrdad,dunan}@gatech.edu
[email protected]
{zha,lsong}@cc.gatech.edu
[email protected]
Abstract
Events in an online social network can be categorized roughly into endogenous
events, where users just respond to the actions of their neighbors within the network, or exogenous events, where users take actions due to drives external to the
network. How much external drive should be provided to each user, such that the
network activity can be steered towards a target state? In this paper, we model
social events using multivariate Hawkes processes, which can capture both endogenous and exogenous event intensities, and derive a time dependent linear relation between the intensity of exogenous events and the overall network activity.
Exploiting this connection, we develop a convex optimization framework for determining the required level of external drive in order for the network to reach a
desired activity level. We experimented with event data gathered from Twitter,
and show that our method can steer the activity of the network more accurately
than alternatives.
1
Introduction
Online social platforms routinely track and record a large volume of event data, which may correspond to the usage of a service (e.g., url shortening service, bit.ly). These events can be categorized
roughly into endogenous events, where users just respond to the actions of their neighbors within
the network, or exogenous events, where users take actions due to drives external to the network.
For instance, a user?s tweets may contain links provided by bit.ly, either due to his forwarding of a
link from his friends, or due to his own initiative to use the service to create a new link.
Can we model and exploit these data to steer the online community to a desired activity level?
Specifically, can we drive the overall usage of a service to a certain level (e.g., at least twice per
day per user) by incentivizing a small number of users to take more initiatives? What if the goal is
to make the usage level of a service more homogeneous across users? What about maximizing the
overall service usage for a target group of users? Furthermore, these activity shaping problems need
to be addressed by taking into account budget constraints, since incentives are usually provided in
the form of monetary or credit rewards.
Activity shaping problems are significantly more challenging than traditional influence maximization problems, which aim to identify a set of users, who, when convinced to adopt a product, shall
influence others in the network and trigger a large cascade of adoptions [1, 2]. First, in influence
maximization, the state of each user is often assumed to be binary, either adopting a product or
not [1, 3, 4, 5]. However, such assumption does not capture the recurrent nature of product usage,
where the frequency of the usage matters. Second, while influence maximization methods identify
a set of users to provide incentives, they do not typically provide a quantitative prescription on how
much incentive should be provided to each user. Third, activity shaping concerns a larger variety of
target states, such as minimum activity and homogeneity of activity, not just activity maximization.
In this paper, we will address the activity shaping problems using multivariate Hawkes processes [6],
which can model both endogenous and exogenous recurrent social events, and were shown to be a
good fit for such data in a number of recent works (e.g., [7, 8, 9, 10, 11, 12]). More importantly,
1
we will go beyond model fitting, and derive a novel predictive formula for the overall network activity given the intensity of exogenous events in individual users, using a connection between the
processes and branching processes [13, 14, 15, 16]. Based on this relation, we propose a convex
optimization framework to address a diverse range of activity shaping problems given budget constraints. Compared to previous methods for influence maximization, our framework can provide
more fine-grained control of network activity, not only steering the network to a desired steady-state
activity level but also do so in a time-sensitive fashion. For example, our framework allows us to
answer complex time-sensitive queries, such as, which users should be incentivized, and by how
much, to steer a set of users to use a product twice per week after one month?
In addition to the novel framework, we also develop an efficient gradient based optimization algorithm, where the matrix exponential needed for gradient computation is approximated using the
truncated Taylor series expansion [17]. This algorithm allows us to validate our framework in a
variety of activity shaping tasks and scale up to networks with tens of thousands of nodes. We also
conducted experiments on a network of 60,000 Twitter users and more than 7,500,000 uses of a popular url shortening services. Using held-out data, we show that our algorithm can shape the network
behavior much more accurately than alternatives.
2
Modeling Endogenous-Exogenous Recurrent Social Events
We model the events generated by m users in a social network as a m-dimensional counting process
N (t) = (N1 (t), N2 (t), . . . , Nm (t))" , where Ni (t) records the total number of events generated by
user i up to time t. Furthermore, we represent each event as a tuple (ui , ti ), where ui is the user identity and ti is the event timing. Let the history of the process up to time t be Ht := {(ui , ti ) | ti ! t},
and Ht? be the history until just before time t. Then the increment of the process, dN (t), in an infinitesimal window [t, t + dt] is parametrized by the intensity ?(t) = (?1 (t), . . . , ?m (t))" " 0, i.e.,
E[dN (t)|Ht? ] = ?(t) dt.
(1)
Intuitively, the larger the intensity ?(t), the greater the likelihood of observing an event in the time
window [t, t + dt]. For instance, a Poisson process in [0, ?) can be viewed as a special counting
process with a constant intensity function ?, independent of time and history. To model the presence
of both endogenous and exogenous events, we will decompose the intensity into two terms
?(t)
!"#$
overall event intensity
=
?(0) (t)
! "# $
exogenous event intensity
+
?? (t)
! "# $
,
(2)
endogenous event intensity
where the exogenous event intensity models drive outside the network, and the endogenous event
intensity models interactions within the network. We assume that hosts of social platforms can
potentially drive up or down the exogenous events intensity by providing incentives to users; while
endogenous events are generated due to users? own interests or under the influence of network peers,
and the hosts do not interfere with them directly. The key questions in the activity shaping context
are how to model the endogenous event intensity which are realistic to recurrent social interactions,
and how to link the exogenous event intensity to the endogenous event intensity. We assume that the
exogenous event intensity is independent of the history and time, i.e., ?(0) (t) = ?(0) .
2.1 Multivariate Hawkes Process
Recurrent endogenous events often exhibit the characteristics of self-excitation, where a user tends
to repeat what he has been doing recently, and mutual-excitation, where a user simply follows what
his neighbors are doing due to peer pressure. These social phenomena have been made analogy to
the occurrence of earthquake [18] and the spread of epidemics [19], and can be well-captured by
multivariate Hawkes processes [6] as shown in a number of recent works (e.g., [7, 8, 9, 10, 11, 12]).
More specifically, a multivariate Hawkes process is a counting process who has a particular form
of intensity. We assume that the strength of influence between users is parameterized by a sparse
nonnegative influence matrix A = (auu! )u,u! ?[m] , where auu! > 0 means user u% directly excites
user u. We also allow A to have nonnegative diagonals to model self-excitation of a user. Then, the
intensity of the u-th dimension is
& t
%
%
?
?u (t) =
auui g(t ? ti ) =
auu!
g(t ? s) dNu! (s),
(3)
i:ti <t
u! ?[m]
0
'?
where g(s) is a nonnegative kernel function such that g(s) = 0 for s ? 0 and 0 g(s) ds <
?; the second equality is obtained by grouping events according to users and use the fact that
2
1
2
1
3
t1
3
1
2
5
4
2
1
3
5
1
5
1
6
1
5
t2
2
3
t3
3
6
5
5
6
3
4
2
2 3 1
1 2 4
t
(a) An example social network
(b) Branching structure of events
Figure 1: In Panel (a), each directed edge indicates that the target node follows, and can be influenced
by, the source node. The activity in this network is modeled using Hawkes processes, which result in
branching structure of events shown in Panel (b). Each exogenous event is the root node of a branch
(e.g., top left most red circle at t1 ), and it occurs due to a user?s own initiative; and each event can
trigger one or more endogenous events (blue square at t2 ). The new endogenous events can create
the next generation of endogenous events (green triangles at t3 ), and so forth. The social network
will constrain the branching structure of events, since an event produced by a user (e.g., user 1) can
only trigger endogenous events in the same user or one or more of her followers (e.g., user 2 or 3).
't
(
?
g(t ? s) dNu! (s) =
ui =u! ,ti <t g(t ? ti ). Intuitively, ?u (t) models the propagation of peer
0
influence over the network ? each event (ui , ti ) occurred in the neighbor of a user will boost her
intensity by a certain amount which itself decays over time. Thus, the more frequent the events
occur in the user?s neighbor, the more likely she will be persuaded to generate a new event.
For simplicity, we will focus on an exponential kernel, g(t ? ti ) = exp(??(t ? ti )) in the reminder
of the paper. However, multivariate Hawkes processes and the branching processed explained in
next section is independent of the kernel choice and can be extended to other kernels such as powerlaw, Rayleigh or any other long tailed distribution over nonnegative real domain. Furthermore, we
can rewrite equation (3) in vectorial format
& t
?
? (t) =
G(t ? s) dN (s),
(4)
0
by defining a m ? m time-varying matrix G(t) = (auu! g(t))u,u! ?[m] . Note that, for multivariate
Hawkes processes, the intensity, ?(t), itself is a random quantity, which depends on the history Ht .
We denote the expectation of the intensity with respect to history as
?(t) := EHt? [?(t)]
(5)
2.2 Connection to Branching Processes
A branching process is a Markov process that models a population in which each individual in
generation k produces some random number of individuals in generation k + 1, according some
distribution [20]. In this section, we will conceptually assign both exogenous events and endogenous
events in the multivariate Hawkes process to levels (or generations), and associate these events with
a branching structure which records the information on which event triggers which other events (see
Figure 1 for an example). Note that this genealogy of events should be interpreted in probabilistic
terms and may not be observed in actual data. Such connection has been discussed in Hawkes?
original paper on one dimensional Hawkes processes [21], and it has recently been revisited in the
context of multivariate Hawkes processes by [11]. The branching structure will play a crucial role in
deriving a novel link between the intensity of the exogenous events and the overall network activity.
More specifically, we assign all exogenous events to the zero-th generation, and record the number
of such events as N (0) (t). These exogenous events will trigger the first generation of endogenous
events whose number will be recorded as N (1) (t). Next these first generation of endogenous events
will further trigger a second generation of endogenous events N (2) (t), and so on. Then the total
number of events in the network is the sum of the numbers of events from all generations
N (t) = N (0) (t) + N (1) (t) + N (2) (t) + . . .
(k?1)
Ht
.
(6)
Furthermore, denote all events in generation k ? 1 as
Then, independently for each event
(k?1)
(ui , ti ) ? Ht
in generation k ? 1, it triggers a Poisson process in its neighbor u independently
with intensity auui g(t?ti ). Due to the superposition theorem of independent Poisson processes [22],
3
(k)
the intensity, ?u (t), of events at node u and generation k is simply the sum of conditional intensities
(
(k)
of the Poisson processes triggered by all its neighbors, i.e., ?u (t) = (ui ,ti )?H(k?1) auui g(t ?
t
't
(
(k?1)
ti ) =
(s). Concatenate the intensity for all u ? [m], and use the
u! ?[m] 0 g(t ? s) dNu!
time-varying matrix G(t) (4), we have
& t
?(k) (t) =
G(t ? s) dN (k?1) (s),
(7)
0
(k)
(k)
(?1 (t), . . . , ?m (t))"
where ? (t) =
is the intensity for counting process N (k) (t) at k-th generation. Again, due to the superposition of independent Poisson processes, we can decompose the
intensity of N (t) into a sum of conditional intensities from different generation
(k)
?(t) = ?(0) (t) + ?(1) (t) + ?(2) (t) + . . .
(8)
Next, based on the above decomposition, we will develop a closed form relation between the expected intensity ?(t) = EHt? [?(t)] and the intensity, ?(0) (t), of the exogenous events. This relation will form the basis of our activity shaping framework.
3
Linking Exogenous Event Intensity to Overall Network Activity
Our strategy is to first link the expected intensity ?(k) (t) := EHt? [?(k) (t)] of events at the k-th
generation with ?(0) (t), and then derive a close form for the infinite series sum
?(t) = ?(0) (t) + ?(1) (t) + ?(2) (t) + . . .
(9)
Define a series of auto-convolution matrices, one for each generation, with G (t) = I and
& t
G(!k) (t) =
G(t ? s) G(!k?1) (s) ds = G(t) # G(!k?1) (t)
(10)
(!0)
0
Then the expected intensity of events at the k-th generation is related to exogenous intensity ?(0) by
Lemma 1 ?(k) (t) = G(!k) (t) ?(0) .
Next, by summing together all auto-convolution matrices,
?(t) := I + G(!1) (t) + G(!2) (t) + . . .
we obtain a linear relation between the expected intensity of the network and the intensity of the
exogenous events, i.e., ?(t) = ?(t)?(0) . The entries in the matrix ?(t) roughly encode the ?influence? between pairs of users. More precisely, the entry ?uv (t) is the expected intensity of events
at node u due to a unit level of exogenous intensity at(node v. We can also derive several other
useful quantities from ?(t). For example, ??v (t) := u ?uv (t) can be thought of as the overall
influence user v has on all users. Surprisingly, for exponential kernel, the infinite sum of matrices
results in a closed form using matrix exponentials. First, let )? denote the Laplace transform of a
function, and we have the following intermediate results on the Laplace transform of G(!k) (t).
) (!k) (z) =
Lemma 2 G
'?
0
G(!k) (t) dt =
1
z
?
Ak
(z+?)k
With Lemma 2, we are in a position
* to prove our main theorem below:
+
Theorem 3 ?(t) = ?(t)?(0) = e(A??I)t + ?(A ? ?I)?1 (e(A??I)t ? I) ?(0) .
Theorem 3 provides us a linear relation between exogenous event intensity and the expected overall
intensity at any point in time but not just stationary intensity. The significance of this result is that
it allows us later to design a diverse range of convex programs to determine the intensity of the
exogenous event in order to achieve a target intensity.
In fact, we can recover the previous results in the stationary case as a special case of our general
result. More specifically, a multivariate Hawkes process is stationary if the spectral radius
,& ?
-.
& ?
/
A
? :=
G(t) dt =
g(t) dt
auu!
=
(11)
!
?
u,u ?[m]
0
0
is strictly smaller than 1 [6]. In this case, the expected intensity is ? = (I ? ?)?1 ?(0) independent
of the time. We can obtain this relation from theorem 3 if we let t ? ?.
?1
Corollary 4 ? = (I ? ?) ?(0) = limt?? ?(t) ?(0) .
Refer to Appendix A for all the proofs.
4
4
Convex Activity Shaping Framework
Given the linear relation between exogenous event intensity and expected overall event intensity, we
now propose a convex optimization framework for a variety of activity shaping tasks. In all tasks
discussed below, we will optimize the exogenous event intensity ?(0) such that the expected overall
event intensity ?(t) is maximized with respect to some concave utility U (?) in ?(t), i.e.,
maximize?(t),?(0) U (?(t))
(12)
subject to
?(t) = ?(t)?(0) , c" ?(0) ! C, ?(0) " 0
where c = (c1 , . . . , cm )" " 0 is the cost per unit event for each user and C is the total budget.
Additional regularization can also be added to ?(0) either to restrict the number of incentivized
users (with $0 norm '?(0) '0 ), or to promote a sparse solution (with $1 norm '?(0) '1 , or to obtain a
smooth solution (with $2 regularization '?(0) '2 ). We next discuss several instances of the general
framework which achieve different goals (their constraints remain the same and hence omitted).
Capped Activity Maximization. In real networks, there is an upper bound (or a cap) on the activity
each user can generate due to limited attention of a user. For example, a Twitter user typically posts
a limited number of shortened urls or retweets a limited number of tweets [23]. Suppose we know
the upper bound, ?u , on a user?s activity, i.e., how much activity each user is willing to generate.
Then we can perform the following capped activity maximization task
(
maximize?(t),?(0)
(13)
u?[m] min {?u (t), ?u }
Minimax Activity Shaping. Suppose our goal is instead maintaining the activity of each user in the
network above a certain minimum level, or, alternatively make the user with the minimum activity
as active as possible. Then, we can perform the following minimax activity shaping task
maximize?(t),?(0) minu ?u (t)
(14)
Least-Squares Activity Shaping. Sometimes we want to achieve a pre-specified target activity
levels, v, for users. For example, we may like to divide users into groups and desire a different level
of activity in each group. Inspired by these examples, we can perform the following least-squares
activity shaping task
maximize?(t),?(0) ?'B?(t) ? v'22
(15)
where B encodes potentially additional constraints (e.g., group partitions). Besides Euclidean distance, the family of Bregman divergences can be used to measure the difference between B?(t)
and v here. That is, given a function f (?) : Rm (? R convex in its argument, we can use
D(B?(t)'v) := f (B?(t)) ? f (v) ? )?f (v), B?(t) ? v+ as our objective function.
Activity Homogenization. Many other concave utility functions can be used. For example, we may
want to steer users activities to a more homogeneous profile. If we measure homogeneity of activity
with Shannon entropy, then we can perform the following activity homogenization task
(
maximize?(t),?(0) ? u?[m] ?u (t) ln ?u (t)
(16)
5
Scalable Algorithm
All the activity shaping problems defined above require an efficient evaluation of the instantaneous
average intensity ?(t) at time t, which entails computing matrix exponentials to obtain ?(t). In
small or medium networks, we can rely on well-known numerical methods to compute matrix exponentials [24]. However, in large networks, the explicit computation of ?(t) becomes intractable.
Fortunately, we can exploit the following key property of our convex activity shaping framework:
the instantaneous average intensity only depends on ?(t) through matrix-vector product operations.
In particular, we start by using Theorem 3* to rewrite the +multiplication of ?(t) and a vector v
as ?(t)v = e(A??I)t v + ?(A ? ?I)?1 e(A??I)t v ? v . We then get a tractable solution by
first computing e(A??I)t v *efficiently, subtracting
v from it, and solving a sparse linear system of
+
equations, (A ? ?I)x = e(A??I)t v ? v , efficiently. The steps are illustrated in Algorithm 1.
Next, we elaborate on two very efficient algorithms for computing the product of matrix exponential
with a vector and for solving a sparse linear system of equations.
For the computation of the product of matrix exponential with a vector, we rely on the iterative
algorithm by Al-Mohy et al. [17], which combines a scaling and squaring method with a truncated
Taylor series approximation to the matrix exponential. For solving the sparse linear system of equa5
Algorithm 1: Average Instantaneous Intensity
Algorithm 2: PGD for Activity Shaping
input : A, ?, t, v
output: ?(t)v
v1 = e(A??I)t v
v2 = v2 ? v;
v3 = (A ? ?I)?1 v2
return v1 + ?v3 ;
Initialize ?(0) ;
repeat
1- Project ?(0) into ?(0) " 0, c! ?(0) ! C;
2- Evaluate the gradient g(?(0) ) at ?(0) ;
3- Update ?(0) using the gradient g(?(0) );
until convergence;
tion, we use the well-known GMRES method [25], which is an Arnoldi process for constructing
an l2 -orthogonal basis of Krylov subspaces. The method solves the linear system by iteratively
minimizing the norm of the residual vector over a Krylov subspace.
Perhaps surprisingly, we will now show that it is possible to compute the gradient of the objective functions of all our activity shaping problems using the algorithm developed above for computing the average instantaneous intensity. We only need to define the vector v appropriately
for each problem, as follows: (i) Activity maximization: g(?(0) ) = ?(t)" v, where v is defined such that vj = 1 if ?j > ?j , and vj = 0, otherwise. (ii) Minimax activity shaping:
g(?(0) ) = ?(t)" e, where e is defined such that ej = 1 *if ?j = ?min , and
+ ej = 0, otherwise. (iii)
Least-squares activity shaping: g(?(0) ) = 2?(t)" B " B?(t)?(0) ? v . (iv) Activity homogenization: g(?(0) ) = ?(t)" ln (?(t)?(0) ) + ?(t)" 1, where ln(?) on a vector is the element-wise
natural logarithm. Since the activity maximization and the minimax activity shaping tasks require
only one evaluation of ?(t) times a vector, Algorithm 1 can be used directly. However, computing
the gradient for least-squares activity shaping and activity homogenization is slightly more involved
and it requires to be careful with the order in which we perform the operations (Refer to Appendix B
for details). Equipped with an efficient way to compute of gradients, we solve the corresponding
convex optimization problem for each activity shaping problem by applying projected gradient descent (PGD) [26] with the appropriate gradient1 . Algorithm 2 summarizes the key steps.
6
Experimental Evaluation
We evaluate our framework using both simulated and real world held-out data, and show that our
approach significantly outperforms several baselines. The appendix contains additional experiments.
Dataset description and network inference. We use data gathered from Twitter as reported in [27],
which comprises of all public tweets posted by 60,000 users during a 8-month period, from January
2009 to September 2009. For every user, we record the times she uses any of six popular url shortening services (refer to Appendix C for details). We evaluate the performance of our framework on
a subset of 2,241 active users, linked by 4,901 edges, which we call 2K dataset, and we evaluate its
scalability on the overall 60,000 users, linked by ? 200,000 edges, which we call 60K dataset. The
2K dataset accounts for 691,020 url shortened service uses while the 60K dataset accounts for ?7.5
million uses. Finally, we treat each service as independent cascades of events.
In the experiments, we estimated the nonnegative influence matrix A and the exogenous intensity
?(0) using maximum log-likelihood, as in previous work [8, 9, 12]. We used a temporal resolution
of one minute and selected the bandwidth ? = 0.1 by cross validation. Loosely speaking, ? = 0.1
corresponds to loosing 70% of the initial influence after 10 minutes, which may be explained by the
rapid rate at which each user? news feed gets updated.
Evaluation schemes. We focus on three tasks: capped activity maximization, minimax activity
shaping, and least square activity shaping. We set the total budget to C = 0.5, which corresponds
to supporting a total extra activity equal to 0.5 actions per unit time, and assume all users entail the
same cost. In the capped activity maximization, we set the upper limit of each user?s intensity, ?,
by adding a nonnegative random vector to their inferred initial intensity. In the least-squares activity
shaping, we set B = I and aim to create three user groups: less-active, moderate, and super-active.
We use three different evaluation schemes, with an increasing resemblance to a real world scenario:
Theoretical objective: We compute the expected overall (theoretical) intensity by applying Theorem 3 on the optimal exogenous event intensities to each of the three activity shaping tasks, as well
as the learned A and ?. We then compute and report the value of the objective functions.
1
For nondifferential objectives, subgradient algorithms can be used instead.
6
K
G
0
?4
4
3
0.4
0.2
0
2
0 1 2 3 4 5 6 7 8 9
logarithm of time
?4
1.2
0 1 2 3 4 5 6 7 8 9
logarithm of time
0.4
0.2
0
D
1.4
0.6
GR
1.6
0.8
LS
LSGRD
OP
1.2
0 1 2 3 4 5 6 7 8 9
logarithm of time
PROP
PR
1.4
LSASH
H
1.6
1.8 x 10
AS
LSGRD
LS
PROP
rank correlation
LSASH
Euclidean distance
?4
D
5
*
0.6
GR
GRD
LP
LP
I
2
MINMU
MU
4
UNI
6
MI
N
MMASH
UN
GRD
AS
H
LP
MM
MINMU
rank correlation
x 10
UNI
minimum activity
minimum activity
0.5
0 1 2 3 4 5 6 7 8 9
logarithm of time
0
0 1 2 3 4 5 6 7 8 9
logarithm of time
Euclidean distance
*
1
PR
0.6
?4
1.8 x 10
PRK
0.65
x 10
MMASH
DEG
0.7
0 1 2 3 4 5 6 7 8 9
logarithm of time
6
WEI
DE
0.6
XMU
U
0.65
CAM
WE
I
0.7
0.75
M
PRK
XM
DEG
CA
WEI
rank correlation
XMU
sum of users? activity
sum of users? activity
CAM
0.75
(a) Theoretical objective
(b) Simulated objective
(c) Held-out data
Figure 2: Row 1: Capped activity maximization. Row 2: Minimax activity shaping. Row 3: Leastsquares activity shaping. * means statistical significant at level of 0.01 with paired t-test between
our method and the second best
Simulated objective: We simulate 50 cascades with Ogata?s thinning algorithm [28], using the optimal exogenous event intensities to each of the three activity shaping tasks, and the learned A and ?.
We then estimate empirically the overall event intensity based on the simulated cascades, by computing a running average over non-overlapping time windows, and report the value of the objective
functions based on this estimated overall intensity. Appendix D provides a comparison between the
simulated and the theoretical objective.
Held-out data: The most interesting evaluation scheme would entail carrying out real interventions
in a social platform. However, since this is very challenging to do, instead, in this evaluation scheme,
we use held-out data to simulate such process, proceeding as follows. We first partition the 8-month
data into 50 five-day long contiguous intervals. Then, we use one interval for training and the
remaining 49 intervals for testing. Suppose interval 1 is used for training, the procedure is as follows:
(0)
1. We estimate A1 , ?1 and ?1 using the events from interval 1. Then, we fix A1 and ?1 ,
(0)
and estimate ?i for all other intervals, i = 2, . . . , 49.
(0)
2. Given A1 and ?1 , we find the optimal exogenous event intensities, ?opt , for each of the
three activity shaping task, by solving the associated convex program. We then sort the
(0)
(0)
estimated ?i (i = 2, . . . , 49) according to their similarity to ?opt , using the Euclidean
(0)
(0)
distance '?opt ? ?i '2 .
3. We estimate the overall event intensity for each of the 49 intervals (i = 2, . . . , 49), as in the
?simulated objective? evaluation scheme, and sort these intervals according to the value of
their corresponding objective function.
4. Last, we compute and report the rank correlation score between the two orderings obtained
in step 2 and 3.2 The larger the rank correlation, the better the method.
We repeat this procedure 50 times, choosing each different interval for training once, and compute
and report the average rank correlations. More details can be found in the appendix.
2
rank correlation = number of pairs with consistent ordering / total number of pairs.
7
Capped activity maximization (CAM). We compare to a number of alternatives. XMU: heuristic
based on ?(t) without optimization; DEG and WEI: heuristics based on the degree of the user;
PRANK: heuristic based on page rank (refer to Appendix C for further details). The first row of
Figure 2 summarizes the results for the three different evaluation schemes. We find that our method
(CAM) consistently outperforms the alternatives. For the theoretical objective, CAM is 11 % better
than the second best, DEG. The difference in overall users? intensity from DEG is about 0.8 which,
roughly speaking, leads to at least an increase of about 0.8 ? 60 ? 24 ? 30 = 34, 560 in the overall
number of events in a month. In terms of simulated objective and held-out data, the results are
similar and provide empirical evidence that, compared to other heuristics, degree is an appropriate
surrogate for influence, while, based on the poor performance of XMU, it seems that high activity
does not necessarily entail being influential. To elaborate on the interpretability of the real-world
experiment on held-out data, consider for example the difference in rank correlation between CAM
and DEG, which is almost 0.1. Then, roughly speaking, this means that incentivizing users based
on our approach accommodates with the ordering of real activity patterns in 0.1 ? 50?49
= 122.5
2
more pairs of realizations.
Minimax activity shaping (MMASH). We compare to a number of alternatives. UNI: heuristic
based on equal allocation; MINMU: heuristic based on ?(t) without optimization; LP: linear programming based heuristic; GRD: a greedy approach to leverage the activity (see Appendix C for
more details). The second row of Figure 2 summarizes the results for the three different evaluation
schemes. We find that our method (MMASH) consistently outperforms the alternatives. For the theoretical objective, it is about 2? better than the second best, LP. Importantly, the difference between
MMASH and LP is not trifling and the least active user carries out 2?10?4 ?60?24?30 = 4.3 more
actions in average over a month. As one may have expected, GRD and LP are the best among the
heuristics. The poor performance of MINMU, which is directly related to the objective of MMASH,
may be because it assigns the budget to a low active user, regardless of their influence. However,
our method, by cleverly distributing the budget to the users whom actions trigger many other users?
actions (like those ones with low activity), it benefits from the budget most. In terms of simulated
objective and held-out data, the algorithms? performance become more similar.
Least-squares activity shaping (LSASH). We compare to two alternatives. PROP: Assigning the
budget proportionally to the desired activity; LSGRD: greedily allocating budget according the difference between current and desired activity (refer to Appendix C for more details). The third row of
Figure 2 summarizes the results for the three different evaluation schemes. We find that our method
(LSASH) consistently outperforms the alternatives. Perhaps surprisingly, PROP, despite its simplicity, seems to perform slightly better than LSGRD. This is may be due to the way it allocates the
budget to users, e.g., it does not aim to strictly fulfill users? target activity but benefit more users by
assigning budget proportionally. Refer to Appendix E for additional experiments.
Sparsity and Activity Shaping. In some applications there is a limitation on the number of users we
can incentivize. In our proposed framework, we can handle this requirement by including a sparsity
constraint on the optimization problem. In order to maintain the convexity of the optimization
problem, we consider a l1 regularization term, where a regularization parameter ? provides the
trade-off between sparsity and the activity shaping goal. Refer to Appendix F for more details and
experimental results for different values of ?.
Scalability. The most computationally demanding part of the proposed algorithm is the evaluation of
matrix exponentials, which we scale up by utilizing techniques from matrix algebra, such as GMRES
and Al-Mohy methods. As a result, we are able to run our methods in a reasonable amount of time
on the 60K dataset, specifically, in comparison with a naive implementation of matrix exponential
evaluations. Refer to Appendix G for detailed experimental results on scalability.
Appendix H discusses the limitations of our framework and future work.
Acknowledgement. This project was supported in part by NSF IIS1116886, NSF/NIH BIGDATA
1R01GM108341, NSF CAREER IIS1350983 and Raytheon Faculty Fellowship to Le Song. Isabel Valera acknowledge the support of Plan Regional-Programas I+D of Comunidad de Madrid
(AGES-CM S2010/BMD-2422), Ministerio de Ciencia e Innovaci?on of Spain (project DEIPRO
TEC2009-14504-C02-00 and program Consolider-Ingenio 2010 CSD2008-00010 COMONSENS).
8
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9
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4,822 | 5,366 | Learning Time-Varying Coverage Functions
Nan Du? , Yingyu Liang? , Maria-Florina Balcan , Le Song?
?
College of Computing, Georgia Institute of Technology
?
Department of Computer Science, Princeton University
School of Computer Science, Carnegie Mellon University
[email protected],[email protected]
[email protected],[email protected]
Abstract
Coverage functions are an important class of discrete functions that capture the
law of diminishing returns arising naturally from applications in social network
analysis, machine learning, and algorithmic game theory. In this paper, we propose a new problem of learning time-varying coverage functions, and develop a
novel parametrization of these functions using random features. Based on the connection between time-varying coverage functions and counting processes, we also
propose an efficient parameter learning algorithm based on likelihood maximization, and provide a sample complexity analysis. We applied our algorithm to the
influence function estimation problem in information diffusion in social networks,
and show that with few assumptions about the diffusion processes, our algorithm
is able to estimate influence significantly more accurately than existing approaches
on both synthetic and real world data.
1
Introduction
Coverage functions are a special class of the more general submodular functions which play important role in combinatorial optimization with many interesting applications in social network analysis [1], machine learning [2], economics and algorithmic game theory [3], etc. A particularly
important example of coverage functions in practice is the influence function of users in information
diffusion modeling [1] ? news spreads across social networks by word-of-mouth and a set of influential sources can collectively trigger a large number of follow-ups. Another example of coverage
functions is the valuation functions of customers in economics and game theory [3] ? customers are
thought to have certain requirements and the items being bundled and offered fulfill certain subsets
of these demands.
Theoretically, it is usually assumed that users? influence or customers? valuation are known in advance as an oracle. In practice, however, these functions must be learned. For example, given past
traces of information spreading in social networks, a social platform host would like to estimate
how many follow-ups a set of users can trigger. Or, given past data of customer reactions to different bundles, a retailer would like to estimate how likely customer would respond to new packages of
goods. Learning such combinatorial functions has attracted many recent research efforts from both
theoretical and practical sides (e.g., [4, 5, 6, 7, 8]), many of which show that coverage functions can
be learned from just polynomial number of samples.
However, the prior work has widely ignored an important dynamic aspect of the coverage functions.
For instance, information spreading is a dynamic process in social networks, and the number of
follow-ups of a fixed set of sources can increase as observation time increases. A bundle of items
or features offered to customers may trigger a sequence of customer actions over time. These real
world problems inspire and motivate us to consider a novel time-varying coverage function, f (S, t),
which is a coverage function of the set S when we fix a time t, and a continuous monotonic function
of time t when we fix a set S. While learning time-varying combinatorial structures has been ex1
plored in graphical model setting (e.g., [9, 10]), as far as we are aware of, learning of time-varying
coverage function has not been addressed in the literature. Furthermore, we are interested in estimating the entire function of t, rather than just treating the time t as a discrete index and learning
the function value at a small number of discrete points. From this perspective, our formulation is the
generalization of the most recent work [8] with even less assumptions about the data used to learn
the model.
Generally, we assume that the historical data are provided in pairs of a set and a collection of timestamps when caused events by the set occur. Hence, such a collection of temporal events associated
with a particular set Si can be modeled principally by a counting process Ni (t), t > 0 which is a
stochastic process with values that are positive, integer, and increasing along time [11]. For instance,
in the information diffusion setting of online social networks, given a set of earlier adopters of some
new product, Ni (t) models the time sequence of all triggered events of the followers, where each
jump in the process records the timing tij of an action. In the economics and game theory setting, the
counting process Ni (t) records the number of actions a customer has taken over time given a particular bundled offer. This essentially raises an interesting question of how to estimate the time-varying
coverage function from the angle of counting processes. We thus propose a novel formulation which
builds a connection between the two by modeling the cumulative intensity function of a counting
process as a time-varying coverage function. The key idea is to parametrize the intensity function
as a weighted combination of random kernel functions. We then develop an efficient learning algorithm TC OVERAGE L EARNER to estimate the parameters of the function using maximum likelihood
approach. We show that our algorithm can provably learn the time-varying coverage function using
only polynomial number of samples. Finally, we validate TC OVERAGE L EARNER on both influence
estimation and maximization problems by using cascade data from information diffusion. We show
that our method performs significantly better than alternatives with little prior knowledge about the
dynamics of the actual underlying diffusion processes.
2
Time-Varying Coverage Function
We will first give a formal definition of the time-varying coverage function, and then explain its
additional properties in details.
Definition. Let U be a (potentially uncountable) domain. We endow U with some ?-algebra A and
denote a probability distribution on U by P. A coverage function is a combinatorial function over a
finite set V of items, defined as
[
f (S) := Z ? P
Us , for all S ? 2V ,
(1)
s?S
where Us ? U is the subset of domain U covered by item s ? V, and Z is the additional normalization constant. For time-varying coverage functions, we let the size of the subset Us to grow
monotonically over time, that is
Us (t) ? Us (? ),
for all t 6 ? and s ? V,
which results in a combinatorial temporal function
[
f (S, t) = Z ? P
Us (t) ,
s?S
for all S ? 2V .
(2)
(3)
In this paper, we assume that f (S, t) is smooth and continuous, and its first order derivative with
respect to time, f 0 (S, t), is also smooth and continuous.
Representation. We now show that a time-varying coverage function, f (S, t), can be represented
as an expectation over random functions based on multidimensional step basis functions. Since
Us (t) is varying over time, we can associate each u ? U with a |V|-dimensional vector ?u of change
points. In particular, the s-th coordinate of ?u records the time that source node s covers u. Let ?
to be a random variable obtained by sampling u according to P and setting ? = ?u . Note that given
all ?u we can compute f (S, t); now we claim that the distribution of ? is sufficient.
We first introduce some notations. Based on ?u we define a |V|-dimensional step function ru (t) :
|V|
R+ 7? {0, 1} , where the s-th dimension of ru (t) is 1 if u is covered by the set Us (t) at time t, and
0 otherwise. To emphasize the dependence of the function ru (t) on ?u , we will also write ru (t) as
ru (t|?u ). We denote the indicator vector of a set S by ?S ? S
{0, 1}|V| where the s-th dimension of
?S is 1 if s ? S, and 0 otherwise. Then u ? U is covered by s?S Us (t) at time t if ?>
S ru (t) > 1.
2
Lemma 1. There exists a distribution Q(? ) over the vector of change points ? , such that the timevarying coverage function can be represented as
f (S, t) = Z ? E? ?Q(? ) ?(?>
(4)
S r(t|? ))
where ?(x) := min {x, 1}, and r(t|? ) is a multidimensional step function parameterized by ? .
S
Proof. Let US := s?S Us (t). By definition (3), we have the following integral representation
Z
Z
>
f (S, t) = Z ?
I {u ? US } dP(u) = Z ?
?(?>
S ru (t)) dP(u) = Z ? Eu?P(u) ?(?S ru (t)) .
U
U
We can define the setR of u having the same ? as U? := {u ? U | ?u = ? } and define a distribution
over ? as dQ(? ) := U? dP(u). Then the integral representation of f (S, t) can be rewritten as
>
Z ? Eu?P(u) ?(?>
S ru (t)) = Z ? E? ?Q(? ) ?(?S r(t|? )) ,
which proves the lemma.
3
Model for Observations
In general, we assume that the input data are provided in the form of pairs, (Si , Ni (t)), where Si is
a set, and Ni (t) is a counting process in which each jump of Ni (t) records the timing of an event.
We first give a brief overview of a counting process [11] and then motivate our model in details.
Counting Process. Formally, a counting process {N (t), t > 0} is any nonnegative, integer-valued
stochastic process such that N (t0 ) 6 N (t) whenever t0 6 t and N (0) = 0. The most common
use of a counting process is to count the number of occurrences of temporal events happening along
time, so the index set is usually taken to be the nonnegative real numbers R+ . A counting process
is a submartingale: E[N (t) | Ht0 ] > N (t0 ) for all t > t0 where Ht0 denotes the history up to time t0 .
By Doob-Meyer theorem [11], N (t) has the unique decomposition:
N (t) = ?(t) + M (t)
(5)
where ?(t) is a nondecreasing predictable process called the compensator (or cumulative intensity),
and M (t) is a mean zero martingale. Since E[dM (t) | Ht? ] = 0, where dM (t) is the increment of
M (t) over a small time interval [t, t + dt), and Ht? is the history until just before time t,
E[dN (t) | Ht? ] = d?(t) := a(t) dt
(6)
where a(t) is called the intensity of a counting process.
Model formulation. We assume that the cumulative intensity of the counting process is modeled
by a time-varying coverage function, i.e., the observation pair (Si , Ni (t)) is generated by
Ni (t) = f (Si , t) + Mi (t)
(7)
in the time window [0, T ] for some T > 0, and df (S, t) = a(S, t)dt. In other words, the timevarying coverage function controls the propensity of occurring events over time. Specifically, for a
fixed set Si , as time t increases, the cumulative number of events observed grows accordingly for
that f (Si , t) is a continuous monotonic function over time; for a given time t, as the set Si changes
to another set Sj , the amount of coverage over domain U may change and hence can result in a
different cumulative intensity. This abstract model can be mapped to real world applications. In
the information diffusion context, for a fixed set of sources Si , as time t increases, the number of
influenced nodes in the social network tends to increase; for a given time t, if we change the sources
to Sj , the number of influenced nodes may be different depending on how influential the sources
are. In the economics and game theory context, for a fixed bundle of offers Si , as time t increases, it
is more likely that the merchant will observe the customers? actions in response to the offers; even
at the same time t, different bundles of offers, Si and Sj , may have very different ability to drive the
customers? actions.
Compared to a regression model yi = g(Si ) + i with i.i.d. input data (Si , yi ), our model outputs
a special random function over time, that is, a counting process Ni (t) with the noise being a zero
mean martingale Mi (t). In contrast to functional regression models, our model exploits much more
interesting structures of the problem. For instance, the random function representation in the last
section can be used to parametrize the model. Such special structure of the counting process allows
us to estimate the parameter of our model using maximum likelihood approach efficiently, and the
martingale noise enables us to use exponential concentration inequality in analyzing our algorithm.
3
4
Parametrization
Based on the following two mild assumptions, we will show how to parametrize the intensity function as a weighted combination of random kernel functions, learn the parameters by maximum
likelihood estimation, and eventually derive a sample complexity.
(A1) a(S, t) is smooth and boundedR on [0, T ]: 0 < amin 6 a 6 amax < ?, and a
? := d2 a/dt2
is absolutely continuous with a
?(t)dt < ?.
(A2) There is a known distribution Q0 (? ) and a constant C with Q0 (? )/C 6 Q(? ) 6 CQ0 (? ).
Kernel Smoothing To facilitate our finite dimensional parameterization, we first convolve the
intensity function with K(t) = k(t/?)/? where ? is the bandwidth parameter and k is a kernel
?
2
function (such as the Gaussian RBF kernel k(t) = e?t /2 / 2?) with
Z
Z
Z
2
0 6 k(t) 6 ?max ,
k(t) dt = 1,
t k(t) dt = 0, and ?k := t2 k(t) dt < ?. (8)
The convolution results in a smoothed intensity aK (S, t) = K(t) ? (df (S, t)/dt) = d(K(t) ?
?(S, t))/dt. By the property of convolution and exchanging derivative with integral, we have that
aK (S, t) = d(Z ? E? ?Q(? ) [K(t) ? ?(?>
by definition of f (?)
S r(t|? )])/dt
>
= Z ? E? ?Q(? ) d(K(t) ? ?(?S r(t|? ))/dt exchange derivative and integral
= Z ? E? ?Q(? ) [K(t) ? ?(t ? t(S, r)]
by property of convolution and function ?(?)
= Z ? E? ?Q(? ) [K(t ? t(S, ? ))]
by definition of ?(?)
?(?>
S r(t|? ))
jumps from 0 to 1. If we choose small enough
where t(S, ? ) is the time when function
kernel bandwidth, aK only incurs a small bias from a. But the smoothed intensity still results in
infinite number of parameters, due to the unknown distribution Q(? ). To address this problem, we
design the following random approximation with finite number of parameters.
Random Function Approximation The key idea is to sample a collection of W random change
points ? from a known distribution Q0 (? ) which can be different from Q(? ). If Q0 (? ) is not very
far way from Q(? ), the random approximation will be close to aK , and thus close to a. More
specifically, we will denote the space of weighted combination of W random kernel function by
(
)
W
X
Z
i.i.d.
K
wi K(t ? t(S, ?i )) : w > 0, 6 kwk1 6 ZC , {?i } ? Q0 (? ). (9)
A = aw (S, t) =
C
i=1
? 2 /(?)2 ), then with probability > 1 ? ?, there exists an e
Lemma 2. If W = O(Z
a ? A such that
RT
ES Et (a(S, t) ? e
a(S, t))2 := ES?P(S) 0 (a(S, t) ? e
a(S, t))2 dt/T = O(2 + ? 4 ).
?
The lemma then suggests to set the kernel bandwidth ? = O( ) to get O(2 ) approximation error.
5
Learning Algorithm
We develop a learning algorithm, referred to as TC OVERAGE L EARNER, to estimate the parameters
of aK
w (S, t) by maximizing the joint likelihood of all observed events based on convex optimization
techniques as follows.
Maximum Likelihood Estimation Instead of directly estimating the time-varying coverage function, which is the cumulative intensity function of the counting process, we turn to estimate
the intensity function a(S, t) = ??(S, t)/?t. Given m i.i.d. counting processes, Dm :=
{(S1 , N1 (t)), . . . , (Sm , Nm (t))} up to observation time T , the log-likelihood of the dataset is [11]
(Z
)
Z T
m
T
X
m
(10)
`(D |a) =
{log a(Si , t)} dNi (t) ?
a(Si , t) dt .
i=1
0
0
Maximizing the log-likelihood with respect to the intensity function a(S, t) then gives us the estimation b
a(S, t). The W -term random kernel function approximation reduces a function optimization
problem to a finite dimensional optimization problem, while incurring only small bias in the estimated function.
4
Algorithm 1 TC OVERAGE L EARNER
INPUT : {(Si , Ni (t))} , i = 1, . . . , m;
Sample W random features ?1 , . . . , ?W from Q0 (? );
Compute {t(Si , ?w )} , {gi } , {k(tij )} , i ? {1, . . . , m} , w = 1, . . . , W, tij < T ;
Initialize w0 ? ? = {w > 0, kwk1 6 1};
Apply projected quasi-newton algorithm [12] to solve 11;
PW
OUTPUT : aK
w (S, t) =
i=1 wi K(t ? t(S, ?i ))
Convex Optimization. By plugging the parametrization aK
w (S, t) (9) into the log-likelihood (10),
we formulate the optimization problem as :
?
?
m ?
?
X
X
min
w> gi ?
log w> k(tij )
subject to w > 0, kwk1 6 1,
(11)
w
?
?
i=1
tij <T
where we define
Z
T
K (t ? t(Si , ?k )) dt and
gik =
kl (tij ) = K(tij ? t(Si , ?l )),
(12)
0
tij when the j-th event occurs in the i-th counting process. By treating the normalization constant
Z as a free variable which will be tuned by cross validation later, we simply require that kwk1 6 1.
By applying the Gaussian RBF kernel, we can derive a closed form of gik and the gradient O` as
?
?
m ?
X
X
T ? t(Si , ?k )
k(tij ) ?
t(Si , ?k )
1
?
? erfc
, O` =
gi ?
.
erfc ? ?
gik =
?
2
w> k(tij ) ?
2h
2h
i=1
tij <T
(13)
A pleasing feature of this formulation is that it is convex in the argument w, allowing us to apply
various convex optimization techniques to solve the problem efficiently. Specifically, we first draw
W random features ?1 , . . . , ?W from Q0 (? ). Then, we precompute the jumping time t(Si , ?w )
m
W
for every source set {Si }i=1 on each random feature {?w }w=1 . Because in general |Si | << n,
this computation costs O(mW ). Based on the achieved m-by-W jumping-time matrix, we preprom
cess the feature vectors {gi }i=1 and k(tij ), i ? {1, . . . , m} , tij < T , which costs O(mW ) and
O(mLW ) where L is the maximum number of events caused by a particular source set before time
T . Finally, we apply the projected quasi-newton algorithm [12] to find the weight w that minimizes
the negative log-likelihood of observing the given event data. Because the evaluation of the objective
function and the gradient, which costs O(mLW ), is much more expensive than the projection onto
the convex constraint set, and L << n, the worst case computation complexity is thus O(mnW ).
Algorithm 1 summarizes the above steps in the end.
Sample Strategy. One important constitution of our parametrization is to sample W random change
points ? from a known distribution Q0 (? ). Because given a set Si , we can only observe the jumping
time of the events in each counting process without knowing the identity of the covered items (which
is a key difference from [8]), the best thing we can do is to sample from these historical data.
Specifically, let the number of counting processes that a single item s ? V is involved to induce
be Ns , and the collection of all the jumping timestamps before time T be Js . Then, for the s-th
entry of ? , with probability |Js |/nNs , we uniformly draw a sample from Js ; and with probability
1 ? |Js |/nNs , we assign a time much greater than T to indicate that the item will never be covered
until infinity. Given the very limited information, although this Q0 (? ) might be quite different from
Q(? ), by drawing sufficiently large number of samples and adjusting the weights, we expect it still
can lead to good results, as illustrated in our experiments later.
6
Sample Complexity
Suppose we use W random features and m training examples to compute an ` -MLE solution b
a, i.e.,
`(Dm |b
a) > max
`(Dm |a0 ) ? ` .
0
a ?A
The goal is to analyze how well the function fb induced by b
a approximates the true function f . This
sections describes the intuition and the complete proof is provided in the appendix.
5
A natural choice for connecting the error between f and fb with the log-likelihood cost used in MLE
is the Hellinger distance [22]. So it suffices to prove an upper bound on the Hellinger distance
h(a, b
a) between b
a and the true intensity a, for which we need to show a high probability bound on
b 2 (a, a0 ) between the two. Here, h and H
b are defined as
the (total) empirical Hellinger distance H
h
i
p
p
2
1
h2 (a, a0 ) := ES Et
a(S, t) ? a0 (S, t) ,
2
m Z T hp
i2
X
p
b 2 (a, a0 ) := 1
a(Si , t) ? a0 (Si , t) dt.
H
2 i=1 0
The key for the analysis is to show that the empirical Hellinger distance can be bounded by a martingale plus some other additive error terms, which we then bound respectively. This martingale is
defined based on our hypotheses and the martingales Mi associated with the counting process Ni :
!
Z t
m Z t
X
X
M (t|g) :=
g(t)d
Mi (t) =
g(t)dMi (t)
0
n
where g ? G = ga0 =
1
2
log
a+a
2a
0
i
i=1
0
o
: a0 ? A . More precisely, we have the following lemma.
Lemma 3. Suppose b
a is an ` -MLE. Then
m 0
b 2 (b
H
a, a) 6 16M (T ; gba ) + 4 `(Dm |a) ? max
`(D
|a
)
+ 4` .
0
a ?A
The right hand side has three terms: the martingale (estimation error), the likelihood gap between
the truth and the best one in our hypothesis class (approximation error), and the optimization error.
We then focus on bounding the martingale and the likelihood gap.
To bound the martingale, we first introduce a notion called (d, d0 )-covering dimension measuring
the complexity of the hypothesis class, generalizing that in [25]. Based on this notion, we prove
a uniform convergence inequality, combining the ideas in classic works on MLE [25] and counting process [13]. Compared to the classic uniform inequality, our result is more general, and the
complexity notion has more clear geometric interpretation and are thus easier to verify. For the likelihood gap, recall that by Lemma 2, there exists an good approximation a
? ? A. The likelihood gap
is then bounded by that between a and a
?, which is small since a and a
? are close.
Combining the two leads to a bound on the Hellinger distance based on bounded dimension of the
hypothesis class. We then show that the dimension of our specific hypothesis class is at most the
b 2 (b
number of random features W , and convert H
a, a) to the desired `2 error bound on f and fb.
5/4
ZT 5/2
ZT
2
? ZT [W + ` ] . Then
?
Theorem 4. Suppose W = O Z
+ amin
and m = O
W
with probability > 1 ? ? over the random sample of {?i }i=1 , we have that for any 0 6 t 6 T ,
h
i2
ES fb(S, t) ? f (S, t) 6 .
The theorem shows that the number of random functions needed to achieve error is roughly
O(?5/2 ), and the sample size is O(?7/2 ). They also depend on amin , which means with more
random functions and data, we can deal with intensities with more extreme values. Finally, they
increase with the time T , i.e., it is more difficult to learn the function values at later time points.
7
Experiments
We evaluate TC OVERAGE L EARNER on both synthetic and real world information diffusion data.
We show that our method can be more robust to model misspecification than other state-of-the-art
alternatives by learning a temporal coverage function all at once.
7.1 Competitors
Because our input data only include pairs of a source set and the temporal information of its trigm
gered events {(Si , Ni (t))}i=1 with unknown identity, we first choose the general kernel ridge regression model as the major baseline, which directly estimates the influence value of a source set
6
15
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
5
5
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
4
10
MAE
10
MAE
20
MAE
10
30
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
MAE
15
5
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
3
2
1
0
1
2
3
4
5 6
Time
7
8
9
10
0
1
2
3
4
5 6
Time
7
8
9
10
0
1
2
3
4
5 6
Time
7
8
9
10
0
1
2
3
4
5 6
Time
7
8
9
10
(a) Weibull (CIC)
(b) Exponential (CIC)
(c) DIC
(d) LT
Figure 1: MAE of the estimated influence on test data along time with the true diffusion model being
continuous-time independent cascade with pairwise Weibull (a) and Exponential (b) transmission
functions, (c) discrete-time independent cascade model and (d) linear-threshold cascade model.
?S by f (?S ) = k(?S )(K + ?I)?1 y where k(?S ) = K(?Si , ?S ), and K is the kernel matrix. We discretize the time into several steps and fit a separate model to each of them. Between
two consecutive time steps, the predictions are simply interpolated. In addition, to further demonstrate the robustness of TC OVERAGE L EARNER, we compare it to the two-stage methods which
must know the identity of the nodes involved in an information diffusion process to first learn
a specific diffusion model based on which they can then estimate the influence. We give them
such an advantage and study three well-known diffusion models : (I) Continuous-time Independent
Cascade model(CIC)[14, 15]; (II) Discrete-time Independent Cascade model(DIC)[1]; and (III)
Linear-Threshold cascade model(LT)[1].
7.2
Influence Estimation on Synthetic Data
We generate Kronecker synthetic networks ([0.9 0.5;0.5 0.3]) which mimic real world information
diffusion patterns [16]. For CIC, we use both Weibull distribution (Wbl) and Exponential distribution (Exp) for the pairwise transmission function associated with each edge, and randomly set their
parameters to capture the heterogeneous temporal dynamics. Then, we use NETRATE [14] to learn
the model by assuming an exponential pairwise transmission function. For DIC, we choose the pairwise infection probability uniformly from 0 to 1 and fit the model by [17]. For LT, we assign the edge
weight wuv between u and v as 1/dv , where dv is the degree of node v following [1]. Finally, 1,024
source sets are sampled with power-law distributed cardinality (with exponent 2.5), each of which
induces eight independent cascades(or counting processes), and the test data contains another 128
independently sampled source sets with the ground truth influence estimated from 10,000 simulated
cascades up to time T = 10. Figure 1 shows the MAE(Mean Absolute Error) between the estimated
influence value and the true value up to the observation window T = 10. The average influence
is 16.02, 36.93, 9.7 and 8.3. We use 8,192 random features and two-fold cross validation on the
train data to tune the normalization Z, which has the best
? value 1130, 1160, 1020, and 1090, respectively. We choose the RBF kernel bandwidth h = 1/ 2? so that the magnitude of the smoothed
approximate function still equals to 1 (or it can be tuned by cross-validation as well), which matches
the original indicator function. For the kernel ridge regression, the RBF kernel bandwidth and the
regularization ? are all chosen by the same two-fold cross validation. For CIC and DIC, we learn
the respective model up to time T for once.
Figure 1 verifies that even though the underlying diffusion models can be dramatically different,
the prediction performance of TC OVERAGE L EARNER is robust to the model changes and consistently outperforms the nontrivial baseline significantly. In addition, even if CIC and DIC are
provided with extra information, in Figure 1(a), because the ground-truth is continuous-time diffusion model with Weibull functions, they do not have good performance. CIC assumes the right
model but the wrong family of transmission functions. In Figure 1(b), we expect CIC should have
the best performance for that it assumes the correct diffusion model and transmission functions.
Yet, TC OVERAGE L EARNER still has comparable performance with even less information. In Figure 1(c), although DIC has assumed the correct model, it is hard to determine the correct step size to
discretize the time line, and since we only learn the model once up to time T (instead of at each time
point), it is harder to fit the whole process. In Figure1(d), both CIC and DIC have the wrong model,
so we have similar trend as Figure synthetic(a). Moreover, for kernel ridge regression, we have to
first partition the timeline with arbitrary step size, fit the model to each of time, and interpolate the
value between neighboring time legs. Not only will the errors from each stage be accumulated to
the error of the final prediction, but also we cannot rely on this method to predict the influence of a
source set beyond the observation window T .
7
5
80
2
6
4
10
influence
10
100
10
8
time(s)
15
0
3
10
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
20
Average MAE
Average MAE
25
1
10
2
3
4
5
6
Groups of Memes
7
0
0
128 256 512 1024 2048 4096 8192
# Random features
10
60
40
2
1
TCoverageLearner
Kernel Ridge Regression
CIC
DIC
128 256 512 1024 2048 4096 8192
# random features
20
1
2
3
4
5 6
Time
7
8
9
10
(a) Average MAE
(b) Features? Effect
(c) Runtime
(d) Influence maximization
Figure 2: (a) Average MAE from time 1 to 10 on seven groups of real cascade data; (b) Improved
estimation with increasing number of random features; (c) Runtime in log-log scale; (d) Maximized
influence of selected sources on the held-out testing data along time.
Overall, compared to the kernel ridge regression, TC OVERAGE L EARNER only needs to be trained
once given all the event data up to time T in a compact and principle way, and then can be used to infer the influence of any given source set at any particular time much more efficiently and accurately.
In contrast to the two-stage methods, TC OVERAGE L EARNER is able to address the more general
setting with much less assumption and information but still can produce consistently competitive
performance.
7.3
Influence Estimation on Real Data
MemeTracker is a real-world dataset [18] to study information diffusion. The temporal flow of information was traced using quotes which are short textual phrases spreading through the websites.
We have selected seven typical groups of cascades with the representative keywords like ?apple and
jobs?, ?tsunami earthquake?, etc., among the top active 1,000 sites. Each set of cascades is split into
60%-train and 40%-test. Because we often can observe cascades only from single seed node, we
rarely have cascades produced from multiple sources simultaneously. However, because our model
can capture the correlation among multiple sources, we challenge TC OVERAGE L EARNER with sets
of randomly chosen multiple source nodes on the independent hold-out data. Although the generation of sets of multiple source nodes is simulated, the respective influence is calculated from the real
test data as follows : Given a source set S, for each node u ? S, let C(u) denote the set of cascades
generated from u on the testing data. We uniformly sample cascades from C(u). The average length
of all sampled cascades is treated as the true influence of S. We draw 128 source sets and report
the average MAE along time in Figure 2(a). Again, we can observe that TC OVERAGE L EARNER
has consistent and robust estimation performance across all testing groups. Figure 2(b) verifies that
the prediction can be improved as more random features are exploited, because the representational
power of TC OVERAGE L EARNER increases to better approximate the unknown true coverage function. Figure 2(c) indicates that the runtime of TC OVERAGE L EARNER is able to scale linearly with
large number of random features. Finally, Figure 2(d) shows the application of the learned coverage
function to the influence maximization problem along time, which seeks to find a set of source nodes
that maximize the expected number of infected nodes by time T . The classic greedy algorithm[19]
is applied to solve the problem, and the influence is calculated and averaged over the seven held-out
test data. It shows that TC OVERAGE L EARNER is very competitive to the two-stage methods with
much less assumption. Because the greedy algorithm mainly depends on the relative rank of the
selected sources, although the estimated influence value can be different, the selected set of sources
could be similar, so the performance gap is not large.
8
Conclusions
We propose a new problem of learning temporal coverage functions with a novel parametrization
connected with counting processes and develop an efficient algorithm which is guaranteed to learn
such a combinatorial function from only polynomial number of training samples. Empirical study
also verifies our method outperforms existing methods consistently and significantly.
Acknowledgments This work was supported in part by NSF grants CCF-0953192, CCF-1451177,
CCF-1101283, and CCF-1422910, ONR grant N00014-09-1-0751, AFOSR grant FA9550-09-10538, Raytheon Faculty Fellowship, NSF IIS1116886, NSF/NIH BIGDATA 1R01GM108341, NSF
CAREER IIS1350983 and Facebook Graduate Fellowship 2014-2015.
8
References
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9
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4,823 | 5,367 | Online and Stochastic Gradient Methods for
Non-decomposable Loss Functions
Purushottam Kar?
Harikrishna Narasimhan?
Prateek Jain?
Microsoft Research, INDIA
?
Indian Institute of Science, Bangalore, INDIA
{t-purkar,prajain}@microsoft.com, [email protected]
?
Abstract
Modern applications in sensitive domains such as biometrics and medicine frequently require the use of non-decomposable loss functions such as precision@k,
F-measure etc. Compared to point loss functions such as hinge-loss, these offer much more fine grained control over prediction, but at the same time present
novel challenges in terms of algorithm design and analysis. In this work we initiate
a study of online learning techniques for such non-decomposable loss functions
with an aim to enable incremental learning as well as design scalable solvers for
batch problems. To this end, we propose an online learning framework for such
loss functions. Our model enjoys several nice properties, chief amongst them being the existence of efficient online learning algorithms with sublinear regret and
online to batch conversion bounds. Our model is a provable extension of existing
online learning models for point loss functions. We instantiate two popular losses,
Prec@k and pAUC, in our model and prove sublinear regret bounds for both of
them. Our proofs require a novel structural lemma over ranked lists which may
be of independent interest. We then develop scalable stochastic gradient descent
solvers for non-decomposable loss functions. We show that for a large family
of loss functions satisfying a certain uniform convergence property (that includes
Prec@k , pAUC, and F-measure), our methods provably converge to the empirical
risk minimizer. Such uniform convergence results were not known for these losses
and we establish these using novel proof techniques. We then use extensive experimentation on real life and benchmark datasets to establish that our method can be
orders of magnitude faster than a recently proposed cutting plane method.
1
Introduction
Modern learning applications frequently require a level of fine-grained control over prediction performance that is not offered by traditional ?per-point? performance measures such as hinge loss.
Examples include datasets with mild to severe label imbalance such as spam classification wherein
positive instances (spam emails) constitute a tiny fraction of the available data, and learning tasks
such as those in medical diagnosis which make it imperative for learning algorithms to be sensitive to class imbalances. Other popular examples include ranking tasks where precision in the top
ranked results is valued more than overall precision/recall characteristics. The performance measures of choice in these situations are those that evaluate algorithms over the entire dataset in a
holistic manner. Consequently, these measures are frequently non-decomposable over data points.
More specifically, for these measures, the loss on a set of points cannot be expressed as the sum of
losses on individual data points (unlike hinge loss, for example). Popular examples of such measures
include F-measure, Precision@k, (partial) area under the ROC curve etc.
Despite their success in these domains, non-decomposable loss functions are not nearly as well
understood as their decomposable counterparts. The study of point loss functions has led to a deep
1
understanding about their behavior in batch and online settings and tight characterizations of their
generalization abilities. The same cannot be said for most non-decomposable losses. For instance, in
the popular online learning model, it is difficult to even instantiate a non-decomposable loss function
as defining the per-step penalty itself becomes a challenge.
1.1
Our Contributions
Our first main contribution is a framework for online learning with non-decomposable loss functions.
The main hurdle in this task is a proper definition of instantaneous penalties for non-decomposable
losses. Instead of resorting to canonical definitions, we set up our framework in a principled way
that fulfills the objectives of an online model. Our framework has a very desirable characteristic that
allows it to recover existing online learning models when instantiated with point loss functions. Our
framework also admits online-to-batch conversion bounds.
We then propose an efficient Follow-the-Regularized-Leader [1] algorithm within our framework.
We show that for loss functions
that satisfy a generic ?stability? condition, our algorithm is able
to offer vanishing O ?1T regret. Next, we instantiate within our framework, convex surrogates
for two popular performances measures namely, Precision at k (Prec@k ) and partial area under the
ROC curve (pAUC) [2] and show, via a stability analysis, that we do indeed achieve sublinear
regret bounds for these loss functions. Our stability proofs involve a structural lemma on sorted
lists of inner products which proves Lipschitz continuity properties for measures on such lists (see
Lemma 2) and might be useful for analyzing non-decomposable loss functions in general.
A key property of online learning methods is their applicability in designing solvers for offline/batch problems. With this goal in mind, we design a stochastic gradient-based solver for
non-decomposable loss functions. Our methods apply to a wide family of loss functions (including
Prec@k , pAUC and F-measure) that were introduced in [3] and have been widely adopted [4, 5, 6]
in the literature. We design several variants of our method and show that our methods provably converge to the empirical risk minimizer of the learning instance at hand. Our proofs involve uniform
convergence-style results which were not known for the loss functions we study and require novel
techniques, in particular the structural lemma mentioned above.
Finally, we conduct extensive experiments on real life and benchmark datasets with pAUC and
Prec@k as performance measures. We compare our methods to state-of-the-art methods that are
based on cutting plane techniques [7]. The results establish that our methods can be significantly
faster, all the while offering comparable or higher accuracy values. For example, on a KDD 2008
challenge dataset, our method was able to achieve a pAUC value of 64.8% within 30ms whereas it
took the cutting plane method more than 1.2 seconds to achieve a comparable performance.
1.2
Related Work
Non-decomposable loss functions such as Prec@k , (partial) AUC, F-measure etc, owing to their
demonstrated ability to give better performance in situations with label imbalance etc, have generated significant interest within the learning community. From their role in early works as indicators
of performance on imbalanced datasets [8], their importance has risen to a point where they have
become the learning objectives themselves. Due to their complexity, methods that try to indirectly
optimize these measures are very common e.g. [9], [10] and [11] who study the F-measure. However, such methods frequently seek to learn a complex probabilistic model, a task arguably harder
than the one at hand itself. On the other hand are algorithms that perform optimization directly via
structured losses. Starting from the seminal work of [3], this method has received a lot of interest
for measures such as the F-measure [3], average precision [4], pAUC [7] and various ranking losses
[5, 6]. These formulations typically use cutting plane methods to design dual solvers.
We note that the learning and game theory communities are also interested in non-additive notions
of regret and utility. In particular [12] provides a generic framework for online learning with nonadditive notions of regret with a focus on showing regret bounds for mixed strategies in a variety of
problems. However, even polynomial time implementation of their strategies is difficult in general.
Our focus, on the other hand, is on developing efficient online algorithms that can be used to solve
large scale batch problems. Moreover, it is not clear how (if at all) can the loss functions considered
here (such as Prec@k ) be instantiated in their framework.
2
Recently, online learning for AUC maximization has received some attention [13, 14]. Although
AUC is not a point loss function, it still decomposes over pairs of points in a dataset, a fact that [13]
and [14] crucially use. The loss functions in this paper do not exhibit any such decomposability.
2
Problem Formulation
Let x1:t := {x1 , . . . , xt }, xi ? Rd and y1:t := {y1 , . . . , yt }, yi ? {?1, 1} be the observed data
points and true binary labels. We will use yb1:t := {b
y1 , . . . , ybt }, ybi ? R to denote the predictions of a
learning algorithm. We shall, for sake of simplicity, restrict ourselves to linear predictors ybi = w> xi
for parameter vectors w ? Rd . A performance measure P : {?1, 1}t ? Rt ? R+ shall be used to
evaluate the the predictions of the learning algorithm against the true labels. Our focus shall be on
non-decomposable performance measures such as Prec@k , partial AUC etc.
Since these measures are typically non-convex, convex surrogate loss functions are used instead (we
will use the terms loss function and performance measure interchangeably). A popular technique for
constructing such loss functions is the structural SVM formulation [3] given below. For simplicity,
we shall drop mention of the training points and use the notation `P (w) := `P (x1:T , y1:T , w).
`P (w) =
max
? ?{?1,+1}T
y
T
X
(?
yi ? yi )x>
y, y).
i w ? P(?
(1)
i=1
Precision@k. The Prec@k measure ranks the data points in order of the predicted scores ybi and then
returns the number of true positives in the top ranked positions. This is valuable in situations where
there are very few positives. To formalize this, for any predictor w and set of points x1:t , define
S(x, w) := {j : w> x > w> xj } to be the set of points which w ranks above x. Then define
1, if |S(x, w)| < d?te,
T?,t (x, w) =
(2)
0, otherwise.
i.e. T?,t (x, w) is non-zero iff x is in the top-? fraction of the set. Then we define1
X
Prec@k (w) :=
I [yj = 1] .
j:Tk,t (xj ,w)=1
The structural surrogate for this measure is then calculated as 2
t
t
X
X
`Prec@k (w) =
max t
(?
yi ? yi )xTi w ?
yi y?i .
? ?{?1,+1}
y
P
i=1
yi +1)=2kt
i (?
(3)
i=1
Partial AUC. This measures the area under the ROC curve with the false positive rate restricted to
the range [0, ?]. This is in contrast to AUC that considers the entire range [0, 1] of false positive
rates. pAUC is useful in medical applications such as cancer detection where a small false positive
rate is desirable. Let us extend notation to use the indicator T?
top ? fraction of
?,t (x, w) to select the
j : yj < 0, w> x > w> xj ? d?t? e where
the negatively labeled points i.e. T?
(x,
w)
=
1
iff
?,t
t? is the number of negatives. Then we define
X X
>
>
pAUC(w) =
T?
(4)
?,t (xj , w) ? I[xi w ? xj w].
i:yi >0 j:yj <0
Let ? : R ? R+ be any convex, monotone, Lipschitz, classification surrogate. Then we can obtain
convex surrogates for pAUC(w) by replacing the indicator functions above with ?(?).
X X
>
>
`pAUC (w) =
T?
(5)
?,t (xj , w) ? ?(xi w ? xj w),
i:yi >0 j:yj <0
It can be shown [7, Theorem 4] that the structural surrogate for pAUC is equivalent to (5) with
?(c) = max(0, 1 ? c), the hinge loss function. In the next section we will develop an online
learning framework for non-decomposable performance measures and instantiate loss functions such
as `Prec@k and `pAUC in our framework. Then in Section 4, we will develop stochastic gradient
methods for non-decomposable loss functions and prove error bounds for the same. There we will
focus on a much larger family of loss functions including Prec@k , pAUC and F-measure.
1
2
An equivalent definition considers k to be the number of top ranked points instead.
[3] uses a slightly modified, but equivalent, definition that considers labels to be Boolean.
3
3
Online Learning with Non-decomposable Loss Functions
We now present our online learning framework for non-decomposable loss functions. Traditional
online learning takes place in several rounds, in each of which the player proposes some wt ? W
while the adversary responds with a penalty function Lt : W ? R and a loss Lt (wt ) is incurred.
PT
PT
The goal is to minimize the regret i.e.
t=1 Lt (w). For point loss
t=1 Lt (wt ) ? arg minw?W
functions, the instantaneous penalty Lt (?) is encoded using a data point (xt , yt ) ? Rd ? {?1, 1}
as Lt (w) = `P (xt , yt , w). However, for (surrogates of) non-decomposable loss functions such as
`pAUC and `Prec@k the definition of instantaneous penalty itself is not clear and remains a challenge.
To guide us in this process we turn to some properties of standard online learning frameworks. For
point losses, we note that the best solution in hindsight is also the batch optimal solution. This is
PT
equivalent to the condition arg minw?W t=1 Lt (w) = arg minw?W `P (x1:T , y1:T , w) for nondecomposable losses. Also, since the batch optimal solution is agnostic to the ordering of points,
PT
we should expect t=1 Lt (w) to be invariant to permutations within the stream. By pruning away
several naive definitions of Lt using these requirements, we arrive at the following definition:
Lt (w) = `P (x1:t , y1:t , w) ? `P (x1:(t?1) , y1:(t?1) , w).
(6)
It turns out that the above is a very natural penalty function as it measures the amount of ?extra?
penalty incurred due to the inclusion of xt into the set of points. It can be readily verified that
PT
generalizes
t=1 Lt (w) = `P (x1:T , y1:T , w) as required. Also, this penalty function seamlessly
Pt
online learning frameworks since for point losses, we have `P (x1:t , y1:t , w) = i=1 `P (xi , yi , w)
and thus Lt (w) = `P (xt , yt , w). We note that our framework also recovers the model for online
AUC maximization used in [13] and [14]. The notion of regret corresponding to this penalty is
R(T ) =
T
1X
1
Lt (wt ) ? arg min `P (x1:T , y1:T , w).
T t=1
w?W T
We note that Lt , being the difference of two loss functions, is non-convex in general and thus, standard online convex programming regret bounds cannot be applied in our framework. Interestingly, as
we show below, by exploiting structural properties of our penalty function, we can still get efficient
low-regret learning algorithms, as well as online-to-batch conversion bounds in our framework.
3.1
Low Regret Online Learning
We propose an efficient Follow-the-Regularized-Leader (FTRL) style algorithm in our framework.
Let w1 = arg minw?W kwk22 and consider the following update:
t
X
?
?
Lt (w) + kwk22 = arg min `P (x1:t , y1:t , w) + kwk22
w?W
w?W
2
2
t=1
wt+1 = arg min
(FTRL)
We would like to stress that despite the non-convexity of Lt , the FTRL objective is strongly convex
if `P is convex and thus the update can be implemented efficiently by solving a regularized batch
problem on x1:t . We now present our regret bound analysis for the FTRL update given above.
Theorem 1. Let `P (?, w) be a convex loss function and W ? Rd be a convex set. Assume w.l.o.g.
kxt k2 ? 1, ?t. Also, for the penalty function Lt in (6), let |Lt (w) ? Lt (w0 )| ? Gt ? kw ? w0 k2 ,
for all t and all w, w0 ? W, for some Gt > 0. Suppose we use the update step given in ((FTRL)) to
obtain wt+1 , 0 ? t ? T ? 1. Then for all w? , we have
q P
T
T
2 t=1 G2t
1X
1
Lt (wt ) ? `P (x1:T , y1:T , w? ) + kw? k2
.
T t=1
T
T
See Appendix A for a proof. The above result requires the penalty function Lt to be Lipschitz
continuous i.e. be ?stable? w.r.t. w. Establishing this for point losses such as hinge loss is relatively
straightforward. However, the same becomes non-trivial for non-decomposable loss functions as
4
Lt is now the difference of two loss functions, both of which involve ? (t) data points. A naive
argument would thus, only be able to show Gt ? O(t) which would yield vacuous regret bounds.
Instead, we now show that for the surrogate loss functions for Prec@k and pAUC, this Lipschitz
continuity property does indeed hold. Our proofs crucially use a structural lemma given below that
shows that sorted lists of inner products are Lipschitz at each fixed position.
Lemma 2 (Structural Lemma). Let x1 , . . . , xt be t points with kxi k2 ? 1 ?t. Let w, w0 ? W be any
two vectors. Let zi = hw, xi i ? ci and zi0 = hw0 , xi i ? ci , where ci ? R are constants independent
of w, w0 . Also, let {i1 , . . . , it } and {j1 , . . . , jt } be ordering of indices such that zi1 ? zi2 ? ? ? ? ?
zit and zj0 1 ? zj0 2 ? ? ? ? ? zj0 t . Then for any 1-Lipschitz increasing function g : R ? R (i.e.
|g(u) ? g(v)| ? |u ? v| and u ? v ? g(u) ? g(v)), we have, ?k |g(zik ) ? g(zj0 k )| ? 3kw ? w0 k2 .
See Appendix B for a proof. Using this lemma
qwecan show that the Lipschitz constant for `Prec@k
1
is bounded by Gt ? 8 which gives us a O
regret bound for Prec@k (see Appendix C for
T
the proof). In Appendix D, we show that the same technique can be used to prove a stability result
for the structural SVM surrogate of the Precision-Recall Break Even Point (PRBEP) performance
measure [3] as well. The case of pAUC is handled similarly. However, since pAUC discriminates
between positives and negatives, our previous analysis cannot be applied directly. Nevertheless, we
can obtain the following regret bound for pAUC (a proof will appear in the full version of the paper).
Theorem 3. Let T+ and T? resp. be the number of positive and negative points in the stream and
let wt+1 , 0 ? t ? T ? 1 be obtained using the FTRL algorithm ((FTRL)). Then we have
s
!
T
X
1
1
1
1
Lt (wt ) ? min
`pAUC (x1:T , y1:T , w) + O
+
.
w?W ?T+ T?
?T+ T? t=1
T+
T?
Notice that the above regret bound depends on both T+ and T? and the regret becomes large even
if one of them is small. This is actually quite intuitive because if, say T+ = 1 and T? = T ? 1,
an adversary may wish to provide the lone positive point in the last round. Naturally the algorithm,
having only seen negatives till now, would not be able to perform well and would incur a large error.
3.2
Online-to-batch Conversion
To present our bounds we generalize our framework slightly: we now consider the stream of T
points to be composed of T /s batches Z1 , . . . , ZT /s of size s each. Thus, the instantaneous penalty
is now defined as Lt (w) = `P (Z1 , . . . , Zt , w) ? `P (Z1 , . . . , Zt?1 , w) for t = 1 . . . T /s and the
PT /s
regret becomes R(T, s) = T1 t=1 Lt (wt ) ? arg minw?W T1 `P (x1:T , y1:T , w). Let RP denote
the population risk for the (normalized) performance measure P. Then we have:
Theorem 4. Suppose the sequence of points (xt , yt ) is generated i.i.d. and let w1 , w2 , . . . , wT /s
be an ensemble of models generated by an online learning algorithm upon receiving these T /s
batches. Suppose the online learning algorithm has a guaranteed regret bound R(T, s). Then for
PT /s
w = T1/s t=1 wt , any w? ? W, ? (0, 0.5] and ? > 0, with probability at least 1 ? ?,
!
r
s
ln(1/?)
??(s2 )
?
?
RP (w) ? (1 + )RP (w ) + R(T, s) + e
+O
.
T
p
?
? T ) and = 4 1/T gives us, with probability at least 1 ? ?,
In particular, setting s = O(
!
r
?
4 ln(1/?)
?
?
RP (w) ? RP (w ) + R(T, T ) + O
.
T
We conclude by noting that for Prec@k and pAUC, R(T,
4
?
T) ? O
p
4
1/
T
(see Appendix E).
Stochastic Gradient Methods for Non-decomposable Losses
The online learning algorithms discussed in the previous section present attractive guarantees in the
sequential prediction model but are required to solve batch problems at each stage. This rapidly
5
Algorithm 1 1PMB: Single-Pass with Mini-batches
Algorithm 2 2PMB: Two-Passes with Mini-batches
Input: Step length scale ?, Buffer B of size s
Input: Step length scale ?, Buffers B+ , B? of size s
Output: A good predictor w ? W
Output: A good predictor w ? W
1: w0 ? 0, B ? ?, e ? 0
Pass 1: B+ ? ?
+
2: while stream not exhausted do
1: Collect random sample of pos. x+
1 , . . . , xs in B+
3:
Collect s data points (xe1 , y1e ), . . . , (xes , yse ) in
Pass 2: w0 ? 0, B? ? ?, e ? 0
buffer B
2: while stream of negative points not exhausted do
e?
4:
Set step length ?e ? ??e
3:
Collect s negative points xe?
1 , . . . , xs in B?
?
e
4:
Set step length ?e ? ?e
5:
we+1 ? ?W [we + ?e ?w `P (xe1:s , y1:s
, we )]
+
//?W projects onto the set W
5:
we+1 ? ?W we + ?e ?w `P (xe?
1:s , x1:s , we )
6:
Flush buffer B
6:
Flush buffer B?
7:
e?e+1
//start a new epoch 7:
e?e+1
//start a new epoch
8: end while
8: end while
P
P
9: return w = 1e ei=1 wi
9: return w = 1e ei=1 wi
becomes infeasible for large scale data. To remedy this, we now present memory efficient stochastic
gradient descent methods for batch learning with non-decomposable loss functions. The motivation
for our approach comes from mini-batch methods used to make learning methods for point loss
functions amenable to distributed computing environments [15, 16], we exploit these techniques to
offer scalable algorithms for non-decomposable loss functions.
Single-pass Method with Mini-batches. The method assumes access to a limited memory buffer
and takes a pass over the data stream. The stream is partitioned into epochs. In each epoch, the
method accumulates points in the stream, uses them to form gradient estimates and takes descent
steps. The buffer is flushed after each epoch. Algorithm 1 describes the 1PMB method. Gradient
computations can be done using Danskin?s theorem (see Appendix H).
Two-pass Method with Mini-batches. The previous algorithm is unable to exploit relationships
between data points across epochs which may help improve performance for loss functions such as
pAUC. To remedy this, we observe that several real life learning scenarios exhibit mild to severe
label imbalance (see Table 2 in Appendix H) which makes it possible to store all or a large fraction
of points of the rare label. Our two pass method exploits this by utilizing two passes over the data:
the first pass collects all (or a random subset of) points of the rare label using some stream sampling
technique [13]. The second pass then goes over the stream, restricted to the non-rare label points,
and performs gradient updates. See Algorithm 2 for details of the 2PMB method.
4.1
Error Bounds
Given a set of n labeled data points (xi , yi ), i = 1 . . . n and a performance measure P, our goal is to
approximate the empirical risk minimizer w? = arg min `P (x1:n , y1:n , w) as closely as possible.
w?W
In this section we shall show that our methods 1PMB and 2PMB provably converge to the empirical
risk minimizer. We first introduce the notion of uniform convergence for a performance measure.
Definition 5. We say that a loss function ` demonstrates
uniform convergence with respect to a set of
?1, . . . , x
? s chosen
predictors W if for some ?(s, ?) = poly 1s , log 1? , when given a set of s points x
randomly from an arbitrary set of n points {(x1 , y1 ), . . . , (xn , yn )} then w.p. at least 1 ? ?, we have
sup |`P (x1:n , y1:n , w) ? `P (?
x1:s , y?1:s , w)| ? ?(s, ?).
w?W
Such uniform convergence results are fairly common for decomposable loss functions such as the
squared loss, logistic loss etc. However, the same is not true for non-decomposable loss functions
barring a few exceptions [17, 10]. To bridge this gap, below we show that a large family of surrogate
loss functions for popular non decomposable performance measures does indeed exhibit uniform
convergence. Our proofs require novel techniques and do not follow from traditional proof progressions. However, we first show how we can use these results to arrive at an error bound.
Theorem 6. Suppose the loss function ` is convex and demonstrates ?(s, ?)-uniform convergence.
Also suppose we have an arbitrary set of n points which are randomly ordered, then the predictor
6
CP
PSG
1PMB
2PMB
0.3
0.2
0.1
0
1
2
3
4
Training time (secs)
0.6
0.4
CP
PSG
1PMB
2PMB
0.2
5
0
(a) PPI
0.2
0.4
0.6
Training time (secs)
0.6
0.4
CP
PSG
1PMB
2PMB
0.2
0.8
0
(b) KDDCup08
Average pAUC in [0, 0.1]
0.4
Average pAUC in [0, 0.1]
0.5
Average pAUC in [0, 0.1]
Average pAUC in [0, 0.1]
0.6
0.6
0.5
0.4
CP
PSG
1PMB
2PMB
0.3
0.2
0.1
0.5
1
1.5
Training time (secs)
0
(c) IJCNN
0.1
0.2
0.3
Training time (secs)
(d) Letter
Figure 1: Comparison of stochastic gradient methods with the cutting plane (CP) and projected
subgradient (PSG) methods on partial AUC maximization tasks. The epoch lengths/buffer sizes for
1PMB and 2PMB were set to 500.
0.1
0
2
0.4
0.3
CP
1PMB
2PMB
0.2
0.1
4
6
8
10
Training time (secs)
0
(a) PPI
10
20
30
Training time (secs)
0.6
0.4
CP
1PMB
2PMB
0.2
0
(b) KDDCup08
Average Prec@k
CP
1PMB
2PMB
Average Prec@k
0.2
Average Prec@k
Average Prec@k
0.5
0.3
0.4
0.3
CP
1PMB
2PMB
0.2
0.1
5
10
Training time (secs)
0
(c) IJCNN
0.2
0.4
0.6
Training time (secs)
0.8
(d) Letter
Figure 2: Comparison of stochastic gradient methods with the cutting plane (CP) method on Prec@k
maximization tasks. The epoch lengths/buffer sizes for 1PMB and 2PMB were set to 500.
w returned by 1PMB with buffer size s satisfies w.p. 1 ? ?,
s?
`P (x1:n , y1:n , w) ? `P (x1:n , y1:n , w? ) + 2? s,
n
r
s
+O
n
We would like to stress that the above result does not assume i.i.d. data and works for arbitrary
datasets so long as they are randomly ordered. We can show similar guarantees for the two pass
method as well (see Appendix F). Using regularized formulations, we can also exploit logarithmic
regret guarantees [18], offered by online gradient descent, to improve this result ? however we do not
explore those considerations here. Instead, we now look at specific instances of loss functions that
possess the desired uniform convergence properties. As mentioned before, due to the combinatorial
nature of these performance measures, our proofs do not follow from traditional methods.
Theorem 7 (Partial Area under the ROC Curve). For any convex, monotone, Lipschitz, classification
surrogate ? : R ? R+ , the surrogate loss function for the (0, ?)-partial
AUC performance
measure
p
log(1/?)/s :
defined as follows exhibits uniform convergence at the rate ?(s, ?) = O
1
d?n? en+
X
X
>
>
T?
?,t (xj , w) ? ?(xi w ? xj w)
i:yi >0 j:yj <0
See Appendix G for a proof sketch. This result covers a large family of surrogate loss functions such
as hinge loss (5), logistic loss etc. Note that the insistence on including only top ranked negative
points introduces a high degree of non-decomposability into the loss function. A similar result for
the special case ? = 1 is due to [17]. We extend the same to the more challenging case of ? < 1.
Theorem 8 (Structural SVM loss for Prec@k ). The structural SVM surrogate forthe Prec@k per
p
formance measure (see (3)) exhibits uniform convergence at the rate ?(s, ?) = O
log(1/?)/s .
We defer the proof to the full version of the paper. The above result can be extended to a large family
of performances measures introduced in [3] that have been widely adopted [10, 19, 8] such as Fmeasure, G-mean, and PRBEP. The above indicates that our methods are expected to output models
that closely approach the empirical risk minimizer for a wide variety of performance measures. In
the next section we verify that this is indeed the case for several real life and benchmark datasets.
5
Experimental Results
We evaluate the proposed stochastic gradient methods on several real-world and benchmark datasets.
7
2PMB
0.15 (69.6)
0.55 (38.7)
CP
0.39 (62.5)
23.25 (40.8)
0.6
0.54
0.52
0.5
0.48
0.46
0.44
0.42
1PMB
0
Table 1: Comparison of training time (secs) and accuracies (in brackets) of 1PMB, 2PMB and cutting plane
methods for pAUC (in [0, 0.1]) and Prec@k maximization tasks on the KDD Cup 2008 dataset.
Average pAUC
1PMB
0.10 (68.2)
0.49 (42.7)
Average pAUC
Measure
pAUC
Prec@k
10
2
10
Epoch length
4
10
0.55
0.5
0.45
2PMB
0
10
2
10
Epoch length
4
10
Figure 3: Performance of 1PMB and
2PMB on the PPI dataset with varying
epoch/buffer sizes for pAUC tasks.
Performance measures: We consider three measures, 1) partial AUC in the false positive range
[0, 0.1], 2) Prec@k with k set to the proportion of positives (PRBEP), and 3) F-measure.
Algorithms: For partial AUC, we compare against the state-of-the-art cutting plane (CP) and projected subgradient methods (PSG) proposed in [7]; unlike the (online) stochastic methods considered
in this work, the PSG method is a ?batch? algorithm which, at each iteration, computes a subgradientbased update over the entire training set. For Prec@k and F-measure, we compare our methods
against cutting plane methods from [3]. We used structural SVM surrogates for all the measures.
Datasets: We used several data sets for our experiments (see Table 2 in Appendix H); of these,
KDDCup08 is from the KDD Cup 2008 challenge and involves a breast cancer detection task [20],
PPI contains data for a protein-protein interaction prediction task [21], and the remaining datasets
are taken from the UCI repository [22].
Parameters: We used 70% of the data set for training and the remaining for testing, with the results
averaged over 5 random train-test splits. Tunable parameters such as step length scale were chosen
using a small validation set. The epoch lengths/buffer sizes were set to 500 in all experiments. Since
a single iteration of the proposed stochastic methods is very fast in practice, we performed multiple
passes over the training data (see Appendix H for details).
The results for pAUC and Prec@k maximization tasks are shown in the Figures 1 and 2. We found
the proposed stochastic gradient methods to be several orders of magnitude faster than the baseline
methods, all the while achieving comparable or better accuracies. For example, for the pAUC task
on the KDD Cup 2008 dataset, the 1PMB method achieved an accuracy of 64.81% within 0.03
seconds, while even after 0.39 seconds, the cutting plane method could only achieve an accuracy
of 62.52% (see Table 1). As expected, the (online) stochastic gradient methods were faster than
the ?batch? projected subgradient descent method for pAUC as well. We found similar trends on
Prec@k (see Figure 2) and F-measure maximization tasks as well. For F-measure tasks, on the KDD
Cup 2008 dataset, for example, the 1PMB method achieved an accuracy of 35.92 within 12 seconds
whereas, even after 150 seconds, the cutting plane method could only achieve an accuracy of 35.25.
The proposed stochastic methods were also found to be robust to changes in epoch lengths (buffer
sizes) till such a point where excessively long epochs would cause the number of updates as well as
accuracy to dip (see Figure 3). The 2PMB method was found to offer higher accuracies for pAUC
maximization on several datasets (see Table 1 and Figure 1), as well as be more robust to changes
in buffer size (Figure 3). We defer results on more datasets and performance measures to the full
version of the paper.
The cutting plane methods were generally found to exhibit a zig-zag behaviour in performance
across iterates. This is because these methods solve the dual optimization problem for a given performance measure; hence the intermediate models do not necessarily yield good accuracies. On the
other hand, (stochastic) gradient based methods directly offer progress in terms of the primal optimization problem, and hence provide good intermediate solutions as well. This can be advantageous
in scenarios with a time budget in the training phase.
Acknowledgements
The authors thank Shivani Agarwal for helpful comments. They also thank the anonymous reviewers
for their suggestions. HN thanks support from a Google India PhD Fellowship.
8
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4,824 | 5,368 | Optimistic planning in Markov decision processes
using a generative model
Bal?azs Sz?or?enyi
INRIA Lille - Nord Europe,
SequeL project, France /
MTA-SZTE Research Group on
Arti?cial Intelligence, Hungary
[email protected]
Gunnar Kedenburg
INRIA Lille - Nord Europe,
SequeL project, France
[email protected]
Remi Munos?
INRIA Lille - Nord Europe,
SequeL project, France
[email protected]
Abstract
We consider the problem of online planning in a Markov decision process with
discounted rewards for any given initial state. We consider the PAC sample complexity problem of computing, with probability 1??, an ?-optimal action using the
smallest possible number of calls to the generative model (which provides reward
and next-state samples). We design an algorithm, called StOP (for StochasticOptimistic Planning), based on the ?optimism in the face of uncertainty? principle. StOP can be used in the general setting, requires only a generative model, and
enjoys a complexity bound that only depends on the local structure of the MDP.
1
Introduction
1.1
Problem formulation
In a Markov decision process (MDP), an agent navigates in a state space X by making decisions
from some action set U . The dynamics of the system are determined by transition probabilities
P : X ? U ? X ? [0, 1] and reward probabilities R : X ? U ? [0, 1] ? [0, 1], as follows: when
the agent chooses action u in state x, then, with probability R(x, u, r), it receives reward r, and with
probability P (x, u, x? ) it makes a transition to a next state x? . This happens independently of all
previous actions, states and rewards?that is, the system possesses the Markov property. See [20, 2]
for a general introduction to MDPs. We do not assume that the transition or reward probabilities
are fully known. Instead, we assume access to the MDP via a generative model (e.g. simulation
software), which, for a state-action (x, u), returns a reward sample r ? R(x, u, ?) and a next-state
sample x? ? P (x, u, ?). We also assume the number of possible next-states to be bounded by N ? N.
We would like to ?nd?
an agent that implements a policy which maximizes the expected cumulative
?
discounted reward E[ t=0 ? t rt ], which we will also refer to as the return. Here, rt is the reward
received at time t and ? ? (0, 1) is the discount factor. Further, we take an online planning approach,
where at each time step, the agent uses the generative model to perform a simulated search (planning)
in the set of policies, starting from the current state. As a result of this search, the agent takes a single
action. An expensive global search for the optimal policy in the whole MDP is avoided.
?
Current af?liation: Google DeepMind
1
To quantify the performance of our algorithm, we consider a PAC (Probably Approximately Correct)
setting, where, given ? > 0 and ? ? (0, 1), our algorithm returns, with probability 1??, an ?-optimal
action (i.e. such that the loss of performing this action and then following an optimal policy instead
of following an optimal policy from the beginning is at most ?). The number of calls to the generative
model required by the planning algorithm is referred to as its sample complexity. The sample and
computational complexities of the planning algorithm introduced here depend on local properties
of the MDP, such as the quantity of near-optimal policies starting from the initial state, rather than
global features like the MDP?s size.
1.2
Related work
The online planning approach and, in particular, its ability to get rid of the dependency on the global
features of the MDP in the complexity bounds (mentioned above, and detailed further below) is
the driving force behind the Monte Carlo Tree Search algorithms [16, 8, 11, 18]. 1 The theoretical analysis of this approach is still far from complete. Some of the earlier algorithms use strong
assumptions, others are applicable only in restricted cases, or don?t adapt to the complexity of the
problem. In this paper we build on ideas used in previous works, and aim at ?xing these issues.
A ?rst related work is the sparse sampling algorithm of [14]. It builds a uniform look-ahead tree of a
given depth (which depends on the precision ?), using for each transition a ?nite number of samples
obtained from a generative model. An estimate of the value function is then built using empirical
averaging instead of expectations in the dynamic programming back-up scheme. This results in an
2 )])
?
? log K+log[1/(?(1??)
log(1/?)
1
algorithm with (problem-independent) sample complexity of order (1??)
3?
(neglecting some poly-logarithmic dependence), where K is the number of actions. In terms of ?,
this bound scales as exp(O([log(1/?)]2 )), which is non-polynomial in 1/?. 2 Another disadvantage
of the algorithm is that the expansion of the look-ahead tree is uniform; it does not adapt to the MDP.
An algorithm which addresses this appears in [21]. It avoids evaluating some unnecessary branches
of the look-ahead tree of the sparse sampling algorithm. However, the provided sample bound does
not improve on the one in [14], and it is possible to show that the bound is tight (for both algorithms).
In fact, the sample complexity turns out to be super-polynomial even in the pure Monte Carlo setting
1
(i.e., when K = 1): 1/?2+(log C)/ log(1/?) , with C ? ?2 (1??)
4.
Close to our contribution are the planning algorithms [13, 3, 5, 15] (see also the survey [18]) that
follow the so-called ?optimism in the face of uncertainty? principle for online planning. This principle has been extensively investigated in the multi-armed bandit literature (see e.g. [17, 1, 4]). In
the planning problem, this approach translates to prioritizing the most promising part of the policy
space during exploration. In [13, 3, 5], the sample complexity depends on a measure of the quantity
of near-optimal policies, which gives a better understanding of the real hardness of the problem than
the uniform bound in [14].
The case of deterministic dynamics and rewards is considered in [13]. The proposed algorithm has
log ?
sample complexity of order (1/?) log(1/?) , where ? ? [1, K] measures (as a branching factor) the
quantity of nodes of the planning tree that belong to near-optimal policies. If all policies are very
good, many nodes need to be explored in order to distinguish the optimal policies from the rest, and
log K
therefore, ? is close to the number of actions K, resulting in the minimax bound of (1/?) log(1/?) .
Now if there is structure in the rewards (e.g. when sub-optimal policies can be eliminated by observing the ?rst rewards along the sequence), then the proportion of near-optimal policies is low,
so ? can be small and the bound is much better. In [3], the case of stochastic rewards have been
considered. However, in that work the performance is not compared to the optimal (closed-loop)
policy, but to the best open-loop policy (i.e. which does not depends on the state but only on the
log(?)
sequence of actions). In that situation, the sample complexity is of order (1/?)max(2, log(1/?) ) .
The deterministic and open-loop settings are relatively simple, since any policy can be identi?ed with
a sequence of actions. In the general MDP case however, a policy corresponds to an exponentially
1
A similar planning approach has been considered in the control literature, such as the model-predictive
control [6] or in the AI community, such as the A? heuristic search [19] and the AO? variant [12].
2
A problem-independent lower bound for the sample complexity, of order (1/?)1/ log(1/?) , is provided too.
2
wide tree, where several branches need to be explored. The closest work to ours in this respect is
[5]. However, it makes the (strong) assumption that a full model of the rewards and transitions is
? ? log(?)
available. The sample complexity achieved is again 1/? log(1/?) , but where ? ? (1, N K] is de?ned
as the branching factor of the set of nodes that simultaneously (1) belong to near-optimal policies,
and (2) whose ?contribution? to the value function at the initial state is non-negligible.
1.3
The main results of the paper
Our main contribution is a planning algorithm, called StOP (for Stochastic Optimistic Planning)
that achieves a polynomial sample complexity in terms of ? (which can be regarded as the leading
parameter in this problem), and which is, in terms of this complexity, competitive to other algorithms
that can exploit more speci?cs of their respective domains. It bene?ts from possible reward or
transition probability structures, and does not require any special restriction or knowledge about the
MDP besides having access to a generative model. The sample complexity bound is more involved
than in previous works, but can be upper-bounded by:
log ?
(1/?)2+ log(1/?) +o(1)
(1)
The important quantity ? ? [1, KN ] plays the role of a branching factor of the set of important
states S ?,? (de?ned precisely later) that ?contribute? in a signi?cant way to near-optimal policies.
These states have a non-negligible probability to be reached when following some near-optimal
policy. This measure is similar (but with some differences illustrated below) to the ? introduced in
the analysis of OP-MDP in [5]. Comparing the two, (1) contains an additional constant of 2 in the
exponent. This is a consequence of the fact that the rewards are random and that we do not have
access to the true probabilities, only to a generative model generating transition and reward samples.
In order to provide intuition about the bound, let us consider several speci?c cases (the derivation of
these bounds can be found in Section E):
? Worst-case. When there is no structure at all, then S ?,? may potentially be the set of
all possible reachable nodes (up to some depth which depends on ?), and its branching
factor is ? = KN . The sample complexity is thus of order (neglecting logarithmic faclog(KN )
tors) (1/?)2+ log(1/?) . This is the same complexity that uniform planning algorithm would
achieve. Indeed, uniform planning would build a tree of depth h with branching factor KN
where from each state-action one would generate m rewards and next-state samples. Then,
dynamic programming would be used with the empirical Bellman operator built from the
samples. Using Chernoff-Hoeffding bound,
? the estimation error is of the order (neglecting
logarithms and (1 ? ?) dependence) of 1/ m. So for a desired error ? we need to choose h
of order log(1/?)/ log(1/?), and m of order 1/?2 leading to a sample complexity of order
log(KN )
m(KN )h = (1/?)2+ log(1/?) . (See also [15]) Note that in the worst-case sense there is no
uniformly better strategy than a uniform planning, which is achieved by StOP. However,
StOP can also do much better in speci?c settings, as illustrated next.
? Case with K0 > 1 actions at the initial state, K1 = 1 actions for all other states, and
arbitrary transition probabilities. Now each branch corresponds to a single policy. In
that case one has ? = 1 (even though N > 1) and the sample complexity of StOP is of
2
?
) with high probability3 . This is the same rate as a Monte-Carlo evalorder O(log(1/?)/?
uation strategy would achieve, by sampling O(log(1/?)/?2 ) random trajectories of length
log(1/?)/ log(1/?). Notice that this result is surprisingly different from OP-MDP which
log N
has a complexity of order (1/?) log(1/?) (in the case when ? = N , i.e., when all transitions
are uniform). Indeed, in the case of uniform transition probabilities, OP-MDP would sample the nodes in breadth-?rst search way, thus achieving this minimax-optimal complexity.
2
?
This does not contradict the O(log(1/?)/?
) bound for StOP (and Monte-Carlo) since this
bound applies to an individual problem and holds in high probability, whereas the bound
for OP-MDP is deterministic and holds uniformly over all problems of this type.
3
We emphasize the dependence on ? here since we want to compare this high-probability bound to the
deterministic bound of OP-MDP.
3
Here we see the potential bene?t of using StOP instead of OP-MDP, even though StOP
only uses a generative model of the MDP whereas OP-MDP requires a full model.
? Highly structured policies. This situation holds when there is a substantial gap between
near optimal policies and other sub-optimal policies. For example if along an optimal
policy, all immediate rewards are 1, whereas as soon as one deviates from it, all rewards
are < 1. Then only a small proportion of the nodes (the ones that contribute to near-optimal
policies) will be expanded by the algorithm. In such cases, ? is very close to 1 and in the
limit, we recover the previous case when K = 1 and the sample complexity is O(1/?)2 .
? Deterministic MDPs. Here N = 1 and we have that ? ? [1, K]. When there is structure in
2
?
). Now when
the rewards (like in the previous case), then ? = 1 and we obtain a rate O(1/?
the MDP is almost deterministic, in the sense that N > 1 but from any state-action, there
is one next-state probability which is close to 1, then we have almost the same complexity
as in the deterministic case (since the nodes that have a small probability to be reached will
not contribute to the set of important nodes S ?,? , which characterizes ?).
? Multi-armed bandit we essentially recover the result of the Action Elimination algorithm
[9] for the PAC setting.
Thus we see that in the worst case StOP is minimax-optimal, and in addition, StOP is able to bene?t
from situations when there is some structure either in the rewards or in the transition probabilities.
We stress that StOP achieves the above mentioned results having no knowledge about ?.
1.4
The structure of the paper
Section 2 describes the algorithm, and introduces all the necessary notions. Section 3 presents the
consistency and sample complexity results. Section 4 discusses run time ef?ciency, and in Section 5
we make some concluding remarks. Finally, the supplementary material provides the missing proofs,
the analysis of the special cases, and the necessary ?xes for the issues with the run-time complexity.
2
StOP: Stochastic Optimistic Planning
Recall that N ? N denotes the number of possible next states. That is, for each state x ? X and each
action u available at x, it holds that P (x, u, x? ) = 0 for all but at most N states x? ? X. Throughout
this section, the state of interest is denoted by x0 , the requested accuracy by ?, and the con?dence
parameter by ?0 . That is, the problem to be solved is to output an action u which is, with probability
at least (1 ? ?0 ), at least ?-optimal in x0 .
The algorithm and the analysis make use of the notion of an (in?nite) planning tree, policies and
trajectories. These notions are introduced in the next subsection.
2.1
Planning trees and trajectories
The in?nite planning tree ?? for a given MDP is a rooted and labeled in?nite tree. Its root is
denoted s0 and is labeled by the state of interest, x0 ? X. Nodes on even levels are called action
nodes (the root is an action node), and have Kd children each on the d-th level of action nodes: each
action u is represented by exactly one child, labeled u. Nodes on odd levels are called transition
nodes and have N children each: if the label of the parent (action) node is x, and the label of the
transition node itself is u, then for each x? ? X with P (x, u, x? ) > 0 there is a corresponding child,
labeled x? . There may be children with probability zero, but no duplicates.
An in?nite policy is a subtree of ?? with the same root, where each action node has exactly one
child and each transition node has N children. It corresponds to an agent having ?xed all its possible
future actions. A (partial) policy ? is a ?nite subtree of ?? , again with the same root, but where
the action nodes have at most one child, each transition node has N children, and all leaves 4 are
on the same level. The number of transition nodes on any path from the root to a leaf is denoted
d(?) and is called the depth of ?. A partial policy corresponds to the agent having its possible
future actions planned for d(?) steps. There is a natural partial order over these policies: a policy
4
Note that leaves are, by de?nition, always action nodes.
4
?? is called descendant policy of a policy ? if ? is a subtree of ?? . If, additionally, it holds that
d(?? ) = d(?) + 1, then ? is called the parent policy of ?? , and ?? the child policy of ?.
A (random) trajectory, or rollout, for some policy ? is a realization ? := (xt , ut , rt )Tt=0 of the
stochastic process that belongs to the policy. A random path is generated from the root by always
following, from a non-leaf action node with label xt , its unique child in ?, then setting ut to the
label of this node, from where, drawing ?rst a label xt+1 from P (xt , ut , ?), one follows the child
with label xt+1 . The reward rt is drawn from the distribution determined by R(xt , ut , ?). The value
?T
of the rollout ? (also called return or payoff in the literature) is v(? ) := t=0 rt ? t , and the value of
?T
the policy ? is v(?) := E[v(? )] = E[ t=0 rt ? t ]. For an action u available at x0 , denote by v(u)
the maximum of the values of the policies having u as the label of the child of root s0 . Denote by v ?
the maximum of these v(u) values. Using this notation, the task of the algorithm is to return, with
high probability, an action u with v(u) ? v ? ? ?.
2.2
The algorithm
StOP (Algorithm 1, see Figure 1 in the supplementary material for an illustration) maintains for each
action u available at x0 a set of active policies Active(u). Initially, it holds that Active(u) = {?u },
where ?u is the shallowest partial policy with the child of the root being labeled u. Also, for each
policy ? that becomes a member of an active set, the algorithm maintains high con?dence lower and
upper bounds for the value v(?) of the policy, denoted ?(?) and b(?), respectively.
In each round t, an optimistic policy ??t,u := argmax??Active(u) b(?) is determined for each action u. Based on this, the current optimistic action u?t := argmaxu b(??t,u ) and secondary action
?
u??
t := argmaxu?=u?t b(?t,u ) are computed. A policy ?t to explore is then chosen: if the policy
that belongs to the secondary action is at least as deeply developed as the policy that belongs to
the optimistic action, the optimistic one is chosen for exploration, otherwise the secondary one.
Note that a smaller depth is equivalent to a larger gap between lower and upper bound, and vice
versa5 . The set Active(ut ) is then updated by replacing the policy ?t by its child policies. Accordingly, the upper and lower bounds for these policies are computed. The algorithm terminates when
?(??t ) + ? ? maxu?=u? b(??t,u )?that is, when, with high con?dence, no policies starting with an
t
action different from u?t have the potential to have signi?cantly higher value.
2.2.1
Number and length of trajectories needed for one partial policy
Fix some integer d > 0 and let ? be a partial policy of depth d. Let, furthermore, ?? be an in?nite
policy that is a descendant of ?. Note that
0 ? v(?? ) ? v(?) ?
?d
1?? .
(2)
d
?
-accurate approximation of the value of ?? . On the other hand, having m
The value of ? is a 1??
trajectories for ?, their average reward v?(?) can be used as an estimate of the value v(?)?
of ?. From
d
ln(1/?)
the Hoeffding bound, this estimate has, with probability at least (1 ? ?), accuracy 1??
1??
2m .
?
d
d
ln(1/?)
?d
( 1??
)2 ? trajectories, 1??
? 1??
holds, so with probWith m := m(d, ?) := ? ln(1/?)
2
1??
2m
?d
?
d
d
ln(1/?)
?
?d
ability at least (1 ? ?), b(?) := v?(?) + 1??
+ 1??
? v?(?) + 2 1??
and ?(?) :=
1??
2m
?
ln(1/?)
1?? d
?d
? v?(?) ? 1?? bound v(?? ) from above and below, respectively. This choice
v?(?) ? 1??
2m
balances the inaccuracy of estimating v(?? ) based on v(?) and the inaccuracy of estimating v(?).
d?
?
6
)/ ln(1/?)?, the smallest integer satisfying 3 1??
? ?/2. Note
Let d? := d? (?, ?) := ?(ln (1??)?
?
that if d(?) = d for any given policy ?, then b(?) ? ?(?) ? ?/2. Because of this, it follows
(see Lemma 3 in the supplementary material) that d? is the maximal length the algorithm ever has
to develop a policy.
5
This approach of using secondary actions is based on the UGapE algorithm [10].
5
Algorithm 1 StOP(s0 , ?0 , ?, ?)
1: for all u available from x0 do
? initialize
2:
?u := smallest policy with the child of s0 labeled u
3:
?1 := (?0 /d? ) ? (K0 )?1
? d(?u ) = 1
4:
(?(?u ), b(?u )) := BoundValue(?u , ?1 )
5:
Active(u) := {?u }
? the set of active policies that follow u in s0
6: for round t=1, 2, . . . do
7:
for all u available at x0 do
8:
??t,u := argmax??Active(u) b(?)
9:
??t := ?? ? , where u?t := argmaxu b(??t,u ),
? optimistic action and policy
t,ut
?
???
t := ?
10:
if
11:
?(??t )
t,u??
t
?
, where u??
t := argmaxu?=u? b(?t,u ),
+? ?
u?t
t
maxu?=u? b(??t,u )
t
then
? secondary action and policy
? termination criterion
return
?
if d(???
? select the policy to evaluate
t ) ? d(?t ) then
?
ut := ut and ?t := ??t
else
??
ut := u??
? action and policy to explore
t and ?t := ?t
Active(ut ) := Active(ut ) \ {?t }
?d(? )?1
?d?1
?
?
? := (?0 /d? ) ? ?=0t (K? )?N
? ?=0 (K? )N = # of policies of depth at most d
?
for all child policy ? of ?t do
(?(?), b(?)) := BoundValue(?? , ?)
Active(ut ) := Active(ut ) ? {?? }
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
2.2.2
Samples and sample trees
Algorithm StOP aims to aggressively reuse every sample for each transition node and every sample
for each state-action pair, in order to keep the sample complexity as low as possible. Each time the
value of a partial policy is evaluated, all samples that are available for any part of it from previous
rounds are reused. That is, if m trajectories are necessary for assessing the value of some policy
?, and there are m? complete trajectories available and m?? that end at some inner node of ?, then
StOP (more precisely, another algorithm, Sample, called from StOP) samples rewards (using
SampleReward) and transitions (SampleTransition) to generate continuations for the m??
incomplete trajectories and to generate (m ? m? ? m?? ) new trajectories, as described in Section 2.1,
where
? SampleReward(s) for some action node s samples a reward from the distribution
R(x, u, ?), where u is the label of the parent of s and x is the label of the grandparent
of s, and
? SampleTransition(s) for some transition node s samples a next state from the distribution P (x, u, ?), where u is the label of s and x is the label of the parent of s.
To compensate for the sharing of the samples, the con?dences of the estimates are increased, so that
with probability at least (1 ? ?0 ), all of them are valid6 . The samples are organized as a collection of
sample trees, where a sample tree T is a (?nite) subtree of ?? with the property that each transition
node has exactly one child, and that each action node s is associated with some reward rT (s). Note
that the intersection of a policy ? and a sample tree T is always a path. Denote this path by ? (T , ?)
and note that it necessarily starts from the root and ends either in a leaf or in an internal node of ?. In
the former case, this path can be interpreted as a complete trajectory for ?, and in the latter case, as
an initial segment. Accordingly,
?m when the value of a new policy ? needs to be estimated/bounded, it
1
is computed as v?(?) := m
i=1 v(? (Ti , ?)) (see Algorithm 2: BoundValue), where T1 , . . . , Tm
are sample trees constructed by the algorithm. For terseness, these are considered to be global
variables, and are constructed and maintained using algorithm Sample (Algorithm 3).
?
?N ?
In particular, the con?dence is set to 1 ? ?d(?) for policy ?, where ?d = (?0 /d? ) d?1
is ?0
?=0 K?
divided by the number of policies of depth at most d, and by the largest possible depth?see section 2.2.1.
6
6
Algorithm 2 BoundValue(?, ?)
Ensure: with
at least
(1 ? ?), interval [?(?), b(?)] contains v(?)
? probability
?
?2 ?
ln(1/?) 1?? d(?)
1: m :=
2
? d(?)
2: Sample(?,?
s0 , m)
m
1
3: v?(?) := m
i=1 v(? (Ti , ?))
4: ?(?) := v?(?) ?
1?? d(?)
1??
5: b(?) := v?(?) +
?
1??
d(?)
+
?
6: return (?(?), b(?))
? Ensure that at least m trajectories exist for ?
? empirical estimate of v(?)
ln(1/?)
2m
d(?)
1??
1??
?
? Hoeffding bound
ln(1/?)
2m
? . . . and (2)
Algorithm 3 Sample(?, s, m)
Ensure: there are m sample trees T1 , . . . , Tm that contain a complete trajectory for ? (i.e. ? (Ti , ?)
ends in a leaf of ? for i = 1, . . . , m)
1: for i := 1, . . . , m do
2:
if sample tree Ti does not yet exist then
3:
let Ti be a new sample tree of depth 0
4:
let s be the last node of ? (Ti , ?)
? s is an action node
5:
while s is not a leaf of ? do
6:
let s? be the child of s in ? and add it to T as a new child of s
7:
s?? := SampleTransition(s? ),
? s? is a transition node
8:
add s?? to T as a new child of s?
9:
s := s??
10:
rT (s?? ) := SampleReward(s?? )
3
Analysis
Recall that v ? denotes the maximal value of any (possibly in?nite) policy tree. The following theorem formalizes the consistency result for StOP (see the proof in Section C).
Theorem 1. With probability at least (1 ? ?0 ), StOP returns an action with value at least v ? ? ?.
Before stating the sample complexity result, some further notation needs to be introduced.
Let u? denote an optimal action available at state x0 . That is, v(u? ) = v ? . De?ne for u ?= u?
?
?
d(?)
d(?)
Pu? := ? : ? follows u from s0 and v(?) + 3 ?1?? ? v ? ? 3 ?1?? + ? ,
and also de?ne
?
?
d(?)
d(?)
Pu? ? := ? : ? follows u? from s0 , v(?) + 3 ?1?? ? v ? and v(?) ? 6 ?1?? + ? ? max? v(u) .
?
u?=u
?
is the set of ?important? policies that potentially need to be evaluated
Then P :=
in order to determine an ?-optimal action. (See also Lemma 8 in the supplementary material.)
?
Pu? ?
?
u?=u? Pu
Let now p(s) denote the product of the probabilities of the transitions on the path from s0 to s. That
is, for any policy tree ? containing s, a trajectory for ? goes through s with probability p(s). When
estimating the value of some policy ? of depth d, the expected number of trajectories going through
some nodes s of it is p(s)m(d, ?d ). The sample complexity therefore has to take into consideration
for each node s (at least for the ones with ?high? p(s) value) the maximum ?(s) = max{d(?) : ? ?
P ? contains s} of the depth of the relevant policies it is included in. Therefore, the expected number
of trajectories going through s in a given run of StOP is
?
?
? ?
ln(1/??(s) ) 1?? ?(s) 2
(3)
p(s) ? m(?(s), ??(s) ) = p(s)
2
? ?(s)
If (3) is ?large? for some s, it can be used to deduce a high con?dence upper bound on the number of
times s gets sampled. To this end, let S ? denote the set of nodes of the trees in P ? , let N ? denote the
7
??
??
smallest positive integer N satisfying N ? ? s ? S ? : p(s) ? m(?(s), ??(s) ) ? (8/3) ln(2N /?0 ) ?
(obviously N ? ? |S ? |), and de?ne
?
?
S ?,? := s ? S ? : p(s) ? m(?(s), ??(s) ) ? (8/3) ln(2N ? /?0 ) .
S ? is the set of ?important? nodes (P ? is the set of ?important? policies), and S ?,? consists of the
important nodes that, with high probability, are not sampled more than twice as often as expected.
?0
(This high probability is 1 ? 2N
? according to the Bernstein bound, so these upper bounds hold
jointly with probability at least (1 ? ?20 ), as N ? = |S ?,? |. See also Appendix D.)
For s? ? S ? \ S ?,? , the number of times s? gets sampled has a variance that is too high compared
to its expected value (3), so in this case, a different approach is needed in order to derive
? high
con?dence upper bounds. To this end, for a transition node s, let p? (s) := p? (s, ?) := {p(s? ) :
s? is a child of s with p(s? ) ? m(?(s? ), ??(s? ) ) < (8/3) ln(2N ? /?0 )}, and de?ne
?
?
0,
if p? (s) ? 2N ? m(?(s),?
?(s) )
B(s) := B(s, ?) :=
?
?
max(6 ln( 2N
),
2p
(s)m(?(s),
?
))
otherwise.
?(s)
?0
As it will be shown in the proof of Theorem 2 (in Section D), this is a high con?dence upper bound
on the number of trajectories that go through some child s? ? S ? \ S ?,? of some s? ? S ?,? .
Theorem 2. With probability at ?
least (1 ? 2?), StOP outputs a policy of value at least (v?? ? ?), af?
?d
??(s)
ter generating at most s?S ?,? 2p(s)m(?(s), ??(s) ) + B(s) d=d(s)+1 ?=d(s)+1 K? samples,
where d(s) = min{d(?) : s appears in policy ?} is the depth of node s.
Finally, the bound discussed in Section 1 is obtained by setting ? := lim sup??0 max(?1 , ?2 ),
??
?1/d?
?2 (1??)2
2p(s)m(?(s),
?
)
and ?2 := ?2 (?, ?0 , ?) :=
where ?1 := ?1 (?, ?0 , ?) :=
?(s)
s?S ?,? ln(1/?0 )
?
?
? 2
1/d
??(s) ?d
? (1??)2 ?
.
s?S ?,? B(s)
?=d(s) K?
d=d(s)
ln(1/?0 )
4
Ef?ciency
StOP, as presented in Algorithm 1, is not ef?ciently executable. First of all, whenever it evaluates
an optimistic policy, it enumerates all its child policies, which typically has exponential time complexity. Besides that, the sample trees are also treated in an inef?cient way. An ef?cient version of
StOP with all these issues ?xed is presented in Appendix F of the supplementary material.
5
Concluding remarks
In this work, we have presented and analyzed our algorithm, StOP. To the best of our knowledge,
StOP is currently the only algorithm for optimal (i.e. closed loop) online planning with a generative
model that provably bene?ts from local structure both in reward as well as in transition probabilities.
It assumes no knowledge about this structure other than access to the generative model, and does
not impose any restrictions on the system dynamics.
One should note though that the current version of StOP does not support domains with in?nite
N . The sparse sampling algorithm in [14] can easily handle such problems (at the cost of a nonpolynomial (in 1/?) sample complexity), however, StOP has much better sample complexity in case
of ?nite N . An interesting problem for future research is to design adaptive planning algorithms
with sample complexity independent of N ([21] presents such an algorithm, but the complexity
bound provided there is the same as the one in [14]).
Acknowledgments
This work was supported by the French Ministry of Higher Education and Research, and by the
European Community?s Seventh Framework Programme (FP7/2007-2013) under grant agreement
no 270327 (project CompLACS). Author two would like to acknowledge the support of the BMBF
project ALICE (01IB10003B).
8
References
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9
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4,825 | 5,369 | Conditional Swap Regret and
Conditional Correlated Equilibrium
Mehryar Mohri
Courant Institute and Google
251 Mercer Street
New York, NY 10012
Scott Yang
Courant Institute
251 Mercer Street
New York, NY 10012
mohri@cims?nyu?edu
yangs@cims?nyu?edu
Abstract
We introduce a natural extension of the notion of swap regret, conditional swap
regret, that allows for action modi?cations conditioned on the player?s action history. We prove a series of new results for conditional swap regret minimization.
We present algorithms for minimizing conditional swap regret with bounded conditioning history. We further extend these results to the case where conditional
swaps are considered only for a subset of actions. We also de?ne a new notion
of equilibrium, conditional correlated equilibrium, that is tightly connected to the
notion of conditional swap regret: when all players follow conditional swap regret
minimization strategies, then the empirical distribution approaches this equilibrium. Finally, we extend our results to the multi-armed bandit scenario.
1
Introduction
On-line learning has received much attention in recent years. In contrast to the standard batch
framework, the online learning scenario requires no distributional assumption. It can be described
in terms of sequential prediction with expert advice [13] or formulated as a repeated two-player
game between a player (the algorithm) and an opponent with an unknown strategy [7]: at each time
step, the algorithm probabilistically selects an action, the opponent chooses the losses assigned to
each action, and the algorithm incurs the loss corresponding to the action it selected.
The standard measure of the quality of an online algorithm is its regret, which is the difference
between the cumulative loss it incurs after some number of rounds and that of an alternative policy.
The cumulative loss can be compared to that of the single best action in retrospect [13] (external
regret), to the loss incurred by changing every occurrence of a speci?c action to another [9] (internal
regret), or, more generally, to the loss of action sequences obtained by mapping each action to some
other action [4] (swap regret). Swap regret, in particular, accounts for situations where the algorithm
could have reduced its loss by swapping every instance of one action with another (e.g. every time
the player bought Microsoft, he should have bought IBM).
There are many algorithms for minimizing external regret [7], such as, for example, the randomized
weighted-majority algorithm of [13]. It was also shown in [4] and [15] that there exist algorithms for
minimizing internal and swap regret. These regret minimization techniques have been shown to be
useful for approximating game-theoretic equilibria: external regret algorithms for Nash equilibria
and swap regret algorithms for correlated equilibria [14].
By de?nition, swap regret compares a player?s action sequence against all possible modi?cations at
each round, independently of the previous time steps. In this paper, we introduce a natural extension
of swap regret, conditional swap regret, that allows for action modi?cations conditioned on the
player?s action history. Our de?nition depends on the number of past time steps we condition upon.
1
As a motivating example, let us limit this history to just the previous one time step, and suppose we
design an online algorithm for the purpose of investing, where one of our actions is to buy bonds
and another to buy stocks. Since bond and stock prices are known to be negatively correlated, we
should always be wary of buying one immediately after the other ? unless our objective was to pay
for transaction costs without actually modifying our portfolio? However, this does not mean that we
should avoid purchasing one or both of the two assets completely, which would be the only available
alternative in the swap regret scenario. The conditional swap class we introduce provides precisely
a way to account for such correlations between actions. We start by introducing the learning set-up
and the key notions relevant to our analysis (Section 2).
2
Learning set?up and model
We consider the standard online learning set-up with a set of actions N = {1? . . . ? N }. At each
round t ? {1? . . . ? T }, T ? 1, the player selects an action xt ? N according to a distribution pt
over N , in response to which the adversary chooses a function f t : N t ? [0? 1] and causes the
player to incur a loss f t ?xt ? xt?1 ? . . . ? x1 ). The objective of the player is to choose a sequence of
?T
actions ?x1 ? . . . ? xT ) that minimizes his cumulative loss t=1 f t ?xt ? xt?1 ? . . . ? x1 ).
A standard metric used to measure the performance of an online algorithm ? over T rounds is its
?expected) external regret, which measures the player?s expected performance against the best ?xed
action in hindsight:
Reg??? T ) =
Ext
T
?
t=1
?
?xt ?..?x1 )?
?pt ?...?p1 )
[f t ?xt ? ..? x1 )] ? min
j?N
T
?
f t ?j? j? ...? j).
t=1
There are several common modi?cations to the above online learning scenario: (1) we may com?T
t
pare regret against stronger competitor classes: Reg? ??? T ) =
t=1 ?pt ?...?p1 f ?xt ? ..? x1 ) ?
?T
t
min??? t=1 ?pt ?...?p1 [f ???xt )? ??xt?1 )? ...? ??x1 ))] for some function class C ? N N ; (2)
the player may have access to only partial information about the loss, i.e. only knowledge
of f t ?xt ? ..? x1 ) as opposed to f t ?a? xt?1 ? . . . ? x1 )?a ? N (also known as the bandit scenario); (3) the loss function may have bounded memory: f t ?xt ? ...? xt?k ? xt?k?1 ? ...? x1 ) =
f t ?xt ? ...? xt?k ? yt?k?1 ? ...? y1 ), ?xj ? yj ? N .
The scenario where C = N N in (1) is called the swap regret case, and the case where k = 0 in (3) is
referred to as the oblivious adversary. (Sublinear) regret minimization is possible for loss functions
against any competitor class of the form described in (1), with only partial information, and with
at least some level of bounded memory. See [4] and [1] for a reference on (1), [2] and [5] for (2),
and [1] for (3). [6] also provides a detailed summary of the best known regret bounds in all of these
scenarios and more.
The introduction of adversaries with bounded memory naturally leads to an interesting question:
what if we also try to increase the power of the competitor class in this way?
While swap regret is a natural competitor class and has many useful game theoretic consequences
(see [14]), one important missing ingredient is that the competitor class of functions does not have
memory. In fact, in most if not all online learning scenarios and regret minimization algorithms
considered so far, the point of comparison has been against modi?cation of the player?s actions
at each point of time independently of the previous actions. But, as we discussed above in the
?nancial markets example, there exist cases where a player should be measured against alternatives
that depend on the past and the player should take into account the correlations between actions.
Speci?cally, we consider competitor functions of the form ?t : N t ? N t . Let Call = {?t : N t ?
?T
t
N t }?
t=1 denote the class of all such functions. This leads us to the expression:
t=1 ?p1 ?...?pt [f ] ?
?T
t
t
min?t ??all t=1 ?p1 ?...?pt [f ? ? ]. Call is clearly a substantially richer class of competitor functions than traditional swap regret. In fact, it is the most comprehensive class, since we can always
?T
?T
reach t=1 ?p1 ?...?pt [f t ] ? t=1 min?x1 ?..?xt ) f t ?x1 ? ..? xt ) by choosing ?t to map all points to
t
argmin?xt ?..?x1 ) f ?xt ? ...? x1 ). Not surprisingly, however, it is not possible to obtain a sublinear
regret bound against this general class.
2
???
????????
????????
??????
??????
??????
?
???
????????
?
?
????????
????????
???
????????
???
????????
????????
????????
?
?
(a)
(b)
Figure 1: (a) unigram conditional swap class interpreted as a ?nite-state transducer. This is the same
as the usual swap class and has only the trivial state; (b) bigram conditional swap class interpreted as
a ?nite-state transducer. The action at time t ? 1 de?nes the current state and in?uences the potential
swap at time t.
Theorem 1. No algorithm can achieve sublinear regret against the class Call , regardless of the loss
function?s memory.
This result is well-known in the on-line learning community, but, for completeness, we include a
proof in Appendix 9. Theorem 1 suggests examining more reasonable subclasses of Call . To simplify
the notation and proofs that follow in the paper, we will henceforth restrict ourselves to the scenario
of an oblivious adversary, as in the original study of swap regret [4]. However, an application of the
batching technique of [1] should produce analogous results in the non-oblivious case for all of the
theorems that we provide.
Now consider the collection of competitor functions Ck = {? : N k ? N }. Then, a player
who has played actions {as }t?1
s=1 in the past should have his performance compared against
??at ? at?1 ? at?2 ? . . . ? at??k?1) ) at time t, where ? ? Ck . We call this class Ck of functions the
k-gram conditional swap regret class, which also leads us to the regret de?nition:
Reg??? T ) =
?k
T
?
t=1
? t [f t ?xt )] ? min
xt ?p
???k
T
?
t=1
? [f t ???xt ? at?1 ? at?2 ? . . . ? at??k?1) ))].
xt ?pt
Note that this is a direct extension of swap regret to the scenario where we allow for swaps conditioned on the history of the previous ?k ? 1) actions. For k = 1, this precisely coincides with swap
regret.
One important remark about the k-gram conditional swap regret is that it is a random quantity that
depends on the particular sequence of actions played. A natural deterministic alternative would be
of the form:
T
?
t=1
? t [f t ?xt )] ? min
xt ?p
???k
T
?
t=1
?
?xt ?...?x1 )??pt ?...?p1 )
[f t ???xt ? xt?1 ? xt?2 ? . . . ? xt??k?1) ))].
However, by taking the expectation of Reg?k ??? T ) with respect to aT ?1 ? aT2 ? . . . ? a1 and applying
Jensen?s inequality, we obtain
T
T
?
?
Reg??? T )?
? t [f t ?xt )]? min
?k
t=1
xt ?p
???k
t=1
?
?xt ?...?x1 )??pt ?...?p1 )
[f t ???xt ? xt?1 ? xt?2 ? . . . ? xt??k?1) ))]?
and so no generality is lost by considering the randomized sequence of actions in our regret term.
Another interpretation of the bigram conditional swap class is in the context of ?nite-state transducers. Taking a player?s sequence of actions ?x1 ? ...? xT ), we may view each competitor function in
the conditional swap class as an application of a ?nite-state transducer with N states, as illustrated
by Figure 1. Each state encodes the history of actions ?xt?1 ? . . . ? xt??k?1) ) and admits N outgoing
transitions representing the next action along with its possible modi?cation. In this framework, the
original swap regret class is simply a transducer with a single state.
3
3
Full Information Scenario
Here, we prove that it is in fact possible to minimize k-gram conditional swap regret against an
oblivious adversary, starting with the easier to interpret bigram scenario. Our proof constructs a
meta-algorithm using external regret algorithms as subroutines, as in [4]. The key is to attribute
a fraction of the loss to each external regret algorithm, so that these losses sum up to our actual
realized loss and also press the subroutines to minimize regret against each of the conditional swaps.
Theorem 2. There? exists
algorithm ? with bigram swap regret bounded as follows:
?
? an online
Reg?2 ??? T ) ? O N T log N .
Proof. Since the distribution pt at round t is ?nite-dimensional, we can represent it as a vector
pt = ?pt1 ? ...? ptN ). Similarly, since oblivious adversaries take only N arguments, we can write f t
t
as the loss vector f t = ?f1t ? ...? fN
). Let {at }Tt=1 be a sequence of random variables denoting the
player?s actions at each time t, and let ?at t denote the (random) Dirac delta distribution concentrated
at at and applied to variable xt . Then, we can rewrite the bigram swap regret as follows:
Reg??? T ) =
?2
T
?
t=1
=
?t [f t ?xt )] ? min
???2
p
T ?
N
?
t=1 i=1
pti fit ? min
???2
T
?
?
t t?1
t=1 p ??at?1
N
T ?
?
[f t ???xt ? xt?1 )]
t?1
pti ?{a
ft
t?1 =j} ??i?j)
t=1 i?j=1
Our algorithm for achieving sublinear regret is de?ned as follows:
1. At t = 1, initialize N 2 external regret minimizing algorithms Ai?k , ?i? k) ? N 2 .
We can view these in the form of N matrices in RN ?N , {Qt?k }N
k=1 , where for each
is
a
row
vector
consisting
of
the
distribution
weights generated
k ? {1? . . . ? N }, Qt?k
i
by algorithm Ai?k at time t based on losses received at times 1? . . . ? t ? 1.
2. At each time t, let at?1 denote the random action played at time t ? 1 and let ?at?1
denote
t?1
the (random) Dirac delta distribution for this action. De?ne the N ? N matrix Qt =
?N
t?1
t?k
t
k=1 ?{at?1 =k} Q . Q is a Markov chain (i.e., its rows sum up to one), so it admits a
t
stationary distribution p which we we will use as our distribution for time t.
3. When we draw from pt , we play a random action at and receive loss f t . Attribute the
t?1
portion of loss pti ?{a
f t to algorithm Ai?k , and generate distributions Qt?k
i . Notice
t?1 =k}
?N
t t?1
t
t
that i?k=1 pi ?{at?1 =k} f = f , so that the actual realized loss is allocated completely.
Recall that an optimal external regret minimizing algorithm ? (e.g.
majority)
??randomized weighted
?
i?k
i?k
admits a regret bound of the form Ri?k = Ri?k ?Lmin ? T? N ) = O
Lmin log?N ) , where Li?k
min =
?
T
t?i?k
minN
for the sequence of loss vectors {f t?i?k }Tt=1 incurred by the algorithm. Since
j=1
t=1 fj
t
t t
p = p Q is a stationary distribution, we can write:
T
?
t=1
pt ? f t =
N
T ?
?
t=1 j=1
ptj fjt =
N ?
N
T ?
?
pti Qti?j fjt =
t=1 j=1 i=1
N ?
N
T ?
?
t=1 j=1 i=1
4
pti
N
?
k=1
t?1
t
?{i
Qt?k
i?j fj .
t?1 =k}
Rearranging leads to
T
?
t=1
pt ? f t =
?
=
T ?
N ?
N
?
t?1
t
pti ?{i
Qt?k
i?j fj
t?1 =k}
i?k=1 t=1 j=1
N
?
i?k=1
N
?
i?k=1
?? T
?
t?1
pti ?{i
ft
t?1 =k} ??i?k)
t=1
? T
?
t?1
pti ?{i
ft
t?1 =k} ??i?k)
t=1
?
?
?2
T
?
t=1
pt ? f t ? min
???2
+ Ri?k ?Lmin ? T? N )
+
N
?
(for arbitrary ? : N 2 ? N )
Ri?k ?Lmin ? T? N ).
i?k=1
Since ? is arbitrary, we obtain
Reg??? T ) =
?
T ?
N
?
t=1 i?k=1
t?1
pti ?{i
ft
?
t?1 =k} ??i?k)
N
?
Ri?k ?Lmin ? T? N ).
i?k=1
?
log?N ) and that we scaled the losses to algorithm Ai?k by
Using the fact that Ri?k = O
?N ?N
t t?1
pi ?{it?1 =k} , the following inequality holds: k=1 j=1 Lk?j
min ? T . By Jensen?s inequality, this
implies
?
?
?
N
N N ?
N ?
? 1 ?
1 ??
T
k?j
k?j
?
Lmin ?
Lmin ?
?
2
N2
N
N
k=1 j=1
k=1 j=1
?
?N ?N ?
or, equivalently, k=1 j=1 Lk?j
min ? N T . Combining this with our regret bound yields
??
?
N
N
?
? ?
?
?
Li?k
Ri?k ?Lmin ? T? N ) =
O
? O N T log N ?
Reg??? T ) ?
min log N
?2
i?k=1
??
Li?k
min
i?k=1
which concludes the proof.
Remark 1. The computational complexity of a standard external regret minimization algorithm such
as randomized weighted majority per round is in O?N ) ?update the distribution on each of the N
actions multiplicatively and then renormalize), which implies that updating the N 2 subroutines will
cost O?N 3 ) per round. Allocating losses to these subroutines and combining the distributions that
they return will cost an additional O?N 3 ) time. Finding the stationary distribution of a stochastic
3
matrix can be done
?via matrix inversion in O?N )3time. Thus, the total computational complexity
of achieving O?N T log?N )) regret is only O?N T ). We remark that in practice, one often uses
iterative methods to compute dominant eigenvalues ?see [16] for a standard reference and [11] for
recent improvements). [10] has also studied techniques to avoid computing the exact stationary
distribution at every iteration step for similar types of problems.
The meta-algorithm above can be interpreted in three equivalent ways: (1) the player draws an
action xt from distribution pt at time t; (2) the player uses distribution pt to choose among the
N subsets of algorithms Qt1 ? ...? QtN , picking one subset Qtj ; next, after drawing j from pt , the
t?N
t?1
to randomly choose among the algorithms Qt?1
player uses ?{a
j ? ...? Qj , picking algorithm
t?1 =k}
t?a
t?a
Qj t?1 ; after locating this algorithm, the player uses the distribution from algorithm Qj t?1 to draw
t t?1
an action; (3) the player chooses algorithm Qt?k
j with probability pj ?{at?1 =k} and draws an action
from its distribution.
The following more general bound can be given for an arbitrary k-gram swap scenario.
Theorem 3. There??
exists an online
? algorithm ? with k-gram swap regret bounded as follows:
Reg?k ??? T ) ? O N k T log N .
The algorithm used to derive this result is a straightforward extension of the algorithm provided in
the bigram scenario, and the proof is given in Appendix 11.
Remark 2. The computational complexity of achieving the above regret bound is O?N k+1 T ).
5
????????
????????
???
?
?
????????
???
????????
???????? ????????
???
???
?
Figure 2: bigram conditional swap class restricted to a ?nite number of active states. When the
action at time t ? 1 is 1 or 2, the transducer is in the same state, and the swap function is the same.
4
State?Dependent Bounds
In some situations, it may not be relevant to consider conditional swaps for every possible action,
either because of the speci?c problem at hand or simply for the sake of computational ef?ciency.
Thus, for any S ? N 2 , we de?ne the following competitor class of functions:
? for ?i? k) ? S where ?? : N ? N }.
C2?S = {? : N 2 ? N |??i? k) = ??i)
See Figure 2 for a transducer interpretation of this scenario.
We will now show that the algorithm above can be easily modi?ed to derive a tighter bound that
is dependent on the number of states in our competitor class. We will focus on the bigram case,
although a similar result can be shown for the general k-gram conditional swap regret.
?
Theorem 4. There exists an online algorithm ? such that Reg?2?? ??? T )
?
c
O? T ?|S | + N ) log?N )).
The proof of this result is given in Appendix 10. Note that when S = ?, we are in the scenario where
all the previous states matter, and our bound coincides with that of the previous section.
Remark 3. The computational complexity of achieving the above regret bound is O??N ?|?1 ?S)| +
|S c |) + N 3 )T ), where ?1 is projection onto the ?rst component. This follows from the fact that
we allocate the same loss to all {Ai?k }k:?i?k)?S ?i ? ?1 ?S), so we effectively only have to manage
|?1 ?S)| + |S c | subroutines.
5
Conditional Correlated Equilibrium and ??Dominated Actions
It is well-known that regret minimization in on-line learning is related to game-theoretic equilibria
[14]. Speci?cally, when both players in a two-player zero-sum game follow external regret minimizing strategies, then the product of their individual empirical distributions converges to a Nash
equilibrium. Moreover, if all players in a general K-player game follow swap regret minimizing
strategies, then their empirical joint distribution converges to a correlated equilibrium [7].
We will show in this section that when all players follow conditional swap regret minimization
strategies, then the empirical joint distribution will converge to a new stricter type of correlated
equilibrium.
?k)
:S ?
De?nition 1. Let Nk = {1? ...? Nk }, for k ? {1? ...? K} and G = ?S = ?K
k=1 Nk ? {l
K
[0? 1]}k=1 ) denote a K-player game. Let s = ?s1 ? ...? sK ) ? S denote the strategies of all players in
one instance of the game, and let s??k) denote the ?K ? 1)-vector of strategies played by all players
aside from player k. A joint distribution P on two rounds of this game is a conditional correlated
equilibrium if for any player k, actions j? j ? ? Nk , and map ?k : Nk2 ? Nk , we have
?
?
?
P ?s? r) l?k) ?sk ? s??k) ) ? l?k) ??k ?sk ? rk )? s??k) ) ? 0.
?s?r)?S 2 : sk =j?rk =j ?
The standard interpretation of correlated equilibrium, which was ?rst introduced by Aumann, is a
scenario where an external authority assigns mixed strategies to each player in such a way that no
player has an incentive to deviate from the recommendation, provided that no other player deviates
6
from his [3]. In the context of repeated games, a conditional correlated equilibrium is a situation
where an external authority assigns mixed strategies to each player in such a way that no player
has an incentive to deviate from the recommendation in the second round, even after factoring in
information from the previous round of the game, provided that no other player deviates from his.
It is important to note that the concept of conditional correlated equilibrium presented here is different from the notions of extensive form correlated equilibrium and repeated game correlated equilibrium that have been studied in the game theory and economics literature [8, 12].
Notice that when the values taken for ?k are indepndent of its second argument, we retrieve the
familiar notion of correlated equilibrium.
Theorem 5. Suppose that all players in a K-player repeated game follow bigram conditional swap
regret minimizing strategies. Then, the joint empirical distribution of all players converges to a
conditional correlated equilibrium.
Proof. Let I t ? S be a random vector denoting the actions played by all K players in the game
at round t. The empirical joint distribution of every two subsequent rounds of a K-player game
?T ?
played repeatedly for T total rounds has the form P?T = T1 t=1 ?s?r)?S 2 ?{I t =s?I t?1 =r} , where
I = ?I1 ? ..? IK ) and Ik ? p?k) denotes the action played by player k using the mixed strategy p?k) .
t?1
? pt?1??k?1) . Then, the conditional swap regret of each player k,
Let q t??k) denote ?{i
t?1 =k}
reg?k? T ), can be bounded as follows since he is playing with a conditional swap regret minimizing
strategy:
reg?k? T ) =
T
T
?
?
1?
1?
l?k) ?sk ? s??k) ) ? min
?
? T
T t=1 stk ?pt??k)
t=1
?
? ?
log?N )
.
?O N
T
?
?stk ?st?1
)
k
??pt??k) ?q t??k) )
?
?
t
l?k) ???stk ? st?1
k )? s??k) )
De?ne the instantaneous conditional swap regret vector as
? ? ?
?
??
?k)
t
r?t?j0 ?j1 = ?{I t =j0 ?I t?1 =j1 } l?k) I t ? l?k) ?k ?j0 ? j1 )? I??k)
?
?k)
?k)
and the expected instantaneous conditional swap regret vector as
? ?
?
?
??
?k)
t
t
rt?j0 ?j1 = P?stk = j0 )?{I t?1 =j1 } l?k) j0 ? I??k)
? l?k) ?k ?j0 ? j1 )? I??k)
.
?k)
Consider the ?ltration Gt = {information of opponents at time t and of the player?s actions up to
?
? ?k)
?k)
?k)
?k)
time t ? 1}. Then, we see that ? r?t?j0 ?j1 |Gt = rt?j0 ?j1 . Thus, {Rt = rt?j0 ?j1 ? r?t?j0 ?j1 }?
t=1 is a
sequence of bounded martingale differences, and by the Hoeffding-Azuma inequality, we can write
?T
for any ? > 0, that P[| t=1 Rt | > ?] ? 2 exp??C?2 /T ) for some constant C > 0.
? ?
?? ?
? 2 ??
?
?
T
. By our concentration bound, we
Now de?ne the sets AT := ? T1 t=1 Rt ? > C
T log ?T
?
have P ?AT ) ? ?T . Setting ?T = exp?? T ) and applying the Borel-Cantelli lemma, we obtain
?T
that lim supT ?? | T1 t=1 Rt | = 0 a.s..
Finally, since each player followed a conditional swap regret minimizing strategy, we can write
?T
?k)
lim supT ?? T1 t=1 r?t?j0 ?j1 ? 0. Now, if the empirical distribution did not converge to a conditional correlated equilibrium, then by Prokhorov?s theorem, there exists a subsequence {P?Tj }j
satisfying the conditional correlated equilibrium inequality but converging to some limit P ? that is
not a conditional correlated equilibrium. This cannot be true because the inequality is closed under
weak limits.
Convergence to equilibria over the course of repeated game-playing also naturally implies the
scarcity of ?very suboptimal? strategies.
7
De?nition 2. An action pair ?sk ? rk ) ? Nk2 played by player k is conditionally ??dominated if
there exists a map ?k : Nk2 ? Nk such that
l?k) ?sk ? s??k) ) ? l?k) ??k ?sk ? rk )? s??k) ) ? ?.
Theorem 6. Suppose player k follows a conditional swap regret minimizing strategy that produces
a regret R over T instances of the repeated game. Then, on average, an action pair of player k is
R
conditionally ?-dominated at most ?T
fraction of the time.
The proof of this result is provided in Appendix 12.
6
Bandit Scenario
As discussed earlier, the bandit scenario differs from the full-information scenario in that the player
only receives information about the loss of his action f t ?xt ) at each time and not the entire loss
function f t . One standard external regret minimizing algorithm is the Exp3 algorithm introduced
by [2], and it is the base learner off of which we will build a conditional swap regret minimizing
algorithm.
To derive a sublinear conditional swap regret bound, we require an external regret bound on Exp3:
T
?
t=1
? [f t ?xt )] ? min
pt
a?N
T
?
t=1
?
f t ?a) ? 2 Lmin N log?N )?
which can be found in Theorem 3.1 of [5]. Using this estimate, we can derive the following result.
??
?
N 3 log?N )T .
Theorem 7. There exists an algorithm ? such that Reg?2 ?bandit ??? T ) ? O
The proof is given in Appendix 13 and is very similar to the proof for the full information setting.
It can also easily be extended in the analogous way to provide a regret bound for the k-gram regret
in the bandit scenario.
??
?
Theorem 8. There exists an algorithm ? such that Reg?k ?bandit ??? T ) ? O N k+1 log?N )T .
See Appendix 14 for an outline of the algorithm.
7
Conclusion
We analyzed the extent to which on-line learning scenarios are learnable. In contrast to some of
the more recent work that has focused on increasing the power of the adversary (see e.g. [1]), we
increased the power of the competitor class instead by allowing history-dependent action swaps and
thereby extending the notion of swap regret. We proved that this stronger class of competitors can
still be beaten in the sense of sublinear regret as long as the memory of the competitor is bounded.
We also provided a state-dependent bound that gives a more favorable guarantee when only some
parts of the history are considered. In the bigram setting, we introduced the notion of conditional
correlated equilibrium in the context of repeated K-player games, and showed how it can be seen
as a generalization of the traditional correlated equilibrium. We proved that if all players follow
bigram conditional swap regret minimizing strategies, then the empirical joint distribution converges
to a conditional correlated equilibrium and that no player can play very suboptimal strategies too
often. Finally, we showed that sublinear conditional swap regret can also be achieved in the partial
information bandit setting.
8
Acknowledgements
We thank the reviewers for their comments, many of which were very insightful. We are particularly
grateful to the reviewer who found an issue in our discussion on conditional correlated equilibrium
and proposed a helpful resolution. This work was partly funded by the NSF award IIS-1117591. The
material is also based upon work supported by the National Science Foundation Graduate Research
Fellowship under Grant No. DGE 1342536.
8
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9
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4,826 | 537 | Adaptive Synchronization of
Neural and Physical Oscillators
Kenji Doya
University of California, San Diego
La Jolla, CA 92093-0322, USA
Shuji Yoshizawa
University of Tokyo
Bunkyo-ku, Tokyo 113, Japan
Abstract
Animal locomotion patterns are controlled by recurrent neural networks
called central pattern generators (CPGs). Although a CPG can oscillate
autonomously, its rhythm and phase must be well coordinated with the
state of the physical system using sensory inputs. In this paper we propose
a learning algorithm for synchronizing neural and physical oscillators with
specific phase relationships. Sensory input connections are modified by the
correlation between cellular activities and input signals. Simulations show
that the learning rule can be used for setting sensory feedback connections
to a CPG as well as coupling connections between CPGs.
1
CENTRAL AND SENSORY MECHANISMS IN
LOCOMOTION CONTROL
Patterns of animal locomotion, such as walking, swimming, and fiying, are generated
by recurrent neural networks that are located in segmental ganglia of invertebrates
and spinal cords of vertebrates (Barnes and Gladden, 1985). These networks can
produce basic rhythms of locomotion without sensory inputs and are called central
pattern generators (CPGs). The physical systems of locomotion, such as legs, fins,
and wings combined with physical environments, have their own oscillatory characteristics. Therefore, in order to realize efficient locomotion, the frequency and
the phase of oscillation of a CPG must be well coordinated with the state of the
physical system. For example, the bursting patterns of motoneurons that drive a
leg muscle must be coordinated with the configuration of the leg, its contact with
the ground, and the state of other legs.
109
110
Doya and Yoshizawa
The oscillation pattern of a ePG is largely affected by proprioceptive inputs. It has
been shown in crayfish (Siller et al., 1986) and lamprey (Grillner et aI, 1990) that the
oscillation of a ePG is entrained by cyclic stimuli to stretch sensory neurons over a
wide range of frequency. Both negative and positive feedback pathways are found in
those systems. Elucidation of the function of the sensory inputs to CPGs requires
computational studies of neural and physical dynamical systems. Algorithms for
the learning of rhythmic patterns in recurrent neural networks have been derived by
Doya and Yoshizawa (1989), Pearlmutter (1989), and Williams and Zipser (1989).
In this paper we propose a learning algorithm for synchronizing a neural oscillator
to rhythmic input signals with a specific phase relationship.
It is well known that a coupling between nonlinear oscillators can entrainment their
frequencies. The relative phase between oscillators is determined by the parameters
of coupling and the difference of their intrinsic frequencies. For example, either
in-phase or anti-phase oscillation results from symmetric coupling between neural
oscillators with similar intrinsic frequencies (Kawato and Suzuki, 1980). Efficient
locomotion involves subtle phase relationships between physical variables and motor
commands. Accordingly, our goal is to derive a learning algorithm that can finely
tune the sensory input connections by which the relative phase between physical
and neural oscillators is kept at a specific value required by the task.
2
LEARNING OF SYNCHRONIZATION
We will deal with the following continuous-time model of a CPG network.
d e s
Ti
dtXi(t) = -Xi(t) +
L Wijgj(Xj(t)) + L Vi1:yA:(t) ,
(1)
j=1
1:=1
where Xi(t) and gi(Xi(t)) (i = 1, ... , C) represent the states and the outputs ofCPG
neurons and Y1:(t) (k = 1, ... , S) represents sensory inputs. We assume that the
connection weights W = {Wij} are already established so that the network oscillates
without sensory inputs. The goal oflearning is to find the input connection weights
V
{Vij} that make the network state x(t) (Xl (t), ... ,xc(t))t entrained to the
input signal yet) = (Yl(t), .. . ,Ys(t))t with a specific relative phase.
=
2.1
=
AN OBJECTIVE FUNCTION FOR PHASE-LOCKING
The standard way to derive a learning algorithm is to find out an objective function
to be minimized. If we can approximate the waveforms of Xi(t) and Y1:(t) by sine
waves, a linear relationship
x(t) = Py(t)
specifies a phase-locked oscillation of x(t) and Yet). For example, if we have Yl =
sin wt and Y2 = cos wt, then a matrix P = (~
specifies Xl = v'2 sinewt +1r /4) and
X2 = 2 sine wt + 1r /3). Even when the waveforms are not sinusoidal, minimization of
fi)
an objective function
1 c
1
E(t)
= "2l1x(t) -
py(t)1I2
s
="2 2: {Xi(t) - L Pi1:Y1:(t)}2
i=l
1:=1
(2)
Adaptive Synchronization of Neural and Physical Oscillators
determines a specific relative phase between x(t) and y(t). Thus we call P = {Pik}
a phase-lock matrix.
2.2
LEARNING PROCEDURE
Using the above objective function, we will derive a learning procedure for phaselocked oscillation of x(t) and y(t). First, an appropriate phase-lock matrix P is
identified while the relative phase between x(t) and y(t) changes gradually in time.
Then, a feedback mechanism can be applied so that the network state x(t) is kept
close to the target waveform P y(t).
Suppose we actually have an appropriate phase relationship between x(t) and y(t),
then the phase-lock matrix P can be obtained by gradient descent of E(t) with
respect to PH: as follows (Widrow and Stearns, 1985).
d
dtPik
= -TJ
{}E(t)
{}.
P,k
= TJ {Xi(t) -
S
LPijYj(t)}Yk(t).
(3)
j=1
If the coupling between x(t) and y(t) are weak enough, their relative phase changes
in time unless their intrinsic frequencies are exactly equal and the systems are
completely noiseless. By modulating the learning coefficient TJ by some performance
index of the total system, for example, the speed of locomotion, it is possible to
obtain a matrix P that satisfies the requirement of the task.
Once a phase-lock matrix is derived, we can control x(t) close to Py(t) using the
gradient of E(t) with respect to the network state
{}E(t)
{} X,.()t = Xi(t) -
S
L PikYk(t).
k=1
The simplest feedback algorithm is to add this term to the CPG dynamics as follows.
d e s
Ti dtXi(t) = -Xi(t) + L Wijgj(Xj(t)) - O'{Xi(t) - LPikYk(t)}.
k=1
j=1
The feedback gain 0' (> 0) must be set small enough so that the feedback term
does not destroy the intrinsic oscillation of the CPG. In that case, by neglecting the
small additional decay term O'Xi(t), we have
Tj
d e s
dt Xi(t) -Xj(t) + L Wijgj(Xj (t)) + L O'PikYk(t),
=
j=1
k=1
which is equivalent to the equation (1) with input weights Vik
= O'Pik.
(4)
111
112
Doya and Yoshizawa
3
DELAYED SYNCHRONIZATION
We tested the above learning scheme on a delayed synchronization task; to find
coupling weights between neural oscillators so that they synchronize with a specific
time delay. We used the following coupled CPG model.
c
c
Tdd xi(t) = -xi(t) + L wijyj(t) + ~ Lpi1:y~-n(t),
t
.
J=1
1:=1
yi(t)
= g(xi(t)),
(5)
(i = 1, . .. , C),
=
where superscripts denote the indices of two CPGs (n 1,2). The goal of learning
was to synchronize the waveforms yHt) and y~(t) with a time delay ~T. We used
z(t) = -Iy~(t - ~T) - y~(t)1
as the performance index. The learning coefficient 7] of equation (3) was modulated
by the deviation of z(t) from its running average z(t) using the following equations.
7](t)
= 7]0 {z(t) -
d
z(t)},
Ta dt z(t)
= -z(t) + z(t).
(6)
a
.....
y2
0.0
4. 0
8. 0
12. 0
16. 0
b
20. 0
d
24. 0
28. 0
32. 0
....
y1
y2~rvl\?
O. 0
4. 0
8. 0
12. 0
0.'-;;'O---;4~:0i\""""'""-"8."A'o---:-l-;!-i"""o-~1S: 0
16. 0
c
e
y1 .
y2~y2
0.0
4.0
8.0
12. 0
16. 0
o....'o-----:4:-'-::o::---~8:. 0---:-1-;:1-i-:-0
..
--1~6: 0
Figure 1: Learning of delayed synchronization of neural oscillators. The dotted and
solid curves represent yf(t) and y;(t) respectively. a:without coupling. b:~T = 0.0.
c:~T = 1.0. c:~T = 2.0. d : ~T = 3.0.
Adaptive Synchronization of Neural and Physical Oscillators
First, two CPGs were trained independently to oscillate with sinusoidal waveforms
of period Tl
4.0 and T2
5.0 using continuous-time back-propagation learning
(Doyaand Yoshizawa, 1989). Each CPG was composed of two neurons (C = 2) with
time constants T
1.0 and output functions g()
tanh() . Instead of following
the two step procedure described in the previous section, the network dynamics (5)
and the learning equations (3) and (6) were simulated concurrently with parameters
a = 0.1, '10 = 0.2, and To = 20.0.
=
=
=
=
Figure 1 a shows the oscillation of two CPGs without coupling. Figures 1 b through
e show the phase-locked waveforms after learning for 200 time units with different
desired delay times.
ZERO-LEGGED LOCOMOTION
4
Next we applied the learning rule to the simplest locomotion system that involves a critical phase-lock between the state of the physical system and the motor
command-a zero-legged locomotion system as shown in Figure 2 a.
The physical system is composed of a wheel and a weight that moves back and
forth on a track fixed radially in the wheel. It rolls on the ground by changing its
balance with the displacement of the weight. In order to move the wheel in a given
direction, the weight must be moved at a specific phase with the rotation angle of
the wheel. The motion equations are shown in Appendix.
First, a CPG network was trained to oscillate with a sinusoidal waveform of period
T
1.0 (Doya and Yoshizawa, 1989). The network consisted of one output and
two hidden units (C = 3) with time constants Ti 0.2 and output functions giO =
tanh(). Next, the output of the CPG was used to drive the weight with a force
/max gl(Xl(t?. The position T and the velocity T of the weight and the rotation
angle (cos 0, sin 0) and the angular velocity of the wheel iJ were used as sensory
feedback inputs Yl:(t) (k 1, .. . ,5) after scaling to [-1,1].
=
=
/=
=
In order to eliminate the effect of biases in x(t) and yet), we used the following
learni~g equations.
d
dtPil:
= '1 ((Xi(t) -
S
Xi(t? -
L Pi; (y;(t) -
y;(t?}(Yl:(t) - Yl:(t?,
;=1
d
Ttl:
dt Xi(t) = -Xi(t) + Xi(t),
Ty
dtYl:(t)
d
(7)
=-Yl:(t) + Yl:(t).
The rotation speed of the wheel was employed as the performance index z(t) after
smoothing by the following equation.
T,
d
.
dt z(t) = -z(t) + OCt).
The learning coefficient '1 was modulated by equations (6). The time constants were
Ttl:
4.0, Ty
1.0, T, = 1.0, and To
4.0. Each training run was started from a
random configuration of the wheel and was finished after ten seconds.
=
=
=
113
114
Doya and Yoshizawa
a
sin90
?
cos9O----
9~
b
pos
vel
cos
SID
rot
0.0
/'
-0.5
,
,
,
,
,
1.0
2.0
3.0
4.0
5.0
/'
,perle6.0 0.0
0.0
,
,
1.0
2.0
/'
;-=
5.0
/'
/'
3. 0
/'
4.0
6.0
0.5
c
pos "------'
vel
cos
sm
~
bidS
O. 0
-0.5
/'
1. 0
2. 0
/'
3. 0
/'
4. 0
/'
5. 0
:r----,. ._. '.___,'-,- - -' , . - - '-:-'
6. 0 O. 0
_ , - 1 - '_-::-I'
1. 0
2. 0
3. 0
0.0
Figure 2: Learning of zero-legged locomotion.
4. 0
5. 0
6. 0
0.5
Adaptive Synchronization of Neural and Physical Oscillators
Figure 2 b is an example of the motion of the wheel without sensory feedback.
The rhythms of the CPG and the physical system were not entrained to each other
and the wheel wandered left and right. Figure 2 c shows an example of the wheel
motion after 40 runs of training with parameters Tlo = 0.1 and (}' = 0.2. At first, the
oscillation of the CPG was slowed down by the sensory inputs and then accelerated
with the rotation of the wheel in the right direction.
We compared the patterns of sensory input connections made after learning with
wheels of different sizes. Table 1 shows the connection weights to the output unit.
The positive connection from sin 0 forces the weight to the right-hand side of the
wheel and stabilize clockwise rotation. The negative connection from cos 0 with
smaller radius fastens the rhythm of the CPG when the wheel rotates too fast and
the weight is lifted up. The positive input from r with larger radius makes the
weight stickier to both ends of the track and slows down the rhythm of the CPG.
Table 1: Sensory input weights to the output unit (Plk; k = 1, ... ,5).
radius
2cm
4cm
6cm
8cm
10cm
5
r
r
0.15
0.28
0.67
0.70
0.90
-0.53
-0.55
-0.21
-0.33
-0.12
cosO
-1.35
-1.09
-0.41
-0.40
-0 .30
sinO
1.32
1.22
0.98
0.92
0.93
0
0.07
0.01
0.00
0.03
-0.02
DISCUSSION
The architectures of CPGs in lower vertebrates and invertebrates are supposed to
be determined by genetic information. Nevertheless, the wayan animal utilizes the
sensory inputs must be adaptive to the characteristics of the physical environments
and the changing dimensions of its body parts.
Back-propagation through forward models of physical systems can also be applied
to the learning of sensory feedback (Jordan and Jacobs, 1990). However, learning of
nonlinear dynamics of locomotion systems is a difficult task; moreover, multi-layer
back-propagation is not appropriate as a biological model of learning. The learning
rule (7) is similar to the covariance learning rule (Sejnowski and Stanton, 1990),
which is a biological model of long term potentiation of synapses.
Acknowledgements
The authors thank Allen Selverston, Peter Rowat, and those who gave comments
to our poster at NIPS Conference. This work was partly supported by grants from
the Ministry of Education, Culture, and Science of Japan.
115
116
Doya and Yoshizawa
References
Barnes, W. J. P. & Gladden, M. H. (1985) Feedback and Motor Control in Invertebrates and Vertebrates. Beckenham, Britain: Croom Helm.
Doya, K. & Yoshizawa, S. (1989) Adaptive neural oscillator using continuous-time
back-propagation learning. Neural Networks, 2, 375-386.
Grillner, S. & Matsushima, T. (1991) The neural network underlying locomotion in
Lamprey-Synaptic and cellular mechanisms. Neuron, 7(July), 1-15.
Jordan, M. I. & Jacobs, R. A. (1990) Learning to control an unstable system with
forward modeling. In Touretzky, D. S. (ed.), Advances in Neural Information Processing Systems 2. San Mateo, CA: Morgan Kaufmann.
Kawato, M. & Suzuki, R. (1980) Two coupled neural oscillators as a model of the
circadian pacemaker. Journal of Theoretical Biology, 86, 547-575.
Pearlmutter, B. A. (1989) Learning state space trajectories in recurrent neural networks. Neural Computation, 1, 263-269.
Sejnowski, T. J. & Stanton, P. K. (1990) Covariance storage in the Hippocampus.
In Zornetzer, S. F. et aI. (eds.), An Introduction to Neural and Electronic Networks,
365-377. San Diego, CA: Academic Press.
Siller, K. T., Skorupski, P., Elson, R. C., & Bush, M. H. (1986) Two identified
afferent neurones entrain a central locomotor rhythm generator. Nature, 323, 440443.
Widrow, B. & Stearns, S. D. (1985) Adaptive Signal Processing. Englewood Cliffs,
NJ: Prentice Hall.
Williams, R. J. & Zipser, D. (1989) A learning algorithm for continually running
fully recurrent neural networks. Neural Computation, 1, 270-280.
Appendix
The dynamics of the zero-legged locomotion system:
..
mr
=
.f.(1
JO
+
mR2 sin2 0)
(0
10
- mgc cos
+
mRsin 2 0(r+RcosO?
10
? Ov+2mr(r+RcosO)0'
0'2
10
+mr ,
+m R sm
-loR sin 0 + mgcsinO(r + RcosO) - (v + 2mr(r + RcosO?O,
100
10
Imax
10
1+ MR2
g(Xl(t? -
ur 3 - /Jr,
+ m(r + RcoSO)2.
Parameters: the masses of the weight m = 0.2[kg) and the wheel M = 0.8[kg);
the radius of the wheel R
0.02throughO.l[m)j the inertial moment of the wheel
I
M R2 j the maximum force to the weight 1max
5[N) j the stiffness of the
3
limiter of the weight u
20/ R3 [N/m ); the damping coefficients of the weight
motion /J 0.2/ R [N/(m/s?) and the wheel rotation v 0.05(M +m)R [N/(rad/s?).
=
=t
=
=
=
=
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4,827 | 5,370 | Efficient Partial Monitoring with Prior Information
Hastagiri P Vanchinathan
Dept. of Computer Science
ETH Z?urich, Switzerland
[email protected]
G?abor Bart?ok
Dept. of Computer Science
ETH Z?urich, Switzerland
[email protected]
Andreas Krause
Dept. of Computer Science
ETH Z?urich, Switzerland
[email protected]
Abstract
Partial monitoring is a general model for online learning with limited feedback: a
learner chooses actions in a sequential manner while an opponent chooses outcomes.
In every round, the learner suffers some loss and receives some feedback based on
the action and the outcome. The goal of the learner is to minimize her cumulative
loss. Applications range from dynamic pricing to label-efficient prediction to dueling
bandits. In this paper, we assume that we are given some prior information about the
distribution based on which the opponent generates the outcomes. We propose BPM, a
family of new efficient algorithms whose core is to track the outcome distribution with
an ellipsoid centered around the estimated distribution. We show that our algorithm
provably enjoys near-optimal regret rate for locally observable partial-monitoring
problems against stochastic opponents. As demonstrated with experiments on synthetic
as well as real-world data, the algorithm outperforms previous approaches, even for very
uninformed priors, with an order of magnitude smaller regret and lower running time.
1 Introduction
We consider Partial Monitoring, a repeated game where in every time step a learner chooses an action while,
simultaneously, an opponent chooses an outcome. Then the player receives a loss based on the action and
outcome chosen. The learner also receives some feedback based on which she can make better decisions in
subsequent time steps. The goal of the learner is to minimize her cumulative loss over some time horizon.
The performance of the learner is measured by the regret, the excess cumulative loss of the learner
compared to that of the best fixed constant action. If the regret scales linearly with the time horizon, it
means that the learner does not approach the performance of the best action, that is, the learner fails to learn
the problem. On the other hand, sublinear regret indicates that the disadvantage of the learner compared
to the best fixed strategy fades with time.
Games in which the learner receives the outcome as feedback after every time step are called online
learning with full information. This special case of partial monitoring has been addressed by (among
others) Vovk [1] and Littlestone and Warmuth [2], who designed the randomized
? algorithm Exponentially
Weighted Averages (EWA) as a learner strategy. This algorithm achieves ?( T logN) expected regret
against any opponent, where N is the number of actions and T is the time horizon. This regret growth
rate is also proven to be optimal.
Another well-studied special case is the so-called multi-armed bandit problem. In this feedback model,
the learner gets to observe the loss she suffered in every time step. That is, the learner does not receive
any information about losses of actions she did not choose. Asymptotically optimal results were obtained
by Audibert and Bubeck
? [3], who designed the1Implicitly Normalized Forecaster (INF) that achieves the
minimax optimal ?( T N) regret growth rate.
1
The algorithm Exp3 due to Auer et al. [4] achieves the same rate up to a logarithmic factor.
1
However, not all online learning problems have one of the above feedback structures. An important
example for a problem that does not fit in either full-information or bandit problems is dynamic pricing.
Consider the problem of a vendor wanting to sell his products to customers for the best possible price.
When a customer comes in, she (secretly) decides on a maximum price she is willing to buy his product
for, while the vendor has to set a price without knowing the customer?s preferences. The loss of the vendor
is some preset constant if the customer did not buy the product, and an ?opportunity loss?, when the
product was sold cheaper than the customer?s maximum. The feedback, on the other hand, is merely an
indicator whether the transaction happened or not.
Dynamic pricing is just one of the practical applications of partial monitoring. Label efficient prediction,
in its simplest form, has three actions: the first two actions are guesses of a binary outcome but provide
no information, while the third action provides information about the outcome for some unit loss as the
price. This can be thought of an abstract form of spam filtering: the first two actions correspond to putting
an email to the inbox and the spam folder, the third action corresponds to asking the user if the email
is spam or not. Another problem that can be cast as partial monitoring is that of dueling bandits [5, 6]
in which the learner chooses a pair of actions in every time step, the loss she suffers is the average loss
of the two actions, and the feedback is which action was ?better?.
In online learning, we distinguish different models of how the opponent generates the outcomes. In the
mildest version called stochastic or stationary memoryless, the opponent chooses an outcome distribution
before the game starts and then selects outcomes in an iid random manner drawn from the chosen
distribution. The oblivious adversarial opponent chooses the outcomes arbitrarily, but without observing the
actions of the learner. This selection method is equivalent to choosing an outcome sequence ahead of time.
Finally, the non-oblivious or adaptive adversarial opponent chooses outcomes arbitrarily with the possibility
of looking at past actions of the learner. In this work, we focus on strategies against stochastic opponents.
Related work Partial monitoring was first addressed in the seminal paper of Piccolboni and Schindelhauer [7], who designed and analyzed the algorithm FeedExp3. The algorithm?s main idea is to maintain
an unbiased estimate for the loss of each action in every time step, and then use these estimates to run
the full-information algorithm (EWA). Piccolboni and Schindelhauer [7] proved an O(T 3/4) upper bound
on the regret (not taking into account the number of actions) for games for which learning is at all possible.
This bound was later improved by Cesa-Bianchi et al. [8] to O(T 2/3), who also constructed an example
of a problem for which this bound is optimal.
From the above bounds it can be seen that not?all partial-monitoring problems have the same level of
difficulty: while bandit problems enjoy an O( T ) regret rate, some partial-monitoring problems have
?(T 2/3) regret. To this end, Bart?ok et al. [9] showed that partial-monitoring problems with finitely
? many
e T ) regret,
actions and outcomes can be classified into four groups: trivial with zero regret, easy with ?(
hard with ?(T 2/3) regret, and hopeless with linear regret. The distinguishing feature between easy and
hard problems is the local observability condition, an algebraic condition on the feedback structure that can
be efficiently verified for any problem. Bart?ok et al. [9] showed the above classification against stochastic
opponents with the help of algorithm BALATON. This algorithm keeps track of estimates of the loss
difference of ?neighboring? action pairs and eliminates actions that are highly likely to be suboptimal.
?
e T ) regret bound for easy
Since then, several algorithms have been proposed that achieve the O(
games [10, 11]. All these algorithms rely on the core idea of estimating the expected loss difference
between pairs of actions.
Our contributions In this paper, we introduce BPM (Bayes-update Partial Monitoring), a new family of
algorithms against iid stochastic opponents that rely on a novel way of the usage of past observations. Our
algorithms maintain a confidence ellipsoid in the space of outcome distributions, and update the ellipsoid
based on observations following a Bayes-like update. Our approach enjoys better empirical performance
and lower computational overhead; another crucial advantage is that we can incorporate prior information
about the outcome distribution by means of an initial confidence ellipsoid. We prove near-optimal minimax
expected regret bounds for our algorithm, and demonstrate its effectiveness on several partial monitoring
problems on synthetic and real data.
2
2 Problem setup
Partial monitoring is a repeated game where in every round, a learner chooses an action while the opponent
chooses an outcome from some finite action and outcome sets. Then, the learner observes a feedback
signal (from some given set of symbols) and suffers some loss, both of which are deterministic functions
of the action and outcome chosen. In this paper we assume that the opponent chooses the outcomes in
an iid stochastic manner. The goal of the learner is to minimize her cumulative loss.
The following definitions and concepts are mostly taken from Bart?ok et al. [9]. An instance of partial
monitoring is defined by the loss matrix L?RN?M and the feedback table H ??N?M , where N and M
are the cardinality of the action set and the outcome set, respectively, while ? is some alphabet of symbols.
That is, if learner chooses action i while the outcome is j, the loss suffered by the learner is L[i,j], and
the feedback received is H[i,j].
For an action 1 ? i ? N, let `i denote the column vector given by the ith row of L. Let ?M denote the
M-dimensional probability simplex. It is easy to see that for any p??M , if we assume that the opponent
uses p to draw the outcomes (that is, p is the opponent strategy), the expected loss of action i can be
expressed as `>
i p.
We measure the performance of an algorithm with its expected regret, defined as the expected difference
of the cumulative loss of the algorithm and that of the best fixed action in hindsight:
RT = max
1?i?N
T
X
(`It ?`i)>p,
t=1
where T is some time horizon, It (t = 1,...,T ) is the action chosen in time step t, and p is the outcome
distribution the opponent uses.
In this paper, we also assume we have some prior knowledge about the outcome distribution in the form
of a confidence ellipsoid: we are given a distribution p0 ? ?M and a symmetric positive semidefinite
covariance matrix ?0 ?RM?M such that the true outcome distribution p? satisfies
q
?
kp0 ?p?k??1 = (p0 ?p?)>??1
0 (p0 ?p )?1.
0
We use the term ?confidence ellipsoid? even though our condition is not probabilistic; we do not assume
that p? is drawn from a Gaussian distribution before the game starts. On the other hand, the way we track
p? is derived by Bayes updates with a Gaussian conjugate prior, hence the name. We would also like
to note that having the above prior knowledge is without loss of generality. For ?large enough? ?0, the
whole probability simplex is contained in the confidence ellipsoid and thus partial monitoring without
any prior information reduces to our setting.
The following definition reveals how we use the loss matrix to recover the structure of a game.
Definition 1 (Cell decomposition, Bart?ok et al. [9, Definition 2]). For any action 1?i?N, let Ci denote
the set of opponent strategies for which action i is optimal:
Ci = p??M : ?1?j ?N,(`i ?`j )>p?0 .
We call the set Ci the optimality cell of action i. Furthermore, we call the set of optimality cells {C1,...,CN }
the cell decomposition of the game.
Every cell Ci is a convex closed polytope, as it is defined by a linear inequality system. Normally, a cell
has dimension M ?1, which is the same as the dimensionality of the probability simplex. It might happen
however, that a cell is of lower dimensionality. Another possible degeneracy is when two actions share
the same cell. In this paper, for ease of presentation, we assume that these degeneracies do not appear.
For an illustration of cell decomposition, see Figure 1(a).
Now that we know the regions of optimality, we can define when two actions are neighbors. Intuitively,
two actions are neighbors if their optimality cells are neighbors in the strong sense that they not only meet
in ?one corner?.
Definition 2 (Neighbors, Bart?ok et al. [9, page 4]). Two actions i and j are neighbors, if the intersection
of their optimality cells Ci ?Cj is an M ?2-dimensional convex polytope.
3
C5
C3
p?
C1
C4
p?
p?
pt
pt?1
pt?1
C2
(a) Cell decomposition
(b) Before the update
(c) After the update
Figure 1: (a) An example for a cell decomposition with M = 3 outcomes. Under the true outcome distribution
p? , action 3 is optimal. Cells C1 and C3 are neighbors, but C2 and C5 are not. (b) The current estimate pt?1 is far away
from the true distribution, the confidence ellipsoid is large. (c) After updating, pt is closer to the truth, the confidence
ellipsoid shrinks.
To optimize performance, the learner?s primary goal is to find out which cell the opponent strategy lies
in. Then, the learner can choose the action associated with that cell to play optimally. Since the feedback
the learner receives is limited, this task of finding the optimal cell may be challenging.
The next definition enables us to utilize the feedback table H.
Definition 3 (Signal matrix, Bart?ok et al. [9, Definition 1]). Let {?1,?2,...,??i }?? be the set of symbols
appearing in row i of the feedback table H. We define the signal matrix Si ?{0,1}?i ?M of action i as
Si[k,j]=I(H[i,j]=?k ).
In words, Si is the indicator table of observing symbols ?1,...,??i under outcomes 1,...,M given that
the action chosen is i. For an example, consider the case when the ith row of H is (a b a c). Then,
!
1 0 1 0
Si = 0 1 0 0 .
0 0 0 1
A very useful property of the signal matrix is that if we represent outcomes with M-dimensional unit vectors,
then Si can be used as a linear transformation to arrive at the unit-vector representation of the observation.
The following condition condition is key in distinguishing easy and hard games:
Definition 4 (Local observability, Bart?ok et al. [9, Definition 3]). Let actions i and j be neighbors. These
actions are said to be locally observable if `i ? `j ? ImSi> ? ImSj>. Furthermore, a game is locally
observable if all of its neighboring action pairs are locally observable.
Bart?ok et
?al. [9] showed that finite stochastic partial-monitoring problems that admit local observability
e T ) minimax expected regret. In the following, we present our new algorithm family that achieves
have ?(
the same regret rate for locally observable games against stochastic opponents.
3
BPM: New algorithms for Partial Monitoring based on Bayes updates
The algorithms we propose can be decomposed into two main building blocks: the first one keeps track
of a belief about the true outcome distribution and provides us with a set of feasible actions in every round.
The second one is responsible for selecting the action to play from this action set. Pseudocode for the
algorithm family is shown in Algorithm 1.
3.1 Update Rule
The method of updating the belief about the true outcome distribution (p?) is based on the idea that
we pretend that the outcomes are generated from a Gaussian distribution with covariance ? = IM and
unknown mean. We also pretend we have a Gaussian prior for tracking the mean. The parameters of
this prior are denoted by p0 (mean) and ?0 (covariance). In every time step, we perform a Gaussian
Bayes-update using the observation received.
4
Algorithm 1 BPM
input: L,H,p0,?0
initialization: Calculate signal matrices Si
for t=1 to T do
Use selection rule (cf., Sec. 3.2) to choose an action It
Observe feedback Yt
?1
?1
>
> ?1
Update posterior: ??1
t =?t?1 +PIt and pt =?t ?t?1 pt?1 +SIt (SIt SIt ) Yt ;
end for
Full-information case As a gentle start, we explain how the update rule would look like if we had full
information about the outcome in each time step. The update in this case is identical with the standard
Gaussian one-step update:
?1
?t =?t?1 ??t?1(?t?1 +I)
?t =?t ??1
t?1 ?t?1 +Xt
?t?1
or equiv.
or equiv.
?1
??1
t =?t?1 +I,
?t =?t?1 +?t(Xt ??t?1).
Here we use subindex t?1 for the prior parameters and t for the posterior parameters in time step t, and
denote by Xt the outcome (observed in this case), encoded by an M-dimensional unit vector.
General case Moving away from the full-information case, we face the problem of not observing the
outcome, only some symbol that is governed by the signal matrix of the action we chose and the outcome
itself. If we denote, as above, the outcome at time step t by an M-dimensional unit vector Xt, then the
observation symbol can be thought of as a unit vector given by Yt = SiXt, provided the chosen action
is i. It follows that what we observe is a linear transformation of the sample from the outcome distribution.
Following the Bayes update rule and assuming we chose action i at time step t, we derive the
corresponding Gaussian posterior given that the likelihood of the observation is ?(Y |p)?N(Sip,SiSi>).
After some algebraic manipulations we get that the posterior
distribution is Gaussian with covariance
?1
?1
?t =(??1
+P
)
and
mean
p
=?
?
p
+P
X
,
where
Pi =Si>(SiSi>)?1Si is the orthogonal
i
t
t
t?1
i
t
t?1
t?1
>
projection to the image space of Si . Note that even though Xt is not observed, the update can be
performed, since PiXi =Si>(SiSi>)?1SiXt =Si>(SiSi>)?1Yt.
A significant advantage of this method of tracking the outcome distribution as opposed to keeping track
of loss difference estimates (as done in previous works), is that feedback from one action can provide
information about losses across all the actions. We believe that this property has a major role in the
empirical performance improvement over existing methods.
An important part in analyzing our algorithm is to show that, despite the fact that the outcome distribution
is not Gaussian, the update tracks the true outcome distribution well. For an illustration of tracking the
true outcome distribution with the above update, see Figures 1(b) and 1(c).
3.2 Selection rules
For selecting actions given the posterior parameters, we propose two versions for the selection rule:
1. Draw a random sample p from the distribution N(pt?1,?t?1), project the sample to the probability simplex, then choose the action that minimizes the loss for outcome distribution p. This
rule is a close relative of Thompson-sampling. We call this version of the algorithm BPM-TS.
2. Use pt?1 and ?t?1 to build a confidence ellipsoid for p?, enumerate all actions whose cells
intersect with this ellipsoid, then choose the action that was chosen the fewest times so far (called
BPM-LEAST).
Our experiments demonstrate the performance of both versions. We analyze version BPM-LEAST.
5
4 Analysis
We now analyze BPM-LEAST that uses the Gaussian updates, and considers a set of feasible actions based
on the criterion that an action is feasible if its optimality cell intersects with the ellipsoid
(
)
r
1
p:kp?ptk??1 ?1+
NlogMT .
t
2
From these feasible actions, it picks the one that has been chosen the fewest times up to time step t. For
this version of the algorithm, the following regret bound holds.
Theorem 1. Given a locally observable partial-monitoring problem (L,H) with prior information p0,?0,
the algorithm BPM-LEAST achieves expected regret
p
RT ?C T Nlog(MT ),
where C is some problem-dependent constant.
The above constant C depends on two main factors, both of them related to the feedback structure.
The first one is the sum of the smallest eigenvalues of SiSi> for every action i. The second is related
to the local observability condition. As the condition says, for every neighboring action pairs i and j,
`i ?`j ?ImSi> ?ImSj>. This means that there exist vij and vji vectors such that `i ?`j =Si>vij ?Sj>vji.
The constant depends on the maximum 2-norm of these vij vectors.
The proof of the theorem is deferred to the supplementary material. In a nutshell, the proof is divided into
two main parts. First we need to show that the update rule?even though the underlying distribution is not
Gaussian?serves as a good tool for tracking the true outcome distribution. After some algebraic manipulations, the problem reduces to a finding a high probability upper bound for norms of weighted sums of noise
vectors. To this end, we used the martingale version of the matrix Hoeffding inequality [12, Theorem 1.3].
Then, we need to show that the confidence ellipsoid shrinks fast enough that if we only choose actions
whose cell intersect with the ellipsoid, we do not suffer a large regret. In the core of proving this, we arrive
at a term where we need to upper bound k`i ? `j k?t , for some neighboring action pairs (i,j), and we
show that due to local observability?
and the speed at which the posterior covariance shrinks, this term
can be upper bounded by roughly 1/ t.
5 Experiments
First, we run extensive evaluations of BPM on various synthetic datasets and compare the performance
against CBP [10] and FeedExp3 [7]. The datasets used in the simulated experiments are identical to the
ones used by Bart?ok et al. [10] and thus allow us to benchmark against the current state of the art. We also
provide results of BPM on a dataset that was collected by Singla and Krause [13] from real interactions
with many users on the Amazon Mechanical Turk (AMT) [14] crowdsourcing platform. We present the
details of the datasets used and the summarize our results and findings in this section.
5.1 Implementation Details
In order to implement BPM, we made the following implementation choices:
1. To use BPM-LEAST (see Section 3.2), we need to recover the current feasible actions. We
do so by sampling multiple (10000) times from concentric Gaussian ellipsoids centred at the
current mean (pt) and collect feasible actions based on which cells the samples lie in. We resort
to sampling for ease of implementation because otherwise we deal with the problem of finding
the intersection between an ellipsoid and a simplex in M-dimensional space.
2. To implement BPM-TS, we draw p from the distribution N(pt?1,?t?1). We then project it
back to the simplex to obtain a probability distribution on the outcome space.
Primarily due to sampling, both our algorithms are computationally more efficient than the existing
approaches. In particular, BPM-TS is ideally suited for real world tasks as it is several orders of magnitude
faster than existing algorithms during all our experiments. In each iteration, BPM-TS only needs to
draw one sample from a multivariate gaussian and does not need any cell decompositions or expensive
computations to obtain high dimensional intersections.
6
40
20
BPM?TS
BPM?Least
10
FeedExp
0
40
BPM?TS
30
10
2
4
6
Time Steps ? 105
8
2
4
6
Time Steps ? 105
8
0
10
FeedExp
18
CBP
14
Regret ? 103
Regret ? 103
Regret ? 103
2
8
6
BPM?TS
4
12
BPM?TS
10
8
CBP
6
BPM?Least
BPM?Least
BPM?TS
4
FeedExp
16
8
10
10
20
10
CBP
5
7.5
Time Steps ? 105
(c) Effects of priors
FeedExp
0
2.5
(b) Minimax (hard)
4
4
2
BPM?Least
2
2
0
0
misspec. p0,wide ?0
0
0
0
10
8
6
5
BPM?Least
(a) Minimax (easy)
6
accurate p0,wide ?0
accurate p0, tight ?0
20
5
0
0
10
Regret ? 103
Minimax Regret ? 104
Minimax Regret ? 103
CBP
25
misspec. p0,tight ?0
CBP
20
FeedExp
30
15
10
30
35
0
2.5
5
Time Steps ? 105
7.5
10
(d) Single opponent (easy).
0
0
2
4
6
Time Steps ? 105
8
0
10
(e) Single opponent (hard).
0.5
1
1.5
2
Time Steps ? 105
2.5
3
(f) Real data (dynamic pricing).
Figure 2: (a,b,d,e) Comparing BPM on the locally non-observable game ((a,d) benign opponent; (c,e) hard opponent).
Hereby, (a,b) show the pointwise maximal regret over 15 scenarios, and (d,e) show the regret against a single opponent
strategy. (c) shows the effect of a misspecified prior. (f) is the performance of the algorithms on the real dynamic
pricing dataset.
5.2 Simulated dynamic pricing games
Dynamic pricing is a classic example of partial monitoring (see the introduction), and we compare the
performance of the algorithms on the small but not locally observable game described in Bart?ok et al.
[10]. The loss matrix and feedback tables for the dynamic pricing game are given by:
?
?
?
?
0 1 ??? N ?1
y y ??? y
?n y ??? y?
?c 0 ??? N ?2?
?
L= ?
;
H =?
.. ?
? ... . . . . . .
?
? ... . . . . . . ... ?.
.
c
???
c
0
n
???
n
y
Recall that the game models a repeated interaction of a seller with buyers in a market. Each buyer can
either buy the product at the price (signal ?y?) or deny the offer (signal ?n?). The corresponding loss to
the seller is either a known constant c (representing opportunity cost) or the difference between offered
price and the outcome of the customer?s latent valuation of the product (willingness-to-pay). A similar
game models procurement processes as well. Note that this game does not satisfy local observability.
While our theoretical results require this condition, in practice, if the opponent does not intentionally select
harsh regions on the simplex, BPM remains applicable. Under this setting, expected individual regret is a
reasonable measure of performance. That is, we measure the expected regret for fixed opponent strategies.
We also consider the minimax expected regret, which measures worst-case performance (pointwise
maximum) against multiple opponent strategies.
Benign opponent While the dynamic pricing game is not locally observable in general, certain opponent
strategies are easier to compete with than others. Specifically, if the stochastic opponent chooses an
outcome distribution that is away from the intersection of the cells that do not have local observability, the
learning happens in ?non-dangerous? or benign regions. We present results under this setting for simulated
dynamic pricing with N = M = 5. The results shown in Figures 2(a) and 2(d) illustrate the benefits of
both variants of BPM over previous approaches. We achieve an order of magnitude reduction in the regret
suffered w.r.t. both the minimax and the individual regret.
7
Harsh opponent For the same problem, with opponent chooses close to the boundary of the cells of
two non-locally observable actions, the problem becomes harder. Still, BPM dramatically outperforms
the baselines and suffers very little regret as shown in Figures 2(b) and 2(e).
Effect of the prior We study the effects of a misspecified prior in Figure 2(c). As long as the initial
confidence interval specified by the prior covariance is large enough to contain the opponent?s distribution,
an incorrectly specified prior mean does not have an adverse effect on the performance of BPM. As
expected, if the prior confidence ellipse used by BPM does not contain the opponent?s outcome distribution,
however, the regret grows linear in time. Further, if the prior is very informative (accurately specified
prior mean and tight confidence ellipse), very little regret is suffered.
5.3 Results on real data
Dataset description We simulate a procurement game based on real data. Parameter estimation was
done by posting a Human Intelligence Task (HIT) on the Amazon Mechanical Turk (AMT) platform.
Motivated by an application in viral marketing, users were asked about the price they would accept for
(hypothetically) letting us post promotional material to their friends on a social networking site. The survey
also collected features like age, geographic region, number of friends in the social network, activity levels
(year of joining, time spent per day etc.). Note that since the HIT was just a survey and the questions
were about a hypothetical scenario, participants had no incentives to misreport their responses. Complete
responses were collected from approx. 800 participants. See [13] for more details.
The procurement game We simulate a procurement auction by playing back these responses offline.
The game is very similar in structure to dynamic pricing, with the optimal action being the best fixed price
that maximized the marketer?s value or equivalently, minimized the loss. We sampled iid from the survey
data and perturbed the samples slightly to simulate a stream of 300000 potential users. At each iteration,
we simulate a user with a private valuation generated as a function of her attributes. We discretized the
offer prices and the private valuations to be one of 11 values and set the opportunity cost of losing a user
due to low pricing to be 0.5. Thus we recover a partial monitoring game with 11 actions and 11 outcomes
with a 0/1 feedback matrix.
Results We present the results of our evaluation on this dataset in Figure 2(f). Notice that although the
game is not locally observable, the outcome distribution does not seem to be in a difficult region of the cell
decomposition as the adaptive algorithms (CBP and both versions of BPM) perform well. We note that
the total regret suffered by BPM-LEAST is a factor of 10 lower than the regret achieved by CBP on this
dataset. The plots are averaged over 30 runs of the competing algorithms on the stream. To the best of our
knowledge, this is the first time partial monitoring has been evaluated on a real world problem of this size.
6 Conclusions and future work
We introduced a new family of algorithms for locally observable partial-monitoring problems against
stochastic opponents. We also enriched the model of partial monitoring with the possibility of incorporating
prior information about the outcome distribution in the form of a confidence ellipsoid. The new insight
of our approach is that instead of tracking loss differences, we explicitly track the true outcome distribution.
This approach not only eases computational overhead but also helps achieve low regret by being able
to transfer information between actions. In particular, BPM-TS runs orders of magnitude faster than any
existing algorithms, opening the path for the model of partial monitoring to be applied on realistic settings
involving large numbers of actions and outcomes.
Future work includes extending our method for adversarial opponents. Bart?ok [11] already uses the idea
of tracking the true outcome distribution with the help of a confidence parallelotope, which is rather close
to our approach, but has the same shortcomings as other algorithms that track loss differences: it can not
transfer information between actions. Extending our results to problems with large action and outcome
spaces is also an important direction: if we have some prior information about the similarities between
outcomes and/or actions, we have a chance for a reasonable regret.
Acknowledgments This research was supported in part by SNSF grant 200021 137971, ERC StG
307036 and a Microsoft Research Faculty Fellowship.
8
References
[1] V. G. Vovk. Aggregating strategies. In COLT, pages 371?386, 1990.
[2] Nick Littlestone and Manfred K. Warmuth. The weighted majority algorithm. Inf. Comput., 108
(2):212?261, 1994.
[3] Jean-Yves Audibert and S?ebastien Bubeck. Minimax policies for adversarial and stochastic bandits.
In COLT, 2009.
[4] Peter Auer, Nicol`o Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic
multiarmed bandit problem. SIAM J. Comput., 32(1):48?77, 2002.
[5] Yisong Yue, Josef Broder, Robert Kleinberg, and Thorsten Joachims. The K-armed dueling bandits
problem. Journal of Computer and System Sciences, 78(5):1538?1556, 2012.
[6] Nir Ailon, Thorsten Joachims, and Zohar Karnin. Reducing dueling bandits to cardinal bandits.
arXiv preprint arXiv:1405.3396, 2014.
[7] Antonio Piccolboni and Christian Schindelhauer. Discrete prediction games with arbitrary feedback
and loss. In COLT/EuroCOLT, pages 208?223, 2001.
[8] Nicol`o Cesa-Bianchi, G?abor Lugosi, and Gilles Stoltz. Regret minimization under partial monitoring.
Math. Oper. Res., 31(3):562?580, 2006.
[9] G?abor Bart?ok, D?avid P?al, and Csaba Szepesv?ari. Minimax regret of finite partial-monitoring games
in stochastic environments. Journal of Machine Learning Research - Proceedings Track (COLT),
19:133?154, 2011.
[10] G?abor Bart?ok, Navid Zolghadr, and Csaba Szepesv?ari. An adaptive algorithm for finite stochastic
partial monitoring. In Proceedings of the 29th International Conference on Machine Learning, ICML
2012, Edinburgh, Scotland, UK, June 26 - July 1, 2012, 2012.
[11] G?abor Bart?ok. A near-optimal algorithm for finite partial-monitoring games against adversarial
opponents. In COLT 2013 - The 26th Annual Conference on Learning Theory, June 12-14, 2013,
Princeton University, NJ, USA, pages 696?710, 2013.
[12] Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational
Mathematics, 12(4):389?434, 2012.
[13] Adish Singla and Andreas Krause. Truthful incentives in crowdsourcing tasks using regret
minimization mechanisms. In International World Wide Web Conference (WWW), 2013.
[14] Amazon Mechanical Turk platform. URL https://www.mturk.com.
9
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4,828 | 5,371 | Nonparametric Bayesian inference on multivariate
exponential families
William Vega-Brown, Marek Doniec, and Nicholas Roy
Massachusetts Institute of Technology
Cambridge, MA 02139
{wrvb, doniec, nickroy}@csail.mit.edu
Abstract
We develop a model by choosing the maximum entropy distribution from the
set of models satisfying certain smoothness and independence criteria; we show
that inference on this model generalizes local kernel estimation to the context of
Bayesian inference on stochastic processes. Our model enables Bayesian inference in contexts when standard techniques like Gaussian process inference are
too expensive to apply. Exact inference on our model is possible for any likelihood function from the exponential family. Inference is then highly efficient,
requiring only O (log N ) time and O (N ) space at run time. We demonstrate our
algorithm on several problems and show quantifiable improvement in both speed
and performance relative to models based on the Gaussian process.
1
Introduction
Many learning problems can be formulated in terms of inference on predictive stochastic models.
These models are distributions p(y|x) over possible observation values y drawn from some observation set Y, conditioned on a known input value x from an input set X . The supervised learning
problem is then to infer a distribution p(y|x? , D) over possible observations for some target input
x? , given a sequence of N independent observations D = {(x1 , y 1 ), . . . , (xN , y N )}.
It is often convenient to associate latent parameters ? ? ? with each input x, where p(y|?) is a
known likelihood function. By inferring a distribution over target parameters ? ? associated with x? ,
we can infer a distribution over y.
Z
p(y|x? , D) =
d? ? p(y|? ? )p(? ? |x? , D)
(1)
?
For instance, regression problems can be formulated as the inference of an unknown but deterministic underlying function ?(x) given noisy observations, so that p(y|x) = N (y; ?(x), ? 2 ), where
? 2 is a known noise variance. If we can specify a joint prior over the parameters corresponding to
different inputs, we can infer p(? ? |x? , D) using Bayes? rule.
"N
#
Z
Y
p(? ? |x? , D) ?
d? i p(y i |? i ) p(? 1:N , ? ? |x? , x1:N )
(2)
?N
i=1
The notation x1:N indicates the sample inputs {x1 , . . . , xN }; this model is depicted graphically in
figure 1a. Although the choice of likelihood is often straightforward, specifying a prior can be more
difficult. Ideally, we want a prior which is expressive, in the sense that it can accurately capture all
prior knowledge, and which permits efficient and accurate inference.
A powerful motivating example for specifying problems in terms of generative models is the Gaussian process [1], which specifies the prior p(? 1:N |x1:N ) as a multivariate Gaussian with a covariance
parameterized by x1:N . This prior can express complex and subtle relationships between inputs and
1
xn
xn
?n
?n
yn
N
yn
?
x?
i, j
??
y?
(b) Inference model
(a) Stochastic process
Figure 1: Figure 1a models any stochastic process with fully connected latent parameters. Figure 1b is our
approximate model, used for inference; we assume that the latent parameters are independent given the target
parameters. Shaded nodes are observed.
observations, and permits efficient exact inference for a Gaussian likelihood with known variance.
Extensions exist to perform approximate inference with other likelihood functions [2, 3, 4, 5].
However, the assumptions of the Gaussian process are not the only set of reasonable assumptions,
and are not always appropriate. Very large datasets require complex sparsification techniques to be
computationally tractable [6]. Likelihood functions with many coupled parameters, such as the parameters of a categorical distribution or of the covariance matrix of a multivariate Gaussian, require
the introduction of large numbers of latent variables which must be inferred approximately. As an
example, the generalized Wishart process developed by Wilson and Ghahramani [7] provides a distribution over covariance matrices using a sum of Gaussian processes. Inference of the the posterior
distribution over the covariance can only be performed approximately: no exact inference procedure
is known.
Historically, an alternative approach to estimation has been to use local kernel estimation techniques
[8,
are often developed from a weighted parameter likelihood of the form p(?|D) =
Q 9, 10], which
wi
i p(y i |?) . Algorithms for determining the maximum likelihood parameters of such a model
are easy to implement and very fast in practice; various techniques, such as dual trees [11] or the
improved fast Gauss transform [12] allow the computation of kernel estimates in logarithmic or
constant time. This choice of model is often principally motivated by the computational convenience
of resulting algorithms. However, it is not clear how to perform Bayesian inference on such models.
Most instantiations instead return a point estimate of the parameters.
In this paper, we bridge the gap between the local kernel estimators and Bayesian inference. Rather
than perform approximate inference on an exact generative model, we formulate a simplified model
for which we can efficiently perform exact inference. Our simplification is to choose the maximum
entropy distribution from the set of models satisfying certain smoothness and independence criteria. We then show that for any likelihood function in the exponential family, our process model
has a conjugate prior, which permits us to perform Bayesian inference in closed form. This motivates many of the local kernel estimators from a Bayesian perspective, and generalizes them to new
problem domains. We demonstrate the usefulness of this model on multidimensional regression
problems with coupled observations and input-dependent noise, a setting which is difficult to model
using Gaussian process inference.
2
The kernel process model
Given a likelihood function, a generative model can be specified by a prior p(? 1:N , ? ? |x? , x1:N ).
For almost all combinations of prior and likelihood, inference is analytically intractable. We relax
the requirement that the model be generative, and instead require only that the prior be well-formed
for a given query x? . To facilitate inference, we make the strong assumption that the latent training
parameters ? 1:N are conditionally independent given the target parameters ? ? .
"N
#
Y
?
?
?
?
p(? 1:N , ? |x1:N , x ) =
p(? i |? , xi , x ) p(? ? |x? )
(3)
i=1
This restricted structure is depicted graphically in figure 1b. Essentially, we assume that interactions between latent parameters are unimportant relative to interactions between the latent and target
parameters; this will allow us to build models based on exponential family likelihood functions
which will permit exact inference. Note that models with this structure will not correspond exactly
to probabilistic generative models; the prior distribution over the latent parameters associated with
the training inputs varies depending on the target input. Instead of approximating inference on our
2
model, we make our approximation at the stage of model selection; in doing so, we enable fast,
exact inference. Note that the class of models with the structure of equation (3) is quite rich, and
as we demonstrate in section 3.2, performs well on many problems. We discuss in section 4 the
ramifications of this assumption and when it is appropriate.
This assumption is closely related to known techniques for sparsifying Gaussian process inference.
Qui?nonero-Candela and Rasmussen [6] provide a summary of many sparsification techniques, and
describe which correspond to generative models. One of the most successful sparsification techniques, the fully independent training conditional approximation of Snelson [13], assumes all training examples are independent given a specified set of inducing inputs. Our assumption extends this
to the case of a single inducing input equal to the target input.
Note that by marginalizing the parameters ? 1:N , we can directly relate the observations y 1:N to the
target parameters ? ? . Combining equations (2) and (3),
"N Z
#
Y
? ?
?
p(? |x , D) ?
d? i p(y i |? i )p(? i |? , xi , x? ) p(? ? |x? )
(4)
i=1
?
and marginalizing the latent parameters ? 1:N , we observe that the posterior factors into a product
over likelihoods p(y i |? ? , x, x? ) and a prior over ? ? .
"N
#
Y
?
?
=
p(y i |? , xi , x ) p(? ? |x? )
(5)
i=1
Note that we can equivalently specify either p(?|? ? , x, x? ) or p(y|? ? , x, x? ), without loss of generality. In other words, we can equivalently describe the interaction between input variables either
in the likelihood function or in the prior.
2.1
The extended likelihood function
By construction, we know the distribution p(y i |? i ). After making the transformation to equation (5), much of the problem of model specification has shifted to specifying the distribution
p(y i |? ? , xi , x? ). We call this distribution the extended likelihood distribution. Intuitively, these distributions should be related; if x? = xi , then we expect ? i = ? ? and p(y i |? ? , xi , x? ) = p(y i |? i ).
We therefore define the extended likelihood function in terms of the known likelihood p(y i |? i ).
Typically, we prefer smooth models: we expect similar inputs to lead to a similar distribution over
outputs. In the absence of a smoothness constraint, any inference method can perform arbitrarily
poorly [14]. However, the notion of smoothness is not well-defined in the context of probability
distributions. Denote g(y i ) = p(y i |? ? , xi , x? ), and f (y i ) = p(y i |? i ). We can formalize a smooth
model as one in which the information divergence of the likelihood distribution f from the extended
likelihood distribution g is bounded by some function ? : X ? X ? R+ .
DKL (gkf ) ? ?(x? , xi )
(6)
Since the divergence is a premetric, ?(?, ?) must also satisfy the properties of a premetric: ?(x, x) =
0 ?x and ?(x1 , x2 ) ? 0 ?x1 , x2 . For example, if X = Rn , we may draw an analogy to Lipschitz
continuity and choose ?(x1 , x2 ) = Kkx1 ? x2 k, with K a positive constant. The class of models
with bounded divergence has the property that g ? f as x0 ? x, and it does so smoothly provided
?(?, ?) is smooth. Note that this bound is a constraint on possible g, not an objective to be minimized;
in particular, we do not minimize the divergence between g and f to develop an approximation, as is
common in the approximate inference literature. Note also that this constraint has a straightforward
information-theoretic interpretation; ?(x1 , x2 ) is a bound on the amount of information we would
lose if we were to assume an observation y 1 were taken at x2 instead of at x1 .
The assumptions of equations (3) and (6) define a class of models for a given likelihood function,
but are insufficient for specifying a well-defined prior. We therefore use the principle of maximum
entropy and choose the maximum entropy distribution from among that class. In our attached supporting material, we prove the following.
Theorem 1 The maximum entropy distribution g satisfying DKL (gkf ) ? ?(x? , x) has the form
?
g(y) ? f (y)k(x ,x)
(7)
where k : X ? X ? [0, 1] is a kernel function which can be uniquely determined from ?(?, ?) and
f (?).
3
There is an equivalence relationship between functions k(?, ?) and ?(?, ?); as either is uniquely determined by the other, it may more convenient to select a kernel function than a smoothness bound,
and doing so implies no loss in generality or correctness. Note it is neither necessary nor sufficient
that the kernel function k(?, ?) be positive definite. It is necessary only that k(x, x) = 1?x and that
k(x, x0 ) ? [0, 1]?x, x0 . This includes the possibility of asymmetric kernel functions. We discuss
in the attached supporting material the mapping between valid kernel functions k(?, ?) and bounding
functions ?(?, ?).
It follows from equation (7) that the maximum entropy distribution satisfying a bound of ?(x, x? )
on the divergence of the observation distribution p(y|? ? , x, x? ) from the known distribution
p(y|?, x, x? ) = p(y|?) is
?
p(y|? ? , x, x? ) ? p(y|?)k(x,x ) .
(8)
By combining equations (5) and (6), we can fully specify a stochastic model with a likelihood
p(y|?), a pointwise marginal prior p(?|x), and a kernel function k : X ? X ? [0, 1]. To perform
inference, we must evaluate
p(?|x, D) ?
N
Y
p(y i |?)k(x,xi ) p(?|x)
(9)
i=1
This can be done in closed form if we can normalize the terms on the right side of the equality.
In certain limiting cases with uninformative priors, our model can be reduced to known frequentist
estimators. For instance, if we employ an uninformative prior p(?|x) ? 1 and choose the maximum? ? = arg max p(? ? |x? , D), we recover the weighted maximumlikelihood target parameters ?
likelihood estimator, detailed by Wang [15]. If the function k(x, x0 ) is local, in the sense that it goes
to zero if the distance kx ? x0 k is large, then choosing maximum likelihood parameter estimates for
an uninformative prior gives the locally weighted maximum-likelihood estimator, described in the
context of regression by Cleveland [16] and for generalized linear models by Tibshirani and Hastie
[10]. However, our result is derived from a Bayesian interpretation of statistics, and we infer a full
distribution over the parameters; we are not limited to a point estimate. The distinction is of both
academic and practical interest; in addition to providing insight into to the meaning of the weighting
function and the validity of the inferred parameters, by inferring a posterior distribution we provide
a principled way to reason about our knowledge and to insert prior knowledge of the underlying
process.
2.2
Kernel inference on the exponential family
Equation (8) is particularly useful if we choose our likelihood model p(y|?) from the exponential
family.
p(y|?) = h(y) exp ? > T (y) ? A(?)
(10)
A member of an exponential family remains in the same family when raised to the power of
k(x, xi ). Because every exponential family has a conjugate prior, we may choose our point-wise
prior p(? ? |x? ) to be conjugate to our chosen likelihood. We denote this conjugate prior p? (?, ?),
where ? and ? are hyperparameters.
p(?|x? ) = p? (?(x? ), ?(x? )) = f (?(x? ), ?(x? )) exp (? ? ?(x? ) ? ?(x? )A(?))
(11)
Therefore, our posterior as defined by equation (9) may be evaluated in closed form.
N
N
X
X
p(? ? |x? , D) = p? (
k(x? , xi )T (y i ) + ?(x? ),
k(x? , xi ) + ?(x? ))
i=1
The prior predictive distribution p(y|x) is given by
Z
p(y|x) = p(y|?)p? (?|?(x? ), ?(x? ))
= h(y)
(12)
i=1
f (?(x? ), ?(x? ))
f (?(x? ) + T (y), ?(x? ) + 1)
4
(13)
(14)
and the posterior predictive distribution is
PN
PN
f ( i=1 k(x? , xi )T (y i ) + ?(x? ), i=1 k(x? , xi ) + ?(x? ))
p(y|x? , D) = h(y) PN
PN
f ( i=1 k(x? , xi )T (y i ) + ?(x? ) + T (y), i=1 k(x? , xi ) + ?(x? ) + 1)
(15)
This is a general formulation of the posterior distribution over the parameters of any likelihood
model belonging to the exponential family. Note that given a function k(x? , x), we may evaluate this
posterior without sampling, in time linear in the number of samples. Moreover, for several choices
of kernels the relevant sums can be evaluates in sub-linear time; a sum over squared exponential
kernels, for instance, can be evaluated in logarithmic time.
3
Local inference for multivariate Gaussian
We now discuss in detail the application of equation (12) to the case of a multivariate Gaussian
likelihood model with unknown mean ? and unknown covariance ?.
p(y|?, ?) = N (y; ?, ?)
(16)
We present the conjugate prior, posterior, and predictive distributions without derivation; see [17],
for example, for a derivation. The conjugate prior for a multivariate Gaussian with unknown mean
and covariance is the normal-inverse Wishart distribution, with hyperparameter functions ?0 (x? ),
?(x? ), ?(x? ), and ?(x? ).
?
?
?
p(?, ?|x ) = N ?; ?0 (x ),
? W ?1 (?; ?(x? ), ?(x? ))
(17)
?(x? )
The hyperparameter functions have intuitive interpretations; ?0 (x? ) is our initial belief of the mean
function, while ?(x? ) is our confidence in that belief, with ?(x? ) = 0 indicating no confidence
in the region near x? , and ?(x? ) ? ? indicating a state of perfect knowledge. Likewise, ?(x? )
indicates the expected covariance, and ?(x? ) represents the confidence in that estimate, much like
?. Given a dataset D, we can compute a posterior over the mean and covariance, represented by
updated parameters ?00 (x? ), ?0 (x? ), ?0 (x? ), and ? 0 (x? ).
?0 (x? ) = ?(x? ) + k(x? )
?(x? )?0 (x? ) + y
?00 (x? ) =
?(x? ) + k(x? )
?0 (x? ) = ?(x? ) + S(x? ) +
? 0 (x? ) = ?(x? ) + k(x? )
(18)
?
?
?(x )k(x )
E(x? )
?(x? ) + k(x? )
where
k(x? ) =
N
X
N
k(x? , xi )
y(x? ) =
i=1
S(x? ) =
N
X
1 X
k(x? , xi )y i
k(x? ) i=1
>
k(x? , xi ) y i ? y(x? ) y i ? y(x? )
(19)
i=1
>
E(x? ) = y(x? ) ? ?0 (x? ) y(x? ) ? ?0 (x? )
The resulting posterior predictive distribution is a multivariate Student-t distribution.
?0 (x? ) + 1
0
?
0
?
?
p(y|x ) = t? 0 (x? ) ?0 (x ), 0 ? 0 ? ? (x )
? (x )? (x )
3.1
(20)
Special cases
Two special cases of the multivariate Gaussian are worth mentioning. First, a fixed, known co?
)
variance ?(x? ) can be described by the hyperparameters ?(x? ) = lim??? ?(x
? . The resulting
posterior distribution is then
1
?
0
?
p(?|x , D) = N ?0 , 0 ? ?(x )
(21)
? (x )
5
with predictive distribution
p(?|x? , D) = N
1 + ?0 (x? )
?
?00 ,
?(x
)
?0 (x? )
(22)
In the limit as ? goes to 0, when the prior is uninformative, the mean and mode of the predictive
distribution approaches the Nadaraya-Watson [8, 9] estimate.
PN
k(x? , xi )yi
?
(23)
?N W (x ) = Pi=1
N
?
i=1 k(x , xi )
The complementary case of known mean ?(x? ) and unknown covariance ?(x? ) is described by the
limit ? ? ?. In this case, the posterior distribution is
N
N
X
X
p(?|x? , D) = W ?1 ?(x? ) +
ki (y i ? ?(x? ))(y i ? ?(x? ))> , ?(x? ) +
ki
(24)
i=1
i=1
with predictive distribution
?
p(y|x ) = t? 0 (x? )
N
X
1
ki (y i ? ?(x? ))(y i ? ?(x? ))>
?(x ), 0 ? ?(x? ) +
? (x )
i=1
?
!
(25)
In the limit as ? goes to 0, the maximum likelihood covariance estimate is
?ML (x? ) =
N
X
ki (y i ? ?(x? ))(y i ? ?(x? ))>
(26)
i=1
which is precisely the result of our prior work [18, 19]. In both cases, our method yields distributions
over parameters, rather than point estimates; moreover, the use of Bayesian inference naturally
handles the case of limited or no available samples.
3.2
Experimental results
We evaluate our approach on several regression problems, and compare the results with alternative nonparametric Bayesian models. In all experiments, we use the squared-exponential kernel
k(y, y 0 ) = exp( 2c ky ? y 0 k2 ). This function meets both the requirements of our algorithm and is
positive-definite and thus a suitable covariance function for models based on the Gaussian process.
We set the kernel scale c by maximum likelihood for each model.
We compare our approach to covariance prediction to the generalized Wishart process (GWP) of [7].
First, we sample a synthetic dataset; the output is a two-dimensional observation set Y = R2 , where
samples are drawn from a zero-mean normal distribution with a covariance that rotates over time.
>
cos(t) ? sin(t)
4 0
cos(t) ? sin(t)
?(t) =
(27)
0 10
sin(t) cos(t)
sin(t) cos(t)
Second, we predict the covariances of the returns on two currency exchanges?the Euro to US dollar,
and the Japanese yen to US dollar?over the past four years. Following Wilson and Ghahramani, we
), where Pt is the exchange rate on day t. Illustrative results are provided
define a return as log( PPt+1
t
in figure 2. To compare these results quantitatively, one natural measure is the mean of the logarithm
of the likelihood of the predicted model given the data.
MLL =
N
1 X 1 > ? ?1
? i)
? (y ? y + log det ?
N i=1 2 i i i
(28)
? i is the maximum likelihood covariance predicted for the ith sample.
Here, ?
In addition to how well our model describes the available data, we may also be interested in how
accurately we recover the distribution used to generate the data. This is a measure of how closely
the inferred ellipses in figure 2 approximate the true covariance ellipses. One measure of the quality
of the inferred distribution is the KL divergence of the inferred distribution from the true distribution
6
?=0.83
?=1.20
?=1.57
?=1.94
?=2.31
5
5
5
5
5
0
?5
?10
?10
0
?5
?5
0
x1
5
10
?10
?10
0
?5
?5
0
5
x1
Ground
Truth
10
?10
?10
x2
10
x2
10
x2
10
x2
10
x2
10
0
?5
?5
0
x1
Inverse Wishart
5
10
?10
?10
0
?5
?5
0
5
x
Generalised Wishart 1Process
10
?10
?10
?5
0
x1
5
10
(a) Synthetic periodic data
0.05
0.05
JPY/USD
2014/1/10
0.05
JPY/USD
2013/3/22
0.05
JPY/USD
2012/6/4
0.05
JPY/USD
2011/8/17
JPY/USD
2010/10/29
0
0
?0.05
?0.05
?0.05?0.025 0 0.025 0.05 ?0.05?0.025
EUR/USD
0
0
?0.05
0.025 0.05 ?0.05?0.025
0
Inverse Wishart
0
?0.05
0.025 0.05 ?0.05?0.025
Generalised Wishart Process
0
0
?0.05
0.025 0.05 ?0.05?0.025 0 0.025 0.05
EUR/USD
(b) Exchange data
Figure 2: Comparison of covariances predicted by our kernel inverse Wishart process and the generalized
Wishart process for the problems described in section 3.2. The true covariance used to generate data is provided
for comparison. The samples used are plotted so that the area of the circle is proportional to the weight assigned
by the kernel. The kernel inverse Wishart process outperforms the generalized Wishart process, both in terms
of the likelihood of the training data, and in terms of the divergence of the inferred distribution from the true
distribution.
used to generate the data. Note we cannot evaluate this quantity on the exchange dataset, as we do
not know the true distribution. We present both the mean likelihood and the KL divergence of both
algorithms, along with running times, in table 1.
By both metrics, our algorithm outperforms the GWP by a significant margin; the running time
advantage of kernel estimation over the GWP is even more dramatic. It is important to note that
running times are difficult to compare, as they depend heavily on implementation and hardware
details; the numbers reported should be considered qualitatively. Both algorithms were implemented
in the MATLAB programming language, with the likelihood functions for the GWP implemented in
heavily optimized c code in an effort to ensure a fair competition. Despite this, the GWP took over
a thousand times longer than our method to generate predictions.
Periodic
Exchange
kNIW
GWP
kNIW
GWP
ttr (s)
0.022
7.08
0.520
15.7
tev (ms)
0.003
0.135
0.020
1.708
MLL
-10.43
-19.79
7.73
7.56
DKL (?
pkp)
0.0138
0.0248
?
?
Table 1: Comparison of the performance of two models of covariance prediction, based on time required to
make predictions at evaluation, the mean log likelihood and the KL divergence between the predicted covariance
and the ground truth covariance.
We next evaluate our approach on heteroscedastic regression problems. First, we generate 100
samples from the distribution described by Yuan and Wahba [20], which has mean ?(x? ) =
2 exp(?30(x? ? 0.25)2 ) + sin(?(x? )2 ) and variance ? 2 (x? ) = exp(2 ? sin(2?x? )). Second,
we test on the motorcycle dataset of Silverman et al. [21]. We compare our approach to a variety
of Gaussian process based regression algorithms, including a standard homoscedastic Gaussian process, the variational heteroscedastic Gaussian process of L?azaro-Gredilla and Titsias [4], and the
maximum likelihood heteroscedastic Gaussian process of Quadrianto et al. [22]. All algorithms are
implemented in MATLAB, using the authors? own code. Running times are presented with the same
caveat as in the previous experiments, and a similar conclusion holds: our method provides results
which are as good or better than methods based upon the Gaussian process, and does so in a fraction
of the time. Figure 3 illustrates the predictions made by our method on the heteroscedastic motor7
100
100
50
50
0
a
a
0
?50
?50
?100
?100
?150
10
15
20
25
30
35
40
45
?150
50
10
15
20
25
t
30
35
40
45
50
t
(a) kNIW
(b) VHGP
Figure 3: Comparison of the distributions inferred using the kernel normal inverse Wishart process and the
variational heteroscedastic Gaussian process to model Silverman?s motorcycle dataset. Both models capture
the time-varying nature of the measurement noise; as is typical, the kernel model is much less smooth and has
more local structure than the Gaussian process model. Both models perform well according to most metrics,
but the kernel model can be computed in a fraction of the time.
cycle dataset of Silverman. For reference, we provide the distribution generated by the variational
heteroscedastic Gaussian process.
Motorcycle
Periodic
kNIW
GP
VHGP
MLHGP
kNIW
GP
VHGP
MLHGP
ttr (s)
0.124
0.52
3.12
2.39
0.68
3.41
26.4
38.3
tev (ms)
2.95
3.52
7.53
5.83
7.94
22
54.4
29.1
NMSE
0.2
0.202
0.202
0.204
0.0708
0.0822
0.0827
0.0827
MLL
-4.04
-4.51
-4.07
-4.03
-2.07
-2.56
-1.85
-2.38
Table 2: Comparison of the performance of various models of heteroscedastic processes, based on time required to train, time required to make predictions at evaluation, the normalized mean squared error, and the
mean log likelihood. Note how the normal-inverse Wishart process obtains performance as good or better than
the other algorithms in a fraction of the time.
4
Discussion
We have presented a family of stochastic models which permit exact inference for any likelihood
function from the exponential family. Algorithms for performing inference on this model include
many local kernel estimators, and extend them to probabilistic contexts. We showed the instantiation
of our model for a multivariate Gaussian likelihood; due to lack of space, we do not present others,
but the approach is easily extended to tasks like classification and counting. The models we develop
are built on a strong assumption of independence; this assumption is critical to enabling efficient
exact inference. We now explore the costs of this assumption, and when it is inappropriate.
First, while the kernel function in our model does not need to be positive definite?or even
symmetric?we lose an important degree of flexibility relative to the covariance functions employed
in a Gaussian process. Covariance functions can express a number of complex concepts, such as a
prior over functions with a specified additive or hierarchical structure [23]; these concepts cannot
be easily formulated in terms of smoothness. Second, by neglecting the relationships between latent
parameters, we lose the ability to extrapolate trends in the data, meaning that in places where data
is sparse we cannot expect good performance. Thus, for a problem like time series forecasting, our
approach will likely be unsuccessful. Our approach is suitable in situations where we are likely to
see similar inputs many times, which is often the case. Moreover, regardless of the family of models
used, extrapolation to regions of sparse data can perform very poorly if the prior does not model the
true process well. Our approach is particularly effective when data is readily available, but computation is expensive; the gains in efficiency due to an independence assumption allow us to scale to
larger much larger datasets, improving predictive performance with less design effort.
Acknowledgements
This research was funded by the Office of Naval Research under contracts N00014-09-1-1052 and
N00014-10-1-0936. The support of Behzad Kamgar-Parsi and Tom McKenna is gratefully acknowledged.
8
References
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9
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4,829 | 5,372 | Fast Kernel Learning for Multidimensional Pattern
Extrapolation
Andrew Gordon Wilson?
CMU
Elad Gilboa?
WUSTL
Arye Nehorai
WUSTL
John P. Cunningham
Columbia
Abstract
The ability to automatically discover patterns and perform extrapolation is an essential quality of intelligent systems. Kernel methods, such as Gaussian processes,
have great potential for pattern extrapolation, since the kernel flexibly and interpretably controls the generalisation properties of these methods. However, automatically extrapolating large scale multidimensional patterns is in general difficult, and developing Gaussian process models for this purpose involves several
challenges. A vast majority of kernels, and kernel learning methods, currently
only succeed in smoothing and interpolation. This difficulty is compounded by
the fact that Gaussian processes are typically only tractable for small datasets, and
scaling an expressive kernel learning approach poses different challenges than
scaling a standard Gaussian process model. One faces additional computational
constraints, and the need to retain significant model structure for expressing the
rich information available in a large dataset. In this paper, we propose a Gaussian
process approach for large scale multidimensional pattern extrapolation. We recover sophisticated out of class kernels, perform texture extrapolation, inpainting,
and video extrapolation, and long range forecasting of land surface temperatures,
all on large multidimensional datasets, including a problem with 383,400 training
points. The proposed method significantly outperforms alternative scalable and
flexible Gaussian process methods, in speed and accuracy. Moreover, we show
that a distinct combination of expressive kernels, a fully non-parametric representation, and scalable inference which exploits existing model structure, are critical
for large scale multidimensional pattern extrapolation.
1
Introduction
Our ability to effortlessly extrapolate patterns is a hallmark of intelligent systems: even with large
missing regions in our field of view, we can see patterns and textures, and we can visualise in our
mind how they generalise across space. Indeed machine learning methods aim to automatically
learn and generalise representations to new situations. Kernel methods, such as Gaussian processes
(GPs), are popular machine learning approaches for non-linear regression and classification [1, 2, 3].
Flexibility is achieved through a kernel function, which implicitly represents an inner product of
arbitrarily many basis functions. The kernel interpretably controls the smoothness and generalisation
properties of a GP. A well chosen kernel leads to impressive empirical performances [2].
However, it is extremely difficult to perform large scale multidimensional pattern extrapolation with
kernel methods. In this context, the ability to learn a representation of the data entirely depends
on learning a kernel, which is a priori unknown. Moreover, kernel learning methods [4] are not
typically intended for automatic pattern extrapolation; these methods often involve hand crafting
combinations of Gaussian kernels (for smoothing and interpolation), for specific applications such
as modelling low dimensional structure in high dimensional data. Without human intervention,
the vast majority of existing GP models are unable to perform pattern discovery and extrapolation.
?
Authors contributed equally.
1
While recent approaches such as [5] enable extrapolation on small one dimensional datasets, it is
difficult to generalise these approaches for larger multidimensional situations. These difficulties
arise because Gaussian processes are computationally intractable on large scale data, and while
scalable approximate GP methods have been developed [6, 7, 8, 9, 10, 11, 12, 13], it is uncertain
how to best scale expressive kernel learning approaches. Furthermore, the need for flexible kernel
learning on large datasets is especially great, since such datasets often provide more information to
automatically learn an appropriate statistical representation.
In this paper, we introduce GPatt, a flexible, non-parametric, and computationally tractable approach
to kernel learning for multidimensional pattern extrapolation, with particular applicability to data
with grid structure, such as images, video, and spatial-temporal statistics. Specifically:
? We extend fast Kronecker-based GP inference (e.g., [14, 15]) to account for non-grid data. Our
experiments include data where more than 70% of the training data are not on a grid. Indeed most
applications where one would want to exploit Kronecker structure involve missing and non-grid
data ? caused by, e.g., water, government boundaries, missing pixels and image artifacts. By
adapting expressive spectral mixture kernels to the setting of multidimensional inputs and KroP +1
necker structure, we achieve exact inference and learning costs of O(P N P ) computations and
2
O(P N P ) storage, for N datapoints and P input dimensions, compared to the standard O(N 3 )
computations and O(N 2 ) storage associated with GPs.
? We show that i) spectral mixture kernels (adapted for Kronecker structure); ii) scalable inference based on Kronecker methods (adapted for incomplete grids); and, iii) truly non-parametric
representations, when used in combination (to form GPatt) distinctly enable large-scale multidimensional pattern extrapolation with GPs. We demonstrate this through a comparison with
various expressive models and inference techniques: i) spectral mixture kernels with arguably the
most popular scalable GP inference method (FITC) [10]; ii) a flexible and efficient recent spectral
based kernel learning method (SSGP) [6]; and, iii) the most popular GP kernels with Kronecker
based inference.
? The information capacity of non-parametric methods grows with the size of the data. A truly
non-parametric GP must have a kernel that is derived from an infinite basis function expansion.
We find that a truly non-parametric representation is necessary for pattern extrapolation on large
datasets, and provide insights into this surprising result.
? GPatt is highly scalable and accurate. This is the first time, as far as we are aware, that highly
expressive non-parametric kernels with in some cases hundreds of hyperparameters, on datasets
exceeding N = 105 training instances, can be learned from the marginal likelihood of a GP, in
only minutes. Such experiments show that one can, to some extent, solve kernel selection, and
automatically extract useful features from the data, on large datasets, using a special combination
of expressive kernels and scalable inference.
? We show the proposed methodology provides a distinct approach to texture extrapolation and inpainting; it was not previously known how to make GPs work for these fundamental applications.
? Moreover, unlike typical inpainting approaches, such as patch-based methods (which work by
recursively copying pixels or patches into a gap in an image, preserving neighbourhood similarities), GPatt is not restricted to spatial inpainting. This is demonstrated on a video extrapolation
example, for which standard inpainting methods would be inapplicable [16]. Similarly, we apply
GPatt to perform large-scale long range forecasting of land surface temperatures, through learning
a sophisticated correlation structure across space and time. This learned correlation structure also
provides insights into the underlying statistical properties of these data.
? We demonstrate that GPatt can precisely recover sophisticated out-of-class kernels automatically.
2
Spectral Mixture Product Kernels for Pattern Discovery
The spectral mixture kernel has recently been introduced [5] to offer a flexible kernel that can learn
any stationary kernel. By appealing to Bochner?s theorem [17] and building a scale mixture of A
Gaussian pairs in the spectral domain, [5] produced the spectral mixture kernel
kSM (? ) =
A
X
wa2 exp{?2? 2 ? 2 ?a2 } cos(2?? ?a ) ,
a=1
2
(1)
which they applied to one-dimensional input data with a small number of points. For tractability
with multidimensional inputs and large data, we propose a spectral mixture product (SMP) kernel:
kSMP (? |?) =
P
Y
kSM (?p |? p ) ,
(2)
p=1
P
where ?p is the pth component of ? = x ? x0 ? R , ? p are the hyperparameters {?a , ?a2 , wa2 }A
a=1 of
the pth spectral mixture kernel in the product of Eq. (2), and ? = {? p }P
are
the
hyperparameters
p=1
of the SMP kernel. The SMP kernel of Eq. (2) has Kronecker structure which we exploit for scalable
and exact inference in section 2.1. With enough components A, the SMP kernel of Eq. (2) can model
any stationary product kernel to arbitrary precision, and is flexible even with a small number of
components, since scale-location Gaussian mixture models can approximate many spectral densities.
We use SMP-A as shorthand for an SMP kernel with A components in each dimension (for a total
of 3P A kernel hyperparameters and 1 noise hyperparameter). Wilson [18, 19] contains detailed
discussions of spectral mixture kernels.
Critically, a GP with an SMP kernel is not a finite basis function method, but instead corresponds to
a finite (A component) mixture of infinite basis function expansions. Therefore such a GP is a truly
nonparametric method. This difference between a truly nonparametric representation ? namely a
mixture of infinite bases ? and a parametric kernel method, a finite basis expansion corresponding
to a degenerate GP, is critical both conceptually and practically, as our results will show.
2.1
Fast Exact Inference with Spectral Mixture Product Kernels
Gaussian process inference and learning requires evaluating (K +? 2 I)?1 y and log |K +? 2 I|, for an
N ? N covariance matrix K, a vector of N datapoints y, and noise variance ? 2 , as described in the
supplementary material. For this purpose, it is standard practice to take the Cholesky decomposition
of (K + ? 2 I) which requires O(N 3 ) computations and O(N 2 ) storage, for a dataset of size N .
However, many real world applications are engineered for grid structure, including spatial statistics,
sensor arrays, image analysis, and time sampling. [14] has shown that the Kronecker structure
2
in product kernels can be exploited for exact inference and hyperparameter learning in O(P N P )
P +1
storage and O(P N P ) operations, so long as the inputs x ? X are on a multidimensional grid,
meaning X = X1 ? ? ? ? ? XP ? RP . Details are in the supplement.
Here we relax this grid assumption. Assuming we have a dataset of M observations which are
not necessarily on a grid, we propose to form a complete grid using W imaginary observations,
yW ? N (f W , ?1 IW ), ? 0. The total observation vector y = [yM , yW ]> has N = M + W
entries: y = N (f , DN ), where the noise covariance matrix DN = diag(DM , ?1 IW ), DM =
? 2 IM . The imaginary observations yW have no corrupting effect on inference: the moments of
the resulting predictive distribution are exactly the same as for the standard predictive distribution,
namely lim?0 (KN + DN )?1 y = (KM + DM )?1 yM (proof in the supplement).
?1
For inference, we must evaluate (KN + DN ) y. Since DN is not a scaled identity (as is the usual
case in Kronecker methods), we cannot efficiently decompose KN + DN , but we can efficiently
take matrix vector products involving KN and DN . We therefore use preconditioned conjugate gra?1
dients (PCG) [20] to compute (KN + DN ) y, an iterative method involving only matrix vector
?1/2
products. We use the preconditioning matrix C = DN
to solve C > (KN + DN ) Cz = C > y.
The preconditioning matrix C speeds up convergence by ignoring the imaginary observations yW .
P +1
Exploiting the fast multiplication of Kronecker matrices, PCG takes O(JP N P ) total operations
?1
(where the number of iterations J N ) to compute (KN + DN ) y to convergence within machine precision (supplement). This procedure can also be used to handle heteroscedastic noise.
For learning (hyperparameter training) we must evaluate the marginal likelihood (supplement). We
cannot efficiently compute the log |KM + DM | complexity penalty in the marginal likelihood, because KM is not a Kronecker matrix. We approximate the complexity penalty as
M
M
X
X
2
?M + ?2 ) ,
log |KM + DM | =
log(?M
+
?
)
?
log(?
(3)
i
i
i=1
2
i=1
We approximate the eigenvalues ?M
i
for noise variance ? .
of KM using the eigenvalues of KN such
? M = M ?N for i = 1, . . . , M , which is particularly effective for large M (e.g. M > 1000)
that ?
i
N i
3
[7]. [21] proves this eigenvalue approximation is asymptotically consistent (e.g., converges in the
limit of large M ), and [22] shows how one can bound the true eigenvalues by their approximation
using PCA. Notably, only the log determinant (complexity penalty) term in the marginal likelihood
undergoes a small approximation, and inference remains exact.
All remaining terms in the marginal likelihood can be computed exactly and efficiently using PCG.
The total runtime cost of hyperparameter learning and exact inference with an incomplete grid is thus
P +1
O(P N P ). In image problems, for example, P = 2, and so the runtime complexity reduces to
1.5
O(N ). Although the proposed inference can handle non-grid data, this inference is most suited
to inputs where there is some grid structure ? images, video, spatial statistics, etc. If there is no
such grid structure (e.g., none of the training data fall onto a grid), then the computational expense
necessary to augment the data with imaginary grid observations can be prohibitive. Although incomplete grids have been briefly considered in, e.g. [23], such approaches generally involve costly
and numerically unstable rank 1 updates, inducing inputs, and separate (and restricted) treatments
of ?missing? and ?extra? data. Moreover, the marginal likelihood, critical for kernel learning, is not
typically considered in alternate approaches to incomplete grids.
3
Experiments
In our experiments we combine the SMP kernel of Eq. (2) with the fast exact inference and learning
procedures of section 2.1, in a GP method we henceforth call GPatt1,2 .
We contrast GPatt with many alternative Gaussian process kernel methods. We are particularly
interested in kernel methods, since they are considered to be general purpose regression methods, but
conventionally have difficulty with large scale multidimensional pattern extrapolation. Specifically,
we compare to the recent sparse spectrum Gaussian process regression (SSGP) [6] method, which
provides fast and flexible kernel learning. SSGP models the kernel spectrum (spectral density)
as a sum of point masses, such that SSGP is a finite basis function (parametric) model, with as
many basis functions as there are spectral point masses. SSGP is similar to the recent models of
Le et al. [8] and Rahimi and Recht [9], except it learns the locations of the point masses through
marginal likelihood optimization. We use the SSGP implementation provided by the authors at
http://www.tsc.uc3m.es/?miguel/downloads.php.
To further test the importance of the fast inference (section 2.1) used in GPatt, we compare to a GP
which uses the SMP kernel of section 2 but with the popular fast FITC [10, 24] inference, which
uses inducing inputs, and is implemented in GPML (http://www.gaussianprocess.org/
gpml). We also compare to GPs with the popular squared exponential (SE), rational quadratic (RQ)
and Mat?ern (MA) (with 3 degrees of freedom) kernels, catalogued in Rasmussen and Williams [1],
respectively for smooth, multi-scale, and finitely differentiable functions. Since GPs with these
kernels cannot scale to the large datasets we consider, we combine these kernels with the same fast
inference techniques that we use with GPatt, to enable a comparison.3 Moreover, we stress test each
of these methods in terms of speed and accuracy, as a function of available data and extrapolation
range, and number of components. All of our experiments contain a large percentage of non-grid
data, and we test accuracy and efficiency as a function of the percentage of missing data.
In all experiments we assume Gaussian noise, to express the marginal likelihood of the data p(y|?)
solely as a function of kernel hyperparameters ?. To learn ? we optimize the marginal likelihood
using BFGS. We use a simple initialisation scheme: any frequencies {?a } are drawn from a uniform
distribution from 0 to the Nyquist frequency (1/2 the sampling rate), length-scales {1/?a } from a
truncated Gaussian distribution, with mean proportional to the range of the data, and weights {wa }
are initialised as the empirical standard deviation of the data divided by the number of components
used in the model. In general, we find GPatt is robust to initialisation, particularly for N > 104
datapoints. We show a representative initialisation in the experiments.
This range of tests allows us to separately understand the effects of the SMP kernel, a non-parametric
representation, and the proposed inference methods of section 2.1; we will show that all are required
for good extrapolation performance.
1
We write GPatt-A when GPatt uses an SMP-A kernel.
Experiments were run on a 64bit PC, with 8GB RAM and a 2.8 GHz Intel i7 processor.
3
We also considered the model of [25], but this model is intractable for the datasets we considered and is
not structured for the fast inference of section 2.1.
2
4
3.1
Extrapolating Metal Tread Plate and Pores Patterns
We extrapolate the missing region, shown in Figure 1a, on a real metal tread plate texture. There
are 12675 training instances (Figure 1a), and 4225 test instances (Figure 1b). The inputs are pixel
locations x ? R2 (P = 2), and the outputs are pixel intensities. The full pattern is shown in Figure
1c. This texture contains shadows and subtle irregularities, no two identical diagonal markings, and
patterns that have correlations across both input dimensions.
(a) Train
(g) GP-SE
(k) Train
(b) Test
(h) GP-MA
(l) GPatt
(c) Full
(d) GPatt
(i) GP-RQ
(m) GP-MA
(e) SSGP
(f) FITC
(j) GPatt Initialisation
(n) Train
(o) GPatt
(p) GP-MA
Figure 1: (a)-(j): Extrapolation on a Metal Tread Plate Pattern. Missing data are shown in black. a)
Training region (12675 points), b) Testing region (4225 points), c) Full tread plate pattern, d) GPatt30, e) SSGP with 500 basis functions, f) FITC with 500 inducing (pseudo) inputs, and the SMP-30
kernel, and GPs with the fast exact inference in section 2.1, and g) squared exponential (SE), h)
Mat?ern (MA), and i) rational quadratic (RQ) kernels. j) Initial and learned hyperparameters using
GPatt using simple initialisation. During training, weights of extraneous components automatically
shrink to zero. (k)-(h) and (n)-(p): Extrapolation on tread plate and pore patterns, respectively, with
added artifacts and non-stationary lighting changes.
To reconstruct the missing and training regions, we use GPatt-30. The GPatt reconstruction shown
in Fig 1d is as plausible as the true full pattern shown in Fig 1c, and largely automatic. Without hand
crafting of kernel features to suit this image, exposure to similar images, or a sophisticated initialisation, GPatt has automatically discovered the underlying structure of this image, and extrapolated
that structure across a large missing region, even though the structure of this pattern is not independent across the two spatial input dimensions. Indeed the separability of the SMP kernel represents
only a soft prior assumption, and does not rule out posterior correlations between input dimensions.
The reconstruction in Figure 1e was produced with SSGP, using 500 basis functions. In principle
SSGP can model any spectral density (and thus any stationary kernel) with infinitely many components (basis functions). However, since these components are point masses (in frequency space),
each component has highly limited expressive power. Moreover, with many components SSGP experiences practical difficulties regarding initialisation, over-fitting, and computation time (scaling
quadratically with the number of basis functions). Although SSGP does discover some interesting
structure (a diagonal pattern), and has equal training and test performance, it is unable to capture
enough information for a convincing reconstruction, and we did not find that more basis functions
improved performance. Likewise, FITC with an SMP-30 kernel and 500 inducing (pseudo) inputs
cannot capture the necessary information to interpolate or extrapolate. On this example, FITC ran
for 2 days, and SSGP-500 for 1 hour, compared to GPatt which took under 5 minutes.
GPs with SE, MA, and RQ kernels are all truly Bayesian nonparametric models ? these kernels
are derived from infinite basis function expansions. Therefore, as seen in Figure 1 g), h), i), these
methods are completely able to capture the information in the training region; however, these kernels
do not have the proper structure to reasonably extrapolate across the missing region ? they simply
act as smoothing filters. Moreover, this comparison is only possible because we have implemented
these GPs using the fast exact inference techniques introduced in section 2.1.
5
(b) Accuracy Stress Test
?
50
1
k3
0.5
0
0
(a) Runtime Stress Test
1
k2
k1
1
0.5
0
0
?
50
True
Recovered
0.5
0
0
?
50
(c) Recovering Sophisticated Kernels
Figure 2: Stress Tests. a) Runtime Stress Test. We show the runtimes in seconds, as a function
of training instances, for evaluating the log marginal likelihood, and any relevant derivatives, for a
standard GP with SE kernel (as implemented in GPML), FITC with 500 inducing (pseudo) inputs
and SMP-25 and SMP-5 kernels, SSGP with 90 and 500 basis functions, and GPatt-100, GPatt-25,
and GPatt-5. Runtimes are for a 64bit PC, with 8GB RAM and a 2.8 GHz Intel i7 processor, on the
cone pattern (P = 2), shown in the supplement. The ratio of training inputs to the sum of imaginary
and training inputs for GPatt is 0.4 and 0.6 for the smallest two training sizes, and 0.7 for all other
training sets. b) Accuracy Stress Test. MSLL as a function of holesize on the metal pattern of
Figure 1. The values on the horizontal axis represent the fraction of missing (testing) data from
the full pattern (for comparison Fig 1a has 25% missing data). We compare GPatt-30 and GPatt-15
with GPs with SE, MA, and RQ kernels (and the inference of section 2.1), and SSGP with 100 basis
functions. The MSLL for GPatt-15 at a holesize of 0.01 is ?1.5886. c) Recovering Sophisticated
Kernels. A product of three kernels (shown in green) was used to generate a movie of 112,500
training points. From this data, GPatt-20 reconstructs these component kernels (the learned SMP-20
kernel is shown in blue). All kernels are a function of ? = x ? x0 and have been scaled by k(0).
Overall, these results indicate that both expressive nonparametric kernels, such as the SMP kernel,
and the specific fast inference in section 2.1, are needed to extrapolate patterns in these images.
We note that the SMP-30 kernel used with GPatt has more components than needed for this problem.
However, as shown in Fig. 1j, if the model is overspecified, the complexity penalty in the marginal
likelihood shrinks the weights ({wa } in Eq. (1)) of extraneous components, as a proxy for model
selection ? an effect similar to automatic relevance determination [26]. Components which do not
significantly contribute to model fit are automatically pruned, as shrinking the weights decreases the
eigenvalues of K and thus minimizes the complexity penalty (a sum of log eigenvalues). The simple
GPatt initialisation in Fig 1j is used in all experiments and is especially effective for N > 104 .
In Figure 1 (k)-(h) and (n)-(p) we use GPatt to extrapolate on treadplate and pore patterns with added
artifacts and lighting changes. GPatt still provides a convincing extrapolation ? able to uncover both
local and global structure. Alternative GPs with the inference of section 2.1 can interpolate small
artifacts quite accurately, but have trouble with larger missing regions.
3.2
Stress Tests and Recovering Complex 3D Kernels from Video
We stress test GPatt and alternative methods in terms of speed and accuracy, with varying datasizes, extrapolation ranges, basis functions, inducing (pseudo) inputs, and components. We assess
accuracy using standardised mean square error (SMSE) and mean standardized log loss (MSLL) (a
scaled negative log likelihood), as defined in Rasmussen and Williams [1] on page 23. Using the
empirical mean and variance to fit the data would give an SMSE and MSLL of 1 and 0 respectively.
Smaller SMSE and more negative MSLL values correspond to better fits of the data.
The runtime stress test in Figure 2a shows that the number of components used in GPatt does not
significantly affect runtime, and that GPatt is much faster than FITC (using 500 inducing inputs) and
SSGP (using 90 or 500 basis functions), even with 100 components (601 kernel hyperparameters).
The slope of each curve roughly indicates the asymptotic scaling of each method. In this experiment,
the standard GP (with SE kernel) has a slope of 2.9, which is close to the cubic scaling we expect. All
other curves have a slope of 1 ? 0.1, indicating linear scaling with the number of training instances.
However, FITC and SSGP are used here with a fixed number of inducing inputs and basis functions.
More inducing inputs and basis functions should be used when there are more training instances ?
and these methods scale quadratically with inducing inputs and basis functions for a fixed number
of training instances. GPatt, on the other hand, can scale linearly in runtime as a function of training
6
Table 1: We compare the test performance of GPatt-30 with SSGP (using 100 basis functions), and
GPs using SE, MA, and RQ kernels, combined with the inference of section 3.2, on patterns with a
train test split as in the metal treadplate pattern of Figure 1. We show the results as SMSE (MSLL).
train, test
GPatt
SSGP
SE
MA
RQ
Rubber mat
Tread plate
Pores
Wood
Chain mail
12675, 4225
12675, 4225
12675, 4225
14259, 4941
14101, 4779
0.31 (?0.57)
0.65 (?0.21)
0.97 (0.14)
0.86 (?0.069)
0.89 (0.039)
0.45 (?0.38)
1.06 (0.018)
0.90 (?0.10)
0.88 (?0.10)
0.90 (?0.10)
0.0038 (?2.8)
1.04 (?0.024)
0.89 (?0.21)
0.88 (?0.24)
0.88 (?0.048)
0.015 (?1.4)
0.19 (?0.80)
0.64 (1.6)
0.43 (1.6)
0.077 (0.77)
0.79 (?0.052)
1.1 (0.036)
1.1 (1.6)
0.99 (0.26)
0.97 (?0.0025)
size, without any deterioration in performance. Furthermore, the fixed 2-3 orders of magnitude
GPatt outperforms the alternatives is as practically important as asymptotic scaling.
The accuracy stress test in Figure 2b shows extrapolation (MSLL) performance on the metal tread
plate pattern of Figure 1c with varying holesizes, running from 0% to 60% missing data for testing
(for comparison the hole in Fig 1a has 25% missing data). GPs with SE, RQ, and MA kernels (and
the fast inference of section 2.1) all steadily increase in error as a function of holesize. Conversely,
SSGP does not increase in error as a function of holesize ? with finite basis functions SSGP cannot
extract as much information from larger datasets as the alternatives. GPatt performs well relative to
the other methods, even with a small number of components. GPatt is particularly able to exploit the
extra information in additional training instances: only when the holesize is so large that over 60%
of the data are missing does GPatt?s performance degrade to the same level as alternative methods.
In Table 1 we compare the test performance of GPatt with SSGP, and GPs using SE, MA, and RQ
kernels, for extrapolating five different patterns, with the same train test split as for the tread plate
pattern in Figure 1. All patterns are shown in the supplement. GPatt consistently has the lowest
SMSE and MSLL. Note that many of these datasets are sophisticated patterns, containing intricate
details which are not strictly periodic, such as lighting irregularities, metal impurities, etc. Indeed
SSGP has a periodic kernel (unlike the SMP kernel which is not strictly periodic), and is capable of
modelling multiple periodic components, but does not perform as well as GPatt on these examples.
We also consider a particularly large example, where we use GPatt-10 to perform learning and exact
inference on the Pores pattern, with 383,400 training points, to extrapolate a large missing region
with 96,600 test points. The SMSE is 0.077, and the total runtime was 2800 seconds. Images of the
successful extrapolation are shown in the supplement.
We end this section by showing that GPatt can accurately recover a wide range of kernels, even using
a small number of components. To test GPatt?s ability to recover ground truth kernels, we simulate
a 50 ? 50 ? 50 movie of data (e.g. two spatial input dimensions, one temporal) using a GP with
kernel k = k1 k2 k3 (each component kernel in this product operates on a different input dimension),
where k1 = kSE + kSE ? kPER , k2 = kMA ? kPER + kMA ? kPER , and k3 = (kRQ + kPER ) ? kPER + kSE .
(kPER (? ) = exp[?2 sin2 (? ? ?)/`2 ], ? = x ? x0 ). We use 5 consecutive 50 ? 50 slices for testing,
leaving a large number N = 112500 of training points, providing much information to learn the true
generating kernels. Moreover, GPatt-20 reconstructs these complex out of class kernels in under 10
minutes, as shown in Fig 2c. In the supplement, we show true and predicted frames from the movie.
3.3
Wallpaper and Scene Reconstruction and Long Range Temperature Forecasting
Although GPatt is a general purpose regression method, it can also be used for inpainting: image
restoration, object removal, etc. We first consider a wallpaper image stained by a black apple mark,
shown in Figure 3. To remove the stain, we apply a mask and then separate the image into its
three channels (red, green, and blue), resulting in 15047 pixels in each channel for training. In each
channel we ran GPatt using SMP-30. We then combined the results from each channel to restore the
image without any stain, which is impressive given the subtleties in the pattern and lighting.
In our next example, we wish to reconstruct a natural scene obscured by a prominent rooftop, shown
in the second row of Figure 3a). By applying a mask, and following the same procedure as for
the stain, this time with 32269 pixels in each channel for training, GPatt reconstructs the scene
without the rooftop. This reconstruction captures subtle details, such as waves, with only a single
7
(a) Inpainting
1
0.8
0.6
0.4
0.2
0.5
0
0
50
X [Km]
0
Correlations
Correlations
1
0.5
50
Y [Km]
0
0
0.8
0.6
0.4
0.2
0
50
100
Time [mon]
(b) Learned GPatt Kernel for Temperatures
0.8
0.6
0.4
0.2
20
40
X [Km]
0
0.8
0.6
0.4
0.2
50
Y [Km]
0
5
Time [mon]
(c) Learned GP-SE Kernel for Temperatures
Figure 3: a) Image inpainting with GPatt. From left to right: A mask is applied to the original image,
GPatt extrapolates the mask region in each of the three (red, blue, green) image channels, and the
results are joined to produce the restored image. Top row: Removing a stain (train: 15047 ? 3).
Bottom row: Removing a rooftop to restore a natural scene (train: 32269?3). We do not extrapolate
the coast. (b)-(c): Kernels learned for land surface temperatures using GPatt and GP-SE.
training image. In fact this example has been used with inpainting algorithms which were given
access to a repository of thousands of similar images [27]. The results emphasized that conventional
inpainting algorithms and GPatt have profoundly different objectives, which are sometimes even at
cross purposes: inpainting attempts to make the image look good to a human (e.g., the example in
[27] placed boats in the water), while GPatt is a general purpose regression algorithm, which simply
aims to make accurate predictions at test input locations, from training data alone. For example,
GPatt can naturally learn temporal correlations to make predictions in the video example of section
3.2, for which standard patch based inpainting methods would be inapplicable [16].
Similarly, we use GPatt to perform long range forecasting of land surface temperatures. After training on 108 months (9 years) of temperature data across North America (299,268 training points; a
71 ? 66 ? 108 completed grid, with missing data for water), we forecast 12 months (1 year) ahead
(33,252 testing points). The runtime was under 30 minutes. The learned kernels using GPatt and GPSE are shown in Figure 3 b) and c). The learned kernels for GPatt are highly non-standard ? both
quasi periodic and heavy tailed. These learned correlation patterns provide insights into features
(such as seasonal influences) which affect how temperatures vary in space and time. Indeed learning
the kernel allows us to discover fundamental properties of the data. The temperature forecasts using
GPatt and GP-SE, superimposed on maps of North America, are shown in the supplement.
4
Discussion
Large scale multidimensional pattern extrapolation problems are of fundamental importance in machine learning, where we wish to develop scalable models which can make impressive generalisations. However, there are many obstacles towards applying popular kernel methods, such as Gaussian processes, to these fundamental problems. We have shown that a combination of expressive
kernels, truly Bayesian nonparametric representations, and inference which exploits model structure, can distinctly enable a kernel approach to these problems. Moreover, there is much promise
in further exploring Bayesian nonparametric kernel methods for large scale pattern extrapolation.
Such methods can be extremely expressive, and expressive methods are most needed for large scale
problems, which provide relatively more information for automatically learning a rich statistical
representation of the data.
Acknowledgements AGW thanks ONR grant N000141410684 and NIH grant R01GM093156. JPC
thanks Simons Foundation grants SCGB #325171, #325233, and the Grossman Center at Columbia.
8
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9
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4,830 | 5,373 | Mind the Nuisance: Gaussian Process
Classification using Privileged Noise
Daniel Hern?andez-Lobato
Universidad Aut?onoma de Madrid
Madrid, Spain
Viktoriia Sharmanska
IST Austria
Klosterneuburg, Austria
[email protected]
[email protected]
Kristian Kersting
TU Dortmund
Dortmund, Germany
Christoph H. Lampert
IST Austria
Klosterneuburg, Austria
[email protected]
[email protected]
Novi Quadrianto
SMiLe CLiNiC, University of Sussex
Brighton, United Kingdom
[email protected]
Abstract
The learning with privileged information setting has recently attracted a lot of attention within the machine learning community, as it allows the integration of additional knowledge into the training process of a classifier, even when this comes
in the form of a data modality that is not available at test time. Here, we show
that privileged information can naturally be treated as noise in the latent function
of a Gaussian process classifier (GPC). That is, in contrast to the standard GPC
setting, the latent function is not just a nuisance but a feature: it becomes a natural
measure of confidence about the training data by modulating the slope of the GPC
probit likelihood function. Extensive experiments on public datasets show that the
proposed GPC method using privileged noise, called GPC+, improves over a standard GPC without privileged knowledge, and also over the current state-of-the-art
SVM-based method, SVM+. Moreover, we show that advanced neural networks
and deep learning methods can be compressed as privileged information.
1
Introduction
Prior knowledge is a crucial component of any learning system as without a form of prior knowledge learning is provably impossible [1]. Many forms of integrating prior knowledge into machine
learning algorithms have been developed: as a preference of certain prediction functions over others,
as a Bayesian prior over parameters, or as additional information about the samples in the training
set used for learning a prediction function. In this work, we rely on the last of these setups, adopting
Vapnik and Vashist?s learning using privileged information (LUPI), see e.g. [2, 3]: we want to learn
a prediction function, e.g. a classifier, and in addition to the main data modality that is to be used for
prediction, the learning system has access to additional information about each training example.
This scenario has recently attracted considerable interest within the machine learning community
because it reflects well the increasingly relevant situation of learning as a service: an expert trains a
machine learning system for a specific task on request from a customer. Clearly, in order to achieve
the best result, the expert will use all the information available to him or her, not necessarily just the
1
information that the system itself will have access to during its operation after deployment. Typical
scenarios for learning as a service include visual inspection tasks, in which a classifier makes realtime decisions based on the input from its sensor, but at training time, additional sensors could be
made use of, and the processing time per training example plays less of a role. Similarly, a classifier
built into a robot or mobile device operates under strong energy constraints, while at training time,
energy is less of a problem, so additional data can be generated and made use of. A third scenario is
when the additional data is confidential, as e.g. in health care applications. Specifically, a diagnosis
system may be improved when more information is available at training time, e.g., specific blood
tests, genetic sequences, or drug trials, for the subjects that form the training set. However, the same
data may not be available at test time, as obtaining it could be impractical, unethical, or illegal.
We propose a novel method for using privileged information based on the framework of Gaussian
process classifiers (GPCs). The privileged data enters the model in form of a latent variable, which
modulates the noise term of the GPC. Because the noise is integrated out before obtaining the final
model, the privileged information is only required at training time, not at prediction time. The most
interesting aspect of the proposed model is that by this procedure, the influence of the privileged
information becomes very interpretable: its role is to model the confidence that the GPC has about
any training example, which can be directly read off from the slope of the probit likelihood. Instances
that are easy to classify by means of their privileged data cause a faster increasing probit, which
means the GP trusts the training example and tried to fit it well. Instances that are hard to classify
result in a slowly increasing slope, so that the GPC considers them less reliable and does not put a
lot of effort in fitting their label well. Our experiments on multiple datasets show that this procedure
leads not just to more interpretable models, but also to better prediction accuracy.
Related work: The LUPI framework was originally proposed by Vapnik and Vashist [2], inspired
by a thought-experiment: when training a soft-margin SVM, what if an oracle would provide us
with the optimal values of the slack variables? As it turns out, this would actually provably reduce
the amount of training data needed, and consequently, Vapnik and Vashist proposed the SVM+
classifier that uses privileged data to predict values for the slack variables, which led to improved
performance on several categorisation tasks and found applications, e.g., in finance [4]. This setup
was subsequently improved, by a faster training algorithm [5], better theoretical characterisation [3],
and it was generalised, e.g., to the learning to rank setting [6], clustering [7], metric learning [8] and
multi-class data classification [9]. Recently, however, it was shown that the main effect of the SVM+
procedure is to assign a data-dependent weight to each training example in the SVM objective [10].
The proposed method, GPC+, constitutes the first Bayesian treatment of classification using privileged information. The resulting privileged noise approach is related to input-modulated noise
commonly done in the regression task, where several Bayesian treatments of heteroscedastic regression using GPs have been proposed. Since the predictive density and marginal likelihood are no
longer analytically tractable, most works deal with approximate inference, i.e., techniques such as
Markov Chain Monte Carlo [11], maximum a posteriori [12], and variational Bayes [13]. To our
knowledge, however, there is no prior work on heteroscedastic classification using GPs ? we will
elaborate the reasons in Section 2.1 ? and this work is the first to develop approximate inference
based on expectation propagation for the heteroscedastic noise case in the context of classification.
2
GPC+: Gaussian process classification with privileged noise
For self-consistency we first review the GPC model [14] with an emphasis on the noise-corrupted
latent Gaussian process view. Then, we show how to treat privileged information as heteroscedastic
noise in this process. An elegant aspect of this view is how the privileged noise is able to distinguish
between easy and hard samples and to re-calibrate the uncertainty on the class label of each instance.
2.1
Gaussian process classifier with noisy latent process
Consider a set of N input-output data points or samples D = {(x1 , y1 ), . . . , (xN , yN )} ? Rd ?
{0, 1}. Assume that the class label yi of the sample xi has been generated as yi = I[ f?(xi ) ? 0 ],
where f?(?) is a noisy latent function and I[?] is the Iverson?s bracket notation, i.e., I[ P ] = 1 when
the condition P is true, and 0 otherwise. Induced by the label generation process, we adopt the
2
following form of likelihood function for ?
f = (f?(x1 ), . . . , f?(xN ))> :
YN
YN
Pr(y|?
f , X = (x1 , . . . , xN )> ) =
Pr(yn = 1|xn , f?) =
n=1
n=1
I[ f?(xn ) ? 0 ],
(1)
where f?(xn ) = f (xn ) + n with f (xn ) being the noise-free latent function. The noise term n
is assumed to be independent and normally distributed with zero mean and variance ? 2 , that is
n ? N (n |0, ? 2 ). To make inference about f?(xn ), we need to specify a prior over this function.
We proceed by imposing a zero mean Gaussian process prior [14] on the noise-free latent function,
that is f (xn ) ? GP(0, k(xn , ?)) where k(?, ?) is a positive-definite kernel function [15] that specifies
prior properties of f (?). A typical kernel function that allows for non-linear smooth functions is the
2
1
squared exponential kernel kf (xn , xm ) = ? exp(? 2l
kxn ? xm k`2 ), where ? controls the prior
amplitude of f (?) and l controls its prior smoothness. The prior and the likelihood are combined
using Bayes? rule to get the posterior of f?(?). Namely, Pr(?
f |X, y) = Pr(y|?
f , X)Pr(?
f )/Pr(y|X).
We can simplify the above noisy latent process view by integrating out the noise term n and writing
down the individual likelihood at sample xn in terms of the noise-free latent function f (?). Namely,
Z
Pr(yn = 1|xn , f ) = I[f?(xn ) ? 0]N (n |0, ? 2 )dn = ?(0,?2 ) (f (xn )),
(2)
where we have used that f?(xn ) = f (xn ) + n and ?(?,?2 ) (?) is a Gaussian cumulative distribution
function (CDF) with mean ? and variance ? 2 . Typically the standard Gaussian CDF is used, that is
?(0,1) (?), in the likelihood of (2). Coupled with a Gaussian process prior on the latent function f (?),
this results in the widely adopted noise-free latent Gaussian process view with probit likelihood.
The equivalence between a noise-free latent process with probit likelihood and a noisy latent process
with step-function likelihood is widely known [14]. It is also widely accepted that the function f?(?)
(or the functionf (?)) is a nuisance function as we do not observe its value and its sole purpose is
for a convenient formulation of the model [14]. However, in this paper, we show that by using
privileged information as the noise term, the latent function f? now plays a crucial role. The latent
function with privileged noise adjusts the slope transition in the Gaussian CDF to be faster or slower
corresponding to more certainty or more uncertainty about the samples in the original input space.
2.2
Introducing privileged information into the nuisance function
In the learning under privileged information (LUPI) paradigm [2], besides input data points
?
{x1 , . . . , xN } and associated labels {y1 , . . . , yN }, we are given additional information x?n ? Rd
about each training instance xn . However, this privileged information will not be available for unseen test instances. Our goal is to exploit the additional data x? to influence our choice of the latent
function f?(?). This needs to be done while making sure that the function does not directly use the
privileged data as input, as it is simply not available at test time. We achieve this naturally by treating
the privileged information as a heteroscedastic (input-dependent) noise in the latent process.
Our classification model with privileged noise is then as follows:
Likelihood model : Pr(yn = 1|xn , f?) = I[ f?(xn ) ? 0 ] ,
Assume : f?(xn ) = f (xn ) + n
where xn ? Rd
(3)
(4)
?
i.i.d.
Privileged noise model : n ? N (n |0, z(x?n ) = exp(g(x?n ))) , where x?n ? Rd
GP prior model : f (xn ) ? GP(0, kf (xn , ?)) and g(x?n ) ? GP(0, kg (x?n , ?)).
(5)
(6)
In the above, the function exp(?) is needed to ensure positivity of the noise variance. The term kg (?, ?)
is a positive-definite kernel function that specifies the prior properties of another latent function g(?),
which is evaluated in the privileged space x? . Crucially, the noise term n is now heteroscedastic,
that is, it has a different variance z(x?n ) at each input point xn . This is in contrast to the standard GPC
approach discussed in Section 2.1 where the noise term is homoscedastic, n ? N (n |0, z(x?n ) =
? 2 ). An input-dependent noise term is very common in regression tasks with continuous output
values yn ? R, resulting in heteroscedastic regression models, which have been proven more flexible
in numerous applications as already touched upon in the section on related work. However, to our
knowledge, there is no prior work on heteroscedastic classification models. This is not surprising as
the nuisance view of the latent function renders a flexible input-dependent noise point-less.
3
Posterior mean of
for an easy instance
= 1.0
= 5.0
0.98
= 0.5
0.8
exp(g(x?n ))
exp(g(x?n ))
exp(g(x?n ))
0.84
0.6
0.6
0.58
0.4
0.4
0.0
?10
0.2
0.2
?5
0
1
5
Posterior mean of
for a difficult instance
0.0
?(0,exp(g(x?n))(f (xn))
0.8
1.0
AwA (DeCAF) / Chimpanzee v. Giant Panda
1.0
10
f (xn)
?2
?1
0
1
2
Figure 1: Effects of privileged noise on the nuisance function. (Left) On synthetic data. Suppose for an input
xn , the latent function value is f (xn ) = 1. Now also assume that the associated privileged information x?n for
the n-th data point deems the sample as difficult, say exp(g(x?n )) = 5.0. Then the likelihood will reflect this
uncertainty Pr(yn = 1|f, g, xn , x?n ) = 0.58. In contrast, if the associated privileged information considers the
sample as easy, say e.g. exp(g(x?n )) = 0.5, the likelihood is very certain Pr(yn = 1|f, g, xn , x?n ) = 0.98.
(Right) On real data taken from our experiments in Sec. 4. The posterior means of the ?(?) function (solid)
and its 1-standard deviation confidence interval (dash-dot) for easy (blue) and difficult (black) instances of the
Chimpanzee v. Giant Panda binary task on the Animals with Attributes (AwA) dataset. (Best viewed in color).
In the context of privileged information heteroscedastic classification is a very sensible idea, which is
best illustrated when investigating the effect of privileged information in the equivalent formulation
of a noise free latent process, i.e., when one integrates out the privileged input-dependent noise term:
Z
Pr(yn = 1|xn , x?n , f, g) = I[ f?(xn ) ? 0 ]N (n |0, exp(g(x?n ))dn
p
= ?(0,exp(g(x?n ))) (f (xn )) = ?(0,1) (f (xn )/ exp(g(x?n )).
(7)
This equation shows that the privileged information adjusts the slope transition of the Gaussian
CDF through the latent function g(?). For difficult samples the latent function g(?) will be high,
the slope transition will be slower, and thus more uncertainty will be in the likelihood Pr(yn =
1|xn , x?n , f, g). For easy samples, however, g(?) will be low, the slope transition will be faster,
and thus less uncertainty will be in the likelihood term. This behaviour is illustrated in Figure 1.
For non-informative samples in the privileged space, the value of g for those samples should be
equal to a global noise value, as in a standard GPC. Thus, privileged information should in principle
never hurt. Proving this theoretically is, however, an interesting and challenging research direction.
Experimentally, however, we observe in the section on experiments the scenario described.
2.3
Posterior and prediction on test data
Define g = (g(x?1 ), . . . , g(x?n ))T and X? = (x?1 , . . . , x?n )T .
Given the likelihood
QN
?
?
Pr(y|X, X , f , g) = n=1 Pr(yn = 1|f, g, xn , xn ) with the individual term Pr(yn |f, g, xn , x?n )
given in (7) and the Gaussian process priors on functions, the posterior for f and g is:
Pr(f , g|y, X, X? ) =
Pr(y|X, X? , f , g)Pr(f )Pr(g)
,
Pr(y|X, X? )
(8)
where Pr(y|X, X? ) can be maximised with respect to a set of hyper-parameter values such as the
amplitude ? and the smoothness l of the kernel functions [14]. For a previously unseen test point
xnew ? Rd , the predictive distribution for its label ynew is given as:
Z
Pr(ynew = 1|y, X, X? ) = I[ f?(xnew ) ? 0 ]Pr(fnew |f )Pr(f , g|y, X, X? )df dgdfnew ,
(9)
where Pr(fnew |f ) is a Gaussian conditional distribution. We note that in (9) we do not consider the
privileged information x?new associated to xnew . The interpretation is that we consider homoscedastic
4
noise at test time. This is a reasonable approach as there is no additional information for increasing
or decreasing our confidence in the newly observed data xnew . Finally, we predict the label for a test
point via Bayesian decision theory: the label being predicted is the one with the largest probability.
3
Expectation propagation with numerical quadrature
Unfortunately, as for most interesting Bayesian models, inference in the GPC+ model is very challenging. Already in the homoscedastic case, the predictive density and marginal likelihood are
not tractable. Here, we therefore adapt Minka?s expectation propagation (EP) [16] with numerical
quadrature for approximate inference. Our choice is supported on the fact that EP is the preferred
method for approximate inference in GPCs, in terms of accuracy and computational cost [17, 18].
Consider the joint distribution of f , g and y, Pr(y|X, X? , f , g)Pr(f )Pr(g), where Pr(f ) and Pr(g)
QN
are Gaussian process priors and the likelihood Pr(y|X, X? , f , g) equals n=1 Pr(yn |xn , x?n , f, g),
with Pr(yn |xn , x?n , f, g) given by (7). EP approximates each non-normal factor in this distribution
by an un-normalised bi-variate normal distribution of f and g (we assume independence between f
and g). The only non-normal factors are those of the likelihood, which are approximated as:
Pr(yn |xn , x?n , f, g) ? ? n (f, g) = z n N (f (xn )|mf , v f )N (g(x?n )|mg , v g ) ,
(10)
where the parameters with the super-script are to be found by EP. The posterior approximation Q
computed by EP results from normalising with respect to f and g the EP approximate joint. That is,
Q is obtained by replacing each likelihood factor by the corresponding approximate factor ? n :
YN
Pr(f , g|X, X? , y) ? Q(f , g) := Z ?1 [
?(f, g)]Pr(f )Pr(g) ,
(11)
n=1
where Z is a normalisation constant that approximates the model evidence, Pr(y|X, X? ). The
normal distribution belongs to the exponential family of probability distributions and is closed under
the product and division. It is hence possible to show that Q is the product of two multi-variate
normals [19]. The first normal approximates the posterior for f and the second the posterior for g.
EP tries to fix the parameters of ? n so that it is similar to the exact factor Pr(yn |xn , x?n , f, g) in
regions of high posterior probability [16]. For this,
EP iteratively updates each ? n until convergence
to hminimise KLi Pr(yn |xn , x?n , f, g)Qold /Zn ||Q , where Qold is a normal distribution proportional
Q
?
to
n0 6=n ? n0 Pr(f )Pr(g) with all variables different from f (xn ) and g(xn ) marginalised out, Zn
is simply a normalisation constant and KL(?||?) denotes the Kullback-Leibler divergence between
probability distributions. Assume Qnew is the distribution minimising the previous divergence. Then,
? n ? Qnew /Qold and the parameter z n of ? n is fixed to guarantee that ? n integrates the same as
the exact factor with respect to Qold . The minimisation of the KL divergence involves matching
expected sufficient statistics (mean and variance) between Pr(yn |xn , x?n , f, g)Qold /Zn and Qnew .
These expectations can be obtained from the derivatives of log Zn with respect to the (natural)
parameters of Qold [19]. Unfortunately, the computation of log Zn in closed form is intractable. We
show here that it can be approximated by a one dimensional quadrature. Denote by mf , vf , mg and
vg the means and variances of Qold for f (xn ) and g(x?n ), respectively. Then,
Z
q
Zn = ?(0,1) yn mf / vf + exp(g(x?n )) N (g(x?n )|mg , vg )dg(x?n ) .
(12)
Thus, EP only requires five quadratures to update each ? n . One to compute log Zn and four extras
to compute its derivatives with respect to mf , vf , mg and vg . After convergence, Q can be used
to approximate predictive distributions and the normalisation constant Z can be maximised to find
good values for the model?s hyper-parameters. In particular, it is possible to compute the gradient
of Z with respect to the parameters of the Gaussian process priors for f and g [19]. An R language
implementation of GPC+ using EP for approximate inference is found in the supplementary material.
4
Experiments
We investigate the performance of GPC+. To this aim we considered three types of binary classification tasks corresponding to different privileged information using two real-world datasets: Attribute
Discovery and Animals with Attributes. We detail these experiments in turn in the following sections.
5
Methods: We compared our proposed GPC+ method with the well-established LUPI method based
on SVM, SVM+ [5]. As a reference, we also fit standard GP and SVM classifiers when learning on
the original space Rd (GPC and SVM baselines). For all four methods, we used a squared exponential
kernel with amplitude parameter ? and smoothness parameter l. For simplicity, we set ? = 1.0 in
all cases. There are two hyper-parameters in GPC (smoothness parameter l and noise variance ? 2 )
and also two in GPC+ (smoothness parameters l of kernel kf (?, ?) and of kernel kg (?, ?)). In GPC
and GPC+, we used type II-maximum likelihood for finding all hyper-parameters. SVM has two
knobs, i.e., smoothness and regularisation, and SVM+ has four knobs, two smoothness and two
regularisation parameters. In SVM we used a grid search guided by cross-validation to set all hyperparameters. However, this procedure was too expensive for finding the best parameters in SVM+.
Thus, we used the performance on a separate validation set to guide the search. This means that we
give a competitive advantage to SVM+ over the other methods, which do not use the validation set.
Evaluation metric: To evaluate the performance of each method we used the classification error
measured on an independent test set. We performed 100 repeats of all the experiments to get the
better statistics of the performance and we report the mean and the standard deviation of the error.
4.1
Attribute discovery dataset
The data set was collected from a website that aggregates product data from a variety of e-commerce
sources and includes both images and associated textual descriptions [20]. The images and texts are
grouped into 4 broad shopping categories: bags, earrings, ties, and shoes. We used 1800 samples
from this dataset. We generated 6 binary classification tasks for each pair of the 4 classes with 200
samples for training, 200 samples for validation, and the rest of the samples for testing performance.
Neural networks on texts as privileged information: We used images as the original domain and
texts as the privileged domain. This setting was also explored in [6]. However, we used a different
dataset because textual descriptions of the images used in [6] are sparse and contain duplicates. More
precisely, we extracted more advanced text features instead of simple term frequency (TF) features.
For the images representation, we extracted SURF descriptors [21] and constructed a codebook of
100 visual words using the k-means clustering. For the text representation, we extracted 200 dimensional continuous word-vectors using a neural network skip-gram architecture [22]1 . To convert this
word representation into a fixed-length sentence representation, we constructed a codebook of 100
word-vectors using again k-means clustering. We note that a more elaborate approach to transform
word to sentence or document features has recently been developed [23], and we are planning to
explore this in the future. We performed PCA for dimensionality reduction in the original and privileged domains and only kept the top 50 principal components. Finally, we standardised the data so
that each feature had zero mean and unit standard deviation.
The experimental results are summarised in Table 1. On average over 6 tasks, SVM with hinge loss
outperforms GPC with probit likelihood. However, GPC+ significantly improves over GPC providing the best results on average. This clearly shows that GPC+ is able to employ the neural network
textual representation as privileged information. In contrast, SVM+ produced the same result as
SVM. We suspect this is due to the fact that that SVM has already shown strong performance on
the original image space coupled with the difficulties of finding the best values of the four hyperparameters of SVM+. Keep in mind that in SVM+ we discretised the hyper-parameter search space
over 625 (5 ? 5 ? 5 ? 5) possible combination values and used a separate validation set to estimate
the resulting prediction performance.
4.2
Animals with attributes (AwA) dataset
The dataset was collected by querying image search engines for each of the 50 animals categories
which have complimentary high level descriptions of their semantic properties such as shape, colour,
or habitat information among others [24]. The semantic attributes per animal class were retrieved
from a prior psychological study. We focused on the 10 categories corresponding to the test set of this
dataset for which the predicted attributes are provided based on the probabilistic DAP model [24].
The 10 classes are: chimpanzee, giant panda, leopard, persian cat, pig, hippopotamus, humpback
whale, raccoon, rat, seal, which have 6180 images associated in total. As in Section 4.1 and also in
1
https://code.google.com/p/word2vec/
6
Table 1: Average error rate in % (the lower the better) on the Attribute Discovery dataset over 100 repetitions.
We used images as the original domain and neural networks word-vector representation on texts as the privileged domain. The best method for each binary task is highlighted in boldface. An average rank equal to one
means that the corresponding method has the smallest error on the 6 tasks.
bags v. earrings
bags v. ties
bags v. shoes
earrings v. ties
earrings v. shoes
ties v. shoes
average error on each task
average ranking
GPC
9.79?0.12
10.36?0.16
9.66?0.13
10.84?0.14
7.74?0.11
15.51?0.16
10.65?0.11
3.0
GPC+ (Ours)
9.50?0.11
10.03?0.15
9.22?0.11
10.56?0.13
7.33?0.10
15.54?0.16
10.36?0.12
1.8
SVM
9.89?0.14
9.44?0.16
9.31?0.12
11.15?0.16
7.75?0.13
14.90?0.21
10.41?0.11
2.7
SVM+
9.89?0.13
9.47?0.13
9.29?0.14
11.11?0.16
7.63?0.13
15.10?0.18
10.42?0.11
2.5
[6], we generated 45 binary classification tasks for each pair of the 10 classes with 200 samples for
training, 200 samples for validation, and the rest of samples for testing the predictive performance.
Neural networks on images as privileged information: Deep learning methods have gained an increased attention within the machine learning and computer vision community over the recent years.
This is due to their capability in extracting informative features and delivering strong predictive performance in many classification tasks. As such, we are interested to explore the use of deep learning
based features as privileged information so that their predictive power can be used even if we do not
have access to them at prediction time. We used the standard SURF features [21] with 2000 visual
words as the original domain and the recently proposed DeCAF features [25] extracted from the
activation of a deep convolutional network trained in a fully supervised fashion as the privileged domain. The DeCAF features have 4096 dimensions. All features are provided with the AwA dataset2 .
We again performed PCA for dimensionality reduction in the original and privileged domains and
only kept the top 50 principal components, as well as standardised the data.
Attributes as privileged information: Following the experimental setting of [6], we also used
images as the original domain and attributes as the privileged domain. Images were represented by
2000 visual words based on SURF descriptors and attributes were in the form of 85 dimensional
predicted attributes based on probabilistic binary classifiers [24]. As previously, we also performed
PCA and kept the top 50 principal components in the original domain and standardised the data.
The results of these experiments are shown in Figure 2 in terms of pairwise comparisons over 45
binary tasks between GPC+ and the main baselines, GPC and SVM+. The complete results with
the error of each method GPC, GPC+, SVM, and SVM+ on each problem are relegated to the
supplementary material. In contrast to the results on the attribute discovery dataset, on the AwA
dataset it is clear that GPC outperforms SVM in almost all of the 45 binary classification tasks
(see the supplementary material). The average error of GPC over 4500 (45 tasks and 100 repeats
per task) experiments is much lower than SVM. On the AwA dataset, SVM+ can take advantage
of privileged information ? be it deep belief DeCAF features or semantic attributes ? and shows
significant performance improvement over SVM. However, GPC+ still shows the best overall results
and further improves the already strong performance of GPC. As illustrated in Figure 1 (right), the
privileged information modulates the slope of the probit likelihood function differently for easy
and difficult examples: easy examples gain slope and hence importance whereas difficult ones lose
importance in the classification. In this dataset we analysed our experimental results using the
multiple dataset statistical comparison method described in [26]3 . The results of the statistical tests
are summarised in Figure 3. When DeCAF attributes are used as privileged information, there is
statistical evidence supporting that GPC+ performs best among the four methods, while when the
semantic attributes are used as privileged information, GPC+ still performs best but there is not
enough evidence to reject that GPC+ performs comparable to GPC.
2
http://attributes.kyb.tuebingen.mpg.de
Note that we are not able to use this method on the results of the attribute discovery dataset in Table 1
because the number of methods compared (i.e., 4) is almost equal to the number of tasks or datasets (i.e., 6).
3
7
(DeCAF as privileged)
(Attributes as privileged)
Figure 2: Pairwise comparison of the proposed GPC+ method and main baselines is shown via the relative
difference of the error rate (top: GPC+ versus GPC, bottom: GPC+ versus SVM+). The length of the 45 bars
corresponds to relative difference of the error rate over 45 cases. Average error rates of each method on the
AwA dataset across each of the 45 tasks are found in the supplementary material. (Best viewed in color).
Critical Distance
GPC
SVM+
GPC+
1
Critical Distance
SVM
2
3
GPC
SVM+
GPC+
SVM
1
4
(DeCAF as privileged)
2
3
4
(Attributes as privileged)
Figure 3: Average rank (the lower the better) of the four methods and critical distance for statistically significant differences (see [26]) on the AwA dataset. An average rank equal to one means that particular method
has the smallest error on the 45 tasks. Whenever the average ranks differ by more than the critical distance,
there is statistical evidence (p-value < 10%) supporting a difference in the average ranks and hence in the
performance. We also link two methods with a solid line if they are not statistically different from each other
(p-value > 10%). When the DeCAF features are used as privileged information, there is statistical evidence
supporting that GPC+ performs best among the four methods considered. When the attributes are used, GPC+
still performs best, but there is not enough evidence to reject that GPC+ performs comparable to GPC.
5
Conclusions and future work
We presented the first treatment of the learning with privileged information paradigm under the
Gaussian process classification (GPC) framework, and called it GPC+. In GPC+ privileged information is used in the latent noise layer, resulting in a data-dependent modulation of the slope of the
likelihood. The training time of GPC+ is about twice times the training time of a standard Gaussian
process classifier. The reason is that GPC+ must train two latent functions, f and g, instead of only
one. Nevertheless, our results show that GPC+ is an effective way to use privileged information,
which manifest itself in significantly better prediction accuracy. Furthermore, to our knowledge,
this is the first time that a heteroscedastic noise term is used to improve GPC. We have also shown
that recent advances in continuous word-vector neural networks representations [23] and deep convolutional networks for image representations [25] can be used as privileged information. For future
work, we plan to extend the GPC+ framework to the multi-class case and to speed up computation
by devising a quadrature-free expectation propagation method, similar to the ones in [27, 28].
Acknowledgement: D. Hern?andez-Lobato is supported by Direcci?on General de Investigaci?on MCyT and by
Consejer??a de Educaci?on CAM (projects TIN2010-21575-C02-02, TIN2013-42351-P and S2013/ICE-2845).
V. Sharmanska is funded by the European Research Council under the ERC grant agreement no 308036.
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4,831 | 5,374 | Automated Variational Inference
for Gaussian Process Models
Edwin V. Bonilla
The University of New South Wales
[email protected]
Trung V. Nguyen
ANU & NICTA
[email protected]
Abstract
We develop an automated variational method for approximate inference in Gaussian process (GP) models whose posteriors are often intractable. Using a mixture
of Gaussians as the variational distribution, we show that (i) the variational objective and its gradients can be approximated efficiently via sampling from univariate Gaussian distributions and (ii) the gradients wrt the GP hyperparameters can
be obtained analytically regardless of the model likelihood. We further propose
two instances of the variational distribution whose covariance matrices can be
parametrized linearly in the number of observations. These results allow gradientbased optimization to be done efficiently in a black-box manner. Our approach is
thoroughly verified on five models using six benchmark datasets, performing as
well as the exact or hard-coded implementations while running orders of magnitude faster than the alternative MCMC sampling approaches. Our method can be
a valuable tool for practitioners and researchers to investigate new models with
minimal effort in deriving model-specific inference algorithms.
1
Introduction
Gaussian processes (GPs, [1]) are a popular choice in practical Bayesian non-parametric modeling.
The most straightforward application of GPs is the standard regression model with Gaussian likelihood, for which the posterior can be computed in closed form. However, analytical tractability is
no longer possible when having non-Gaussian likelihoods, and inference must be carried out via approximate methods, among which Markov chain Monte Carlo (MCMC, see e.g. [2]) and variational
inference [3] are arguably the two techniques most widely used.
MCMC algorithms provide a flexible framework for sampling from complex posterior distributions
of probabilistic models. However, their generality comes at the expense of very high computational
cost as well as cumbersome convergence analysis. Furthermore, methods such as Gibbs sampling
may perform poorly when there are strong dependencies among the variables of interest. Other
algorithms such as the elliptical slice sampling (ESS) developed in [4] are more effective at drawing
samples from strongly correlated Gaussians. Nevertheless, while improving upon generic MCMC
methods, the sampling cost of ESS remains a major challenge for practical usages.
Alternative to MCMC is the deterministic approximation approach via variational inference, which
has been used in numerous applications with some empirical success ( see e.g. [5, 6, 7, 8, 9, 10, 11]).
The main insight from variational methods is that optimizing is generally easier than integrating.
Indeed, they approximate a posterior by optimizing a lower bound of the marginal likelihood, the
so-called evidence lower bound (ELBO). While variational inference can be considerably faster
than MCMC, it lacks MCMC?s broader applicability as it requires derivations of the ELBO and its
gradients on a model-by-model basis.
This paper develops an automated variational inference technique for GP models that not only reduces the overhead of the tedious mathematical derivations inherent to variational methods but also
1
allows their application to a wide range of problems. In particular, we consider Gaussian process
models that satisfy the following properties: (i) factorization across latent functions and (ii) factorization across observations. The former assumes that, when there is more than one latent function,
they are generated from independent GPs. The latter assumes that, given the latent functions, the
observations are conditionally independent. Existing GP models, such as regression [1], binary and
multi-class classification [6, 12], warped GPs [13], log Gaussian Cox process [14], and multi-output
regression [15], all fall into this class of models. We note, however, that our approach goes beyond
standard settings for which elaborate learning machinery has been developed, as we only require
access to the likelihood function in a black-box manner.
Our automated deterministic inference method uses a mixture of Gaussians as the approximating
posterior distribution and exploits the decomposition of the ELBO into a KL divergence term and
an expected log likelihood term. In particular, we derive an analytical lower bound for the KL term;
and we show that the expected log likelihood term and its gradients can be computed efficiently by
sampling from univariate Gaussian distributions, without explicitly requiring gradients of the likelihood. Furthermore, we optimize the GP hyperparameters within the same variational framework by
using their analytical gradients, irrespective of the specifics of the likelihood models.
Additionally, we exploit the efficient parametrization of the covariance matrices in the models, which
is linear in the number of observations, along with variance-reduction techniques in order to provide an automated inference framework that is useful in practice. We verify the effectiveness of
our method with extensive experiments on 5 different GP settings using 6 benchmark datasets. We
show that our approach performs as well as exact GPs or hard-coded deterministic inference implementations, and that it can be up to several orders of magnitude faster than state-of-the-art MCMC
approaches.
Related work
Black box variational inference (BBVI, [16]) has recently been developed for general latent variable
models. Due to this generality, it under-utilizes the rich amount of information available in GP models that we previously discussed. For example, BBVI approximates the KL term of the ELBO, but
this is computed analytically in our method. A clear disadvantage of BBVI is that it does not provide
an analytical or practical way of learning the covariance hyperparameters of GPs ? in fact, these are
set to fixed values. In principle, these values can be learned in BBVI using stochastic optimization, but experimentally, we have found this to be problematic, ineffectual, and time-consuming. In
contrast, our method optimizes the hyperparameters using their exact gradients.
An approach more closely related to ours is in [17], which investigates variational inference for
GP models with one latent function and factorial likelihood. Their main result is an efficient
parametrization when using a standard variational Gaussian distribution. Our method is more general in that it allows multiple latent functions, hence being applicable to settings such as multi-class
classification and multi-output regression. Furthermore, our variational distribution is a mixture of
Gaussians, with the full Gaussian distribution being a particular case. Another recent approach to
deterministic approximate inference is the Integrated Nested Laplace Approximation (INLA, [18]).
INLA uses numerical integration to approximate the marginal likelihood, which makes it unsuitable
for GP models that contain a large number of hyperparameters.
2
A family of GP models
We consider supervised learning problems with a dataset of N training inputs x = {xn }N
n=1 and
their corresponding targets y = {yn }N
n=1 . The mapping from inputs to outputs is established via
Q underlying latent functions, and our objective is to reason about these latent functions from the
observed data. We specify a class of GP models for which the priors and the likelihoods have the
following structure:
Q
Q
Y
Y
p(f |?0 ) =
p(f?j |?0 ) =
N (f?j ; 0, Kj ),
(1)
j=1
p(y|f , ?1 ) =
N
Y
j=1
p(yn |fn? , ?1 ),
n=1
2
(2)
where f is the set of all latent function values; f?j = {fj (xn )}N
n=1 denotes the values of the latent
Q
function j; fn? = {fj (xn )}j=1 is the set of latent function values which yn depends upon; Kj is
the covariance matrix evaluated at every pair of inputs induced by the covariance function kj (?, ?);
and ?0 and ?1 are covariance hyperparameters and likelihood parameters, respectively.
In other words, the class of models specified by Equations (1) and (2) satisfy the following two criteria: (a) factorization of the prior over the latent functions and (b) factorization of the conditional
likelihood over the observations. Existing GP models including GP regression [1], binary classification [6, 12], warped GPs [13], log Gaussian Cox processes [14], multi-class classification [12], and
multi-output regression [15] all belong to this family of models.
3
Automated variational inference for GP models
This section describes our automated inference framework for posterior inference of the latent functions for the given family of models. Apart from Equations (1) and (2), we only require access to
the likelihood function in a black-box manner, i.e. specific knowledge of its shape or its gradient is
not needed. Posterior inference for general (non-Gaussian) likelihoods is analytically intractable.
We build our posterior approximation framework upon variational inference principles. This entails
positing a tractable family of distributions and finding the member of the family that is ?closest?
to the true posterior in terms of their KL divergence. Herein we choose the family of mixture of
Gaussians (MoG) with K components, defined as
K
K Q
1 X
1 XY
q(f |?) =
qk (f |mk , Sk ) =
N (f?j ; mkj , Skj ),
K
K
j=1
k=1
? = {mkj , Skj },
(3)
k=1
where qk (f |mk , Sk ) is the component k with variational parameters mk = {mkj }Q
j=1 and Sk =
{Skj }Q
.
Less
general
MoG
with
isotropic
covariances
have
been
used
with
variational
inference
j=1
in [7, 19]. Note that within each component, the posteriors over the latent functions are independent.
Minimizing the divergence KL[q(f |?)||p(f |y)] is equivalent to maximizing the evidence lower
bound (ELBO) given by:
K
1 X
?
Eq [log p(y|f )] = L.
log p(y) ? Eq [? log q(f |?)] + Eq [log p(f )] +
{z
} K k=1 k
|
(4)
?KL[q(f |?)||p(f )]
Observe that the KL term in Equation (4) does not depend on the likelihood. The remaining term,
called the expected log likelihood (ELL), is the only contribution of the likelihood to the ELBO. We
can thus address the technical difficulties regarding each component and their derivatives separately
using different approaches. In particular, we can obtain a lower bound of the first term (KL) and
approximate the second term (ELL) via sampling. Due to the limited space, we only show the main
results and refer the reader to the supplementary material for derivation details.
3.1
A lower bound of ?KL[q(f |?)||p(f )]
The first component of the KL divergence term is the entropy of a Gaussian mixture which is not
analytically tractable. However, a lower bound of this entropy can be obtained using Jensen?s inequality (see e.g. [20]) giving:
Eq [? log q(f |?)] ? ?
K
K
X
X
1
1
log
N (mk ; ml , Sk + Sl ).
K
K
k=1
(5)
l=1
The second component of the KL term is a negative cross-entropy between a Gaussian mixture and
a Gaussian, which can be computed analytically giving:
Eq [log p(f )] = ?
K Q
1 XX
?1
N log 2? + log |Kj | + mTkj K?1
j mkj + tr (Kj Skj ) .
2K
j=1
(6)
k=1
The gradients of the two terms in Equations (5) and (6) wrt the variational parameters can be computed analytically and are given in the supplementary material.
3
3.2
An approximation to the expected log likelihood (ELL)
It is clear from Equation (4) that the ELL can be obtained via the ELLs of the individual mixture
components Eqk [log p(y|f )]. Due to the factorial assumption of p(y|f ), the expectation becomes:
Eqk [log p(y|f )] =
N
X
Eqk(n) [log p(yn |fn? )],
(7)
n=1
where qk(n) = qk(n) (fn? |?k(n) ) is the marginal posterior with variational parameters ?k(n) that
correspond to fn? . The gradients of these individual ELL terms wrt the variational parameters ?k(n)
are given by:
??k(n) Eqk(n) [log p(yn |fn? )] =Eqk(n) ??k(n) log qk(n) (fn? |?k(n) ) log p(yn |fn? ).
(8)
Using Equations (7) and (8) we establish the following theorem regarding the computation of the
ELL and its gradients.
Theorem 1. The expected log likelihood and its gradients can be approximated using samples from
univariate Gaussian distributions.
The proof is in Section 1 of the supplementary material. A less general result, for the case of
one latent function and the variational Gaussian posterior, was obtained in [17] using a different
derivation. Note that when Q > 1, qk(n) is not a univariate marginal. Nevertheless, it has a diagonal
covariance matrix due to the factorization of the latent posteriors so the theorem still holds.
3.3
Learning of the variational parameters and other model parameters
In order to learn the parameters of the model we use gradient-based optimization of the ELBO. For
this we require the gradients of the ELBO wrt all model parameters.
Variational parameters. The noisy gradients of the ELBO w.r.t. the variational means mk(n) and
variances Sk(n) corresponding to data point n are given by:
S
X
i
? m L ? ?m Lent + ?m Lcross + 1 s?1 ?
?
(f i ? mk(n) ) log p(yn |fn?
),
k(n)
k(n)
k(n)
KS k(n) i=1 n?
(9)
? S L ? ?S Lent + ?S Lcross
?
k(n)
k(n)
k(n)
S
X
1
?1
?1
?1
i
i
i
+
dg
s
?s
? (fn? ? mk(n) ) ? (fn? ? mk(n) ) ? sk(n) log p(yn |fn?
) (10)
2KS i=1 k(n) k(n)
i
where ? is the entrywise Hadamard product; {fn?
}Si=1 are samples from qk(n) (fn? |mk(n) , sk(n) );
?1
sk(n) is the diagonal of Sk(n) and sk(n) is the element-wise inverse of sk(n) ; dg turns a vector to a
diagonal matrix; and Lent = Eq [? log q(f |?)] and Lcross = Eq [log p(f )] are given by Equations (5)
and (6). The control variates technique described in [16] is also used to further reduce the variance
of these estimators.
Covariance hyperparameters. The ELBO in Equation (4) reveals a remarkable property: the hyperparameters depend only on the negative cross-entropy term Eq [log p(f )] whose exact expression
was derived in Equation (6). This has a significant practical implication: despite using black-box
inference, the hyperparameters are optimized wrt the true evidence lower bound (given fixed variational parameters). This is an additional and crucial advantage of our automated inference method
over other generic inference techniques [16] that seem incapable of hyperparameter learning, in
part because there are not yet techniques for reducing the variance of the gradient estimators. The
gradient of the ELBO wrt any hyperparameter ? of the j-th covariance function is given by:
?? L = ?
K
1 X
?1
?1
T
tr K?1
j ?? Kj ? Kj ?? Kj Kj (mkj mkj + Sj ) .
2K
k=1
4
(11)
Likelihood parameters The noisy gradients w.r.t. the likelihood parameters can also be estimated
via samples from univariate marginals:
??1 L ?
K N
S
1 XXX
k,i
k,i
??1 log p(yn |f(n)
, ?1 ), where f(n)
? qk(n) (fn? |mk(n) , sk(n) ).
KS
n=1 i=1
(12)
k=1
3.4
Practical variational distributions
The gradients from the previous section may be used for automated variational inference for GP
models. However, the mixture of Gaussians (MoG) requires O(N 2 ) variational parameters for each
covariance matrix, i.e. we need to estimate a total of O(QKN 2 ) parameters. This causes difficulties
for learning when these parameters are optimized simultaneously. This section introduces two special members of the MoG family that improve the practical tractability of our inference framework.
Full Gaussian posterior. This instance is the mixture with only 1 component and is thus a Gaussian distribution. Its covariance matrix has block diagonal structure, where each block is a full
covariance corresponding to that of a single latent function posterior. We thus refer to it as the
full Gaussian posterior. As stated in the following theorem, full Gaussian posteriors can still be
estimated efficiently in our variational framework.
Theorem 2. Only O(QN ) variational parameters are required to parametrize the latent posteriors
with full covariance structure.
The proof is given Section 2 of the supplementary material. This result has been stated previously
(see e.g. [6, 7, 17]) but for specific models that belong to the class of GP models considered here.
Mixture of diagonal Gaussians posterior. Our second practical variational posterior is a Gaussian mixture with diagonal covariances, yielding two immediate benefits. Firstly, only O(QN )
parameters are required for each mixture component. Secondly, computation is more efficient as
inverting a diagonal covariance can be done in linear time. Furthermore, as a result of the following
theorem, optimization will typically converge faster when using a mixture of diagonal Gaussians.
Theorem 3. The estimator of the gradients wrt the variational parameters using the mixture of
diagonal Gaussians has a lower variance than the full Gaussian posterior?s.
The proof is in Section 3 of the supplementary material and is based on the Rao-Blackwellization
technique [21]. We note that this result is different to that in [16]. In particular, our variational
distribution is a mixture, thus multi-modal. The theorem is only made possible due to the analytical
tractability of the KL term in the ELBO.
Given the noisy gradients, we use off-the-shelf, gradient-based optimizers, such as conjugate gradient, to learn the model parameters. Note that stochastic optimization may also be used, but it may
require significant time and effort in tuning the learning rates.
3.5
Prediction
Given the MoG posterior, the predictive distribution for new test points x? is given by:
p(Y? |x? ) =
Z
K Z
1 X
p(Y? |f? ) p(f? |f )qk (f )df df? .
K
(13)
k=1
The inner integral is the predictive distribution of the latent values f? and it is a Gaussian since
both qk (f ) and p(f? |f ) are Gaussian. The probability of the test points taking values y? (e.g. in
classification) can thus be readily estimated via Monte Carlo sampling. The predictive means and
variances of a MoG can be obtained from that of the individual mixture components as described in
Section 6 of the supplementary material.
5
Table 1: Datasets, their statistics, and the corresponding likelihood functions and models used in the
experiments, where Ntrain , Ntest , and D are the training size, testing size, and the input dimension,
respectively. See text for detailed description of the models.
Dataset Ntrain Ntest D Likelihood p(y|f )
Model
Mining disasters 811
0
1
?y exp(??)/y!
Log Gausian Cox process
Boston housing 300
206
13 N (y; f, ? 2 )
Standard regression
Creep 800
1266 30 ?y t(y)N (t(y); f, ? 2 ) Warped Gaussian processes
Abalone 1000
3177 8
same as above
Warped Gaussian processes
Breast cancer 300
383
9
1/(1 + exp(?f
Binary classification
P ))
USPS 1233
1232 256 exp(fc )/ i=1 exp(fi ) Multi-class classification
4
Experiments
We perform experiments with five GP models: standard regression [1], warped GPs [13], binary
classification [6, 12], multi-class classification [12], and log Gaussian Cox processes [14] on six
datasets (see Table 1) and repeat the experiments five times using different data subsets.
Experimental settings. The squared exponential covariance function with automatic relevance determination (see Ch. 4 in [1]) is used with the GP regression and warped GPs. The isotropic covariance is used with all other models. The noisy gradients of the ELBO are approximated with
2000 samples and 200 samples are used with control variates to reduce the variance of the gradient
estimators. The model parameters (variational, covariance hyperparameters and likelihood parameters) are learned by iteratively optimizing one set while fixing the others until convergence, which is
determined when changes are less than 1e-5 for the ELBO or 1e-3 for the variational parameters.
Evaluation metrics. To assess the predictive accuracy, we use the standardized squared error (SSE)
for the regression tasks and the classification error rates for the classification tasks. The negative log
predictive density (NLPD) is also used to evaluate the confidence of the prediction. For all of the
metrics, smaller figures are better.
Notations. We call our method AGP and use AGP-FULL, AGP-MIX and AGP-MIX2 when
using the full Gaussian and the mixture of diagonal Gaussians with 1 and 2 components, respectively.
Details of these two posteriors were given in Section 3.4. On the plots, we use the shorter notations,
FULL, MIX, and MIX2 due to the limited space.
Reading the box plots. We used box plots to give a more complete picture of the predictive performance. Each plot corresponds to the distribution of a particular metric evaluated at all test points
for a given task. The edges of a box are the q1 = 25th and q3 = 75th percentiles and the central
mark is the median. The dotted line marks the limit of extreme points that are greater than the 97.5th
percentile. The whiskers enclose the points in the range (q1 ? 1.5(q3 ? q1 ), q3 + 1.5(q3 ? q1 )), which
amounts to approximately ?2.7? if the data is normally distributed. The points outside the whiskers
and below the dotted line are outliers and are plotted individually.
4.1
Standard regression
First we consider the standard Gaussian process regression for which the predictive distribution can
be computed analytically. We compare with this exact inference method (GPR) using the Boston
housing dataset [22]. The results in Figure 1 show that AGP-FULL achieves nearly identical performance as GPR. This is expected as the analytical posterior is a full Gaussian. AGP-MIX and
AGP-MIX2 also give comparable performance in terms of the median SSE and NLPD.
4.2
Warped Gaussian processes (WGP)
The WGP allows for non-Gaussian processes and non-Gaussian noises. The likelihood for each
target yn is attained by warping it through a nonlinear monotonic transformation t(y) giving
p(yn |fn ) = ?yn t(yn )N (t(yn )|fn , ? 2 ). We used the same neural net style transformation as in
[13]. We fixed the warp parameters and used the same procedure for making analytical approximations to the predicted means and variances for all methods.
6
Boston housing
Boston housing
8
0.8
7
NLPD
SSE
0.6
0.4
6
5
4
3
0.2
2
0
FULL
MIX
MIX2
GPR
FULL
MIX
MIX2
GPR
Figure 1: The distributions of SSE and NLPD of all methods on the regression task. Compared to the
exact inference method GPR, the performance of AGP-FULL is identical while that of AGP-MIX
and AGP-MIX2 are comparable.
Creep
Abalone
Creep
Abalone
5
0.4
3
7
0.1
5
1.5
4
0.5
2
FULL
MIX
MIX2
GPR
WGP
3
2
1
3
0
NLPD
2
SSE
0.2
4
2.5
6
NLPD
SSE
0.3
1
0
FULL
MIX
MIX2
GPR
FULL
WGP
MIX
MIX2
GPR
WGP
FULL
MIX
MIX2
GPR
WGP
Figure 2: The distributions of SSE and NLPD of all methods on the regression task with warped
GPs. The AGP methods (FULL, MIX and MIX 2) give comparable performance to exact inference
with WGP and slightly outperform GPR which has narrower ranges of predictive variances.
We compare with the exact implementation of [13] and the standard GP regression (GPR) on the
Creep [23] and Abalone [22] datasets. The results in Figure 2 show that the AGP methods give
comparable performance to the exact method WGP and slightly outperform GPR. The prediction
by GPR exhibits characteristically narrower ranges of predictive variances which can be attributed
to its Gaussian noise assumption.
4.3
Classification
For binary classification, we use the logistic likelihood and experiment with the Wisconsin breast
cancer dataset [22]. We compare with the variational bounds (VBO) and the expectation propagation
(EP) methods. Details of VBO and EP can be found in [6]. All methods use the same analytical
approximations when making prediction.
For multi-class classification, we use the softmax likelihood and experiment with a subset of the
USPS dataset [1] containing the digits 4, 7, and 9. We compare with a variational inference method
(VQ) which constructs the ELBO via a quadratic lower bound to the likelihood terms [5]. Prediction
is made by squashing the samples from the predictive distributions of the latent values at test points
through the softmax likelihood for all methods.
Breast cancer
USPS
0.06
1
1
0.05
0.03
0.02
0.8
0.6
NLPD
Error rates
VQ
FULL
MIX
MIX2
VBO
EP
NLPD
0.8
0.04
0.4
0.2
0.6
0.4
0.2
0.01
0
0
Breast cancer
USPS
0
FULL
MIX
MIX2
VBO
EP
FULL
MIX
MIX2
VQ
Figure 3: Left plot: classification error rates averaged over 5 runs (the error bars show two standard
deviations). The AGP methods have classification errors comparable to the hard-coded implementations. Middle and right plots: the distribution of NLPD of all methods on the binary and multi-class
classification tasks, respectively. The hard-coded methods are slightly better than AGP.
7
4
Posteriors of the latent intensity
0.5
Intensity
Event counts
3
2
0.4
FULL
MIX
HMC & ESS
0.3
0.2
1
0.1
0
1860 1880 1900 1920 1940 1960
Time
0
1860 1880 1900 1920 1940 1960
Time
2.5
Log10 speed?up factor
0.6
2
1.5
1
FULL
MIX
ESS
0.5
0
Time comparison against HMC
Figure 4: Left plot: the true event counts during the given time period. Middle plot: the posteriors
(estimated intensities) inferred by all methods. For each method, the middle line is the posterior
mean and the two remaining lines enclose 90% interval. AGP-FULL infers the same posterior as
HMC and ESS while AGP-MIX obtains the same mean but underestimates the variance. Right
plot: speed-up factors against the HMC method. The AGP methods run more than 2 orders of
magnitude faster than the sampling methods.
The classification error rates and the NLPD are shown in Figure 3 for both tasks. For binary classification, the AGP methods give comparable performance to the hard-coded implementations, VBO
and EP. The latter is often considered the best approximation method for this task [6]. Similar
results can be observed for the multi-class classification problem.
We note that the running times of our methods are comparable to that of the hard-coded methods.
For example, the average training times for VBO, EP, MIX, and FULL are 76s, 63s, 210s, and 480s
respectively, on the Wisconsin dataset.
4.4
Log Gaussian Cox process (LGCP)
The LGCP is an inhomogeneous Poisson process with the log-intensity function being a shifted
?yn exp(?? )
draw from a Gaussian process. Following [4], we use the likelihood p(yn |fn ) = n yn ! n ,
where ?n = exp(fn + m) is the mean of a Poisson distribution and m is the offset to the log mean.
The data concerns coal-mining disasters taken from a standard dataset for testing point processes
[24]. The offset m and the covariance hyperparameters are set to the same values as in [4].
We compare AGP with the Hybrid Monte Carlo (HMC, [25]) and elliptical slice sampling (ESS,
[4]) methods, where the latter is designed specifically for GP models. We collected every 100th
sample for a total of 10k samples after a burn-in period of 5k samples; the Gelman-Rubin potential
scale reduction factors [26] are used to check for convergence. The middle plot of Figure 4 shows
the posteriors learned by all methods. We see that the posterior by AGP-FULL is similar to that
by HMC and ESS. AGP-MIX obtains the same posterior mean but it underestimates the variance.
The right plot shows the speed-up factors of all methods against the slowest method HMC. The
AGP methods run more than two orders of magnitude faster than HMC, thus confirming the computational advantages of our method to the sampling approaches. Training time was measured on a
desktop with Intel(R) i7-2600 3.40GHz CPU with 8GB of RAM using Matlab R2012a.
5
Discussion
We have developed automated variational inference for Gaussian process models (AGP). AGP performs as well as the exact or hard-coded implementations when testing on five models using six real
world datasets. AGP has the potential to be a powerful tool for GP practitioners and researchers
when devising models for new or existing problems for which variational inference is not yet available. In the future we will address the scalability of AGP to deal with very large datasets.
Acknowledgements
NICTA is funded by the Australian Government through the Department of Communications and
the Australian Research Council through the ICT Centre of Excellence Program.
8
References
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9
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4,832 | 5,375 | Variational Gaussian Process State-Space Models
Roger Frigola, Yutian Chen and Carl E. Rasmussen
Department of Engineering
University of Cambridge
{rf342,yc373,cer54}@cam.ac.uk
Abstract
State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian
processes. The result of learning is a tractable posterior over nonlinear dynamical
systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst
avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo. We also present stochastic
variational inference and online learning approaches for fast learning with long
time series.
1
Introduction
State-space models (SSMs) are a widely used class of models that have found success in applications
as diverse as robotics, ecology, finance and neuroscience (see, e.g., Brown et al. [3]). State-space
models generalize other popular time series models such as linear and nonlinear auto-regressive
models: (N)ARX, (N)ARMA, (G)ARCH, etc. [21].
In this article we focus on Bayesian learning of nonparametric nonlinear state-space models. In
particular, we use sparse Gaussian processes (GPs) [19] as a convenient method to encode general
assumptions about the dynamical system such as continuity or smoothness. In contrast to conventional parametric methods, we allow the user to easily trade off model capacity and computation
time. Moreover, we present a variational training procedure that allows very complex models to be
learned without risk of overfitting.
Our variational formulation leads to a tractable approximate posterior over nonlinear dynamical
systems. This approximate posterior can be used to compute fast probabilistic predictions of future
trajectories of the dynamical system. The computational complexity of our learning approach is
linear in the length of the time series. This is possible thanks to the use of variational sparse GPs [22]
which lead to a smoothing problem for the latent state trajectory in a simpler auxiliary dynamical
system. Smoothing in this auxiliary system can be carried out with any conventional technique (e.g.
sequential Monte Carlo). In addition, we present a stochastic variational inference procedure [10] to
accelerate learning for long time series and we also present an online learning scheme.
This work is useful in situations where: 1) it is important to know how uncertain future predictions
are, 2) there is not enough knowledge about the underlying nonlinear dynamical system to create
a principled parametric model, and 3) it is necessary to have an explicit model that can be used
to simulate the dynamical system into the future. These conditions arise often in engineering and
finance. For instance, consider an autonomous aircraft adapting its flight control when carrying a
large external load of unknown weight and aerodynamic characteristics. A model of the nonlinear
dynamics of the new system can be very useful in order to automatically adapt the control strategy.
When few data points are available, there is high uncertainty about the dynamics. In this situation,
1
a model that quantifies its uncertainty can be used to synthesize control laws that avoid the risks of
overconfidence.
The problem of learning flexible models of nonlinear dynamical systems has been tackled from
multiple perspectives. Ghahramani and Roweis [9] presented a maximum likelihood approach to
learn nonlinear SSMs based on radial basis functions. This work was later extended by using a
parameterized Gaussian process point of view and developing tailored filtering algorithms [6, 7, 23].
Approximate Bayesian learning has also been developed for parameterized nonlinear SSMs [5, 24].
Wang et al. [25] modeled the nonlinear functions in SSMs using Gaussian processes (GP-SSMs) and
found a MAP estimate of the latent variables and hyperparameters. Their approach preserved the
nonparametric properties of Gaussian processes. Despite using MAP learning over state trajectories,
overfitting was not an issue since it was applied in a dimensionality reduction context where the
latent space of the SSM was much smaller than the observation space. In a similar vein, [4, 12]
presented a hierarchical Gaussian process model that could model linear dynamics and nonlinear
mappings from latent states to observations. More recently, Frigola et al. [8] learned GP-SSMs
in a fully Bayesian manner by employing particle MCMC methods to sample from the smoothing
distribution. However, their approach led to predictions with a computational cost proportional to
the length of the time series.
In the rest of this article, we present an approach to variational Bayesian learning of flexible nonlinear state-space models which leads to a simple representation of the posterior over nonlinear
dynamical systems and results in predictions having a low computational complexity.
2
Gaussian Process State-Space Models
We consider discrete-time nonlinear state-space models built with deterministic functions and additive noise
xt+1 = f (xt ) + vt ,
(1a)
yt = g(xt ) + et .
(1b)
The dynamics of the system are defined by the state transition function f (xt ) and independent
additive noise vt (process noise). The states xt ? RD are latent variables such that all future
variables are conditionally independent on the past given the present state. Observations yt ? RE
are linked to the state via another deterministic function g(xt ) and independent additive noise et
(observation noise). State-space models are stochastic dynamical processes that are useful to model
time series y , {y1 , ..., yT }. The deterministic functions in (1) can also take external known inputs
(such as control signals) as an argument but, for conciseness, we will omit those in our notation.
A traditional approach to learn f and g is to restrict them to a family of parametric functions. This is
particularly appropriate when the dynamical system is very well understood, e.g. orbital mechanics
of a spacecraft. However, in many applications, it is difficult to specify a class of parametric models
that can provide both the ability to model complex functions and resistance to overfitting thanks to an
easy to specify prior or regularizer. Gaussian processes do have these properties: they can represent
functions of arbitrary complexity and provide a straightforward way to specify assumptions about
those unknown functions, e.g. smoothness. In the light of this, it is natural to place Gaussian process
priors over both f and g [25]. However, the extreme flexibility of the two Gaussian processes
leads to severe nonidentifiability and strong correlations between the posteriors of the two unknown
functions. In the rest of this paper we will focus on a model with a GP prior over the transition
function and a parametric likelihood. However, our variational formulation can also be applied to
the double GP case (see supplementary material).
A probabilistic state-space model with a Gaussian process prior over the transition function and a
parametric likelihood is specified by
f (x) ? GP mf (x), kf (x, x0 ) ,
(2a)
xt | ft ? N (xt | ft , Q),
x0 ? p(x0 )
yt | xt ? p(yt | xt , ? y ),
(2b)
(2c)
(2d)
where we have used ft , f (xt?1 ). Since f (x) ? RD , we use the convention that the covariance
function kf returns a D ? D matrix. We group all hyperparameters into ? , {? f , ? y , Q}. Note that
2
states
0
0
time
0
time
0
time
time
Figure 1: State trajectories from four 2-state nonlinear dynamical systems sampled from a GP-SSM
prior with fixed hyperparameters. The same prior generates systems with qualitatively different
behaviors, e.g. the leftmost panel shows behavior similar to that of a non-oscillatory linear system
whereas the rightmost panel appears to have arisen from a limit cycle in a nonlinear system.
we are not restricting the likelihood (2d) to any particular form. The joint distribution of a GP-SSM
is
T
Y
p(y, x, f ) = p(x0 )
p(yt |xt )p(xt |ft )p(ft |f1:t?1 , x0:t?1 ),
(3)
t=1
where we use the convention f1:0 = ? and omit the conditioning on ? in the notation. The GP on
the transition function induces a distribution over the latent function values with the form of a GP
predictive:
p(ft |f1:t?1 , x0:t?1 ) = N mf (xt?1 ) + Kt?1,0:t?2 K?1
0:t?2,0:t?2 (f1:t?1 ? mf (x0:t?2 )),
?1
Kt?1,t?1 ? Kt?1,0:t?2 K0:t?2,0:t?2 K>
(4)
t?1,0:t?2 ,
where the subindices of the kernel matrices indicate the arguments to the covariance function necessary to build each matrix, e.g. Kt?1,0:t?2 = [kf (xt?1 , x0 ) . . . kf (xt?1 , xt?2 )]. When t = 1, the
distribution is that of a GP marginal p(f1 |x0 ) = N (mf (x0 ), kf (x0 , x0 )).
Equation (3) provides a sequential procedure to sample state trajectories and observations. GPSSMs are doubly stochastic models in the sense that one could, at least notionally, first sample a
state transition dynamics function from eq. (2a) and then, conditioned on that function, sample the
state trajectory and observations.
GP-SSMs are a very rich prior over nonlinear dynamical systems. In Fig. 1 we illustrate this concept
by showing state trajectories sampled from a GP-SSM with fixed hyperparameters. The dynamical
systems associated with each of these trajectories are qualitatively very different from each other. For
instance, the leftmost panel shows the dynamics of an almost linear non-oscillatory system whereas
the rightmost panel corresponds to a limit cycle in a nonlinear system. Our goal in this paper is
to use this prior over dynamical systems and obtain a tractable approximation to the posterior over
dynamical systems given the data.
3
Variational Inference in GP-SSMs
Since the GP-SSM is a nonparametric model, in order to define a posterior distribution over f (x) and
make probabilistic predictions it is necessary to first find the smoothing distribution p(x0:T |y1:T ).
Frigola et al. [8] obtained samples from the smoothing distribution that could be used to define a
predictive density via Monte Carlo integration. This approach is expensive since it requires averaging over L state trajectory samples of length T . In this section we present an alternative approach
that aims to find a tractable distribution over the state transition function that is independent of the
length of the time series. We achieve this by using variational sparse GP techniques [22].
3.1
Augmenting the Model with Inducing Variables
As a first step to perform variational inference in a GP-SSM, we augment the model with M inducing
points u , {ui }M
i=1 . Those inducing points are jointly Gaussian with the latent function values. In
the case of a GP-SSM, the joint probability density becomes
p(y, x, f , u) = p(x, f |u) p(u)
T
Y
t=1
3
p(yt |xt ),
(5)
where
p(u) = N (u | 0, Ku,u )
p(x, f |u) = p(x0 )
T
Y
(6a)
p(ft |f1:t?1 , x0:t?1 , u)p(xt |ft ),
(6b)
t=1
T
Y
?1
>
p(ft |f1:t?1 , x0:t?1 , u) = N f1:T | K0:T ?1,u K?1
u,u u, K0:T ?1 ? K0:T ?1,u Ku,u K0:T ?1,u . (6c)
t=1
Kernel matrices relating to the inducing points depend on a set of inducing inputs {zi }M
i=1 in such
a way that Ku,u is an M D ? M D matrix formed with blocks kf (zi , zj ) having size D ? D. For
brevity, we use a zero mean function and we omit conditioning on the inducing inputs in the notation.
3.2
Evidence Lower Bound of an Augmented GP-SSM
Variational inference [1] is a popular method for approximate Bayesian inference based on making
assumptions about the posterior over latent variables that lead to a tractable lower bound on the
evidence of the model (sometimes referred to as ELBO). Maximizing this lower bound is equivalent
to minimizing the Kullback-Leibler divergence between the approximate posterior and the exact
one. Following standard variational inference methodology, [1] we obtain the evidence lower bound
of a GP-SSM augmented with inducing points
QT
Z
p(u)p(x0 ) t=1 p(ft |f1:t?1 , x0:t?1 , u)p(yt |xt )p(xt |ft )
log p(y|?) ?
q(x, f , u) log
.
(7)
q(x, f , u)
x,f ,u
In order to achieve tractability, we use a variational distribution that factorizes as
q(x, f , u) = q(u)q(x)
T
Y
p(ft |f1:t?1 , x0:t?1 , u),
(8)
t=1
where q(u) and q(x) can take any form but the terms relating to f are taken to match those of the
prior (3). As a consequence, the difficult p(ft |...) terms inside the log cancel out and lead to the
following lower bound
Z
L(q(u), q(x),?) = ?KL(q(u)kp(u)) + H(q(x)) + q(x) log p(x0 )
x
+
T Z
X
t=1
q(x)q(u)
x,u
ft
|
Z
Z
p(ft |xt?1 , u) log p(xt |ft ) +
{z
}
q(x) log p(yt |xt )
(9)
x
?(xt ,xt?1 ,u)
where KL denotes the Kullback-Leibler divergence and H the entropy. The integral with respect to
ft can be solved analytically: ?(xt , xt?1 , u) = ? 21 tr(Q?1 Bt?1 ) + log N (xt |At?1 u, Q) where
?1
At?1 = Kt?1,u K?1
u,u , and Bt?1 = Kt?1,t?1 ? Kt?1,u Ku,u Ku,t?1 .
As in other variational sparse GP methods, the choice of variational distribution (8) gives the ability to precisely learn the latent function at the locations of the inducing inputs. Away from those
locations, the posterior takes the form of the prior conditioned on the inducing variables. By increasing the number of inducing variables, the ELBO can only become tighter [22]. This offers a
straightforward trade-off between model capacity and computation cost without increasing the risk
of overfitting.
3.3
Optimal Variational Distribution for u
The optimal distribution of q(u) can be found by setting to zero the functional derivative of the
evidence lower bound with respect to q(u)
q ? (u) ? p(u)
T
Y
exp{hlog N (xt |At?1 u, Q)iq(x) },
t=1
4
(10)
where h?iq(x) denotes an expectation with respect to q(x). The optimal variational distribution
q ? (u) is, conveniently, a multivariate Gaussian distribution. If, for simplicity of notation, we restrict
ourselves to D = 1 the natural parameters of the optimal distribution are
?1
?1 = Q
T
X
hATt?1 xt iq(xt ,xt?1 ) ,
t=1
1
?2 = ?
2
K?1
uu
?1
+Q
T
X
hATt?1 At?1 iq(xt?1 )
!
. (11)
t=1
The mean and covariance matrix of q ? (u), denoted as ? and ? respectively, can be computed as
? = ?? 1 and ? = (?2? 2 )?1 . Note that the optimal q(u) depends on the sufficient statistics
PT
PT
?1 = t=1 hKTt?1,u xt iq(xt ,xt?1 ) and ?2 = t=1 hKTt?1,u Kt?1,u iq(xt?1 ) .
3.4
Optimal Variational Distribution for x
In an analogous way as for q ? (u), we can obtain the optimal form of q(x)
q ? (x) ? p(x0 )
T
Y
1
p(yt |xt ) exp{? tr Q?1 (Bt?1 + At?1 ?ATt?1 ) } N (xt |At?1 ?, Q), (12)
2
t=1
where, in the second equation, we have used q(u) = N (u|?, ?).
The optimal distribution q ? (x) is equivalent to the smoothing distribution of an auxiliary parametric
state-space model. The auxiliary model is simpler than the original one in (3) since the latent states
factorize with a Markovian structure. Equation (12) can be interpreted as a nonlinear state-space
model with a Gaussian state transition density, N (xt |At?1 ?,
Q), and a likelihood augmented with
an additional term: exp{? 21 tr Q?1 (Bt?1 + At?1 ?ATt?1 ) }.
Smoothing in nonlinear Markovian state-space models is a standard problem in the context of time
series modeling. There are various existing strategies to find the smoothing distribution which could
be used depending on the characteristics of each particular problem [20]. For instance, in a mildly
nonlinear system with Gaussian noise, an extended Kalman smoother can have very good performance. On the other hand, problems with severe nonlinearities and/or non-Gaussian likelihoods can
lead to heavily multimodal smoothing distributions that are better represented using particle methods. We note that the application of sequential Monte Carlo (SMC) is particularly straightforward
in the present auxiliary model.
3.5
Optimizing the Evidence Lower Bound
Algorithm 1 presents a procedure to maximize the evidence lower bound by alternatively sampling
from the smoothing distribution and taking steps both in ? and in the natural parameters of q ? (u).
We propose a hybrid variational-sampling approach whereby approximate samples from q ? (x) are
obtained with a sequential Monte Carlo smoother. However, as discussed in section 3.4, depending
on the characteristics of the dynamical system, other smoothing methods could be more appropriate
[20]. As an alternative to smoothing on the auxiliary dynamical system in (12), one could force a
q(x) from a particular family of distributions and optimise the evidence lower bound with respect
to its variational parameters. For instance, we could posit a Gaussian q(x) with a sparsity pattern in
the covariance matrix assuming zero covariance between non-neighboring states and maximize the
ELBO with respect to the variational parameters.
We use stochastic gradient descent [10] to maximize the ELBO (where we have plugged in the
optimal q ? (u) [22]) by using its gradient with respect to the hyperparameters. Both quantities are
stochastic in our hybrid approach due to variance introduced by the sampling of q ? (x). In fact,
vanilla sequential Monte Carlo methods will result in biased estimators of the gradient and the
parameters of q ? (u). However, in our experiments this has not been an issue. Techniques such as
particle MCMC would be a viable alternative to conventional sequential Monte Carlo [13].
5
Algorithm 1 Variational learning of GP-SSMs with particle smoothing. Batch mode (i.e. non-SVI)
is the particular case where the mini-batch is the whole dataset.
Require: Observations y1:T . Initial values for ?, ? 1 and ? 2 . Schedules for ? and ?. i = 1.
repeat
y? :? 0 ? S AMPLE M INI BATCH(y1:T )
{x? :? 0 }L
sample from eq. (12)
l=1 ? G ET S AMPLES O PTIMAL QX(y? :? 0 , ?, ? 1 , ? 2 )
?? L ? G ET T HETAG RADIENT({x? :? 0 }L
,
?)
supp. material
l=1
? ?1 , ? ?2 ? G ET O PTIMAL QU({x? :? 0 }L
,
?)
eq.
(11) or (14)
l=1
? 1 ? ? 1 + ?i (? ?1 ? ? 1 )
? 2 ? ? 2 + ?i (? ?2 ? ? 2 )
? ? ? + ? i ?? L
i?i+1
until ELBO convergence
3.6
Making Predictions
One of the most appealing properties of our variational approach to learning GP-SSMs is that the
approximate predictive distribution of the state transition function can be cheaply computed
Z
Z
p(f? |x? , y) =
p(f? |x? , x, u) p(x|u, y) p(u|y) ?
p(f? |x? , u) p(x|u, y) q(u)
x,u
x,u
Z
=
p(f? |x? , u) q(u) = N (f? |A? ?, B? + A? ?A>
(13)
? ).
u
The derivation in eq. (13) contains two approximations: 1) predictions at new test points are considered to depend only on the inducing variables, and 2) the posterior distribution over u is approximated by a variational distribution.
After pre-computations, the cost of each prediction is O(M ) for the mean and O(M 2 ) for the
variance. This contrasts with the O(T L) and O(T 2 L) complexity
of approaches based on sampling
R
from the smoothing distribution where p(f? |x? , y) = x p(f? |x? , x) p(x|y) is approximated with
L samples from p(x|y) [8]. The variational approach condenses the learning of the latent function
on the inducing points u and does not explicitly need the smoothing distribution p(x|y) to make
predictions.
4
Stochastic Variational Inference
Stochastic variational inference (SVI) [10] can be readily applied using our evidence lower bound.
When the observed time series is long, it can be expensive to compute q ? (u) or the gradient of L with
?L
respect to the hyperparameters and inducing inputs. Since both q ? (u) and ??/z
depend linearly
1:M
on q(x) via sufficient statistics that contain a summation over all elements in the state trajectory,
we can obtain unbiased estimates of these sufficient statistics by using one or multiple segments
of the sequence that are sampled uniformly at random. However, obtaining q(x) also requires a
time complexity of O(T ). Yet, in practice, q(x) can be approximated by running the smoothing
algorithm locally around those segments. This can be justified by the fact that in a time series
context, the smoothing distribution at a particular time is not largely affected by measurements that
are far into the past or the future [20]. The natural parameters of q ? (u) can be estimated by using a
portion of the time series of length S
?
?
?0
?0
X
X
1
T
?1 T
? 1 = Q?1
hAT xt iq(xt ,xt?1 ) , ? 2 = ? ?K?1
hAT At?1 iq(xt?1 ) ? . (14)
uu + Q
S t=? t?1
2
S t=? t?1
5
Online Learning
Our variational approach to learn GP-SSMs also leads naturally to an online learning implementation. This is of particular interest in the context of dynamical systems as it is often the case that data
arrives in a sequential manner, e.g. a robot learning the dynamics of different objects by interacting
6
Table 1: Experimental evaluation of 1D nonlinear system. Unless otherwise stated, training times
are reported for a dataset with T = 500 and test times are given for a test set with 105 data points. All
pre-computations independent on test data are performed before timing the ?test time?. Predictive
log likelihoods are the average over the full test set. * our PMCMC code did not use fast updatesdowndates of the Cholesky factors during training. This does not affect test times.
Variational GP-SSM
Var. GP-SSM (SVI, T = 104 )
PMCMC GP-SSM [8]
GP-NARX [17]
GP-NARX + FITC [17, 18]
Linear (N4SID, [16])
Test RMSE
test
tr
log p(xtest
t+1 |xt , y0:T )
1.15
1.07
1.12
1.46
1.47
2.35
-1.61
-1.47
-1.57
-1.90
-1.90
-2.30
Train time
Test time
2.14 min
4.12 min
547 min*
0.22 min
0.17 min
0.01 min
0.14 s
0.14 s
421 s
3.85 s
0.23 s
0.11 s
with them. Online learning in a Bayesian setting consists in sequential application of Bayes rule
whereby the posterior after observing data up to time t becomes the prior at time t + 1 [2, 15].
In our case, this involves replacing the prior p(u) = N (u|0, Ku,u ) by the approximate posterior
N (u|?, ?) obtained in the previous step. The expressions for the update of the natural parameters
of q ? (u) with a new mini batch y? :? 0 are
?0
?0
X
1 ?1 X T
0
?1
T
0
hAt?1 At?1 iq(xt?1 ) .
(15)
?1 = ?1 + Q
hAt?1 xt iq(xt ,xt?1 ) , ? 2 = ? 2 ? Q
2
t=?
t=?
6
Experiments
The goal of this section is to showcase the ability of variational GP-SSMs to perform approximate
Bayesian learning of nonlinear dynamical systems. In particular, we want to demonstrate: 1) the
ability to learn the inherent nonlinear dynamics of a system, 2) the application in cases where the
latent states have higher dimensionality than the observations, and 3) the use of non-Gaussian likelihoods.
6.1 1D Nonlinear System
We apply our variational learning procedure presented above to the one-dimensional nonlinear system described by p(xt+1 |xt ) = N (f (xt ), 1) and p(yt |xt ) = N (xt , 1) where the transition function
is xt + 1 if x < 4 and ?4xt + 21 if x ? 4. Its pronounced kink makes it challenging to learn. Our
goal is to find a posterior distribution over this function using a GP-SSM with Mat?ern covariance
function. To solve the expectations with respect to the approximate smoothing distribution q(x) we
use a bootstrap particle fixed-lag smoother with 1000 particles and a lag of 10.
In Table 1, we compare our method (Variational GP-SSM) against the PMCMC sampling procedure from [8] taking 100 samples and 10 burn in samples. As in [8], the sampling exhibited very
good mixing with 20 particles. We also compare to an auto-regressive model based on Gaussian
process regression [17] of order 5 with Mat?ern ARD covariance function with and without FITC
approximation. Finally, we use a linear subspace identification method (N4SID, [16]) as a baseline for comparison. The PMCMC training offers the best test performance from all methods using
500 training points at the cost of substantial train and test time. However, if more data is available
(T = 104 ) the stochastic variational inference procedure can be very attractive since it improves test
performance while having a test time that is independent of the training set size. The reported SVI
performance has been obtained with mini-batches of 100 time-steps.
6.2 Neural Spike Train Recordings
We now turn to the use of SSMs to learn a simple model of neural activity in rats? hippocampus. We
use data in neuron cluster 1 (the most active) from experiment ec013.717 in [14]. In some regions
of the time series, the action potential spikes show a clear pattern where periods of rapid spiking
are followed by periods of very little spiking. We wish to model this behaviour as an autonomous
nonlinear dynamical system (i.e. one not driven by external inputs). Many parametric models of
nonlinear neuron dynamics have been proposed [11] but our goal here is to learn a model from data
7
30
20
0
10
940
940.5
941
940
time [s]
940.5
30
states
spike counts
40
states
spike counts
40
20
0
10
941
0
time [s]
0.5
1
0
prediction time [s]
0.5
1
prediction time [s]
x(1)
0
x(2)
0
x(2)
x(2)
x(2)
Figure 2: From left to right: 1) part of the observed spike count data, 2) sample from the corresponding smoothing distribution, 3) predictive distribution of spike counts obtained by simulating
the posterior dynamical from an initial state, and 4) corresponding latent states.
0
x(1)
0
x(1)
(2)
xt+1
x(1)
(1)
(2)
f (xt , xt ),
Figure 3: Contour plots of the state transition function
=
and trajectories in
state space. Left: mean posterior function and trajectory from smoothing distribution. Other three
panels: transition functions sampled from the posterior and trajectories simulated conditioned on
the corresponding sample. Those simulated trajectories start inside the limit cycle and are naturally
attracted towards it. Note how function samples are very similar in the region of the limit cycle.
without using any biological insight. We use a GP-SSM with a structure such that it is the discretetime analog of a second order nonlinear ordinary differential equation: two states one of which
is the derivative of the other. The observations are spike counts in temporal bins of 0.01 second
width. We use a Poisson likelihood relating the spike counts to the second latent state yt |xt ?
(2)
Poisson(exp(?xt + ?)).
We find a posterior distribution for the state transition function using our variational GP-SSM approach. Smoothing is done with a fixed-lag particle smoother and training until convergence takes
approximately 50 iterations of Algorithm 1. Figure 2 shows a part of the raw data together with an
approximate sample from the smoothing distribution during the same time interval. In addition, we
show the distribution over predictions made by chaining 1-step-ahead predictions. To make those
predictions we have switched off process noise (Q = 0) to show more clearly the effect of uncertainty in the state transition function. Note how the frequency of roughly 6 Hz present in the data is
well captured. Figure 3 shows how the limit cycle corresponding to a nonlinear dynamical system
has been captured (see caption for details).
7
Discussion and Future Work
We have derived a tractable variational formulation to learn GP-SSMs: an important class of models of nonlinear dynamical systems that is particularly suited to applications where a principled
parametric model of the dynamics is not available. Our approach makes it possible to learn very expressive models without risk of overfitting. In contrast to previous approaches [4, 12, 25], we have
demonstrated the ability to learn a nonlinear state transition function in a latent space of greater
dimensionality than the observation space. More crucially, our approach yields a tractable posterior
over nonlinear systems that, as opposed to those based on sampling from the smoothing distribution
[8], results in a computation time for the predictions that does not depend on the length of the time
series.
Given the interesting capabilities of variational GP-SSMs, we believe that future work is warranted.
In particular, we want to focus on structured variational distributions q(x) that could eliminate the
need to solve the smoothing problem in the auxiliary dynamical system at the cost of having more
variational parameters to optimize. On a more theoretical side, we would like to better characterize
GP-SSM priors in terms of their dynamical system properties: stability, equilibria, limit cycles, etc.
8
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9
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4,833 | 5,376 | Gaussian Process Volatility Model
Jos?e Miguel Hern?andez Lobato
Cambridge University
[email protected]
Yue Wu
Cambridge University
[email protected]
Zoubin Ghahramani
Cambridge University
[email protected]
Abstract
The prediction of time-changing variances is an important task in the modeling of
financial data. Standard econometric models are often limited as they assume rigid
functional relationships for the evolution of the variance. Moreover, functional
parameters are usually learned by maximum likelihood, which can lead to overfitting. To address these problems we introduce GP-Vol, a novel non-parametric
model for time-changing variances based on Gaussian Processes. This new model
can capture highly flexible functional relationships for the variances. Furthermore,
we introduce a new online algorithm for fast inference in GP-Vol. This method
is much faster than current offline inference procedures and it avoids overfitting
problems by following a fully Bayesian approach. Experiments with financial data
show that GP-Vol performs significantly better than current standard alternatives.
1
Introduction
Time series of financial returns often exhibit heteroscedasticity, that is the standard deviation or
volatility of the returns is time-dependent. In particular, large returns (either positive or negative) are
often followed by returns that are also large in size. The result is that financial time series frequently
display periods of low and high volatility. This phenomenon is known as volatility clustering [1].
Several univariate models have been proposed in the literature for capturing this property. The best
known and most popular is the Generalised Autoregressive Conditional Heteroscedasticity model
(GARCH) [2]. An alternative to GARCH are stochastic volatility models [3]. However, there is no
evidence that SV models have better predictive performance than GARCH [4, 5, 6].
GARCH has further inspired a host of variants and extensions. A review of many of these models
can be found in [7]. Most of these GARCH variants attempt to address one or both limitations of
GARCH: a) the assumption of a linear dependency between current and past volatilities, and b)
the assumption that positive and negative returns have symmetric effects on volatility. Asymmetric
effects are often observed, as large negative returns often send measures of volatility soaring, while
this effect is smaller for large positive returns [8, 9]. Finally, there are also extensions that use
additional data besides daily closing prices to improve volatility predictions [10].
Most solutions proposed in these variants of GARCH involve: a) introducing nonlinear functional
relationships for the evolution of volatility, and b) adding asymmetric effects in these functional
relationships. However, the GARCH variants do not fundamentally address the problem that the
specific functional relationship of the volatility is unknown. In addition, these variants can have a
high number of parameters, which may lead to overfitting when using maximum likelihood learning.
More recently, volatility modeling has received attention within the machine learning community,
with the development of copula processes [11] and heteroscedastic Gaussian processes [12]. These
1
models leverage the flexibility of Gaussian Processes [13] to model the unknown relationship between the variances. However, these models do not address the asymmetric effects of positive and
negative returns on volatility.
We introduce a new non-parametric volatility model, called the Gaussian Process Volatility Model
(GP-Vol). This new model is more flexible, as it is not limited by a fixed functional form. Instead, a
non-parametric prior distribution is placed on possible functions, and the functional relationship is
learned from the data. This allows GP-Vol to explicitly capture the asymmetric effects of positive
and negative returns on volatility. Our new volatility model is evaluated in a series of experiments
with real financial returns, and compared against popular econometric models, namely, GARCH,
EGARCH [14] and GJR-GARCH [15]. In these experiments, GP-Vol produces the best overall
predictions. In addition to this, we show that the functional relationship learned by GP-Vol often
exhibits the nonlinear and asymmetric features that previous models attempt to capture.
The second main contribution of the paper is the development of an online algorithm for learning
GP-Vol. GP-Vol is an instance of a Gaussian Process State Space Model (GP-SSM). Previous work
on GP-SSMs [16, 17, 18] has mainly focused on developing approximation methods for filtering
and smoothing the hidden states in GP-SSM, without jointly learning the GP transition dynamics.
Only very recently have Frigola et al. [19] addressed the problem of learning both the hidden states
and the transition dynamics by using Particle Gibbs with Ancestor Sampling (PGAS) [20]. In this
paper, we introduce a new online algorithm for performing inference on GP-SSMs. Our algorithm
has similar predictive performance as PGAS on financial data, but is much faster.
2
Review of GARCH and GARCH variants
The standard variance model for financial data is GARCH. GARCH assumes a Gaussian observation
model and a linear transition function for the variance: the time-varying variance ?t2 is linearly
dependent on p previous variance values and q previous squared time series values, that is,
Pq
Pp
2
xt ? N (0, ?t2 ) ,
and
?t2 = ?0 + j=1 ?j x2t?j + i=1 ?i ?t?i
,
(1)
where xt are the values of the return time series being modeled. This model is flexible and can
produce a variety of clustering behaviors of high and low volatility periods for different settings
of ?1 , . . . , ?q and ?1 , . . . , ?p . However, it has several limitations. First, only linear relationships
2
between ?t?p:t?1
and ?t2 are allowed. Second, past positive and negative returns have the same
effect on ?t2 due to the quadratic term x2t?j . However, it is often observed that large negative returns
lead to larger rises in volatility than large positive returns [8, 9].
A more flexible and often cited GARCH extension is Exponential GARCH (EGARCH) [14]. The
equation for ?t2 is now:
Pq
Pp
2
log(?t2 ) = ?0 + j=1 ?j g(xt?j ) + i=1 ?i log(?t?i
) , where g(xt ) = ?xt + ? |xt | . (2)
Asymmetry in the effects of positive and negative returns is introduced through the function g(xt ). If
the coefficient ? is negative, negative returns will increase volatility, while the opposite will happen
if ? is positive. Another GARCH extension that models asymmetric effects is GJR-GARCH [15]:
Pq
Pp
Pr
2
?t2 = ?0 + j=1 ?j x2t?j + i=1 ?i ?t?i
+ k=1 ?k x2t?k It?k ,
(3)
where It?k = 0 if xt?k ? 0 and It?k = 1 otherwise. The asymmetric effect is now captured by
It?k , which is nonzero if xt?k < 0.
3
Gaussian process state space models
GARCH, EGARCH and GJR-GARCH can be all represented as General State-Space or Hidden
Markov models (HMM) [21, 22], with the unobserved dynamic variances being the hidden states.
Transition functions for the hidden states are fixed and assumed to be linear in these models. The
linear assumption limits the flexibility of these models.
More generally, a non-parametric approach can be taken where a Gaussian Process (GP) prior is
placed on the transition function, so that its functional form can be learned from data. This Gaussian
Process state space model (GP-SSM) is a generalization of HMM. GP-SSM and HMM differ in two
main ways. First, in HMM the transition function has a fixed functional form, while in GP-SSM
2
3
2.5
truth
GP?Vol 5%
GP?Vol 95%
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
0
20
40
60
80
Number of Observations
truth
GP?Vol 5%
GP?Vol 95%
2
100
?0.5
0
20
40
60
80
100
Number of Observations
Figure 1: Left, graphical model for GP-Vol. The transitions of the hidden states vt is represented by
the unknown function f . f takes as inputs the previous state vt?1 and previous observation xt?1 .
Middle, 90% posterior interval for a. Right, 90% posterior interval for b.
it is represented by a GP. Second, in GP-SSM the states do not have Markovian structure once the
transition function is marginalized out.
The flexibility of GP-SSMs comes at a cost: inference in GP-SSMs is computationally challenging.
Because of this, most of the previous work on GP-SSMs [16, 17, 18] has focused on filtering and
smoothing the hidden states in GP-SSM, without jointly learning the GP dynamics. Note that in
[18], the authors learn the dynamics, but using a separate dataset in which both input and target
values for the GP model are observed. A few papers considered learning both the GP dynamics and
the hidden states for special cases of GP-SSMs. For example, [23] applied EM to obtain maximum
likelihood estimates for parametric systems that can be represented by GPs. A general method has
been recently proposed for joint inference on the hidden states and the GP dynamics using Particle
Gibbs with Ancestor Sampling (PGAS) [20, 19]. However, PGAS is a batch MCMC inference
method that is computationally very expensive.
4
Gaussian process volatility model
Our new Gaussian Process Volatility Model (GP-Vol) is an instance of GP-SSM:
xt ? N (0, ?t2 ) ,
vt := log(?t2 ) = f (vt?1 , xt?1 ) + t ,
t ? N (0, ?n2 ) .
(4)
Note that we model the logarithm of the variance, which has real support. Equation (4) defines
a GP-SMM. We place a GP prior on the transition function f . Let zt = (vt , xt ). Then f ?
GP(m, k) where m(zt ) and k(zt , zt0 ) are the GP mean and covariance functions, respectively. The
mean function can encode prior knowledge of the system dynamics. The covariance function gives
the prior covariance between function values: k(zt , zt0 ) = Cov(f (zt ), f (zt0 )) . Intuitively if zt and zt0
are close to each other, the covariances between the corresponding function values should be large:
f (zt ) and f (zt0 ) should be highly correlated.
The graphical model for GP-Vol is given in Figure 1. The explicit dependence of transition function
values on the previous return xt?1 enables GP-Vol to model the asymmetric effects of positive and
negative returns on the variance evolution. GP-Vol can be extended to depend on p previous log
variances and q past returns like in GARCH(p,q). In this case, the transition would be of the form
vt = f (vt?1 , vt?2 , ..., vt?p , xt?1 , xt?2 , ..., xt?q ) + t .
5
Bayesian inference in GP-Vol
In the standard GP regression setting, the inputs and targets are fully observed and f can be learned
using exact Bayesian inference [13]. However, this is not the case in GP-Vol, where the unknown
{vt } form part of the inputs and all the targets. Let ? denote the model hyper-parameters and let
f = [f (v1 ), . . . , f (vT )]. Directly learning the joint posterior of the unknown variables f , v1:T and
? is a challenging task. Fortunately, the posterior p(vt |?, x1:t ), where f has been marginalized out,
can be approximated with particles [24]. We first describe a standard sequential Monte Carlo (SMC)
particle filter to learn this posterior.
i
Let {v1:t?1
}N
i=1 be particles representing chains of states up to t ? 1 with corresponding normalized
i
weights Wt?1 . The posterior p(v1:t?1 |?, x1:t?1 ) is then approximated by
PN
i
i
p?(v1:t?1 |?, x1:t?1 ) = i=1 Wt?1
?v1:t?1
(v1:t?1 ) .
(5)
3
The corresponding posterior for v1:t can be approximated by propagating these particles forward.
For this, we propose new states from the GP-Vol transition model and then we importance-weight
j
them according to the GP-Vol observation model. Specifically, we resample particles v1:t?1
from
j
(5) according to their weights Wt?1 , and propagate the samples forward. Then, for each of the
j
, x1:t?1 ), which is the GP predictive
particles propagated forward, we propose vtj from p(vt |?, v1:t?1
distribution. The proposed particles are then importance-weighted according to the observation
model, that is, Wtj ? p(xt |?, vtj ) = N (xt |0, exp{vtj }).
The above setup assumes that ? is known. To learn these hyper-parameters, we can also encode them
in particles and filter them together with the hidden states. However, since ? is constant across time,
naively filtering such particles without regeneration will fail due to particle impoverishment, where
a few or even one particle receives all the weight. To solve this problem, the Regularized Auxiliary
Particle Filter (RAPF) regenerates parameter particles by performing kernel smoothing operations
[25]. This introduces artificial dynamics and estimation bias. Nevertheless, RAPF has been shown
to produce state-of-the-art inference in multivariate parametric financial models [6].
RAPF was designed for HMMs, but GP-Vol is non-Markovian once f is marginalized out. Therefore,
we design a new version of RAPF for non-Markovian systems and refer to it as the Regularized
Auxiliary Particle Chain Filter (RAPCF), see Algorithm 1. There are two main parts in RAPCF.
First, there is the Auxiliary Particle Filter (APF) part in lines 5, 6 and 7 of the pseudocode [26].
This part selects particles associated with high expected likelihood, as given by the new expected
state in (7) and the corresponding resampling weight in (8). This bias towards particles with high
expected likelihood is eliminated when the final importance weights are computed in (9). The most
promising particles are propagated forward in lines 8 and 9. The main difference between RAPF and
i
RAPCF is in the effect that previous states v1:t?1
have in the propagation of particles. In RAPCF
all the previous states determine the probabilities of the particles being propagated, as the model is
i
non-Markovian, while in RAPF these probabilities are only determined by the last state vt?1
. The
second part of RAPCF avoids particle impoverishment in ?. For this, new particles are generated
in line 10 by sampling from a Gaussian kernel. The over-dispersion introduced by these artificial
dynamics is eliminated in (6) by shrinking the particles towards their empirical average. We fix the
shrinking parameter ? to be 0.95. In practice, we found little difference in predictions when we
varied ? from 0.99 to 0.95.
RAPCF has limitations similar to those of RAPF. First, it introduces bias as sampling from the
kernel adds artificial dynamics. Second, RAPCF only filters forward and does not smooth backward.
Consequently, there will be impoverishment in distant ancestors vt?L , since these states are not
regenerated. When this occurs, GP-Vol will consider the collapsed ancestor states as inputs with
little uncertainty and the predictive variance near these inputs will be underestimated. These issues
can be addressed by adopting a batch MCMC approach. In particular, Particle Markov Chain Monte
Carlo (PMCMC) procedures [24] established a framework for learning the states and the parameters
in general state space models. Additionally, [20] developed a PMCMC algorithm called Particle
Gibbs with ancestor sampling (PGAS) for learning non-Markovian state space models. PGAS was
applied by [19] to learn GP-SSMs. These batch MCMC methods are computationally much more
expensive than RAPCF. Furthermore, our experiments show that in the GP-Vol model, RAPCF and
PGAS have similar empirical performance, while RAPCF is orders of magnitude faster than PGAS.
This indicates that the aforementioned issues have limited impact in practice.
6
Experiments
We performed three sets of experiments. First, we tested on synthetic data whether we can jointly
learn the hidden states and transition dynamics in GP-Vol using RAPCF. Second, we compared
the performance of GP-Vol against standard econometric models GARCH, EGARCH and GJRGARCH on fifty real financial time series. Finally, we compared RAPCF with the batch MCMC
method PGAS in terms of accuracy and execution time. The code for RAPCF in GP-Vol is publicly
available at http://jmhl.org.
6.1
Experiments with synthetic data
We generated ten synthetic datasets of length T = 100 according to (4). The transition function f is
sampled from a GP prior specified with a linear mean function and a squared exponential covariance
4
Algorithm 1 RAPCF
1: Input: data x1:T , number of particles N , shrinkage parameter 0 < ? < 1, prior p(?).
2: Sample N parameter particles from the prior: {?0i }i=1,...,N ? p(?).
3: Set initial importance weights, W0i = 1/N .
4: for t = 1 to T do
PN
i
i
5:
Shrink parameter particles towards their empirical mean ??t?1 = i=1 Wt?1
?t?1
by setting
i
i
e
?
? = ??
+ (1 ? ?)?t?1 .
(6)
t
t?1
Compute the new expected states:
i
?it = E(vt |?eti , v1:t?1
, x1:t?1 ) .
(7)
Compute importance weights proportional to the likelihood of the new expected states:
i
gti ? Wt?1
p(xt |?it , ?eti ) .
(8)
Resample N auxiliary indices {j} according to weights {gti }.
j
}j?J .
Propagate the corresponding chains of hidden states forward, that is, {v1:t?1
Add jitter: ?tj ? N (?etj , (1 ? ?2 )Vt?1 ), where Vt?1 is the empirical covariance of ?t?1 .
j
Propose new states vtj ? p(vt |?tj , v1:t?1
, x1:t?1 ).
Compute importance weights adjusting for the modified proposal:
(9)
Wtj ? p(xt |vtj , ?tj )/p(xt |?jt , ?etj ) ,
end for
j
Output: particles for chains of states v1:T
, particles for parameters ?tj and particle weights Wtj .
6:
7:
8:
9:
10:
11:
12:
13:
14:
function. The linear mean function is E(vt ) = m(vt?1 , xt?1 ) = avt?1 + bxt?1 . The squared
exponential covariance function is k(y, z) = ? exp(?0.5|y ? z|2 /l2 ) where l is the length-scale
parameter and ? is the amplitude parameter.
We used RAPCF to learn the hidden states v1:T and the hyper-parameters ? = (a, b, ?n , ?, l) using
non-informative diffuse priors for ?. In these experiments, RAPCF successfully recovered the state
and the hyper-parameter values. For the sake of brevity, we only include two typical plots of the 90%
posterior intervals for hyper-parameters a and b in the middle and right of Figures 1. The intervals
are estimated from the filtered particles for a and b at each time step t. In both plots, the posterior
intervals eventually concentrate around the true parameter values, shown as dotted blue lines.
6.2
Experiments with real data
We compared the predictive performances of GP-Vol, GARCH, EGARCH and GJR-GARCH on real
financial datasets. We used GARCH(1,1), EGARCH(1,1) and GJR-GARCH(1,1,1) models since
these variants have the least number of parameters and are consequently less affected by overfitting
problems. We considered fifty datasets, consisting of thirty daily Equity and twenty daily foreign
exchange (FX) time series. For the Equity series, we used daily closing prices. For FX, which
operate 24h a day, with no official daily closing prices, we cross-checked different pricing sources
and took the consensus price up to 4 decimal places at 10am New York, which is the time with
most market liquidity. Each of the resulting time series contains a total of T = 780 observations
from January 2008 to January 2011. The price data p1:T was pre-processed to eliminate prices
corresponding to times when markets were closed or not liquid. After this, prices were converted
into logarithmic returns, xt = log(pt /pt?1 ). Finally, the resulting returns were standardized to have
zero mean and unit standard deviation.
During the experiments, each method receives an initial time series of length 100. The different
models are trained on that data and then a one-step forward prediction is made. The performance of
each model is measured in terms of the predictive log-likelihood on the first return out of the training
set. Then the training set is augmented with the new observation and the training and prediction steps
are repeated. The whole process is repeated sequentially until no further data is received.
GARCH, EGARCH and GJR-GARCH were implemented using numerical optimization routines
provided by Kevin Sheppard 1 . A relatively long initial time series of length 100 was needed to
to train these models. Using shorter initial data resulted in wild jumps in the maximum likelihood
1
http:///www.kevinsheppard.com/wiki/UCSD_GARCH/
5
Nemenyi Test
CD
GJR
GP?VOL
GARCH
EGARCH
1
2
3
4
Figure 2: Comparison between GP-Vol, GARCH, EGARCH and GJR-GARCH via a Nemenyi test.
The figure shows the average rank across datasets of each method (horizontal axis). The methods
whose average ranks differ more than a critical distance (segment labeled CD) show significant
differences in performance at this confidence level. When the performances of two methods are
statistically different, their corresponding average ranks appear disconnected in the figure.
estimates of the model parameters. These large fluctuations produced very poor one-step forward
predictions. By contrast, GP-Vol is less susceptible to overfitting since it approximates the posterior
distribution using RAPCF instead of finding point estimates of the model parameters. We placed
broad non-informative priors on ? = (a, b, ?n , ?, l) and used N = 200 particles and shrinkage
parameter ? = .95 in RAPCF.
Dataset
GARCH EGARCH GJR
AUDUSD
BRLUSD
CADUSD
CHFUSD
CZKUSD
EURUSD
GBPUSD
IDRUSD
JPYUSD
KRWUSD
MXNUSD
MYRUSD
NOKUSD
NZDUSD
PLNUSD
SEKUSD
SGDUSD
TRYUSD
TWDUSD
ZARUSD
?1.303
?1.203
?1.402
?1.375
?1.422
?1.418
?1.382
?1.223
?1.350
?1.189
?1.220
?1.394
?1.416
?1.369
?1.395
?1.403
?1.382
?1.224
?1.384
?1.318
?1.514
?1.227
?1.409
?1.404
?1.473
?2.120
?3.511
?1.244
?2.704
?1.168
?3.438
?1.412
?1.567
?3.036
?1.385
?3.705
?2.844
?1.461
?1.377
?1.344
?1.305
?1.201
?1.402
?1.404
?1.422
?1.426
?1.386
?1.209
?1.355
?1.209
?1.278
?1.395
?1.419
?1.379
?1.382
?1.402
?1.398
?1.238
?1.388
?1.301
Table 1: FX series.
GP-Vol
?1.297
?1.180
?1.386
?1.359
?1.456
?1.403
?1.385
?1.039
?1.347
?1.154
?1.167
?1.392
?1.416
?1.389
?1.393
?1.407
?1.393
?1.236
?1.294
?1.304
Dataset GARCH EGARCH GJR
A
AA
AAPL
ABC
ABT
ACE
ADBE
ADI
ADM
ADP
ADSK
AEE
AEP
AES
AET
?1.304
?1.228
?1.234
?1.341
?1.295
?1.084
?1.335
?1.373
?1.228
?1.229
?1.345
?1.292
?1.151
?1.237
?1.285
?1.449
?1.280
?1.358
?1.976
?1.527
?2.025
?1.501
?1.759
?1.884
?1.720
?1.604
?1.282
?1.177
?1.319
?1.302
?1.281
?1.230
?1.219
?1.344
?1.3003
?1.106
?1.386
?1.352
?1.223
?1.205
?1.340
?1.263
?1.146
?1.234
?1.269
GP-Vol
?1.282
?1.218
?1.212
?1.337
?1.302
?1.073
?1.302
?1.356
?1.223
?1.211
?1.316
?1.166
?1.142
?1.197
?1.246
Table 2: Equity series 1-15.
Dataset GARCH EGARCH
AFL
AGN
AIG
AIV
AIZ
AKAM
AKS
ALL
ALTR
AMAT
AMD
AMGN
AMP
AMT
AMZN
?1.057
?1.270
?1.151
?1.111
?1.423
?1.230
?1.030
?1.339
?1.286
?1.319
?1.342
?1.191
?1.386
?1.206
?1.206
?1.126
?1.338
?1.256
?1.147
?1.816
?1.312
?1.034
?3.108
?1.443
?1.465
?1.348
?1.542
?1.444
?1.820
?1.567
GJR
GP-Vol
?1.061
?1.261
?1.195
?1.1285
?1.469
?1.229
?1.052
?1.316
?1.277
?1.332
?1.332
?1.1772
?1.365
?1.3658
?1.3537
?0.997
?1.274
?1.069
?1.133
?1.362
?1.246
?1.015
?1.327
?1.282
?1.310
?1.243
?1.189
?1.317
?1.210
?1.342
Table 3: Equity series 16-30.
We show the average predictive log-likelihood of GP-Vol, GARCH, EGARCH and GJR-GARCH in
tables 1, 2 and 3 for the FX series, the first 15 Equity series and the last 15 Equity series, respectively.
The results of the best performing method in each dataset have been highlighted in bold. These tables
show that GP-Vol obtains the highest predictive log-likelihood in 29 of the 50 analyzed datasets. We
perform a statistical test to determine whether differences among GP-Vol, GARCH, EGARCH and
GJR-GARCH are significant. These methods are compared against each other using the multiple
comparison approach described by [27]. In this comparison framework, all the methods are ranked
according to their performance on different tasks. Statistical tests are then applied to determine
whether the differences among the average ranks of the methods are significant. In our case, each of
the 50 datasets analyzed represents a different task. A Friedman rank sum test rejects the hypothesis
that all methods have equivalent performance at ? = 0.05 with p-value less than 10?15 . Pairwise
comparisons between all the methods with a Nemenyi test at a 95% confidence level are summarized
in Figure 2. The Nemenyi test shows that GP-Vol is significantly better than the other methods.
The other main advantage of GP-Vol over existing models is that it can learn the functional relationship f between the new log variance vt and the previous log variance vt?1 and previous return xt?1 .
We plot a typical log variance surface in the left of Figure 3. This surface is generated by plotting the
mean predicted outputs vt against a grid of inputs for vt?1 and xt?1 . For this, we use the functional
dynamics learned with RAPCF on the AUDUSD time series. AUDUSD stands for the amount of
US dollars that an Australian dollar can buy. The grid of inputs is designed to contain a range of
values experienced by AUDUSD from 2008 to 2011, which is the period covered by the data. The
surface is colored according to the standard deviation of the posterior predictive distribution for the
log variance. Large standard deviations correspond to uncertain predictions, and are redder.
6
Cross section vt vs xt?1
Cross section vt vs vt?1
Log Variance Surface for AUDUSD
0.35
4
4
2
0.25
0
0.2
?2
0.15
?4
1.5
1
0
vt
2
vt
output, vt
2
3
0.3
1
?1
?2
0.5
?3
5
0
input, xt?1
?5
?2
0
2
4
input, vt?1
0.1
?4
0
?5
?6
?4
?2
0
vt?1
2
4
6
?6
?4
?2
0
2
4
6
xt?1
Figure 3: Left, surface generated by plotting the mean predicted outputs vt against a grid of inputs
for vt?1 and xt?1 . Middle, predicted vt ? 2 s.d. for inputs (0, xt?1 ). Right, predicted vt ? 2 s.d.
for inputs (0, xt?1 ).
The plot in the left of Figure 3 shows several patterns. First, there is an asymmetric effect of positive
and negative previous returns xt?1 . This can be seen in the skewness and lack of symmetry of the
contour lines with respect to the vt?1 axis. Second, the relationship between vt?1 and vt is slightly
non-linear because the distance between consecutive contour lines along the vt?1 axis changes as we
move across those lines, especially when xt?1 is large. In addition, the relationship between xt?1
and vt is nonlinear, but some sort of skewed quadratic function. These two patterns confirm the
asymmetric effect and the nonlinear transition function that EGARCH and GJR-GARCH attempt
to model. Third, there is a dip in predicted log variance for vt?1 < ?2 and ?1 < xt?1 < 2.5.
Intuitively this makes sense, as it corresponds to a calm market environment with low volatility.
However, as xt?1 becomes more extreme the market becomes more turbulent and vt increases.
To further understand the transition function f we study cross sections of the log variance surface.
First, vt is predicted for a grid of vt?1 and xt?1 = 0 in the middle plot of Figure 3. Next, vt is
predicted for various xt?1 and vt?1 = 0 in the right plot of Figure 3. The confidence bands in the
figures correspond to the mean prediction ?2 standard deviations. These cross sections confirm the
nonlinearity of the transition function and the asymmetric effect of positive and negative returns on
the log variance. The transition function is slightly non-linear as a function of vt?1 as the band in
the middle plot of Figure 3 passes through (?2, ?2) and (0, 0), but not (2, 2). Surprisingly, we
observe in the right plot of Figure 3 that large positive xt?1 produces larger vt when vt?1 = 0 since
the band is slightly higher at xt?1 = 6 than at xt?1 = ?6. However, globally, the highest predicted
vt occurs when vt?1 > 5 and xt?1 < ?5, as shown in the surface plot.
6.3
Comparison between RAPCF and PGAS
We now analyze the potential shortcomings of RAPCF that were discussed in Section 5. For this,
we compare RAPCF against PGAS on the twenty FX time series from the previous section in terms
of predictive log-likelihood and execution times. The RAPCF setup is the same as in Section 6.2.
For PGAS, which is a batch method, the algorithm is run on initial training data x1:L , with L = 100,
and a one-step forward prediction is made. The predictive log-likelihood is evaluated on the next
observation out of the training set. Then the training set is augmented with the new observation
and the batch training and prediction steps are repeated. The process is repeated sequentially until
no further data is received. For these experiments we used shorter time series with T = 120 since
PGAS is computationally very expensive. Note that we cannot simply learn the GP-SSM dynamics
on a small set of training data and then predict on a large test dataset, as it was done in [19]. These
authors were able to predict forward as they were using synthetic data with known ?hidden? states.
We analyze different settings of RAPCF and PGAS. In RAPCF we use N = 200 particles since that
number was used to compare against GARCH, EGARCH and GJR-GARCH in the previous section.
PGAS has two parameters: a) N , the number of particles and b) M , the number of iterations.
Three combinations of these settings were used. The resulting average predictive log-likelihoods for
RAPCF and PGAS are shown in Table 4. On each dataset, the results of the best performing method
7
have been highlighted in bold. The average rank of each method across the analyzed datasets is
shown in Table 5. From these tables, there is no evidence that PGAS outperforms RAPCF on these
financial datasets, since there is no clear predictive edge of any PGAS setting over RAPCF.
Dataset
AUDUSD
BRLUSD
CADUSD
CHFUSD
CZKUSD
EURUSD
GBPUSD
IDRUSD
JPYUSD
KRWUSD
MXNUSD
MYRUSD
NOKUSD
NZDUSD
PLNUSD
SEKUSD
SGDUSD
TRYUSD
TWDUSD
ZARUSD
RAPCF PGAS.1
N = 200 N = 10
M = 100
?1.1205 ?1.0571
?1.0102 ?1.0043
?1.4174 ?1.4778
?1.8431 ?1.8536
?1.2263 ?1.2357
?1.3837 ?1.4586
?1.1863 ?1.2106
?0.5446 ?0.5220
?2.0766 ?1.9286
?1.0566 ?1.1212
?0.2417 ?0.2731
?1.4615 ?1.5464
?1.3095 ?1.3443
?1.2254 ?1.2101
?0.8972 ?0.8704
?1.0085 ?1.0085
?1.6229 ?1.9141
?1.8336 ?1.8509
?1.7093 ?1.7178
?1.3236 ?1.3326
PGAS.2
N = 25
M = 100
?1.0699
?0.9959
?1.4514
?1.8453
?1.2424
?1.3717
?1.1790
?0.5388
?2.1585
?1.2032
?0.2271
?1.4745
?1.3048
?1.2366
?0.8708
?1.0505
?1.7566
?1.8352
?1.8315
?1.3440
PGAS.3
N = 10
M = 200
?1.0936
?0.9759
?1.4077
?1.8478
?1.2093
?1.4064
?1.1729
?0.5463
?2.1658
?1.2066
?0.2538
?1.4724
?1.3169
?1.2373
?0.8704
?1.0360
?1.7837
?1.8553
?1.7257
?1.3286
Method
RAPCF
PGAS.1
PGAS.2
PGAS.3
Configuration
N = 200
N = 10, M = 100
N = 25, M = 100
N = 10, M = 200
Rank
2.025
2.750
2.550
2.675
Table 5: Average ranks.
Method
RAPCF
PGAS.1
PGAS.2
PGAS.3
Configuration
N = 200
N = 10, M = 100
N = 25, M = 100
N = 10, M = 200
Avg. Time
6
732
1832
1465
Table 6: Avg. running time.
Table 4: Results for RAPCF vs. PGAS.
As mentioned above, there is little difference between the predictive accuracies of RAPCF and
PGAS. However, PGAS is computationally much more expensive. We show average execution times
in minutes for RAPCF and PGAS in Table 6. Note that RAPCF is up to two orders of magnitude
faster than PGAS. The cost of this latter method could be reduced by using fewer particles N or
fewer iterations M , but this would also reduce its predictive accuracy. Even after doing so, PGAS
would still be more costly than RAPCF. RAPCF is also competitive with GARCH, EGARCH and
GJR, whose average training times are in this case 2.6, 3.5 and 3.1 minutes, respectively. A naive
implementation of RAPCF has cost O(N T 4 ), since at each time step t there is a O(T 3 ) cost from
the inversion of the GP covariance matrix. On the other hand, the cost of applying PGAS naively is
O(N M T 5 ), since for each batch of data x1:t there is a O(N M T 4 ) cost. These costs can be reduced
to be O(N T 3 ) and O(N M T 4 ) for RAPCF and PGAS respectively by doing rank one updates of
the inverse of the GP covariance matrix at each time step. The costs can be further reduced by a
factor of T 2 by using sparse GPs [28].
7
Summary and discussion
We have introduced a novel Gaussian Process Volatility model (GP-Vol) for time-varying variances
in financial time series. GP-Vol is an instance of a Gaussian Process State-Space model (GP-SSM)
which is highly flexible and can model nonlinear functional relationships and asymmetric effects of
positive and negative returns on time-varying variances. In addition, we have presented an online
inference method based on particle filtering for GP-Vol called the Regularized Auxiliary Particle
Chain Filter (RAPCF). RAPCF is up to two orders of magnitude faster than existing batch Particle
Gibbs methods. Results for GP-Vol on 50 financial time series show significant improvements in
predictive performance over existing models such as GARCH, EGARCH and GJR-GARCH. Finally,
the nonlinear transition functions learned by GP-Vol can be easily analyzed to understand the effect
of past volatility and past returns on future volatility.
For future work, GP-Vol can be extended to learn the functional relationship between a financial
instrument?s volatility, its price and other market factors, such as interest rates. The functional
relationship thus learned can be useful in the pricing of volatility derivatives on the instrument.
Additionally, the computational efficiency of RAPCF makes it an attractive choice for inference in
other GP-SSMs different from GP-Vol. For example, RAPCF could be more generally applied to
learn the hidden states and the dynamics in complex control systems.
8
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9
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4,834 | 5,377 | Bandit Convex Optimization: Towards Tight Bounds
Kfir Y. Levy
Technion?Israel Institute of Technology
Haifa 32000, Israel
[email protected]
Elad Hazan
Technion?Israel Institute of Technology
Haifa 32000, Israel
[email protected]
Abstract
Bandit Convex Optimization (BCO) is a fundamental framework for decision
making under uncertainty, which generalizes many problems from the realm of online and statistical learning. While the special case of linear cost functions is well
understood, a gap on the attainable regret for BCO with nonlinear losses remains
an important open question. In this paper we take a step towards understanding
the best attainable regret bounds for BCO: we give an efficient and near-optimal
regret algorithm for BCO with strongly-convex and smooth loss functions. In contrast to previous works on BCO that use time invariant exploration schemes, our
method employs an exploration scheme that shrinks with time.
1
Introduction
The power of Online Convex Optimization (OCO) framework is in its ability to generalize many
problems from the realm of online and statistical learning, and supply universal tools to solving
them. Extensive investigation throughout the last decade has yield efficient algorithms with worst
case guarantees. This has lead many practitioners to embrace the OCO framework in modeling and
solving real world problems.
One of the greatest challenges in OCO is finding tight bounds to the problem of Bandit Convex
Optimization (BCO). In this ?bandit? setting the learner observes the loss function only at the point
that she has chosen. Hence, the learner has to balance between exploiting the information she has
gathered and between exploring the new data. The seminal work of [5] elegantly resolves this
?exploration-exploitation? dilemma by devising a combined explore-exploit gradient descent algorithm. They obtain a bound of O(T 3/4 ) on the expected regret for the general case of an adversary
playing bounded and Lipschitz-continuous convex losses.
In this paper we investigate the BCO setting assuming that the adversary is limited to inflicting
strongly-convex and smooth losses and the player may choose points from a constrained
decision
?
? T ). This rate is
set. In this setting we devise an efficient algorithm that achieves a regret of O(
the best possible up?to logarithmic factors as implied by a recent work of [11], cleverly obtaining a
lower bound of ?( T ) for the same setting.
During our analysis, we develop a full-information algorithm that takes advantage of the strongconvexity of loss functions and uses a self-concordant barrier as a regularization term. This algorithm enables us to perform ?shrinking exploration? which is a key ingredient in our BCO algorithm.
Conversely, all previous works on BCO use a time invariant exploration scheme.
This paper is organized as follows. In Section 2 we introduce our setting and review necessary
preliminaries regarding self-concordant barriers. In Section 3 we discuss schemes to perform single1
Setting
Full-Info.
BCO
Convex
? 3/4 )
O(T
Linear
?
?( ?T )
? T)
O(
Smooth
Str.-Convex
? 2/3 )
O(T
?
?( T )
Str.-Convex & Smooth
?(log
? T)
? T ) [Thm. 10]
O(
Table 1: Known regret bounds in the Full-Info./ BCO setting. Our new result is highlighted, and
? 2/3 ) bound.
improves upon the previous O(T
point gradient estimations, then we define first-order online methods and analyze the performance
of such methods receiving noisy gradient estimates. Our main result is described and analyzed in
Section 4; Section 5 concludes.
1.1
Prior work
For BCO with general convex loss functions, almost simultaneously to [5], a bound of O(T 3/4 )
was also obtained by [7] for the setting of?
Lipschitz-continuous convex losses. Conversely, the best
known lower bound for this problem is ?( T ) proved for the easier full-information setting.
In case the adversary is limited to using linear losses, it can be shown that the player does not
?pay? for exploration; this property was
? used by [4] to devise the Geometric Hedge algorithm that
? T ). Later [1], inspired by interior point methods, devised the
achieves an optimal regret rate of O(
first efficient algorithm that attains the same nearly-optimal regret rate for this setup of bandit linear
optimization.
For some special classes of nonlinear convex losses, there are several works that lean on ideas
from [5] to achieve improved upper bounds for BCO. In the case of convex and smooth losses [9]
? 2/3 ). The same regret rate of O(T
? 2/3 ) was achieved by [2] in the
attained an upper bound of O(T
case of strongly-convex losses. For the special?case of unconstrained BCO with strongly-convex
? T ). A recent paper by Shamir [11], significantly
and smooth losses, [2] obtained a regret of O(
?
advanced our understanding of BCO by devising a lower bound of ?( T ) for the setting of stronglyconvex and smooth BCO. The latter implies the tightness of our bound.
A comprehensive survey by Bubeck and Cesa-Bianchi [3], provides a review of the bandit optimization literature in both stochastic and online setting.
2
Setting and Background
Notation: During this paper we denote by || ? || the `2 norm when referring to vectors, and use
the same notation for the spectral norm when referring to matrices. We denote by Bn and Sn the
n-dimensional euclidean unit ball and unit sphere, and by v ? Bn and u ? Sn random variables
chosen uniformly from these sets. The symbol I is used for the identity matrix (its dimension will
be clear from the context). For a positive definite matrix A 0 we denote by A1/2 the matrix B
such that B > B = A, and by A?1/2 the inverse of B. Finally, we denote [N ] := {1, . . . , N }.
2.1
Bandit Convex Optimization
We consider a repeated game of T rounds between a player and an adversary, at each round t ?
[T ]
1. player chooses a point xt ? K.
2. adversary independently chooses a loss function ft ? F.
3. player suffers a loss ft (xt ) and receives a feedback Ft .
2
In the OCO (Online Convex Optimization) framework we assume that the decision set K is convex and that all functions in F are convex. Our paper focuses on adversaries limited to choosing
functions from the set F?,? ; the set off all ?-strongly-convex and ?-smooth functions.
We also limit ourselves to oblivious adversaries where the loss sequence {ft }Tt=1 is predetermined
and is therefore independent of the player?s choices. Mind that in this case the best point in hindsight
is also independent of the player?s choices. We also assume that the loss functions are defined over
the entire space Rn and are strongly-convex and smooth there; yet the player may only choose points
from a constrained set K.
Let us define the regret of A, and its regret with respect to a comparator w ? K:
RegretA
T =
T
X
t=1
ft (xt ) ? min
?
w ?K
T
X
RegretA
T (w) =
ft (w? ),
t=1
T
X
t=1
ft (xt ) ?
T
X
ft (w)
t=1
A player aims at minimizing his regret, and we are interested in players that ensure an o(T ) regret
for any loss sequence that the adversary may choose.
The player learns through the feedback Ft received in response to his actions. In the full informations
setting, he receives the loss function ft itself as a feedback, usually by means of a gradient oracle i.e. the decision maker has access to the gradient of the loss function at any point in the decision set.
Conversely, in the BCO setting the given feedback is ft (xt ), i.e., the loss function
only
at the point
that he has chosen; and the player aims at minimizing his expected regret, E RegretA
T .
2.2
Strong Convexity and Smoothness
As mentioned in the last subsection we consider an adversary limited to choosing loss functions
from the set F?,? , the set of ?-strongly convex and ?-smooth functions, here we define these properties.
Definition 1. (Strong Convexity) We say that a function f : Rn ? R is ?-strongly convex over the
set K if for all x, y ? K it holds that,
f (y) ? f (x) + ?f (x)> (y ? x) +
?
||x ? y||2
2
(1)
Definition 2. (Smoothness) We say that a convex function f : Rn ? R is ?-smooth over the set K
if the following holds:
f (y) ? f (x) + ?f (x)> (y ? x) +
2.3
?
||x ? y||2 ,
2
?x, y ? K
(2)
Self Concordant Barriers
Interior point methods are polynomial time algorithms to solving constrained convex optimization
programs. The main tool in these methods is a barrier function that encodes the constrained set and
enables the use of a fast unconstrained optimization machinery. More on this subject can be found
in [8].
Let K ? Rn be a convex set with a non empty interior int(K)
Definition 3. A function R : int(K) ? R is called ?-self-concordant if:
1. R is three times continuously differentiable and convex, and approaches infinity along any
sequence of points approaching the boundary of K.
2. For every h ? Rn and x ? int(K) the following holds:
|?3 R(x)[h, h, h]| ? 2(?2 R(x)[h, h])3/2
3
and
|?R(x)[h]| ? ? 1/2 (?2 R(x)[h, h])1/2
here, ?3 R(x)[h, h, h] :=
?3
?t1 ?t2 ?t3 R(x
+ t1 h + t2 h + t3 h)
t1 =t2 =t3 =0
.
?
Our algorithm requires a ?-self-concordant barrier over K, and its regret depends on ?. It is well
n
known that any convex set in R admits a ? = O(n) such barrier (? might be much smaller), and that
most interesting convex sets admit a self-concordant barrier that is efficiently represented.
The Hessian of a self-concordant barrier induces a local norm at every x ? int(K), we denote this
norm by || ? ||x and its dual by || ? ||?x and define ?h ? Rn :
q
q
||h||x = h> ?2 R(x)h,
||h||?x = h> (?2 R(x))?1 h
we assume that ?2 R(x) always has a full rank.
The following fact is a key ingredient in the sampling scheme of BCO algorithms [1, 9]. Let R is
be self-concordant barrier and x ? int(K) then the Dikin Ellipsoide,
W1 (x) := {y ? Rn : ||y ? x||x ? 1}
(3)
i.e. the || ? ||x -unit ball centered around x, is completely contained in K.
Our regret analysis requires a bound on R(y) ? R(x); hence, we will find the following lemma
useful:
Lemma 4. Let R be a ?-self-concordant function over K, then:
R(y) ? R(x) ? ? log
1
,
1 ? ?x (y)
where ?x (y) = inf{t ? 0 : x + t?1 (y ? x) ? K},
?x, y ? int(K)
?x, y ? int(K)
Note that ?x (y) is called the Minkowsky function and it is always in [0, 1]. Moreover, as y approaches the boundary of K then ?x (y) ? 1.
3
3.1
Single Point Gradient Estimation and Noisy First-Order Methods
Single Point Gradient Estimation
A main component of BCO algorithms is a randomized sampling scheme for constructing gradient estimates. Here, we survey the previous schemes as well as the more general scheme that we
use.
Spherical estimators: Flaxman et al. [5] introduced a method that produces single point gradient
estimates through spherical sampling. These estimates are then inserted into a full-information procedure that chooses the next decision point for the player. Interestingly, these gradient estimates are
unbiased predictions for the gradients of a smoothed version function which we next define.
Let ? > 0 and v ? Bn , the smoothed version of a function f : Rn ? R is defined as follows:
f?(x) = E[f (x + ?v)]
(4)
The next lemma of [5] ties between the gradients of f? and an estimate based on samples of f :
Lemma 5. Let u ? Sn , and consider the smoothed version f? defined in Equation (4), then the
following applies:
n
(5)
?f?(x) = E[ f (x + ?u)u]
?
Therefore, n? f (x + ?u)u is an unbiased estimator for the gradients of the smoothed version.
4
x
x
x
t
K
K
(a) Eigenpoles Sampling
(b) Continuous Sampling
K
(c) Shrinking Sampling
Figure 1: Dikin Ellipsoide Sampling Schemes
Ellipsoidal estimators: Abernethy et al. [1] introduced the idea of sampling from an ellipsoid
(specifically the Dikin ellipsoid) rather than a sphere in the context of BCO. They restricted the
sampling to the eigenpoles of the ellipsoid (Fig. 1a). A more general method of sampling continuously from an ellipsoid was introduced in [9] (Fig. 1b). We shall see later that our?algorithm
? T ) regret
uses a ?shrinking-sampling? scheme (Fig. 1c), which is crucial in achieving the O(
bound.
The following lemma of [9] shows that we can sample f non uniformly over all directions and create
an unbiased gradient estimate of a respective smoothed version:
Corollary 6. Let f : Rn ? R be a continuous function, let A ? Rn?n be invertible, and v ? Bn ,
u ? Sn . Define the smoothed version of f with respect to A:
f?(x) = E[f (x + Av)]
(6)
Then the following holds:
?f?(x) = E[nf (x + Au)A?1 u]
(7)
Note that if A 0 then {Au : u ? Sn } is an ellipsoid?s boundary.
Our next lemma shows that the smoothed version preserves the strong-convexity of f , and that we
can measure the distance between f? and f using the spectral norm of A2 :
Lemma 7. Consider a function f : Rn ? R, and a positive definite matrix A ? Rn?n . Let f? be
the smoothed version of f with respect to A as defined in Equation (6). Then the following holds:
? If f is ?-strongly convex then so is f?.
? If f is convex and ?-smooth, and ?max be the largest eigenvalue of A then:
?
?
0 ? f?(x) ? f (x) ? ||A2 ||2 = ?2max
2
2
(8)
Remark: Lemma 7 also holds if we define the smoothed version of f as f?(x) = Eu?Sn [f (x + Au)]
i.e. an average of the original function values over the unit sphere rather than the unit ball as defined
in Equation (6). Proof is similar to the one of Lemma 7.
3.2
Noisy First-Order Methods
Our algorithm utilizes a full-information online algorithm, but instead of providing this method with
exact gradient values we insert noisy estimates of the gradients. In what follows we define first-order
online algorithms, and present a lemma that analyses the regret of such algorithm receiving noisy
gradients.
5
Definition 8. (First-Order Online Algorithm) Let A be an OCO algorithm receiving an arbitrary
sequence of differential convex loss functions f1 , . . . , fT , and providing points x1 ? A and xt ?
A(f1 , . . . , ft?1 ). Given that A requires all loss functions to belong to some set F0 . Then A is called
first-order online algorithm if the following holds:
? Adding a linear function to a member of F0 remains in F0 ; i.e., for every f ? F0 and
a ? Rn then also f + a> x ? F0
? The algorithm?s choices depend only on its gradient values taken in the past choices of A,
i.e. :
A(f1 , . . . , ft?1 ) = A(?f1 (x1 ), . . . , ?ft?1 (xt?1 )),
?t ? [T ]
The following is a generalization of Lemma 3.1 from [5]:
Lemma 9. Let w be a fixed point in K. Let A be a first-order online algorithm receiving a sequence
of differential convex loss functions f1 , . . . , fT : K ? R (ft+1 possibly depending on z1 , . . . zt ).
Where z1 . . . zT are defined as follows: z1 ? A, zt ? A(g1 , . . . , gt?1 ) where gt ?s are vector valued
random variables such that:
E[gt z1 , f1 , . . . , zt , ft ] = ?ft (zt )
Then if A ensures a regret bound of the form: RegretA
T ? BA (?f1 (x1 ), . . . , ?fT (xT )) in the full
information case then, in the case of noisy gradients it ensures the following bound:
T
T
X
X
E[
ft (zt )] ?
ft (w) ? E[BA (g1 , . . . , gT )]
t=1
4
t=1
Main Result and Analysis
Following is the main theorem of this paper:
Theorem 10. Let K be a convex set with diameter DK and R be a ?-self-concordant barrier over
K. Then in the BCO setting where the adversary is limited to choosing ?-smooth and ?-stronglyconvex
functions and |ft (x)| ? L, ?x ? K, then the expected regret of Algorithm 1 with ? =
q
(?+2?/?) log T
2n2 L2 T
is upper bounded as
s
E[RegretT ] ? 4nL
2?
?+
?
2
?DK
T log T + 2L +
=O
2
r
??
T log T
?
!
whenever T / log T ? 2 (? + 2?/?).
Algorithm 1 BCO Algorithm for Str.-convex & Smooth losses
Input: ? > 0, ? > 0, ?-self-concordant barrier R
Choose x1 = arg minx?K R(x)
for t = 1, 2 . . . T do
?1/2
Define Bt = ?2 R(xt ) + ??tI
Draw u ? Sn
Play yt = xt + Bt u
Observe ft (xt + Bt u) and define gt= nft (xt + Bt u)Bt?1
u ?1
Pt
?
>
2
Update xt+1 = arg minx?K ? =1 g? x + 2 ||x ? x? || + ? R(x)
end for
Algorithm 1 shrinks the exploration magnitude with time (Fig. 1c); this is enabled thanks to the
strong-convexity of the losses. It also updates according to a full-information first-order algorithm
6
denoted FTARL-?, which is defined below. This algorithm is a variant of the FTRL methodology
as defined in [6, 10].
Algorithm 2 FTARL-?
Input: ? > 0, ?-self concordant barrier R
Choose x1 = arg minx?K R(x)
for t = 1, 2 . . . T do
Receive ?ht (xt )
Pt
Output xt+1 = arg minx?K ? =1 ?h? (x? )> x + ?2 ||x ? x? ||2 + ? ?1 R(x)
end for
Next we give a proof sketch of Theorem 10
Proof sketch of Therorem 10. Let us decompose the expected regret of Algorithm 1 with respect to
w ? K:
PT
E [RegretT (w)] := t=1 E [ft (yt ) ? ft (w)]
PT
= t=1 E [ft (yt ) ? ft (xt )]
(9)
h
i
PT
+ t=1 E ft (xt ) ? f?t (xt )
(10)
h
i
PT
? t=1 E ft (w) ? f?t (w)
(11)
h
i
PT
+ t=1 E f?t (xt ) ? f?t (w)
(12)
where expectation is taken with respect to the player?s choices, and f?t is defined as
f?t (x) = E[ft (x + Bt v)],
?x ? K
here v ? Bn and the smoothing matrix Bt is defined in Algorithm 1.
The sampling scheme used by Algorithm 1 yields an unbiased gradient estimate gt of the smoothed
version f?t , which is then inserted to FTARL-? (Algorithm 2). We can therefore interpret Algorithm 1 as performing noisy first-order method (FTARL-?) over the smoothed versions. The xt ?s
in Algorithm 1 are the outputs of FTARL-?, thus the term in Equation (12) is associated with ?exploitation?. The other terms in Equations (9)-(11) measure the cost of sampling away from xt , and
the distance between the smoothed version and the original function, hence these term are associated
with ?exploration?.
In what follows we analyze these terms separately and show that Algorithm 1
?
? T ) regret.
achieves O(
The Exploration Terms: The next hold by the remark that follows Lemma 7 and by the lemma
itself:
E[ft (yt ) ? ft (xt )] = E Eu [ft (xt + Bt u)] ? ft (xt ) xt ] ? 0.5?E ||Bt2 ||2 ? ?/2??t (13)
h
i
? E[ft (w) ? f?t (w)] = E E[f?t (w) ? ft (w) xt ] ? 0.5?E ||Bt2 ||2 ? ?/2??t
(14)
h
i
E[ft (xt ) ? f?t (xt )] = E E[ft (xt ) ? f?t (xt ) xt ] ? 0
(15)
where ||Bt2 ||2 ? 1/??t follows by the definition of Bt and by the fact that ?2 R(xt ) is positive
definite.
7
The Exploitation Term: The next Lemma bounds the regret of FTARL-? in the full-information
setting:
Lemma 11. Let R be a self-concordant barrier over a convex set K, and ? > 0. Consider an
online player receiving ?-strongly-convex loss functions h1 , . . . , hT and choosing points according
to FTARL-? (Algorithm 2), and ?||?ht (xt )||?t ? 1/2, ?t ? [T ]. Then the player?s regret is upper
bounded as follows:
T
X
t=1
ht (xt ) ?
T
X
t=1
ht (w) ? 2?
T
X
t=1
2
(||?ht (xt )||?t ) + ? ?1 R(w),
?z ? K
here (||a||?t )2 = aT (?2 R(xt ) + ??tI)?1 a
Note that Algorithm 1 uses the estimates gt as inputs into FTARL-?. Using Corollary 6 we can
show that the gt ?s are unbiased estimates for the gradients of the smoothed versions f?t ?s. Using the
regret bound of the above lemma, and the unbiasedness of the gt ?s, Lemma 9 ensures us:
T
X
t=1
T
i
X
?
?
E ft (xt ) ? ft (w) ? 2?
E[(||gt ||?t )2 ] + ? ?1 R(w)
h
(16)
t=1
By the definitions of gt and Bt , and recalling |ft (x)| ? L, ?x ? K, we can bound:
h
?1 ?1 i
2
E[(||gt ||?t )2 xt ] = E n2 (ft (xt + Bt u)) u> Bt?1 ?2 R(xt ) + ??tI
Bt u xt ? (nL)2
Concluding: Plugging the latter into Equation (16) and combining Equations (9)-(16) we get:
E[RegretT (w)] ? 2?(nL)2 T + ? ?1 R(w) + 2?? ?1 log T
(17)
Recall that x1 = arg minx?K R(x) and assume w.l.o.g. that R(x1 ) = 0 (we can always add
R a constant). Thus, for a point w ? K such that ?x1 (w) ? 1 ? T ?1 Lemma 4 ensures us that
R(w) ? ? log T . Combining the latter
p with Equation (17) and the choice of ? in Theorem 10 assures
an expected regret bounded by 4nL (? + 2?? ?1 ) T log T . For w ? K such that ?x1 (w) > 1?T ?1
we can always find w0 ? K such that ||w ? w0 || ? O(T ?1 ) and ?x1 (w0 ) ? 1 ? T ?1 , using the
Lipschitzness of the ft ?s, Theorem 10 holds.
Correctness:
Note that Algorithm 1 chooses points from the set {xt +
?1/2
2
? R(xt ) + ??tI
u, u ? Sn } which is inside the Dikin ellipsoid and therefore belongs to K
(the Dikin Eliipsoid is always in K).
5
Summary and open questions
We have presented an efficient algorithm that attains near optimal regret for the setting of BCO with
strongly-convex and smooth losses, advancing our understanding of optimal regret rates for bandit
learning.
Perhaps the most important question in bandit learning remains the resolution of the attainable regret
bounds for smooth but non-strongly-convex, or vice versa, and generally convex cost functions (see
Table 1). Ideally, this should be accompanied by an efficient algorithm, although understanding the
optimal rates up to polylogarithmic factors would be a significant advancement by itself.
Acknowledgements
The research leading to these results has received funding from the European Union?s Seventh Framework Programme (FP7/2007-2013) under grant agreement n? 336078 ? ERCSUBLRN.
8
References
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[3] S?ebastien Bubeck and Nicolo Cesa-Bianchi. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends in Machine Learning, 5(1):1?122,
2012.
[4] Varsha Dani, Thomas P. Hayes, and Sham Kakade. The price of bandit information for online
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9
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specifically:1 uniformly:2 lemma:19 called:3 concordant:13 player:16 latter:3 alexander:1 |
4,835 | 5,378 | Stochastic Multi-Armed-Bandit Problem
with Non-stationary Rewards
Yonatan Gur
Stanford University
Stanford, CA
[email protected]
Omar Besbes
Columbia University
New York, NY
[email protected]
Assaf Zeevi
Columbia University
New York, NY
[email protected]
Abstract
In a multi-armed bandit (MAB) problem a gambler needs to choose at each round
of play one of K arms, each characterized by an unknown reward distribution.
Reward realizations are only observed when an arm is selected, and the gambler?s
objective is to maximize his cumulative expected earnings over some given horizon of play T . To do this, the gambler needs to acquire information about arms
(exploration) while simultaneously optimizing immediate rewards (exploitation);
the price paid due to this trade off is often referred to as the regret, and the main
question is how small can this price be as a function of the horizon length T . This
problem has been studied extensively when the reward distributions do not change
over time; an assumption that supports a sharp characterization of the regret, yet is
often violated in practical settings. In this paper, we focus on a MAB formulation
which allows for a broad range of temporal uncertainties in the rewards, while still
maintaining mathematical tractability. We fully characterize the (regret) complexity of this class of MAB problems by establishing a direct link between the extent
of allowable reward ?variation? and the minimal achievable regret, and by establishing a connection between the adversarial and the stochastic MAB frameworks.
1
Introduction
Background and motivation. In the presence of uncertainty and partial feedback on rewards, an
agent that faces a sequence of decisions needs to judiciously use information collected from past
observations when trying to optimize future actions. A widely studied paradigm that captures this
tension between the acquisition cost of new information (exploration) and the generation of instantaneous rewards based on the existing information (exploitation), is that of multi armed bandits
(MAB), originally proposed in the context of drug testing by [1], and placed in a general setting
by [2]. The original setting has a gambler choosing among K slot machines at each round of play,
and upon that selection observing a reward realization. In this classical formulation the rewards
are assumed to be independent and identically distributed according to an unknown distribution
that characterizes each machine. The objective is to maximize the expected sum of (possibly discounted) rewards received over a given (possibly infinite) time horizon. Since their inception, MAB
problems with various modifications have been studied extensively in Statistics, Economics, Operations Research, and Computer Science, and are used to model a plethora of dynamic optimization
problems under uncertainty; examples include clinical trials ([3]), strategic pricing ([4]), investment
in innovation ([5]), packet routing ([6]), on-line auctions ([7]), assortment selection ([8]), and on1
line advertising ([9]), to name but a few. For overviews and further references cf. the monographs
by [10], [11] for Bayesian / dynamic programming formulations, and [12] that covers the machine
learning literature and the so-called adversarial setting. Since the set of MAB instances in which one
can identify the optimal policy is extremely limited, a typical yardstick to measure performance of a
candidate policy is to compare it to a benchmark: an oracle that at each time instant selects the arm
that maximizes expected reward. The difference between the performance of the policy and that of
the oracle is called the regret. When the growth of the regret as a function of the horizon T is sublinear, the policy is long-run average optimal: its long run average performance converges to that of
the oracle. Hence the first order objective is to develop policies with this characteristic. The precise
rate of growth of the regret as a function of T provides a refined measure of policy performance.
[13] is the first paper that provides a sharp characterization of the regret growth rate in the context of
the traditional (stationary random rewards) setting, often referred to as the stochastic MAB problem.
Most of the literature has followed this path with the objective of designing policies that exhibit the
?slowest possible? rate of growth in the regret (often referred to as rate optimal policies).
In many application domains, several of which were noted above, temporal changes in the reward
distribution structure are an intrinsic characteristic of the problem. These are ignored in the traditional stochastic MAB formulation, but there have been several attempts to extend that framework.
The origin of this line of work can be traced back to [14] who considered a case where only the state
of the chosen arm can change, giving rise to a rich line of work (see, e.g., [15], and [16]). In particular, [17] introduced the term restless bandits; a model in which the states (associated with reward
distributions) of arms change in each step according to an arbitrary, yet known, stochastic process.
Considered a hard class of problems (cf. [18]), this line of work has led to various approximations
(see, e.g., [19]), relaxations (see, e.g., [20]), and considerations of more detailed processes (see, e.g.,
[21] for irreducible Markov process, and [22] for a class of history-dependent rewards).
Departure from the stationarity assumption that has dominated much of the MAB literature raises
fundamental questions as to how one should model temporal uncertainty in rewards, and how to
benchmark performance of candidate policies. One view, is to allow the reward realizations to be
selected at any point in time by an adversary. These ideas have their origins in game theory with the
work of [23] and [24], and have since seen significant development; [25] and [12] provide reviews
of this line of research. Within this so called adversarial formulation, the efficacy of a policy over a
given time horizon T is often measured relative to a benchmark defined by the single best action one
could have taken in hindsight (after seeing all reward realizations). The single best action benchmark
represents a static oracle, as it is constrained to a single (static) action. This static oracle can perform
quite poorly relative to a dynamic oracle that follows the optimal dynamic sequence of actions, as
the latter optimizes the (expected) reward at each time instant over all possible actions.1 Thus, a
potential limitation of the adversarial framework is that even if a policy has a ?small? regret relative
to a static oracle, there is no guarantee with regard to its performance relative to the dynamic oracle.
Main contributions. The main contribution of this paper lies in fully characterizing the (regret)
complexity of a broad class of MAB problems with non-stationary reward structure by establishing
a direct link between the extent of reward ?variation? and the minimal achievable regret. More
specifically, the paper?s contributions are along four dimensions. On the modeling side we formulate
a class of non-stationary reward structure that is quite general, and hence can be used to realistically
capture a variety of real-world type phenomena, yet is mathematically tractable. The main constraint
that we impose on the evolution of the mean rewards is that their variation over the relevant time
horizon is bounded by a variation budget VT ; a concept that was recently introduced in [26] in the
context of non-stationary stochastic approximation. This limits the power of nature compared to
the adversarial setup discussed above where rewards can be picked to maximally affect the policy?s
performance at each instance within {1, . . . , T }. Nevertheless, this constraint allows for a rich
class of temporal changes, extending most of the treatment in the non-stationary stochastic MAB
literature, which mainly focuses on a finite number of changes in the mean rewards, see, e.g., [27]
and references therein. We further discuss connections with studied non-stationary instances in ?6.
The second dimension of contribution lies in the analysis domain. For a general class of
non-stationary reward distributions we establish lower bounds on the performance of any nonanticipating policy relative to the dynamic oracle, and show that these bounds can be achieved,
1
Under non-stationary rewards it is immediate that the single best action may be sub-optimal in many
decision epochs, and the performance gap between the static and the dynamic oracles can grow linearly with T .
2
uniformly over the class of admissible reward distributions, by a suitable policy construction. The
term ?achieved? is meant in the sense of the order of the regret as a function of the time horizon T ,
the variation budget VT , and the number of arms K. Our policies are shown to be minimax optimal
up to a term that is logarithmic in the number of arms, and the regret is sublinear and is of order
1/3
(KVT ) T 2/3 . Our analysis complements studied non-stationary instances by treating a broad and
flexible class of temporal changes in the reward distributions, yet still establishing optimality results
and showing that sublinear regret is achievable. Our results provide a spectrum of orders of the
minimax regret ranging between order T 2/3 (when VT is a constant independent of T ) and order T
(when VT grows linearly with T ), mapping allowed variation to best achievable performance.
With the analysis described above we shed light on the exploration-exploitation trade off that characterizes the non-stationary reward setting, and the change in this trade off compared to the stationary
setting. In particular, our results highlight the tension that exists between the need to ?remember?
and ?forget.? This is characteristic of several algorithms that have been developed in the adversarial MAB literature, e.g., the family of exponential weight methods such as EXP3, EXP3.S and the
like; see, e.g., [28], and [12]. In a nutshell, the fewer past observations one retains, the larger the
stochastic error associated with one?s estimates of the mean rewards, while at the same time using
more past observations increases the risk of these being biased.
One interesting observation drawn in this paper connects between the adversarial MAB setting, and
the non-stationary environment studied here. In particular, as in [26], it is seen that an optimal policy
in the adversarial setting may be suitably calibrated to perform near-optimally in the non-stationary
stochastic setting. This will be further discussed after the main results are established.
2
Problem Formulation
Let K = {1, . . . , K} be a set of arms. Let T = {1, 2, . . . , T } denote a sequence of decision epochs
faced by a decision maker. At any epoch t ? T , the decision-maker pulls one of the K arms. When
pulling arm k ? K at epoch t ? T , a reward Xtk ? [0, 1] is obtained, where Xtk is a random
variable with expectation ?kt = E Xtk . We denote the best possible expected reward at decision
epoch t by ??t , i.e., ??t = maxk?K ?kt .
Changes in the expected rewards of the arms. We assume the expected reward of each arm ?kt
may change at any decision epoch. We denote by ?k the sequence of expected rewards of arm k:
T
?k = ?kt t=1 . In addition, we denote by ? the sequence of vectors of all K expected rewards:
k K
? = ? k=1 . We assume that the expected reward of each arm can change an arbitrary number of
times, but bound the total variation of the expected rewards:
T
?1
X
sup ?kt ? ?kt+1 .
(1)
t=1 k?K
Let {Vt : t = 1, 2, . . .} be a non-decreasing sequence of positive real numbers such that V1 = 0,
KVt ? t for all t, and for normalization purposes set V2 = 2 ? K ?1 . We refer to VT as the variation
budget over T . We define the corresponding temporal uncertainty set, as the set of reward vector
sequences that are subject to the variation budget VT over the set of decision epochs {1, . . . , T }:
(
)
T
?1
X
k
K?T
k
V = ? ? [0, 1]
:
sup ?t ? ?t+1 ? VT .
t=1 k?K
The variation budget captures the constraint imposed on the non-stationary environment faced by
the decision-maker. While limiting the possible evolution in the environment, it allows for numerous forms in which the expected rewards may change: continuously, in discrete shocks, and of a
changing rate (Figure 1 depicts two different variation patterns that correspond to the same variation
budget). In general, the variation budget VT is designed to depend on the number of pulls T .
Admissible policies, performance, and regret. Let U be a random variable defined over a probability space (U, U, Pu ). Let ?1 : U ? K and ?t : [0, 1]t?1 ? U ? K for t = 2, 3, . . . be measurable
functions. With some abuse of notation we denote by ?t ? K the action at time t, that is given by
?1 (U )
t = 1,
?t =
?
?t Xt?1
, . . . , X1? , U
t = 2, 3, . . . ,
3
Figure 1: Two instances of variation in the mean rewards: (Left) A fixed variation budget (that equals 3) is
?spent? over the whole horizon. (Right) The same budget is ?spent? in the first third of the horizon.
The mappings {?t : t = 1, . . . , T } together with the distribution Pu define the class of admissible
policies. We denote this class by P. We further denote by {H
1, . . . ,T } the filtration associt , t =
t?1
ated with a policy ? ? P, such that H1 = ? (U ) and Ht = ? Xj? j=1 , U for all t ? {2, 3, . . .}.
Note that policies in P are non-anticipating, i.e., depend only on the past history of actions and observations, and allow for randomized strategies via their dependence on U .
We define the regret under policy ? ? P compared to a dynamic oracle as the worst-case difference
between the expected performance of pulling at each epoch t the arm which has the highest expected
reward at epoch t (the dynamic oracle performance) and the expected performance under policy ?:
( T
" T
#)
X
X
?
?
?
?
R (V, T ) = sup
?t ? E
?t
,
??V
t=1
t=1
?
where the expectation E [?] is taken with respect to the noisy rewards, as well as to the policy?s
actions. In addition, we denote by R? (V, T ) the minimal worst-case regret that can be guaranteed
by an admissible policy ? ? P, that is, R? (V, T ) = inf ??P R? (V, T ). Then, R? (V, T ) is the best
achievable performance. In the following sections we study the magnitude of R? (V, T ). We analyze
the magnitude of this quantity by establishing upper and lower bounds; in these bounds we refer to
a constant C as absolute if it is independent of K, VT , and T .
3
Lower bound on the best achievable performance
We next provide a lower bound on the the best achievable performance.
Theorem 1 Assume that rewards have a Bernoulli distribution. Then, there is some
absolute con
stant C > 0 such that for any policy ? ? P and for any T ? 1, K ? 2 and VT ? K ?1 , K ?1 T ,
1/3
R? (V, T ) ? C (KVT )
T 2/3 .
We note that when reward distributions
are stationary, there are known policies such as UCB1 ([29])
?
that achieve regret of order T in the stochastic setup. When the reward structure is non-stationary
and defined by the class V, then no policy may achieve such a performance and the best performance
must incur a regret of at least order T 2/3 . This additional complexity embedded in the non-stationary
stochastic MAB problem compared to the stationary one will be further discussed in ?6. We note
that Theorem 1 also holds when VT is increasing with T . In particular, when the variation budget is
linear in T , the regret grows linearly and long run average optimality is not achievable.
The driver of the change in the best achievable performance relative to the one established in a
stationary environment, is a second tradeoff (over the tension between exploring different arms and
capitalizing on the information already collected) introduced by the non-stationary environment,
between ?remembering? and ?forgetting?: estimating the expected rewards is done based on past
observations of rewards. While keeping track of more observations may decrease the variance of
mean rewards estimates, the non-stationary environment implies that ?old? information is potentially
less relevant due to possible changes in the underlying rewards. The changing rewards give incentive
to dismiss old information, which in turn encourages enhanced exploration. The proof of Theorem 1
emphasizes the impact of these tradeoffs on the achievable performance.
4
Key ideas in the proof. At a high level the proof of Theorem 1 builds on ideas of identifying
a worst-case ?strategy? of nature (e.g., [28], proof of Theorem 5.1) adapting them to our setting.
While the proof is deferred to the online companion (as supporting material), we next describe the
key ideas when VT = 1.2 We define a subset of vector sequences V 0 ? V and show that when
1/3
? is drawn randomly from V 0 , any admissible policy must incur regret of order (KVT ) T 2/3 .
? T each (except,
We define a partition of the decision horizon T into batches T1 , . . . , Tm of size ?
possibly the last batch):
n
n
oo
? T + 1 ? t ? min j ?
?T,T
Tj = t : (j ? 1)?
, for all j = 1, . . . , m,
(2)
? T e is the number of batches. In V 0 , in every batch there is exactly one ?good?
where m = dT /?
arm with expected reward 1/2 + ? for some 0 < ? ? 1/4, and all the other arms have expected
reward 1/2. The ?good? arm is drawn independently in the beginning of each batch according to
a discrete uniform distribution over {1, . . . , K}. Thus, the identity of the ?good? arm can change
? T ? VT , any ? ? V 0 is composed of expected
only between batches. By selecting ? such that ?T /?
reward sequences with a variation of at most VT , and therefore V 0 ? V. Given the draws under which
expected reward sequences are generated, nature prevents any accumulation of information from one
batch to another, since at the beginning of each batch a new ?good?parm is drawn independently of
? T no admissible policy can
the history. The proof of Theorem 1 establishes that when ? ? 1/ ?
? T epochs in each
identify the ?good? arm with high probability within a batch. Since there are p
?
?
?
batch, the regret that
p any policy must incur
p along a batch is of order ?T ? ? ? ?T , which yields
?
?
?
a regret of order ?T ? T /?T ? T / ?T throughout the whole horizon. Selecting the smallest
? T such that the variation budget constraint is satisfied leads to ?
? T ? T 2/3 , yielding a
feasible ?
2/3
regret of order T
throughout the horizon.
4
A near-optimal policy
We apply the ideas underlying the lower bound in Theorem 1 to develop a rate optimal policy for
the non-stationary stochastic MAB problem with a variation budget. Consider the following policy:
Rexp3. Inputs: a positive number ?, and a batch size ?T .
1. Set batch index j = 1
2. Repeat while j ? dT /?T e:
(a) Set ? = (j ? 1) ?T
(b) Initialization: for any k ? K set wtk = 1
(c) Repeat for t = ? + 1, . . . , min {T, ? + ?T }:
? For each k ? K, set
wk
pkt = (1 ? ?) PK t
k0
k0 =1 wt
+
?
K
K
? Draw an arm k 0 from K according to the distribution pkt k=1
0
? Receive a reward Xtk
? k0 = X k0 /pk0 , and for any k 6= k 0 set X
? k = 0. For all k ? K update:
? For k 0 set X
t
t
t
t
(
)
? tk
?
X
k
wt+1
= wtk exp
K
(d) Set j = j + 1, and return to the beginning of step 2
Clearly ? ? P. The Rexp3 policy uses Exp3, a policy introduced by [30] for solving a worst-case
sequential allocation problem, as a subroutine, restarting it every ?T epochs.
2
For the sake of simplicity, the discussion in this paragraph assumes a variation budget that is fixed and
independent of T ; the proof of Theorem 3 details a general treatment for a budget that depends on T .
5
l
m
1/3
2/3
Theorem 2 Let ? be the Rexp3 policy with a batch size ?T = (K log K) (T /VT )
and
q
n
o
K log K
?
with ? = min 1 ,
(e?1)?T . Then, there is some absolute constant C such that for every T ? 1,
?1
K ? 2, and VT ? K , K ?1 T :
1/3
R? (V, T ) ? C? (K log K ? VT ) T 2/3 .
Theorem 2 is obtained by establishing a connection between the regret relative to the single best
action in the adversarial setting, and the regret with respect to the dynamic oracle in non-stationary
stochastic setting with variation budget. Several classes of policies, such as exponential-weight
?
(including Exp3) and polynomial-weight policies, have been shown to achieve regret of order T
with respect to the single best action in the adversarial setting (see chapter 6 of [12] for a review).
While in general these policies tend to perform well numerically, there is no guarantee for their
performance relative to the dynamic oracle studied in this paper, since the single best action itself
may incur linear regret relative to the dynamic oracle; see also [31] for a study of the empirical
performance of?
one class of algorithms. The proof of Theorem 2 shows that any policy that achieves
regret of order T with respect to the single best action in the adversarial setting, can be used as a
subroutine to obtain near-optimal performance with respect to the dynamic oracle in our setting.
Rexp3 emphasizes the two tradeoffs discussed in the previous section. The first tradeoff, information
acquisition versus capitalizing on existing information, is captured by the subroutine policy Exp3. In
fact, any policy that achieves a good performance compared to a single best action benchmark in the
adversarial setting must balance exploration and exploitation. The second tradeoff, ?remembering?
versus ?forgetting,? is captured by restarting Exp3 and forgetting any acquired information every
?T pulls. Thus, old information that may slow down the adaptation to the changing environment
is being discarded. Theorem 1 and Theorem 2 together characterize the minimax regret (up to a
multiplicative factor, logarithmic in the number of arms) in a full spectrum of variations VT :
R? (V, T ) (KVT )
1/3
T 2/3 .
Hence, we have quantified the impact of the extent of change in the environment on the best achievable performance in this broad class of problems. For example, for the case in which VT = C ? T ? ,
for some absolute constant C and 0 ? ? < 1 the best achievable regret is of order T (2+?)/3 .
We finally note that restarting is only one way of adapting policies from the adversarial MAB setting
to achieve near optimality in the non-stationary stochastic setting; a way that articulates well the
principles leading to near optimality. In the online companion we demonstrate that near optimality
can be achieved by other adaptation methods, showing that the Exp3.S policy (given in [28]) can be
1/3
to achieve near optimality in our setting, without restarting.
tuned by ? = T1 and ? ? (KVT /T )
5
Proof of Theorem 2
The structure of the proof is as follows. First, we break the horizon to a sequence of batches of size
?T each, and analyze the performance gap between the single best action and the dynamic oracle
in each batch. Then, we plug in a known performance guarantee for Exp3 relative to the single best
action, and sum over batches to establish the regret of Rexp3 relative to the dynamic oracle.
Step 1 (Preliminaries). Fix T ? 1, K ? 2, and VT ? K ?1 , K ?1 T . Let ? be the Rexp3 policy,
o
n q
K log K
and ?T ? {1, . . . , T } (to be specified later on). We break the
tuned by ? = min 1 ,
(e?1)?T
horizon T into a sequence of batches T1 , . . . , Tm of size ?T each (except, possibly Tm ) according
to (2). Let ? ? V, and fix j ? {1, . . . , m}. We decomposition the regret in batch j:
?
?
E ?
?
X
t?Tj
(??t
?
??t )?
?
=
X
t?Tj
|
??t
? E ?max
k?K
?
?X
?
Xtk
t?Tj
{z
J1,j
??
?
??
?
?
?
?X
?
?
X ?
k
?
? + E ?max
Xt ? ? E ?
?t ? .
k?K ?
?
?
t?Tj
t?Tj
} |
{z
}
J2,j
(3)
The first component, J1,j , is the expected loss associated with using a single action over batch j.
The second component, J2,j , is the expected regret relative to the best static action in batch j.
6
Step 2 (Analysis of J1,j and J2,j ). Defining ?kT +1 = ?kT for all k ? K,
we denote the variation in
P
k
k
expected rewards along batch Tj by Vj = t?Tj maxk?K ?t+1 ? ?t . We note that:
m
X
Vj =
j=1
m X
X
j=1 t?Tj
max ?kt+1 ? ?kt ? VT .
(4)
k?K
o
nP
k
. Then,
?
Let k0 be an arm with best expected performance over Tj : k0 ? arg maxk?K
t
t?Tj
?
?
??
?
?
?
?
?X ?
?
?X
X
X
max
?kt
=
?kt 0 = E ?
(5)
Xtk0 ? ? E ?max
Xtk ? ,
?
?
k?K ?
k?K ?
t?Tj
t?Tj
t?Tj
t?Tj
and therefore, one has:
?
J1,j
X
=
??t ? E ?max
k?K
t?Tj
?
?X
?
t?Tj
??
? (a) X
??t ? ?kt 0
Xtk ? ?
?
t?Tj
n
o (b)
? ?T max ??t ? ?kt 0 ? 2Vj ?T ,
(6)
t?Tj
for any ? ? V and j ? {1, . . . , m}, where (a) holds by (5) and (b) holds by the following argument:
otherwise there is an epoch t0 ? Tj for which ??t0 ? ?kt00 > 2Vj . Indeed, let k1 = arg maxk?K ?kt0 .
In such case, for all t ? Tj one has ?kt 1 ? ?kt01 ? Vj > ?kt00 + Vj ? ?kt 0 , since Vj is the maximal
variation in batch Tj . This however, contradicts the optimality of k0 at epoch t, and thus (6) holds.
In addition,
q Corollary
n
o 3.2 in [28] points out that the regret incurred by Exp3 (tuned by ? =
K log K
min 1 ,
(e?1)?T ) along ?T batches, relative to the single best action, is bounded by
?
?
2 e ? 1 ?T K log K. Therefore, for each j ? {1, . . . , m} one has
?
?
?
?
??
?X
?
X
(a)
p
?
J2,j = E ?max
Xtk ? E? ?
(7)
??t ?? ? 2 e ? 1 ?T K log K,
?
k?K ?
t?Tj
t?Tj
for any ? ? V, where (a) holds since within each batch arms are pulled according to Exp3(?).
Step 3 (Regret throughout the horizon). Summing over m = dT /?T e batches we have:
( T
" T
#)
m
X
X
(a) X
p
?
?
?
?
?
R (V, T ) = sup
?t ? E
?t
?
2 e ? 1 ?T K log K + 2Vj ?T
??V
(b)
?
=
t=1
t=1
j=1
?
p
T
+ 1 ? 2 e ? 1 ?T K log K + 2?T VT .
?T
?
?
p
?
2 e ? 1 K log K ? T
?
+ 2 e ? 1 ?T K log K + 2?T VT ,
?T
where: l (a) holds by (3), (6),m and (7); and (b) follows from (4).
1/3
2/3
, we establish:
?T = (K log K) (T /VT )
R? (V, T )
?
(a)
?
Finally, selecting
?
1/3
2 e ? 1 (K log K ? VT ) T 2/3
r
?
1/3
2/3
+2 e ? 1
(K log K) (T /VT ) + 1 K log K
1/3
2/3
+2 (K log K) (T /VT ) + 1 VT
? ?
1/3
2+2 2
e ? 1 + 4 (K log K ? VT ) T 2/3 ,
where (a) follows from T ? K ? 2, and VT ? K ?1 , K ?1 T . This concludes the proof.
7
(8)
6
Discussion
Unknown variation budget. The Rexp3 policy relies on prior knowledge of VT , but predictions
of VT may be inaccurate (such estimation can be maintained from historical data if actions are
occasionally randomized, for example, by fitting VT = T ? ). Denoting the ?true? variation budget
by VT and the estimate that is used by the agent when tuning Rexp3 by V?T , one may observe that
the analysis in the proof of Theorem 2 holds until equation (8), but then ?T will be tuned using V?T .
This implies that when VT and V?T are ?close,? Rexp3 still guarantees long-run average optimality.
For example, suppose that Rexp3 is tuned by V?T = T ? , but the variation is VT = T ?+? . Then
sublinear regret (of order T 2/3+?/3+? ) is guaranteed as long as ? < (1 ? ?)/3; e.g., if ? = 0 and
? = 1/4, Rexp3 guarantees regret of order T 11/12 (accurate tuning would have guaranteed order
T 3/4 ). Since there are no restrictions on the rate at which the variation budget can be spent, an
interesting and potentially challenging open problem is to delineate to what extent it is possible to
design adaptive policies that do not use prior knowledge of VT , yet guarantee ?good? performance.
Contrasting with traditional (stationary) ?
MAB problems. The characterized minimax regret in
the stationary stochastic setting is of order T when expected rewards can be arbitrarily close to
each other, and of order log T when rewards are ?well separated? (see [13] and [29]). Contrast1/3
ing the minimax regret (of order VT T 2/3 ) we have established in the stochastic non-stationary
MAB problem with those established in stationary settings allows one to quantify the ?price of nonstationarity,? which mathematically captures the added complexity embedded in changing rewards
versus stationary ones (as a function of the allowed variation). Clearly, additional complexity is
introduced even when the allowed variation is fixed and independent of the horizon length.
Contrasting with other non-stationary MAB instances. The class of MAB problems with nonstationary rewards that is formulated in the current chapter extends other MAB formulations that
allow rewards to change in a more structured manner. For example, [32] consider a setting where
rewards evolve according to a Brownian motion and regret is linear in T ; our results (when VT is
linear in T ) are consistent with theirs. Two other representative studies are those of [27], that study a
stochastic MAB problems in which expected rewards may change a finite number of times, and [28]
that formulate an adversarial MAB problem in which the identity of the best arm may change a finite
number of times. Both studies suggest policies?that, utilizing the prior knowledge that the number of
changes must be finite, achieve regret of order T relative to the best sequence of actions. However,
the performance of these policies can deteriorate to regret that is linear in T when the number of
changes is allowed to depend on T . When there is a finite variation (VT is fixed and independent of
T ) but not necessarily a finite number of changes, we establish that the best achievable performance
deteriorate to regret of order T 2/3 . In that respect, it is not surprising that the ?hard case? used to
establish the lower bound in Theorem 1 describes a nature?s strategy that allocates variation over a
large (as a function of T ) number of changes in the expected rewards.
Low variation rates. While our formulation focuses on ?significant? variation in the mean rewards,
our established bounds also hold for ?smaller? variation scales;
when VT decreases from O(1) to
?
O(T ?1/2 ) the minimax regret rate decreases from T 2/3 to T . Indeed, when the variation scale is
O(T ?1/2 ) or smaller, the rate of regret coincides with that of the classical stochastic MAB setting.
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4,836 | 5,379 | Extreme bandits
Alexandra Carpentier
Statistical Laboratory, CMS
University of Cambridge, UK
Michal Valko
SequeL team
INRIA Lille - Nord Europe, France
[email protected]
[email protected]
Abstract
In many areas of medicine, security, and life sciences, we want to allocate limited resources to different sources in order to detect extreme values. In this paper,
we study an efficient way to allocate these resources sequentially under limited
feedback. While sequential design of experiments is well studied in bandit theory,
the most commonly optimized property is the regret with respect to the maximum
mean reward. However, in other problems such as network intrusion detection, we
are interested in detecting the most extreme value output by the sources. Therefore, in our work we study extreme regret which measures the efficiency of an algorithm compared to the oracle policy selecting the source with the heaviest tail.
We propose the E XTREME H UNTER algorithm, provide its analysis, and evaluate
it empirically on synthetic and real-world experiments.
1
Introduction
We consider problems where the goal is to detect outstanding events or extreme values in domains
such as outlier detection [1], security [18], or medicine [17]. The detection of extreme values is
important in many life sciences, such as epidemiology, astronomy, or hydrology, where, for example,
we may want to know the peak water flow. We are also motivated by network intrusion detection
where the objective is to find the network node that was compromised, e.g., by seeking the one
creating the most number of outgoing connections at once. The search for extreme events is typically
studied in the field of anomaly detection, where one seeks to find examples that are far away from
the majority, according to some problem-specific distance (cf. the surveys [8, 16]).
In anomaly detection research, the concept of anomaly is ambiguous and several definitions exist [16]: point anomalies, structural anomalies, contextual anomalies, etc. These definitions are
often followed by heuristic approaches that are seldom analyzed theoretically. Nonetheless, there
exist some theoretical characterizations of anomaly detection. For instance, Steinwart et al. [19]
consider the level sets of the distribution underlying the data, and rare events corresponding to rare
level sets are then identified as anomalies. A very challenging characteristic of many problems in
anomaly detection is that the data emitted by the sources tend to be heavy-tailed (e.g., network traffic [2]) and anomalies come from the sources with the heaviest distribution tails. In this case, rare
level sets of [19] correspond to distributions? tails and anomalies to extreme values. Therefore, we
focus on the kind of anomalies that are characterized by their outburst of events or extreme values,
as in the setting of [22] and [17].
Since in many cases, the collection of the data samples emitted by the sources is costly, it is important to design adaptive-learning strategies that spend more time sampling sources that have a
higher risk of being abnormal. The main objective of our work is the active allocation of the sampling resources for anomaly detection, in the setting where anomalies are defined as extreme values.
Specifically, we consider a variation of the common setting of minimal feedback also known as
the bandit setting [14]: the learner searches for the most extreme value that the sources output by
probing the sources sequentially. In this setting, it must carefully decide which sources to observe
1
because it only receives the observation from the source it chooses to observe. As a consequence,
it needs to allocate the sampling time efficiently and should not waste it on sources that do not have
an abnormal character. We call this specific setting extreme bandits, but it is also known as max-k
problem [9, 21, 20]. We emphasize that extreme bandits are poles apart from classical bandits, where
the objective is to maximize the sum of observations [3]. An effective algorithm for the classical
bandit setting should focus on the source with the highest mean, while an effective algorithm for the
extreme bandit problem should focus on the source with the heaviest tail. It is often the case that
a heavy-tailed source has a small mean, which implies that the classical bandit algorithms perform
poorly for the extreme bandit problem.
The challenging part of our work dwells in the active sampling strategy to detect the heaviest tail
under the limited bandit feedback. We proffer E XTREME H UNTER, a theoretically founded algorithm, that sequentially allocates the resources in an efficient way, for which we prove performance
guarantees. Our algorithm is efficient under a mild semi-parametric assumption common in extreme value theory, while known results by [9, 21, 20] for the extreme bandit problem only hold in
a parametric setting (see Section 4 for a detailed comparison).
2
Learning model for extreme bandits
In this section, we formalize the active (bandit) setting and characterize the measure of performance
for any algorithm ?. The learning setting is defined as follows. Every time step, each of the K arms
(sources) emits a sample Xk,t ? Pk , unknown to the learner. The precise characteristics of Pk are
defined in Section 3. The learner ? then chooses some arm It and then receives only the sample
XIt ,t . The performance of ? is evaluated by the most extreme value found and compared to the
most extreme value possible. We define the reward of a learner ? as:
G?n = max XIt ,t
t?n
The optimal oracle strategy is the one that chooses at each time the arm with the highest potential
revealing the highest value, i.e., the arm ? with the heaviest tail. Its expected reward is then:
?
E [Gn ] = max E max Xk,t
k?K
t?n
The goal of learner ? is to get as close as possible to the optimal oracle strategy. In other words, the
aim of ? is to minimize the expected extreme regret:
Definition 1. The extreme regret in the bandit setting is defined as:
?
?
?
E [Rn ] = E [Gn ] ? E [Gn ] = max E max Xk,t ? E max XIt ,t
k?K
3
t?n
t?n
Heavy-tailed distributions
In this section, we formally define our observation model. Let X1 , . . . , Xn be n i.i.d. observations
from a distribution P . The behavior of the statistic maxi?n Xi is studied by extreme value theory.
One of the main results is the Fisher-Tippett-Gnedenko theorem [11, 12] that characterizes the limiting distribution of this maximum as n converges to infinity. Specifically, it proves that a rescaled
version of this maximum converges to one of the three possible distributions: Gumbel, Fr?echet, or
Weibull. This rescaling factor depends on n. To be concise, we write ?maxi?n Xi converges to a
distribution? to refer to the convergence of the rescaled version to a given distribution. The Gumbel distribution corresponds to the limiting distribution of the maximum of ?not too heavy tailed?
distributions, such as sub-Gaussian or sub-exponential distributions. The Weibull distribution coincides with the behaviour of the maximum of some specific bounded random variables. Finally,
the Fr?echet distribution corresponds to the limiting distribution of the maximum of heavy-tailed
random variables. As many interesting problems concern heavy-tailed distributions, we focus on
Fr?echet distributions in this work. The distribution function of a Fr?echet random variable is defined
for x ? m, and for two parameters ?, s as:
?
P (x) = exp ? x?m
.
s
2
In this work, we consider positive distributions P : [0, ?) ? [0, 1]. For ? > 0, the FisherTippett-Gnedenko theorem also states that the statement ?P converges to an ?-Fr?echet distribution?
is equivalent to the statement ?1 ? P is a ?? regularly varying function in the tail?. These statements
are slightly less restrictive than the definition of approximately ?-Pareto distributions1 , i.e., that there
exists C such that P verifies:
|1 ? P (x) ? Cx?? |
lim
= 0,
(1)
x??
x??
or equivalently that P (x) = 1 ? Cx?? + o(x?? ). If and only if 1 ? P is ?? regularly varying in
the tail, then the limiting distribution of maxi Xi is an ?-Fr?echet distribution. The assumption of
?? regularly varying in the tail is thus the weakest possible assumption that ensures that the (properly rescaled) maximum of samples emitted by a heavy tailed distributions has a limit. Therefore,
the very related assumption of approximate Pareto is almost minimal, but it is (provably) still not
restrictive enough to ensure a convergence rate. For this reason, it is natural to introduce an assumption that is slightly stronger than (1). In particular, we assume, as it is common in the extreme value
literature, a second order Pareto condition also known as the Hall condition [13].
Definition 2. A distribution P is (?, ?, C, C 0 )-second order Pareto (?, ?, C, C 0 > 0) if for x ? 0:
1 ? P (x) ? Cx?? ? C 0 x??(1+?)
By this definition, P (x) = 1 ? Cx?? + O x??(1+?) , which is stronger than the assumption
P (x) = 1 ? Cx?? + o(x?? ), but similar for small ?.
Remark 1. In the definition above, ? defines the rate of the convergence (when x diverges to infinity)
of the tail of P to the tail of a Pareto distribution 1 ? Cx?? . The parameter ? characterizes the
heaviness of the tail: The smaller the ?, the heavier the tail. In the reminder of the paper, we will be
therefore concerned with learning the ? and identifying the smallest one among the sources.
4
Related work
There is a vast body of research in offline anomaly detection which looks for examples that deviate
from the rest of the data, or that are not expected from some underlying model. A comprehensive
review of many anomaly detection approaches can be found in [16] or [8]. There has been also some
work in active learning for anomaly detection [1], which uses a reduction to classification. In online
anomaly detection, most of the research focuses on studying the setting where a set of variables is
monitored. A typical example is the monitoring of cold relief medications, where we are interested
in detecting an outbreak [17]. Similarly to our focus, these approaches do not look for outliers in a
broad sense but rather for the unusual burst of events [22].
In the extreme values settings above, it is often assumed, that we have full information about each
variable. This is in contrast to the limited feedback or a bandit setting that we study in our work.
There has been recently some interest in bandit algorithms for heavy-tailed distributions [4]. However the goal of [4] is radically different from ours as they maximize the sum of rewards and not
the maximal reward. Bandit algorithms have been already used for network intrusion detection [15],
but they typically consider classical or restless setting. [9, 21, 20] were the first to consider the
extreme bandits problem, where our setting is defined as the max-k problem. [21] and [9] consider a fully parametric setting. The reward distributions are assumed to be exactly generalized
extreme value distributions. Specifically, [21] assumes that the distributions are exactly Gumbel,
P (x) = exp(?(x ? m)/s)), and [9], that the distributions are exactly of Gumbel or Fr?echet
P (x) = exp(?(x ? m)? /(s?))). Provided that these assumptions hold, they propose an algorithm for which the regret is asymptotically negligible when compared to the optimal oracle reward.
These results are interesting since they are the first for extreme bandits, but their parametric assumption is unlikely to hold in practice and the asymptotic nature of their bounds limits their impact.
Interestingly, the objective of [20] is to remove the parametric assumptions of [21, 9] by offering
the T HRESHOLDA SCENT algorithm. However, no analysis of this algorithm for extreme bandits is
provided. Nonetheless, to the best of our knowledge, this is the closest competitor for E XTREME H UNTER and we empirically compare our algorithm to T HRESHOLDA SCENT in Section 7.
1
We recall the definition of the standard Pareto distribution as a distribution P , where for some constants ?
and C, we have that for x ? C 1/? , P = 1 ? Cx?? .
3
In this paper we also target the extreme bandit setting, but contrary to [9, 21, 20], we only make a
semi-parametric assumption on the distribution; the second order Pareto assumption (Definition 2),
which is standard in extreme value theory (see e.g., [13, 10]). This is light-years better and significantly weaker than the parametric assumptions made in the prior works for extreme bandits.
Furthermore, we provide a finite-time regret bound for our more general semi-parametric setting
(Theorem 2), while the prior works only offer asymptotic results. In particular, we provide an upper bound on the rate at which the regret becomes negligible when compared to the optimal oracle
reward (Definition 1).
5
Extreme Hunter
In this section, we present our main results. In particular, we present the algorithm and the main
theorem that bounds its extreme regret. Before that, we first provide an initial result on the expectation of the maximum of second order Pareto random variables which will set the benchmark for the
oracle regret. We first characterize the expectation of the maximum of second order Pareto distributions. The following lemma states that the expectation of the maximum of i.i.d. second order Pareto
samples is equal, up to a negligible term, to the expectation of the maximum of i.i.d. Pareto samples.
This result is crucial for assessing the benchmark for the regret, in particular the expected value of
the maximal oracle sample. Theorem 1 is based on Lemma 3, both provided in the appendix.
Theorem 1. Let X1 , . . . , Xn be n i.i.d. samples drawn according to (?, ?, C, C 0 )-second order
Pareto distribution P (see Definition 2). If ? > 1, then:
2C 0 D
1/?
1/?
2
(nC)1/? + B = o (nC)
,
+ C ?+1?+1
E(max Xi ) ? (nC)1/? ? 1? ?1 ? 4D
n (nC)
n?
i
where D2 , D1+? > 0 are some universal constants, and B is defined in the appendix (9).
Theorem 1 implies that the optimal strategy in hindsight attains the following expected reward:
h
i
1/?
E [G?n ] ? max (Ck n) k ? 1? ?1
k
Our objective is therefore to find a learner ? Algorithm 1 E XTREME H UNTER
such that E [G?n ] ? E [G?n ] is negligible when
Input:
compared to E[G?n ], i.e., when compared to
K: number of arms
?
?
(nC ? )1/? ? 1? ?1? ? n1/? where ? is the
n: time horizon
optimal arm.
b: where b ? ?k for all k ? K
N : minimum number of pulls of each arm
From the discussion above, we know that the
Initialize:
minimization of the extreme regret is linked
Tk ? 0 for all k ? K
with the identification of the arm with the heav? ? exp(? log2 n)/(2nK)
iest tail. Our E XTREME H UNTER algorithm is
Run:
based on a classical idea in bandit theory: opfor t = 1 to n do
timism in the face of uncertainty. Our stratfor k = 1 to K do
egy is to estimate E [maxt?n Xk,t ] for any k
if Tk ? N then
and to pull the arm which maximizes its upBk,t ? ?
per bound. From Definition 2, the estimation
else
of this quantity relies heavily on an efficient esestimate b
hk,t that verifies (2)
timation of ?k and Ck , and on associated confibk,t using (3)
estimate C
dence widths. This topic is a classic problem in
update Bk,t using (5) with (2) and (4)
extreme value theory, and such estimators exist
end if
provided that one knows a lower bound b on ?k
end for
[10, 6, 7]. From now on we assume that a conPlay arm kt ? arg maxk Bk,t
stant b > 0 such that b ? mink ?k is known
Tkt ? Tkt + 1
to the learner. As we argue in Remark 2, this
end for
assumption is necessary .
Since our main theoretical result is a finite-time upper bound, in the following exposition we carefully describe all the constants and stress what quantities they depend on. Let Tk,t be the number of
samples drawn from arm k at time t. Define ? = exp(? log2 n)/(2nK) and consider an estimator
4
b
hk,t of 1/?k at time t that verifies the following condition with probability 1 ? ?, for Tk,t larger than
some constant N2 that depends only on ?k , Ck , C 0 and b:
p
1
?b/(2b+1)
hk,t ? D log(1/?)Tk,t
= B1 (Tk,t ),
(2)
?k ? b
where D is a constant that also depends only on ?k , Ck , C 0 , and b. For instance, the estimator
in [6] (Theorem 3.7) verifies this property and provides D and N2 but other estimators are possible.
Consider the associated estimator for Ck :
?
?
Tk,t n
o
X
b
1
h /(2b+1) ?
bk,t = T 1/(2b+1) ?
C
1 Xk,u ? Tk,tk,t
(3)
k,t
Tk,t u=1
For this estimator, we know [7] with probability 1 ? ? that for Tk,t ? N2 :
q
bk,t ? E log(Tk,t /?) log(Tk,t )T ?b/(2b+1) = B2 (Tk,t ),
Ck ? C
k,T
(4)
where E is derived in [7] in the proof of Theorem 2. Let N = max A log(n)2(2b+1)/b , N2 where
A depends on (?k , Ck )k , b, D, E, and C 0 , and is such that:
(2b+1)/b
?
2D log(n)2
max (2B1 (N ), 2B2 (N )/Ck ) ? 1, N ? (2D log2 n)(2b+1)/b , and N > 1?maxk 1/?k
This inspires Algorithm 1, which first pulls each arm N times and then, at each time t > KN , pulls
the arm that maximizes Bk,t , which we define as:
bhk,t +B1 (Tk,t )
? b
bk,t + B2 (Tk,t ) n
?
hk,t , B1 (Tk,t ) ,
(5)
C
? y) = ?(1
? ? x ? y), where we set ?
? = ? for any x > 0 and +? otherwise.
where ?(x,
Remark 2. A natural question is whether it is possible to learn ?k as well. In fact, this is not possible
for this model and a negative result was proved by [7]. The result states that in this setting it is not
possible to test between two fixed values of ? uniformly over the set of distributions. Thereupon, we
define b as a lower bound for all ?k . With regards to the Pareto distribution, ? = ? corresponds to
the exact Pareto distribution, while ? = 0 for such distribution that is not (asymptotically) Pareto.
We show that this algorithm meets the desired properties. The following theorem states our main
result by upper-bounding the extreme regret of E XTREME H UNTER.
Theorem 2. Assume that the distributions of the arms are respectively (?k , ?k , Ck , C 0 ) second
order Pareto (see Definition 2) with mink ?k > 1. If n ? Q, the expected extreme regret of E X TREME H UNTER is bounded from above as:
?
(2b+1)/b
? log(n)(1?1/?? )
?b/((b+1)?? )
E [Rn ] ? L(nC ? )1/? K
log(n)
+
n
+
n
= E [G?n ] o(1),
n
where L, Q > 0 are some constants depending only on (?k , Ck )k , C 0 , and b (Section 6).
Theorem 2 states that the E XTREME H UNTER strategy performs almost as well as the best (oracle)
strategy, up to a term that is negligible when compared to the performance of the oracle strategy.
?
Indeed, the regret is negligible when compared to (nC ? )1/? , which is the order of magnitude of the
performance of the best oracle strategy E [G?n ] = maxk?K E [maxt?n Xk,t ]. Our algorithm thus
detects the arm that has the heaviest tail.
For n large enough (as a function of (?k , ?k , Ck )k , C 0 and K), the two first terms in the regret
become negligible when compared to the third one, and the regret is then bounded as:
?
E [Rn ] ? E [G?n ] O n?b/((b+1)? )
We make two observations: First, the larger the b, the tighter this bound is, since the model is then
closer to the parametric case. Second, smaller ?? also tightens the bound, since the best arm is then
very heavy tailed and much easier to recognize.
5
6
Analysis
In this section, we prove an upper bound on the extreme regret of Algorithm 1 stated in Theorem 2.
Before providing the detailed proof, we give a high-level overview and the intuitions.
In Step 1, we define the (favorable) high probability event ? of interest, useful for analyzing the
mechanism of the bandit algorithm. In Step 2, given ?, we bound the estimates of ?k and Ck , and
use them to bound the main upper confidence bound. In Step 3, we upper-bound the number of pulls
of each suboptimal arm: we prove that with high probability we do not pull them too often. This
enables us to guarantee that the number of pulls of the optimal arms ? is on ? equal to n up to a
negligible term.
The final Step 4 of the proof is concerned with using this lower bound on the number of pulls of
the optimal arm in order to lower bound the expectation of the maximum of the collected samples.
Such step is typically straightforward in the classical (mean-optimizing) bandits by the linearity of
the expectation. It is not straightforward in our setting. We therefore prove Lemma 2, in which we
show that the expected value of the maximum of the samples in the favorable event ? will be not too
far away from the one that we obtain without conditioning on ?.
Step 1: High probability event. In this step, we define the favorable event ?. We set
def
? = exp(? log2 n)/(2nK) and consider the event ? such that for any k ? K, N ? T ? n:
p
1
? k (T ) ? D log(1/?)T ?b/(2b+1) ,
?k ? h
p
Ck ? C?k (T ) ? E log(T /?)T ?b/(2b+1) ,
? k (T ) and C?k (T ) are the estimates of 1/?k and Ck respectively using the first T samples.
where h
bk,t which are the estimates of the same quantities at time
Notice, they are not the same as b
hk,t and C
t for the algorithm, and thus with Tk,t samples. The probability of ? is larger than 1 ? 2nK? by a
union bound on (2) and (4).
Step 2: Bound on Bk,t . The following lemma holds on ? for upper- and lower-bounding Bk,t .
Lemma 1. (proved in the appendix) On ?, we have that for any k ? K, and for Tk,t ? N :
p
1
1
?b/(2b+1)
(Ck n) ?k ? 1? ?1k ? Bk,t ? (Ck n) ?k ? 1? ?1k
1 + F log(n) log(n/?)Tk,t
(6)
Step 3: Upper bound on the number of pulls of a suboptimal arm. We proceed by using the
bounds on Bk,t from the previous step to upper-bound the number of suboptimal pulls. Let ? be the
best arm. Assume that at round t, some arm k 6= ? is pulled. Then by definition of the algorithm
B?,t ? Bk,t , which implies by Lemma 1:
p
?b/(2b+1)
1/??
1/?
(C ? n)
? 1? ?1? ? (Ck n) k ? 1? ?1k
1 + F log(n) log(n/?)Tk,t
Rearranging the terms we get:
1/??
(C ? n)
(Ck n)
1/?k
? 1? ?1?
? 1? ?1k
p
?b/(2b+1)
? 1 + F log(n) log(n/?)Tk,t
(7)
We now define ?k which is analogous to the gap in the classical bandits:
1/??
(C ? n)
? 1? ?1?
?k =
?1
1/?
(Ck n) k ? 1? ?1k
Since Tk,t ? n, (7) implies for some problem dependent constants G and G0 dependent only on
(?k , Ck )k , C 0 and b, but independent of ? that:
2
(2b+1)/(2b)
(2b+1)(2b)
log(n/?)
Tk,t ? N + G0 log n ?
? N + G log2 n log(n/?)
2
k
6
This implies that number T ? of pulls of arm ? is with probability 1 ? ? 0 , at least
X
(2b+1)/(2b)
n?
G log2 n log(2nK/? 0 )
? KN,
k6=?
where ? 0 = 2nK?. Since n is larger than
Q ? 2KN + 2GK log2 n log (2nK/? 0 )
we have that T ? ?
n
2
(2b+1)/(2b)
,
as a corollary.
Step 4: Bound on the expectation. We start by lower-bounding the expected gain:
E[Gn ] = E max XIt ,Tk,t ? E max XIt ,Tk,t 1{?} ? E max X?,T?,t 1{?} = E max? Xi 1{?}
t?n
t?n
t?n
i?T
The next lemma links the expectation of maxt?T ? X?,t with the expectation of maxt?T ? X?,t 1{?}.
Lemma 2. (proved in the appendix) Let X1 , . . . , XT be i.i.d. samples from an (?, ?, C, C 0 )-second
order Pareto distribution F . Let ? 0 be an event of probability larger than 1 ? ?. Then for ? < 1/2
and for T ? Q large enough so that c max 1/T, 1/T ? ? 1/4 for a given constant c > 0, that
1/?
depends only on C, C 0 and ?, and also for T ? log(2) max C (2C 0 )
, 8 log (2) :
1/? 1?1/?
1/?
8
(T C)
?
E max Xt 1{?} ? (T C)
? 1? ?1 ? 4 + ??1
t?T
2C 0 D1+?
1/?
1/?
2
? 2 4D
(T
C)
+
(T
C)
+
B
.
1+?
?
T
C
T
Since n is large enough so that 2n2 K? 0 = 2n2 K exp ? log2 n ? 1/2, where ? 0 = exp ? log2 n ,
and the probability of ? is larger than 1 ? ? 0 , we can use Lemma 2 for the optimal arm:
01? 1 8D2
1
0
Dmax
8
2B
? ?? ? T ? ? (C4C
,
E max? X?,t 1{?} ? (T ? C ? ) ?? ? 1? ?1? ? 4+ ??1
?
1
? )1+b (T ? )b
?
? ?
t?T
(T C ) ?
def
where Dmax = maxi D1+?i . Using Step 3, we bound the above with a function of n. In particular,
?
we lower-bound the last three terms in the brackets using T ? ? n2 and the (T ? C ? )1/? factor as:
?
(T ? C ? )1/? ? (nC ? )1/?
?
1?
GK
n
2b+1
log(2n2 K/? 0 ) 2b ? KN
n
We are now ready to relate the lower bound on the gain of E XTREME H UNTER with the upper bound
of the gain of the optimal policy (Theorem 1), which brings us the upper bound for the regret:
E [Rn ] = E [G?n ] ? E [Gn ] ? E [G?n ] ? E max? Xi ? E [G?n ] ? E max? X?,t 1{?}
i?T
t?T
2b+1
?
2
0
01?1/??
KN
B
2b
,
? H(nC ? )1/? n1 + (nC1? )b + GK
log(2n
K/?
)
+
+
?
+
?
n
n
(nC ? )1/?
where H is a constant that depends on (?k , Ck )k , C 0 , and b. To bound the last term, we use the
?
?
?
?
definition of B (9) to get the n?? /((? +1)? ) term, upper-bounded by n?b/((b+1)? ) as b ? ? ? .
?1
?b
?
Notice that this final term also eats up n and n terms since b/((b + 1)? ) ? min(1, b).
We finish by using ? 0 = exp ? log2 n and grouping the problem-dependent constants into L to get
the final upper bound:
?
?
?
(2b+1)/b
E [Rn ] ? L(nC ? )1/? K
+ n? log(n)(1?1/? ) + n?b/((b+1)? )
n log(n)
7
Comparison of extreme bandit strategies (K=3)
Comparison of extreme bandit strategies on the network data K=5
250
ExtremeHunter
UCB
ThresholdAscent
ExtremeHunter
UCB
ThresholdAscent
9000
Comparison of extreme bandit strategies (K=3)
2500
10000
ExtremeHunter
UCB
ThresholdAscent
200
2000
8000
6000
5000
4000
extreme regret
extreme regret
extreme regret
7000
1500
1000
150
100
3000
2000
50
500
1000
0
0
1000
2000
3000
4000
5000
6000
time t
7000
8000
9000
10000
0
0
1000
2000
3000
4000
5000
time t
6000
7000
8000
9000
10000
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
time t
Figure 1: Extreme regret as a function of time for the exact Pareto distributions (left), approximate
Pareto (middle) distributions, and the network traffic data (right).
7
Experiments
In this section, we empirically evaluate E XTREME H UNTER on synthetic and real-world data. The
measure of our evaluation is the extreme regret from Definition 1. Notice that even thought we
evaluate the regret as a function of time T , the extreme regret is not cumulative and it is more in the
spirit of simple regret [5]. We compare our E XTREME H UNTER with T HRESHOLDA SCENT [20].
Moreover, we also compare to classical U CB [3], as an example of the algorithm that aims for the
arm with the highest mean as opposed to the heaviest tail. When the distribution of a single arm
has both the highest mean and the heaviest-tail, both E XTREME H UNTER and U CB are expected to
perform the same with respect to the extreme regret. In the light of Remark 2, we set b = 1 to
consider a wide class of distributions.
Exact Pareto Distributions In the first experiment, we consider K = 3 arms with the distributions
Pk (x) = 1?x??k , where ? = [5, 1.1, 2]. Therefore, the most heavy-tailed distribution is associated
with the arm k = 2. Figure 1 (left) displays the averaged result of 1000 simulations with the time
horizon T = 104 . We observe that E XTREME H UNTER eventually keeps allocating most of the
pulls to the arm of the interest. Since in this case, the arm with the heaviest tail is also the arm
with the largest mean, U CB also performs well and it is even able to detect the best arm earlier.
T HRESHOLDA SCENT, on the other way, was not always able to allocate the pulls properly in 104
steps. This may be due to the discretization of the rewards that this algorithm is using.
Approximate Pareto Distributions For the exact Pareto distributions, the smaller the tail index
the higher the mean and even UCB obtains a good performance. However, this is no longer necessarily the case for the approximate Pareto distributions. For this purpose, we perform the second
experiment where we mix an exact Pareto distribution with a Dirac distribution in 0. We consider
K = 3 arms. Two of the arms follow the exact Pareto distributions with ?1 = 1.5 and ?3 = 3.
On the other hand, the second arm has a mixture weight of 0.2 for the exact Pareto distribution with
?2 = 1.1 and 0.8 mixture weight of the Dirac distribution in 0. For this setting, the second arm
is the most heavy-tailed but the first arms has the largest mean. Figure 1 (middle) shows the result. We see that U CB performs worse since it eventually focuses on the arm with the largest mean.
T HRESHOLDA SCENT performs better than U CB but not as good as E XTREME H UNTER.
Computer Network Traffic Data In this experiment, we evaluate E XTREME H UNTER on heavytailed network traffic data which was collected from user laptops in the enterprise environment [2].
The objective is to allocate the sampling capacity among the computer nodes (arms), in order to find
the largest outbursts of the network activity. This information then serves an IT department to further
investigate the source of the extreme network traffic. For each arm, a sample at the time t corresponds to the number of network activity events for 4 consecutive seconds. Specifically, the network
events are the starting times of packet flows. In this experiment, we selected K = 5 laptops (arms),
where the recorded sequences were long enough. Figure 1 (right) shows that E XTREME H UNTER
again outperforms both T HRESHOLDA SCENT and U CB.
Acknowledgements We would like to thank John Mark Agosta and Jennifer Healey for the network traffic data. The research presented in this paper was supported by Intel Corporation, by
French Ministry of Higher Education and Research, and by European Community?s Seventh Framework Programme (FP7/2007-2013) under grant agreement no 270327 (CompLACS).
8
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9
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4,837 | 538 | Iterative Construction of
Sparse Polynomial Approximations
Terence D. Sanger
Massachusetts Institute
of Technology
Room E25-534
Cambridge, MA 02139
[email protected]
Richard S. Sutton
GTE Laboratories
Incorporated
40 Sylvan Road
Waltham, MA 02254
[email protected]
Christopher J. Matheus
GTE Laboratories
Incorporated
40 Sylvan Road
Waltham, MA 02254
[email protected]
Abstract
We present an iterative algorithm for nonlinear regression based on construction of sparse polynomials. Polynomials are built sequentially from
lower to higher order. Selection of new terms is accomplished using a novel
look-ahead approach that predicts whether a variable contributes to the
remaining error. The algorithm is based on the tree-growing heuristic in
LMS Trees which we have extended to approximation of arbitrary polynomials of the input features. In addition, we provide a new theoretical
justification for this heuristic approach. The algorithm is shown to discover a known polynomial from samples, and to make accurate estimates
of pixel values in an image-processing task.
1
INTRODUCTION
Linear regression attempts to approximate a target function by a model that is a
linear combination of the input features. Its approximation ability is thus limited
by the available features. We describe a method for adding new features that are
products or powers of existing features. Repeated addition of new features leads
to the construction of a polynomial in the original inputs, as in (Gabor 1961).
Because there is an infinite number of possible product terms, we have developed
a new method for predicting the usefulness of entire classes of features before they
are included. The resulting nonlinear regression will be useful for approximating
functions that can be described by sparse polynomials.
1064
Iterative Construction of Sparse Polynomial Approximations
f
Xn
Figure 1: Network depiction of linear regression on a set of features Xi.
2
THEORY
Let {xdi=l be the set of features already included in a model that attempts to
predict the function f . The output of the model is a linear combination
n
i = LCiXi
i=l
where the Ci'S are coefficients determined using linear regression. The model can
also be depicted as a single-layer network as in figure 1. The approximation error
is e
f - j, and we will attempt to minimize E[e 2 ] where E is the expectation
operator.
=
The algorithm incrementally creates new features that are products of existing
features. At each step, the goal is to select two features xp and Xq already in the
model and create a new feature XpXq (see figure 2). Even if XpXq does not decrease
the approximation error, it is still possible that XpXqXr will decrease it for some
X r . So in order to decide whether to create a new feature that is a product with
x p , the algorithm must "look-ahead" to determine if there exists any polynomial a
in the xi's such that inclusion ofax p would significantly decrease the error. If no
such polynomial exists, then we do not need to consider adding any features that
are products with xp.
=
Define the inner product between two polynomials a and b as (alb)
E[ab] where
the expected value is taken with respect to a probability measure I-" over the (zeroE[a 2 ], and let P be the set of
mean) input values. The induced norm is IIal12
polynomials with finite norm. {P, (?I?)} is then an infinite-dimensional linear vector
space. The Weierstrass approximation theorem proves that P is dense in the set of
all square-integrable functions over 1-", and thus justifies the assumption that any
function of interest can be approximated by a member of P.
=
Assume that the error e is a polynomial in P. In order to test whether
ipates in e for any polynomial a E P, we write
e
=
apxp
+ bp
axp
partic-
1065
1066
Sanger, Sutton, and Matheus
f
Figure 2: Incorporation of a new product term into the model.
where ap and bp are polynomials, and ap is chosen to minimize lIapxp - ell 2
E[( apxp - e )2]. The orthogonality principle then shows that apxp is the projection
of the polynomial e onto the linear subspace of polynomials xpP. Therefore, bp is
orthogonal to xpP, so that E[bpg] = 0 for all g in xpP.
We now write
E[e 2]
= E[a;x;] + 2E[apxpbp] + E[b;] = E[a;x;] + E[b;]
since E[apxpbp] = 0 by orthogonality. If apxp were included in the model, it would
thus reduce E[e 2] by E[a;x;], so we wish to choose xp to maximize E[a;x;]. Unfortunately, we have no dIrect measurement of ap ?
3
METHODS
Although E[a;x;] cannot be measured directly, Sanger (1991) suggests choosing xp
to maximize E[e2x~] instead, which is directly measurable. Moreover, note that
E[e 2x;]
=
E[a;x;] + 2E[apx;bp] + E[x;b;]
=
E[a;x;]
and thus E[e 2x;] is related to the desired but unknown value E[a;x;]. Perhaps
better would be to use
E[e 2x 2]
E[x~] -
~=-::-:p-
E[a 2x4 ]
p
p
E[x~]
which can be thought of as the regression of (a;x~)xp against xp'
More recently, (Sutton and Matheus 1991) suggest using the regression coefficients
of e2 against
for all i as the basis for comparison. The regression coefficients Wi
are called "potentials", and lead to a linear approximation of the squared error:
xr
(1)
Iterative Construction of Sparse Polynomial Approximations
If a new term apxp were included in the model of f, then the squared error would
be b; which is orthogonal to any polynomial in xpP and in particular to x;. Thus
the coefficient of x; in (1) would be zero after inclusion of apxp, and wpE[x;] is an
approximation to the decrease in mean-squared error E[e 2 ] - E[b;] which we can
expect from inclusion of apxp. We thus choose xp by maximizing wpE[x;].
This procedure is a form of look-ahead which allows us to predict the utility of a
high-order term apxp without actually including it in the regression. This is perhaps
most useful when the term is predicted to make only a small contribution for the
optimal a p , because in this case we can drop from consideration any new features
that include xp.
We can choose a different variable Xq similarly, and test the usefulness of incorporating the product XpXq by computing a "joint potential" Wpq which is the regression of
the squared error against the model including a new term x~x~. The joint potential
attempts to predict the magnitude of the term E[a~qx;xi].
We now use this method to choose a single new feature XpXq to include in the model.
For all pairs XiXj such that Xi and Xj individually have high potentials, we perform
a third regression to determine the joint potentials of the product terms XiXj. Any
term with a high joint potential is likely to participate in f. We choose to include the
new term XpXq with the largest joint potential. In the network model, this results in
the construction of a new unit that computes the product of xp and x q, as in figure
2. The new unit is incorporated into the regression, and the resulting error e will
be orthogonal to this unit and all previous units. Iteration of this technique leads
to the successive addition of new regression terms and the successive decrease in
mean-squared error E[e 2 ]. The process stops when the residual mean-squared error
drops below a chosen threshold, and the final model consists of a sparse polynomial
in the original inputs.
We have implemented this algorithm both in a non-iterative version that computes
coefficients and potentials based on a fixed data set, and in an iterative version that
uses the LMS algorithm (Widrow and Hoff 1960) to compute both coefficients and
potentials incrementally in response to continually arriving data. In the iterative
version, new terms are added at fixed intervals and are chosen by maximizing over
the potentials approximated by the LMS algorithm. The growing polynomial is
efficiently represented as a tree-structure, as in (Sanger 1991a).
Although the algorithm involves three separate regressions, each is over only O( n)
terms, and thus the iterative version of the algorithm is only of O(n) complexity
per input pattern processed.
4
RELATION TO OTHER ALGORITHMS
Approximation of functions over a fixed monomial basis is not a new technique
(Gabor 1961, for example) . However, it performs very poorly for high-dimensional
input spaces, since the set of all monomials (even of very low order) can be prohibitively large. This has led to a search for methods which allow the generation of
sparse polynomials. A recent example and bibliography are provided in (Grigoriev
et al. 1990), which describes an algorithm applicable to finite fields (but not to
1067
1068
Sanger, Sutton, and Matheus
j
Figure 3: Products of hidden units in a sigmoidal feedforward network lead to a
polynomial in the hidden units themselves.
real-valued random variables).
The GMDH algorithm (Ivakhnenko 1971, Ikeda et al. 1976, Barron et al. 1984)
incrementally adds new terms to a polynomial by forming a second (or higher)
order polynomial in 2 (or more) of the current terms, and including this polynomial
as a new term if it correlates with the error. Since GMDH does not use look-ahead,
it risks avoiding terms which would be useful at future steps. For example, if the
polynomial to be approximated is xyz where all three variables are independent,
then no polynomial in x and y alone will correlate with the error, and thus the
term xy may never be included. However, x 2y2 does correlate with x 2y2 Z2, so
the look-ahead algorithm presented here would include this term, even though the
error did not decrease until a later step. Although GMDH can be extended to
test polynomials of more than 2 variables, it will always be testing a finite-order
polynomial in a finite number of variables, so there will always exist target functions
which it will not be able to approximate.
Although look-ahead avoids this problem, it is not always useful. For practical
purposes, we may be interested in the best Nth-order approximation to a function,
so it may not be helpful to include terms which participate in monomials of order
greater than N, even if these monomials would cause a large decrease in error.
For example, the best 2nd-order approximation to x 2 + ylOOO + zlOOO may be x 2 ,
even though the other two terms contribute more to the error. In practice, some
combination of both infinite look-ahead and GMDH-type heuristics may be useful.
5
APPLICATION TO OTHER STRUCTURES
These methods have a natural application to other network structures. The inputs
to the polynomial network can be sinusoids (leading to high-dimensional Fourier
representations), Gaussians (leading to high-dimensional Radial Basis Functions)
or other appropriate functions (Sanger 1991a, Sanger 1991b). Polynomials can
Iterative Construction of Sparse Polynomial Approximations
even be applied with sigmoidal networks as input, so that
Xi
=
(T
(I:
SijZj )
where the z;'s are the original inputs, and the Si;'S are the weights to a sigmoidal
hidden unit whose value is the polynomial term Xi. The last layer of hidden units
in a multilayer network is considered to be the set of input features Xi to a linear
output unit, and we can compute the potentials of these features to determine the
hidden unit xp that would most decrease the error if apxp were included in the
model (for the optimal polynomial ap ). But a p can now be approximated using a
subnetwork of any desired type. This subnetwork is used to add a new hidden unit
C&pxp that is the product of xp with the subnetwork output C&p, as in figure 3.
In order to train the C&p subnetwork iteratively using gradient descent, we need to
compute the effect of changes in C&p on the network error ? E[(f - j)2]. We have
=
where S 4pXp is the weight from the new hidden unit to the outpu t. Without loss of
1 by including this factor within C&p. Thus the error
generality we can set S4pXp
term for iteratively training the subnetwork ap is
=
which can be used to drive a standard backpropagation-type gradient descent algorithm. This gives a method for constructing new hidden nodes and a learning
algorithm for training these nodes. The same technique can be applied to deeper
layers in a multilayer network.
6
EXAMPLES
We have applied the algorithm to approximation of known polynomials in the presence of irrelevant noise variables, and to a simple image-processing task.
Figure 4 shows the results of applying the algorithm to 200 samples of the polynomial 2 + 3XIX2 + 4X3X4X5 with 4 irrelevant noise variables. The algorithm correctly
finds the true polynomial in 4 steps, requiring about 5 minutes on a Symbolics Lisp
Machine. Note that although the error did not decrease after cycle 1, the term X4X5
was incorporated since it would be useful in a later step to reduce the error as part
of X3X4X5 in cycle 2.
The image processing task is to predict a pixel value on the succeeding scan line
from a 2x5 block of pixels on the preceding 2 scan lines. If successful, the resulting
polynomial can be used as part of a DPCM image coding strategy. The network
was trained on random blocks from a single face image, and tested on a different
image. Figure 5 shows the original training and test images, the pixel predictions,
and remaining error . Figure 6 shows the resulting 55-term polynomial. Learning
this polynomial required less than 10 minutes on a Sun Sparcstation 1.
1069
1070
Sanger, Sutton, and Matheus
+
+
200 sa.mples of IJ = 2
3z1 z2
4x3 z4 Zs
with 4 additional irrelevant inputs, z6-z9
Original MSE: 1.0
Cycle 1 :
MSE:
Terms:
Coeffs:
Po ten tials:
Top Pairs:
New Term:
0.967
X2
X4
Xl
X3
-0.19
0.14
0.24
0.31
0.22
0.24
0.2S
0 . 32
(S 4) (5 3) (43) (4 4)
XIO =X4 X S
Cycle 2:
MSE:
Terms:
Coeffs:
Potentials:
Top Pairs:
New Term:
0.966
Xl
X2
X3
X4
-0.19
0.14
0.24
0.30
0.25
0.22
0.2S
O.OS
(103) (101) (102) (10 10)
Xu =X10 X 3 =X3 X 4 X S
Cycle 3:
MSE:
Terms:
Coeffs:
Potentials:
Top Pairs:
New Term:
0.349
Xl
X2
X4
X3
0. 04 -0.26
0.09
0.37
0.02
0.S2
0.S9
0.03
(2 1) (2 9) (22) (1 9)
Xu =X1 X 2
Cycle 4:
MSE:
Terms:
Coeffs:
Solution:
0.000
Xl
X2
-0. 00
-0.00
2
X3
-0.00
X4
0.00
Xs
0. 17
0 .33
X6
0.48
0.01
X7
0.03
0.08
X8
O.OS
0. 01
X9
0.S8
0.05
Xs
0.18
0 .02
X6
0.48
0.03
X7
0.03
0.08
X8
O.OS
0.02
X9
0.S7
0.03
XlO
O.OS
0 .47
Xs
-0.04
-0.08
X6
0.27
0. 03
X7
0.10
-O.OS
X8
0 .22
-0.06
X9
0.42
0.05
X10
-0.26
-O.OS
Xll
4.07
O. OS
Xs
-0.00
X6
0 .00
X7
0.00
X8
0.00
X9
0.00
X10
-0.00
Xu
4.00
X l2
3.00
+ 3X1 X2 + 4X3X4X5
Figure 4: A simple example of polynomial learning.
Figure 5: Original, predicted, and error images. The top row is the training image
(RMS error 8.4), and the bottom row is the test image (RMS error 9.4).
Iterative Construction of Sparse Polynomial Approximations
-40? 1z0 + -23.9z1 + -5.4z2 + -17?1z3+
(1.1z 5 + 2.4z8 + -1.1z2 + -1.5z0 + -2.0Z1 + 1.3z 4 + 2.3z6 + 3?1z7 + -25 .6)z4 +
(
(-2.9z9 + 3.0z 8 + -2.9z4 + -2.8z3 + -2 .9z2 + -1.9z5 + -6. 3%0 + -5.2%1 + 2.5z6 + 6.7z7 + 1.1)z9+
(3 . 9z 8 + Z5 + 3.3z4 + 1. 6z3 + 1.1z2 + 2 .9z 6 + 5.0Z7 + 16 .1)z8+
-2.3%3 + -2 .1%2 + -1.6.%1 + 1.1z4 + 2?1z6 + 3.5%7 + 28 .6)z5+
87 ?1z6 + 128.1%7 + 80 .5%8+
(
(-2?6.%9 + -2.4%5 + -4.5%0 + -3 .9%1 + 3.4%6 + 7 .3%7 + -2.5)%9+
21.7%8 + -16 . 0%4 + -12?1z3 + -8.8%2 + 31.4)%9+
2.6
Figure 6: 55-term polynomial used to generate figure 5.
Acknowledgments
We would like to thank Richard Brandau for his helpful comments and suggestions
on an earlier draft of this paper. This report describes research done both at
GTE Laboratories Incorporated, in Waltham MA, and at the laboratory of Dr.
Emilio Bizzi in the department of Brain and Cognitive Sciences at MIT. T. Sanger
was supported during this work by a National Defense Science and Engineering
Graduate Fellowship, and by NIH grants 5R37 AR26710 and 5R01NS09343 to Dr.
Bizzi.
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Record, Part 4, pages 96-104.
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4,838 | 5,380 | Discovering, Learning and Exploiting Relevance
Mihaela van der Schaar
Electrical Engineering Department
University of California Los Angeles
[email protected]
Cem Tekin
Electrical Engineering Department
University of California Los Angeles
[email protected]
Abstract
In this paper we consider the problem of learning online what is the information
to consider when making sequential decisions. We formalize this as a contextual
multi-armed bandit problem where a high dimensional (D-dimensional) context
vector arrives to a learner which needs to select an action to maximize its expected
reward at each time step. Each dimension of the context vector is called a type. We
assume that there exists an unknown relation between actions and types, called the
relevance relation, such that the reward of an action only depends on the contexts
of the relevant types. When the relation is a function, i.e., the reward of an action
only depends on the context of a single type, and the expected reward of an action
is Lipschitz continuous in the context of its relevant type, we propose an algo?
? ? ) regret with a high probability, where ? = 2/(1 + 2).
rithm that achieves O(T
Our algorithm achieves this by learning the unknown relevance relation, whereas
prior contextual bandit algorithms that do not exploit the existence of a relevance
? (D+1)/(D+2) ) regret. Our algorithm alternates between exrelation will have O(T
ploring and exploiting, it does not require reward observations in exploitations,
and it guarantees with a high probability that actions with suboptimality greater
than are never selected in exploitations. Our proposed method can be applied to
a variety of learning applications including medical diagnosis, recommender systems, popularity prediction from social networks, network security etc., where at
each instance of time vast amounts of different types of information are available
to the decision maker, but the effect of an action depends only on a single type.
1
Introduction
In numerous learning problems the decision maker is provided with vast amounts of different types
of information which it can utilize to learn how to select actions that lead to high rewards. The
value of each type of information can be regarded as the context on which the learner acts, hence
all the information can be encoded in a context vector. We focus on problems where this context
vector is high dimensional but the reward of an action only depends on a small subset of types. This
dependence is given in terms of a relation between actions and types, which is called the relevance
relation. For an action set A and a type set D, the relevance relation is given by R = {R(a)}a?A ,
where R(a) ? D. Expected reward of an action a only depends on the values of the relevant
types of contexts. Hence, for a context vector x, action a?s expected reward is equal to ?(a, xR(a) ),
where xR(a) is the context vector corresponding to the types in R(a). Several examples of relevance
relations and their effect on expected action rewards are given in Fig. 1. The problem of finding
the relevance relation is important especially when maxa?A |R(a)| << |D|.1 In this paper we
consider the case when the relevance relation is a function, i.e., |R(a)| = 1, for all a ? A, which is
an important special case. We discuss the extension of our framework to the more general case in
Section 3.3.
1
For a set A, |A| denotes its cardinality.
1
Figure 1: Examples of relevance relations: (i) general relevance relation, (ii) linear relevance relation, (iii) relevance function. In this paper we only consider (iii), while our methods can easily be
generalized to (i) and (ii).
Relevance relations exists naturally in many practical applications. For example, when sequentially
treating patients with a particular disease, many types of information (contexts) are usually available
- the patients? age, weight, blood tests, scans, medical history etc. If a drug?s effect on a patient is
caused by only one of the types, then learning the relevant type for the drug will result in significantly
faster learning for the effectiveness of the drug for the patients.2 Another example is recommender
systems, where recommendations are made based on the high dimensional information obtained
from the browsing and purchase histories of the users. A user?s response to a product recommendation will depend on the user?s gender, occupation, history of past purchases etc., while his/her
response to other product recommendations may depend on completely different information about
the user such as the age and home address.
Traditional contextual bandit solutions disregard existence of such relations, hence have regret
bounds that scale exponentially with the dimension of the context vector [1, 2]. In order to solve the
curse of dimensionality problem, a new approach which learns the relevance relation in an online
way is required. The algorithm we propose simultaneously learns the relevance relation (when it is a
function) and the action rewards by comparing sample mean rewards of each action for context pairs
of different types that are calculated based on the context and reward observations so far. The only
assumption we make about actions and contexts is the Lipschitz continuity of expected reward of an
action in the context of its relevant type. Our main contributions can be summarized as follows:
? We propose the Online Relevance Learning with Controlled Feedback (ORL-CF) algorithm
that alternates between exploration
and exploitation phases, which achieves a regret bound
?
? 3
?
of O(T ), with ? = 2/(1 + 2), when the relevance relation is a function.
? We derive separate bounds on the regret incurred in exploration and exploitation phases.
ORL-CF only needs to observe the reward in exploration phases, hence the reward feedback
is controlled. ORL-CF achieves the same time order of regret even when observing the
reward has a non-zero cost.
? Given any ? > 0, which is an input to ORL-CF, suboptimal actions will never be selected
in exploitation steps with probability at least 1 ? ?. This is very important, perhaps vital in
numerous applications where the performance needs to be guaranteed, such as healthcare.
Due to the limited space, numerical results on the performance of our proposed algorithm is included
in the supplementary material.
2
Even when there are multiple relevant types for each action, but there is one dominant type whose effect
on the reward of the action is significantly larger than the effects of other types, assuming that the relevance
relation is a function will be a good approximation.
3
? is the same as O(?) except it hides terms that have polylogarithmic growth.
O(?) is the Big O notation, O(?)
2
2
Problem Formulation
A is the set of actions, D is the dimension of the context vector, D := {1, 2, . . . , D} is the set of
types, and R = {R(a)}a?A : A ? D is the relevance function, which maps every a ? A to a
unique d ? D. At each time step t = 1, 2, . . ., a context vector xt arrives to the learner. After
observing xt the learner selects an action a ? A, which results in a random reward rt (a, xt ). The
learner may choose to observe this reward by paying cost cO ? 0. The goal of the learner is to
maximize the sum of the generated rewards minus costs of observations for any time horizon T .
Each xt consists of D types of contexts, and can be written as xt = (x1,t , x2,t , . . . , xD,t ) where xi,t
is called the type i context. Xi denotes the space of type i contexts and X := X1 ? X2 ? . . . ? XD
denotes the space of context vectors. At any t, we have xi,t ? Xi for all i ? D. For the sake of
notational simplicity we take Xi = [0, 1] for all i ? D, but all our results can be generalized to
the case when Xi is a bounded subset of the real line. For x = (x1 , x2 , . . . , xD ) ? X , rt (a, x)
is generated according to an i.i.d. process with distribution F (a, xR(a) ) with support in [0, 1] and
expected value ?(a, xR(a) ).
The following assumption gives a similarity structure between the expected reward of an action and
the contexts of the type that is relevant to that action.
Assumption 1. For all a ? A, x, x0 ? X , we have |?(a, xR(a) )??(a, x0R(a) )| ? L|xR(a) ?x0R(a) |,
where L > 0 is the Lipschitz constant.
We assume that the learner knows the L given in Assumption 1. This is a natural assumption in
contextual bandit problems [1, 2]. Given a context vector x = (x1 , x2 , . . . , xD ), the optimal action
is a? (x) := arg maxa?A ?(a, xR(a) ), but the learner does not know it since it does not know R,
F (a, xR(a) ) and ?(a, xR(a) ) for a ? A, x ? X a priori. In order to assess the learner?s loss due to
unknowns, we compare its performance with the performance of an oracle benchmark which knows
a? (x) for all x ? X . Let ?t (a) := ?(a, xR(a),t ). The action chosen by the learner at time t is
denoted by ?t . The learner also decides whether to observe the reward or not, and this decision of
the learner at time t is denoted by ?t ? {0, 1}, where ?t = 1 implies that the learner chooses to
observe the reward and ?t = 0 implies that the learner does not observe the reward. The learner?s
performance loss with respect to the oracle benchmark is defined as the regret, whose value at time
T is given by
T
T
X
X
(?t (?t ) ? cO ?t ).
(1)
?t (a? (xt )) ?
R(T ) :=
t=1
t=1
A regret that grows sublinearly in T , i.e., O(T ? ), ? < 1, guarantees convergence in terms of the
average reward, i.e., R(T )/T ? 0. We are interested in achieving sublinear growth with a rate
independent of D.
3
3.1
Online Relevance Learning with Controlled Feedback
Description of the algorithm
In this section we propose the algorithm Online Relevance Learning with Controlled Feedback
(ORL-CF), which learns the best action for each context vector by simultaneously learning the relevance relation, and then estimating the expected reward of each action. The feedback, i.e., reward
observations, is controlled based on the past context vector arrivals, in a way that reward observations are only made for actions for which the uncertainty in the reward estimates are high for
the current context vector. The controlled feedback feature allows ORL-CF to operate as an active
learning algorithm. Operation of ORL-CF can be summarized as follows:
? Adaptively discretize (partition) the context space of each type to learn action rewards of
similar contexts together.
? For an action, form reward estimates for pairs of intervals corresponding to pairs of types.
Based on the accuracy of these estimates, either choose to explore and observe the reward
or choose to exploit the best estimated action for the current context vector.
? In order to choose the best action, compare the reward estimates for pairs of intervals for
which one interval belongs to type i, for each type i and action a. Conclude that type i
3
is relevant to a if the variation of the reward estimates does not greatly exceed the natural
variation of the expected reward of action a over the interval of type i (calculated using
Assumption 1).
Online Relevance Learning with Controlled Feedback (ORL-CF):
1: Input: L, ?, ?.
2: Initialization: Pi,1 = {[0, 1]}, i ? D. Run Initialize(i, Pi,1 , 1), i ? D.
3: while t ? 1 do
4:
Observe xS
t , find pt that xt belongs to.
5:
Set Ut := i?D Ui,t , where Ui,t (given in (3)), is the set of under explored actions for type i.
6:
if Ut 6= ? then
7:
(Explore) ?t = 1, select ?t randomly from Ut , observe rt (?t , xt ).
8:
Update pairwise sample means: for all q ? Qt , given in (2).
r?ind(q) (q, ?t ) = (S ind(q) (q, ?t )?
rind(q) (q, ?t ) + rt (?t , xt ))/(S ind(q) (q, ?t ) + 1).
9:
Update counters: for all q ? Qt , S ind(q) (q, ?t ) + +.
10:
else
11:
(Exploit) ?t = 0, for each a ? A calculate the set of candidate relevant contexts Relt (a) given
in (4).
12:
for a ? A do
13:
if Relt (a) = ? then
14:
Randomly select c?t (a) from D.
15:
else
16:
For each i ? Relt (a), calculate Vart (i, a) given in (5).
17:
Set c?t (a) = arg mini?Relt (a) Vart (i, a).
18:
end if
c
? (a)
19:
Calculate r?t t (a) as given in (6).
20:
end for
c
? (a)
21:
Select ?t = arg maxa?A r?t t (pc?t (a),t , a).
22:
end if
23:
for i ? D do
24:
N i (pi,t ) + +.
25:
if N i (pi,t ) ? 2?l(pi,t ) then
26:
Create two new level l(pi,t ) + 1 intervals p, p0 whose union gives pi,t .
27:
Pi,t+1 = Pi,t ? {p, p0 } ? {pi,t }.
28:
Run Initialize(i, {p, p0 }, t).
29:
else
30:
Pi,t+1 = Pi,t .
31:
end if
32:
end for
33:
t=t+1
34: end while
Initialize(i, B, t):
1: for p ? B do
2:
Set N i (p) = 0, r?i,j (p, pj , a) = r?j,i (pj , p, a) = 0, S i,j (p, pj , a) = S j,i (pj , p, a) = 0 for all a ? A,
j ? D?i and pj ? Pj,t .
3: end for
Figure 2: Pseudocode for ORL-CF.
Since the number of contexts is infinite, learning the reward of an action for each context is not
feasible. In order to learn fast, ORL-CF exploits the similarities between the contexts of the relevant
type given in Assumption 1 to estimate the rewards of the actions. The key to success of our algorithm is that this estimation is good enough. ORL-CF adaptively forms the partition of the space
for each type in D, where the partition for the context space of type i at time t is denoted by Pi,t .
All the elements of Pi,t are disjoint intervals of Xi = [0, 1] whose lengths are elements of the set
{1, 2?1 , 2?2 , . . .}.4 An interval with length 2?l , l ? 0 is called a level l interval, and for an interval
p, l(p) denotes its level, s(p) denotes its length. By convention, intervals are of the form (a, b], with
the only exception being the interval containing 0, which is of the form [0, b].5 Let pi,t ? Pi,t be the
interval that xi,t belongs to, pt := (p1,t , . . . , pD,t ) and P t := (P1,t , . . . , PD,t ).
4
Setting interval lengths to powers of 2 is for presentational simplicity. In general, interval lengths can be
set to powers of any real number greater than 1.
5
Endpoints of intervals will not matter in our analysis, so our results will hold even when the intervals have
common endpoints.
4
The pseudocode of ORL-CF is given in Fig. 2. ORL-CF starts with Pi,1 = {Xi } = {[0, 1]} for
each i ? D. As time goes on and more contexts arrive for each type i, it divides Xi into smaller and
smaller intervals. The idea is to combine the past observations made in an interval to form sample
mean reward estimates for each interval, and use it to approximate the expected rewards of actions
for contexts lying in these intervals. The intervals are created in a way to balance the variation of
the sample mean rewards due to the number of past observations that are used to calculate them and
the variation of the expected rewards in each interval.
We also call Pi,t the set of active intervals for type i at time t. Since the partition of each type is
adaptive, as time goes on, new intervals become active while old intervals are deactivated, based on
how contexts arrive. For a type i interval p, let Nti (p) be the number of times xi,t0 ? p ? Pi,t0 for
t0 ? t. The duration of time that an interval remains active, i.e., its lifetime, is determined by an input
parameter ? > 0, which is called the duration parameter. Whenever the number of arrivals to an
interval p exceeds 2?l(p) , ORL-CF deactivates p and creates two level l(p)+1 intervals, whose union
gives p. For example, when pi,t = (k2?l , (k + 1)2?l ] for some 0 < k ? 2l ? 1 if Nti (pi,t ) ? 2?l ,
ORL-CF sets
Pi,t+1 = Pi,t ? {(k2?l , (k + 1/2)2?l ], ((k + 1/2)2?l , (k + 1)2?l ]} ? {pi,t }.
Otherwise Pi,t+1 remains the same as Pi,t . It is easy to see that the lifetime of an interval increases
exponentially in its duration parameter.
We next describe the counters, control numbers and sample mean rewards the learner keeps for each
pair of intervals corresponding to a pair of types to determine whether to explore or exploit and how
to exploit. Let D?i := D ? {i}. For type i, let Qi,t := {(pi,t , pj,t ) : j ? D?i } be the pair of
intervals that are related to type i at time t, and let
[
Qt :=
Qi,t .
(2)
i?D
To denote an element of Qi,t or Qt we use index q. For any q ? Qt , the corresponding pair of types
is denoted by ind(q). For example, ind((pi,t , pj,t )) = i, j. The decision to explore or exploit at time
t is solely
T based on pt . For events A1 , . . . , AK , let I(A1 , . . . , Ak ) denote the indicator function of
event k=1:K Ak . For p ? Pi,t , p0 ? Pj,t , let
Sti,j (p, p0 , a) :=
t?1
X
I (?t0 = a, ?t = 1, pi,t0 = p, pj,t0 = p0 ) ,
t0 =1
be the number of times a is selected and the reward is observed when the type i context is in p and
type j context is in p0 , summed over times when both intervals are active. Also for the same p and
p0 let
!
t?1
X
i,j
0
0
r?t (p, p , a) :=
rt (a, xt )I (?t0 = a, ?t = 1, pi,t0 = p, pj,t0 = p ) /(Sti,j (p, p0 , a)),
t0 =1
be the pairwise sample mean reward of action a for pair of intervals (p, p0 ).
At time t, ORL-CF assigns a control number to each i ? D denoted by
Di,t :=
2 log(tD|A|/?)
,
(Ls(pi,t ))2
which depends on the cardinality of A, the length of the active interval that type i context is in
at time t and a confidence parameter ? > 0, which controls the accuracy of sample mean reward
estimates. Then, it computes the set of under-explored actions for type i as
ind(q)
Ui,t := {a ? A : St
(q, a) < Di,t for some q ? Qi (t)},
(3)
S
and then, the set of under-explored actions as Ut := i?D Ui,t . The decision to explore or exploit is
based on whether or not Ut is empty.
(i) If Ut 6= ?, ORL-CF randomly selects an action ?t ? Ut to explore, and observes its reward
rt (?t , xt ). Then, it updates the pairwise sample mean rewards and pairwise counters for all q ? Qt ,
ind(q)
r?t+1 (q, ?t ) =
ind(q)
St
ind(q)
(q,?t )?
rt+1 (q,?t )+rt (?t ,xt )
ind(q)
St
(q,?t )+1
ind(q)
ind(q)
, St+1 (q, ?t ) = St
5
(q, ?t ) + 1.
(ii) If Ut = ?, ORL-CF exploits by estimating the relevant type c?t (a) for each a ? A and forming
sample mean reward estimates for action a based on c?t (a). It first computes the set of candidate
relevant types for each a ? A,
Relt (a) := {i ? D : |?
rti,j (pi,t , pj,t , a) ? r?ti,k (pi,t , pk,t , a)| ? 3Ls(pi,t ), ?j, k ? D?i }.
(4)
The intuition is that if i is the type that is relevant to a, then independent of the values of the contexts
of the other types, the variation of the pairwise sample mean reward of a over pi,t must be very close
to the variation of the expected reward of a in that interval.
If Relt (a) is empty, this implies that ORL-CF failed to identify the relevant type, hence c?t (a) is
randomly selected from D. If Relt (a) is nonempty, ORL-CF computes the maximum variation
Vart (i, a) := max |?
rti,j (pi,t , pj,t , a) ? r?ti,k (pi,t , pk,t , a)|,
j,k?D?i
(5)
for each i ? Relt (a). Then it sets c?t (a) = mini?Relt (a) Vart (i, a). This way, whenever the type
relevant to action a is in Relt (a), even if it is not selected as the estimated relevant type, the sample
mean reward of a calculated based on the estimated relevant type will be very close to the sample
mean of its reward calculated according to the true relevant type. After finding the estimated relevant
types, the sample mean reward of each action is computed based on its estimated relevant type as
P
c? (a),j
c? (a),j
r? t
(pc?t (a),t , pj,t , a)St t
(pc?t (a),t , pj,t , a)
j?D??
ct (a) t
c?t (a)
r?t
(a) :=
.
(6)
P
c? (a),j
St t
(pc?t (a),t , pj,t , a)
j?D??
c (a)
t
c? (a)
arg maxa?A r?t t (pc?t (a),t , a).
Then, ORL-CF selects ?t =
Since the reward is not observed in
exploitations, pairwise sample mean rewards and counters are not updated.
3.2
Regret analysis of ORL-CF
Let ? (T ) ? {1, 2, . . . , T } be the set of time steps in which ORL-CF exploits by time T . ? (T ) is a
random set which depends on context arrivals and the randomness of the action selection of ORLCF. The regret R(T ) defined in (1) can be written as a sum of the regret incurred during explorations
(denoted by RO (T )) and the regret incurred during exploitations (denoted by RI (T )). The following
theorem gives a bound on the regret of ORL-CF in exploitation steps.
Theorem 1. Let ORL-CF run with duration parameter ? > 0, confidence parameter ? > 0 and
control numbers Di,t := 2 log(t|A|D/?)
(Ls(pi,t ))2 , for i ? D. Let Rinst (t) be the instantaneous regret at time t,
which is the loss in expected reward at time t due to not selecting a? (xt ). Then, with probability at
least 1 ? ?, we have
Rinst (t) ? 8L(s(pR(?t ),t ) + s(pR(a? (xt )),t )),
for all t ? ? (T ), and the total regret in exploitation steps is bounded above by
X
RI (T ) ? 8L
(s(pR(?t ),t + s(pR(a? (xt )),t )) ? 16L22? T ?/(1+?) ,
t?? (T )
for arbitrary context vectors x1 , x2 , . . . , xT .
Theorem 1 provides both context arrival process dependent and worst case bounds on the exploitation regret of ORL-CF. By choosing ? arbitrarily close to zero, RI (T ) can be made O(T ? ) for any
? > 0. While this is true, the reduction in regret for smaller ? not only comes from increased accuracy, but it is also due to the reduction in the number of time steps in which ORL-CF exploits, i.e.,
|? (T )|. By definition time t is an exploitation step if
Sti,j (pi,t , pj,t , a) ?
L2
22 max{l(pi,t ),l(pj,t )}+1 log(t|A|D/?)
2 log(t|A|D/?)
=
,
2
2
min{s(pi,t ) , s(pj,t ) }
L2
for all q = (pi,t , pj,t ) ? Qt , i, j ? D. This implies that for any q ? Qi,t which has the interval
? 2l ) explorations are required before any exploitation can take
with maximum level equal to l, O(2
place. Since the time a level l interval can stay active is 2?l , it is required that ? ? 2 so that ? (T ) is
nonempty.
The next theorem gives a bound on the regret of ORL-CF in exploration steps.
6
Theorem 2. Let ORL-CF run with ?, ? and Di,t , i ? D values as stated in Theorem 1. Then,
RO (T ) ?
960D2 (cO + 1) log(T |A|D/?) 4/? 64D2 (cO + 1) 2/?
T
+
T ,
7L2
3
with probability 1, for arbitrary context vectors x1 , x2 , . . . , xT .
Based on the choice of the duration parameter ?, which determines how long an interval will stay
active, it is possible to get different regret bounds for explorations and exploitations. Any ? > 4 will
give a sublinear regret bound for both explorations and exploitations. The regret in exploitations
increases in ? while the regret in explorations decreases in ?.
?
Theorem 3. Let ORL-CF run with ? and Di,t , i ? D values as stated in Theorem 1 and ? = 2+2 2.
Then, the time order of exploration and exploitation regrets are?balanced up to logaritmic orders.
?
? 2/(1+ 2) ) and RO (T ) = O(T
? 2/(1+ 2) ) .
With probability at least 1 ? ? we have both RI (T ) = O(T
Remark 1. Prior work on contextual bandits focused on balancing the regret due to exploration
and exploitation. For example in [1, 2], for a D-dimensional context vector algorithms are shown
? (D+1)/(D+2) ) regret.6 Also in [1] a O(T (D+1)/(D+2) ) lower bound on the regret
to achieve O(T
is proved. An interesting question is to find the tightest lower bound for contextual bandits with
relevance function. One trivial lower bound is O(T 2/3 ), which corresponds to D = 1. However,
since finding the action with the highest expected reward for a context vector requires comparisons
of estimated rewards of actions with different relevant types, which requires accurate sample mean
reward estimates for 2 dimensions of the context space corresponding to those types, we conjecture
that a tighter lower bound is O(T 3/4 ). Proving this is left as future work.
Another interesting case is when actions with suboptimality greater than > 0 must never be chosen
in any exploitation step by time T . When such a condition is imposed, ORL-CF can start with
partitions Pi,1 that have sets with high levels such that it explores more at the beginning to have
more accurate reward estimates before any exploitation. The following theorem gives the regret
bound of ORL-CF for this case.
Theorem 4. Let ORL-CF run with duration parameter ? > 0, confidence parameter ? > 0, control
numbers Di,t := 2 log(t|A|D/?)
(Ls(pi,t ))2 , and with initial partitions Pi,1 , i ? D consisting of intervals of
length lmin = dlog2 (3L/(2))e. Then, with probability 1 ? ?, Rinst (t) ? for all t ? ? (T ),
RI (T ) ? 16L22? T ?/(1+?) and
81L4 960D2 (cO + 1) log(T |A|D/?) 4/? 64D2 (cO + 1) 2/?
RO (T ) ? 4
T
+
T
,
7L2
3
for arbitrary
context vectors x1 , x2 , . . . , xT . Bounds on RI (T ) and RO (T ) are balanced for ? =
?
2 + 2 2.
3.3
Future Work
In this paper we only considered the relevance relations that are functions. Similar learning methods
can be developed for more general relevance relations such as the ones given in Fig. 1 (i) and (ii).
For example, for the general case in Fig. 1 (i), if |R(a)| ? Drel << D, for all a ? A, and Drel is
known by the learner, the following variant of ORL-CF can be used to achieve regret whose time
order depends only on Drel but not on D.
? Instead of keeping pairwise sample mean reward estimates, keep sample mean reward estimates of actions for Drel + 1 tuples of intervals of Drel + 1 types.
? For a Drel tuple of types i, let Qi,t be the Drel + 1 tuples of intervals that are related to i
at time t, and Qt be the union of Qi,t over all Drel tuples of types. Similar to ORL-CF,
compute the set of under-explored actions Ui,t , and the set of candidate relevant Drel tuples
of types Relt (a), using the newly defined sample mean reward estimates.
6
The results are shown in terms of the covering dimension which reduces to Euclidian dimension for our
problem.
7
? In exploitation, set c?t (a) to be the Drel tuple of types with the minimax variation, where the
variation of action a for a tuple i is defined similar to (5), as the maximum of the distance
between the sample mean rewards of action a for Drel +1 tuples that are in Qi,t .
Another interesting case is when the relevance relation is linear as given in Fig. 1 (ii). For example,
for action a if there is a type i that is much more relevant compared to other types j ? D?i , i.e.,
wa,i >> wa,j , where the weights wa,i are given in Fig. 1, then ORL-CF is expected to have good
performance (but not sublinear regret with respect to the benchmark that knows R).
4
Related Work
Contextual bandit problems are studied by many others in the past [3, 4, 1, 2, 5, 6]. The problem we consider in this paper is a special case of the Lipschitz contextual bandit problem [1, 2],
where the only assumption is the existence of a known similarity metric between the expected rewards of actions for different contexts. It is known that the lower bound on regret for this problem
? (D+1)/(D+2) ) regret [1, 2].
is O(T (D+1)/(D+2) ) [1], and there exists algorithms that achieve O(T
Compared to the prior work above, ORL-CF only needs to observe rewards in explorations and has
a regret whose time order is independent of D. Hence it can still learn the optimal actions fast
enough in settings where observations are costly and context vector is high dimensional.
Examples of related works that consider limited observations are KWIK learning [7, 8] and label
efficient learning [9, 10, 11]. For example, [8] considers a bandit model where the reward function
comes from a parameterized family of functions and gives bound on the average regret. An online
prediction problem is considered in [9, 10, 11], where the predictor (action) lies in a class of linear
predictors. The benchmark of the context is the best linear predictor. This restriction plays a crucial
role in deriving regret bounds whose time order does not depend on D. Similar to these works,
ORL-CF can guarantee with a high probability that actions with large suboptimalities will never
be selected in exploitation steps. However, we do not have any assumptions on the form of the
expected reward function other than the Lipschitz continuity and that it depends on a single type for
each action.
In [12] graphical bandits are proposed where the learner takes an action vector a which includes
actions from several types that consitute a type set T . The expected reward of a for context vector x
can be decomposed into sum of reward functions each of which only depends on a subset of D ? T .
However, it is assumed that the form of decomposition is known but the functions are not known.
Another work [13] proposes a fast learning algorithm for an i.i.d. contextual bandit problem in
which the rewards for contexts and actions are sampled from a joint probability distribution. In this
work the authors consider learning
? the best policy from a finite set of policies with oracle access,
and prove a regret bound of O( T ) which is also logarithmic in the size of the policy space. In
contrast, in our problem (i) contexts arrive according to an arbitrary exogenous process, and the
action rewards are sampled from an i.i.d. distribution given the context value, (ii) the set of policies
that the learner can adopt is not restricted.
Large dimensional action spaces, where the rewards depend on a subset of the types of actions are
considered in [14] and [15]. [14] considers the problem when the reward is H?older continuous in
an unknown low-dimensional tuple of types, and uses a special discretization of the action space
to achieve dimension independent bounds on the regret. This discretization can be effectively used
since the learner can select the actions, as opposed to our case where the learner does not have any
control over contexts. [15] considers the problem of optimizing high dimensional functions that
have an unknown low dimensional structure from noisy observations.
5
Conclusion
In this paper we formalized the problem of learning the best action through learning the relevance
relation between types of contexts and actions. For the case when the relevance relation is a function,
we proposed an algorithm that (i) has sublinear regret with time order independent of D, (ii) only
requires reward observations in explorations, (iii) for any > 0, does not select any suboptimal
actions in exploitations with a high probability. In the future we will extend our results to the linear
and general relevance relations illustrated in Fig. 1.
8
References
[1] T. Lu, D. P?al, and M. P?al, ?Contextual multi-armed bandits,? in International Conference on
Artificial Intelligence and Statistics (AISTATS), 2010, pp. 485?492.
[2] A. Slivkins, ?Contextual bandits with similarity information,? in Conference on Learning Theory (COLT), 2011.
[3] E. Hazan and N. Megiddo, ?Online learning with prior knowledge,? in Learning Theory.
Springer, 2007, pp. 499?513.
[4] J. Langford and T. Zhang, ?The epoch-greedy algorithm for contextual multi-armed bandits,?
Advances in Neural Information Processing Systems (NIPS), vol. 20, pp. 1096?1103, 2007.
[5] W. Chu, L. Li, L. Reyzin, and R. E. Schapire, ?Contextual bandits with linear payoff functions,?
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214.
[6] M. Dudik, D. Hsu, S. Kale, N. Karampatziakis, J. Langford, L. Reyzin, and T. Zhang, ?Efficient
optimal learning for contextual bandits,? arXiv preprint arXiv:1106.2369, 2011.
[7] L. Li, M. L. Littman, T. J. Walsh, and A. L. Strehl, ?Knows what it knows: a framework for
self-aware learning,? Machine Learning, vol. 82, no. 3, pp. 399?443, 2011.
[8] K. Amin, M. Kearns, M. Draief, and J. D. Abernethy, ?Large-scale bandit problems and KWIK
learning,? in International Conference on Machine Learning (ICML), 2013, pp. 588?596.
[9] N. Cesa-Bianchi, C. Gentile, and F. Orabona, ?Robust bounds for classification via selective
sampling,? in International Conference on Machine Learning (ICML), 2009, pp. 121?128.
[10] S. M. Kakade, S. Shalev-Shwartz, and A. Tewari, ?Efficient bandit algorithms for online multiclass prediction,? in International Conference on Machine Learning (ICML), 2008, pp. 440?
447.
[11] E. Hazan and S. Kale, ?Newtron: an efficient bandit algorithm for online multiclass prediction.?
in Advances in Neural Information Processing Systems (NIPS), 2011, pp. 891?899.
[12] K. Amin, M. Kearns, and U. Syed, ?Graphical models for bandit problems,? in Conference on
Uncertainty in Artificial Intelligence (UAI), 2011.
[13] A. Agarwal, D. Hsu, S. Kale, J. Langford, L. Li, and R. E. Schapire, ?Taming the monster: A
fast and simple algorithm for contextual bandits,? arXiv preprint arXiv:1402.0555, 2014.
[14] H. Tyagi and B. Gartner, ?Continuum armed bandit problem of few variables in high dimensions,? in Workshop on Approximation and Online Algorithms (WAOA), 2014, pp. 108?119.
[15] J. Djolonga, A. Krause, and V. Cevher, ?High-dimensional Gaussian process bandits,? in Advances in Neural Information Processing Systems (NIPS), 2013, pp. 1025?1033.
9
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4,839 | 5,381 | Online combinatorial optimization with stochastic
decision sets and adversarial losses
Gergely Neu
Michal Valko
SequeL team, INRIA Lille ? Nord Europe, France
{gergely.neu,michal.valko}@inria.fr
Abstract
Most work on sequential learning assumes a fixed set of actions that are available
all the time. However, in practice, actions can consist of picking subsets of readings from sensors that may break from time to time, road segments that can be
blocked or goods that are out of stock. In this paper we study learning algorithms
that are able to deal with stochastic availability of such unreliable composite actions. We propose and analyze algorithms based on the Follow-The-PerturbedLeader prediction method for several learning settings differing in the feedback
provided to the learner. Our algorithms rely on a novel loss estimation technique
that we call Counting Asleep Times. We deliver regret bounds for our algorithms
for the previously studied full information and (semi-)bandit settings, as well as a
natural middle point between the two that we call the restricted information setting. A special consequence of our results is a significant improvement of the best
known performance guarantees achieved by an efficient algorithm for the sleeping
bandit problem with stochastic availability. Finally, we evaluate our algorithms
empirically and show their improvement over the known approaches.
1
Introduction
In online learning problems [4] we aim to sequentially select actions from a given set in order to
optimize some performance measure. However, in many sequential learning problems we have to
deal with situations when some of the actions are not available to be taken. A simple and wellstudied problem where such situations arise is that of sequential routing [8], where we have to select
every day an itinerary for commuting from home to work so as to minimize the total time spent
driving (or even worse, stuck in a traffic jam). In this scenario, some road segments may be blocked
for maintenance, forcing us to work with the rest of the road network. This problem is isomorphic to
packet routing in ad-hoc computer networks where some links might not be always available because
of a faulty transmitter or a depleted battery. Another important class of sequential decision-making
problems where the decision space might change over time is recommender systems [11]. Here,
some items may be out of stock or some service may not be applicable at some time (e.g., a movie
not shown that day, bandwidth issues in video streaming services). In these cases, the advertiser may
refrain from recommending unavailable items. Other reasons include a distributor being overloaded
with commands or facing shipment problems.
Learning problems with such partial-availability restrictions have been previously studied in the
framework of prediction with expert advice. Freund et al. [7] considered the problem of online
prediction with specialist experts, where some experts? predictions might not be available from time
to time, and the goal of the learner is to minimize regret against the best mixture of experts. Kleinberg
et al. [15] proposed a stronger notion of regret measured against the best ranking of experts and gave
efficient algorithms that work under stochastic assumptions on the losses, referring to this setting as
prediction with sleeping experts. They have also introduced the notion of sleeping bandit problems
where the learner only gets partial feedback about its decisions. They gave an inefficient algorithm
1
for the non-stochastic case, with some hints that it might be difficult to learn efficiently in this
general setting. This was later reaffirmed by Kanade and Steinke [14], who reduce the problem of
PAC learning of DNF formulas to a non-stochastic sleeping experts problem, proving the hardness
of learning in this setup. Despite these negative results, Kanade et al. [13] have shown that there
is still hope to obtain efficient algorithms in adversarial environments, if one introduces a certain
stochastic a assumption on the decision set.
In this paper, we extend the work of Kanade et al. [13] to combinatorial settings where the action
set of the learner is possibly huge, but has a compact representation. We also assume stochastic
action availability: in each decision period, the decision space is drawn from a fixed but unknown
probability distribution independently of the history of interaction between the learner and the environment. The goal of the learner is to minimize the sum of losses associated with its decisions. As
usual in online settings, we measure the performance of the learning algorithm by its regret defined
as the gap between the total loss of the best fixed decision-making policy from a pool of policies
and the total loss of the learner. The choice of this pool, however, is a rather delicate question in our
problem: the usual choice of measuring regret against the best fixed action is meaningless, since not
all actions are available in all time steps. Following Kanade et al. [13] (see also [15]), we consider
the policy space composed of all mappings from decision sets to actions within the respective sets.
We study the above online combinatorial optimization setting under three feedback assumptions.
Besides the full-information and bandit settings considered by Kanade et al. [13], we also consider a
restricted feedback scheme as a natural middle ground between the two by assuming that the learner
gets to know the losses associated only with available actions. This extension (also studied by [15])
is crucially important in practice, since in most cases it is unrealistic to expect that an unavailable
expert would report its loss. Finally, we also consider a generalization of bandit feedback to the
combinatorial case known as semi-bandit feedback.
Our main contributions in this paper are two algorithms called S LEEPING C AT and S LEEPING C ATBANDIT that work in the restricted and semi-bandit information schemes, respectively. The best
known competitor of our algorithms is the BSFPL algorithm of Kanade et al. [13] that works in
two phases. First, an initial phase is dedicated to the estimation of the distribution of the available
actions. Then, in the main phase, BSFPL randomly alternates between exploration and exploitation.
Our technique improves over the FPL-based method of Kanade et al. [13] by removing the costly exploration phase dedicated to estimate the availability probabilities, and also the explicit exploration
steps in their main phase. This is achieved by a cheap alternative loss estimation procedure called
Counting Asleep Times (or CAT) that does not require estimating the distribution of the action sets.
This technique improves the regret bound of [13] ?
after T steps from O(T 4/5 ) to O(T 2/3 ) in their
setting, and also provides a regret guarantee of O( T ) in the restricted setting.1
2
Background
We now give the formal definition of the learning problem. We consider a sequential interaction
scheme between a learner and an environment where in each round t ? [T ] = {1, 2, . . . , T }, the
d
learner has to choose an action Vt from a subset St of a known decision set S ? {0, 1} with
kvk1 ? m for all v ? S. We assume that the environment selects St according to some fixed (but
unknown) distribution P, independently of the interaction history. Unaware of the learner?s decision,
the environment also decides on a loss vector `t ? [0, 1]d that will determine the loss suffered by the
learner, which is of the form Vt> `t . We make no assumptions on how the environment generates
the sequence of loss vectors, that is, we are interested in algorithms that work in non-oblivious (or
adaptive) environments. At the end of each round, the learner receives some feedback based on the
loss vector and the action of the learner. The goal of the learner is pick its actions so as to minimize
the losses it accumulates by the end of the T ?th round. This setup generalizes the setting of online
combinatorial optimization considered by Cesa-Bianchi and Lugosi [5], Audibert et al. [1], where
the decision set is assumed to be fixed throughout the learning procedure. The interaction protocol
is summarized on Figure 1 for reference.
1
While not explicitly proved by Kanade et al. [13], their technique can be extended to work in the restricted
setting, where it can be shown to guarantee a regret of O(T 3/4 ).
2
Parameters:
d
full set of decision vectors S = {0, 1} , number of rounds T , unknown distribution P ? ?2S
For all t = 1, 2, . . . , T repeat
1. The environment draws a set of available actions St ? P and picks a loss vector
`t ? [0, 1]d .
2. The set St is revealed to the learner.
3. Based on its previous observations (and possibly some source of randomness), the
learner picks an action Vt ? St .
4. The learner suffers loss Vt> `t and gets some feedback:
(a) in the full information setting, the learner observes `t ,
(b) in the restricted setting, the learner observes `t,i for all i ? Dt ,
(c) in the semi-bandit setting, the learner observes `t,i for all i such that Vt,i = 1.
Figure 1: The protocol of online combinatorial optimization with stochastic action availability.
We distinguish between three different feedback schemes, the simplest one being the full information
scheme where the loss vectors are completely revealed to the learner at the end of each round. In
the restricted-information scheme, we make a much milder assumption that the learner is informed
about the losses of the available actions. Precisely, we define the set of available components as
Dt = {i ? [d] : ?v ? St : vi = 1}
and assume that the learner can observe the i-th component of the loss vector `t if and only if i ? Dt .
This is a sensible assumption in a number of practical applications, e.g., in sequential routing problems where components are associated with links in a network. Finally, in the semi-bandit scheme,
we assume that the learner only observes losses associated with the components of its own decision,
that is, the feedback is `t,i for all i such that Vt,i = 1. This is the case in in online advertising settings where components of the decision vectors represent customer-ad allocations. The observation
history Ft is defined as the sigma-algebra generated by the actions chosen by the learner and the
decision sets handed out by the environment by the end of round t: Ft = ?(Vt , St , . . . , V1 , S1 ).
The performance of the learner is measured with respect to the best fixed policy (otherwise known
as a choice function in discrete choice theory [16]) of the form ? : 2S ? S. In words, a policy
? ? S? whenever the environment selects action set S.
? The (total expected)
? will pick action ?(S)
regret of the learner is defined as
RT = max
?
T
h
i
X
>
E (Vt ? ?(St )) `t .
(1)
t=1
Note that the above expectation integrates over both the randomness injected by the learner and the
stochastic process generating the decision sets. The attentive reader might notice that this regret
criterion is very similar to that of Kanade et al. [13], who study the setting of prediction with expert
advice (where m = 1) and measure regret against the best fixed ranking of experts. It is actually
easy to show that the optimal policy in their setting belongs to the set of ranking policies, making
our regret definition equivalent to theirs.
3
Loss estimation by Counting Asleep Times
In this section, we describe our method used for estimating unobserved losses that works without
having to explicitly learn the availability distribution P. To explain the concept on a high level, let
us now consider our simpler partial-observability setting, the restricted-information setting. For the
formal treatment of the problem, let us fix any component i ? [d] and define At,i = 1{i?Dt } and
ai = E [At,i |Ft?1 ]. Had we known the observation probability ai , we would be able to estimate
the i?th component of the loss vector `t by `??t,i = (`t,i At,i )/ai , as the quantity `t,i At,i is observable.
It is easy to see that the estimate `??t,i is unbiased by definition ? but, unfortunately, we do not
know ai , so we have no hope to compute it. A simple idea used by Kanade et al. [13] is to devote
3
the first T0 rounds of interaction solely to the purpose of estimating ai by the sample mean a
?i =
PT0
( t=1 At,i )/T0 . While this trick gets the job done, it is obviously wasteful as we have to throw
away all loss observations before the estimates are sufficiently concentrated. 2
We take a much simpler approach based on the observation that the ?asleep-time? of component i
is a geometrically distributed random variable with parameter ai . The asleep-time of component i
starting from time t is formally defined as
Nt,i = min {n > 0 : i ? Dt+n } ,
which is the number of rounds until the next observation of the loss associated with component i.
Using the above definition, we construct our loss estimates as the vector `?t whose i-th component is
`?t,i = `t,i At,i Nt,i .
(2)
It is easy to see that the above loss estimates are unbiased as
1
= `t,i
ai
for any i. We will refer to this loss-estimation method as Counting Asleep Times (CAT).
E [`t,i At,i Nt,i |Ft?1 ] = `t,i E [At,i |Ft?1 ] E [Nt,i |Ft?1 ] = `t,i ai ?
Looking at the definition (2), the attentive reader might worry that the vector `?t depends on future
realizations of the random decision sets and thus could be useless for practical use. However, observe that there is no reason that the learner should use the estimate `?t,i before component i wakes
up in round t + Nt,i ? which is precisely the time when the estimate becomes well-defined. This
suggests a very simple implementation of CAT: whenever a component is not available, estimate its
loss by the last observation from that component! More formally, set
(
`t,i ,
if i ? Dt
`?t,i = ?
`t?1,i , otherwise.
It is easy to see that at the beginning of any round t, the two alternative definitions match for all
components i ? Dt . In the next section, we confirm that this property is sufficient for running our
algorithm.
4
Algorithms & their analyses
For all information settings, we base our learning algorithms on the Follow-the-Perturbed-Leader
(FPL) prediction method of Hannan [9], as popularized by Kalai and Vempala [12]. This algorithm
works by additively perturbing the total estimated loss of each component, and then running an optimization oracle over the perturbed losses to choose the next action. More precisely, our algorithms
b t = Pt `?t and pick the action
maintain the cumulative sum of their loss estimates L
s=1
b t?1 ? Zt ,
Vt = arg min v T ? L
v?St
where Zt is a perturbation vector with independent exponentially distributed components with unit
expectation, generated independently of the history, and ? > 0 is a parameter of the algorithm. Our
algorithms for the different information settings will be instances of FPL that employ different loss
estimates suitable for the respective settings. In the first part of this section, we present the main
tools of analysis that will be used for each resulting method.
As usual for analyzing FPL-based methods [12, 10, 18], we start by defining a hypothetical foree with standard exponential components
caster that uses a time-independent perturbation vector Z
and peeks one step into the future. However, we need an extra trick to deal with the randomness of
the decision set: we introduce the time-independent decision set Se ? P (drawn independently of
the filtration (Ft )t ) and define
bt ? Z
e .
Vet = arg min v T ? L
v?Se
2
Notice that we require ?sufficient concentration? from 1/?
ai and not only from a
?i ! The deviation of such
quantities is rather difficult to control, as demonstrated by the complicated analysis of Kanade et al. [13].
4
Clearly, this forecaster is infeasible as it uses observations from the future. Also observe that
Vet?1 ? Vt given Ft?1 . The following two lemmas show how analyzing this forecaster can help in
establishing the performance of our actual algorithms.
Lemma 1. For any sequence of loss estimates, the expected regret of the hypothetical forecaster
against any fixed policy ? : 2S ? S satisfies
" T
#
T
X
e `?t ? m (log d + 1) .
E
Vet ? ?(S)
?
t=1
The statement is easily proved by applying
the follow-the-leader/be-the-leader
lemma3 (see, e.g., [4,
e
Lemma 3.1]) and using the upper bound E
Z
? log d + 1.
?
The following result can be extracted from the proof of Theorem 1 of Neu and Bart?ok [18].
Lemma 2. For any sequence of nonnegative loss estimates,
i
h
2
T
E (Vet?1 ? Vet )T `?t Ft?1 ? ? E Vet?1
`?t Ft?1 .
In the next subsections, we apply these results to obtain bounds for the three information settings.
4.1
Algorithm for full information
In the simplest setting, we can use `?t = `t , which yields the following theorem:
Theorem 1. Define
(
" T
#
)
X
?
T
LT = max min E
?(St ) `t , 4(log d + 1) .
?
Setting ? =
p
t=1
(log d + 1)/L?T , the regret of FPL in the full information scheme satisfies
q
RT ? 2m 2L?T (log d + 1).
As this result is comparable to the best available bounds for FPL [10, 18] in the full information
setting and a fixed decision set, it reinforces the observation of Kanade et al. [13], who show that the
sleeping experts problem with full information and stochastic availability is no more difficult than
the standard experts problem. The proof of Theorem 1 follows directly from combining Lemmas 1
and 2 with some standard tricks. For completeness, details are provided in Appendix A.
4.2
Algorithm for restricted feedback
In this section, we use the CAT loss estimate defined in Equation (2) as `?t in FPL, and call the
resulting method S LEEPING C AT. The following theorem gives the performance guarantee for this
algorithm.
Pd
Theorem 2. Define Qt =
i=1 E [ Vt,i | i ? Dt ]. The total expected regret of S LEEPING C AT
against the best fixed policy is upper bounded as
T
X
m(log d + 1)
RT ?
+ 2?m
Qt .
?
t=1
e T `?t = E [?(St )T `t ], where we used that `?t is independent of
Proof. We start by observing E ?(S)
e The proof is completed by combining
Se and is an unbiased estimate of `t , and also that St ? S.
this with Lemmas 1 and 2, and the bound
2
T
?
e
E Vt?1 `t Ft?1 ? 2mQt .
The proof of this last statement follows from a tedious calculation that we defer to Appendix B.
3
e allowing the necessary
This lemma can be proved in the current case by virtue of the fixed decision set S,
recursion steps to go through.
5
Below, we provide two ways of further bounding the regret under various assumptions. The first one
provides a universal upper bound that holds without any further assumptions.
p
Corollary 1. Setting ? = (log d + 1)/(2dT ), the regret of S LEEPING C AT against the best fixed
policy is bounded as
p
RT ? 2m 2dT (log d + 1).
The proof follows from the fact that
? Qt ? d no matter what P is. A somewhat surprising feature
of our bound is its scaling with d log d, which is much worse than the logarithmic dependence
exhibited in the full information case. It is easy to see, however, that this bound is not improvable in
general ? see Appendix D for a simple example. The next bound, however, shows that it is possible
to improve this bound by assuming that most components are reliable in some sense, which is the
case in many practical settings.
p
Corollary 2. Assuming ai ? ? for all i, we have Qt ? 1/?, and setting ? = ?(log d + 1)/(2T )
guarantees that the regret of S LEEPING C AT against the best fixed policy is bounded as
s
2T (log d + 1)
RT ? 2m
.
?
4.3
Algorithm for semi-bandit feedback
We now turn our attention to the problem of learning with semi-bandit feedback where the learner
only gets to observe the losses associated with its own decision. Specifically, we assume that the
learner observes all components i of the loss vector such that Vt,i = 1. The extra difficulty in this
setting is that our actions influence the feedback that we receive, so we have to be more careful when
defining our loss estimates. Ideally, we would like to work with unbiased estimates of the form
X
`t,i
?
?
P(S)E
Vt,i Ft?1 , St = S? . (3)
`??t,i = ? Vt,i ,
where
qt,i
= E [ Vt,i | Ft?1 ] =
qt,i
? S
S?2
for all i ? [d]. Unfortunately though, we are in no position to compute these estimates, as this would
require perfect knowledge of the availability distribution P! Thus we have to look for another way
to compute reliable loss estimates. A possible idea is to use
qt,i ? ai = E [ Vt,i | Ft?1 , St ] ? P [i ? Dt ] .
?
qt,i
instead of
in Equation 3 to normalize the observed losses. This choice yields another unbiased
loss estimate as
`t,i Vt,i
`t,i
Vt,i
`t,i
E
F
F
,
S
F
=
E
E
E [ At,i | Ft?1 ] = `t,i ,
(4)
t?1
t?1
t t?1 =
qt,i ai
ai
qt,i
ai
which leaves us with the problem of computing qt,i and ai . While this also seems to be a tough
challenge, we now show to estimate this quantity by generalizing the CAT technique presented in
Section 3.
Besides our trick used for estimating the 1/ai ?s by the random variables Nt,i , we now also have to
face the problem of not being able to find a closed-form expression for the qt,i ?s. Hence, we follow
the geometric resampling approach of Neu and Bart?ok [18] and draw an additional sequence of M
perturbation vectors Zt0 (1), . . . , Zt0 (M ) and use them to compute
n
o
b t?1 ? Z 0 (k)
Vt0 (k) = arg min ? L
t
v?St
for all k ? [M ]. Using these simulated actions, we define
0
Kt,i = min k ? [M ] : Vt,i
(k) = Vt,i ? {M } .
and
`?t,i = `t,i Kt,i Nt,i Vt,i
(5)
`t,i Vt,i
qt,i ai
in expectation, yielding yet
for all i. Setting M = ? makes this expression equivalent to
another unbiased estimator for the losses. Our analysis, however, crucially relies on setting M to
6
a finite value so as to control the variance of the loss estimates. We are not aware of any other
work that achieves a similar variance-reduction effect without explicitly exploring the action space
[17, 6, 5, 3], making this alternative bias-variance tradeoff a unique feature of our analysis. We call
the algorithm resulting from using the loss estimates above S LEEPING C AT BANDIT. The following
theorem gives the performance guarantee for this algorithm.
Pd
Theorem 3. Define Qt = i=1 E [ Vt,i | i ? Dt ]. The total expected regret of S LEEPING C AT BAN DIT against the best fixed policy is bounded as
RT ?
T
X
m(log d + 1)
dT
+ 2?M m
.
Qt +
?
eM
t=1
Proof. First, observe that E `?t,i Ft?1
? `t,i as E [ Kt,i Vt,i | Ft?1 , St ] ? At,i and
e T `?t ? E [?(St )T `t ] by a simiE [ At,i Nt,i | Ft?1 ] = 1 by definition. Thus, we can get E ?(S)
lar argument that we used in the proof of Theorem 2. After yet another long and tedious calculation
(see Appendix C), we can prove
2
T
?
e
E Vt?1 `t Ft?1 ? 2M mQt .
(6)
The proof is concluded by combining this bound with Lemmas 1 and 2 and the upper bound
h
i
d
T
,
E [ VtT `t | Ft?1 ] ? E Vet?1
`?t Ft?1 +
eM
which can be proved by following the proof of Theorem 1 in Neu and Bart?ok [18].
?
2/3
1/3
m(log d+1)
dT
and M = ?1e ? ?2m(log
guarantees that the
Corollary 3. Setting ? =
2dT
d+1)
regret of S LEEPING C AT BANDIT against the best fixed policy is bounded as
RT ? (2mdT )2/3 ? (log d + 1)1/3 .
The proof of the corollary follows from bounding Qt ? d and plugging the parameters into the
bound of Theorem 3. Similarly to the improvement of Corollary 2, it is possible to replace the factor
d2/3 by (d/?)1/3 if we assume that ai ? ? for all i and some ? > 0.
This corollary implies that S LEEPING C AT BANDIT achieves a regret of (2KT )2/3 ? (log K + 1)1/3
in the case when S = [K], that is, in the K-armed sleeping bandit problem considered by Kanade
et al. [13]. This improves their bound of O((KT )4/5 log T ) by a large margin, thanks to the fact
that we did not have to explicitly learn the distribution P.
5
Experiments
In this section we present the empirical evaluation of our algorithms for bandit and semi-bandit
settings, and compare them to its counterparts [13]. We demonstrate that the wasteful exploration of
BSFPL does not only result in worse regret bounds but also degrades its empirical performance.
For the bandit case, we evaluate S LEEPING C AT BANDIT using the same setting as Kanade et al. [13].
We consider an experiment with T = 10, 000 and 5 arms, each of which are available independenly
of each other with probability p. Losses for each arm are constructed as random walks with Gaussian
increments of standard deviation 0.002, initialized uniformly on [0, 1]. Losses outside [0, 1] are truncated. In our first experiment (Figure 2, left), we study the effect of changing p on the performance
of BSFPL and S LEEPING C AT BANDIT. Notice that when p is very low, there are few or no arms to
choose from. In this case the problems are easy by design and all algorithms suffer low regret. As
p increases, the policy space starts to blow up and the problem becomes more difficult. When p approaches one, it collapses into the set of single arms and the problem gets easier again. Observe that
the behavior of S LEEPING C AT BANDIT follows this trend. On the other hand, the performance of
BSFPL steadily decreases with increasing availability. This is due to the explicit exploration rounds
in the main phase of BSFPL, that suffers the loss of the uniform policy scaled by the exploration
probability. The performance of the uniform policy is plotted for reference.
7
sleeping bandits, 5 arms, varying availabity, average over 20 runs
cumulative regret at time T = 10000
BSFPL
0.25
SleepingCat
RandomGuess
0.2
0.15
0.1
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
availabity
Figure 2: Left: Multi-arm bandits - varying availabilities. Middle: Shortest paths on a 3 ? 3 grid.
Right: Shortest paths on a 10 ? 10 grid.
To evaluate S LEEPING C AT BANDIT in the semi-bandit setting, we consider the shortest path problem on grids of 3 ? 3 and 10 ? 10 nodes, which amounts to 12 and 180 edges respectively. For
each edge, we generate a random-walk loss sequence in the same way as in our first experiment.
In each round t, the learner has to choose a path from the lower left corner to the upper right one
composed from available edges. The individual availability of each edge is sampled with probability
0.9, independently of the others. Whenever an edge gets disconnected from the source, it becomes
unavailable itself, resuling in a quite complicated action-availability distribution. Once a learner
chooses a path, the losses of chosen road segments are revealed and the learner suffers their sum.
Since [13] does not provide a combinatorial version of their approach, we compare against C OMB BSFPL, a straightforward extension of BSFPL. As in BSFPL, we dedicate an initial phase to estimate the availabilities of each component, requiring d oracle calls per step. In the main phase, we
follow BSFPL and alternate between exploration and exploitation. In exploration rounds, we test
for the reachability of a randomly sampled edge and update the reward estimates as in BSFPL.
Figure 2 (middle and right) shows the performance of C OMB BSFPL and S LEEPING C AT BANDIT
for a fixed loss sequence, averaged over 20 samples of the component availabilities. We also plot the
performance of a random policy that follows the perturbed leader with all-zero loss estimates. First
observe that the initial exploration phase sets back the performance of C OMB BSFPL significantly.
The second drawback of C OMB BSFPL is the explicit separation of exploration and the exploitation
rounds. This drawback is far more apparent when the number of components increases, as it is the
case for the 10 ? 10 grid graph with 180 components. As C OMB BSFPL only estimates the loss of
one edge per exploration step, sampling each edge as few as 50 times eats up 9, 000 rounds from
the available 10, 000. S LEEPING C AT BANDIT does not suffer from this problem as it uses all its
observations in constructing the loss estimates.
6
Conclusions & future work
In this paper, we studied the problem of online combinatorial optimization with changing decision
sets. Our main contribution is a novel loss-estimation technique that enabled us to prove strong
regret bounds under various partial-feedback schemes. In particular, our results largely improve on
the best known results for the sleeping bandit problem [13], which suffers large losses from both
from an initial exploration phase and from explicit exploration rounds in the main phase. These
findings are also supported by our experiments.
?
Still, one might ask if it is possible to efficiently achieve a regret of order T under semi-bandit
feedback. While the E XP 4 algorithm of Auer et al. [2] can be used to obtain such regret guarantee,
running this algorithm is out of question as its time and space complexity can be double-exponential
in d (see also the comments in [15]). Had we had access to the loss estimates (3), we would be able
to control the regret of FPL as the term on the right hand side of
? Equation (6) could be replaced
by md, which is sufficient for obtaining a regret bound of O(m dT log d). In fact, it seems that
learning in the bandit setting requires significantly more knowledge about P than the knowledge of
the ai ?s. The question if we can extend the CAT technique to estimate all the relevant quantities of
P is an interesting problem left for future investigation.
Acknowledgements The research presented in this paper was supported by French Ministry
of Higher Education and Research, by European Community?s Seventh Framework Programme
(FP7/2007-2013) under grant agreement no 270327 (CompLACS), and by FUI project Herm`es.
8
References
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9
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4,840 | 5,382 | Multilabel Structured Output Learning with Random
Spanning Trees of Max-Margin Markov Networks
Hongyu Su
Helsinki Institute for Information Technology
Dept of Information and Computer Science
Aalto University, Finland
[email protected]
Mario Marchand
D?epartement d?informatique et g?enie logiciel
Universit?e Laval
Qu?ebec (QC), Canada
[email protected]
Emilie Morvant?
LaHC, UMR CNRS 5516
Univ. of St-Etienne, France
[email protected]
Juho Rousu
Helsinki Institute for Information Technology
Dept of Information and Computer Science
Aalto University, Finland
[email protected]
John Shawe-Taylor
Department of Computer Science
University College London
London, UK
[email protected]
Abstract
We show that the usual score function for conditional Markov networks can be
written as the expectation over the scores of their spanning trees. We also show
that a small random sample of these output trees can attain a significant fraction
of the margin obtained by the complete graph and we provide conditions under
which we can perform tractable inference. The experimental results confirm that
practical learning is scalable to realistic datasets using this approach.
1
Introduction
Finding an hyperplane that minimizes the number of misclassifications is N P-hard. But the support
vector machine (SVM) substitutes the hinge for the discrete loss and, modulo a margin assumption,
can nonetheless efficiently find a hyperplane with a guarantee of good generalization. This paper
investigates whether the problem of inference over a complete graph in structured output prediction
can be avoided in an analogous way based on a margin assumption.
We first show that the score function for the complete output graph can be expressed as the expectation over the scores of random spanning trees. A sampling result then shows that a small random
sample of these output trees can attain a significant fraction of the margin obtained by the complete
graph. Together with a generalization bound for the sample of trees, this shows that we can obtain
good generalization using the average scores of a sample of trees in place of the complete graph.
We have thus reduced the intractable inference problem to a convex optimization not dissimilar to
a SVM. The key inference problem to enable learning with this ensemble now becomes finding the
maximum violator for the (finite sample) average tree score. We then provide the conditions under
which the inference problem is tractable. Experimental results confirm this prediction and show that
?
Most of this work was carried out while E. Morvant was affiliated with IST Austria, Klosterneurburg.
1
practical learning is scalable to realistic datasets using this approach with the resulting classification
accuracy enhanced over more naive ways of training the individual tree score functions.
The paper aims at exploring the potential ramifications of the random spanning tree observation
both theoretically and practically. As such, we think that we have laid the foundations for a fruitful
approach to tackle the intractability of inference in a number of scenarios. Other attractive features
are that we do not require knowledge of the output graph?s structure, that the optimization is convex,
and that the accuracy of the optimization can be traded against computation. Our approach is firmly
rooted in the maximum margin Markov network analysis [1]. Other ways to address the intractability
of loopy graph inference have included using approximate MAP inference with tree-based and LP
relaxations [2], semi-definite programming convex relaxations [3], special cases of graph classes for
which inference is efficient [4], use of random tree score functions in heuristic combinations [5].
Our work is not based on any of these approaches, despite superficial resemblances to, e.g., the
trees in tree-based relaxations and the use of random trees in [5]. We believe it represents a distinct
approach to a fundamental problem of learning and, as such, is worthy of further investigation.
2
Definitions and Assumptions
We consider supervised learning problems where the input space X is arbitrary and the output space
def
Y consists of the set of all `-dimensional multilabel vectors (y1 , . . . , y` ) = y where each yi ?
{1, . . . , ri } for some finite positive integer ri . Each example (x, y) ? X ?Y is mapped to a joint
feature vector ? (x, y). Given a weight vector w in the space of joint feature vectors, the predicted
output yw (x) at input x ? X , is given by the output y maximizing the score F (w, x, y), i.e.,
def
yw (x) = argmax F (w, x, y)
;
where
def
F (w, x, y) = hw, ? (x, y)i ,
(1)
y?Y
and where h?, ?i denotes the inner product in the joint feature space. Hence, yw (x) is obtained by
solving the so-called inference problem, which is known to be N P-hard for many output feature
maps [6, 7]. Consequently, we aim at using an output feature map for which the inference problem can be solved by a polynomial time algorithm such as dynamic programming. The margin
?(w, x, y) achieved by predictor w at example (x, y) is defined as,
def
?(w, x, y) =
min [F (w, x, y) ? F (w, x, y0 )] .
y0 6=y
We consider the case where the feature map ? is a potential function for a Markov network defined
by a complete graph G with ` nodes and `(` ? 1)/2 undirected edges. Each node i of G represents
an output variable yi and there exists an edge (i, j) of G for each pair (yi , yj ) of output variables.
For any example (x, y) ? X ? Y, its joint feature vector is given by
? (x, y) = ? i,j (x, yi , yj ) (i,j)?G = ? (x) ? ? i,j (yi , yj ) (i,j)?G ,
where ? is the Kronecker product. Hence, any predictor w can be written as w = (wi,j )(i,j)?G
where wi,j is w?s weight on ? i,j (x, yi , yj ). Therefore, for any w and any (x, y), we have
X
X
F (w, x, y) = hw, ? (x, y)i =
hwi,j , ? i,j (x, yi , yj )i =
Fi,j (wi,j , x, yi , yj ) ,
(i,j)?G
(i,j)?G
where we denote by Fi,j (wi,j , x, yi , yj ) = hwi,j , ? i,j (x, yi , yj ) the score of labeling the edge (i, j)
by (yi , yj ) given input x.
For any vector a, let kak denote its L2 norm. Throughout the paper, we make the assumption that
?(x, y)k = 1 for all (x, y)
we have a normalized joint feature space such that k?
? X ? Y and
?i,j (x, yi , yj )k is the same for all (i, j) ? G. Since the complete graph G has 2` edges, it follows
k?
?1
?i,j (x, yi , yj )k2 = 2`
that k?
for all (i, j) ? G.
def
We also have a training set S = {(x1 , y1 ), . . . , (xm , ym )} where each example is generated independently according to some unknown distribution D. Mathematically, we do not assume the
existence of a predictor w achieving some positive margin ?(w, x, y) on each (x, y) ? S. Indeed,
2
for some S, there might not exist any w where ?(w, x, y) > 0 for all (x, y) ? S. However, the
generalization guarantee will be best when w achieves a large margin on most training points.
Given any ? > 0, and any (x, y) ? X ? Y, the hinge loss (at scale ?) incurred on (x, y) by a unit L2
norm predictor w that achieves a (possibly negative) margin ?(w, x, y) is given by L? (?(w, x, y)),
def
where the so-called hinge loss function L? is defined as L? (s) = max (0, 1 ? s/?) ?s ? R . We
def
will also make use of the ramp loss function A? defined by A? (s) = min(1, L? (s)) ?s ? R .
The proofs of all the rigorous results of this paper are provided in the supplementary material.
3
Superposition of Random Spanning Trees
Given a complete graph G of ` nodes (representing the Markov network), let S(G) denote the set of
all ``?2 spanning trees of G. Recall that each spanning tree of G has ` ? 1 edges. Hence, forany
edge (i, j) ? G, the number of trees in S(G) covering that edge (i, j) is given by ``?2 (`?1)/ 2` =
(2/`)``?2 . Therefore, for any function f of the edges of G we have
X X
2 X
f ((i, j)) = ``?2
f ((i, j)) .
`
T ?S(G) (i,j)?T
(i,j)?G
Given any spanning tree T of G and given any predictor w, let wT denote the projection of w on the
edges of T . Namely, (wT )i,j = wi,j if (i, j) ? T , and (wT )i,j = 0 otherwise. Let us also denote
?T (x, y))i,j = ? i,j (x, yi , yj )
by ? T (x, y), the projection of ? (x, y) on the edges of T . Namely, (?
?1
?T (x, y))i,j = 0 otherwise. Recall that k?
?i,j (x, yi , yj )k2 = 2`
if (i, j) ? T , and (?
?(i, j) ? G.
Thus, for all (x, y) ? X ? Y and for all T ? S(G), we have
X
`?1
2
?T (x, y)k2 =
?i,j (x, yi , yj )k2 = ` = .
k?
k?
`
2
(i,j)?T
We now establish how F (w, x, y) can be written as an expectation over all the spanning trees of G.
def
def
? =
?T k. Let U(G) denote the uniform distribution on
? T = wT /kwT k, ?
Lemma 1. Let w
? T /k?
T
S(G). Then, we have
r
`
def
?
? T , ? T (x, y)i, where aT =
kwT k .
F (w, x, y) =
E aT hw
2
T ?U (G)
Moreover, for any w such that kwk = 1, we have:
E
T ?U (G)
a2T = 1, and
E
T ?U (G)
aT ? 1 .
def
Let T = {T1 , . . . , Tn } be a sample of n spanning trees of G where each Ti is sampled independently
according to U(G). Given any unit L2 norm predictor w on the complete graph G, our task is to
investigate how the margins ?(w, x, y), for each (x, y) ? X ?Y, will be modified if we approximate
the (true) expectation over all spanning trees by an average over the sample T .
For this task, we consider any (x, y) and any w of unit L2 norm. Let FT (w, x, y) denote the
estimation of F (w, x, y) on the tree sample T ,
n
1X
def
? (x, y)i ,
? Ti , ?
FT (w, x, y) =
aT hw
Ti
n i=1 i
and let ?T (w, x, y) denote the estimation of ?(w, x, y) on the tree sample T ,
def
?T (w, x, y) = min
[FT (w, x, y) ? FT (w, x, y0 )] .
0
y 6=y
The following lemma states how ?T relates to ?.
Lemma 2. Consider any unit L2 norm predictor w on the complete graph G that achieves a margin
of ?(w, x, y) for each (x, y) ? X ? Y, then we have
?T (w, x, y) ? ?(w, x, y) ? 2 ?(x, y) ? X ? Y ,
whenever we have |FT (w, x, y) ? F (w, x, y)| ? for all (x, y) ? X ? Y.
3
Lemma 2 has important consequences whenever |FT (w, x, y) ? F (w, x, y)| ? for all (x, y) ?
X ? Y. Indeed, if w achieves a hard margin ?(w, x, y) ? ? > 0 for all (x, y) ? S, then we have
that w also achieves a hard margin of ?T (w, x, y) ? ? ? 2 on each (x, y) ? S when using the tree
sample T instead of the full graph G. More generally, if w achieves a ramp loss of A? (?(w, x, y))
for each (x, y) ? X ? Y, then w achieves a ramp loss of A? (?T (w, x, y)) ? A? (?(w, x, y) ? 2)
for all (x, y) ? X ? Y when using the tree sample T instead of the full graph G. This last property
follows directly from the fact that A? (s) is a non-increasing function of s.
?
The next lemma tells us that, apart from a slow ln2 ( n) dependence, a sample of n ? ?(`2 /2 )
spanning trees is sufficient to assure that the condition of Lemma 2 holds with high probability for all
(x, y) ? X ? Y. Such a fast convergence rate was made possible by using PAC-Bayesian methods
which, in our case, prevented us of using the union bound over all possible y ? Y.
Lemma 3. Consider any > 0 and any unit L2 norm predictor w for the complete graph G acting
on a normalized joint feature space. For any ? ? (0, 1), let
? 2
`2 1
1 8 n
n ? 2
+ ln
.
(2)
16 2
?
Then with probability of at least 1 ? ?/2 over all samples T generated according to U(G)n , we
have, simultaneously for all (x, y) ? X ? Y, that |FT (w, x, y) ? F (w, x, y)| ? .
def
? T1 , . . . , w
? Tn }
Given a sample T of n spanning trees of G, we now consider an arbitrary set W = {w
? (x, y).
? Ti operates on a unit L2 norm feature vector ?
of unit L2 norm weight vectors where each w
Ti
For any T and any such set W, we consider an arbitrary unit L2 norm conical combination of each
Pn
def
weight in W realized by a n-dimensional weight vector q = (q1 , . . . , qn ), where i=1 qi2 = 1 and
each qi ? 0. Given any (x, y) and any T , we define the score FT (W, q, x, y) achieved on (x, y)
by the conical combination (W, q) on T as
n
X
def 1
? (x, y)i ,
? Ti , ?
qi hw
FT (W, q, x, y) = ?
Ti
n i=1
(3)
?
where
n denominator ensures
Pthe
? that we always have FT (W, q, x, y) ? 1 in view of the fact
n
that i=1 qi can be as large as n. Note also that FT (W, q, x, y) is the score of the feature vector
obtained by the concatenation of all the weight vectors in W (and weighted by q) acting on a feature
? multiplied by 1/?n. Hence, given T , we define the
vector obtained by concatenating each ?
Ti
margin ?T (W, q, x, y) achieved on (x, y) by the conical combination (W, q) on T as
def
?T (W, q, x, y) = min
[FT (W, q, x, y) ? FT (W, q, x, y0 )] .
0
y 6=y
(4)
For any unit L2 norm predictor w that achieves a margin of ?(w, x, y) for all (x, y) ? X ? Y, we
now show that there exists, with high probability, a unit L2 norm conical combination (W, q) on T
achieving margins that are not much smaller than ?(w, x, y).
Theorem 4. Consider any unit L2 norm predictor w for the complete graph G, acting on a normalized joint feature space, achieving a margin of ?(w, x, y) for each (x, y) ? X ? Y. Then for any
> 0, and any n satisfying Lemma 3, for any ? ? (0, 1], with probability of at least 1 ? ? over all
samples T generated according to U(G)n , there exists a unit L2 norm conical combination (W, q)
on T such that, simultaneously for all (x, y) ? X ? Y, we have
?T (W, q, x, y) ? ?
1
[?(w, x, y) ? 2] .
1+
From Theorem 4, and since A? (s) is a non-increasing function of s, it follows that, with probability at least 1 ? ? over the random draws of T ? U(G)n , there exists (W, q) on T such that,
simultaneously for all ?(x, y) ? X ? Y, for any n satisfying Lemma 3 we have
A? (?T (W, q, x, y)) ? A? [?(w, x, y) ? 2] (1 + )?1/2 .
Hence, instead of searching for a predictor w for the complete graph G that achieves a small expected ramp loss E(x,y)?D A? (?(w, x, y), Theorem 4 tells us that we can settle the search for a
4
unit L2 norm conical combination (W, q) on a sample T of randomly-generated spanning trees of
G that achieves small E(x,y)?D A? (?T (W, q, x, y)). But recall that ?T (W, q, x, y)) is the margin
of a weight vector obtained by the concatenation of all the weight vectors in W (weighted by q) on
? ?
a feature vector obtained by the concatenation of the n feature vectors (1/ n)?
Ti . It thus follows
that any standard risk bound for the SVM applies directly to E(x,y)?D A? (?T (W, q, x, y)). Hence,
by adapting the SVM risk bound of [8], we have the following result.
Theorem 5. Consider any sample T of n spanning trees of the complete graph G. For any ? > 0
and any 0 < ? ? 1, with probability of at least 1 ? ? over the random draws of S ? Dm ,
simultaneously for all unit L2 norm conical combinations (W, q) on T , we have
r
m
1 X ?
2
ln(2/?)
?
A (?T (W, q, xi , yi )) + ? + 3
.
E A (?T (W, q, x, y)) ?
m i=1
2m
? m
(x,y)?D
Hence, according to this theorem, the conical combination (W, q) having the best generalization
guarantee is the one which minimizes the sum of the first two terms on the right hand side of
the inequality. Note that the theorem is still valid if we replace, in the empirical risk term, the
non-convex ramp loss A? by the convex hinge loss L? . This provides the theoretical basis of the
proposed optimization problem for learning (W, q) on the sample T .
4
A L2 -Norm Random Spanning Tree Approximation Approach
def
If we introduce the usual
variables ?k = ? ? L? (?T (W, q, xk , yk ), Theorem 5 suggests that
Pslack
m
1
we should minimize ? k=1 ?k for some fixed margin value ? > 0. Rather than performing this
task for several values of ?, we show in the supplementary material that we can, equivalently, solve
the following optimization problem for several values of C > 0.
Definition 6. Primal L2 -norm Random Tree Approximation.
m
n
min
wTi ,?k
s.t.
X
1X
2
?k
||wTi ||2 + C
2 i=1
k=1
n
X
? (xk , yk )i ? max
hwTi , ?
Ti
y6=yk
i=1
n
X
? (xk , y)i ? 1 ? ?k ,
hwTi , ?
Ti
i=1
?k ? 0 , ? k ? {1, . . . , m},
where {wTi |Ti ? T } are the feature weights to be learned on each tree, ?k is the margin slack
allocated for each xk , and C is the slack parameter that controls the amount of regularization.
This primal form has the interpretation of maximizing the joint margins from individual trees between (correct) training examples and all the other (incorrect) examples.
The key for the efficient optimization is solving the ?argmax? problem efficiently. In particular, we
note that the space of all multilabels is exponential in size, thus forbidding exhaustive enumeration
over it. In the following, we show how exact inference over a collection T of trees can be implemented in ?(Kn`) time per data point, where K is the smallest number such that the average score
Pn
def
? (x, y)i.
of the K?th best multilabel for each tree of T is at most FT (x, y) = n1 i=1 hwTi , ?
Ti
Whenever K is polynomial in the number of labels, this gives us exact polynomial-time inference
over the ensemble of trees.
4.1
Fast inference over a collection of trees
It is well known that the exact solution to the inference problem
def
? (x, y)i,
? Ti (x) = argmax FwTi (x, y) = argmax hwTi , ?
y
Ti
y?Y
(5)
y?Y
on an individual tree Ti can be obtained in ?(`) time by dynamic programming. However, there is
? Ti of Equation (5) is also a maximizer of FT . In practice, y
? Ti
no guarantee that the maximizer y
5
can differ for each spanning tree Ti ? T . Hence, instead of using only the best scoring multil? Ti from each individual Ti ? T , we consider the set of the K highest scoring multilabels
abel y
? Ti ,K } of FwTi (x, y). In the supplementary material we describe a dynamic
YTi ,K = {?
yTi ,1 , ? ? ? , y
programming to find the K highest multilabels in ?(K`) time. Running this algorithm for all of the
trees gives us a candidate set of ?(Kn) multilabels YT ,K = YT1 ,K ? ? ? ? ? YTn ,K . We now state a
key lemma that will enable us to verify if the candidate set contains the maximizer of FT .
?
Lemma 7. Let yK
= argmax FT (x, y) be the highest scoring multilabel in YT ,K . Suppose that
y?YT ,K
n
1X
def
FwTi (x, yTi ,K ) = ?x (K).
n i=1
?
It follows that FT (x, yK
) = maxy?Y FT (x, y).
?
FT (x, yK
)?
?
We can use any K satisfying the lemma as the length of K-best lists, and be assured that yK
is a
maximizer of FT .
We now examine the conditions under which the highest scoring multilabel is present in our candef
def
? = yw (x) =
didate set YT ,K with high probability. For any x ? X and any predictor w, let y
argmax F (w, x, y) be the highest scoring multilabel in Y for predictor w on the complete graph G.
y?Y
def
For any y ? Y, let KT (y) be the rank of y in tree T and let ?T (y) = KT (y)/|Y| be the normalized
rank of y in tree T . We then have 0 < ?T (y) ? 1 and ?T (y0 ) = miny?Y ?T (y) whenever y0 is a
highest scoring multilabel in tree T . Since w and x are arbitrary and fixed, let us drop them momendef
def
tarily from the notation and let F (y) = F (w, x, y), and FT (y) = FwT (x, y). Let U(Y) denote the
def
def
uniform distribution of multilabels on Y. Then, let ?T = Ey?U (Y) FT (y) and ? = ET ?U (G) ?T .
Let T ? U(G)n be a sample of n spanning trees of G. Since the scoring function FT of each tree
T of G is bounded in absolute value, it follows that FT is a ?T -sub-Gaussian random variable for
some ?T > 0. We now show that, with high probability, there exists a tree T ? T such that ?T (?
y)
def
is decreasing exponentially rapidly with (F (?
y) ? ?)/?, where ? 2 = ET ?U (G) ?T2 .
Lemma 8. Let the scoring function FT of each spanning tree of G be a ?T -sub-Gaussian random
variable under the uniform distribution of labels; i.e., for each T on G, there exists ?T > 0 such
that for any ? > 0 we have
?2
def
Let ? 2 =
E
T ?U (G)
2
e?(FT (y)??T ) ? e 2 ?T .
y?U (Y)
def
?T2 , and let ? = Pr
?T ? ? ? FT (?
y) ? F (?
y) ? ?T2 ? ? 2 . Then,
E
T ?U (G)
Pr
T ?U (G)n
? 12
?T ? T : ?T (?
y) ? e
(F (?
y)??)2
?2
? 1 ? (1 ? ?)n .
Thus, even for very small ?, when n is large enough, there exists, with high probability, a tree T ? T
? has a small ?T (?
such that y
y) whenever [F (?
y) ? ?]/? is large for G. For example, when |Y| = 2`
n
(the multiple binary classification case), we have with probability
? of at least 1 ? (1 ? ?) , that there
exists T ? T such that KT (?
y) = 1 whenever F (?
y) ? ? ? ? 2` ln 2.
4.2
Optimization
To optimize the L2 -norm RTA problem (Definition 6) we convert it to the marginalized dual form
(see the supplementary material for the derivation), which gives us a polynomial-size problem (in
the number of microlabels) and allows us to use kernels to tackle complex input spaces efficiently.
Definition 9. L2 -norm RTA Marginalized Dual
1 X
1 X
maxm
?(k, e, ue ) ?
?(k, e, ue )KTe (xk , ue ; x0k , u0e )?(k 0 , e, u0e ) ,
? ?M
|ET |
2
e,k,ue
e,k,ue ,
k0 ,u0e
where ET is the union of the sets of edges appearing in T , and ? ? Mm are the marginal dual
def
variables ? = (?(k, e, ue ))k,e,ue , with the triplet (k, e, ue ) corresponding to labeling the edge
6
DATASET
M ICROLABEL L OSS (%)
SVM
E MOTIONS
Y EAST
S CENE
E NRON
C AL 500
22.4
20.0
9.8
6.4
13.7
F INGERPRINT 10.3
NCI60
15.3
M EDICAL
2.6
C IRCLE 10
4.7
C IRCLE 50
5.7
0/1 L OSS (%)
MTL MMCRF MAM
RTA
SVM
MTL
20.2
20.7
11.6
6.5
13.8
17.3
16.0
2.6
6.3
6.2
18.8
19.8
8.8
5.3
13.8
10.7
14.9
2.1
0.6
3.8
77.8
85.9
47.2
99.6
100.0
99.0
56.9
91.8
28.9
69.8
74.5
88.7
55.2
99.6
100.0
100.0
53.0
91.8
33.2
72.3
20.1
21.7
18.4
6.2
13.7
10.5
14.6
2.1
2.6
1.5
19.5
20.1
17.0
5.0
13.7
10.5
14.3
2.1
2.5
2.1
MMCRF MAM
71.3
93.0
72.2
92.7
100.0
99.6
63.1
63.8
20.3
38.8
69.6
86.0
94.6
87.9
100.0
99.6
60.0
63.1
17.7
46.2
RTA
66.3
77.7
30.2
87.7
100.0
96.7
52.9
58.8
4.0
52.8
Table 1: Prediction performance of each algorithm in terms of microlabel loss and 0/1 loss. The best
performing algorithm is highlighted with boldface, the second best is in italic.
e = (v, v 0 ) ? ET of the output graph by ue = (uv , uv0 ) ? Yv ?Yv0 for the training example xk . Also,
Mm is the marginal dual feasible set and
def
KTe (xk , ue ; xk0 , u0e ) =
NT (e)
K(xk , xk0 ) ? e (ykv , ykv0 ) ? ? e (uv , uv0 ), ? e (yk0 v , yk0 v0 ) ? ? e (u0v , u0v0 )
|ET |2
is the joint kernel of input features and the differences of output features of true and competing
multilabels (yk , u), projected to the edge e. Finally, NT (e) denotes the number of times e appears
among the trees of the ensemble.
The master algorithm described in the supplementary material iterates over each training example
until convergence. The processing of each training example xk proceeds by finding the worst violating multilabel of the ensemble defined as
def
? k = argmax FT (xk , y) ,
y
(6)
y6=yk
using the K-best inference approach of the previous section, with the modification that the correct
? k is mapped to a vertex
multilabel is excluded from the K-best lists. The worst violator y
? (xk ) = C ? ([?
?
ye = ue ])e,ue ? Mk
corresponding to the steepest feasible ascent direction (c.f, [9]) in the marginal dual feasible set Mk
of example xk , thus giving us a subgradient of the objective of Definition 9. An exact line search is
?.
used to find the saddle point between the current solution and ?
5
Empirical Evaluation
We compare our method RTA to Support Vector Machine (SVM) [10, 11], Multitask Feature Learning (MTL) [12], Max-Margin Conditional Random Fields (MMCRF) [9] which uses the loopy belief propagation algorithm for approximate inference on the general graph, and Maximum Average
Marginal Aggregation (MAM) [5] which is a multilabel ensemble model that trains a set of random
tree based learners separately and performs the final approximate inference on a union graph of the
edge potential functions of the trees. We use ten multilabel datasets from [5]. Following [5], MAM
is constructed with 180 tree based learners, and for MMCRF a consensus graph is created by pooling edges from 40 trees. We train RTA with up to 40 spanning trees and with K up to 32. The linear
kernel is used for methods that require kernelized input. Margin slack parameters are selected from
{100, 50, 10, 1, 0.5, 0.1, 0.01}. We use 5-fold cross-validation to compute the results.
Prediction performance. Table 1 shows the performance in terms of microlabel loss and 0/1 loss.
The best methods are highlighted in ?boldface? and the second best in ?italics? (see supplementary
material for full results). RTA quite often improves over MAM in 0/1 accuracy, sometimes with
noticeable margin except for Enron and Circle50. The performances in microlabel accuracy are
quite similar while RTA is slightly above the competition. This demonstrates the advantage of RTA
that gains by optimizing on a collection of trees simultaneously rather than optimizing on individual
trees as MAM. In addition, learning using approximate inference on a general graph seems less
7
?
?
1
3
?
10
?
?
32
K (% of |Y|)
100
316
1000
?
?
1
3
?
?
10
?
32
?
?
K (% of |Y|)
100
?
100
316
1000
?
?
??
?
60
80
?
40
60
80
?
20
?
Y* being verified (% of data)
?
??
Emotions
Yeast
Scene
Enron
Cal500
Fingerprint
NCI60
Medical
Circle10
Circle50
??
?
0
20
40
?
?
40
60
80
?
|T| = 40
?
?
20
?
Y* being verified (% of data)
??
0
?
100
|T| = 10
?
?
0
Y* being verified (% of data)
100
|T| = 5
1
3
?
10
?
?
32
?
K (% of |Y|)
?
?
100
316
1000
Figure 1: Percentage of examples with provably optimal y? being in the K-best lists plotted as a
function of K, scaled with respect to the number of microlabels in the dataset.
favorable as the tree-based methods, as MMCRF quite consistently trails to RTA and MAM in
both microlabel and 0/1 error, except for Circle50 where it outperforms other models. Finally, we
notice that SVM, as a single label classifier, is very competitive against most multilabel methods for
microlabel accuracy.
Exactness of inference on the collection of trees. We now study the empirical behavior of the
inference (see Section 4) on the collection of trees, which, if taken as a single general graph, would
call for solving an N P-hard inference problem. We provide here empirical evidence that we can
perform exact inference on most examples in most datasets in polynomial time.
We ran the K-best inference on eleven datasets where the RTA models were trained with different
amounts of spanning trees |T | = {5, 10, 40} and values for K = {2, 4, 8, 16, 32, 40, 60}. For each parameter combination and for each example, we recorded whether the K-best inference was provably
exact on the collection (i.e., if Lemma 7 was satisfied). Figure 1 plots the percentage of examples
where the inference was indeed provably exact. The values are shown as a function of K, expressed
as the percentage of the number of microlabels in each dataset. Hence, 100% means K = `, which
denotes low polynomial (?(n`2 )) time inference in the exponential size multilabel space.
We observe, from Figure 1, on some datasets (e.g., Medical, NCI60), that the inference task is very
easy since exact inference can be computed for most of the examples even with K values that are
below 50% of the number of microlabels. By setting K = ` (i.e., 100%) we can perform exact
inference for about 90% of the examples on nine datasets with five trees, and eight datasets with
40 trees. On two of the datasets (Cal500, Circle50), inference is not (in general) exact with low
values of K. Allowing K to grow superlinearly on ` would possibly permit exact inference on these
datasets. However, this is left for future studies.
Finally, we note that the difficulty of performing provably exact inference slightly increases when
more spanning trees are used. We have observed that, in most cases, the optimal multilabel y? is
still on the K-best lists but the conditions of Lemma 7 are no longer satisfied, hence forbidding us
to prove exactness of the inference. Thus, working to establish alternative proofs of exactness is a
worthy future research direction.
6
Conclusion
The main theoretical result of the paper is the demonstration that if a large margin structured output
predictor exists, then combining a small sample of random trees will, with high probability, generate
a predictor with good generalization. The key attraction of this approach is the tractability of the
inference problem for the ensemble of trees, both indicated by our theoretical analysis and supported
by our empirical results. However, as a by-product, we have a significant added benefit: we do not
need to know the output structure a priori as this is generated implicitly in the learned weights
for the trees. This is used to significant advantage in our experiments that automatically leverage
correlations between the multiple target outputs to give a substantive increase in accuracy. It also
suggests that the approach has enormous potential for applications where the structure of the output
is not known but is expected to play an important role.
8
References
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learning. Machine Learning, 73(3):243?272, 2008.
[13] Yevgeny Seldin, Franc?ois Laviolette, Nicol`o Cesa-Bianchi, John Shawe-Taylor, and Peter
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[14] Andreas Maurer. A note on the PAC Bayesian theorem. CoRR, cs.LG/0411099, 2004.
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9
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4,841 | 5,383 | Metric Learning for Temporal Sequence Alignment
Damien Garreau ? ?
ENS
[email protected]
R?emi Lajugie ? ?
INRIA
[email protected]
Sylvain Arlot ?
CNRS
[email protected]
Francis Bach ?
INRIA
[email protected]
Abstract
In this paper, we propose to learn a Mahalanobis distance to perform alignment
of multivariate time series. The learning examples for this task are time series for
which the true alignment is known. We cast the alignment problem as a structured
prediction task, and propose realistic losses between alignments for which the
optimization is tractable. We provide experiments on real data in the audio-toaudio context, where we show that the learning of a similarity measure leads to
improvements in the performance of the alignment task. We also propose to use
this metric learning framework to perform feature selection and, from basic audio
features, build a combination of these with better alignment performance.
1
Introduction
The problem of aligning temporal sequences is ubiquitous in applications ranging from bioinformatics [5, 1, 23] to audio processing [4, 6]. The goal is to align two similar time series that have the
same global structure, but local temporal differences. Most alignments algorithms rely on similarity measures, and having a good metric is crucial, especially in the high-dimensional setting where
some features of the signals can be irrelevant to the alignment task. The goal of this paper is to show
how to learn this similarity measure from annotated examples in order to improve the relevance of
the alignments.
For example, in the context of music information retrieval, alignment is used in two different cases:
(1) audio-to-audio alignment and (2) audio-to-score alignment. In the first case, the goal is to match
two audio interpretations of the same piece that are potentially different in rythm, whereas audio-toscore alignment focuses on matching an audio signal to a symbolic representation of the score. In
the second case, there are some attempts to learn from annotated data a measure for performing the
alignment. Joder et al. [12] propose to fit a generative model in that context, and Keshet et al. [13]
learn this measure in a discriminative setting.
Similarly to Keshet et al. [13], we use a discriminative loss to learn the measure, but our work focuses
on audio-to-audio alignment. In that context, the set of authorized alignments is much larger, and we
explicitly cast the problem as a structured prediction task, that we solve using off-the-shelf stochastic
optimization techniques [15] but with proper and significant adjustments, in particular in terms of
losses. The ideas of alignment are also very relevant to the community of speech recognition since
the pioneering work of Sakoe and Chiba [19].
?
?
Contributed equally
SIERRA project-team, D?epartement d?Informatique de l?Ecole Normale Sup?erieure (CNRS, INRIA, ENS)
1
The need for metric learning goes far beyond unsupervised partitioning problems. Weinberger and
Saul [26] proposed a large-margin framework for learning a metric in nearest-neighbour algorithms
based on sets of must-link/must-not-link constraints. Lajugie et al. [16] proposed to use a large
margin framework to learn a Mahalanobis metric in the context of partitioning problems. Since
structured SVM have been proposed by Tsochantaridis et al. [25] and Taskar et al. [22], they have
successfully been used to solve many learning problems, for instance to learn weights for graph
matching [3] or a metric for ranking tasks [17]. They have also been used to learn graph structures
using graph cuts [21].
We make the following five contributions:
? We cast the learning of a Mahalanobis metric in the context of alignment as a structured prediction
problem.
? We show that on real musical datasets this metric improves the performance of alignment algorithms using high-level features.
? We propose to use the metric learning framework to learn combinations of basic audio features
and get good alignment performances.
? We show experimentally that the standard Hamming loss, although tractable computationnally,
does not permit to learn a relevant similarity measure in some real world settings.
? We propose a new loss, closer to the true evaluation loss for alignments, leading to a tractable
learning task, and derive an efficient Frank-Wolfe-based algorithm to deal with this new loss.
That loss solves some issues encountered with the Hamming loss.
2
Matricial formulation of alignment problems
2.1 Notations
In this paper, we consider the alignment problem between two multivariate time series sharing the
same dimension p, but possibly of different lengths TA and TB , namely A ? RTA ?p and B ?
RTB ?p . We refer to the rows of A as a1 , . . . , aTA ? Rp and those of B as b1 , . . . , bTB ? Rp as
column vectors. From now on, we denote by X the pair of signals (A, B).
Let C(X) ? RTA ?TB be an arbitrary pairwise affinity matrix associated to the pair X, that is,
C(X)i,j encodes the affinity between ai and bj . Note that our framework can be extended to the
case where A and B are multivariate signals of different dimensions, as long as C(X) is welldefined. The goal of the alignment task is to find two non-decreasing sequences of indices ? and ?
of same length u ? max(TA , TB ) and to match each
index ?(i) in the time series A to the time
Ptime
u
index ?(i) in the time series B, in such a way that i=1 C(X)?(i),?(i) is maximal, and that (?, ?)
satisfies:
?
?(1) = ?(1) = 1
(matching beginnings)
?
?(u) = TA , ?(u) = TB
(matching endings)
(1)
?
?i, (?(i + 1), ?(i + 1)) ? (?(i), ?(i)) ? {(1, 0), (0, 1), (1, 1)} (three type of moves)
For a given (?, ?), we define the binary matrix Y ? {0, 1}TA ?TB such that Y?(i),?(i) = 1 for every
i ? {1, . . . , u} and 0 otherwise. We denote by Y(X) the set of such matrices, which is uniquely
determined by TA and TB . An example is given in Fig. 1. A vertical move in the Y matrix means
that the signal B is waiting for A, whereas an horizontal one means that A is waiting for B, and a
diagonal move means that they move together. In this sense the time reference is ?warped?.
When C(X) is known, the alignment task can be cast as the following linear program (LP) over the
set Y(X):
max Tr(C(X)> Y ).
(2)
Y ?Y(X)
Our goal is to learn how to form the affinity matrix: once we have learned C(X), the alignment is
obtained from Eq. (2). The optimization problem in Eq. (2) will be referred to as the decoding of
our model.
Dynamic time warping. Given the affinity matrix C(X) associated with the pair of signals X =
(A, B), finding the alignment that solves the LP of Eq. (2) can be done efficiently in O(TA TB ) using
2
Figure 1: Example of two valid alignments encoded by matrices Y 1 and Y 2 . Red upper triangles
1
2
show the (i, j) such that Yi,j
= 1, and the blue lower ones show the (i, j) such that Yi,j
= 1. The
1
2
grey zone corresponds to the area loss ?abs between Y and Y .
a dynamic programming algorithm. It is often referred to as dynamic time warping [5, 18]. This
algorithm is described in Alg. 1 of the supplementary material. Various additional constraints may
be used in the dynamic time warping algorithm [18], which we could easily add to Alg. 1.
The cardinality of the set Y(X) is huge: it corresponds to the number of paths on a rectangular grid
from the southwest (1, 1) to the northeast corner (TA , TB ) with vertical, horizontal and diagonal
moves allowed. This is the definition of the Delannoy numbers?[2]. As noted in [24], when t =
t
? ?2) .
TA = TB goes to infinity, and one can show that #Yt,s ? ?(3+2
?t
2.2
3 2?4
The Mahalanobis metric
In many applications (see, e.g., [6]), for a pair X = (A, B), the affinity matrix is computed by
C(A, B)i,j = ?kai,k ? bj,k k2 . In this paper we propose to learn the metric to compare ai and bj
instead of using the plain Euclidean metric. That is, C(X) is parametrized by a matrix W ? W ?
Rp?p , where W ? Rp?p is the set of semi-definite positive matrices, and we use the corresponding
Mahalanobis metric to compute the pairwise affinity between ai and bj :
C(X; W )i,j = ?(ai ? bj )> W (ai ? bj ).
(3)
Note that the decoding of Eq. (2) is the maximization of a linear function in the parameter W :
max Tr(C(X; W )> Y )
Y ?Y(X)
?
max Tr(W > ?(X, Y )),
Y ?Y(X)
(4)
if we define the joint feature map
?(X, Y ) = ?
TA X
TB
X
Yi,j (ai ? bj )(ai ? bj )> ? Rp?p .
(5)
i=1 j=1
3
Learning the metric
From now on, we assume that we are given n pairs of training instances1 (X i , Y i ) =
i
i
i
i
((Ai , B i ), Y i ) ? RTA ?p ? RTB ?p ? {0, 1}TA ?TB , i = 1, . . . , n. Our goal is to find a matrix
W such that the predicted alignments are close to the groundtruth on these examples, as well as
on unseen examples. We first define a loss between alignments, in order to quantify the proximity
between alignments.
1
We will see that it is necessary to have fully labelled instances, which means that for each pair X i we need
an exact alignment Y i between Ai and B i . Partial alignment might be dealt with by alternating between metric
learning and constrained alignment.
3
3.1 Losses between alignments
In our framework, the alignments are encoded by matrices in Y(X), thus we P
are interested in func2
tions ` : Y(X) ? Y(X) ? R+ . The Frobenius norm is defined by kM k2F = i,j Mi,j
.
Hamming loss. A simple loss between matrices is the Frobenius norm of their difference, which
turns out to be the unnormalized Hamming loss [9] for 0/1-valued matrices. For two matrices
Y1 , Y2 ? Y(X), it is defined as:
`H (Y1 , Y2 ) = kY1 ? Y2 k2F = Tr(Y1> Y1 ) + Tr(Y2> Y2 ) ? 2 Tr(Y1> Y2 )
>
>
= Tr(Y1 1TB 1>
TA ) + Tr(Y2 1TB 1TA ) ? 2 Tr(Y1 Y2 ),
(6)
T
where 1T is the vector of R with all coordinates equal to 1. The last line of Eq. (6) comes from
the fact that the Yi have 0/1-values; that makes the Hamming loss affine in Y1 and Y2 . This loss is
often used in other structured prediction tasks [15]; in the audio-to-score setting, Keshet et al. [13]
use a modified version of this loss, which is the average number of times the difference between the
two alignments is greater than a fixed threshold.
This loss is easy to optimize since, it is linear in our parametrization of the alignement problem, but
not optimal for audio-to-audio alignment. Indeed, a major drawback of the Hamming loss is that, for
alignments of fixed length, it depends only on the number of ?crossings? between alignment paths:
one can easily find Y1 , Y2 , Y3 such that `H (Y2 , Y1 ) = `H (Y3 , Y1 ) but Y2 is much closer to Y1 than
Y3 (see Fig. 2). It is important to notice this is often the case when the length of the signals grows.
Area loss. A more natural loss can be computed as the mean distance beween the paths depicted by
two matrices Y 1 , Y 2 ? Y(X). This loss corresponds to the area between the paths of two matrices
Y , as represented by the grey zone on Fig. 1.
1
2
Formally, as in Fig. 1, for each t ? {1, . . . , TB } we put ?t = | min{k, Yt,k
= 1}?min{k, Yt,k
= 1}|.
Then the area loss is the mean of the ?t . In the audio literature [14], this loss is sometimes called the
?mean absolute deviation? loss and is noted ?abs (Y 1 , Y 2 ).
Unfortunately, for the general alignment problem, ?abs is not linear in the matrices Y . But in the
context of alignment of sequences of two different natures, one of the signal is a reference and
thus the index sequence ? defined in Eq. (1) is increasing, e.g., for the audio-to-partition alignment
problem [12]. This loss is then linear in each of its arguments. More precisely, if we introduce the
matrix LTA ? RTA ?TA which is lower triangular with ones (including on the diagonal), we can
write the loss as
`O = kLTA (Y1 ? Y2 )k2F
(7)
>
> >
= Tr(LTA Y1 1TB 1>
TA ) + Tr(LTA Y2 1TB 1TA ) ? 2 Tr(LTA Y1 Y2 LTA ).
We now prove that this loss corresponds to the area loss in this special case. Let Y be an alignment,
P
Pi
then it is easy see that (LTA Y )i,j = k (LTA )i,k Yk,j = k=1 Yk,j . If Y does not have vertical
moves, i.e., for eachPj there is an unique kj such that Ykj ,j = 1, we have that (LTA Y )i,j = 1 if and
only if i ? kj . So i,j (LTA Y )i,j = #{(i, j), i ? kj }, which is exactly the area under the curve
determined by the path of Y . In all our experiments, we use ?abs for evaluation but not for training.
Approximation of the area loss: the symmetrized area loss. In many real world applications [14], a meaningful loss to assess the quality of an alignment is the area loss. As shown by
our experiments, if the Hamming loss is sufficient in some simple situations and allows to learn a
metric that leads to good alignment performance in terms of area loss, on more challenging datasets
it does not work at all (see Sec. 5). This is due to the fact that two alignments that are very close
in terms of area loss can suffer a big Hamming loss (cf. Fig. 2). Thus it is natural to extend the
formulation of Eq. (7) to matrices in Y(X). We start by symmetrizing the formulation of Eq. (7) to
overcome problems of overpenalization of vertical vs. horizontal moves. We define, for any couple
of binary matrices (Y 1 , Y 2 ),
1
(8)
`S (Y1 , Y2 ) = kLTA (Y1 ? Y2 )k2F + k(Y1 ? Y2 )LTB )k2F
2
h
1
>
> >
=
Tr(Y1> L>
TA LTA Y1 ) + Tr(LTA Y2 1TB 1TA ) ? 2 Tr(Y2 LTA LTA Y1 )
2
i
>
>
>
>
+ Tr(Y1 LTB L>
.
TB Y ) + Tr(Y2 1TA 1TB LTB LTB Y2 ) ? 2 Tr(Y2 LTB LTB Y1
4
1600
Most violated constraint for Hamming Loss
1400
1200
Most violated constraint for lS
Groundruth alignment
tB
1000
800
600
400
200
0
0
200
400
600
800
tA
1000
1200
1400
1600
1800
Figure 2: On the real world Bach chorales dataset, we have represented a groundtruth alignment
together with two others. In term of Hamming loss, both alignments are as far from the groundtruth
whereas for the area loss, they are not. In the structured prediction setting described in Sec. 4,
the depicted alignment are the so-called ?most violated constraint?, namely the output of the loss
augmented decoding step (see Sec. 4).
We propose now to make this loss concave over the convex hull of Y(X) that we denote from now
2
on Y(X). Let us introduce DT = ?max (L>
T LT )IT ?T with ?max (U ) the largest eigenvalue of U .
For any binary matrices Y1 , Y2 , we have
1
>
>
`S (Y1 , Y2 ) =
Tr(Y1> (L>
TA LTA ? DTA )Y1 ) + Tr(DTA Y1 1TB 1TA ) + Tr(LTA Y2 1TB 1TA )
2
>
? 2 Tr(Y2> (L>
TA L ? DTA )Y1 ) + Tr(Y1 (LTB LTB ? DTB )Y )
i
>
>
>
>
+ Tr(Y1 DTB 1TB 1>
)
+
Tr(Y
L
L
Y
)
?
2
Tr(Y
L
L
Y
)
,
T
2
2
T
TA
2
TB
TB 1
B
B
and we get a concave function over Y(X) that coincides with `S on Y(X).
3.2 Empirical loss minimization
Recall that we are given n alignment examples (X i , Y i )1?i?n . For a fixed loss `, our goal is now
to solve the following minimization problem in W :
?
?
n
?
?1 X
min
` Y i , argmax Tr(C(X i ; W )> Y ) + ??(W ) ,
(9)
W ?W ? n
?
Y ?YT i ,T i
i=1
A
B
where ? = ?2 kW k2F is a convex regularizer preventing from overfitting, with ? ? 0.
4
Large margin approach
In this section we describe a large margin approach to solve a surrogate to the problem in Eq. (9),
which is untractable. As shown in Eq. (4), the decoding task is the maximum of a linear function
in the parameter W and aims at predicting an output over a large and discrete space (the space
of potential alignments with respect to the constraints in Eq. (1)). Learning W thus falls into the
structured prediction framework [25, 22]. We define the hinge loss, a convex surrogate, by
n
o
L(X, Y ; W ) = 0max
`(Y, Y 0 ) ? Tr(W > [?(X, Y ) ? ?(X, Y 0 )]) .
(10)
Y ?Y(X)
2
For completeness, in our experiments, we also try to set the matrices DT with minimal trace that dominate
L>
T LT by solving a semidefinite program (SDP). We report the associated result in Fig 4. Note also that
other matrices could have been chosen. In particular, since our matrices LT are pointwise positive, the matrix
>
Diag(L>
T LT ) ? LT LT is such that the loss is concave.
5
The evaluation of L is usually referred to as ?loss-augmented decoding?, see [25]. If we define Yb i
as the argmax in Eq. (10) when (X, Y ) = (X i , Y i ), then elementary computations show that
Yb i = argmin Tr((U > ? 2Y i> ? C(X i ; W )> )Y ),
Y ?Y(X)
TA ?TB
where U = 1TB 1>
.
TB ? R
We now aim at solving the following problem, sometimes called the margin-rescaled problem:
min
W ?W
n
n
o
?
1X
kW k2F +
max `(Y, Y i ) ? Tr(W > ?(X i , Y i ) ? ?(X i , Y ) ) .
2
n i=1 Y ?Y(X)
(11)
Hamming loss case. From Eq. (4), one can notice that our joint feature map is linear in Y . Thus,
if we take a loss that is linear in the first argument of `, for instance the Hamming loss, the lossaugmented decoding is the maximization of a linear function over the spaces Y(X) that we can
solve efficiently using dynamic programming algorithms (see Sec. 2.1 and supplementary material).
That way, plugging the Hamming loss (Eq. (6)) in Eq. (11) leads to a convex structured prediction
problem. This problem can be solved using standard techniques such as cutting plane methods [11],
stochastic gradient descent [20], or block-coordinate Frank-Wolfe in the dual [15]. Note that we
adapted the standard unconstrained optimization methods to our setting, where W 0.
Optimization using the symmetrized area loss. The symmetrized area loss is concave in its first
argument, thus the problem of Eq. (11) is in a min/max form and deriving a dual is straightforward.
Details can be found in the supplementary material. If we plug the symmetrized area loss `S (SAL)
defined in Eq. (8) into our problem (11), we can show that the dual of (11) has the following form:
Pn
Pn
P
1
1
i
i
T 2
min
i=1 ?
i=1 `S (Z, Z ), (12)
j,k (Yi ? Z )j,k (aj ? bk )(aj ? bk ) kF ? n
2?n2 k
(Z 1 ,...,Z n )?Y
if we denote by Y(X i ) the convex hull of the sets Y(X i ), and by Y the cartesian product over all
the training examples i of such sets. Note that we recover a similar result as [15]. Since the SAL
loss is concave, the aforementioned problem is convex.
The problem (12) is a quadratic program over the compact set Y. Thus we can use a Frank-Wolfe [7]
algorithm. Note that it is similar to the one proposed by Lacoste-Julien et al. [15] but with an
additional term due to the concavity of the loss.
5
Experiments
We applied our method to the task of learning a good similarity measure for aligning audio signals. In this field researchers have spent a lot of efforts in designing well-suited and meaningful
features [12, 4]. But the problem of combining these features for aligning temporal sequences is
still challenging. For simplicity, we took W diagonal for our experiments.
5.1
Dataset of Kirchhoff and Lerch [14]
Dataset description. First, we applied our method on the dataset of Kirchhoff and Lerch [14]. In
this dataset, pairs of aligned examples (Ai , B i ) are artificially created by stretching an original audio
signal. That way, the groundtruth alignment Y i is known and thus the data falls into our setting A
more precise description of the dataset can be found in [14].
The N = 60 pairs are stretched along two different tempo curves. Each signal is made of 30s
of music divided in frames of 46ms with a hopsize of 23ms, thus leading to a typical length of the
signals of T ? 1300 in our setting. We keep p = 11 features that are simple to implement and known
to perform well for alignment tasks [14]. Those were: five MFCC [8] (labeled M 1, . . . , M 5 in
Fig. 3), the spectral flatness (SF), the spectral centroid (SC), the spectral spread (SS), the maximum
of the envelope (Max), and the power level of each frame (Pow), see [14] for more details on the
computation of the features. We normalize each feature by subtracting the median value and dividing
by the standard deviation to the median, as audio data are subject to outliers.
6
0.2
?abs (s)
0.15
0.1
0.05
0
W M PowM1 SC M4 SR SF M3 Max SS M2 M5
Figure 3: Comparison of performance between individual features and the learned metric. Error
bars for the performance of the learned metric were determined with the best and the worst performance on 5 different experiments. W denotes the learned combination using our method, and M
the best MFCC combination.
Experiments. We conducted the following experiment: for each individual feature, we perform
alignment using dynamic time warping algorithm and evaluate the performance of this single feature
in terms of losses typically used to asses performance in this setting [14]. In Fig. 3, we report the
results of these experiments.
Then, we plug these data into our method, using the Hamming loss to learn a linear positive combination of these features. The result is reported in Fig. 3. Thus, combining these features on this
dataset yields to better performances than only considering a single feature.
For completeness, we also conducted the experiments using the standard 13 first MFCCs coefficients
and their first and second order derivatives as features. These results competed with the best learned
combination of the handcrafted features. Namely, in terms of the ?abs loss, they perform at 0.046
seconds. Note that these results are slightly worse than the best single handcrafted feature, but better
than the best MFCC coefficient used as a feature.
As a baseline, we also compared ourselves against the uniform combination of handcrafted features
(the metric being the identity matrix). The results are off the charts on Fig. 3 with ?abs at 4.1 seconds
(individual values ranging from 1.4 seconds to 7.4 seconds).
5.2
Chorales dataset
Dataset. The Bach 10 dataset3 consists in ten J. S. Bach?s Chorales (small quadriphonic pieces).
For each Chorale, a MIDI reference file corresponding to the ?score?, or basically a representation
of the partition. The alignments between the MIDI files and the audio file are given, thus we have
converted these MIDI files into audio following what is classically done for alignment (see e.g, [10]).
That way we fall into the audio-to-audio framework in which our technique apply. Each piece of
music is approximately 25s long, leading to similar signal length (T ? 1300).
Experiments. We use the same features as in Sec. 5.1. As depicted in Fig. 4, the optimization with
Hamming loss performs poorly on this dataset. In fact, the best individual feature performance is far
better than the performance of the learned W . Thus metric learning with the ?practical? Hamming
loss performs much worse than the best single feature.
Then, we conducted the same learning experiment with the symetrized area loss `S . The resulting
learned parameter is far better than the one learned using the Hamming loss. We get a performance
that is similar to the one of the best feature. Note that these features were handcrafted and reaching
their performance on this hard task with only a few training instances is already challenging.
3
http://music.cs.northwestern.edu/data/Bach10.html.
7
8
4
?
abs
(s)
6
2
0
(1) (2) (3) (4) (5) (6)
Figure 4: Performance of our algorithms on the Chorales dataset. From left to right: (1) Best
single feature, (2) Best learned combination of features using the symmetrized area loss `S , (3)
Best combination of MFCC using SAL and DT obtained via SDP (see footnote in section 3) (4)
Best combination of MFCC and derivatives learned with `S , (5) Best combination of MFCCs and
derivatives learned with Hamming loss, (6) Best combination of features of [14] using Hamming
loss.
In Fig. 2, we have depicted the result, for a learned parameter W , of the loss augmented decoding
performed either using the area. As it is known for structured SVM, this represents the most violated
constraint [25]. We can see that the most violated constraint for the Hamming loss leads to an alignment which is totally unrelated to the groundtruth alignment whereas the one for the symmetrized
area loss is far closer and much more discriminative.
5.3
Feature selection
Last, we conducted feature selection experiments over the same datasets. Starting from low level
features, namely the 13 leading MFCCs coefficients and their first two derivatives, we learn a linear
combination of these that achieves good alignment performance in terms of the area loss. Note
that very little musical prior knowledge is put into these. Moreover we either improve on the best
handcrafted feature on the dataset of [14] or perform similarly. On both datasets, the performance
of learned combination of handcrafted features performed similarly to the combination of these 39
MFCCs coefficients.
6
Conclusion
In this paper, we have presented a structured prediction framework for learning the metric for temporal alignment problems. We are able to combine hand-crafted features, as well as building automatically new state-of-the-art features from basic low-level information with little expert knowledge.
Technically, this is made possible by considering a loss beyond the usual Hamming loss which is
typically used because it is ?practical? within a structured prediction framework (linear in the output
representation).
The present work may be extended in several ways, the main one being to consider cases where
only partial information about the alignments is available. This is often the case in music [4] or
bioinformatics applications. Note that, similarly to Lajugie et al. [16] a simple alternating optimization between metric learning and constrained alignment provide a simple first solution, which could
probably be improved upon.
Acknowledgements. The authors acknowledge the support of the European Research Council
(SIERRA project 239993), the GARGANTUA project funded by the Mastodons program of CNRS
and the Airbus foundation through a PhD fellowship. Thanks to Piotr Bojanowski, for helpful
discussions. Warm thanks go to Arshia Cont and Philippe Cuvillier for sharing their knowledge
about audio processing, and to Holger Kirchhoff and Alexander Lerch for their dataset.
8
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9
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4,842 | 5,384 | Proximal Quasi-Newton for Computationally
Intensive `1-regularized M -estimators
Kai Zhong 1
Ian E.H. Yen 2
Inderjit S. Dhillon 2
Pradeep Ravikumar 2
2
Institute for Computational Engineering & Sciences
Department of Computer Science
University of Texas at Austin
[email protected], {ianyen,inderjit,pradeepr}@cs.utexas.edu
1
Abstract
We consider the class of optimization problems arising from computationally intensive `1 -regularized M -estimators, where the function or gradient values are
very expensive to compute. A particular instance of interest is the `1 -regularized
MLE for learning Conditional Random Fields (CRFs), which are a popular class
of statistical models for varied structured prediction problems such as sequence
labeling, alignment, and classification with label taxonomy. `1 -regularized MLEs
for CRFs are particularly expensive to optimize since computing the gradient values requires an expensive inference step. In this work, we propose the use of a
carefully constructed proximal quasi-Newton algorithm for such computationally
intensive M -estimation problems, where we employ an aggressive active set selection technique. In a key contribution of the paper, we show that the proximal
quasi-Newton method is provably super-linearly convergent, even in the absence
of strong convexity, by leveraging a restricted variant of strong convexity. In our
experiments, the proposed algorithm converges considerably faster than current
state-of-the-art on the problems of sequence labeling and hierarchical classification.
1
Introduction
`1 -regularized M -estimators have attracted considerable interest in recent years due to their ability
to fit large-scale statistical models, where the underlying model parameters are sparse. The optimization problem underlying these `1 -regularized M -estimators takes the form:
min f (w) := ?kwk1 + `(w),
w
(1)
where `(w) is a convex differentiable loss function. In this paper, we are particularly interested in the
case where the function or gradient values are very expensive to compute; we refer to these functions
as computationally intensive functions, or CI functions in short. A particular case of interest are `1 regularized MLEs for Conditional Random Fields (CRFs), where computing the gradient requires
an expensive inference step.
There has been a line of recent work on computationally efficient methods for solving (1), including
[2, 8, 13, 21, 23, 4]. It has now become well understood that it is key to leverage the sparsity
of the optimal solution by maintaining sparse intermediate iterates [2, 5, 8]. Coordinate Descent
(CD) based methods, like CDN [8], maintain the sparsity of intermediate iterates by focusing on an
active set of working variables. A caveat with such methods is that, for CI functions, each coordinate
update typically requires a call of inference oracle to evaluate partial derivative for single coordinate.
One approach adopted in [16] to address this is using Blockwise Coordinate Descent that updates
a block of variables at a time by ignoring the second-order effect, which however sacrifices the
convergence guarantee. Newton-type methods have also attracted a surge of interest in recent years
[5, 13], but these require computing the exact Hessian or Hessian-vector product, which is very
1
expensive for CI functions. This then suggests the use of quasi-Newton methods, popular instances
of which include OWL-QN [23], which is adapted from `2 -regularized L-BFGS, as well as Projected
Quasi-Newton (PQN) [4]. A key caveat with OWL-QN and PQN however is that they do not exploit
the sparsity of the underlying solution. In this paper, we consider the class of Proximal QuasiNewton (Prox-QN) methods, which we argue seem particularly well-suited to such CI functions, for
the following three reasons. Firstly, it requires gradient evaluations only once in each outer iteration.
Secondly, it is a second-order method, which has asymptotic superlinear convergence. Thirdly, it
can employ some active-set strategy to reduce the time complexity from O(d) to O(nnz), where d
is the number of parameters and nnz is the number of non-zero parameters.
While there has been some recent work on Prox-QN algorithms [2, 3], we carefully construct an
implementation that is particularly suited to CI `1 -regularized M -estimators. We carefully maintain the sparsity of intermediate iterates, and at the same time reduce the gradient evaluation time.
A key facet of our approach is our aggressive active set selection (which we also term a ?shrinking strategy?) to reduce the number of active variables under consideration at any iteration, and
correspondingly the number of evaluations of partial gradients in each iteration. Our strategy is
particularly aggressive in that it runs over multiple epochs, and in each epoch, chooses the next
working set as a subset of the current working set rather than the whole set; while at the end of an
epoch, allows for other variables to come in. As a result, in most iterations, our aggressive shrinking
strategy only requires the evaluation of partial gradients in the current working set. Moreover, we
adapt the L-BFGS update to the shrinking procedure such that the update can be conducted without
any loss of accuracy caused by aggressive shrinking. Thirdly, we store our data in a feature-indexed
structure to combine data sparsity as well as iterate sparsity.
[26] showed global convergence and asymptotic superlinear convergence for Prox-QN methods under the assumption that the loss function is strongly convex. However, this assumption is known to
fail to hold in high-dimensional sampling settings, where the Hessian is typically rank-deficient, or
indeed even in low-dimensional settings where there are redundant features. In a key contribution
of the paper, we provide provable guarantees of asymptotic superlinear convergence for Prox-QN
method, even without assuming strong-convexity, but under a restricted variant of strong convexity, termed Constant Nullspace Strong Convexity (CNSC), which is typically satisfied by standard
M -estimators.
To summarize, our contributions are twofold. (a) We present a carefully constructed proximal quasiNewton method for computationally intensive (CI) `1 -regularized M -estimators, which we empirically show to outperform many state-of-the-art methods on CRF problems. (b) We provide the
first proof of asymptotic superlinear convergence for Prox-QN methods without strong convexity,
but under a restricted variant of strong convexity, satisfied by typical M -estimators, including the
`1 -regularized CRF MLEs.
2
Proximal Quasi-Newton Method
A proximal quasi-Newton approach to solve M -estimators of the form (1) proceeds by iteratively
constructing a quadratic approximation of the objective function (1) to find the quasi-Newton direction, and then conducting a line search procedure to obtain the next iterate.
Given a solution estimate wt at iteration t, the proximal quasi-Newton method computes a descent
direction by minimizing the following regularized quadratic model,
1
dt = arg min g Tt ? + ?T Bt ? + ?kwt + ?k1
?
2
(2)
where g t = g(wt ) is the gradient of `(wt ) and Bt is an approximation to the Hessian of `(w). Bt
is usually formulated by the L-BFGS algorithm. This subproblem (2) can be efficiently solved by
randomized coordinate descent algorithm as shown in Section 2.2.
The next iterate is obtained from the backtracking line search procedure, wt+1 = wt + ?t dt , where
the step size ?t is tried over {? 0 , ? 1 , ? 2 , ...} until the Armijo rule is satisfied,
f (wt + ?t dt ) ? f (wt ) + ?t ??t ,
where 0 < ? < 1, 0 < ? < 1 and ?t = g Tt dt + ?(kwt + dt k1 ? kwt k1 ).
2
2.1
BFGS update formula
Bt can be efficiently updated by the gradients of the previous iterations according to the BFGS
update [18],
Bt?1 st?1 sTt?1 Bt?1
y t?1 y Tt?1
Bt = Bt?1 ?
+
(3)
sTt?1 Bt?1 st?1
y Tt?1 st?1
where st = wt+1 ? wt and y t = g t+1 ? g t
We use the compact formula for Bt [18],
?
Bt = B0 ? QRQT = B0 ? QQ,
where
?1
StT B0 St
Lt
? := RQT
,Q
Q := [ B0 St Yt ] , R :=
LTt
?Dt
St = [s0 , s1 , ..., st?1 ] , Yt = y 0 , y 1 , ..., y t?1
T
si?1 y j?1 if i > j
T
T
Dt = diag[s0 y 0 , ..., st?1 y t?1 ] and (Lt )i,j =
0
otherwise
In practical implementation, we apply Limited-memory-BFGS. It only uses the information of the
? have only size, d ? 2m and 2m ? d, respectively. B0 is
most recent m gradients, so that Q and Q
usually set as ?t I for computing Bt , where ?t = y Tt?1 st?1 /sTt?1 st?1 [18]. As will be discussed in
? is updated just on the rows(columns) corresponding to the working set, A. The
Section 2.3, Q(Q)
time complexity for L-BFGS update is O(m2 |A| + m3 ).
2.2
Coordinate Descent for Inner Problem
Randomized coordinate descent is carefully employed to solve the inner problem (2) by Tang and
Scheinberg [2]. In the update for coordinate j, d ? d + z ? ej , z ? is obtained by solving the onedimensional problem,
1
z ? = arg min (Bt )jj z 2 + ((g t )j + (Bt d)j )z + ?|(wt )j + dj + z|
z 2
This one-dimensional problem has a closed-form solution, z ? = ?c + S(c ? b/a, ?/a) ,where S is
the soft-threshold function and a = (Bt )jj , b = (g t )j + (Bt d)j and c = (wt )j + dj . For B0 = ?t I,
? j , where q Tj is the j-th row of Q and q
?j
the diagonal of Bt can be computed by (Bt )jj = ?t ? q Tj q
?
is the j-th column of Q. And the second term in b, (Bt d)j can be computed by,
?
? = ?t dj ? q T d,
(Bt d)j = ?t dj ? q Tj Qd
j
? := Qd.
? has only 2m dimension, it is fast to update (Bt d)j by q and d.
? In each
? Since d
where d
j
? d
??d
?+q
?j z?.
inner iteration, only dj is updated, so we have the fast update of d,
Since we only update the coordinates in the working set, the above algorithm has only computation
complexity O(m|A| ? inner iter), where inner iter is the number of iterations used for solving
the inner problem.
2.3
Implementation
In this section, we discuss several key implementation details used in our algorithm to speed up the
optimization.
Shrinking Strategy
In each iteration, we select an active or working subset A of the set of all variables: only the variables
in this set are updated in the current iteration. The complementary set, also called the fixed set, has
only values of zero and is not updated. The use of such a shrinking strategy reduces the overall
complexity from O(d) to O(|A|). Specifically, we (a) update the gradients just on the working set,
? just on the rows(columns) corresponding to the working set, and (c) compute the
(b) update Q (Q)
latest entries in Dt , ?t , Lt and StT St by just using the corresponding working set rather than the
whole set.
3
The key facet of our ?shrinking strategy? however is in aggressively shrinking the active set: at the
next iteration, we set the active set to be a subset of the previous active set, so that At ? At?1 . Such
an aggressive shrinking strategy however is not guaranteed to only weed out irrelevant variables.
Accordingly, we proceed in epochs. In each epoch, we progressively shrink the active set as above,
till the iterations seem to converge. At that time, we then allow for all the ?shrunk? variables to
come back and start a new epoch. Such a strategy was also called an -cooling strategy by Fan et
al. [14], where the shrinking stopping criterion is loose at the beginning, and progressively becomes
more strict each time all the variables are brought back. For L-BFGS update, when a new epoch
starts, the memory of L-BFGS is cleaned to prevent any loss of accuracy.
Because at the first iteration of each new epoch, the entire gradient over all coordinates is evaluated, the computation time for those iterations accounts for a significant portion of the total time
complexity. Fortunately, our experiments show that the number of epochs is typically between 3-5.
Inexact inner problem solution
Like many other proximal methods, e.g. GLMNET and QUIC, we solve the inner problem inexactly.
This reduces the time complexity of the inner problem dramatically. The amount of inexactness is
based on a heuristic method which aims to balance the computation time of the inner problem in each
outer iteration. The computation time of the inner problem is determined by the number of inner
iterations and the size of working set. Thus, we let the number of inner iterations, inner iter =
min{max inner, bd/|A|c}, where max inner = 10 in our experiment.
Data Structure for both model sparsity and data sparsity
In our implementation we take two sparsity patterns into consideration: (a) model sparsity, which
accounts for the fact that most parameters are equal to zero in the optimal solution; and (b) data
sparsity, wherein most feature values of any particular instance are zeros. We use a feature-indexed
data structure to take advantage of both sparsity patterns. Computations involving data will be timeconsuming if we compute over all the instances including those that are zero. So we leverage the
sparsity of data in our experiment by using vectors of pairs, whose members are the index and its
value. Traditionally, each vector represents an instance and the indices in its pairs are the feature
indices. However, in our implementation, to take both model sparsity and data sparsity into account,
we use an inverted data structure, where each vector represents one feature (feature-indexed) and
the indices in its pairs are the instance indices. This data structure facilitates the computation of the
gradient for a particular feature, which involves only the instances related to this feature.
We summarize these steps in the algorithm below. And a detailed algorithm is in Appendix 2.
Algorithm 1 Proximal Quasi-Newton Algorithm (Prox-QN)
Input: Dataset {x(i) , y (i) }i=1,2,...,N , termination criterion , ? and L-BFGS memory size m.
Output: w? converging to arg minw f (w).
? ? ?.
1: Initialize w ? 0, g ? ?`(w)/?w, working set A ? {1, 2, ...d}, and S, Y , Q, Q
2: while termination criterion is not satisfied or working set doesn?t contain all the variables do
3:
Shrink working set.
4:
if Shrinking stopping criterion is satisfied then
5:
Take all the shrunken variables back to working set and clean the memory of L-BFGS.
6:
Update Shrinking stopping criterion and continue.
7:
end if
8:
Solve inner problem (2) over working set and obtain the new direction d.
9:
Conduct line search based on Armijo rule and obtain new iterate w.
? and related matrices over working set.
10:
Update g, s, y, S, Y , Q, Q
11: end while
3
Convergence Analysis
In this section, we analyze the convergence behavior of proximal quasi-Newton method in the superlinear convergence phase, where the unit step size is chosen. To simplify the analysis, in this section,
we assume the inner problem is solved exactly and no shrinking strategy is employed. We also
provide the global convergence proof for Prox-QN method with shrinking strategy in Appendix 1.5.
In current literature, the analysis of proximal Newton-type methods relies on the assumption of
4
strongly convex objective function to prove superlinear convergence [3]; otherwise, only sublinear
rate can be proved [25]. However, our objective (1) is not strongly convex when the dimension is
very large or there are redundant features. In particular, the Hessian matrix H(w) of the smooth
function `(w) is not positive-definite. We thus leverage a recently introduced restricted variant of
strong convexity, termed Constant Nullspace Strong Convexity (CNSC) in [1]. There the authors
analyzed the behavior of proximal gradient and proximal Newton methods under such a condition.
The proximal quasi-Newton procedure in this paper however requires a subtler analysis, but in a key
contribution of the paper, we are nonetheless able to show asymptotic superlinear convergence of
the Prox-QN method under this restricted variant of strong convexity.
Definition 1 (Constant Nullspace Strong Convexity (CNSC)). A composite function (1) is said to
have Constant Nullspace Strong Convexity restricted to space T (CNSC-T ) if there is a constant
vector space T s.t. `(w) depends only on projT (w), i.e. `(w) = `(projT (w)), and its Hessian
satisfies
(4)
mkvk2 ? v T H(w)v ? M kvk2 , ?v ? T , ?w ? Rd
for some M ? m > 0, and
(5)
H(w)v = 0, ?v ? T ? , ?w ? Rd ,
where projT (w) is the projection of w onto T and T ? is the complementary space orthogonal to
T.
This condition can be seen to be an algebraic condition that is satisfied by typical M -estimators considered in high-dimensional settings. In this paper, we will abuse the use of CNSC-T for symmetric
matrices. We say a symmetric matrix H satisfies CNSC-T condition if H satisfies (4) and (5). In
?
the following theorems, we will denote the orthogonal basis of T as U ? Rd?d , where d? ? d is
the dimensionality of T space and U T U = I. Then the projection to T space can be written as
projT (w) = U U T w.
Theorem 1 (Asymptotic Superlinear Convergence). Assume ?2 `(w) and ?`(w) are Lipschitz continuous. Let Bt be the matrices generated by BFGS update (3). Then if `(w) and Bt satisfy CNSC-T
condition, the proximal quasi-Newton method has q-superlinear convergence:
kz t+1 ? z ? k ? o (kz t ? z ? k) ,
T
?
T ?
where z t = U wt , z = U w and w? is an optimal solution of (1).
The proof is given in Appendix 1.4. We prove it by exploiting the CNSC-T property. First, we
?
re-build our problem and algorithm on the reduced space Z = {z ? Rd |z = U T w}, where
the strong-convexity property holds. Then we prove the asymptotic superlinear convergence on Z
following Theorem 3.7 in [26].
Theorem 2. For Lipschitz continuous `(w), the sequence {wt } produced by the proximal quasiNewton Method in the super-linear convergence phase has
f (wt ) ? f (w? ) ? Lkz t ? z ? k,
(6)
?
T
?
T ?
where L = L` + ? d, L` is the Lipschitz constant of `(w), z t = U wt and z = U w .
The proof is also in Appendix 1.4. It is proved by showing that both the smooth part and the nondifferentiable part satisfy the modified Lipschitz continuity.
4
Application to Conditional Random Fields with `1 Penalty
In CRF problems, we are interested in learning a conditional distribution of labels y ? Y given
observation x ? X , where y has application-dependent structure such as sequence, tree, or table in
which label assignments have inter-dependency. The distribution is of the form
( d
)
X
1
Pw (y|x) =
exp
wk fk (y, x) ,
Zw (x)
k=1
where fk is the feature functions, wk is the associated weight, d is the number of feature functions
and Zw (x) is the partition function. Given a training data set {(xi , y i )}N
i=1 , our goal is to find the
optimal weights w such that the following `1 -regularized negative log-likelihood is minimized.
min f (w) = ?kwk1 ?
w
N
X
i=1
5
log Pw (y (i) |x(i) )
(7)
Since |Y|, the number of possible values y takes, can be exponentially large, the evaluation of
`(w) and the gradient ?`(w) needs application-dependent oracles to conduct the summation over
Y. For example, in sequence labeling problem, a dynamic programming oracle, forward-backward
algorithm, is usually employed to compute ?`(w). Such an oracle can be very expensive. In ProxQN algorithm for sequence labeling problem, the forward-backward algorithm takes O(|Y |2 N T ?
exp) time, where exp is the time for the expensive exponential computation, T is the sequence
length and Y is the possible label set for a symbol in the sequence. Then given the obtained oracle,
the evaluation of the partial gradients over the working set A has time complexity, O(Dnnz |A|T ),
where Dnnz is the average number of instances related to a feature. Thus when O(|Y |2 N T ? exp +
Dnnz |A|T ) > O(m3 + m2 |A|), the gradients evaluation time will dominate.
The following theorem gives that the `1 -regularized CRF MLEs satisfy the CNSC-T condition.
PN
Theorem 3. With `1 penalty, the CRF loss function, `(w) = ? i=1 log Pw (y (i) |x(i) ), satisfies
the CNSC-T condition with T = N ? , where N = {v ? Rd |?T v = 0} is a constant subspace of
Rd and ? ? Rd?(N |Y|) is defined as below,
h
i
?kn = fk (y l , x(i) ) ? E fk (y, x(i) )
where n = (i ? 1)|Y| + l, l = 1, 2, ...|Y| and E is the expectation over the conditional probability
Pw (y|x(i) ).
According to the definition of CNSC-T condition, the `1 -regularized CRF MLEs don?t satisfy
the classical strong-convexity condition when N has non-zero members, which happens in the
following two cases: (i) the exponential representation is not minimal [27], i.e. for any instance i there exist a non-zero vector a and a constant bi such that ha, ?(y, x(i) )i = bi , where
?(y, x) = [f1 (y, x(i) ), f2 (y, x(i) ), ..., fd (y, x(i) )]T ; (ii) d > N |Y|, i.e., the number of feature
functions is very large. The first case holds in many problems, like the sequence labeling and hierarchical classification discussed in Section 6, and the second case will hold in high-dimensional
problems.
5
Related Methods
There have been several methods proposed for solving `1 -regularized M -estimators of the form in
(7). In this section, we will discuss these in relation to our method.
Orthant-Wise Limited-memory Quasi-Newton (OWL-QN) introduced by Andrew and Gao [23]
extends L-BFGS to `1 -regularized problems. In each iteration, OWL-QN computes a generalized
gradient called pseudo-gradient to determine the orthant and the search direction, then does a line
search and a projection of the new iterate back to the orthant. Due to its fast convergence, it is
widely implemented by many software packages, such as CRF++, CRFsuite and Wapiti. But OWLQN does not take advantage of the model sparsity in the optimization procedure, and moreover Yu
et al. [22] have raised issues with its convergence proof.
Stochastic Gradient Descent (SGD) uses the gradient of a single sample as the search direction
at each iteration. Thus, the computation for each iteration is very fast, which leads to fast convergence at the beginning. However, the convergence becomes slower than the second-order method
when the iterate is close to the optimal solution. Recently, an `1 -regularized SGD algorithm proposed by Tsuruoka et al.[21] is claimed to have faster convergence than OWL-QN. It incorporates
`1 -regularization by using a cumulative `1 penalty, which is close to the `1 penalty received by the
parameter if it had been updated by the true gradient. Tsuruoka et al. do consider data sparsity, i.e.
for each instance, only the parameters related to the current instance are updated. But they too do
not take the model sparsity into account.
Coordinate Descent (CD) and Blockwise Coordinate Descent (BCD) are popular methods for `1 regularized problem. In each coordinate descent iteration, it solves an one-dimensional quadratic
approximation of the objective function, which has a closed-form solution. It requires the second
partial derivative with respect to the coordinate. But as discussed by Sokolovska et al., the exact
second derivative in CRF problem is intractable. So they instead use an approximation of the second
derivative, which can be computed efficiently by the same inference oracle queried for the gradient
evaluation. However, pure CD is very expensive because it requires to call the inference oracle for
the instances related to the current coordinate in each coordinate update. BCD alleviates this problem by grouping the parameters with the same x feature into a block. Then each block update only
6
needs to call the inference oracle once for the instances related to the current x feature. However,
it cannot alleviate the large number of inference oracle calls unless the data is very sparse such that
every instance appears only in very few blocks.
Proximal Newton method has proven successful on problems of `1 -regularized logistic regression
[13] and Sparse Invariance Covariance Estimation [5], where the Hessian-vector product can be
cheaply re-evaluated for each update of coordinate. However, the Hessian-vector product for CI
function like CRF requires the query of the inference oracle no matter how many coordinates are
updated at a time [17], which then makes the coordinate update on quadratic approximation as expensive as coordinate update in the original problem. Our proximal quasi-Newton method avoids
such problem by replacing Hessian with a low-rank matrix from BFGS update.
6
Numerical Experiments
We compare our approach, Prox-QN, with four other methods, Proximal Gradient (Prox-GD), OWLQN [23], SGD [21] and BCD [16]. For OWL-QN, we directly use the OWL-QN optimizer developed by Andrew et al.1 , where we set the memory size as m = 10, which is the same as that in
Prox-QN. For SGD, we implement the algorithm proposed by Tsuruoka et al. [21], and use cumulative `1 penalty with learning rate ?k = ?0 /(1 + k/N ), where k is the SGD iteration and N is
the number of samples. For BCD, we follow Sokolovska et al. [16] but with three modifications.
First, we add a line search procedure in each block update since we found it is required for convergence. Secondly, we apply shrinking strategy as discussed in Section 2.3. Thirdly, when the second
derivative for some coordinate is less than 10?10 , we set it to be 10?10 because otherwise the lack
of `2 -regularization in our problem setting will lead to a very large new iterate.
We evaluate the performance of Prox-QN method on two problems, sequence labeling and hierarchical classification. In particular, we plot the relative objective difference (f (wt )?f (w? ))/f (w? )
and the number of non-zero parameters (on a log scale) against time in seconds. More experiment
results, for example, the testing accuracy and the performance for different ??s, are in Appendix
5. All the experiments are executed on 2.8GHz Intel Xeon E5-2680 v2 Ivy Bridge processor with
1/4TB memory and Linux OS.
6.1
Sequence Labeling
In sequence labeling problems, each instance (x, y) = {(xt , yt )}t=1,2...,T is a sequence of T pairs
of observations and the corresponding labels. Here we consider the optical character recognition
(OCR) problem, which aims to recognize the handwriting words. The dataset 2 was preprocessed by
Taskar et al. [19] and was originally collected by Kassel [20], and contains 6877 words (instances).
We randomly divide the dataset into two part: training part with 6216 words and testing part with 661
words. The character label set Y consists of 26 English letters and the observations are characters
which are represented by images of 16 by 8 binary pixels as shown in Figure 1(a). We use degree
2 pixels as the raw features, which means all pixel pairs are considered. Therefore, the number of
raw features is J = 128 ? 127/2 + 128 + 1, including a bias. For degree 2 features, xtj = 1
only when both pixels are 1 and otherwise xtj = 0, where xtj is the j-th raw feature of xi . For
the feature functions, we use unigram feature functions 1(yt = y, xtj = 1) and bigram feature
functions 1(yt = y, yt+1 = y 0 ) with their associated weights, ?y,j and ?y,y0 , respectively. So
2
w = {?, ?} for ? ? R|Y |?J and ? ? R|Y |?|Y | and the total number of parameters, d = |Y | +
|Y | ? J = 215, 358.
the above feature functions, the potential
function can be specified as,
n Using
o
PT
PT ?1
P?w (y, x) = exp h?, t=1 (eyt xTt )i + h?, t=1 (eyt eTyt+1 )i ,where h?, ?i is the sum of elementwise product and ey ? R|Y | is an unit vector with 1 at y-th entry and 0 at other entries. The gradient
and the inference oracle are given in Appendix 4.1.
In our experiment, ? is set as 100, which leads to a relative high testing accuracy and an optimal
solution with a relative small number of non-zero parameters (see Appendix 5.2). The learning rate
?0 for SGD is tuned to be 2 ? 10?4 for best performance. In BCD, the unigram parameters are
grouped into J blocks according to the x features while the bigram parameters are grouped into one
block. Our proximal quasi-Newton method can be seen to be much faster than the other methods.
1
2
http://research.microsoft.com/en-us/downloads/b1eb1016-1738-4bd5-83a9-370c9d498a03/
http://www.seas.upenn.edu/ taskar/ocr/
7
Sequence?Labelling?nnz?100
Sequence?Labelling?100
BCD
OWL?QN
Prox?GD
Prox?QN
SGD
5
?2
10
nnz
Relative?objective?difference
10
?4
10
BCD
OWL?QN
Prox?GD
Prox?QN
SGD
?6
10
?8
10
(a) Graphical model of OCR
4
10
0
500
1000
time(s)
3
10
1500
0
(b) Relative Objective Difference
500
1000
time(s)
1500
(c) Non-zero Parameters
Figure 1: Sequence Labeling Problem
6.2
Hierarchical Classification
In hierarchical classification problems, we have a label taxonomy, where the classes are grouped
into a tree as shown in Figure 2(a). Here y ? Y is one of the leaf nodes. If we have totally K
classes (number of nodes) and J raw features, then the number of parameters is d = K ? J. Let
W ? RK?J denote the weights. The feature function corresponding to Wk,j is fk,j (y, x) = 1[k ?
Path(y)]xj , wherenk ? Path(y) means
o class k is an ancestor of y or y itself. The potential function is
P
T
?
PW (y, x) = exp
w x where wT is the weight vector of k-th class, i.e. the k-th row
k
k?Path(y)
k
of W . The gradient and the inference oracle are given in Appendix 4.2.
The dataset comes from Task1 of the dry-run dataset of LSHTC13 . It has 4,463 samples, each with
J=51,033 raw features. The hierarchical tree has 2,388 classes which includes 1,139 leaf labels.
Thus, the number of the parameters d =121,866,804. The feature values are scaled by svm-scale
program in the LIBSVM package. We set ? = 1 to achieve a relative high testing accuracy and
high sparsity of the optimal solution. The SGD initial learning rate is tuned to be ?0 = 10 for best
performance. In BCD, parameters are grouped into J blocks according to the raw features.
Hierarchicial?Classification?nnz?1
BCD
OWL?QN
Prox?GD
Prox?QN
SGD
0
10
BCD
OWL?QN
Prox?GD
Prox?QN
SGD
5
10
?2
10
nnz
Relative?objective?difference
Hierarchical?Classification?1
?4
10
4
10
?6
10
2000
(a) Label Taxonomy
4000
6000
time(s)
8000
500
10000
(b) Relative Objective Difference
1000 1500 2000 2500 3000 3500
time(s)
(c) Non-zero Parameters
Figure 2: Hierarchical Classification Problem
As both Figure 1(b),1(c) and Figure 2(b),2(c) show, Prox-QN achieves much faster convergence and
moreover obtains a sparse model in much less time.
Acknowledgement
This research was supported by NSF grants CCF-1320746 and CCF-1117055. P.R. acknowledges
the support of ARO via W911NF-12-1-0390 and NSF via IIS-1149803, IIS-1320894, IIS-1447574,
and DMS-1264033. K.Z. acknowledges the support of the National Initiative for Modeling and
Simulation fellowship
3
http://lshtc.iit.demokritos.gr/node/1
8
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4,843 | 5,385 | Discriminative Metric Learning by
Neighborhood Gerrymandering
Shubhendu Trivedi, David McAllester, Gregory Shakhnarovich
Toyota Technological Institute
Chicago, IL - 60637
{shubhendu,mcallester,greg}@ttic.edu
Abstract
We formulate the problem of metric learning for k nearest neighbor classification
as a large margin structured prediction problem, with a latent variable representing
the choice of neighbors and the task loss directly corresponding to classification
error. We describe an efficient algorithm for exact loss augmented inference, and
a fast gradient descent algorithm for learning in this model. The objective drives
the metric to establish neighborhood boundaries that benefit the true class labels
for the training points. Our approach, reminiscent of gerrymandering (redrawing
of political boundaries to provide advantage to certain parties), is more direct in
its handling of optimizing classification accuracy than those previously proposed.
In experiments on a variety of data sets our method is shown to achieve excellent
results compared to current state of the art in metric learning.
1
Introduction
Nearest neighbor classifiers are among the oldest and the most widely used tools in machine learning. Although nearest neighor rules are often successful, their performance tends to be limited by
two factors: the computational cost of searching for nearest neighbors and the choice of the metric
(distance measure) defining ?nearest?. The cost of searching for neighbors can be reduced with efficient indexing, e.g., [1, 4, 2] or learning compact representations, e.g., [13, 19, 16, 9]. We will not
address this issue here. Here we focus on the choice of the metric. The metric is often taken to be
Euclidean, Manhattan or ?2 distance. However, it is well known that in many cases these choices are
suboptimal in that they do not exploit statistical regularities that can be leveraged from labeled data.
This paper focuses on supervised metric learning. In particular, we present a method of learning a
metric so as to optimize the accuracy of the resulting nearest neighbor classifier.
Existing works on metric learning formulate learning as an optimization task with various constraints
driven by considerations of computational feasibility and reasonable, but often vaguely justified
principles [23, 8, 7, 22, 21, 14, 11, 18]. A fundamental intuition is shared by most of the work
in this area: an ideal distance for prediction is distance in the label space. Of course, that can not
be measured, since prediction of a test example?s label is what we want to use the similarities to
begin with. Instead, one could learn a similarity measure with the goal for it to be a good proxy
for the label similarity. Since the performance of kNN prediction often is the real motivation for
similarity learning, the constraints typically involve ?pulling? good neighbors (from the correct class
for a given point) closer while ?pushing? the bad neighbors farther away. The exact formulation of
?good? and ?bad? varies but is defined as a combination of proximity and agreement between labels.
We give a formulation that facilitates a more direct attempt to optimize for the kNN accuracy as
compared to previous work as far as we are aware. We discuss existing methods in more detail in
section 2, where we also place our work in context.
1
In the kNN prediction problem, given a point and a chosen metric, there is an implicit hidden
variable: the choice of k ?neighbors?. The inference of the predicted label from these k examples is
trivial, by simple majority vote among the associated labels. Given a query point, there can possibly
exist a very large number of choices of k points that might correspond to zero loss: any set of k
points with the majority of correct class will do. We would like a metric to ?prefer? one of these
?good? example sets over any set of k neighbors which would vote for a wrong class. Note that to
win, it is not necessary for the right class to account for all the k neighbors ? it just needs to get
more votes than any other class. As the number of classes and the value of k grow, so does the space
of available good (and bad) example sets.
These considerations motivate our approach to metric learning. It is akin to the common, albeit
negatively viewed, practice of gerrymandering in drawing up borders of election districts so as to
provide advantages to desired political parties, e.g., by concentrating voters from that party or by
spreading voters of opposing parties. In our case, the ?districts? are the cells in the Voronoi diagram
defined by the Mahalanobis metric, the ?parties? are the class labels voted for by the neighbors
falling in each cell, and the ?desired winner? is the true label of the training points associated with
the cell. This intuition is why we refer to our method as neighborhood gerrymandering in the title.
Technically, we write kNN prediction as an inference problem with a structured latent variable being
the choice of k neighbors. Thus learning involves minimizing a sum of a structural latent hinge loss
and a regularizer [3]. Computing structural latent hinge loss involves loss-adjusted inference ? one
must compute loss-adjusted values of both the output value (the label) and the latent items (the set
of nearest neighbors). The loss augmented inference corresponds to a choice of worst k neighbors
in the sense that while having a high average similarity they also correspond to a high loss (?worst
offending set of k neighbors?). Given the inherent combinatorial considerations, the key to such a
model is efficient inference and loss augmented inference. We give an efficient algorithm for exact
inference. We also design an optimization algorithm based on stochastic gradient descent on the
surrogate loss. Our approach achieves kNN accuracy higher than state of the art for most of the data
sets we tested on, including some methods specialized for the relevant input domains.
Although the experiments reported here are restricted to learning a Mahalanobis distance in an explicit feature space, the formulation allows for nonlinear similarity measures, such as those defined
by nonlinear kernels, provided computing the gradients of similarities with respect to metric parameters is feasible. Our formulation can also naturally handle a user-defined loss matrix on labels.
2
Related Work and Discussion
There is a large body of work on similarity learning done with the stated goal of improving kNN
performance. In much of the recent work, the objective can be written as a combination of some
sort of regularizer on the parameters of similarity, with loss reflecting the desired ?purity? of the
neighbors under learned similarity. Optimization then balances violation of these constraints with
regularization. The main contrast between this body of work and our approach here is in the form
of the loss.
A well known family of methods of this type is based on the Large Margin Nearest Neighbor
(LMNN) algorithm [22] . In LMNN, the constraints for each training point involve a set of predefined ?target neighbors? from correct class, and ?impostors? from other classes. The set of target
neighbors here plays a similar role to our ?best correct set of k neighbors? (h? in Section 4). However the set of target neighbors are chosen at the onset based on the euclidean distance (in absence
of a priori knowledge). Moreover as the metric is optimized, the set of ?target neighbors? is not dynamically updated. There is no reason to believe that the original choice of neighbors based on the
euclidean distance is optimal while the metric is updated. Also h? represents the closest neighbors
that have zero loss but they are not necessarily of the same class. In LMNN the target neighbors are
forced to be of the same class. In doing so it does not fully leverage the power of the kNN objective.
The role of imposters is somewhat similar to the role of the ?worst offending set of k neighbors? in
our method (b
h in Section 4). See Figure 2 for an illustration. Extensions of LMNN [21, 11] allow
for non-linear metrics, but retain the same general flavor of constraints. There is another extension
to LMNN that is more aligned to our work [20], in that they lift the constraint of having a static set
of neighbors chosen based on the euclidean distance and instead learn the neighborhood.
2
d
d
b
a
h
b
a
h
e
e
c
c
f
f
x
x
i
g
g
j
i
j
Figure 1: Illustration of objectives of LMNN (left) and our structured approach (right) for k = 3.
The point x of class blue is the query point. In LMNN, the target points are the nearest neighbors of
the same class, which are points a, b and c (the circle centered at x has radius equal to the farthest
of the target points i.e. point b). The LMNN objective will push all the points of the wrong class
that lie inside this circle out (points e, f, h, i, andj), while pulling in the target points to enforce the
margin. For our structured approach (right), the circle around x has radius equal to the distance of
the farthest of the three nearest neighbors irrespective of class. Our objective only needs to ensure
zero loss which is achieved by pushing in point a of the correct class (blue) while pushing out the
point having the incorrect class (point f ). Note that two points of the incorrect class lie inside the
circle (e, andf ), both being of class red. However f is pushed out and not e since it is farther from
x. Also see section 2.
The above family of methods may be contrasted with methods of the flavor as proposed in [23].
Here ?good? neighbors are defined as all similarly labeled points and each class is mapped into a
ball of a fixed radius, but no separation is enforced between the classes. The kNN objective does
not require that similarly labeled points be clustered together and consequently such methods try to
optimize a much harder objective for learning the metric.
In Neighborhood Component Analysis (NCA) [8], the piecewise-constant error of the kNN rule
is replaced by a soft version. This leads to a non-convex objective that is optimized via gradient
descent. This is similar to our method in the sense that it also attempts to directly optimize for
the choice of the nearest neighbor at the price of losing convexity. This issue of non-convexity
was partly remedied in [7], by optimization of a similar stochastic rule while attempting to collapse
each class to one point. While this makes the optimization convex, collapsing classes to distinct
points is unrealistic in practice. Another recent extension of NCA [18] generalizes the stochastic
classification idea to kNN classification with k > 1.
In Metric Learning to Rank (MLR)[14], the constraints involve all the points: the goal is to push
all the correct matches in front of all the incorrect ones. This again is not the same as requiring
correct classification. In addition to global optimization constraints on the rankings (such as mean
average precision for target class), the authors allow localized evaluation criteria such as Precision
at k, which can be used as a surrogate for classification accuracy for binary classification, but is a
poor surrogate for multi-way classification. Direct use of kNN accuracy in optimization objective is
briefly mentioned in [14], but not pursued due to the difficulty in loss-augmented inference. This is
because the interleaving technique of [10] that is used to perform inference with other losses based
inherently on contingency tables, fails for the multiclass case (since the number of data interleavings
could be exponential). We take a very different approach to loss augmented inference, using targeted
inference and the classification loss matrix, and can easily extend it to arbitrary number of classes.
A similar approach is taking in [15], where the constraints are derived from triplets of points formed
by a sample, correct and incorrect neighbors. Again, these are assumed to be set statically as an
input to the algorithm, and the optimization focuses on the distance ordering (ranking) rather than
accuracy of classification.
3
Problem setup
We are given N training examples X = {x1 , . . . , xN }, represented by a ?native? feature map,
xi ? Rd , and their class labels y = [y1 , . . . , yN ]T , with yi ? [R], where [R] stands for the set
3
{1, . . . , R}. We are also given the loss matrix ? with ?(r, r0 ) being the loss incurred by predicting
r0 when the correct class is r. We assume ?(r, r) = 0, and ?(r, r0 ), ?(r, r0 ) ? 0.
We are interested in Mahalanobis metrics
T
DW (x, xi ) = (x ? xi ) W (x ? xi ) ,
(1)
parameterized by positive semidefinite d ? d matrices W. Let h ? X be a set of examples in X.
For a given W we define the distance score of h w.r.t. a point x as
X
SW (x, h) = ?
DW (x, xj )
(2)
xj ?h
Hence, the set of k nearest neighbors of x in X is
hW (x) = argmax SW (x, h).
(3)
|h|=k
For the remainder we will assume that k is known and fixed. From any set h of k examples from X,
we can predict the label of x by (simple) majority vote:
yb (h) = majority{yj : xj ? h},
with ties resolved by a heuristic, e.g., according to 1NN vote. In particular, the kNN classifier
predicts yb(hW (x)). Due to this deterministic dependence between yb and h, we can define the
classification loss incured by a voting classifier when using the set h as
?(y, h) = ? (y, yb(h)) .
4
(4)
Learning and inference
P
One might want to learn W to minimize training loss i ? (yi , hW (xi )). However, this fails due
to the intractable nature of classification loss ?. We will follow the usual remedy: define a tractable
surrogate loss.
Here we note that in our formulation, the output of the prediction is a structured object hW , for
which we eventually report the deterministically computed yb. Structured prediction problems usually involve loss which is a generalization of the hinge loss; intuitively, it penalizes the gap between
score of the correct structured output and the score of the ?worst offending? incorrect output (the
one with the highest score and highest ?).
However, in our case there is no single correct output h, since in general many choices of h would
lead to correct yb and zero classification loss: any h in which the majority votes for the right class.
Ideally, we want SW to prefer at least one of these correct hs over all incorrect hs. This intuition
leads to the following surrogate loss definition:
L(x, y, W) = max [SW (x, h) + ?(y, h)]
h
?
max
h:?(y,h)=0
SW (x, h).
(5)
(6)
This is a bit different in spirit from the notion of margin sometimes encountered in ranking problems
where we want all the correct answers to be placed ahead of all the wrong ones. Here, we only care
to put one correct answer on top; it does not matter which one, hence the max in (6).
5
Structured Formulation
Although we have motivated this choice of L by intuitive arguments, it turns out that our problem is
an instance of a familiar type of problems: latent structured prediction [24], and thus our choice of
loss can be shown to form an upper bound on the empirical task loss ?.
First, we note that the score SW can be written as
*
+
X
T
SW (x, h) = W, ?
(x ? xj )(x ? xj )
,
xj ?h
4
(7)
where h?, ?i stands for the Frobenius inner product. Defining the feature map
X
?(x, h) , ?
(x ? xj )(x ? xj )T ,
(8)
xj ?h
we get a more compact expression hW, ?(x, h)i for (7).
Furthermore, we can encode the deterministic dependence between y and h by a ?compatibility?
function A(y, h) = 0 if y = yb(h) and A(y, h) = ?? otherwise. This allows us to write the joint
inference of y and (hidden) h performed by kNN classifier as
ybW (x), b
hW (x) = argmax [A(y, h) + hW, ?(x, h)i] .
(9)
h,y
This is the familiar form of inference in a latent structured model [24, 6] with latent variable h. So,
despite our model?s somewhat unusual property that the latent h completely determines the inferred
y, we can show the equivalence to the ?normal? latent structured prediction.
5.1
Learning by gradient descent
We define the objective in learning W as
min kWk2F + C
X
W
L (xi , yi , W) ,
(10)
i
where k ? k2F stands for Frobenius norm of a matrix.1 The regularizer is convex, but as in other
latent structured models, the loss L is non-convex due to the subtraction of the max in (6). To
optimize (10), one can use the convex-concave procedure (CCCP) [25] which has been proposed
specifically for latent SVM learning [24]. However, CCCP tends to be slow on large problems.
Furthermore, its use is complicated here due to the requirement that W be positive semidefinite
(PSD). This means that the inner loop of CCCP includes solving a semidefinite program, making
the algorithm slower still. Instead, we opt for a simpler choice, often faster in practice: stochastic
gradient descent (SGD), described in Algorithm 1.
Algorithm 1: Stochastic gradient descent
Input: labeled data set (X, Y ), regularization parameter C, learning rate ?(?)
initialize W(0) = 0
for t = 0, . . ., while not converged do
sample i ? [N ]
b
hi = argmaxh [SW(t) (xi , h) + ?(yi , h)]
h?i = argmaxh:?(yi ,h)=0 SW(t) (xi , h)
#
"
?SW (xi , h?i )
?SW (xi , b
hi )
?
?W =
(t)
?W
?W
W
W(t+1) = (1 ? ?(t))W(t) ? C?W
project W(t+1) to PSD cone
The SGD algorithm requires solving two inference problems (b
h and h? ), and computing the gradient
2
of SW which we address below.
5.1.1
Targeted inference of h?i
Here we are concerned with finding the highest-scoring h constrained to be compatible with a given
target class y. We give an O(N log N ) algorithm in Algorithm 2. Proof of its correctness and
complexity analysis is in Appendix.
1
We discuss other choices of regularizer in Section 7.
We note that both inference problems over h are done in leave one out settings, i.e., we impose an additional
constraint i ?
/ h under the argmax, not listed in the algorithm explicitly.
2
5
Algorithm 2: Targeted inference
Input: x, W, target class y, ? , Jties forbiddenK
Output: argmaxh:by(h)=y SW (x)
Let n? = d k+? (R?1)
e
// min.
R
h := ?
for j = 1, . . . , n? do
h := h ? argmin DW (x, xi )
required number of neighbors from y
xi : yi =y,i?h
/
for l = n? + 1, . . . , k do
define #(r) , |{i : xi ? h, yi = r}| // count selected neighbors from class r
h := h ?
argmin
DW (x, xi )
xi : yi =y, or #(yi )<#(y)??, i?h
/
return h
The intuition behind Algorithm 2 is as follows. For a given combination of R (number of classes)
and k (number of neighbors), the minimum number of neighbors from the target class y required to
allow (although not guarantee) zero loss, is n? (see Proposition 1 in the App. The algorithm first
includes n? highest scoring neighbors from the target class. The remaining k ? n? neighbors are
picked by a greedy procedure that selects the highest scoring neighbors (which might or might not
be from the target class) while making sure that no non-target class ends up in a majority.
When using Alg. 2 to find an element in H ? , we forbid ties, i.e. set ? = 1.
5.1.2
Loss augmented inference b
hi
Calculating the max term in (5) is known as loss augmented inference. We note that
n
o
0
0
0
0
hW,
?(x,
h
)i
+
?(y,
h
)
=
max
max
hW,
?(x,
h
)i
max
+
?(y,
y
)
0
0
0
?
0
h
y
h ?H (y )
(11)
= hW,?(x,h? (x,y0 ))i
which immediately leads to Algorithm 3, relying on Algorithm 2. The intuition: perform targeted
inference for each class (as if that were the target class), and the choose the set of neighbors for the
class for which the loss-augmented score is the highest. In this case, in each call to Alg. 2 we set
? = 0, i.e., we allow ties, to make sure the argmax is over all possible h?s.
Algorithm 3: Loss augmented inference
Input: x, W,target class y
Output: argmaxh [SW (x, h) + ?(y, h)]
for r ? {1, . . . , R} do
h(r) := h? (x, W, r, 1)
Let Value (r) := SW (x, h(r) ), + ?(y, r)
Let r? = argmaxr Value (r)
?
return h(r )
5.1.3
// using Alg. 2
Gradient update
Finally, we need to compute the gradient of the distance score. From (7), we have
X
?SW (x, h)
= ?(x, h) = ?
(x ? xj )(x ? xj )T .
?W
(12)
xj ?h
Thus, the update in Alg 1 has a simple interpretation, illustrated in Fig 2 on the right. For every
xi ? h? \ b
h, it ?pulls? xi closer to x. For every xi ? b
h \ h? , it ?pushes? it farther from x; these push
and pull refer to increase/decrease of Mahalanobis distance under the updated W. Any other xi ,
including any xi ? h? ? b
h, has no influence on the update. This is a difference of our approach from
6
LMNN, MLR etc. This is illustrated in Figure 2. In particular h? corresponds to points a, c and e,
whereas b
h corresponds to points c, e and f . Thus point a is pulled while point f is pushed.
Since the update does not necessarily preserve W as a PSD matrix, we enforce it by projecting W
onto the PSD cone, by zeroing negative eigenvalues. Note that since we update (or ?downdate?)
W each time by matrix of rank at most 2k, the eigendecomposition can be accomplished more
efficiently than the na??ve O(d3 ) approach, e.g., as in [17].
Using first order methods, and in particular gradient methods for optimization of non-convex functions, has been common across machine learning, for instance in training deep neural networks.
Despite lack (to our knowledge) of satisfactory guarantees of convergence, these methods are often
successful in practice; we will show in the next section that this is true here as well.
One might wonder if this method is valid for our objective that is not differentiable; we discuss this
briefly before describing experiments. A given x imposes a Voronoi-type partition of the space of
W into a finite number of cells; each cell is associated with a particular combination of b
h(x) and
h? (x) under the values of W in that cell. The score SW is differentiable (actually linear) on the
interior of the cell, but may be non-differentiable (though continuous) on the boundaries. Since the
boundaries between a finite number of cells form a set of measure zero, we see that the score is
differentiable almost everywhere.
6
Experiments
We compare the error of kNN classifiers using metrics learned with our approach to that with other
learned metrics. For this evaluation we replicate the protocol in [11], using the seven data sets in
Table 1. For all data sets, we report error of kNN classifier for a range of values of k; for each
k, we test the metric learned for that k. Competition to our method includes Euclidean Distance,
LMNN [22], NCA, [8], ITML [5], MLR [14] and GB-LMNN [11]. The latter learns non-linear
metrics rather than Mahalanobis.
For each of the competing methods, we used the code provided by the authors. In each case we tuned
the parameters of each method, including ours, in the same cross-validation protocol. We omit a few
other methods that were consistently shown in literature to be dominated by the ones we compare
to, such as ?2 distance, MLCC, M-LMNN. We also could not include ?2 -LMNN since code for it is
not available; however published results for k = 3 [11] indicate that our method would win against
?2 -LMNN as well.
Isolet and USPS have a standard training/test partition, for the other five data sets, we report the mean
and standard errors of 5-fold cross validation (results for all methods are on the same folds). We
experimented with different methods for initializing our method (given the non-convex objective),
including the euclidean distance, all zeros etc. and found the euclidean initialization to be always
worse. We initialize each fold with either the diagonal matrix learned by ReliefF [12] (which gives a
scaled euclidean distance) or all zeros depending on whether the scaled euclidean distance obtained
using ReliefF was better than unscaled euclidean distance. In each experiment, x are scaled by mean
and standard deviation of the training portion.3 The value of C is tuned on on a 75%/25% split of
the training portion. Results using different scaling methods are attached in the appendix.
Our SGD algorithm stops when the running average of the surrogate loss over most recent epoch no
longer descreases substantially, or after max. number of iterations. We use learning rate ?(t) = 1/t.
The results show that our method dominates other competitors, including non-linear metric learning
methods, and in some cases achieves results significantly better than those of the competition.
7
Conclusion
We propose a formulation of the metric learning for kNN classifier as a structured prediction problem, with discrete latent variables representing the selection of k neighbors. We give efficient algorithms for exact inference in this model, including loss-augmented inference, and devise a stochastic
gradient algorithm for learning. This approach allows us to learn a Mahalanobis metric with an objective which is a more direct proxy for the stated goal (improvement of classification by kNN rule)
3
For Isolet we also reduce dimensionality to 172 by PCA computed on the training portion.
7
k=3
DSLR
Amazon
Webcam
Caltech
800
157
10
800
958
10
800
295
10
800
1123
10
75.20 ?3.0
24.17 ?4.5
21.65 ?4.8
36.93 ?2.6
19.07 ?4.9
31.90 ?4.9
17.18 ?4.7
k=7
DSLR
60.13 ?1.9
26.72 ?2.1
26.72 ?2.1
24.01 ?1.8
33.83 ?3.3
30.27 ?1.3
21.34 ?2.5
?2.5
80.5 ?4.6
46.93 ?3.9
46.11 ?3.9
46.76 ?3.4
48.78 ?4.5
46.66 ?1.8
43.37 ?2.4
62.21
29.23
29.12
23.17
31.42
29.22
22.44
letters
76.45 ?6.2
25.44 ?4.3
25.44 ?4.3
33.73 ?5.5
22.32 ?2.5
36.94 ?2.6
21.61 ?5.9
k = 11
DSLR
5.89 ?0.4
4.09 ?0.1
2.86 ?0.2
15.54 ?6.8
6.52 ?0.8
6.04 ?2.8
3.05 ?0.1
73.87 ?2.8
23.64 ?3.4
23.64 ?3.4
36.25 ?13.1
22.28 ?3.1
40.06 ?6.0
22.28 ?4.9
64.61 ?4.2
30.12 ?2.9
30.07 ?3.0
24.32 ?3.8
30.48 ?1.4
30.69 ?2.9
24.11 ?3.2
Dataset
Isolet
USPS
letters
d
N
C
Euclidean
LMNN
GB-LMNN
MLR
ITML
NCA
ours
170
7797
26
8.66
4.43
4.13
6.61
7.89
6.16
4.87
256
9298
10
6.18
5.48
5.48
8.27
5.78
5.23
5.18
16
20000
26
4.79 ?0.2
3.26 ?0.1
2.92 ?0.1
14.25 ?5.8
4.97 ?0.2
4.71 ?2.2
2.32 ?0.1
Dataset
Isolet
USPS
letters
Euclidean
LMNN
GB-LMNN
MLR
ITML
NCA
ours
7.44
3.78
3.54
5.64
7.57
6.09
4.61
6.08
4.9
4.9
8.27
5.68
5.83
4.9
5.40 ?0.3
3.58 ?0.2
2.66 ?0.1
19.92 ?6.4
5.37 ?0.5
5.28 ?2.5
2.54 ?0.1
Dataset
Isolet
USPS
Euclidean
LMNN
GB-LMNN
MLR
ITML
NCA
ours
8.02
3.72
3.98
5.71
7.77
5.90
4.11
6.88
4.78
4.78
11.11
6.63
5.73
4.98
Amazon
?2.2
?2.0
?2.1
?2.1
?1.9
?2.7
?1.3
Amazon
56.27
15.59
13.56
23.05
13.22
16.27
10.85
?2.2
?1.9
?2.8
?4.6
?1.5
?3.1
Webcam
57.29
14.58
12.45
18.98
10.85
22.03
11.19
?6.3
?2.2
?4.6
?2.9
?3.1
?6.5
?3.3
Webcam
59.66
13.90
13.90
17.97
11.86
26.44
11.19
?5.5
?2.2
?1.0
?4.1
?5.6
?6.3
?4.4
Caltech
80.76
46.75
46.17
46.85
51.74
45.50
41.61
?3.7
?2.9
?2.8
?4.1
?2.8
?3.0
?2.6
Caltech
81.39
49.06
49.15
44.97
50.76
46.48
40.76
?4.2
?2.3
?2.8
?2.6
?1.9
?4.0
?1.8
Table 1: kNN error,for k=3, 7 and 11. Features were scaled by z-scoring. Mean and standard
deviation are shown for data sets on which 5-fold partition was used. Best performing methods are
shown in bold. Note that the only non-linear metric learning method in the above is GB-LMNN.
than previously proposed similarity learning methods. Our learning algorithm is simple yet efficient,
converging on all the data sets we have experimented upon in reasonable time as compared to the
competing methods.
Our choice of Frobenius regularizer is motivated by desire to control model complexity without
biasing towards a particular form of the matrix. We have experimented with alternative regularizers,
both the trace norm of W and the shrinkage towards Euclidean distance, kW ? Ik2F , but found both
to be inferior to kWk2F . We suspect that often the optimal W corresponds to a highly anisotropic
scaling of data dimensions, and thus bias towards I may be unhealthy.
The results in this paper are restricted to Mahalanobis metric, which is an appealing choice for a
number of reasons. In particular, learning such metrics is equivalent to learning linear embedding
of the data, allowing very efficient methods for metric search. Still, one can consider non-linear
embeddings x ? ?(x; w) and define the distance D in terms of the embeddings, for example, as
D(x, xi ) = k?(x) ? ?(xi )k or as ??(x)T ?(xi ). Learning S in the latter form can be seen as
learning a kernel with discriminative objective of improving kNN performance. Such a model would
be more expressive, but also more challenging to optimize. We are investigating this direction.
Acknowledgments
This work was partly supported by NSF award IIS-1409837.
8
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2:1033?1040, 2002.
9
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4,844 | 5,386 | Fundamental Limits of Online and Distributed
Algorithms for Statistical Learning and Estimation
Ohad Shamir
Weizmann Institute of Science
[email protected]
Abstract
Many machine learning approaches are characterized by information constraints
on how they interact with the training data. These include memory and sequential
access constraints (e.g. fast first-order methods to solve stochastic optimization
problems); communication constraints (e.g. distributed learning); partial access
to the underlying data (e.g. missing features and multi-armed bandits) and more.
However, currently we have little understanding how such information constraints
fundamentally affect our performance, independent of the learning problem semantics. For example, are there learning problems where any algorithm which
has small memory footprint (or can use any bounded number of bits from each
example, or has certain communication constraints) will perform worse than what
is possible without such constraints? In this paper, we describe how a single set
of results implies positive answers to the above, for several different settings.
1
Introduction
Information constraints play a key role in machine learning. Of course, the main constraint is the
availability of only a finite data set to learn from. However, many current problems in machine
learning can be characterized as learning with additional information constraints, arising from the
manner in which the learner may interact with the data. Some examples include:
? Communication constraints in distributed learning: There has been much recent work on learning
when the training data is distributed among several machines. Since the machines may work
in parallel, this potentially allows significant computational speed-ups and the ability to cope
with large datasets. On the flip side, communication rates between machines is typically much
slower than their processing speeds, and a major challenge is to perform these learning tasks with
minimal communication.
? Memory constraints: The standard implementation of many common learning tasks requires
memory which is super-linear in the data dimension. For example, principal component analysis
(PCA) requires us to estimate eigenvectors of the data covariance matrix, whose size is quadratic
in the data dimension and can be prohibitive for high-dimensional data. Another example is kernel
learning, which requires manipulation of the Gram matrix, whose size is quadratic in the number
of data points. There has been considerable effort in developing and analyzing algorithms for
such problems with reduced memory footprint (e.g. [20, 7, 27, 24]).
? Online learning constraints: The need for fast and scalable learning algorithms has popularised
the use of online algorithms, which work by sequentially going over the training data, and incrementally updating a (usually small) state vector. Well-known special cases include gradient
descent and mirror descent algorithms. The requirement of sequentially passing over the data
can be seen as a type of information constraint, whereas the small state these algorithms often
maintain can be seen as another type of memory constraint.
1
? Partial-information constraints: A common situation in machine learning is when the available
data is corrupted, sanitized (e.g. due to privacy constraints), has missing features, or is otherwise
partially accessible. There has also been considerable interest in online learning with partial
information, where the learner only gets partial feedback on his performance. This has been
used to model various problems in web advertising, routing and multiclass learning. Perhaps
the most well-known case is the multi-armed bandits problem with many other variants being
developed, such as contextual bandits, combinatorial bandits, and more general models such as
partial monitoring [10, 11].
Although these examples come from very different domains, they all share the common feature
of information constraints on how the learning algorithm can interact with the training data. In
some specific cases (most notably, multi-armed bandits, and also in the context of certain distributed
protocols, e.g. [6, 29]) we can even formalize the price we pay for these constraints, in terms of
degraded sample complexity or regret guarantees. However, we currently lack a general informationtheoretic framework, which directly quantifies how such constraints can impact performance. For
example, are there cases where any online algorithm, which goes over the data one-by-one, must
have a worse sample complexity than (say) empirical risk minimization? Are there situations where
a small memory footprint provably degrades the learning performance? Can one quantify how a
constraint of getting only a few bits from each example affects our ability to learn?
In this paper, we make a first step in developing such a framework. We consider a general class of
learning processes, characterized only by information-theoretic constraints on how they may interact
with the data (and independent of any specific problem semantics). As special cases, these include
online algorithms with memory constraints, certain types of distributed algorithms, as well as online
learning with partial information. We identify cases where any such algorithm must perform worse
than what can be attained without such information constraints. The tools developed allows us to
establish several results for specific learning problems:
? We prove a new and generic
p regret lower bound for partial-information online learning with expert
advice, of the form ?( (d/b)T ), where T is the number of rounds, d is the dimension of the
loss/reward vector, and b is the number of bits b extracted from each loss vector. It is optimal
up to log-factors (without further assumptions), and holds no matter what these b bits are ? a
single coordinate (as in multi-armed bandits), some information on several coordinates (as in
semi-bandit feedback), a linear projection (as in bandit linear optimization), some feedback signal
from a restricted set (as in partial monitoring) etc. Interestingly, it holds even if the online learner
is allowed to adaptively choose which bits of the loss vector it can retain at each round. The lower
bound quantifies directly how information constraints in online learning degrade the attainable
regret, independent of the problem semantics.
? We prove that for some learning and estimation problems - in particular, sparse PCA and sparse
covariance estimation in Rd - no online algorithm can attain statistically optimal performance (in
? 2 ) memory. To the best of our knowledge, this is
terms of sample complexity) with less than ?(d
the first formal example of a memory/sample complexity trade-off in a statistical learning setting.
? We show that for similar types of problems, there are cases where no distributed algorithm (which
is based on a non-interactive or serial protocol on i.i.d. data) can attain optimal performance with
? 2 ) communication per machine. To the best of our knowledge, this is the first formal
less than ?(d
example of a communication/sample complexity trade-off, in the regime where the communication
budget is larger than the data dimension, and the examples at each machine come from the same
underlying distribution.
? We demonstrate the existence of (synthetic) stochastic optimization problems where any algorithm which uses memory linear in the dimension (e.g. stochastic gradient descent or mirror
descent) cannot be statistically optimal.
Related Work
In stochastic optimization, there has been much work on lower bounds for sequential algorithms
(e.g. [22, 1, 23]). However, these results all hold in an oracle model, where data is assumed to be
made available in a specific form (such as a stochastic gradient estimate). As already pointed out in
2
[22], this does not directly translate to the more common setting, where we are given a dataset and
wish to run a simple sequential optimization procedure.
In the context of distributed learning and statistical estimation, information-theoretic lower bounds
were recently shown in the pioneering work [29], which identifies cases where communication constraints affect statistical performance. These results differ from ours (in the context of distributed
learning) in two important ways. First, they pertain to parametric estimation in Rd , where the communication budget per machine is much smaller than what is needed to even specify the answer
with constant accuracy (O(d) bits). In contrast, our results pertain to simpler detection problems,
where the answer requires only O(log(d)) bits, yet lead to non-trivial lower bounds even when the
budget size is much larger (in some cases, much larger than d). The second difference is that their
work focuses on distributed algorithms, while we address a more general class of algorithms, which
includes other information-constrained settings. Strong lower bounds in the context of distributed
learning have also been shown in [6], but they do not apply to a regime where examples across machines come from the same distribution, and where the communication budget is much larger than
what is needed to specify the output.
There are well-known lower bounds for multi-armed bandit problems and other online learning with
partial-information settings. However, they crucially depend on the semantics of the information
feedback considered. For example, the standard multi-armed bandit lower bound [5] pertain to a
setting where we can view a single coordinate of the loss vector, but doesn?t apply as-is when we
can view more than one coordinate (e.g. [4, 25]), get side-information (e.g. [19]), receive a linear
or non-linear projection (as in bandit linear and convex optimization), or receive a different type of
partial feedback (e.g. partial monitoring [11]). In contrast, our results are generic and can directly
apply to any such setting.
Memory and communication constraints have been extensively studied within theoretical computer
science (e.g. [3, 21]). Unfortunately, almost all these results pertain to data which was either adversarially generated, ordered (in streaming algorithms) or split (in distributed algorithms), and do
not apply to statistical learning tasks, where the data is drawn i.i.d. from an underlying distribution.
[28, 15] do consider i.i.d. data, but focus on problems such as detecting graph connectivity and
counting distinct elements, and not learning problems such as those considered here. Also, there are
works on provably memory-efficient algorithms for statistical problems (e.g. [20, 7, 17, 13]), but
these do not consider lower bounds or provable trade-offs.
Finally, there has been a line of works on hypothesis testing and statistical estimation with finite
memory (see [18] and references therein). However, the limitations shown in these works apply
when the required precision exceeds the amount of memory available. Due to finite sample effects,
this regime is usually relevant only when the data size is exponential in the memory size. In contrast,
we do not rely on finite precision considerations.
2
Information-Constrained Protocols
We begin with a few words about notation. We use bold-face letters (e.g. x) to denote vectors, and
let ej ? Rd denote j-th standard basis vector. When convenient, we use the standard asymptotic
?
notation O(?), ?(?), ?(?) to hide constants, and an additional ? sign (e.g. O(?))
to also hide logfactors. log(?) refers to the natural logarithm, and log2 (?) to the base-2 logarithm.
Our main object of study is the following generic class of information-constrained algorithms:
Definition 1 ((b, n, m) Protocol). Given access to a sequence of mn i.i.d. instances (vectors in Rd ),
an algorithm is a (b, n, m) protocol if it has the following form, for some functions ft returning an
output of at most b bits, and some function f :
? For t = 1, . . . , m
? Let X t be a batch of n i.i.d. instances
? Compute message W t = ft (X t , W 1 , W 2 , . . . W t?1 )
? Return W = f (W 1 , . . . , W m )
3
Note that the functions {ft }m
t=1 , f are completely arbitrary, may depend on m and can also be
randomized. The crucial assumption is that the outputs W t are constrained to be only b bits.
As the definition above may appear quite abstract, let us consider a few specific examples:
? b-memory online protocols: Consider any algorithm which goes over examples one-by-one, and
incrementally updates a state vector W t of bounded size b. We note that a majority of online
learning and stochastic optimization algorithms have bounded memory. For example, for linear
predictors, most gradient-based algorithms maintain a state whose size is proportional to the size
of the parameter vector that is being optimized. Such algorithms correspond to (b, n, m) protocols
where W t is the state vector after round t, with an update function ft depending only on W t?1 ,
and f depends only on W m . n = 1 corresponds to algorithms which use one example at a time,
whereas n > 1 corresponds to algorithms using mini-batches.
? Non-interactive and serial distributed algorithms: There are m machines and each machine receives an independent sample X t of size n. It then sends a message W t = ft (X t ) (which here
depends only on X t ). A centralized server then combines the messages to compute an output
f (W 1 . . . W m ). This includes for instance divide-and-conquer style algorithms proposed for distributed stochastic optimization (e.g. [30]). A serial variant of the above is when there are m
machines, and one-by-one, each machine t broadcasts some information W t to the other machines, which depends on X t as well as previous messages sent by machines 1, 2, . . . , (t ? 1).
? Online learning with partial information: Suppose we sequentially receive d-dimensional loss
vectors, and from each of these we can extract and use only b bits of information, where b d.
This includes most types of bandit problems [10].
In our work, we contrast the performance attainable by any algorithm corresponding to such a protocol, to constraint-free protocols which are allowed to interact with the data in any manner.
3
Basic Results
Our results are based on a simple ?hide-and-seek? statistical estimation problem, for which we show
a strong gap between the performance of information-constrained protocols and constraint-free protocols. It is parameterized by a dimension d, bias ?, and sample size mn, and defined as follows:
Definition 2 (Hide-and-seek Problem). Consider the set of product distributions {Prj (?)}dj=1 over
{?1, 1}d defined via Ex?Prj (?) [xi ] = 2? 1i=j for all coordinates i = 1, . . . d. Given an i.i.d. sample
of mn instances generated from Prj (?), where j is unknown, detect j.
In words, Prj (?) corresponds to picking all coordinates other than j to be ?1 uniformly
at random,
and independently picking coordinate j to be +1 with a higher probability 21 + ? . The goal is to
detect the biased coordinate j based on a sample.
First, we note that without information constraints, it is easy to detect the biased coordinate with
O(log(d)/?2 ) instances. This is formalized in the following theorem, which is an immediate consequence of Hoeffding?s inequality and a union bound:
Theorem 1. Consider the hide-and-seek problem defined earlier. Given mn samples, if J? is the
coordinate with the highest empirical average, then Prj (J? = j) ? 1 ? 2d exp ? 21 mn?2 .
We now show that for this hide-and-seek problem, there is a large regime where detecting j is
information-theoretically possible (by Thm. 1), but any information-constrained protocol will fail to
do so with high probability. We first show this for (b, 1, m) protocols (i.e. protocols which process
one instance at a time, such as bounded-memory online algorithms, and distributed algorithms where
each machine holds a single instance):
Theorem 2. Consider the hide-and-seek problem on d > 1 coordinates, with some bias ? ? 1/4
and sample size m. Then for any estimate J? of the biased coordinate returned by any (b, 1, m)
protocol, there exists some coordinate j such that
r
3
?2 b
?
Prj (J = j) ? + 21 m
.
d
d
4
The theorem implies that any algorithm corresponding to (b, 1, m) protocols requires sample size
m ? ?((d/b)/?2 ) to reliably detect some j. When b is polynomially smaller than d (e.g. a constant),
we get an exponential gap compared to constraint-free protocols, which only require O(log(d)/?2 )
instances.
Moreover, Thm. 2 is tight up to log-factors: Consider a b-memory online algorithm, which splits
the d coordinates into O(d/b) segments of O(b) coordinates each, and sequentially goes over the
2
?
segments, each time using O(1/?
) independent instances to determine if one of the coordinates in
each segment is biased by ? (assuming ? is not exponentially smaller than b, this can be done with
O(b) memory by maintaining the empirical average of each coordinate). This will allow to detect
2
?
the biased coordinate, using O((d/b)/?
) instances.
We now turn to provide an analogous result for general (b, n, m) protocols (where n is possibly
greater than 1). However, it is a bit weaker in terms of the dependence on the bias parameter1 :
Theorem 3. Consider the hide-and-seek problem on d > 1 coordinates, with some bias ? ? 1/4n
and sample size mn. Then for any estimate J? of the biased coordinate returned by any (b, n, m)
protocol, there exists some coordinate j such that
s
3
10?b 2
?
Prj (J = j) ? + 5 mn min
,? .
d
d
The
theorem
n implies
othat any (b, n, m) protocol will require a sample size mn which is at least
(d/b) 1
? max
in order to detect the biased coordinate. This is larger than the O(log(d)/?2 )
? , ?2
instances required by constraint-free protocols whenever ? > b log(d)/d, and establishes trade-offs
between sample complexity and information complexities such as memory and communication.
Due to lack of space, all our proofs appear in the supplementary material. However, the technical
details may obfuscate the high-level intuition, which we now turn to explain.
From an information-theoretic viewpoint, our results are based on analyzing the mutual information
between j and W t in a graphical model as illustrated in figure 1. In this model, the unknown message
j (i.e. the identity of the biased coordinate) is correlated with one of d independent binary-valued
random vectors (one for each coordinate across the data instances X t ). All these random vectors
are noisy, and the mutual information in bits between Xjt and j can be shown to be on the order of
n?2 . Without information constraints, it follows that given m instantiations of X t , the total amount
of information conveyed on j by the data is ?(mn?2 ), and if this quantity is larger than log(d), then
there is enough information to uniquely identify j. Note that no stronger bound can be established
with standard statistical lower-bound techniques, since these do not consider information constraints
internal to the algorithm used.
Indeed, in our information-constrained setting there is an added complication, since the output W t
can only contain b bits. If b d, then W t cannot convey all the information on X1t , . . . , Xdt .
Moreover, it will likely convey only little information if it doesn?t already ?know? j. For example,
W t may provide a little bit of information on all d coordinates, but then the amount of information
conveyed on each (and in particular, the random variable Xjt which is correlated with j) will be
very small. Alternatively, W t may provide accurate information on O(b) coordinates, but since
the relevant coordinate j is not known, it is likely to ?miss? it. The proof therefore relies on the
following components:
? No matter what, a (b, n, m) protocol cannot provide more than b/d bits of information (in expectation) on Xjt , unless it already ?knows? j.
? Even if the mutual information between W t and Xjt is only b/d, and the mutual information between Xjt and j is n?2 , standard information-theoretic tools such as the data processing inequality
only implies that the mutual information between W t and j is bounded by min{n?2 , b/d}. We
essentially prove a stronger information contraction bound, which is the product of the two terms
1
The proof of Thm. 2 also applies in the case n > 1, but the dependence on n is exponential - see the proof
for details.
5
?1?
Figure 1:
Illustration of the relationship between j, the coordinates
1, 2, . . . , j, . . . , d of the sample X t , and
the message W t . The coordinates are independent of each other, and most of them
just output ?1 uniformly at random. Only
Xjt has a slightly different distribution and
hence contains some information on j.
?2?
?
?
??
???
?
???
O(?2 b/d) when n = 1, and O(n?b/d) for general n. At a technical level, this is achieved by
considering the relative entropy between the distributions of W t with and without a biased coordinate j, relating it to the ?2 -divergence between these distributions (using relatively recent
analytic results on Csisz?ar f-divergences [16], [26]), and performing algebraic manipulations to
upper bound it by ?2 times the mutual information between W t and Xjt , which is on average b/d
as discussed earlier. This eventually leads to the m?2 b/d term in Thm. 2, as well as Thm. 3 using
somewhat different calculations.
4
4.1
Applications
Online Learning with Partial Information
Consider the setting of learning with expert advice, defined as a game over T rounds, where each
round t a loss vector `t ? [0, 1]d is chosen, and the learner (without knowing `t ) needs to pick an
action it from a fixed set {1, . . . , d}, after which the learner suffers loss `t,it . The goal of the learner
PT
PT
is to minimize the regret with respect to any fixed action i, t=1 `t,it ? t=1 `t,i . We are interested
in variants where the learner only gets some partial information on `t . For example, in multi-armed
bandits, the learner can only view `t,it . The following theorem is a simple corollary of Thm. 2:
Theorem 4. Suppose d > 3. For any
there
over loss
hP(b, 1, T ) protocol,
i is an i.i.d.
n distribution
o
p
PT
T
d
vectors `t ? [0, 1] for which minj E
(d/b)/T , where
t=1 `t,jt ?
t=1 `t,j ? c min T,
c > 0 is a numerical constant.
As a result, we get that for any algorithm with any partial information feedback model (where b
bitspare extracted from each d-dimensional loss vector), it is impossible to get regret lower than
assumptions on the feedback model, the
?( (d/b)T ) for sufficiently large T . Without further
p
bound is optimal up to log-factors, as shown by O( (d/b)T ) upper bounds for linear or coordinate
measurements (where b is the number of measurements or coordinates seen2 ) [2, 19, 25]. However,
the lower bound extends beyond these specific settings, and include cases such as arbitrary nonlinear measurements of the loss vector, or receiving feedback signals of bounded size (although
some setting-specific lower bounds may be stronger). It also simplifies previous lower bounds,
tailored to specific types of partial information feedback, or relying on careful reductions to multiarmed bandits (e.g. [12, 25]). Interestingly, the bound holds even if the algorithm is allowed to
examine each loss vector `t and adaptively choose which b bits of information it wishes to retain.
4.2
Stochastic Optimization
We now turn to consider an example from stochastic optimization, where our goal is to approximately minimize F (h) = EZ [f (h; Z)] given access to m i.i.d. instantiations of Z, whose distribution is unknown. This setting has received much attention in recent years, and can be used to
model many statistical learning problems. In this section, we demonstrate a stochastic optimization
problem where information-constrained protocols provably pay a performance price compared to
non-constrained algorithms. We emphasize that it is a simple toy problem, and not meant to represent anything realistic. We present it for two reasons: First, it illustrates another type of situation
2
Strictly speaking, if the losses are continuous-valued, these require arbitrary-precision measurements, but
in any practical implementation we can assume the losses and measurements are discrete.
6
where information-constrained protocols may fail (in particular, problems involving matrices). Second, the intuition of the construction is also used in the more realistic problem of sparse PCA and
covariance estimation, considered in the next section.
Specifically, suppose we wish to solve min(w,v) F (w, v) = EZ [f ((w, v); Z)], where
f ((w, v); Z) = w> Zv , Z ? [?1, +1]d?d
Pd
Pd
and w, v range over all vectors in the simplex (i.e. wi , vi ? 0 and i=1 wi =
i=1 vi = 1).
A minimizer of F (w, v) is (ei? , ej ? ), where (i? , j ? ) are indices of the matrix entry with minimal mean. Moreover, by a standard concentration of measure argument, given m i.i.d. instan? J)
? =
tiations Z 1 , . P
. . , Z m from any distribution over Z, then the solution (eI?, eJ?), where (I,
m
1
t
arg mini,j m t=1 Zi,j are the indices of the entry with empirically smallest mean, satisfies
p
F (eI?, eJ?) ? minw,v F (w, v) + O
log(d)/m with high probability.
? J)
? as above requires us to track d2 empirical means, which may be exHowever, computing (I,
pensive when d is large. If instead we constrain ourselves to (b, 1, m) protocols where b = O(d)
(e.g. any sort of stochastic gradient method optimization algorithm, whose memory
pis linear in the
number of parameters), then we claim that we have
a
lower
bound
of
?(min{1,
d/m}) on the
p
expected error, which is much higher than the O( log(d)/m) upper bound for constraint-free protocols. This claim is a straightforward consequence of Thm. 2: We consider distributions where
Z ? {?1, +1}d?d with probability 1, each of the d2 entries
p is chosen independently, and E[Z] is
zero except some coordinate (i? p
, j ? ) where it equals O( d/m). For such distributions, getting optimization error smaller than O( d/m) reduces to detecting (i? , j ? ), and this inp
turn reduces to the
hide-and-seek problem defined earlier, over d2 coordinates and a bias ? = O( d/m). However,
Thm. 2 shows that no (b, 1, m) protocol (where b = O(d)) will succeed if md?2 d2 , which
indeed happens if ? is small enough.
Similar kind of gaps can be shown using Thm. 3 for general (b, n, m) protocols, which apply to any
special case such as non-interactive distributed learning.
4.3
Sparse PCA, Sparse Covariance Estimation, and Detecting Correlations
The sparse PCA problem ([31]) is a standard and well-known statistical estimation problem, defined
as follows: We are given an i.i.d. sample of vectors x ? Rd , and we assume that there is some
direction, corresponding to some sparse vector v (of cardinality at most k), such that the variance
E[(v> x)2 ] along that direction is larger than at any other direction. Our goal is to find that direction.
We will focus here on the simplest possible form of this problem, where the maximizing direction v
is assumed to be 2-sparse, i.e. there are only 2 non-zero coordinates vi , vj . In that case, E[(v> x)2 ] =
v12 E[x21 ] + v22 E[x22 ] + 2v1 v2 E[xi xj ]. Following previous work (e.g. [8]), we even assume that
E[x2i ] = 1 for all i, in which case the sparse PCA problem reduces to detecting a coordinate pair
(i? , j ? ), i? < j ? for which xi? , xj ? are maximally correlated. A special case is a simple and natural
sparse covariance estimation problem [9], where we assume that all covariates are uncorrelated
(E[xi xj ] = 0) except for a unique correlated pair (i? , j ? ) which we need to detect.
This setting bears a resemblance to the example seen in the context of stochastic optimization in section 4.2: We have a d ? d stochastic matrix xx> , and we need to detect an off-diagonal biased entry
at location (i? , j ? ). Unfortunately, these stochastic matrices are rank-1, and do not have independent
entries as in the example considered in section 4.2. Instead, we use a more delicate construction,
relying on distributions supported on sparse vectors. The intuition is that then each instantiation of
xx> is sparse, and the situation can be reduced to a variant of our hide-and-seek problem where only
a few coordinates are non-zero at a time. The theorem below establishes performance gaps between
constraint-free protocols (in particular, a simple plug-in estimator), and any (b, n, m) protocol for a
specific choice of n, or any b-memory online protocol (See Sec. 2).
Theorem 5. Consider the class of 2-sparse PCA (or covariance estimation) problems in d ? 9
dimensions as described above, and all distributions such that E[x2i ] = 1 for all i, and:
1. For a unique pair of distinct coordinates (i? , j ? ), it holds that E[xi? xj ? ] = ? > 0, whereas
E[xi xj ] = 0 for all distinct coordinate pairs (i, j) 6= (i? , j ? ).
7
2. For any i < j, if xg
i xj is the
empirical average
of xi xj over m i.i.d. instances, then
?
2
Pr |xg
i xj ? E[xi xj ]| ? 2 ? 2 exp ?m? /6 .
Then the following holds:
? ?
? J)
? = arg maxi<j xg
? ?
? Let (I,
i xj . Then for any distribution as above, Pr((I, J) = (i , j )) ?
2
2
1 ? d exp(?m? /6). In particular, when the bias ? equals ?(1/d log(d)),
m
? ?
2
?
?
Pr((I, J) = (i , j )) ? 1 ? d exp ??
.
d2 log2 (d)
? J)
? of (i? , j ? ) returned by any b-memory online protocol using m instances,
? For any estimate (I,
m
c) protocol, there exists a distribution with bias ? = ?(1/d log(d))
or any (b, d(d ? 1), b d(d?1)
as above such that
r
m
? J)
? = (i? , j ? ) ? O 1 +
Pr (I,
.
d2
d4 /b
The theorem implies that in the regime where b d2 / log2 (d), we can choose any m such that
2
d4
2
? ?
b m d log (d), and get that the chances of the protocol detecting (i , j ) are arbitrarily
? ?
small, even though the empirical average reveals (i , j ) with arbitrarily high probability. Thus, in
this sparse PCA / covariance estimation setting, any online algorithm with sub-quadratic memory
cannot be statistically optimal for all sample sizes. The same holds for any (b, n, m) protocol in an
appropriate regime of (n, m), such as distributed algorithms as discussed earlier.
To the best of our knowledge, this is the first result which explicitly shows that memory constraints
can incur a statistical cost for a standard estimation problem. It is interesting that sparse PCA was
also shown recently to be affected by computational constraints on the algorithm?s runtime ([8]).
The proof appears in the supplementary material. Besides using a somewhat different hide-andseek construction as mentioned earlier, it also relies on the simple but powerful observation that
any b-memory online protocol is also a (b, ?, bm/?c) protocol for arbitrary ?. Therefore, we only
need to prove the theorem for (b, ?, bm/?c) for some ? (chosen to equal d(d ? 1) in our case) to
automatically get the same result for b-memory protocols.
5
Discussion and Open Questions
In this paper, we investigated cases where a generic type of information-constrained algorithm has
strictly inferior statistical performance compared to constraint-free algorithms. As special cases,
we demonstrated such gaps for memory-constrained and communication-constrained algorithms
(e.g. in the context of sparse PCA and covariance estimation), as well as online learning with
partial information and stochastic optimization. These results are based on explicitly considering
the information-theoretic structure of the problem, and depend only on the number of bits extracted
from each data batch.
Several questions remain open. One question is whether Thm. 3 can be improved. We conjecture
this is true, and that the bound should actually depend on mn?2 b/d rather than mn min{?b/d, ?2 }.
This would allow, for instance, to show the same type of performance gaps for (b, 1, m) protocols
and (b, n, m) protocols. A second open question is whether there are convex stochastic optimization
problems, for which online or distributed algorithms are provably inferior to constraint-free algorithms (the example discussed in section 4.2 refers to an easily-solvable yet non-convex problem). A
third open question is whether our results for distributed algorithms can be extended to more interactive protocols, where the different machines can communicate over several rounds. There is a rich
literature on the subject within theoretical computer science, but it is not clear how to ?import? these
results to a statistical setting based on i.i.d. data. A fourth open question is whether the performance
gap that we demonstrated for sparse-PCA / covariance estimation can be extended to a ?natural?
distribution (e.g. Gaussian), as our result uses a tailored distribution, which has a sufficiently controlled tail behavior but is ?spiky? and not sub-Gaussian uniformly in the dimension. More generally,
it would be interesting to extend the results to other learning problems and information constraints.
Acknowledgements: This research is supported by the Intel ICRI-CI Institute, Israel Science Foundation grant 425/13, and an FP7 Marie Curie CIG grant. We thank John Duchi, Yevgeny Seldin and
Yuchen Zhang for helpful comments.
8
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9
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4,845 | 5,387 | Optimal rates for k-NN density and mode estimation
Sanjoy Dasgupta
University of California, San Diego, CSE
[email protected]
Samory Kpotufe ?
Princeton University, ORFE
[email protected]
Abstract
We present two related contributions of independent interest: (1) high-probability
finite sample rates for k-NN density estimation, and (2) practical mode estimators
? based on k-NN ? which attain minimax-optimal rates under surprisingly general
distributional conditions.
1
Introduction
We prove finite sample bounds for k-nearest neighbor (k-NN) density estimation, and subsequently
apply these bounds to the related problem of mode estimation. These two main results, while related,
are interesting on their own.
First, k-NN density estimation [1] is one of the better known and simplest density estimation procedures. The estimate fk (x) of an unknown density f (see Definition 1 of Section 3) is a simple
n
functional of the distance rk (x) from x to its k-th nearest neighbor in a sample X[n] , {Xi }i=1 .
As such it is intimately related to other functionals of rk (x), e.g. the degree of vertices x in k-NN
graphs and their variants used in modeling communities and in clustering applications (see e.g. [2]).
While this procedure has been known for a long time, its convergence properties are still not fully
understood. The bulk of research in the area has concentrated on establishing its asymptotic convergence, while its finite sample properties have received little attention in comparison. Our finite
sample bounds are concisely derived once the proper tools are identified. The bounds hold with high
probability, under general conditions on the unknown density f . This generality proves quite useful
as shown in our subsequent application to the problem of mode estimation.
The basic problem of estimating the modes (local maxima) of an unknown density f has also been
studied for a while (see e.g. [3] for an early take on the problem). It arises in various unsupervised
problems where modes are used as a measure of typicality of a sample X. In particular, in modern
applications, mode estimation is often used in clustering, with the modes representing cluster centers
(see e.g. [4, 5] and general applications of the popular mean-shift procedure).
While there exists a rich literature on mode estimation, the bulk of theoretical work concerns estimators of a single mode (highest maximum of f ), and often concentrates on procedures that are
hard to implement in practice. Given the generality of our first result on k-NN density estimation,
we can prove that some simple implementable procedures yield optimal estimates of the modes of
an unknown density f , under surprisingly general conditions on f .
Our results are overviewed in the following section, along with an overview of the rich literature on
k-NN density estimation and mode estimation. This is followed by our theoretical setup in Section 3;
our rates for k-NN density estimation are detailed in Section 4, while the results on mode estimation
are given in Section 5.
?
Much of this work was conducted when this author was at TTI-Chicago.
1
2
2.1
Overview of results and related Work
Rates for k-NN density estimates
The k-NN density estimator dates back perhaps to the early work of [1] where it is shown to be
consistent when the unknown density f is continuous on Rd . While one of the best known and
simplest procedure for density estimation, it has proved more cumbersome to analyze than its smooth
counterpart, the kernel density estimator.
More general consistency results such as [6, 7] have been established since its introduction. In
particular [6] shows that, for f Lipschitz in a neighborhood of a point x, where f (x) > 0, and
k = k(n) satisfying k ? ? and k/n2/(2+d) ? 0, the estimator is asymptotically normal, i.e.
?
D
k(fk (x) ? f (x))/f (x) ?? N (0, 1). The recent work of [8], concerning generalized weighted
variants of k-NN, shows that asymptotic normality holds under the weaker restriction k/n4/(4+d) ?
0 if f is twice differentiable at x.
Asymptotic normality as stated above yields
? some insight into the rate of convergence of fk : we
can expect that |fk (x) ? f (x)| . f (x)/ k under the stated conditions on k. In fact, [8] shows
that such a result can be obtained in expectation for n = n(x) sufficiently large. In particular,
their conditions on k allows for a setting of k ? n4/(4+d) (not allowed under the above conditions)
2
yielding a minimax-optimal l2 risk E |fk (x) ? f (x)| . f (x)2 /k = O(n?4/(4+d) ).
While consistency results and bounds on expected error are now well understood, we still don?t have
a clear understanding of the conditions under which high probability bounds on |fk (x) ? f (x)| are
possible. This is particularly important given the inherent instability of nearest neighbors estimates
which are based on order-statistics rather than the more stable average statistics at the core of kerneldensity estimates. The recent result of [9] provides an initial answer: they obtain a high-probability
bound uniformly over x taking value in the sample X[n] , however under conditions not allowing for
optimal settings of k (where f is assumed Lipschitz).
The bounds in the present paper hold with high-probability, simultaneously for all x in the support
of f . Rather than requiring smoothness conditions on f , we simply give the bounds in terms of the
modulus of continuity of f at any x, i.e. how much f can change in a neighborhood of x. This
allows for a useful degree of flexibility in applying these bounds. In particular, optimal bounds
under various degrees of smoothness of f at x easily follow. More importantly, for our application
to mode estimation, the bounds allow us to handle |fk (x) ? f (x)| at different x ? Rd with varying
smoothness in f . As a result we can derive minimax-optimal mode estimation rates for practical
procedures under surprisingly weak assumptions.
2.2
Mode estimation
There is an extensive literature on mode estimation and we unfortunately can only overview some
of the relevant work. Most of the literature covers the case of a unimodal distribution, or one where
there is a single maximizer x0 of f .
Early work on estimating the (single) mode of a distribution focused primarily on understanding the consistency and rates achievable by various approaches, with much less emphasis on the
ease of implementation of these approaches. The common approaches consist of estimating x0 as
x
? , arg supx?Rd fn (x) where fn is an estimate of f , usually a kernel density estimate. Various
work such as [3, 10, 11] establish consistency properties of the approach and achievable rates under
various Euclidean settings and regularity assumptions on the distribution F. More recent work such
as [12, 13] address the problem of optimal choice of bandwidth and kernel to adaptively achieve
the minimax risk for mode estimation. Essentially, under smoothness ? (e.g. f is ? times differentiable), the minimax risk (inf x? supf Ef k?
x ? x0 k) is of the form n?(??1)/(2?+d) , as independently
established in [14] and [15].
As noticed early in [16], the estimator arg supx?Rd fn (x), while yielding much insight into the
problem, is hard to implement in practice. Hence, other work, apparently starting with [16, 14]
have looked into so-called recursive estimators of the (single) mode which are practical and easy
to update as the sample size increases. These approaches can be viewed as some form of gradient2
ascent of fn with carefully chosen step sizes. The later versions of [14] are shown to be minimaxoptimal. Another line of work is that of so-called direct mode estimators which estimate the mode
from practical statistics of the data [17, 18]. In particular, [18] shows that the simple and practical
estimator arg maxx?X[n] fn (x), where fn is a kernel-density estimator, is a consistent estimator of
the mode. We show in the present paper that arg maxx?X[n] fk (x), where fk is a k-NN density
estimator, is not only consistent, but converges at a minimax-optimal rate under surprisingly mild
distributional conditions.
The more general problem of estimating all modes of distribution has received comparatively little
attention. The best known practical approach for this problem is the mean-shift procedure and its
variants [19, 4, 20, 21], quite related to recursive-mode-estimators, as they essentially consist of
gradient ascent of fn starting from every sample point, where fn is required to be appropriately
smooth to ascend (e.g. a smooth kernel estimate). While mean-shift is popular in practice, it has
proved quite difficult to analyze. A recent result of [22] comes close to establishing the consistency
of mean-shift, as it establishes the convergence of the procedure to the right gradient lines (essentially the ascent path to the mode) if it is seeded from fixed starting points rather than the random
samples themselves. It remains unclear however whether mean-shift produces only true modes,
given the inherent variability in estimating f from sample. This question was recently addressed by
[23] which proposes a hypothesis test to detect false modes based on confidence intervals around
Hessians estimated at the modes returned by any procedure.
Interestingly, while a k-NN density estimate fk is far from smooth, in fact not even continuous, we
show a simple practical procedure that identifies any mode of the unknown density f under mild
conditions: we mainly require that f is well approximated by a quadratic in a neighborhood of
each mode. Our finite sample rates (on k?
x ? x0 k, for an estimate x
? of any mode x0 ) are of the
form O(k ?1/4 ), hold with high-probability and are minimax-optimal for an appropriate choice of
k = ?(n4/(4+d) ).
If in addition f is Lipschitz or more generally H?older-continuous (in principle uniform continuity
of f is enough), all the modes returned above a level set ? of fk can be optimally assigned to
n??
separate modes of the unknown f . Since ? ????? 0, the procedure consistently prunes false modes.
This feature is made intrinsic to the procedure by borrowing from insights of [9, 24] on identifying
false clusters by inspecting levels sets of fn . These last works concern the related area of level set
estimation, and do not study mode estimation rates.
As alluded to so far, our results are given in terms of local assumptions on modes rather than
global distributional conditions. We show that any mode that is sufficiently salient (this is locally
parametrized) w.r.t. the finite sample size n, is optimally estimated, while false modes are pruned
away. In particular our results allow for f having a countably infinite number of modes.
3
Preliminaries
n
Throughout the analysis, we assume access to a sample X[n] = {Xi }i=1 drawn i.i.d. from an
absolutely continuous distribution F over Rd , with Lebesgue-density function f . We let X denote
the support of the density function f .
The k-NN density estimate at a point x is defined as follows.
Definition 1 (k-NN density estimate). For every x ? Rd , let rk (x) denote the distance from x to its
k-th nearest neighbor in X[n] . The density estimate is given as:
fk (x) ,
k
,
n ? vd ? rk (x)d
where vd denotes the volume of the unit sphere in Rd .
All balls considered in the analysis are closed Euclidean balls of Rd .
3
4
k-NN density estimation rates
In this section we bound the error in estimating f (x) as fk (x) at every x ? X . The main results of
the section are Lemmas 3 and 4. These lemmas are easily obtained given the right tools: uniform
concentration bounds on the empirical mass of balls in Rd , using relative Vapnik-Chervonenkis
bounds, i.e. Bernstein?s type bounds rather than Chernoff type bounds (see e.g. Theorem 5.1 of
[25]). We next state a form of these bounds for completion.
Lemma 1. Let G be a class of functions from X to {0, 1} with VC dimension d < ?, and P a
probability distribution on X . Let E denote expectation with respect to P. Suppose n points are
drawn independently at random from P; let En denote expectation with respect to this sample. Then
for any ? > 0, with probability at least 1 ? ?, the following holds for all g ? G:
p
p
p
p
? min(?n En g, ?n2 + ?n Eg) ? Eg ? En g ? min(?n2 + ?n En g, ?n Eg),
p
where ?n = (4/n)(d ln 2n + ln(8/?)).
These sort of relative VC bounds allows for a tighter relation (than Chernoff type bounds) between
empirical and true mass of sets (En g and Eg) in those situations where these quantities are small,
?
above. This is particularly useful since the balls we have to deal
i.e. of the order of ?n2 = O(1/n)
with are those containing approximately k points, and hence of (small) mass approximately k/n.
A direct result of the above lemma is the following lemma of [26]. This next lemma essentially
reworks Lemma 1 above into a form we can use more directly. We re-use C?,n below throughout
the analysis.
?
Lemma 2 ([26]). Pick 0 < ? < 1. Let C?,n , 16 log(2/?) d log n. Assume k ? d log n. With
probability at least 1 ? ?, for every ball B ? Rd we have,
?
d log n
F(B) ? C?,n
=? Fn (B) > 0,
? n
k
k
k
F(B) ? + C?,n
=? Fn (B) ? , and
n
n ?
n
k
k
k
F(B) ? ? C?,n
=? Fn (B) < .
n
n
n
The main idea in bounding fk (x) is to bound the random term rk (x) in terms of f (x) using Lemma
2 above. We can deduce from the lemma that if a ball B(x, r) centered has mass roughly k/n, then
its empirical mass is likely to be of the order k/n; hence rk (x) is likely to be close to the radius r
of B(x, r). Now if f does not vary too much in B(x, r), then we can express the mass of B(x, r) in
terms of f (x), and thus get our desired bound on rk (x) and fk (x) in terms of f (x).
Our results are given in terms of how f varies in a neighborhood of x, captured as follows.
n
o
Definition 2. For x ? Rd and > 0, define r?(, x) , sup r : supkx?x0 k?r f (x0 ) ? f (x) ? ,
n
o
and r?(, x) , sup r : supkx?x0 k?r f (x) ? f (x0 ) ? .
The continuity parameters r?(, x) and r?(, x) (related to the modulus of continuity of f at x) are easily bounded under smoothness assumptions on f at x. Our high-probability bounds on the estimates
fk (x) in terms of f (x) and the continuity parameters are given as follows.
2
Lemma 3 (Upper-bound on fk ). Suppose k ? 4C?,n
. Then, with probability at least 1 ? ?, for all
x ? Rd and all > 0,
C?,n
fk (x) < 1 + 2 ?
(f (x) + ) ,
k
provided k satisfies vd ? r?(, x)d ? (f (x) + ) ?
k
n
4
?
? C?,n
k
n .
Lemma 4 (Lower-bound on fk ). Then, with probability at least 1 ? ?, for all x ? Rd and all > 0,
C?,n
fk (x) ? 1 ? ?
(f (x) ? ) ,
k
provided k satisfies vd ? r?(, x)d ? (f (x) ? ) ?
k
n
?
+ C?,n
k
n .
The proof of these results are concise applications of Lemma 2 above. They are given in the appendix
(long version). The trick is in showing that, under the conditions on k, there exists an r ? (k/(n ?
f (x)))1/d which is at most r?(, x) or r?(, x) as appropriate; hence, f does not vary much on B(x, r)
so we must have
k
F (B(x, r)) ? volume (B(x, r)) ? f (x) = vd ? rd ? f (x) ? .
n
?
Using Lemma 2 we get rk (x) ? r; plug this value into fk (x) to obtain fk (x) ? (1 + 1/ k)f (x).
Lemmas 3 and 4 allow a great deal of flexibility as we will soon see with their application to mode
estimation. In particular we can consider various smoothness conditions simultaneously at different
x for different biases .
Suppose for instance that f is locally H?older at x, i.e. ?r, L, ? > 0 s.t. for all x0 ?
0 ?
B(x, r), |f (x) ? f (x0 )| ? L
?kx ? x k . Then for small , both r?(, x) and r?(, x) are at least
1/?
(/L) ; pick = O(f (x)/
? k) for n sufficiently large, then by both lemmas
? we have, w.h.p.,
|fk (x) ? f (x)| ? O(f (x)/ k) provided k = ?(log2 n) and satisfies vd (1/L k)d/? f (x) ? Ck/n
for some constant C. This allows for a setting of k = ? n2?/(2?+d) for a minimax-optimal rate
of |fk (x) ? f (x)| = O n??/(2?+d .
The ability to consider various biases would prove particularly helpful in the next section on
mode estimation where we have to consider different approximations in different parts of space with
varying smoothness in f . In particular, at a mode x, we will essentially have ? = 2 (f is twice
differentiable) while elsewhere on X we might not have much smoothness in f .
5
Mode estimation
We start with the following definition of modes.
Definition 3. We denote the set of modes of f by M ? {x : ?r > 0, ?x0 ? B(x, r), f (x0 ) < f (x)} .
We need the following assumption at modes.
Assumption 1. f is twice differentiable in a neighborhood of every x ? M. We denote the gradient
and Hessian of f by ?f and ?2 f . Furthermore, ?2 f (x) is negative definite at all x ? M.
Assumption 1 excludes modes at the boundary of the support of f (where f cannot be continuously
differentiable). We note that most work on the subject consider only interior modes as we are
doing here. Modes on the boundary can however be handled under additional boundary smoothness
assumptions to ensure that f puts sufficient mass on any ball around such modes. This however only
complicates the analysis, while the main insights remain the same as for interior modes.
An implication of Assumption 1 is that for all x ? M, ?f is continuous in a neighborhood of x,
with ?f (x) = 0. Together with ?2 f (x) ? 0 (i.e. negative definite), f is well-approximated by a
quadratic in a neighborhood of a mode x ? M. This is stated in the following lemma.
Lemma 5. Let f satisfy Assumption 1. Consider any x ? M. Then there exists a neighborhood
B(x, r), r > 0, and constants C?x , C?x > 0 such that, for all x0 ? B(x, r), we have
2
2
C?x kx0 ? xk ? f (x) ? f (x0 ) ? C?x kx0 ? xk .
(1)
We can therefore parametrize a mode x ? M locally as follows:
Definition 4 (Critical radius rx around mode x). For every mode x ? M, there exists rx > 0, such
that B(x, rx ) is contained in a set Ax , satisfying the following conditions:
(i) Ax is a connected component of a level set X ? , {x0 ? X : f (x0 ) > ?} for some ? > 0.
2
2
(ii) ?C?x , C?x > 0, ?x0 ? Ax , C?x kx0 ? xk ? f (x) ? f (x0 ) ? C?x kx0 ? xk . (So Ax ? M = {x}.)
5
Return arg maxx?X[n] fk (x).
Figure 1: Estimate the mode of a unimodal density f from X[n] .
Figure 2: The analysis argues over different regions (depicted) around a mode x.
Finally, we assume that every hill in f corresponds to a mode in M:
Assumption 2. Each connected component of any level set X ? , ? > 0, contains a mode in M.
5.1
Single mode
We start with the simple but common assumption that |M| = 1. This case has been extensively
studied to get a handle on the inherent difficulty of mode estimation. The usual procedures in the
statistical literature are known to be minimax-optimal but are not practical: they invariably return the
maximizer of some density estimator (usually a kernel estimate) over the entire space Rd . Instead
we analyze the practical procedure of Figure 1 where we pick the maximizer of fk out of the finite
sample X[n] . The rates of Theorem 1 are optimal (O(n?1/(4+d) )) for a setting of k = O(n4/(4+d) ).
Theorem 1. Let ? > 0. Assume f has a single mode x0 and satisfies Assumptions 1, 2. There exists
Nx0 ,? such that the following holds for n ? Nx0 ,? . Let C?x0 , C?x0 be as in Definition 4. Suppose k
satisfies
!4d/(4+d)
!2
s
4/(4+d)
24C?,n f (x0 )
1 C?,n
(2d+4)/(4+d) vd
f
(x
)
?
k
?
n
.
(2)
0
2 C?x0
4
C?x0 rx20
Let x be the mode returned in the procedure of Figure 1. With probability at least 1 ? 2? we have
s
C?,n
1
f (x0 ) ? 1/4 .
kx ? x0 k ? 5
k
C?x
0
Proof. Let rx0 be the critical radius of Definition 4. Let rn (x0 ) ? inf r : B(x0 , r) ? X[n] 6= ? .
Let 0 < ? < 1 to be later specified, and assume the event that rn (x0 ) ? ?2 rx0 . We will bound the
probability of this event once the proper setting of ? becomes clear.
Consider r? satisfying rx0 ? r? ? 2rn (x0 )/? (see Figure 2). We will first upper bound fk for any x
outside B(x0 , r?), then lower-bound fk for x ? B(x0 , rn (x0 )).
Recall Ax0 from Definition 4. By equation (1) we have
sup
f (x) ? f (x0 ) ? C?x0 (?
r/2)2 , F? .
(3)
x?Ax0 \B(x0 ,?
r /2)
The above allows us to apply Lemma 3 as follows. First note that for any x ? X \B(x0 , r?/2), f (x) ?
F? since Ax0 is a level set of the unimodal f , i.e. supx?A
/ x0 f (x) ? inf x?Ax0 f (x). Therefore, for
. ?
any x ? X \ B(x0 , r?) let = F ? f (x). By equation (3) the modulus of continuity r?(, x) is at least
6
Initialize: Mn ? ?.
For ? = maxx?X[n] fn (x) down to 0:
?
? Let ? , ? ? C?,n / k.
n om
be the CCs of G (? ? ? ? ?) disjoint from Mn .
? Let A?i
i=1
n
om
? Mn ? Mn ? xi , arg maxx?A?i ?X ? fn (x)
.
[n]
i=1
Return the estimated modes Mn .
Figure 3: Estimate the modes of a multimodal f from X[n] . The parameter ? serves to prune.
r?/2. Therefore, if k satisfies
?
k
k
2
?
vd ? (?
r/2) ? f (x0 ) ? Cx0 (?
r/2) ? ? C?,n
,
n
n
we have with probability at least 1 ? ?
C?,n
sup
fk (x) < 1 + 2 ?
f (x0 ) ? C?x0 (?
r/2)2 .
k
x?X \B(x0 ,?
r)
d
(4)
(5)
Now we turn to x ? B(x0 , rn (x0 )). We have again by equation (1) that inf x?B(x,? r?) f (x) ?
f (x0 ) ? C?x0 (? r?)2 , F? . Therefore, for x ? B(x0 , rn (x0 )) let = f (x) ? F? , we have r?(, x) ?
? r? ? rn (x0 ) ? ? r?/2. It follows that, if k satisfies
?
k
k
d
,
(6)
vd ? ((? /2)?
r) ? f (x0 ) ? C?x0 (? r?)2 ? + C?,n
n
n
we have by Lemma 4 that, with probability at least 1 ? ? (under the same event used in Lemma 3)
C?,n
?
inf
fk (x) ? 1 ?
f (x0 ) ? C?x0 (? r?)2 .
(7)
x?B(x,rn (x0 ))
k
Next, with a bit of algebra, we can pick ? and r? so that the l.h.s. of (5) is less
? than the l.h.s.
2
2
?
?
?
of equation (7). It suffices to pick ? = Cx0 /8Cx0 and r? ? 24f (x0 )C?,n /Cx0 k. Given these
settings, equations (4) and (6) are satisfied whenever k satisfies equation (2) of the lemma statement.
It follows that, with probability at least 1 ? ?, inf x?B(x,rn (x0 )) fk (x) > supx?X \B(x0 ,?r) fk (x).
Therefore, ther
empirical mode chosen by the procedureis in B(x0 , r?). We are free to choose r? as
?
small as max
24f (x0 )C?,n / C?x0 k , 2rn (x0 )/? .
We?ve assumed so far the event that rn (x0 ) ? ?2 rx0 . We bound the probability of this event as
q
?
follows. Let r , 24f (x0 )C?,n /C?x0 k. Under the above setting of ? , the Theorem?s assumptions
?
d
on k imply that r ? rx0 , and that vd ? ((? /2)r) ? f (x0 ) ? C?x0 ((? /2)r)2 ? nk + C?,n nk . Again,
?
by equation (1), this implies that F(B(x0 , (? /2)r)) ? nk + C?,n nk . By Lemma, 2, with probability
at least 1 ? ?, Fn (B(x0 , (? /2)r)) ? k/n and therefore rn (x0 ) ? (? /2)r ? (? /2)rx0 . It now
becomes clear that we can just pick r? = r.
5.2
Multiple modes
In this section we turn to the problem of estimating the modes of a more general density f with an
unknown number of modes.
The algorithm of Figure 3 operates on the following set of nested graphs G(?). These are subgraphs
of a mutual k-NN graph on the sample X[n] , where vertices are connected if they are in each other?s
nearest neighbor sets. The connected components (CCs) of these graphs G(?) are known to be good
estimates of the CCs of corresponding level sets of the unknown density f [9, 26, 27].
7
Definition
Given ? ? R, let G(?) denote the graph with vertices in
5 (k-NN level set G(?)).
?
X[n] , x ? X[n] : fn (x) ? ? , and where vertices x, x0 are connected by an edge when and only
?
when kx ? x0 k ? ? ? min {rk (x), rk (x0 )}, for some ? ? 2.
We will show that for a given n, any sufficiently salient mode is optimally recovered; furthermore,
if f is uniformly continuous on Rd , then the procedure returns no false mode above a level ?n ? 0.
5.2.1
Optimal Recovery for Any Mode
The guarantees of this section would be given in terms of salient modes as defined below. Essentially
a mode x0 is salient if it is separated from other modes by a sufficiently wide and deep valley.
We define saliency in a way similar to [9], but simpler: we only require a wide valley since the
smoothness of f at the mode (as expressed in equation 1) takes care of the depth.
We start with a notion of separation between sets inspired from [26].
Definition 6 (r-separation). A, A0 ? X are r-separated if there exists a (separating) set S ? Rd
such that: every path from A to A0 crosses S, and supx?S+B(0,r) f (x) < inf x?A?A0 f (x).
Our notion of mode saliency follows: for a mode x, we require the critical set Ax of Definition 4 to
be well separated from all components at the level where it appears.
Definition 7 (r-salient Modes). A mode x of f is said to be r-salient for r > 0 if the following
holds. There exist Ax as in Definition 4 (with the corresponding rx , C?x and C?x ), which is a CC of
say X ?x , {x ? X : f (x) ? ?x }. Ax is r-separated from X ?x \ Ax .
The next theorem again yields the optimal rates O(n?1/(4+d) ) for k = O(n4/(4+d) ).
Theorem 2 (Recovery of salient modes). Assume f satisfies Assumptions 1, 2. Suppose
? =
n??
2
?(n) ????? 0. Let x0 be an r-salient mode for some r > 0. Assume k = ? C?,n . Then
there exist N = N (x0 , {?
(n)}) depending on x0 and ?(n) such that the following holds for n ? N .
?
?
Let Ax0 , Cx0 , Cx0 be as in Definition 4, and let ?x0 , inf x?Ax0 f (x). Let ? > 0. Suppose k further
satisfies
!4d/(4+d)
!2
s
4/(4+d)
24C?,n f (x0 )
1 C?,n
(2d+4)/(4+d) vd
.
?
n
?
k
?
x
0
2 C?x0
4
C?x0 min rx20 /4, (r/?)2
Let Mn be the modes returned by the procedure of Figure 3. With probability at least 1 ? 2?, there
exists x ? Mn such that
s
C?,n
1
kx ? x0 k ? 5
f (x0 ) ? 1/4 .
?
k
C x0
5.2.2
Pruning guarantees
The proof of the main theorem of this section is based on Lemma 7.4 of [24].
Theorem 3. Let ? , supx f (x) and r() , supx?Rd max {?
r(, x), r?(, x)}. Assume f satisfies
1/d
Assumption 2. Suppose r(?
) = ? (k/n) , which is feasible whenever f is uniformly continuous
on Rd . In particular, if f is H?older continuous, i.e.
?x, x0 ? Rd ,
?
|f (x) ? f (x0 )| ? L kx ? x0 k , for some L > 0, 0 < ? ? 1,
?/d
since r(?
) ? (?
/L)1/? . Define
)
? !
? 2
k
k
2
+ C?,n
?0 = max 2?
, 8 C?,n ,
.
k
n
n
vd r(?
)d
then we can just let ? = ? (k/n)
(
2
Assume k ? 9C?,n
. The following holds with probability at least 1 ? ?. Pick any ? ? 2?0 , and
?
let ?f = inf x?X ? f (x). All estimated modes in Mn ? X[n]
can be assigned to distinct modes in
[n]
M ? X ?f .
8
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9
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4,846 | 5,388 | Learning on graphs using Orthonormal
Representation is Statistically Consistent
Chiranjib Bhattacharyya
Department of CSA
Indian Institute of Science
Bangalore, 560012, INDIA
[email protected]
Rakesh S
Department of Electrical Engineering
Indian Institute of Science
Bangalore, 560012, INDIA
[email protected]
Abstract
Existing research [4] suggests that embedding graphs on a unit sphere can be beneficial in learning labels on the vertices of a graph. However the choice of optimal
embedding remains an open issue. Orthonormal representation of graphs, a class
of embeddings over the unit sphere, was introduced by Lov?asz [2]. In this paper,
we show that there exists orthonormal representations which are statistically consistent over a large class of graphs, including power law and random graphs. This
result is achieved by extending the notion of consistency designed in the inductive
setting to graph transduction. As part of the analysis, we explicitly derive relationships between the Rademacher complexity measure and structural properties
of graphs, such as the chromatic number. We further show the fraction of vertices
of a graph G, on n nodes, that need to be labelled for the learning algorithm to be
14
consistent, also known as labelled sample complexity, is ? ?(G)
where ?(G)
n
is the famous Lov?asz ? function of the graph. This, for the first time, relates labelled sample complexity to graph connectivity properties, such as the density of
graphs. In the multiview setting, whenever individual views are expressed by a
graph, it is a well known heuristic that a convex combination of Laplacians [7]
tend to improve accuracy. The analysis presented here easily extends to Multiple graph transduction, and helps develop a sound statistical understanding of the
heuristic, previously unavailable.
1
Introduction
In this paper we study the problem of graph transduction on a simple, undirected graph G = (V, E),
with vertex set V = [n] and edge set E ? V ?V . We consider individual vertices to be labelled with
binary values, ?1. Without loss of generality we assume that the first f n vertices are labelled, i.e.,
the set of labelled vertices is given by S = [f n], where f ? (0, 1). Let S? = V \S be the unlabelled
vertex set, and let yS and yS? be the labels corresponding to subgraphs S and S? respectively.
? ? Rn , such that erS0-1
Given G and yS , the goal of graph transduction is to learn predictions y
y] =
? [?
P
? = sgn(?
?j , y
y) is small. To aid further discussion we introduce some notations.
? 1 yj 6= y
j?S
Notation Let S n?1 = {u ? Rn |kuk2 = 1} denote a (n ? 1) dimensional sphere. Let Dn , Sn
and S+
n denote a set of n ? n diagonal, square symmetric and square symmetric positive semidefinite matrices respectively. Let Rn+ be a non-negative orthant. Let 1n ? Rn denote a vector
of all 1?s. Let [n] := {1, . . . , n}. For any M ? Sn , let ?1 (M) ? . . . ? ?n (M) denote the
eigenvalues and Mi denote the ith row of M, ?i ? [n]. We denote the adjacency matrix of a
graph G by A. Let di denote the degree of vertex i ? [n], di := A>
i 1n . Let D ? Dn , where
1
1
1
Dii = di , ?i ? [n]. We refer I ? D? 2 AD? 2 as the Laplacian, where I denotes the identity matrix.
? = 1n 1n > ? I ? A. For
? denote the complement graph of G, with the adjacency matrix A
Let G
n
K ? S+
and
y
?
{?1}
,
the
dual
formulation
of
Support
vector
machine
(SVM) is given by
n
P
n
n
P
?(K, y) = max??Rn+ g(?, K, y) =
?i ? 21
?i ?j yi yj Kij . Let Y = Y? = {?1}, Yb ? R
i=1
i,j=1
be the label, prediction and soft-prediction spaces over V . Given a graph G and labels y ? Y n
P
b
on V , let cut(A, y) :=
yi 6=yj Aij . We use ` : Y ? Y ? R+ to denote any loss function.
0-1
b let ` (a, b) = 1[ab < 0], `hinge (a, b) = (1 ? ab)+ 1 and
In particular, for a ? Y, b ? Y,
ramp
`
(a, b) = min(1, (1 ? ab)+ ) denote the 0-1, hinge and ramp loss respectively. The notations
O, o, ?, ? will denote standard measures defined in asymptotic analysis [14].
Motivation Regularization framework is a widely used tool for learning labels on the vertices of
a graph [23, 4]
1 X
?
minn
`(yi , y?i ) + ??
y> K?1 y
(1)
? ?Y |S|
y
i?S
where K is a kernel matrix and ? > 0 is an appropriately chosen regularization parameter. It was
? ? satisfies the following generalization bound
shown in [4] that the optimal y
tr (K) p
p
? + c2
y] + ??
y> K?1 y
ES erS0-1
y? ] ? c1 inf n erV [?
? [?
? ?Y
y
?|S|
1/p
P
Pn
(?)
1
(?)
?i ), H ? V 2 ; trp (K) = n1 i=1 Kpii
where erH [?
y] := |H|
, p > 0 and
i?H ` (yi , y
c1 , c2 are dependent on `. [4] argued that for good generalization, trp (K) should be a constant,
which motivated them to normalize the diagonal entries of K. It is important to note that the set of
normalized kernels is quite big and the above presented analysis gives little insight in choosing the
optimal kernel from such a set.
The important problem of consistency erS? ? 0 as n ? ?, to be
formally defined in Section 3 of graph transduction algorithms was introduced in [5]. [5] showed
that the formulation (1), when
q used
with a laplacian dependent kernel, achieves a generalization
q
3
error of ES [erS? [?
y? ]] = O
nf , where q is the number of pure components . Though [5]?s
algorithm is consistent for a small number of pure components, they achieve the above convergence
rate by choosing ? dependent on true labels of the unlabeled nodes, which is not practical [6].
In this paper, we formalize the notion of consistency of graph transduction algorithms and derive
novel graph-dependent statistical estimates for the following formulation.
X
X
1
?C (K, yS ) = min
minn ?> K? + C
` y?i , yi + C
` y?j , y?j
(2)
?
?
??R
? j ?Y,j?S
y
+ 2
?
i?S
P
j?S
P
where y?k = i?S Kik yi ?i + j?S? Kjk y?j ?j , ?k ? V . If all the labels are observed then [22]
showed that the above formulation is equivalent to (1). We note that the normalization step considered by [4] is equivalent to finding an embedding of a graph on a sphere. Thus, we study orthonormal representations of graphs [2], which define a rich class of graph embeddings on an unit
sphere. We show that the formulation (2) working with orthonormal representations of graphs is
statistically consistent over a large class of graphs, including random and power law graphs. In the
sequel, we apply Rademacher complexity to orthonormal representations of graphs and derive novel
graph-dependent transductive error bound. We also extend our analysis to study multiple graph
transduction. More specifically, we make the following contributions.
Contributions The main contribution of this paper is that we show there exists orthonormal representations of graphs that are statistically consistent on a large class of graph families Gc . For a
special orthonormal representation?LS labelling, we show consistency on Erd?os R?enyi random
graphs. Given a graph G ? Gc , with a constant fraction of nodes labelled f = O(1), we derive
1
(a)+ = max(a, 0).
? , when implicit from the context.
We drop the argument y
3
Pure component is a connected subgraph, where all the nodes in the subgraph have the same label.
2
2
an error convergence rate of erS0-1
? = O
?(G)
n
14
, with high probability; where ?(G) is the Lov?asz
pq
? function of the graph G. Existing work [5] showed an expected convergence rate of O
n ,
however q is dependent on the true labels of the unlabelled nodes. Hence their bound cannot be
computed explicitly [6]. We also apply Rademacher complexity measure to the function class associated with orthonormal representations and derive a tight bound relating to ?(G), the chromatic
number of the graph G. We show that the Laplacian inverse [4] has O(1) complexity on graphs with
1
high connectivity, whereas LS labelling exhibits a complexity of ?(n 4 ). Experiments demonstrate
superior performance of LS labelling on several real world datasets. We derive novel transductive
error bound, relating to graph structural measures. Using our analysis, we show that observing labels
41
fraction of the nodes is sufficient to achieve consistency. We also propose an effiof ? ?(G)
n
cient Multiple Kernel Learning (MKL) based algorithm, with generalization guarantees for multiple
graph transduction. Experiments demonstrate improved performance in combining multiple graphs.
2
Preliminaries
Orthonormal Representation: [2] introduced the idea of orthonormal representations for the problem of embedding a graph on a unit sphere. More formally, an orthonormal representation of a
simple, undirected graph G = (V, E) with V = [n], is a matrix U = [u1 , . . . , un ] ? Rd?n such
that uTi uj = 0 whenever (i, j) ?
/ E and ui ? S d?1 ?i ? [n].
Let Lab(G)denote the set of all possible orthonormal
representations of the graph G given by
Lab(G) := U|U is an Orthonormal Representation . [1] recently introduced the notion of graph
embedding to Machine Learning community and showed connections to graph kernel matrices. Con/ E}.
sider the set of graph kernels K(G) := {K ? S+
n |Kii = 1, ?i ? [n]; Kij = 0, ?(i, j) ?
[1] showed that for every valid kernel K ? K(G), there exists an orthonormal representation
U ? Lab(G); and it is easy to see the other way, K = U> U ? K(G). Thus, the two sets,
Lab(G) and K(G) are equivalent. Orthonormal representation is also associated
with an interesting
quantity, the Lov?asz number [2], defined as: ?(G) = 2 minK?K(G) ?(K, 1n ) [1]. ? function is a
fundamental tool for combinatorial optimization and approximation algorithms for graphs.
? ?? G
? ? ?(G);
Lov?asz Sandwich
Theorem:
[2]
Given
an
undirected
graph
G
=
(V,
E),
I
G
? is the independent number of the complement graph G.
?
where I G
3
Statistical Consistency of Graph Transduction Algorithms
In this section, we formalize the notion of consistency of graph transduction algorithms. Given
a graph Gn = (Vn , En ) of n nodes, with labels of subgraph Sn ? Vn observable, let erS??n :=
y] denote the minimal unlabelled node set error. Consistency is a measure of the
inf y? ?Y? n erS?n [?
? are the predictions made
quality of the learning algorithm A, comparing erS?n [?
y] to er?S?n , where y
by A. A related notion of loss consistency has been extensively studied in literature [3, 12], which
only show that the difference erS?n [?
y] ? erSn [?
y] ? 0 as n ? ? [6]. This does not confirm the
optimality of A, that is erS?n [?
y] ? erS??n . Hence, a notion stronger than loss consistency is needed.
Let Gn belong to a graph family G, ?n. Let ?f be the uniform distribution over the random draw of
the labelled subgraph Sn ? Vn , such that |Sn | = f n, f ? (0, 1). As discussed earlier, we want the
`-regret, RSn [A] = erS?n [?
y] ? erS??n to be small. Since, the labelled nodes are drawn randomly, there
is a small probability that one gets an unrepresentative subgraph Sn . However, for large n, we want
`-regret to be close to zero with high probability4 . In other words, for every finite and fixed n, we
want to have an estimate on the `-regret, which decreases as n increases. We define the following
notion of consistency of graph transduction algorithms to capture this requirement
Definition 1. Let G be a graph family and f ? (0, 1) be fixed. Let V = {(vi , yi , Ei )}?
i=1 be an
infinite sequence of labelled node set, where yi ? Y and Ei is the edge information of node vi
with the previously observed nodes v1 , . . . , vi?1 , ?i ? 2. Let Vn be the first n nodes in V, and let
4
If G is not deterministic (e.g., Erd?os R?eyni), then there is small probability that one gets an unrepresentative
graph, in which case we want the `-regret to be close to zero with high probability over Gn ? G.
3
Gn ? G be the graph defined by (Vn , E1 , . . . , En ). Let Sn ? Vn , and let yn , ySn be the labels of
? is
Vn , Sn respectively. A learning algorithm A when given Gn and ySn returns soft-predictions y
said to be `-consistent w.r.t G if, when the labelled subgraph Sn are random drawn from ?f , the
`-regret converges in probability to zero, i.e., ? > 0
PrSn ??f [RSn [A] ? ] ? 0
as
n??
In Section 6 we show that the kernel learning style algorithm (2) working with orthonormal representations is consistent on a large class of graph families. To the best of our knowledge, we are
not aware of any literature which provide an explicit empirical error convergence rate and prove
consistency of the graph transduction algorithm considered. Before we prove our main result, we
gather useful tools?a) complexity measure, which reacts to the structural properties of the graph
(Section 4); b) generalization analysis to bound erS? (Section 5). In the interest of space, we defer
most of the proofs to the supplementary material5 .
4
Graph Complexity Measures
In this section we apply Rademacher complexity to orthonormal representations of graphs, and relate
to the chromatic number. In particular, we study LS labelling, whose class complexity can be shown
to be greater than that of the Laplacian inverse on a large class of graphs.
Let (2) be solved for K ? K(G), and let U ? Lab(G) be the orthonormal representation corresponding to K (Section 2). Then by Representer?s theorem, the classifier learnt by (2) is of the
form h = U?, ? ? Rn . We define Rademacher complexity of the function class associated with
orthonormal representations
Definition 2(Rademacher Complexity).
Given a graph G = (V, E), with V = [n]; let U ? Lab(G)
? U = h|h = U?, ? ? Rn be the function class associated with U. For p ? (0, 1/2], let
and H
? = (?1 , . . . , ?n ) be a vector of i.i.d. random variables such that ?i ? {+1, ?1, 0} w.p. p, p and
? U is given by
1 ? 2p respectively. The Rademacher complexity of the graph G defined by U, H
h
i
n
P
? U , p) = 1 E? suph?H?
R(H
?i hh, ui i
n
U
i=1
The above definition is motivated from [9, 3]. This is an empirical complexity measure, suited for
the transductive settings. We derive the following novel tight Rademacher bound
Theorem
4.1. Let G = (V, E)
= [n], U ? Lab(G) and
be a simple, undirected graph with
? V
p ? 1/n, 1/2 . Let HU = h h = U?, ? ? Rn , k?k2 ? tC n , C > 0, t ? [0, 1] and let
K = U> U ? K(G) be the graph-kernel corresponding
to U. The Rademacher
complexity
of graph
p
?
?
G defined by U is given by R(HU , p) = c0 tC p?1 (K), where 1/2 2 ? c0 ? 2 is a constant.
The above result provides a lower bound for the Rademacher complexity for any unit sphere graph
embedding. While upper-bounds maybe available [9, 3] but, to the best of our knowledge, this is the
first attempt at establishing lower bounds. The use of orthonormal representations allow us to relate
class complexity measure to graph-structural properties.
p
Corollary 4.2. For C, t, p = O(1), R(HU , p) = O( ?(G)). (Suppl.)
Such connections between learning theory complexity measures and graph properties was previously
unavailable [9,p3]. Corollary 4.2 suggests that there exists graph regularizers with class complexity
as large as O( ?(G)), which motivate us to find substantially better regularizers. In particular, we
investigate LS labelling [16]; given a graph G, LS labelling KLS ? K(G) is defined as
KLS =
A
+ I, ? ? |?n (A)|
?
(3)
LS labellinghas high Rademacher complexity on a large class of graphs, in particular
Corollary 4.3. For a random graph G(n, q), q ? [0, 1), where each edge is present independently
w.p. q; for C, t, q = O(1) the Rademacher complexity of the function class associated with LS
1
labelling (3) is ?(n 4 ), with high probability. (Suppl.)
5
mllab.csa.iisc.ernet.in/rakeshs/nips14/suppl.pdf
4
For the limiting case of complete graphs, we can show that Laplacian inverse [4], the most widely
used graph regularizer has O(1) complexity (Claim 2, Suppl.), thus indicating that it may be suboptimal for graphs with high connectivity. Experimental results illustrate our observation.
We derive a class complexity measure for unit sphere graph embeddings, which indicates the richness of the function class, and helps the learning algorithm to choose an effective embedding.
5
Generalization Error Bound
In the previous section, we applied Rademacher complexity to orthonormal representations. In
this section we derive novel graph-dependent generalization error bounds, which will be used in
Section 6. Following a similar proof technique as in [3], we propose the following error bound?
Theorem 5.1. Given a graph G = (V, E), V = [n] with y ? Y n being the unknown binary labels
? U = {h|h = U?, ? ?
over V ; let U ? Lab(G), and K ? K(G) be the corresponding kernel. Let H
Rn , k?k? ? C}, C > 0. Let ` be any loss function, bounded in [0, B] and L-Lipschitz in its
second argument. For f ? (0, 1/2]6 , let labels of subgraph S ? V be observable, |S| = nf . Let
? U , with probability ? 1 ? ? over S ? ?f
S? = V \S. For any ? > 0 and h ? H
s
r
2?1 (K)
1
c1 B
1
y] ? erS [?
y] + LC
erS? [?
+
log
(4)
f (1 ? f ) 1 ? f nf
?
? = U> h and c1 > 0 is a constant. (Suppl.)
where y
Discussion Note that from [2], ?1 (K) ? ?(G) and ?(G) is in-turn bounded by the maximum
degree of the graph [21]. Thus, if ?
L, B, f = O(1), then for sparse, degree bounded graphs; for
the choice of parameter C = ?(1/ n), the slack term and the complexity term goes to zero as n
increases. Thus, making the bound useful. Examples include tree, cycle, path, star and d-regular
(with d = O(1)). Such connections relating generalization error to graph properties was not available before. We exploit this novel connection to analyze graph transduction algorithms in Section 6.
Also, in Section 7, we extend the above result to the problem of multiple graph transduction.
5.1
Max-margin Orthonormal Representation
To analyze erS0-1 relating to graph structural measure, the ? function, we study the maximum margin
induced by any orthonormal representation, in an oracle setting.
We study a fully ?labelled graph? G = (V, E, y), where y ? Y n are the binary labels on the vertices
V . Given any U ? Lab(G), the maximum margin classifier is computed by solving ?(K, y) =
g(?? , K, y) where K = U> U ? K(G). It is interesting to note that knowing all the labels, the
max-margin orthonormal representation can be computed by solving an SDP. More formally
Definition 3. Given aS
labelled graph G = (V, E, y), where V = [n] and y ? Y n are the binary
?
? U , where H
? U = {h|h = U?, ? ? Rn }. Let K ? K(G) be
labels on V , let H = U?Lab(G) H
the kernel corresponding to U ? Lab(G). The max-margin orthonormal representation associated
with the kernel function is given by Kmm = argminK?K(G) ?(K, y).
By definition, Kmm induces the largest margin amongst other orthonormal representations, and
hence is optimal. The optimal margin has interesting connections to the Lov?asz ? function ?
Theorem 5.2. Given a labelled graph G = (V, E, y), with V = [n] and y ? Y n being the binary
labels on vertices. Let Kmm be as in Definition 3, then ?(Kmm , y) = ?(G)/2. (Suppl.)
Thus, knowing all the labels, computing Kmm is equivalent to solving the ? function. However,
in the transductive setting, Kmm cannot be computed. Alternatively, we explore LS labelling (3),
which gives a constant factor approximation to the optimal margin on a large class of graphs.
Definition 4. A class of labelled graphs G = {G = (V, E, y)} is said to be a Labelled SVM-? graph
family, if there exist a constant ? > 1 such that ?G ? G, ?(KLS , y) ? ??(Kmm , y).
6
We can generalize our result for f ? (0, 1), but for the simplicity of the proof we assume f ? (0, 1/2].
This is also true in practice, where the number of labelled examples is usually very small.
5
Algorithm 1
Input: U, yS and C > 0.
? S?? by solving ?C (K, yS ) (2) for `hinge and K = U> U.
Get ?? , y
? = U> hS , where hS = UY?? ; Y ? Dn , Y = yi , if i ? S, otherwise y?i? .
Return: y
Such class of graphs are interesting, because one can get a constant factor approximation to the
optimal margin, without the knowledge of the
? true labels e.g., Mixture of random graphs: G =
(V, E, y), with y> 1n = 0, cut(A, y) ? c n, for c > 1 being a constant and the subgraphs
corresponding to the two classes form G(n/2, 1/2) random graphs (Claim 3, Suppl.).
We relate the maximum geometric margin induced by orthonormal representations to the ? function
of the graph. This allows us to derive novel graph dependent learning theory estimates.
6
Consistency of Orthonormal Representation of Graphs
Aggregating results from Section 4 and 5, we show that Algorithm 1 working with orthonormal
representations of graphs is consistent on a large class of graph families. For every finite and fixed
n, we derive an estimate on erS0-1
?n .
? be the predictions
Theorem 6.1. For the setting as in Definition 1, let f ? (0, 1/2] be fixed. Let y
14
?2 (Gn )(1?f )
?
learnt by Algorithm 1 with inputs Un ? Lab(Gn ), ySn and C = 23 n2 f ? G?
. Then ?Un ?
( n)
Lab(Gn ), ?Gn such that with probability atleast 1 ? n1 over Sn ? ?f
s
!
41
?(G
)
1
log n
n
0-1
y] = O
+
erS?n [?
f 3 (1 ? f )n
1?f
nf
Proof. Let Kn ? K(Gn ) be the max-margin kernel associated with Gn (Definition 3), and let
Un ? Lab(G) be the corresponding orthonormal representation. Since `ramp is an upper bound on
`0-1 , we concentrate on bounding erSramp
[?
y]. Note that for any C > 0
?n
C|Sn | ? erSramp
[?
y] ? C|Sn | ? erShinge
[?
y] ? ?C (Kn , ySn )
n
n
?(Gn )
2
The last inequality follows from Theorem 5.2. Note that for ramp loss L = B = 1; using Theorem 5.1 for ? = n1 , it follows that with probability atleast 1 ? n1 over the random draw of Sn ? ?f ,
s
s
?(Gn )
2?1 (Kn )
c1
log n
ramp
erS?n [?
y] ?
+C
+
(5)
2Cnf
f (1 ? f ) 1 ? f
nf
? ?C (Kn , yn ) ? ?(Kn , yn ) =
? n ) [2] and optimizing RHS for C, we get C ? =
where c1 = O(1). Using ?1 (Kn ) ? ?(G
2
41
? (Gn )(1?f )
? n = n [2] proves the claim.
. Plugging back C ? and using ?(Gn )? G
?n)
23 n2 f ?(G
pq
[5] showed that ES erS?n = O
n . However, as noted in Section 1, the quantity q is dependent
on yS?n , and hence their bounds cannot be computed explicitly [6].
We assume that the graph does not contain duplicate nodes with opposite labels, erS??n = 0. Thus,
consistency follows from the fact that ?(G) ? n and for large families of graphs it is O(nc ) where
0 ? c < 1. This theorem points to the fact that if f = O(1), then by Definition 1, Algorithm 1 is
`0-1 - consistency over such class of graph families. Examples include
Power-law graphs:
Graphs where the degree sequence follows a power law distribution. We show
? = O(?n) for naturally occurring power law graphs (Claim 4, Suppl.). Thus, working
that ?(G)
? , makes Algorithm 1 consistent.
with the complement graph G
6
?
Random graphs: For G(n, q) graphs, q = O(1); with high probability ?(G(n, q)) = ?( n) [13].
Note that choosing Kn for various graph families is difficult. Alternatively, for Labelled SVM-?
graph family (Definition 4), if Lov?asz ? function is sub-linear, then for the choice of LS labelling,
Algorithm 1 is `0-1 consistent. Examples include Mixture of random graphs (Section 5.1). Furthermore, we analyze the fraction of labelled nodes to be observed, such that Algorithm 1 is consistent.
Corollary 6.2 (Labelled Sample Complexity). Given a graph family Gc , such that ?(Gn ) =
1/3??
n)
O(nc ), ?Gn ? Gc where 0 ? c < 1. For C = C ? as in Theorem 6.1; 21 ?(G
, ?>0
n
fraction of labelled nodes is sufficient for Algorithm 1 to be `0-1 -consistent w.r.t. Gc .
The proof directly follows from Theorem 6.1. As a consequence of the above result, we can argue that for sparse graphs (?(G) is large) one would need a larger fraction of nodes labelled, but
for denser graphs (?(G) is small) even a smaller fraction of nodes being labelled suffices. Such
connections relating sample complexity and graph properties was not available before.
To end this section, we discuss on the possible extensions to the inductive setting (Claim 5, Suppl.)?
we can show that that the uniform convergence of erS? to erS in the transductive setting (for f = 1/2)
is a necessary and sufficient condition for the uniform convergence of erS to the generalization error.
Thus, the results presented here can be extended to the supervised setting. Furthermore, combining
Theorem 5.1 with the results of [9], we can also extend our results to the semi-supervised setting.
7
Multiple Graph Transduction
Many real world problems can be posed as learning on multiple graphs [19, ?]. Existing algorithms
for single graph transduction [10, 15] cannot be trivially extended to the new setting. It is a well
known heuristic that taking a convex combination of Laplacian improves classification performance
[7], however the underlying principle is not well understood. We propose an efficient MKL style
algorithm with generalization guarantees. Formally, the problem of multiple graph transduction is?
Problem 1. Given G = {G(1) , . . . , G(m) } a set of simple, undirected graphs G(k) = V, E (k) ,
defined on a common vertex set V = [n]. Without loss of generality we assume that the first f n
vertices are labelled, i.e., the set of labelled vertices is given by S = [f n], where f ? (0, 1). Let
S? = V \S be the unlabelled node set. Let yS , yS? be the labels of S, S? respectively. Given G and
labels yS , the goal is to accurately predict the labels of yS? .
Let K = {K(1) , . . . , K(m) } be the set of kernels corresponding to graphs G; K(k) ? K(G(k) ), ?k ?
[m]. We propose the following MKL style formulation for multiple graph transduction
X
m
(k)
?
]
(6)
?C (K, yS ) =
min
min
max
g
?,
?
K
,
[y
,
y
?
k
S
S
m
n
?
? ??R ,k?k? ?C
??R+ ,k?k1 =1 y?j ?Y,?j?
S
+
k=1
Extending our analysis from Section 5, we propose the following error bound
Theorem 7.1. For the setting as in Problem 1, let f ? (0, 1/2]7 and K =
? S?? be the solution to ?C (K, yS )
{K(1) , . . . , K(m) }, K(k) ? K(G(k) ), ?k ? [m]. Let ?? , ? ? , y
m
P ? (k) ? ?
? ? Dn , Y
? ii = yi if i ? S, otherwise y?? . Then, for any
? =
(6). Let y
?k K Y? , where Y
i
i=1
? > 0, with probability ? 1 ? ? over the choice of S ? V such that |S| = nf
s
r
?
??)
?(K,
y)
2?(G
c1
1
1
0-1
erS? [?
y] ?
+C
+
log
Cnf
f (1 ? f ) 1 ? f nf
?
?
where c1 = O(1), ?(K,
y) = mink?[m] ?(K(k) , y) and G? is the union of graphs G8 . (Suppl.)
The above result gives us the ability for the first time to analyze generalization performance of multi?
ple graph transduction algorithms. The expression ?(K,
y) suggests that combining multiple graphs
should improve performance over considering individual graphs separately. Similar to Section 6,
7
8
As in Theorem 5.1, we can generalize our results for f ? (0, 1).
G? = (V, E ? ), where (i, j) ? E ? if edge (i, j) is present in atleast one of the graphs G(k) ? G, k ? [m].
7
(l)
we can show that if one of the graph families G (l) , l ? [m] of G obey ?(Gn ) = O(nc ), 0 ? c < 1;
(l)
Gn ? G (l) , then there exists orthonormal representations K, such that the MKL style algorithm
optimizing for (6) is `0-1 -consistent over G (Claim 6, Suppl.). We can also show that combining
graphs improves labelled sample complexity (Claim 7, Suppl.). This is a first attempt in developing
a statistical understanding for the problem of multiple graph transduction.
8
Experimental results
Table 1: Superior performance of LS labelling.
We conduct two sets of experiments9 .
LS-lab Un-Lap N-Lap KS-Lap
Superior performance of LS labelling: We Dataset
?
?
AuralSonar
76.5
68.1
66.7
69.2
use two datasets?similarity matrices from
?
54.1
52.9
53.3
[11] and RBF kernel10 as similarity matrices for Yeast-SW-5-7 ? 60.4
61.2
60.5
64.3
the UCI datasets? [8]. We built an unweighted Yeast-SW-5-12? 78.6
Yeast-SW-7-12
76.5
64.0
59.5
63.1
graph by thresholding the similarity matrices
?
73.1
68.3
68.6
68.5
about the mean. Let L = D ? A. For the reg- Diabetes
?
73.3
69.3
71.2
71.8
ularized formulation (1), with 10% of labelled Fourclass
nodes observable, we test four types of kernel
matrices?LS labelling(LS-lab), (?1 I + L)?1 (Un-Lap), (?2 I + D?1/2 LD?1/2 )?1 (N-Lap) and
K-Scaling (KS-Lap) [4]. We choose the parameters ?, ?1 and ?2 by cross validation. Table 1
summarizes the results. Each entry is accuracy in % w.r.t. 0-1 loss, and the results were averaged
over 100 iterations. Since we are thresholding by mean, the graphs have high connectivity. Thus,
from Corollary 4.3, the function class associated with LS labellingis rich and expressive, and hence
it outperforms previously proposed regularizers.
Graph transduction across Multiple-views: Table 2: Multiple Graphs Transduction.
Learning on mutli-view data has been of recent Each entry is accuracy in %.
interest [18]. Following a similar line of attack, we
pose the problem of classification on multi-view Graph 1vs2 1vs3 1vs4 2vs3 2vs4 3vs4
data as multiple graph transduction. We investigate Aud 62.8 64.8 68.3 59.3 50.8 61.5
68.9 65.6 68.9 69.1 70.3 75.1
the recently launched Google dataset [17], which Vis
68.7 59.2 64.8 64.6 60.9 65.4
contains multiple views of video game YouTube Txt
videos, consisting of 13 feature types of auditory Unn 69.7 60.3 52.7 62.7 67.4 62.5
(Aud), visual (Vis) and textual (Txt) description. Maj 72.7 75.2 80.5 65.4 62.6 77.4
80.6 83.6 86.0 90.9 75.3 91.8
Each video is labelled one of 30 classes. For each Int
of the views we construct similarity matrices using MV 98.9 93.4 95.6 97.7 87.7 98.8
cosine distance and threshold about the mean to obtain
unweighted graphs. We considered 20% of the data to be labelled. We show results on pair-wise
classification for the first four classes. As a natural way of combining graphs, we compared our
algorithm (6) (MV) with union (Unn), intersection (Int) and majority (Maj)11 of graphs. We used
LS labelling as the graph-kernel and (2) was used to solve single graph transduction. Table 2
summarizes the results, averaged over 20 iterations. We also state top accuracy in each of the views
for comparison. As expected from our analysis in Theorem 7.1, we observe that combining multiple
graphs significantly improves classification accuracy.
9
Conclusion
For the problem of graph transduction, we show that there exists orthonormal representations that
are consistent over a large class of graphs. We also note that the Laplacian inverse regularizer
is suboptimal on graphs with high connectivity, and alternatively show that LS labellingis not only
consistent, but also exhibits high Rademacher complexity on a large class of graphs. Using our analysis, we also develop a sound statistical understanding of the improved classification performance
in combining multiple graphs.
9
10
11
Relevant resources at: mllab.csa.iisc.ernet.in/rakeshs/nips14
?kxi ?xj k2
The (i, j)th entry of an RBF kernel is given by exp
. We set ? to the mean distance.
2? 2
Majority graph is a graph where an edge (i, j) is present, if a majority of the graphs have the edge (i, j).
8
References
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9
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4,847 | 5,389 | Optimal prior-dependent neural population codes
under shared input noise
?
Agnieszka Grabska-Barwinska
Gatsby Computational Neuroscience Unit
University College London
[email protected]
Jonathan W. Pillow
Princeton Neuroscience Institute
Department of Psychology
Princeton University
[email protected]
Abstract
The brain uses population codes to form distributed, noise-tolerant representations of sensory and motor variables. Recent work has examined the theoretical
optimality of such codes in order to gain insight into the principles governing
population codes found in the brain. However, the majority of the population
coding literature considers either conditionally independent neurons or neurons
with noise governed by a stimulus-independent covariance matrix. Here we analyze population coding under a simple alternative model in which latent ?input
noise? corrupts the stimulus before it is encoded by the population. This provides
a convenient and tractable description for irreducible uncertainty that cannot be
overcome by adding neurons, and induces stimulus-dependent correlations that
mimic certain aspects of the correlations observed in real populations. We examine prior-dependent, Bayesian optimal coding in such populations using exact
analyses of cases in which the posterior is approximately Gaussian. These analyses extend previous results on independent Poisson population codes and yield
an analytic expression for squared loss and a tight upper bound for mutual information. We show that, for homogeneous populations that tile the input domain,
optimal tuning curve width depends on the prior, the loss function, the resource
constraint, and the amount of input noise. This framework provides a practical
testbed for examining issues of optimality, noise, correlation, and coding fidelity
in realistic neural populations.
1
Introduction
A substantial body of work has examined the optimality of neural population codes [1?19]. However, the classical literature has focused mostly on codes with independent Poisson noise, and on
Fisher information-based analyses of unbiased decoding. Real neurons, by contrast, exhibit dependencies beyond those induced by the stimulus (i.e., ?noise correlations?), and Fisher information
does not accurately quantify information when performance is close to threshold [7, 15, 18], or
when biased decoding is optimal. Moreover, the classical population codes with independent Poisson noise predict unreasonably good performance with even a small number of neurons. A variety
of studies have shown that the information extracted from independently recorded neurons (across
trials or even animals) outperforms the animal itself [20, 21]. For example, a population of only 220
Poisson neurons with tuning width of 60 deg (full width at half height) and tuning amplitude of 10
spikes can match the human orientation discrimination threshold of ? 1 deg. (See Supplement S1
for derivation.) Note that even fewer neurons would be required if peak spike counts were higher.
The mismatch between this predicted efficiency and animals? actual behaviour has been attributed
to the presence of information-limiting correlations between neurons [22, 23]. However, deviation
1
stimulus prior
0
stimulus
stimulus
posterior
p(stimulus)
spike count
p(stimulus)
population response
Poisson
noise
spike count
tuning curves
input noise
preferred stimulus
likelihood
stimulus
+
Figure 1: Bayesian formulation of neural population coding with input noise.
from independence renders most analytical treatments infeasible, necessitating the use of numerical
methods (Monte Carlo simulations) for quantifying the performance of such codes [7, 15].
Here we examine a family of population codes for which the posterior is Gaussian, which makes it
feasible to perform a variety of analytical treatments. In particular, when tuning curves are Gaussian
and ?tile? the input domain, we obtain codes for which the likelihood is proportional to a Gaussian
[2, 16]. Combined with a Gaussian stimulus prior, this results in a Gaussian posterior whose variance
depends only on the total spike count. This allows us to derive tractable expressions for neurometric
functions such as mean squared error (MSE) and mutual information (MI), and to analyze optimality
without resorting to Fisher information, which can be inaccurate for short time windows or small
spike counts [7, 15, 18]. Secondly, we extend this framework to incorporate shared ?input noise? in
the stimulus variable of interest (See Fig. 1). This form of noise differs from many existing models,
which assume noise to arise from shared connectivity, but with no direct relationship to the stimulus
coding [5, 15, 18, 24] (although see [16, 25] for related approaches).
This paper is organised as follows. In Sec. 2, we describe an idealized Poisson population code with
tractable posteriors, and review classical results based on Fisher Information. In Sec. 3, we provide
a Bayesian treatment of these codes, deriving expressions for mean squared error (MSE) and mutual
information (MI). In Sec. 4, we extend these analyses to a population with input noise. Finally, in
Sec. 5 we examine the patterns of pairwise dependencies introduced by input noise.
2
Independent Poisson population codes
Consider an idealized population of Poisson neurons that encode a scalar stimulus s with Gaussianshaped tuning curves. Under this (classical) model, the response vector r = (r1 , . . . rN )> is conditionally Poisson distributed:
ri |s ? Poiss(fi (s)),
p(r|s) =
N
Y
ri
fi (s)
1
,
ri ! fi (s) e
(Poisson encoding) (1)
i=1
where tuning curves fi (s) take the form
?
fi (s) = ? A exp
1
2
?
?
s i )2 ,
2 (s
t
?
(tuning curves) (2)
with equally-spaced preferred stimuli s = ( s 1 , . . . s N ), tuning width t , amplitude A, and time
window for counting spikes ? . We assume that the tuning curves ?tile?, i.e., sum to a constant over
the relevant stimulus range:
N
X
?
?
(tiling property) (3)
fi (s) =
i=1
We
determine by integrating
the summed tuning curves (eq. 3) over the stimulus space, giving
p
R can
PN
ds i=1 fi (s) = N A 2? t = S , with solution:
(expected total spike count) (4)
p
where
= S/N is the spacing between tuning curve centers, and a = 2?A? is a constant that
we will refer to as the ?effective amplitude?, since it depends on true tuning curve amplitude and
= a t/
2
the time window for integrating spikes. Note, that tiling holds almost perfectly if tuning curves are
broad compared to their spacing (e.g. t > ). However, our results hold on average for a much
broader range of t . (See Supplementary Figs S2 and S3 for a numerical analysis.)
P
Let R =
ri denote the total spike count from the entire population. Due to tiling, R is a Poisson
1 R
random variable with rate , regardless of the stimulus: p(R|s) = R!
e .
For simplicity, we will consider stimuli drawn from a zero-mean Gaussian prior with variance
Q
s ? N (0,
2
s ),
p(s) =
p 1
2?
s
e
s2
2
2 s
.
2
s:
(stimulus prior) (5)
Since i e fi (s) = e due to the tiling assumption, the likelihood (eq. 1 as a function of s) and
posterior both take Gaussian forms:
Y
?
p(r|s) /
fi (s)ri / N s R1 r >s, R1 t2
(likelihood) (6)
i
p(s|r) = N
? r >s?
2 ?
t
,
,
R+? R+?
(posterior) (7)
where ? = t2 / s2 denotes the ratio of the tuning curve variance to prior variance. The maximum of
?
the likelihood (eq. 6) is the so-called center-of-mass estimator estimator, R1 r >s, while the mean of
the posteror (eq. 7) is biased toward zero by an amount that depends on ?. Note that the posterior
variance does not depend on which neurons emitted spikes, only the total spike count R, a fact that
will be important for our analyses below.
2.1
Capacity constraints for defining optimality
Defining optimality for a population code requires some form of constraint on the capacity of the
neural population, since clearly we can achieve arbitrarily narrow posteriors if we allow arbitrarily
large total spike count R. In the following, we will consider two different biologically plausible
constraints:
? A space constraint, in which we constrain only the number of neurons. This means that
increasing the tuning width t will increase the expected population spike count (see
eq. 4), since more neurons will respond as tuning curves grow wider.
? An energy constraint, in which we fix while allowing t and amplitude A to vary. Here,
we can make tuning curves wider but must reduce the amplitude so that total expected spike
count remains fixed.
We will show that the optimal tuning depends strongly on which kind of constraint we apply.
2.2
Analyses based on Fisher Information
The Fisher information provides a popular, tractable metric for quantifying the efficiency of a neural
@2
code, given by E[ @s
2 log p(r|s)], where expectation is taken with respect to encoding distribution
p(r|s). For our idealized Poisson population, the total Fisher information is:
N
N
?
? (s s? )2 ?
X
X
fi0 (s)2
(s s i )2
a
i
IF (s) =
=
A
exp
=
= 2 , (Fisher info) (8)
4
2
f
(s)
2
i
t
t
t
t
i=1
i=1
which we can derive, as before, using the tiling property (eq. 3). (See also Supplemental Sec. S2).
The first of the two expressions at right reflects IF for the space constraint, where varies implicitly
as we vary t . The second expresses IF under the energy constraint, where is constant so that a
varies implicitly with t . For both constraints, IF increases with increasing a and decreasing t [5].
Fisher information provides a well-known bound on the variance of an unbiased estimator s?(r)
known as the Cram?er-Rao (CR) bound, namely var(?
s|s) 1/IF (s). Since FI is constant over s in
our idealized setting, this leads to a bound on the mean squared error ([7, 12]):
?
2
?
?
1
t
MSE , E (?
s(r) s)2 p(r,s) E
=
= t,
(9)
IF (s) p(s)
a
3
effects of prior stdev
effects of time window (ms)
3
MSE
0
10 50
0
10
CR bound
200
10
CR bound
1
10
MSE
energy
constraint
= 25
400
10
2
10
2
1
10
16
8
4
2
1
=3
space
constraint
10
0
10
10
10
10
0
2
4
tuning width
6
8
10
0
2
4
tuning width
6
8
Figure 2: Mean squared error as a function of the tuning width t , under space constraint (top row)
and energy constraint (bottom row), for spacing
= 1 and amplitude A = 20 sp/s. and Top
left: MSE for different prior widths s (with A=2,? = 200ms), showing that optimal t increases
with larger prior variance. Cram?er-Rao bound (gray solid) is minimized at t = 0, whereas bound
(eq. 12, gray dashed) accurately captures shape and location of the minimum. Top right: Similar
curves for different time windows ? for counting spikes (with s =32), showing that optimal t increases for lower spike counts. p
Bottom row: Similar traces under energy constraint (where A scales
inversely with t so that = 2?? A t is constant). Although the CR bound grossly understates
the true MSE for small counting windows (right), the optimal tuning is maximally narrow in this
configuration, consistent with the CR curve.
which is simply the inverse of Fisher Information (eq. 8).
Fisher information also provides a (quasi) lower bound on the mutual information, since an efficient
estimator (i.e., one that achieves the CR bound) has entropy upper-bounded by that of a Gaussian
with variance 1/IF (see [3]). In our setting this leads to the lower bound:
?
?
?
?
2 a
2
1
1
MI(s, r) , H(s) H(s|r)
log
=
log
(10)
s
s
2 .
2
2
t
t
Note that neither of these FI-based bounds apply exactly to the Bayesian setting we consider here,
since Bayesian estimators are generally biased, and are inefficient in the regime of low spike counts
[7]. We examine them here nonetheless (gray traces in Figs. 2 and 3) due to their prominence in the
prior literature ([5, 12, 14]), and to emphasize their limitations for characterizing optimal codes.
2.3
Exact Bayesian analyses
In our idealized population, the total spike count R is a Poisson random variable with mean , which
allows us to compute the MSE and MI by taking expectations w.r.t. this distribution.
Mean Squared Error (MSE)
The mean squared error, which equals the average posterior variance (eq. 7), can be computed
analytically for this model:
? 2
? R
1 ?
2
X
t
t
MSE = E
=
e = t2 e
(?) ? (?,
),
(11)
R + ? p(R)
R + ? R!
R=0
Rz a 1 t
1
where ? = t2 / s2 and ? (a, z) = z a (a)
t
e dt is the holomorphic extension of the lower
0
incomplete gamma function [26] (see SI for derivation). When the tuning curve is narrower than the
prior (i.e., t2 ? s2 ), we can obtain a relatively tight lower bound:
MSE
2
t
1
e
4
+(
2
s
2
t )e
.
(12)
effects of prior stdev
MI (bits)
space
constraint
6
effects of time window (ms)
6
FI-based bound
4
16
8
4
2
1
MI (bits)
0
energy
constraint
4
= 32
2
= 400
0
6
6
4
4
2
2
0
= 25
2
0
2
4
tuning width
6
8
2
4
tuning width
6
8
Figure 3: Mutual information as a function of tuning width t , directly analogous to plots in Fig. 2.
Note the problems with the lower bound on MI derived from Fisher information (top, gray traces)
and the close match of the derived bound (eq. 14, dashed gray traces). The effects are similar
to Fig. 2, except that MI-optimal tuning widths are slightly smaller (upper left and right) than for
MSE-optimal codes. For both loss functions, optimal width is minimal under an energy constraint.
Figure 2 shows the MSE (and derived bound) as a function of the tuning width t over the range
where tiling approximately holds. Note the high accuracy of the approximate formula (12, dashed
gray traces) and that the FI-based bound does not actually lower-bound the MSE in the case of
narrow priors (darker traces).
For the space-constrained setting (top row, obtained by substituting = a t / in eqs. 11 and
12), we observe substantial discrepancies between the true MSE and FI-based analysis. While FI
suggests that optimal tuning width is near zero (down to the limits of tiling), analyses reveal that the
optimal t grows with prior variance (left) and decreasing time window (right). These observations
agree well with the existing literature (e.g. [15, 16]). However, if we restrict the average population
firing rate (energy constraint, bottom plots), the optimal tuning curves once again approach zero. In
this case, FI provides correct intuitions and better approximation of the true MSE.
Mutual Information (MI)
For a tiling population and Gaussian prior, mutual information between the stimulus and response
is:
h ?
?i
2
MI(s, r) = 12 E log 1 + R s2
,
(13)
t
P (R)
which has no closed-form solution, but can
p be calculated efficiently with a discrete sum over R from
0 to some large integer (e.g., R = + n
to capture n standard deviations above the mean). We
can derive an upper bound using the Taylor expansion to log while preserving the exact zeroth order
term:
?
?
?
?
2
2
a t/
MI(s, r) ? 1 2e log 1 + ( 1 e ) s2 = 1 e 2
log 1 + 1 e aa t / ts
(14)
t
Once again, we investigate the efficiency of population coding for neurons, now in terms of the
maximal MI. Figure 3 shows MI as a function of the neural tuning width t . We observe a similar
effect as for the MSE: the optimal tuning widths are now different from zero,but only for the space
constraint. The energy constraint, as well as implications from FI indicate optimum near t =0.
5
3
Poisson population coding with input noise
We can obtain a more general family of correlated population codes by considering ?input noise?,
where the stimulus s is corrupted by an additive noise n (see Fig. 1):
s ? N (0,
2
s)
2
n)
(prior) (15)
n ? N (0,
ri |s, n ? Poiss(fi (s + n))
(input noise) (16)
(population response) (17)
The use of Gaussians allows us to marginalise over n analytically, resulting in a Gaussian form for
the likelihood and Gaussian posterior:
p(r|s) / N s R1 r >s, R1 t2 + n2
?
?
r >s
( t2 + R n2 )
p(s|r) = N
2 / 2 + R( 2 / 2 + 1) , 2 + R( 2 +
t
t
s
n
s
n
?
2
s
2
s)
?
(likelihood) (18)
(posterior) (19)
Note that even in the limit of large spike counts, the posterior variance is non-zero, converging to
2 2
2
2
n s /( n + s ).
3.1
Population coding characteristics: FI, MSE, & MI
Fisher information for a population with input noise can be using the fact that the likelihood (eq. 18)
is Gaussian: Eq. (18):
? 2
?
e
d log p(r|s)
R
=
E
=
(1 + ?) ? (1 + ?,
) (20)
IF (s) , E
2
2
2
2
ds
+
R
t
n
n
p(r|s)
p(R)
where ? = t2 / n2 and ? (?, ?) once again denotes holomorphic extension of lower incomplete
gamma function. Note that for n = 0, this reduces to (eq. 8).
It is straightforward to employ the results from Sec. 2.3 for the exact Bayes analyses of a Gaussian
posterior (19):
?
?
?
2
2
2 2
1
R
t +R n
2
2
s n
MSE = s E 2
= s? E
+ 2+ 2 E
2 + 2)
s
n
+
R(
?
+
R
?
+
R p(R)
t
n
s
p(R)
p(R)
2
?
?
= ? (?) ? (?,
) + 2 +n 2 (1 + ?) ? (1 + ?,
) s2 e ,
(21)
s
?
?
MI = 12 E log 1 +
R s2
2
t +R
n
2
n
?
(22)
,
p(R)
where ? = t2 /( s2 + n2 ). Although we could not determine closed-form analytical expressions
for MI, it can be computed efficiently by summing over a range of integers [0, . . . Rmax ] for which
P (R) has sufficient support. Note this is still a much faster procedure than estimating these values
from Monte Carlo simulations.
3.2
Optimal tuning width under input noise
Fig. 4 shows the optimal tuning width under the space constraint: the value of t minimizing MSE
(left) or maximising MI (right) as a function of the prior width s , for selected time windows of
integration ? . Blue traces show results for a Poisson population, while green traces correspond to a
population with input noise ( n = 1).
For both MSE and MI loss functions, optimal tuning width decreases for narrower priors. However,
under input noise (green traces), the optimal tuning width saturates at the value that depends on
the available number of spikes. As the prior grows wider, the growth of the optimal tuning width
depends strongly on the choice of loss function: optimal t grows approximately logarithmically
with s for minimizing MSE (left), but it grows much slower for maximizing MI (right). Note that
for realistic prior widths (i.e. for s > n ), the effects of input noise on optimal tuning width are far
more substantial under MI than under MSE.
6
mutual information
MSE
optimal TC width
8
8
6
Poisson
noise only
4
2
w/ input noise
=
0
20
6
00
=1
4
= 50
2
= 25
0
0.1
1
prior stdev
0
10
0.1
1
prior stdev
10
Figure 4: Optimal tuning width t (under space constraint only) as a function of prior width s ,
for classic Poisson populations (blue) and populations with input-noise (green, n2 = 1). Different
traces correspond to different time windows of integration, for
= 1 and A = 20 sp/s. As n
increases, the optimal tuning width increases under MI, and under MSE when s < n (traces not
shown). For MSE, predictions of the Poisson and input-noise model converge for priors s > n .
We have not shown plots for energy-constrained population codes because the optimal tuning width
sits at the minimum of the range over which tiling can be said to hold, regardless of prior width,
input noise level, time window, or choice of loss function. This can be seen easily in the expressions
for MI (eqs. 13 and 22), in which each term in the expectation is a decreasing function of t for
all R > 0. This suggests that, contrary to some recent arguments (e.g., [15, 16]), narrow tuning (at
least down to the limit of tiling) really is best if the brain has a fixed energetic budget for spiking, as
opposed to a mere constraint on the number of neurons.
4
Correlations induced by input noise
Input noise alters the mean, variance, and pairwise correlations of population responses in a systematic manner that we can compute directly (see Supplement for derivations). In Fig. 5 we show
the effects input noise with standard deviation n = 0.5 , for neurons with the tuning amplitude
of A = 10. The tuning curve (mean response) becomes slightly flatter (A), while the variance increases, especially at the flanks (B). Fig. 5C shows correlations between the two neurons with tuning
curves and variance are shown in panels A-B: one pair with the same preferred orientation at zero
(red) and a second with a 2 degree difference in preferred orientation (blue). From these plots, it is
clear that the correlation structure depends on both the tuning as well as the stimulus. Thus, in order
to describe such correlations one needs to consider the entire stimulus range, not simply the average
correlation marginalized over stimuli.
Figure 5D shows the pairwise correlations across an entire population of 21 neurons given a stimulus
at s = 0. Although we assumed Gaussian tuning curves here, one obtain similar plots for arbitrary
unimodal tuning curves (see Supplement), which should make it feasible to test our predictions
in real data. However, the time scale of the input noise and basic neural computations is about
10 ms. At such short spike count windows, available number of spikes is low, and so are correlations
induced by input noise. With other sources of second order statistics, such as common input gains
(e.g. by contrast or adaptation), these correlations might be too subtle to recover [23].
5
Discussion
We derived exact expressions for mean squared error and mutual information in a Bayesian analysis
of: (1) an idealized Poisson population coding model; and (2) a correlated, conditionally Poisson
population coding model with shared input noise. These expressions allowed us to examine the
optimal tuning curve width under both loss functions, under two kinds of resource constraints. We
have confirmed that optimal t diverges from predictions based on Fisher information, if the overall
spike count allowed is allowed to grow with tuning width (i.e., because more neurons respond to
the stimulus when tuning curves become broader). We referred to this as a ?space constraint? to
differentiate it from an ?energy constraint?, in which tuning curve amplitude scales down with tuning
7
B
mean
variance
(sp / s)2
sp / s
A
5
5
stimulus s
stimulus s
C
D
r
r
preferred stim
correlation
5
stimulus s
preferred stim
Figure 5: Response statistics of neural population with input noise, for standard deviation n = 0.5.
?
?
(A) Expected spike responses of two neurons: s 1 = 0 (red) and s 2 = 2 (blue). The common
noise effectively smooths blurs the tuning curves with a Gaussian kernel of width n . (B) Variance
of neuron 1, its tuning curve replotted in black for reference. Input noise has largest influence on
variance at the steepest parts of the tuning curve. (C) Cross-correlation of the neuron 1 with two
?
others: one sharing the same preference (red), and one with s = 2 (blue). Note that correlation
of two identically tuned neurons is largest at the steepest part of the tuning curve. (D) Spike count
correlations for entire population of 21 neurons given a fixed stimulus s = 0, illustrating that the
pattern of correlations is signal dependent.
width so that average total spike count is invariant to tuning width. In this latter scenario, predictions
from Fisher information are no longer inaccurate, and we find that optimal tuning width should be
narrow (down to the limit at which the tiling assumption applies), regardless of the duration, prior
width, or input noise level.
We also derived explicit predictions for the dependencies (i.e., response correlations) induced by
the input noise. These depend on the shape (and scale) of tuning responses, and on the amount of
noise ( n ). However, for a reasonable assumption that noise distribution is much narrower than the
width of the prior (and tuning curves), under which the mean firing rate changes little, we can derive
predictions for the covariances directly from the measured tuning curves. An important direction
for future work will be to examine the detailed structure of correlations measured in large populations. We feel that the input noise model ? which describes exactly those correlations that are most
harmful for decoding ? has the potential to shed light on the factors affecting the coding capacity
in optimal neural populations [23].
Finally, if we return to our example from the Introduction to see how the number of neurons necessary to reach the human discrimination threshold of s=1 degree changes in the presence of input
noise. As n approaches s, the number of neurons required goes rapidly to infinity (See Supplementary Fig. S1).
Acknowledgments
This work was supported by the McKnight Foundation (JP), NSF CAREER Award IIS-1150186
(JP), NIMH grant MH099611 (JP) and the Gatsby Charitable Foundation (AGB).
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9
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4,848 | 539 | Nonlinear Pattern Separation in Single Hippocampal
Neurons with Active Dendritic Membrane
Anthony M. Zador t
t
Brenda J. Claiborne?
Depts. of Psychology and Cellular
& Molecular Physiology
Yale University
New Haven, CT 06511
[email protected]
Thomas H. Brown
t
?Division of Life Sciences
University of Texas
San Antonio, TX 78285
ABSTRACT
The dendritic trees of cortical pyramidal neurons seem ideally suited to
perfonn local processing on inputs. To explore some of the implications
of this complexity for the computational power of neurons, we simulated
a realistic biophysical model of a hippocampal pyramidal cell in which a
"cold spot"-a high density patch of inhibitory Ca-dependent K channels
and a colocalized patch of Ca channels-was present at a dendritic
branch point. The cold spot induced a non monotonic relationship between the strength of the synaptic input and the probability of neuronal
fIring. This effect could also be interpreted as an analog stochastic XOR.
1 INTRODUCTION
Cortical neurons consist of a highly branched dendritic tree that is electrically coupled to
the soma. In a typical hippocampal pyramidal cell, over 10,000 excitatory synaptic inputs
are distributed across the tree (Brown and Zador, 1990). Synaptic activity results in current
flow through a transient conductance increase at the point of synaptic contact with the
membrane. Since the primary means of rapid intraneuronal signalling is electrical, information flow can be characterized in tenns of the electrical circuit defIned by the synapses,
the dendritic tree, and the soma.
Over a dozen nonlinear membrane channels have been described in hippocampal pyramidal neurons (Brown and Zador, 1990). There is experimental evidence for a heterogeneous
distribution of some of these channels in the dendritic tree (e.g. Jones et al .? 1989). In the
absence of these dendritic channels, the input-output function can sometimes be reasonably
approximated by a modifIed sigmoidal model. Here we report that introducing a cold spot
51
52
Zador, Claiborne, and Brown
at the junction of two dendritic branches can result in a fundamentally different, nonmonotonic input-output function.
2 MODEL
The biophysical details of the circuit class defined by dendritic trees have been well characterized (reviewed in RaIl, 1977; Jack et al., 1983). The fundamental circuit consists of a
linear and a nonlinear component The linear component can be approximated by a set of
electrical compartments coupled in series (Fig. 1C), each consisting of a resistor and capacitor in parallel (Fig. 1B). The nonlinear component consists of a set of nonlinear resistors
in parallel with the capacitance.
The model is summarized in Fig. 1A. Briefly, simulations were performed on a
3000-compartment anatomical reconstruction of a region CAl hippocampal neuron (Claiborne et aI., 1992; Brown et al., 1992). All dendritic membrane was passive, except at the
cold spot (Fig. 1A). At the soma, fast K and Na channels (cf. Hodgkin-Huxley, 1952) generated action potentials in response to stimuli. The parameters for these channels were
modified from Lytton and Sejnowski (1991; cf. Borg-Graham, 1991).
A
Synapti_c......;+
input-~
Cold spot
Fast somatic
andNa
c
I
r-------------------------------~
I
,
,
"
t
,
" ___________________ .. _____________ ~ I
<
I
)
Radial and longitudinal Ca+2 diffusion
Fig. 1 The model. (A) The 3000-compartrnent electrical model used in these simulations was obtained from a 3-dimensional reconstruction of a hippocampal region CAl pyramidal neuron (Clai?
borne et al, 1992). Each synaptic pathway (A-D) consisted of an adjustable number of synapses
arrayed along the single branch indicated (see text). Random background activity was generated with
a spatially uniform distribution of synapses firing according to Poisson statistics. The neuronal mem?
brane was completely passive (linear), except at the indicated cold spot and at the soma. (B) In the
nonlinear circuit associated with a patch a neuronal membrane containing active channels, each chan?
nel is described by a voltage-dependent conductance in series with its an ionic battery (see text). In
the present model the channels were spatially localized, so no single patch contained all of the non?
linearities depicted in this hypothetical illustration. (Cl. A dendritic segment is illustrated in which
both electrical and ca2+ dynamics were modelled. Ca +buffering, and both radial and longitudinal
Ca2+ diffusion were simulated.
Nonlinear Panern Separation in Single Hippocampal Neurons
We distinguished four synaptic pathways A-D (see Fig. lA). Each pathway consisted of a
population of synapses activated synchronously. The synapses were of the fast AMPA type
(see Brown et. al., 1992). In addition. random background synaptic activity distributed uniformly across the dendritic tree fIred according to Poisson statistics.
The cold spot consisted of a high density of a Ca-activated K channel. the AHP current
(Lancaster and Nicoll. 1987; Lancaster et. aI., 1991) colocalized with a low density patch
ofN-type Ca channels (Lytton and Sejnowski, 1991; cf. Borg-Graham, 1991). Upon localized depolarization in the region of the cold ~t. influx of Ca2+ through the Ca channel resulted in a transient increase in the local rCa +]. The model included Ca2+ buffering, and
both radial and longitudinal diffusion in the region of the cold spot. The increased [Ca2+]
activated the inhibitory AHP current. The interplay between the direct excitatory effect of
synaptic input, and its inhibitory effect via the AHP channels formed the functional basis
of the cold spot.
3 RESULTS
3.1 DYNAMIC BEHAVIOR
Representative behavior of the model is illustrated in Fig. 2. The somatic potential is plotted as a function of time in a series of simulations in which the number of activated synapses in pathway AlB was increased from 0 to about 100. For the fIrst 100 msec of each
simulation, background synaptic activity generated a noisy baseline. At t = 100 msec, the
indicated number of synapses fired synchronously five times at 100 Hz. Since the background activity was noisy, the outcome of the each simulation was a random process.
The key effect of the cold spot was to impose a limit on the maximum stimulus amplitude
that caused firing, resulting in a window of stimulus strengths that triggered an action potential. In the absence of the cold spot a greater synaptic stimulus invariably increased the
likelihood that a spike fIred. This limit resulted from the relative magnitude of the AHP
Sample Soma
tic Voltage Tra celll
60
0
~
.
I
0
>-
-60
<:>
Fig. 2 Sample runs. The membrane voltage at the soma is plotted as a f'wtction of time and synaptic
stimulus intensity. At t = 100 msec, a synaptic stimulus consisting of 5 pulses was activitated. The
noisy baseline resulted from random synaptic input. A single action potential resulted for input intensities within a range determined by the kinetics of the cold spot
53
54
Zador, Claiborne, and Brown
current "threshold" to the threshold for somatic spiking. The AHP current required a relatively high level of activity for its activation. This AHP current "threshold" reflected the
sigmoidal voltage dependence of N-type Ca current activation (V1I2 = -28 mV), since only
as the dendritic voltage approached V1I2 did dendritic [Ca2+] rise enough to activate the
AHP current. Because V1I2 was much higher than the threshold for somatic spiking (about
-55 mV under current clamp), there was a window of stimulus strengths sufficient to trigger
a somatic action potential but insufficient to activate the AHP current Only within this
window of between about 20 and 60 synapses (Fig. 2) did an action potential occur.
3.2 LOCAL NON-MONOTONIC RESPONSE FUNCTION
Because the background activity was random, the outcome of each simulation (e.g. Fig. 2)
represented a sample of a random process. This random process can be used to defme many
different random variables. One variable of interest is whether a spike fired in response to
a stimulus. Although this measure ignores the dynamic nature of neuronal activity, it was
still relatively informative because in these simulations no more than one spike fired per
experiment
Fig. 3A shows the dependence of firing probability on stimulus strength. It was obtained
by averaging over a population of simulations of the type illustrated in Fig. 2. In the absence of AHP current (dotted line), the fIring probability was a sigmoidal function of activity. In its presence, the firing probability was a smooth nonmonotonic function of the
activity (solid line). The firing probability was maximum at about 35 synapses, and occurred only in the range between about 10 and 80 synapses. The statistics illustrated in Fig.
3A quantify the nonmonotonicity that is implied by the single sample shown in Fig. 2.
Spikes required the somatic Hodgkin-Huxley-like Na and K channels. To a first approximation, the effect of these channels was to convert a continuous variable-the somatic voltage-into a discrete variable-the presence or absence of a spike. Although this
approximation ignores the complex interactions between the soma and the cold spot, it is
useful for a qualitative analysis. The nonmonotonic dependence of somatic activity on syn-
A
B
1.0
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-
Cold SpolCold Spol+
..... 0 .6
;0
III
,ll
0
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Po.
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0.4
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~
C
.~
I
I
I
I
r;: 0.2
0.0 '--__' - - _........_ ........--.i::o.........~"__._.......J
o 20 40 60 80 100 120
Number of active synpases
..
-56
Cold SpotCold Spot+
> -56
E
cu -60
".-
tIO
CI
0p
..
.- .'-
"
.-
.. --"
"
-62
-64
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CI
cu
Po. -66
o
20 40 60 60 100 120
Number of active synpases
Fig. 3 Nonmonotonic input-output relation. (A) Each point represents the probability that at least
one spike was fIred at the indicated activity level. In the absence of a cold spot, the fIring probability increased sharply and monotonically as the number of synapses in pathway C/D increased (dotted Une). In contrast, the fIring probability reached amaximumforpathwayA/B and then decreased
(solid line). (B) Each point represents the peak somatic voltage for a single simulation at the indicated activity level in the presence (pathway AlB; solid line) and absence (pathway C/D,? dotted
Une) of a cold spot Because each point represents the outcome of a single simulation, in contrast
to the average used in (A), the points reflect the variance due to the random background activity.
Nonlinear Pattern Separation in Single Hippocampal Neurons
aptic activity was preserved even when active channels at the soma were eliminated (Fig.
3B). This result emphasizes that the critical nonlinearity was the cold spot itself.
3.3 NONLINEAR PATTERN SEPARATION
So far. we have treated the output as a function of a scalar-the total activity in pathway
AlB (or CID). In Fig. 3 for example. the total activity was defmed as the sum of the activities in pathway A and B. The spatial organization of the afferents onto 2 pairs of branches-A & B and C & D (Fig. I)-suggested considering the output as a function of the
activity in the separate elements of each pair.
The effect of the cold spot can be viewed in terms of the dependence of fIring as a function
of separate activity in pathways A and B (Fig. 4). Each fIlled circle indicates that the neuron
fIred for the indicated input intensity of pathways A and B. while a small dot indicates that
it did not fire. As suggested by (Fig. 3). the fIring probability was highest when the total
activity in the two pathways was at some intennediate level. The neuron did not fIre when
the total activity in the two pathways was too large or too small. In the absence of the cold
spot, only a minimum activity level was required.
In our model the probability of fIring was a continuous function of the inputs. In the presence of the dendritic cold spot, the corners of this function suggested the logical operation
XOR. The probability of fIring was high if only one input was activated and low if both or
neither was activated.
4 DISCUSSION
4.1 ASSUMPTIONS
Neuronal morphology in the present model was based on a precise reconstruction of a region CAl pyramidal neuron. The main additional assumptions involved the kinetics and
distribution of the four membrane channels. and the dynamics of Ca2+in the neighborhood
of influx. The forms assumed for these mechanisms were biophysically plausible. and the
kinetic parameters were based on estimates from a collection of experimental studies (listed
in Lytton and Sejnowski. 1991; Zador et aI .? 1990). Variation within the range of uncertainty of these parameters did not alter the main conclusions. The chief untested assumption of this model was the existence of cold spots. Although there is experimental evidence
...... ....... ..
_
......
.
.....
_..... ?... .
t --..... .. .. .
- . e .??. . . . . . .
~
~
==' =::::~ ? ????
p...
~
.. :. iq:I:I!'! :
iiiilli Iii iI fifl iii
~::~:~
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...... :::::::;Z!Zi
::::: :::::
Input A
-+
Fig.4 Nonlinear pattern separation Neuronal fIring is represented as a joint nmction of two input
pathways (AlB). Filled circles indicate that the neuron fIred for the indicated stimulus parameters.
Some indication of the stochastic nature of this function. resulting fonn the noisy background, is given by the density of interdigitation of points and circles.
55
56
Zador, Claiborne, and Brown
supporting the presence of both Ca and AHP channels in the dendrites. there is at present
no direct evidence regarding their colocalization.
4.2 COMPUTATIONS IN SINGLE NEURONS
4.2.1 Neurons and Processing Elements
The limitations of the McCulloch and Pitts (1943) PE as a neuron model have long been
recognized. Their threshold PEt in which the output is the weighted sum of the inputs
passed through a threshold, is static, deterministic and treats all inputs equivalently. This
model ignores at least three key complexities of neurons: temporal, spatial and stochastic.
In subsequent years, augmented models have attempted to capture aspects of these complexities. For example, the leaky integrator (Caianiello, 1961; Hopfield, 1984) incorporates the temporal dynamics implied by the linear RC component of the circuit element
pictured in Fig. IB. We have demonstrated that the input-output function of a realistic neuron model can have qualitatively different behavior from that of a single processing element(pE).
4.2.2 Interactions Within The Dendritic Tree
The early work ofRall (1964) stressed the spatial complexity of even linear dendritic models. He noted that input from different synapses cannot be considered to arrive at a single
point, the soma. Koch et al. (1982) extended this observation by exploring the nonlinear
interactions between synaptic inputs to different regions of the dendritic tree. They emphasized that these interactions can be local in the sense that they effect subpopulations of synapses and suggested that the entire dendritic tree can be considered in terms of electrically
isolated subunits. They proposed a specific role for these subunits in computing a vetoan analog AND-NOT---that might underlie directional selectivity in retinal ganglion cells.
The veto was achieved through inhibitory inputs.
The underlying neuron models of Koch et al. (1982) and Rall (1964) were time-varying but
linear, so it is not surprising that the resulting nonlinearities were monotonic. Much steeper
nonlinearities were achieved by Shepherd and Brayton (1987) in a model that assumed excitable spines with fast Hodgkin-Huxley K and Na channels. These channels alone could
implement the digital logic operations AND and OR. With the addition of extrinsic inhibitory inputs, they showed that a neuron could implement a full complement of digital logic
operations, and concluded that a dendritic tree could in principle implement arbitrarily
complex logic operations.
The emphasis of the present model differs from that of both the purely linear and of the digital approaches, although it shares their emphasis on the locality of dendritic computation.
Because the cold spot involved strongly nonlinear channels, it implemented a non mono tonic response function, in contrast to strictly linear dendritic models. At the same time, the
present model retained the essentially analog nature of intraneuronal signalling, in contrast
to the digital dendritic models. This analog mode seems better suited to processing large
numbers of noisy inputs because it preserves the uncertainties rather than making an immediate decision. Focussing on the analog nature of the response eliminated the requirement
for operating within the digital range of channel dynamics.
The NMDA receptor-gated channel can give rise to an analog AND with a weaker voltagedependence than that induced by fast Na and K channels. Mel (1992) described a model in
which synapses mediating increases to both the NMDA and AMPA conductances were distributed across the dendritic tree of a cortical neuron. When the synaptic activity was dis-
Nonlinear Pattern Separation in Single Hippocampal Neurons
tributed in appropriately sized clusters, the resulting neuronal response function was
reminiscent of that of a sigma-pi unit With suitable preprocessing of inputs. the neuron
could perform complex pattern discrimination.
A unique feature of the present model is that functional inhibition arose from purely excitatory inputs. This mechanism underlying this inhibition -the AHP current-was intrinsic
to the membrane. In both the Koch et ale (1982) and Brayton and Shepherd (1987) models.
the veto or NOT operation was achieved through extrinsic synaptic inhibition. This requires
additional neuronal circuitry. In the case of a dedicated sensory system like the directionally selective retinal granule cell. it is not unreasonable to imagine that the requisite neuronal circuitry is hardwired. In the limiting case of the digital model, the requisite circuitry
would involve a separate inhibitory interneuron for each NOT-gate.
4.2.3 Adaptive Dendritic Computation
What algorithms can harness the computational potential of the dendritic tree? Adaptive
dendritic computation is a very new subject. Brown et ale (1991, 1992) developed a model
in which synapses distributed across the dendritic tree showed interesting forms of spatial
self-organization. Synaptic plasticity was governed by a local biophysically-motivated
Hebb rule (Zador et al' 1990). When temporally correlated but spatially uncorrelated inputs were presented to the neuron, spatial clusters of strengthened synapses emerged within
the dendritic tree. The neuron converted a temporal correlation into a spatial correlation.
J
The computational role of clusters of strengthened synapses within the dendritic tree becomes important in the presence of nonlinear membrane. If the dendrites are purely passive. then saturation ensures that the current injected per synapse actually decreases as the
clustering increases. If purely regenerative nonlinearities are present (Brayton and Shepherd. 1987; Mel. 1992), then the response increases. The cold spot extends the range of
local dendritic computations.
What might control the formation and distribution of the cold spot itself? Cold spots might
arise from the fortuitous colocalization of Ca and K AHP channels. Another possibility is
that some specific biophysical mechanism creates cold spots in a use-dependent manner.
Candidate mechanisms might involve local changes in second messengers such as [Ca2+]
or longitudinal potential gradients (if. Poo, 1985). Bell (1992) has shown that this second
mechanism can induce computationally interesting distributions of membrane channels.
4.3 WHY STUDY SINGLE NEURONS?
We have illustrated an important functional difference between a single neuron and aPE.
A neuron with cold spots can perform extensive local processing in the dendritic tree, giving rise to a complex mapping between input and output. A neuron may perhaps be likened
to a "micronet" of simpler PEs. since any mapping can be approximated by a sufficiently
complex network of sigmoidal units (Cybenko, 1989). This raises the objection that since
micronets represent just a subset of all neural networks, there may be little to be gained by
studying the properties of the special case of neurons.
The intuitive justification for studying single neurons is that they represent a large but highly constrained subset that may have very special properties. Knowledge of the properties
general to all sufficiently complex PE networks may provide little insight into the properties specific to single neurons. These properties may have implications for the behavior of
circuits of neurons. It is not unreasonable to suppose that adaptive mechanisms in biological circuits will utilize the specific strengths of single neurons.
57
58
Zador, Claiborne, and Brown
Acknowledgments
We thank Michael Hines for providing NEURON-MODL assisting with new membrane
mechanisms. This research was supported by grants from the Office of Naval Research,
the Defense Advanced Research Projects Agency, and the Air Force Office of Scientific
Research.
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4,849 | 5,390 | Optimal Neural Codes for Control and Estimation
Alex Susemihl1 , Manfred Opper
Methods of Artificial Intelligence
Technische Universit?at Berlin
1
Current affiliation: Google
Ron Meir
Department of Electrical Engineering
Technion - Haifa
Abstract
Agents acting in the natural world aim at selecting appropriate actions based on
noisy and partial sensory observations. Many behaviors leading to decision making and action selection in a closed loop setting are naturally phrased within a
control theoretic framework. Within the framework of optimal Control Theory,
one is usually given a cost function which is minimized by selecting a control
law based on the observations. While in standard control settings the sensors are
assumed fixed, biological systems often gain from the extra flexibility of optimizing the sensors themselves. However, this sensory adaptation is geared towards
control rather than perception, as is often assumed. In this work we show that sensory adaptation for control differs from sensory adaptation for perception, even for
simple control setups. This implies, consistently with recent experimental results,
that when studying sensory adaptation, it is essential to account for the task being
performed.
1
Introduction
Biological systems face the difficult task of devising effective control strategies based on partial information communicated between sensors and actuators across multiple distributed networks. While
the theory of Optimal Control (OC) has become widely used as a framework for studying motor control, the standard framework of OC neglects many essential attributes of biological control [1, 2, 3].
The classic formulation of closed loop OC considers a dynamical system (plant) observed through
sensors which transmit their output to a controller, which in turn selects a control law that drives
actuators to steer the plant. This standard view, however, ignores the fact that sensors, controllers
and actuators are often distributed across multiple sub-systems, and disregards the communication
channels between these sub-systems. While the importance of jointly considering control and communication within a unified framework was already clear to the pioneers of the field of Cybernetics
(e.g., Wiener and Ashby), it is only in recent years that increasing effort is being devoted to the
formulation of a rigorous systems-theoretic framework for control and communication (e.g., [4]).
Since the ultimate objective of an agent is to select appropriate actions, it is clear that sensation and
communication must subserve effective control, and should be gauged by their contribution to action
selection. In fact, given the communication constraints that plague biological systems (and many
current distributed systems, e.g., cellular networks, sensor arrays, power grids, etc.), a major concern
of a control design is the optimization of sensory information gathering and communication (consistently with theories of active perception). For example, recent theoretical work demonstrated a sharp
communication bandwidth threshold below which control (or even stabilization) cannot be achieved
(for a summary of such results see [4]). Moreover, when informational constraints exists within a
control setting, even simple (linear and Gaussian) problems become nonlinear and intractable, as
exemplified in the famous Witsenhausen counter-example [5].
The inter-dependence between sensation, communication and control is often overlooked both in
control theory and in computational neuroscience, where one assumes that the overall solution to
the control problem consists of first estimating the state of the controlled system (without reference
1
to the control task), followed by constructing a controller based on the estimated state. This idea,
referred to as the separation principle in Control Theory, while optimal in certain restricted settings
(e.g., Linear Quadratic Gaussian (LQG) control) is, in general, sub-optimal [6]. Unfortunately, it is
in general very difficult to provide optimal solutions in cases where separation fails. A special case
of the separation principle, referred to as Certainty Equivalence (CE), occurs when the controller
treats the estimated state as the true state, and forms a controller assuming full state information. It
is generally overlooked, however, that although the optimal control policy does not depend directly
on the observation model at hand, the expected future costs do depend on the specifics of that model
[7]. In this sense, even when CE holds, costs still arise from uncertain estimates of the state and one
can optimise the sensory observation model to minimise these costs, leading to sensory adaptation.
At first glance, it might seem that the observation model that will minimise the expected future cost
will be the observation model that minimises the estimation error. We will show, however, that this
is not generally the case.
A great deal of the work in computational neuroscience has dealt independently with the problem
of sensory adaptation and control, while, as stated above, these two issues are part and parcel of
the same problem. In fact, it is becoming increasingly clear that biological sensory adaptation is
task-dependent [8, 9]. For example, [9] demonstrates that task-dependent sensory adaptation takes
place in purely motor tasks, explaining after-effect phenomena seen in experiments. In [10], the
authors show that specific changes occur in sensory regions, implying sensory plasticity in motor
learning. In this work we consider a simple setting for control based on spike time sensory coding,
and study the optimal coding of sensory information required in order to perform a well-defined
motor task. We show that even if CE holds, the optimal encoder strategy, minimising the control cost,
differs from the optimal encoder required for state estimation. This result demonstrates, consistently
with experiments, that neural encoding must be tailored to the task at hand. In other words, when
analyzing sensory neural data, one must pay careful care to the task being performed. Interestingly,
work within the distributed control community dealing with optimal assignment and selection of
sensors, leads to similar conclusions and to specific schemes for sensory adaptation.
The interplay between information theory and optimal control is a central pillar of modern control
theory, and we believe it must be accounted for in the computational neuroscience community.
Though statistical estimation theory has become central in neural coding issues, often through the
Cram?er-Rao bound, there have been few studies bridging the gap between partially observed control
and neural coding. We hope to narrow this gap by presenting a simple example where control
and estimation yield different conclusions. The remainder of the paper is organised as follows:
In section 1.1 we introduce the notation and concepts; In section 2 we derive expressions for the
cost-to-go of a linear-quadratic control system observed through spikes from a dense populations of
neurons; in section 3 we present the results and compare optimal codes for control and estimation
with point-process filtering, Kalman filtering and LQG control; in section 4 we discuss the results
and their implications.
1.1
Optimal Codes for Estimation and Control
We will deal throughout this paper with a dynamic system with state Xt , observed through noisy
sensory observations Zt , whose conditional distribution can be parametrised by a set of parameters
?, e.g., the widths and locations of the tuning curves of a population of neurons or the noise properties of the observation process. The conditional distribution is then given by P? (Zt |Xt = x).
Zt could stand for a diffusion process dependent on Xt (denoted Yt ) or a set of doubly-stochastic
Poisson processes dependent on Xt (denoted Ntm ). In that sense, the optimal Bayesian encoder for
an estimation problem, based on the Mean Squared Error (MSE) criterion, can be written as
2
??e = argmin E z E Xt Xt ? X?t (Zt ) Zt = z ,
?
? t (Zt ) = E [Xt |Zt ] is the posterior mean, computable, in the linear Gaussian case, by the
where X
Kalman filter. We will throughout this paper consider the MMSE in the equilibrium, that is, the
error in estimating Xt from long sequences of observations Z[0,t] . Similarly, considering a control
problem with a cost given by
Z T
C(X 0 , U 0 ) =
c(Xs , Us , s)ds + cT (XT ),
0
2
where X t = {Xs |s ? [t, T ]}, U t = {Us |s ? [t, T ]}, and so forth. We can define
??c = argmin E z min [E X t [C(X 0 , U 0 )|Z t = z]] .
Ut
?
The certainty equivalence principle states that given a control policy ? ? : X ? U which minimises
the cost C,
? ? = argmin C(X 0 , ?(X 0 )),
?
the optimal control policy for the partially observed problem given by noisy observations Z 0 of X 0
is given by
?CE (Z t ) = ? ? (E [X 0 |Z t ]) .
Note that we have used the notation ?(X 0 ) = {?(Xs ), s ? [0, T ]}.
2
Stochastic Optimal Control
In stochastic optimal control we seek to minimize the expected future cost incurred by a system
with respect to a control variable applied to that system. We will consider linear stochastic systems
governed by the SDE
dXt = (AXt + BUt ) dt + D1/2 dWt ,
(1a)
with a cost given by
Z T
C(X t , U t , t) =
Xs> QXs + Us> RUs ds + XT> QT XT .
(1b)
t
From Bellman?s optimality principle or variational analysis [11], it is well known that the optimal
control is given by Ut? = ?R?1 B > St Xt , where St is the solution of the Riccati equation
?S? t = Q + ASt + St A> ? St B > R?1 BSt ,
(2)
with boundary condition ST = QT . The expected future cost at time t and state x under the optimal
control is then given by
Z T
1 >
J(x, t) = min E [C(X t , U t , t)|Xt = x] = x St x +
Tr (DSs ) ds.
Ut
2
t
This is usually called the optimal cost-to-go. However, the system?s state is not always directly
accessible and we are often left with noisy observations of it. For a class of systems e.g. LQG
control, CE holds and the optimal control policy for the indirectly observed control problem is
simply the optimal control policy for the original control problem applied to the Bayesian estimate of
the system?s state. In that sense, if the CE were to hold for the system above observed through noisy
observations Yt of the state at time t, the optimal control would be given simply by the observationdependent control Ut? = ?R?1 B > St E [Xt |Yt ] [7].
Though CE, when applicable, gives us a simple way to determine the optimal control, when considering neural systems we are often interested in finding the optimal encoder, or the optimal observation model for a given system. That is equivalent to finding the optimal tuning function for a
given neuron model. Since CE neatly separates the estimation and control steps, it would be tempting to assume the optimal codes obtained for an estimation problem would also be optimal for an
associated control problem. We will show here that this is not the case.
As an illustration, let us consider the case of LQG with incomplete state information. One could,
for example, take the observations to be a secondary process Yt , which itself is a solution to
dYt = F Xt dt + G1/2 dVt ,
the optimal cost-to-go would then be given by [11]
J(y, t) = min E C(X t , U t , t) Y[0,t] = y
(3)
Ut
=?t> St ?t + Tr (Kt St ) +
Z
T
Z
Tr (DSs ) ds +
t
t
3
T
Tr Ss BR?1 B > Ss Ks ds,
where we have defined Y[0,t] = {Ys , s ? [0, t]}, ?t = E[Xt |Y[0,t] ] and Kt = cov[Xt |Y[0,t] ]. We
give a demonstration of these results in the SI, but for a thorough review see [11]. Note that through
the last term in equation (3) the cost-to-go now depends on the parameters of the Yt process. More
precisely, the variance of the distribution of Xs given Yt , for s > t obeys the ODE
K? t = AKt + Kt A> + D ? Kt F > G?1 F Kt .
(4)
One could then choose the matrices F and G in such a way as to minimise the contribution of
the rightmost term in equation (3). Note that in the LQG case this is not particularly interesting,
as the conclusion is simply that we should strive to make Kt as small as possible, by making the
term F > G?1 F as large as possible. This translates to choosing an observation process with very
strong steering from the unobserved process (large F ) and a very small noise (small G). One case
that provides some more interesting situations is if we consider a two-dimensional system, where
we are restricted to a noise covariance with constant determinant. That means the hypervolume
spanned by the eigenvectors of the covariance matrix is constant. We will compare this case with
the Poisson-coded case below.
2.1
LQG Control with Dense Gauss-Poisson Codes
Let us now consider the case of the system given by equation (1a), but instead of observing the
system directly we observe a set of doubly-stochastic Poisson processes {Ntm } with rates given by
1
>
?m (x) = ? exp ? (x ? ?m ) P ? (x ? ?m ) .
(5)
2
To clarify, the process Ntm is a counting process which counts how many spikes the neuron m
has fired up to time t. In that sense, the differential of the counting process dNtm will give the
spike train process, a sum of Dirac delta functions placed at the times of spikes fired by neuron
m. Here P ? denotes the pseudo-inverse of P , which is used to allow for tuning functions that
do not depend on certain coordinates of the stimulus x. Furthermore, we will assume that the
tuning centre ?m are such that the probability of observing a spike of any neuron at a given time
? = P ?m (x) is independent of the specific value of the world state x. This can be a consequence
?
m
of either a dense packing of the tuning centres ?m along a given dimension of x, or of an absolute
insensitivity to that aspect of x through a null element in the diagonal of P ? . This is often called
the dense coding hypothesis [12]. It can be readily be shown that the filtering distribution is given
by P (Xt |{N[0,t) }) = N (?t , ?t ), where the mean and covariance are solutions to the stochastic
differential equations (see [13])
X
?1 ?
d?t = (A?t + BUt ) dt +
?t I + P ? ?t
P (?m ? ?t ) dNtm ,
(6a)
m
?1
d?t = A?t + ?t A> + D dt ? ?t P ? ?t I + P ? ?t
dNt ,
(6b)
m
m
where we have defined ?t = E[Xt |{N[0,t]
}] and ?t = cov[Xt |{N[0,t]
}]. Note that we have also
P
m
m
m
defined N[0,t] = {Ns |s ? [0, t]}, the history of the process Ns up to time t, and Nt = m Ntm .
Using Lemma 7.1 from [11] provides a simple connection between the cost function and the solution
of the associated Ricatti equation for a stochastic process. We have
Z T
>
C(X t , U t , t) = XT> QT XT +
Xs QXs + Us> RUs ds
=Xt> St Xt
Z
+
t
T
(Us + R?1 B > Ss Xs )> R(Us + R?1 B > Ss Xs )ds
t
Z
+
T
Z
Tr(DSs )ds +
t
T
dWs> D>/2 Ss Xs ds
t
Z
+
T
Xs> Ss D1/2 dWs .
t
We can average over P (X t , N t |{N[0,t) }) to obtain the expected future cost. That gives us
"Z
# Z
T
T
>
?1 >
>
?1 >
?t St ?t +Tr(?t St )+E
(Us + R B Ss Xs ) R(Us + R B Ss Xs )ds {N[0,t) } +
Tr(DSs )ds
t
t
4
m
We can evaluate the average over P (X t , {N m
t }|{N[0,t) }) in two steps, by first averaging over the
m
Gaussian densities P (Xs |{N[0,s] }) and then over P ({N[0,s] }|{N[0,t) }). The average gives
Z
E
t
T
h
i
(Us + R?1 B > Ss ?s )> R(Us + R?1 B > Ss ?s ) + Tr Ss BR?1 B > Ss ?s ({N[0,s] }) ds {N[0,t) } ,
where ?s and ?s are the mean and variance associated with the distribution P (Xs |{N[0,s) }). Note
that choosing Us = ?R?1 B > Ss ?s will minimise the expression above, consistently with CE. The
optimal cost-to-go is therefore given by
J({N[0,t) }, t) =?>
t St ?t + Tr(?t St )
Z
Z T
Tr (DSs ) ds +
+
T
Tr Ss BR?1 B > Ss E ?s ({N[0,s] })|{N[0,t) } ds
t
t
(7)
Note that the only term in the cost-to-go function that depends on the parameters of the encoders is
the rightmost term and it depends on it only through the average over future paths of the filtering
variance ?s . The average of the future covariance matrix is precisely the MMSE for the filtering
problem conditioned on the belief state at time t [13]. We can therefore analyse the quality of an
encoder for a control task by looking at the values of the term on the right for different encoding
parameters. Furthermore,
since the
dynamics of ?t given by equation (6b) is Markovian, we can
write the average E ?s |{N[0,t) } as E [?s |?t ]. We will define then the function f (?, t) which
gives us the uncertainty-related expected future cost for the control problem as
Z T
f (?, t) =
Tr Ss BR?1 B > Ss E [?s |?t = ?] ds.
(8)
t
2.2
Mutual Information
Many results in information theory are formulated in terms of the mutual information of the communication channel P? (Y |X). For example, the maximum cost reduction achievable with R bits of
information about an unobserved variable X has been shown to be a function of the rate-distortion
function with the cost as the distortion function [14]. More recently there has also been a lot of
interest in the so-called I-MMSE relations, which provide connections between the mutual information of a channel and the minimal mean squared error of the Bayes estimator derived from the
same channel [15, 16]. The mutual information for the cases we are considering is not particularly
complex, as all distributions are Gaussians. Let us denote by ?0t the covariance of of the unobserved
process Xt conditioned on some initial Gaussian distribution P0 = N (?0 , ?0 ) at time 0. We can
then consider the Mutual Information between the stimulus at time t, Xt , and the observations up to
time t, Y[0,t] or N[0,t] . For the LQG/Kalman case we have simply
Z
I(Xt ; Y[0,t] |P0 ) = dx dyP (x, y) [log P (x|y) ? log P (x)] = log |?0t | ? log |?t |,
where ?t is a solution of equation (4). For the Dense Gauss-Poisson code, we can also write
Z
I(Xt ; Nt |P0 ) = dx dn P (x, n) [log P (x|n) ? log P (x)] = log |?0t | ? E N[0,t] log |?t (N[0,t] )| ,
where ?t (N[0,t] ) is a solution to the stochastic differential equation (6b) for the given value of N[0,t] .
3
Optimal Neural Codes for Estimation and Control
What could be the reasons for an optimal code for an estimation problem to be sub-optimal for a
control problem? We present examples that show two possible reasons for different optimal coding
strategies in estimation and control. First, one should note that control problems are often defined
over a finite time horizon. One set of classical experiments involves reaching for a target under
time constraints [3]. If we take the maximal firing rate of the neurons (?) to be constant while
varying the width of the tuning functions, this will lead the number of observed spikes to be inversely
proportional to the precision of those spikes, forcing a trade-off between the number of observations
5
and their quality. This trade-off can be tilted to either side in the case of control depending on the
information available at the start of the problem. If we are given complete information on the system
state at the initial time 0, the encoder needs fewer spikes to reliably estimate the system?s state
throughout the duration of the control experiment, and the optimal encoder will be tilted towards
a lower number of spikes with higher precision. Conversely, if at the beginning of the experiment
we have very little information about the system?s state, reflected in a very broad distribution, the
encoder will be forced towards lower precision spikes with higher frequency. These results are
discussed in section 3.1.
Secondly, one should note that the optimal encoder for estimation does not take into account the
differential weighting of different dimensions of the system?s state. When considering a multidimensional estimation problem, the optimal encoder will generally allocate all its resources equally
between the dimensions of the system?s state. In the framework presented we can think of the dimensions as the singular vectors of the tuning matrix P and the resources allocated to it are the singular
values. In this sense, we will consider a set of coding strategies defined by matrices P of constant
determinant in section 3.2. This constrains the overall firing rate of the population of neurons to be
constant, and we can then consider how the population will best allocate its observations between
these dimensions. Clearly, if we have an anisotropic control problem, which places a higher importance in controlling one dimension, the optimal encoder for the control problem will be expected to
allocate more resources to that dimension. This is indeed shown to be the case for the Poisson codes
considered, as well as for a simple LQG problem when we constrain the noise covariance to have
the same structure.
We do not mean our analysis to be exhaustive as to the factors leading to different optimal codes
in estimation and control settings, as the general problem is intractable, and indeed, is not even
separable. We intend this to be a proof of concept showing two cases in which the analogy between
control and estimation breaks down.
3.1
The Trade-off Between Precision and Frequency of Observations
In this section we consider populations of neurons with tuning functions as given by equation (5)
with tuning centers ?m distributed along a one- dimensional line. In the case of the OrnsteinUhlenbeck process these will be simply one-dimensional values ?m whereas in the case of the
stochastic oscillator, we will consider tuning centres of the form ?m = (?m , 0)> , filling only the
first dimension of the stimulus space. Note
? that in both cases the (dense) population firing rate
? = P ?m (x) will be given by ?
? = 2?p?/|??|, where ?? is the separation between neigh?
m
bouring tuning centres ?m .
The Ornstein-Uhlenbeck (OU) process controlled by a process Ut is given by the SDE
dXt = (bUt ? ?Xt )dt + D1/2 dWt .
Equation (7) can then be solved by simulating the dynamics of ?s . This has been considered extensively in [13] and we refer to the results therein. Specifically, it has been found that the dynamics of
the average can be approximated in a mean-field approach yielding surprisingly good results. The
evolution of the average posterior variance is given by the average of equation (6b), which involves
nonlinear averages over the covariances. These are intractable, but a simple mean-field approach
yields the approximate equation for the evolution of the average h?s i = E [?s |?0 ]
d h?s i
>
? h?s i P ? h?s i I + P ? h?s i ?1 .
= A h?s i + h?s i A> + D ? ?
ds
The alternative is to simulate the stochastic dynamics of ?t for a large number of samples and
compute numerical averages. These results can be directly employed to evaluate the optimal costto-go in the control problem f (?, t).
Alternatively, we can look at a system with more complex dynamics, and we take as an example the
stochastic damped harmonic oscillator given by the system of equations
X? t = Vt , dVt = bUt ? ?Vt ? ? 2 Xt dt + ? 1/2 dWt .
(9)
Furthermore, we assume that the tuning functions only depend on the position of the oscillator,
therefore not giving us any information about the velocity. The controller in turn seeks to keep the
6
0.40
0.35
0.30
0.12
M M SE
M M SE
0.18
0.16
0.14
0.10
0.08
0.25
0.20
0.06
0.15
0.04
0.10
0.02
0.05
0.007
1.35
0.006
1.30
1.25
0 ,t0 )
0.003
1.20
f(
0 ,0)
f(
0.005
0.004
b)
1.10
1.15
0.002
1.05
0.001
1.00
0.95
0.0
0.000
0
1
2
3
4
5
0.2
0.4
0.6
p
0.8
1.0
1.2
Figure 1: The trade-off between the precision and the frequency of spikes is illustrated for the OU process
(a) and the stochastic oscillator (b). In both figures, the initial condition has a very uncertain estimate of the
system?s state, biasing the optimal tuning width towards higher values. This forces the encoder to amass the
maximum number of observations within the duration of the control experiment. Parameters for figure (a) were:
T = 2, ? = 1.0, ? = 0.6, b = 0.2, ? = 0.1, ?? = 0.05, Q = 0.1, QT = 0.001, R = 0.1. Parameters for
figure (b) were T = 5, ? = 0.4, ? = 0.8, ? = 0.4, r = 0.4, q = 0.4, QT = 0, ? = 0.5, ?? = 0.1.
oscillator close to the origin while steering only the velocity. This can be achieved by the choice of
matrices A = (0, 1; ?? 2 , ??), B = (0, 0; 0, b), D = (0, 0; 0, ? 2 ), R = (0, 0; 0, r), Q = (q, 0; 0, 0)
and P = (p2 , 0; 0, 0).
In figure 1 we provide the uncertainty-dependent costs for LQG control, for the Poisson observed
control, as well as the MMSE for the Poisson filtering problem and for a Kalman-Bucy filter with the
same noise covariance matrix P . This illustrates nicely the difference between Kalman filtering and
the Gauss-Poisson filtering considered here. The Kalman filter MSE has a simple, monotonically
increasing dependence on the noise covariance, and one should simply strive to design sensors with
the highest possible precision (p = 0) to minimise the MMSE and control costs. The Poisson
case leads to optimal performance at a non-zero value of p. Importantly the optimal values of p
for estimation and control differ. Furthermore, in view of section 2.2, we also plotted the mutual
information between the process Xt and the observation process Nt , to illustrate that informationbased arguments would lead to the same optimal encoder as MMSE-based arguments.
3.2
Allocating Observation Resources in Anisotropic Control Problems
A second factor that could lead to different optimal encoders in estimation and control is the structure of the cost function C. Specifically, if the cost functions depends more strongly on a certain
coordinate of the system?s state, uncertainty in that particular coordinate will have a higher impact
on expected future costs than uncertainty in other coordinates. We will here consider two simple
linear control systems observed by a population of neurons restricted to a certain firing rate. This
can be thought of as a metabolic constraint, since the regeneration of membrane potential necessary
for action potential generation is one of the most significant metabolic expenditures for neurons
[17]. This will lead to a trade-off, where an increase in precision in one coordinate will result in a
decrease in precision in the other coordinate.
We consider a population of neurons whose tuning functions cover a two-dimensional space. Taking
a two-dimensional isotropic OU system with state Xt = (X1,t , X2,t )> where both dimensions are
uncoupled, we can consider a population with tuning centres ?m = (?1m , ?2m )> densely covering
the stimulus space. To consider a smoother class of stochastic systems we will also consider a
two-dimensional stochastic oscillator with state Xt = (X1,t , V1,t , X2,t , V2,t )> , where again, both
dimensions are uncoupled, and the tuning centres of the form ?m = (?1m , 0, ?2m , 0)> , covering
densely the position space, but not the velocity space.
Since we are interested in the case of limited resources, we will restrict ourselves to populations with a tuning matrix P yielding a constant population firing rate. We can parametrise
these simply as POU (?) = p2 Diag(tan(?), cotan(?)), for the OU case and POsc (?) =
7
estimation
kalman filter
mean field
stochastic
LQG control
0.80
1.2
0.75
1.0
0.8
0.70
0.6
0.65
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0.2
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0.0
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f (?0 , t0 )
f (?0 , t0 )
MMSE
0.85
0.2
0.4
0.6
0.8
?
1.0
1.2
1.4
Poisson MMSE
Kalman MMSE
Mean Field f
Stochastic f
LQG f
1.4
MMSE
0.90
1.6
b)
0.2
0.4
0.6
0.8
?
1.0
1.2
1.4
1.6
Figure 2: The differential allocation of resources in control and estimation for the OU process (left) and the
stochastic oscillator (right). Even though the estimation MMSE leads to a symmetric optimal encoder both in
the Poisson and in the Kalman filtering problem, the optimal encoders for the control problem are asymmetric,
allocating more resources to the first coordinate of the stimulus.
p2 Diag(tan(?), 0, cotan(?), 0) for the stochastic oscillator, where ? ? (0, ?/2). Note that this
? = 2?p?/(??)2 , independent of the specifics of the matrix P .
will yield the firing rate ?
We can then compare the performance of all observers with the same firing rate in both control and
estimation tasks. As mentioned, we are interested in control problems where the cost functions are
anisotropic, that is, one dimension of the system?s state vector contributes more heavily to the cost
function. To study this case we consider cost functions of the type
2
2
2
2
c(Xt , Ut ) = Q1 X1,t
+ Q2 X2,t
+ R1 U1,t
+ R2 U2,t
.
This again, can be readily cast into the formalism introduced above, with a suitable choice of matrices Q and R for both the OU process as for the stochastic oscillator. We will also consider the case
where the first dimension of Xt contributes more strongly to the state costs (i.e., Q1 > Q2 ).
The filtering error can be obtained from the formalism developed in [13] in the case of Poisson
observations and directly from the Kalman-Bucy equations in the case of Kalman filtering [18]. For
LQG control, one can simply solve the control problem for the system mentioned using the standard
methods (see e.g. [11]). The Poisson-coded version of the control problem can be solved using
either direct simulation of the dynamics of ?s or by a mean-field approach which has been shown
to yield excellent results for the system at hand. These results are summarised in figure 2, with
similar notation to that in figure 1. Note the extreme example of the stochastic oscillator, where the
optimal encoder is concentrating all the resources in one dimension, essentially ignoring the second
dimension.
4
Conclusion and Discussion
We have here shown that the optimal encoding strategies for a partially observed control problem is
not the same as the optimal encoding strategy for the associated state estimation problem. Note that
this is a natural consequence of considering noise covariances with a constant determinant in the
case of Kalman filtering and LQG control, but it is by no means trivial in the case of Poisson-coded
processes. For a class of stochastic processes for which the certainty equivalence principle holds we
have provided an exact expression for the optimal cost-to-go and have shown that minimising this
cost provides us with an encoder that in fact minimises the incurred cost in the control problem.
Optimality arguments are central to many parts of computational neuroscience, but it seems that
partial observability and the importance of combining adaptive state estimation and control have
rarely been considered in this literature, although supported by recent experiments. We believe the
present work, while treating only a small subset of the formalisms used in neuroscience, provides a
first insight into the differences between estimation and control. Much emphasis has been placed on
tracing the parallels between the two (see [19, 20], for example), but one must not forget to take into
account the differences as well.
8
References
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9
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4,850 | 5,391 | The Large Margin Mechanism
for Differentially Private Maximization
Kamalika Chaudhuri
UC San Diego
La Jolla, CA
[email protected]
Daniel Hsu
Columbia University
New York, NY
[email protected]
Shuang Song
UC San Diego
La Jolla, CA
[email protected]
Abstract
A basic problem in the design of privacy-preserving algorithms is the private maximization problem: the goal is to pick an item from a universe that (approximately)
maximizes a data-dependent function, all under the constraint of differential privacy. This problem has been used as a sub-routine in many privacy-preserving
algorithms for statistics and machine learning.
Previous algorithms for this problem are either range-dependent?i.e., their utility
diminishes with the size of the universe?or only apply to very restricted function
classes. This work provides the first general purpose, range-independent algorithm for private maximization that guarantees approximate differential privacy.
Its applicability is demonstrated on two fundamental tasks in data mining and machine learning.
1
Introduction
Differential privacy [17] is a cryptographically motivated definition of privacy that has recently
gained significant attention in the data mining and machine learning communities. An algorithm
for processing sensitive data enforces differential privacy by ensuring that the likelihood of any
outcome does not change by much when a single individual?s private data changes. Privacy is
typically guaranteed by adding noise either to the sensitive data, or to the output of an algorithm
that processes the sensitive data. For many machine learning tasks, this leads to a corresponding
degradation in accuracy or utility. Thus a central challenge in differentially private learning is to
design algorithms with better tradeoffs between privacy and utility for a wide variety of statistics
and machine learning tasks.
In this paper, we study the private maximization problem, a fundamental problem that arises while
designing privacy-preserving algorithms for a number of statistical and machine learning applications. We are given a sensitive dataset D ? X n comprised of records from n individuals. We are
also given a data-dependent objective function f : U ? X n ? R, where U is a universe of K items
to choose from, and f (i, ?) is (1/n)-Lipschitz for all i ? U. That is, |f (i, D0 ) ? f (i, D00 )| ? 1/n for
all i and for any D0 , D00 ? X n differing in just one individual?s entry. Always selecting an item that
exactly maximizes f (?, D) is generally non-private, so the goal is to select, in a differentially private
manner, an item i ? U with as high an objective f (i, D) as possible. This is a very general algorithmic problem that arises in many applications, include private PAC learning [25] (choosing the most
accurate classifier), private decision tree induction [21] (choosing the most informative split), private
frequent itemset mining [5] (choosing the most frequent itemset), private validation [12] (choosing
the best tuning parameter), and private multiple hypothesis testing [32] (choosing the most likely
hypothesis).
The most common algorithms for this problem are the exponential mechanism [28], and a computationally efficient alternative from [5], which we call the max-of-Laplaces mechanism. These
1
algorithms are general?they do not require any additional conditions on f to succeed?and hence
have been widely applied. However, a major limitation of both algorithms is that their utility suffers from an explicit range-dependence: the utility deteriorates with increasing universe size. The
range-dependence persists even when there is a single clear maximizer of f (?, D), or a few near
maximizers, and even when the maximizer remains the same after changing the entries of a large
number of individuals in the data. Getting around range-dependence has therefore been a goal for
designing algorithms for this problem.
This problem has also been addressed by recent algorithms of [31, 3], who provide algorithms that
are range-independent and satisfy approximate differential privacy, a relaxed version of differential
privacy. However, none of these algorithms is general; they explicitly fail unless additional special
conditions on f hold. For example, the algorithm from [31] provides a range-independent result only
when there is a single clear maximizer i? such that f (i? , D) is greater than the second highest value
by some margin; the algorithm from [3] also has restrictive conditions that limit its applicability (see
Section 2.2). Thus, a challenge is to develop a private maximization algorithm that is both rangeindependent and free of additional conditions; this is necessary to ensure that an algorithm is widely
applicable and provides good utility when the universe size is large.
In this work, we provide the first such general purpose range-independent private maximization
algorithm. Our algorithm is based on two key insights. The first is that private maximization is
easier when there is a small set of near maximizing items j ? U for which f (j, D) is close to the
maximum value maxi?U f (i, D). A plausible algorithm based on this insight is to first find a set
of near maximizers, and then run the exponential mechanism on this set. However, finding this
set directly in a differentially private manner is very challenging. Our second insight is that only
the number ` of near maximizers needs to be found in a differentially private manner?a task that
is considerably easier. Provided there is a margin between the maximum value and the (` + 1)-th
maximum value of f (i, D), running the exponential mechanism on the items with the top ` values
of f (i, D) results in approximate differential privacy as well as good utility.
Our algorithm, which we call the large margin mechanism, automatically exploits large margins
when they exist to simultaneously (i) satisfy approximate differential privacy (Theorem 2), as well as
(ii) provide a utility guarantee that depends (logarithmically) only on the number of near maximizers,
rather than the universe size (Theorem 3). We complement our algorithm with a lower bound,
showing that the utility of any approximate differentially private algorithm must deteriorate with
the number of near maximizers (Theorem 1). A consequence of our lower bound is that rangeindependence cannot be achieved with pure differential privacy (Proposition 1), which justifies our
relaxation to approximate differential privacy.
Finally, we show the applicability of our algorithm to two problems from data mining and machine
learning: frequent itemset mining and private PAC learning. For the first problem, an application
of our method gives the first algorithm for frequent itemset mining that simultaneously guarantees
approximate differential privacy and utility independent of the itemset universe size. For the second
problem, our algorithm achieves tight sample complexity bounds for private PAC learning analogous
to the shell bounds of [26] for non-private learning.
2
Background
This section reviews differential privacy and introduces the private maximization problem.
2.1
Definitions of Differential Privacy and Private Maximization
For the rest of the paper, we consider randomized algorithms A : X n ? ?(S) that take as input
datasets D ? X n comprised of records from n individuals, and output values in a range S. Two
datasets D, D0 ? X n are said to be neighbors if they differ in a single individual?s entry. A function
? : X n ? R is L-Lipschitz if |?(D) ? ?(D0 )| ? L for all neighbors D, D0 ? X n .
The following definitions of (approximate) differential privacy are from [17] and [20].
Definition 1 (Differential Privacy). A randomized algorithm A : X n ? ?(S) is said to be (?, ?)approximate differentially private if, for all neighbors D, D0 ? X n and all S ? S,
Pr(A(D) ? S) ? e? Pr(A(D0 ) ? S) + ?.
2
The algorithm A is ?-differentially private if it is (?, 0)-approximate differentially private.
Smaller values of the privacy parameters ? > 0 and ? ? [0, 1] imply stronger guarantees of privacy.
Definition 2 (Private Maximization). In the private maximization problem, a sensitive dataset
D ? X n comprised of records from n individuals is given as input; there is also a universe
U := {1, . . . , K} of K items, and a function f : U ? X n ? R such that f (i, ?) is (1/n)-Lipschitz
for all i ? U. The goal is to return an item i ? U such that f (i, D) is as large as possible while
satisfying (approximate) differential privacy.
Always returning the exact maximizer of f (?, D) is non-private, as changing a single individuals?
private values can potentially change the maximizer. Our goal is to design a randomized algorithm
that outputs an approximate maximizer with high probability. (We loosely refer to the expected
f (?, D) value of the chosen item as the utility of the algorithm.)
Note that this problem is different from private release of the maximum value of f (?, D); a solution for the latter is easily obtained by adding Laplace noise with standard deviation O(1/(?n)) to
maxi?U f (i, D) [17]. Privately returning a nearly maximizing item itself is much more challenging.
Private maximization is a core problem in the design of differentially private algorithms, and arises
in numerous statistical and machine learning tasks. The examples of frequent itemset mining and
PAC learning are discussed in Sections 4.1 and 4.2.
2.2
Previous Algorithms for Private Maximization
The standard algorithm for private maximization is the exponential mechanism [28]. Given a privacy
parameter ? > 0, the exponential mechanism randomly draws an item i ? U with probability
pi ? en?f (i,D)/2 ; this guarantees ?-differential privacy. While the exponential mechanism is widely
used because of its generality, a major limitation is its range-dependence?i.e., its utility diminishes
with the universe size K. To be more precise, consider the following example where X := U = [K]
and
1
f (i, D) := |{j ? [n] : Dj ? i}|
(1)
n
(where Dj is the j-th entry in the dataset D). When D = (1, 1, . . . , 1), there is a clear maximizer
i? = 1, which only changes when the entries of at least n/2 individuals in D change. It stands
to reason that any algorithm should report i = 1 in this case with high probability. However, the
exponential mechanism outputs i = 1 only with probability en?/2 /(K ? 1 + en?/2 ), which is small
unless n = ?(log(K)/?). This implies that the utility of the exponential mechanism deteriorates
with K.
Another general purpose algorithm is the max-of-Laplaces mechanism from [5]. Unfortunately, this
algorithm is also range-dependent. Indeed, our first observation is that all ?-differentially private
algorithms that succeed on a wide class of private maximization problems share this same drawback.
Proposition 1 (Lower bound for differential privacy). Let A be any ?-differentially private algorithm for private maximization, ? ? (0, 1), and n ? 2. There exists a domain X , a function
f : U ? X n ? R such that f (i, ?) is (1/n)-Lipschitz for all i ? U, and a dataset D ? X n such that:
!
log K?1
1
2
Pr f (A(D), D) > max f (i, D) ?
< .
i?U
?n
2
We remark that results similar to Proposition 1 have appeared in [23, 2, 10, 11, 7]; we simply reframe those results here in the context of private maximization.
Proposition 1 implies that in order to remove range-dependence, we need to relax the privacy notion.
We consider a relaxation of the privacy constraint to (?, ?)-approximate differential privacy with
? > 0.
The approximate differentially private algorithm from [31] applies in the case where there is a single
clear maximizer whose value is much larger than that of the rest. This algorithm adds Laplace noise
with standard deviation O(1/(?n)) to the difference between the largest and the second-largest values of f (?, D), and outputs the maximizer if this noisy difference is larger than O(log(1/?)/(?n));
3
otherwise, it outputs Fail. Although this solution has high utility for the example in (1) with
D = (1, 1, . . . , 1), it fails even when there is a single additional item j ? U with f (j, D) close to
the maximum value; for instance, D = (2, 2, . . . , 2).
[3] provides an approximate differentially private algorithm that applies when f satisfies a condition
called `-bounded growth. This condition entails the following: first, for any i ? U, adding a single
individual to any dataset D can either keep f (i, D) constant, or increase it by 1/n; and second,
f (i, D) can only increase in this case for at most ` items i ? U. The utility of this algorithm depends
only on log `, rather than log K. In contrast, our algorithm does not require the first condition.
Furthermore, to ensure that our algorithm only depends on log `, it suffices that there only be ?`
near maximizers, which is substantially less restrictive than the `-bounded growth condition.
As mentioned earlier, we avoid range-dependence with an algorithm that finds and optimizes over
near maximizers of f (?, D). We next specify what we mean by near maximizers using a notion of
margin.
3
The Large Margin Mechanism
We now our new algorithm for private maximization, called the large margin mechanism, along with
its privacy and utility guarantees.
3.1
Margins
We first introduce the notion of margin on which our algorithm is based. Given an instance of the
private maximization problem and a positive integer ` ? N, let f (`) (D) denote the `-th highest value
of f (?, D). We adopt the convention that f (K+1) (D) = ??.
Condition 1 ((`, ?)-margin condition). For any ` ? N and ? > 0, we say a dataset D ? X n satisfies
the (`, ?)-margin condition if
f (`+1) (D) < f (1) (D) ? ?
(i.e., there are at most ` items within ? of the top item according to f (?, D)).1
By convention, every dataset satisfies the (K, ?)-margin condition. Intuitively, a (`, ?)-margin condition with a relatively large ? implies that there are ?` near maximizers, so the private maximization
problem is easier when D satisfies an (`, ?)-margin condition with small `.
How large should ? be for a given `? The following lower bound suggests that in order to have
n = O(log(`)/?), we need ? to be roughly log(`)/(?n).
Theorem 1 (Lower bound for approximate differential privacy). Fix any ? ? (0, 1), ` > 1, and ? ?
[0, (1 ? exp(??))/(2(` ? 1))]; and assume n ? 2. Let A be any (?, ?)-approximate differentially
private algorithm, and ? := min{1/2, log((` ? 1)/2)/(n?)}. There exists a domain X , a function
f : U ? X n ? R such that f (i, ?) is (1/n)-Lipschitz for all i ? U, and a dataset D ? X n such that:
1. D satisfies the (`, ?)-margin condition.
1
2. Pr f (A(D), D) > f (1) (D) ? ? < .
2
A consequence of Theorem 1 is that complete range-independence for all (1/n)-Lipschitz functions f is not possible, even with approximate differential privacy. For instance, if D satisfies an
(`, ?(log(`)/(?n)))-margin condition only when ` = ?(K), then n must be ?(log(K)/?) in order
for an approximate differentially private algorithm to be useful.
3.2
Algorithm
The lower bound in Theorem 1 suggests the following algorithm. First, privately determine a pair
(`, ?), with ` is as small as possible and ? = ?(log(`)/(?n)), such that D satisfies the (`, ?)-margin
1
Our notion of margins here is different from the usual notion of margins from statistical learning that
underlies linear prediction methods like support vector machines and boosting. In fact, our notion is more
closely related to the shell decomposition bounds of [26], which we discuss in Section 4.2.
4
Algorithm 1 The large margin mechanism LMM(?, ?, D)
input Privacy parameters ? > 0 and ? ? (0, 1), database D ? X n .
output Item I ? U.
1: For each r = 1, 2, . . . , K, let
6
ln(3r/?)
1
1
r
(r)
t :=
1+
=O
,
+
log
n
?
n n?
?
3
6
3
12
3r(r + 1)
1
r
3
1
(r)
(r)
T :=
ln
+
ln +
ln
+t =O
+
log
.
n? 2?
n? ?
n?
?
n n?
?
2: Draw Z ? Lap(3/?).
3: Let m := f (1) (D) + Z/n. {Estimate of max value.}
iid
Draw G ? Lap(6/?) and Z1 , Z2 , . . . , ZK?1 ? Lap(12/?).
Let ` := 1. {Adaptively determine value ` such that D satisfies (`, t(`) )-margin condition.}
while ` < K do
if m ? f (`+1) (D) > (Z` + G)/n + T (`) then
Break out of while-loop with current value of `.
else
Let ` := ` + 1.
end if
end while
Let U` be the set of ` items in U with highest f (i, D) value (ties broken arbitrarily).
Draw I ? p where pi ? 1{i ? U` } exp(n?f (i, D)/6). {Exponential mechanism on top `
items.}
15: return I.
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
condition. Then, run the exponential mechanism on the set U` ? U of items with the ` highest
f (?, D) values. This sounds rather natural and simple, but a knee-jerk reaction to this approach is
that the set U` itself depends on the sensitive dataset D, and it may have high sensitivity in the sense
that membership of many items in U` can change when a single individual?s private value is changed.
Thus differentially private computation of U` appears challenging.
It turns out we do not need to guarantee the privacy of the set U` , but rather just of a valid (`, ?)
pair. This is essentially because when D satisfies the (`, ?)-margin condition, the probability that
the exponential mechanism picks an item i that occurs in U` when the sensitive dataset is D but not
in U` when the sensitive dataset is its neighbor D0 is very small.
Moreover, we can find such a valid (`, ?) pair using a differentially private search procedure based
on the sparse vector technique [22]. Combining these ideas gives a general (and adaptive) algorithm whose loss of utility due to privacy is only O(log(`/?)/?n) when the dataset satisfies a
(`, O(log(`/?)/(?n))-margin condition. We call this general algorithm the large margin mechanism (Algorithm 1), or LMM for short.
3.3
Privacy and Utility Guarantees
We first show that LMM satisfies approximate differential privacy.
Theorem 2 (Privacy guarantee). LMM(?, ?, ?) satisfies (?, ?)-approximate differential privacy.
The proof of Theorem 2 is in Appendix A.1. The following theorem, proved in Appendix A.2,
provides a guarantee on the utility of LMM.
Theorem 3 (Utility guarantee). Pick any ? ? (0, 1). Suppose D ? X n satisfies the (`? , ? ? )-margin
condition with
?
21
3
ln + T (` ) .
?? =
n? ?
Then with probability at least 1 ? ?, I := LMM(?, ?, D) satisfies
f (I, D) ? f (1) (D) ?
5
6 ln(2`? /?)
.
n?
?
(Above, T (` ) is as defined in Algorithm 1.)
Remark 1. Fix some ?, ? ? (0, 1). Theorem 3 states that if the dataset D satisfies the (`? , ? ? )margin condition, for some positive integer `? and ? ? = C log(`? /?)/(n?) for some universal
constant C > 0, then the value f (I, D) of the item I returned by LMM is within O(log(`? )/(n?))
of the maximum, with high probability. There is no explicit dependence on the cardinality K of the
universe U.
4
Illustrative Applications
We now describe applications of LMM to problems from data mining and machine learning.
4.1
Private Frequent Itemset Mining
Frequent Itemset Mining (FIM) is the following popular data mining problem: given the purchase
lists of users (say, for an online grocery store), the goal is to find the sets of items that are purchased together most often. The work of [5] provides the first differentially private algorithms
for FIM. However, as these algorithms rely on the exponential mechanism and the max-of-Laplaces
mechanism, their utilities degrade with the total number of possible itemsets. Subsequent algorithms
exploit other properties of itemsets or avoid directly finding the most frequent itemset [34, 27, 15, 8].
Let I be the set of items that can be purchased, and let B be the maximum length of an user?s
purchase list. Let U ? 2I be the family of itemsets of interest. For simplicity, we let U := Ir ?
i.e., all itemsets of size r?and consider the problem of picking the itemset with the (approximately)
highest frequency. This is a private maximization problem where D is the users? lists of purchased
items, and f (i, D) is the fraction of users who purchase an itemset i ? U. Let fmax be the highest
frequency of an itemset in D. Let L be the total number of itemsets with non-zero frequency, so
L ? n Br , which is |I|r whenever B |I|. Applying LMM gives the following guarantee.
Corollary 1. Suppose we use LMM(?, ?, ?) on the FIM problem above. Then there exists a constant
C > 0 such that the following holds. If fmax ? C ? log(L/?)/(n?), then with probability ? 1 ? ?,
the frequency of the itemset ILMM output by LMM is
log(L/?)
f (ILMM , D) ? fmax ? O
.
n?
In contrast, the itemset IEM returned by the exponential mechanism is only guaranteed to satisfy
r log(|I|/?)
f (IEM , D) ? fmax ? O
,
n?
which is significantly worse than Corollary 1 whenever L |I|r (as is typically the case). Second,
to ensure differential privacy by running the exponential mechanism, one needs a priori knowledge
of the set U (and thus the universe of items I) independently of the observed data; otherwise the
process will not be end-to-end differentially private. In contrast, our algorithm does not need to
know I in order to provide end-to-end differential privacy. Finally, unlike [31], our algorithm does
not require a gap between the top two itemset frequencies.
4.2
Private PAC Learning
We now consider private PAC learning with a finite hypothesis class H with bounded VC dimension
d [25]. Here, the dataset D consists of n labeled training examples drawn iid from a fixed distribution. The error err(h) of a hypothesis h ? H is the probability that it misclassifies a random
example drawn from the same distribution. The goal is to return a hypothesis h ? H with error
as low as possible. A standard procedure that has been well-studied in the literature simply returns
? ? H of the empirical error err(h,
the minimizer h
c
D) computed on the training data D, but this
does not guarantee (approximate) differential privacy. The work of [25] instead uses the exponential
mechanism to select a hypothesis hEM ? H. With probability ? 1 ? ?0 ,
!
r
d log(n/?0 ) log |H| + log(1/?0 )
+
.
(2)
err(hEM ) ? min err(h) + O
h?H
n
?n
6
The dependence on log |H| is improved to d log |?| by [7] when the data entries come from a finite set ?. The subsequent work of [4] introduces the notion of representation dimension, and
shows how it relates to differentially private learning in the discrete and finite case, and [3] provides improved convergence bounds with approximate differential privacy that exploit the structure
of some specific hypothesis classes. For the case of infinite hypothesis classes and continuous data
distributions, [10] shows that distribution-free private PAC learning is not generally possible, but
distribution-dependent learning can be achieved under certain conditions.
We provide a sample complexity bound of a rather different character compared to previous work.
Our bound only relies on uniform convergence properties of H, and can be significantly tighter
than the bounds from [25] when the number of hypotheses with error close to minh?H err(h) is
small. Indeed, the bounds are a private analogue of the shell bounds of [26], which characterize
the structure of the hypothesis class as a function of the properties of a decomposition based on
hypotheses? error rates. In many situation, these bounds are significantly tighter than those that do
not involve the error distributions.
p
Following [26], we divide the hypothesis class H into R = O( n/(d log n)) shells; the t-th shell
H(t) is defined by
(
)
r
d log(n/?0 )
0
H(t) := h ? H : err(h) ? min
err(h ) + C0 t
.
h0 ?H
n
Above, C0 > 0 is the constant from uniform convergence bounds?i.e., C0 is the smallest
c > 0 such
p
that for all h ? H, with probability ? 1 ? ?0 , we have |err(h,
c
D) ? err(h)| ? c d log(n/?0 )/n.
Observe that H(t + 1) ? H(t); and moreover,
p with probability ? 1 ? ?0 , all h ? H(t) have
0
err(h,
c
D) ? minh0 ?H err(h ) + C0 ? (t + 1) d log(n/?0 )/n.
Let t? (n) as the smallest integer t ? N such that
?
log(|H(t + 1)|) + log(1/?)
C0 ? dn log n
?
t
C
where C > 0 is the constant from Remark 1. Then, with probability ? 1 ? ?0 , the dataset D with
f = 1? err
c satisfies the (`, ?)-margin condition, with ` = |H(t? (n)+1)| and ? = C log(|H(t? (n)+
1)|/?)/(?n). Therefore, we have the following guarantee for applying LMM to this problem.
Corollary 2. Suppose we use LMM(?, ?, ?) on the learning problem above (with U = H and f =
1 ? err).
c Then, with probability ? 1 ? ?0 ? ?, the hypothesis hLMM returned by LMM satisfies
!
r
d log(n/?0 ) log(|H(t? (n) + 1)|/?)
err(hLMM ) ? min err(h) + O
+
.
h?H
n
?n
The dependence on log |H| from (2) is replaced here by log(|H(t? (n) + 1)|/?), which can be vastly
smaller, as discussed in [26].
5
Additional Related Work
There has been a large amount of work on differential privacy for a wide range of statistical and machine learning tasks over the last decade [6, 30, 13, 21, 33, 24, 1]; for overviews, see [18] and [29]. In
particular, algorithms for the private maximization problem (and variants) have been used as subroutines in many applications; examples include PAC learning [25], principle component analysis [14],
performance validation [12], and multiple hypothesis testing [32].
A separation between pure and approximate differential privacy has been shown in several previous
works [19, 31, 3]. The first approximate differentially private algorithm that achieves a separation is
the Propose-Test-Release (PTR) framework [19]. Given a function, PTR determines an upper bound
on its local sensitivity at the input dataset through a search procedure; noise proportional to this
upper bound is then added to the actual function value. We note that the PTR framework does not
directly apply to our setting as the sensitivity is not generally defined for a discrete universe.
In the context of private PAC learning, the work of [3] gives the first separation between pure and
approximate differential privacy. In addition to using the algorithm from [31], they devise two
7
additional algorithmic techniques: a concave maximization procedure for learning intervals, and an
algorithm for the private maximization problem under the `-bounded growth condition discussed
in Section 2.2. The first algorithm is specific to their problem and does not appear to apply to
general private maximization problems. The second algorithm has a sample complexity bound of
n = O(log(`)/?) when the function f satisfies the `-bounded growth condition.
Lower bounds for approximate differential privacy have been shown by [7, 16, 11, 9], and the proof
of our Theorem 1 borrows some techniques from [11].
6
Conclusion and Future Work
In this paper, we have presented the first general and range-independent algorithm for approximate
differentially private maximization. The algorithm automatically adapts to the available large margin
properties of the sensitive dataset, and reverts to worst-case guarantees when such properties are
lacking. We have illustrated the applicability of the algorithm in two fundamental problems from
data mining and machine learning; in future work, we plan to study other applications where rangeindependence is a substantial boon.
Acknowledgments. We thank an anonymous reviewer for suggesting the simpler variant of LMM
based on the exponential mechanism. (The original version of LMM used a max of truncated exponentials mechanism, which gives the same guarantees up to constant factors.) This work was
supported in part by the NIH under U54 HL108460 and the NSF under IIS 1253942.
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9
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4,851 | 5,392 | Extremal Mechanisms for Local Differential Privacy
Peter Kairouz1
Sewoong Oh2
Pramod Viswanath1
1
Department of Electrical & Computer Engineering
2
Department of Industrial & Enterprise Systems Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801, USA
{kairouz2,swoh,pramodv}@illinois.edu
Abstract
Local differential privacy has recently surfaced as a strong measure of privacy
in contexts where personal information remains private even from data analysts.
Working in a setting where the data providers and data analysts want to maximize
the utility of statistical inferences performed on the released data, we study the
fundamental tradeoff between local differential privacy and information theoretic
utility functions. We introduce a family of extremal privatization mechanisms,
which we call staircase mechanisms, and prove that it contains the optimal privatization mechanism that maximizes utility. We further show that for all information
theoretic utility functions studied in this paper, maximizing utility is equivalent
to solving a linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive
since it has a number of variables that is exponential in the data size. To account
for this, we show that two simple staircase mechanisms, the binary and randomized response mechanisms, are universally optimal in the high and low privacy
regimes, respectively, and well approximate the intermediate regime.
1
Introduction
In statistical analyses involving data from individuals, there is an increasing tension between the
need to share the data and the need to protect sensitive information about the individuals. For
example, users of social networking sites are increasingly cautious about their privacy, but still find
it inevitable to agree to share their personal information in order to benefit from customized services
such as recommendations and personalized search [1, 2]. There is a certain utility in sharing data for
both data providers and data analysts, but at the same time, individuals want plausible deniability
when it comes to sensitive information.
For such systems, there is a natural core optimization problem to be solved. Assuming both the
data providers and analysts want to maximize the utility of the released data, how can they do so
while preserving the privacy of participating individuals? The formulation and study of an optimal
framework addressing this tradeoff is the focus of this paper.
Local differential privacy. The need for data privacy appears in two different contexts: the local
privacy context, as in when individuals disclose their personal information (e.g., voluntarily on
social network sites), and the global privacy context, as in when institutions release databases of
information of several people or answer queries on such databases (e.g., US Government releases
census data, companies like Netflix release proprietary data for others to test state of the art data
analytics). In both contexts, privacy is achieved by randomizing the data before releasing it. We
study the setting of local privacy, in which data providers do not trust the data collector (analyst).
Local privacy dates back to Warner [29], who proposed the randomized response method to provide
plausible deniability for individuals responding to sensitive surveys.
1
A natural notion of privacy protection is making inference of information beyond what is released
hard. Differential privacy has been proposed in the global privacy context to formally capture this
notion of privacy [11, 13, 12]. In a nutshell, differential privacy ensures that an adversary should
not be able to reliably infer whether or not a particular individual is participating in the database
query, even with unbounded computational power and access to every entry in the database except
for that particular individual?s data. Recently, the notion of differential privacy has been extended
to the local privacy context [10]. Formally, consider a setting where there are n data providers each
owning a data Xi defined on an input alphabet X . In this paper, we shall deal, almost exclusively,
with finite alphabets. The Xi ?s are independently sampled from some distribution P? parameterized
by ? ? {0, 1}. A statistical privatization mechanism Qi is a conditional distribution that maps
Xi ? X stochastically to Yi ? Y, where Y is an output alphabet possibly larger than X . The
Yi ?s are referred to as the privatized (sanitized) views of Xi ?s. In a non-interactive setting where
the individuals do not communicate with each other and the Xi ?s are independent and identically
distributed, the same privatization mechanism Q is used by all individuals. For a non-negative ?, we
follow the definition of [10] and say that a mechanism Q is ?-locally differentially private if
sup
S??(Y),x,x0 ?X
Q(S|Xi = x)
? e? ,
Q(S|Xi = x0 )
(1)
where ?(Y) denotes an appropriate ?-field on Y.
Information theoretic utilities for statistical analyses. The data analyst is interested in the statistics of the data as opposed to individual samples. Naturally, the utility should also be measured in
terms of the distribution rather than sample quantities. Concretely, consider a client-server setting,
where each client with data Xi sends a privatized version of the data Yi , via an ?-locally differentially private privatization mechanism Q. Given the privatized views {Yi }ni=1 , the data analyst wants
to make inferences based on the induced marginal distribution
Z
M? (S) ?
Q(S|x)dP? (x) ,
(2)
for S ? ?(Y) and ? ? {0, 1}. The power to discriminate data generated from P0 to data generated
from P1 depends on the ?distance? between the marginals M0 and M1 . To measure the ability of
such statistical discrimination, our choice of utility of a particular privatization mechanism Q is an
information theoretic quantity called Csisz?ar?s f -divergence defined as
Z
dM0
dM1 ,
(3)
Df (M0 ||M1 ) = f
dM1
for some convex function f such that f (1) = 0. The Kullback-Leibler (KL) divergence
Dkl (M0 ||M1 ) is a special case with f (x) = x log x, and so is the total variation kM0 ? M1 kTV
with f (x) = (1/2)|x ? 1|. Such f -divergences capture the quality of statistical inference, such as
minimax rates of statistical estimation or error exponents in hypothesis testing [28]. As a motivating
example, suppose a data analyst wants to test whether the data is generated from P0 or P1 based on
privatized views Y1 , . . . , Yn . According to Chernoff-Stein?s lemma, for a bounded type I error probability, the best type II error probability scales as e?n Dkl (M0 ||M1 ) . Naturally, we are interested in
finding a privatization mechanism Q that minimizes the probability of error by solving the following
constraint maximization problem
maximize
Q?D?
Dkl (M0 ||M1 ) ,
(4)
where D? is the set of all ?-locally differentially private mechanisms satisfying (1). Motivated by
such applications in statistical inference, our goal is to provide a general framework for finding
optimal privatization mechanisms that maximize the f -divergence between the induced marginals
under local differential privacy.
Contributions. We study the fundamental tradeoff between local differential privacy and f divergence utility functions. The privacy-utility tradeoff is posed as a constrained maximization
problem: maximize f -divergence utility functions subject to local differential privacy constraints.
This maximization problem is (a) nonlinear: f -divergences are convex in Q; (b) non-standard: we
are maximizing instead of minimizing a convex function; and (c) infinite dimensional: the space
of all differentially private mechanisms is uncountable. We show, in Theorem 2.1, that for all f divergences, any ?, and any pair of distributions P0 and P1 , a finite family of extremal mechanisms
2
(a subset of the corner points of the space of privatization mechanisms), which we call staircase
mechanisms, contains the optimal privatization mechanism. We further prove, in Theorem 2.2, that
solving the original problem is equivalent to solving a linear program, the outcome of which is the
optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has 2|X | variables. To account for this, we show that two simple staircase mechanisms
(the binary and randomized response mechanisms) are optimal in the high and low privacy regimes,
respectively, and well approximate the intermediate regime. This contributes an important progress
in the differential privacy area, where the privatization mechanisms have been few and almost no exact optimality results are known. As an application, we show that the effective sample size reduces
from n to ?2 n under local differential privacy in the context of hypothesis testing.
Related work. Our work is closely related to the recent work of [10] where an upper bound on
Dkl (M0 ||M1 ) was derived under the same local differential privacy setting. Precisely, Duchi et. al.
proved that the KL-divergence maximization problem in (4) is at most 4(e? ? 1)2 kP1 ? P2 k2T V .
This bound was further used to provide a minimax bound on statistical estimation using information
theoretic converse techniques such as Fano?s and Le Cam?s inequalities.
In a similar spirit, we are also interested in maximizing information theoretic quantities of the
marginals under local differential privacy. We generalize the results of [10], and provide stronger
results in the sense that we (a) consider a broader class of information theoretic utilities; (b) provide explicit constructions of the optimal mechanisms; and (c) recover the existing result of [10,
Theorem 1] (with a stronger condition on ?).
While there is a vast literature on differential privacy, exact optimality results are only known for a
few cases. The typical recipe is to propose a differentially private mechanism inspired by [11, 13,
26, 20], and then establish its near-optimality by comparing the achievable utility to a converse, for
example in principal component analysis [8, 5, 19, 24], linear queries [21, 18], logistic regression [7]
and histogram release [25]. In this paper, we take a different route and solve the utility maximization
problem exactly.
Optimal differentially private mechanisms are known only in a few cases. Ghosh et. al. showed
that the geometric noise adding mechanism is optimal (under a Bayesian setting) for monotone
utility functions under count queries (sensitivity one) [17]. This was generalized by Geng et. al.
(for a worst-case input setting) who proposed a family of mechanisms and proved its optimality
for monotone utility functions under queries with arbitrary sensitivity [14, 16, 15]. The family of
optimal mechanisms was called staircase mechanisms because for any y and any neighboring x and
x0 , the ratio of Q(y|x) to Q(y|x0 ) takes one of three possible values e? , e?? , or 1. Since the optimal
mechanisms we develop also have an identical property, we retain the same nomenclature.
2
Main results
In this section, we give a formal definition for staircase mechanisms and show that they are the
optimal solutions to maximization problems of the form (5). Using the structure of staircase mechanisms, we propose a combinatorial representation for this family of mechanisms. This allows us
to reduce the nonlinear program of (5) to a linear program with 2|X | variables. Potentially, for
any instance of the problem, one can solve this linear program to obtain the optimal privatization
mechanism, albeit with significant computational challenges since the number of variables scales
exponentially in the alphabet size. To address this, we prove that two simple staircase mechanisms,
which we call the binary mechanism and the randomized response mechanism, are optimal in high
and low privacy regimes, respectively. We also show how our results can be used to derive upper
bounds on f -divergences under privacy. Finally, we give a concrete example illustrating the exact
tradeoff between privacy and statistical inference in the context of hypothesis testing.
2.1
Optimality of staircase mechanisms
Consider a random variable X ? X generated according to P? , ? ? {0, 1}. The distribution of the
privatized output Y , whenever X is distributed according to P? , is represented by M? and given by
(2). We are interested in characterizing the optimal solution of
maximize
Q?D?
Df (M0 ||M1 ) ,
3
(5)
where D? is the set of all ?-differentially private mechanisms satisfying, for all x, x0 ? X and y ? Y,
Q(y|x)
(6)
0 ? ln
? ?.
Q(y|x0 )
This includes maximization over information theoretic quantities of interest in statistical estimation
and hypothesis testing such as total variation, KL-divergence, and ?2 -divergence [28]. In general
this is a complicated nonlinear program: we are maximizing a convex function in Q; further, the
dimension of Q might be unbounded: the optimal privatization mechanism Q? might produce an
infinite output alphabet Y. The following theorem proves that one never needs an output alphabet
larger than the input alphabet in order to achieve the maximum divergence, and provides a combinatorial representation of the optimal solution.
Theorem 2.1. For any ?, any pair of distributions P0 and P1 , and any f -divergence, there exists an
optimal mechanism Q? maximizing the f -divergence in (5) over all ?-locally differentially private
mechanisms, such that
Q? (y|x)
(7)
ln
? {0, ?} ,
Q? (y|x0 )
for all y ? Y, x, x0 ? X and the output alphabet size is at most equal to the input alphabet size:
|Y| ? |X |.
The optimal solution is an extremal mechanism, since the absolute value of the log-likelihood ratios
can only take one of the two extremal values (see Figure 1). We refer to such a mechanism as a
staircase mechanism, and define the family of staircase mechanisms as
S? ? {Q | satisfying (7)} .
This family includes all the optimal mechanisms (for all choices of ? ? 0, P0 , P1 and f ), and since
(7) implies (6), staircase mechanisms are locally differentially private.
e?
3+e?
e?
1+e?
y=1
1
3+e?
1
1+e?
y=1
2
3
2
4
x=1
2
3
4
5
x=1
2
3
4
Figure 1: Examples of staircase mechanisms: the binary and randomized response mechanisms.
For global differential privacy, we can generalize the definition of staircase mechanisms to hold
for all neighboring database queries x, x0 (or equivalently within some sensitivity), and recover all
known existing optimal mechanisms. Precisely, the geometric mechanism shown to be optimal in
[17], and the mechanisms shown to be optimal in [14, 16] (also called staircase mechanisms) are
special cases of the staircase mechanisms defined above. We believe that the characterization of
these extremal mechanisms and the analysis techniques developed in this paper can be of independent interest to researchers interested in optimal mechanisms for global privacy and more general
utilities.
Combinatorial representation of the staircase mechanisms. Now that we know staircase mechanisms are optimal, we can try to combinatorially search for the best staircase mechanism for any
fixed ?, P0 , P1 , and f . To this end, we give a simple representation of all staircase mechanisms,
exploiting the fact that they are scaled copies of a finite number of patterns.
Let Q ? R|X |?|Y| be a staircase mechanism and k = |X | denote the input alphabet size. Then,
using the definition of staircase mechanisms, Q(y|x)/Q(y|x0 ) ? {e?? , 1, e? } and each column
Q(y|?) must be proportional to one of the canonical staircase patterns. For example, when k = 3,
4
k
there are 2k = 8 canonical patterns. Define a staircase pattern matrix S (k) ? {1, e? }k?(2 ) taking
values either 1 or e? , such that the i-th column of S (k) has a staircase pattern corresponding to the
binary representation of i ? 1 ? {0, . . . , 2k ? 1}. We order the columns of S (k) in this fashion for
notational convenience. For example,
#
"
1 1 1 1 e? e? e? e?
(3)
?
?
?
?
1 1 e e .
S = 1 1 e e
1 e? 1 e? 1 e? 1 e?
For all values of k, there are exactly 2k such patterns, and any column of Q(y|?) is a scaled version
of one of the columns of S (k) . Using this ?pattern? matrix, we will show that we can represent (an
equivalence class of) any staircase mechanism Q as
Q = S (k) ? ,
2k ?2k
(8)
(k)
where ? ? R
is a diagonal matrix representing the scaling of the columns of S . We can
now formulate the problem of maximizing the divergence between the induced marginals as a linear
program and prove that it is equivalent the original nonlinear program.
Theorem 2.2. For any ?, any pair of distributions P0 and P1 , and any f -divergence, the nonlinear
program of (5) and the following linear program have the same optimal value
k
maximize
??R2k ?2k
2
X
(k)
?(Si )?ii
(9)
i=1
(k)
S ?1 = 1 ,
? is a diagonal matrix ,
??0,
P
P
(k)
(k)
(k) P
(k)
(k)
where ?(Si ) = ( x?X P1 (x)Sxi )f ( x?X P0 (x)Sxi / x?X P1 (x)Sxi ) and Si is the i-th
k
P2
(k)
column of S (k) , such that Df (M0 ||M1 ) =
i=1 ?(Si )?ii . The solutions of (5) and (9) are
related by (8).
subject to
The infinite dimensional nonlinear program of (5) is now reduced to a finite dimensional linear
program. The first constraint ensures that we get a valid probability transition matrix Q = S (k) ?
with a row sum of one. One could potentially solve this LP with 2k variables but its computational
complexity scales exponentially in the alphabet size k = |X |. For practical values of k this might
not always be possible. However, in the following section, we give a precise description for the
optimal mechanisms in the high privacy and low privacy regimes.
In order to understand the above theorem, observe that both the f -divergences and the differential
privacy constraints are invariant under permutation (or relabelling) of the columns of a privatization
mechanism Q. For example, the KL-divergence Dkl (M0 ||M1 ) does not change if we permute the
columns of Q. Similarly, both the f -divergences and the differential privacy constraints are invariant
under merging/splitting of outputs with the same pattern. To be specific, consider a privatization
mechanism Q and suppose there exist two outputs y and y 0 that have the same pattern, i.e. Q(y|?) =
C Q(y 0 |?) for some positive constant C. Then, we can consider a new mechanism Q0 by merging the
two columns corresponding to y and y 0 . Let y 00 denote this new output. It follows that Q0 satisfies
the differential privacy constraints and the resulting f -divergence is also preserved. Precisely, using
the fact that Q(y|?) = C Q(y 0 |?), it follows that
P
P
(Q(y|x) + Q(y 0 |x))P0 (x)
(1 + C) x Q(y|x)P0 (x)
M00 (y 00 )
M0 (y)
M0 (y 0 )
x
P
P
=
=
=
=
,
0
M10 (y 00 )
(1 + C) x Q(y|x)P1 (x)
M1 (y)
M1 (y 0 )
x (Q(y|x) + Q(y |x))P1 (x)
and thus the corresponding f -divergence is invariant:
M (y)
M (y 0 )
M 0 (y 00 )
0
0
0
f
M1 (y) + f
M1 (y 0 ) = f
M10 (y 00 ) .
0
M1 (y)
M1 (y )
M10 (y 00 )
We can naturally define equivalence classes for staircase mechanisms that are equivalent up to a
permutation of columns and merging/splitting of columns with the same pattern:
[Q] = {Q0 ? S? | exists a sequence of permutations and merge/split of columns from Q0 to Q} . (10)
5
To represent an equivalence class, we use a mechanism in [Q] that is ordered and merged to match
the patterns of the pattern matrix S (k) . For any staircase mechanism Q, there exists a possibly
different staircase mechanism Q0 ? [Q] such that Q0 = S (k) ? for some diagonal matrix ? with
nonnegative entries. Therefore, to solve optimization problems of the form (5), we can restrict our
attention to such representatives of equivalent classes. Further, for privatization mechanisms of the
form Q = S (k) ?, the f -divergences take the form given in (9), a simple linear function of ?.
2.2
Optimal mechanisms in high and low privacy regimes
For a given P0 and P1 , the binary mechanism is defined as a staircase mechanism with only two
outputs y ? {0, 1} satisfying (see Figure 1)
e?
e?
if P0 (x) ? P1 (x) ,
if P0 (x) < P1 (x) ,
1+e?
1+e?
Q(0|x) =
Q(1|x)
=
(11)
1
1
if
P
(x)
<
P
(x)
.
if P0 (x) ? P1 (x) .
?
?
0
1
1+e
1+e
Although this mechanism is extremely simple, perhaps surprisingly, we will establish that this is the
optimal mechanism when high level of privacy is required. Intuitively, the output is very noisy in the
high privacy regime, and we are better off sending just one bit of information that tells you whether
your data is more likely to have come from P0 or P1 .
Theorem 2.3. For any pair of distributions P0 and P1 , there exists a positive ?? that depends on P0
and P1 such that for any f -divergences and any positive ? ? ?? , the binary mechanism maximizes
the f -divergence between the induced marginals over all ?-local differentially private mechanisms.
This implies that in the high privacy regime, which is a typical setting studied in much of differential
privacy literature, the binary mechanism is a universally optimal solution for all f -divergences in (5).
In particular this threshold ?? is universal, in that it does not depend on the particular choice of which
f -divergence we are maximizing. This is established by proving a very strong statistical dominance
using Blackwell?s celebrated result on comparisons of statistical experiments [4]. In a nutshell, we
prove that for sufficiently small ?, the output of any ?-locally differentially private mechanism can be
simulated from the output of the binary mechanism. Hence, the binary mechanism dominates over
all other mechanisms and at the same time achieves the maximum divergence. A similar idea has
been used previously in [27] to exactly characterize how much privacy degrades under composition.
The optimality of binary mechanisms is not just for high privacy regimes. The next theorem shows
that it is the optimal solution of (5) for all ?, when the objective function is the total variation
Df (M0 ||M1 ) = kM0 ? M1 kTV .
Theorem 2.4. For any pair of distributions P0 and P1 , and any ? ? 0, the binary mechanism maximizes total variation between the induced marginals M0 and M1 among all ?-local differentially
private mechanisms.
When maximizing the KL-divergence between the induced marginals, we show that the binary
mechanism still achieves a good performance for all ? ? C where C ? ?? now does not depend on
P0 and P1 . For the special case of KL-divergence, let OPT denote the maximum value of (5) and
BIN denote the KL-divergence when the binary mechanism is used. The next theorem shows that
BIN
?
2(e?
1
OPT .
+ 1)2
Theorem 2.5. For any ? and for any pair of distributions P0 and P1 , the binary mechanism is an
1/(2(e? + 1)2 ) approximation of the maximum KL-divergence between the induced marginals M0
and M1 among all ?-locally differentially private mechanisms.
Note that 2(e? + 1)2 ? 32 for ? ? 1, and ? ? 1 is a common regime of interest in differential
privacy. Therefore, we can always use the simple binary mechanism in this regime and the resulting
divergence is at most a constant factor away from the optimal one.
The randomized response mechanism is defined as a staircase mechanism with the same set of
outputs as the input, Y = X , satisfying (see Figure 1)
(
e?
if y = x ,
|X |?1+e?
Q(y|x) =
1
if y 6= x .
|X |?1+e?
6
It is a randomization over the same alphabet where we are more likely to give an honest response.
We view it as a multiple choice generalization of the randomized response proposed by Warner [29],
assuming equal privacy level for all choices. We establish that this is the optimal mechanism when
low level of privacy is required. Intuitively, the noise is small in the low privacy regime, and we
want to send as much information about our current data as allowed, but no more. For a special case
of maximizing KL-divergence, we show that the randomized response mechanism is the optimal
solution of (5) in the low privacy regime (? ? ?? ).
Theorem 2.6. There exists a positive ?? that depends on P0 and P1 such that for any P0 and P1 , and
all ? ? ?? , the randomized response mechanism maximizes the KL-divergence between the induced
marginals over all ?-locally differentially private mechanisms.
2.3
Lower bounds in differential privacy
In this section, we provide converse results on the fundamental limit of differentially private mechanisms. These results follow from our main theorems and are of independent interest in other applications where lower bounds in statistical analysis are studied [3, 21, 6, 9]. For example, a bound
similar to (12) was used to provide converse results on the sample complexity for statistical estimation with differentially private data in [10].
Corollary 2.7. For any ? ? 0, let Q be any conditional distribution that guarantees ?-local differential privacy. Then, for any pair of distributions P0 and P1 , and any positive ? > 0, there exists a
positive ?? that depends on P0 , P1 , and ? such that for any ? ? ?? , the induced marginals M0 and
M1 satisfy the bound
Dkl M0 ||M1 + Dkl M1 ||M0
?
2(1 + ?)(e? ? 1)2
P0 ? P1
2 .
?
TV
(e + 1)
(12)
This follows from Theorem 2.3 and the fact that under the binary mechanism, Dkl M0 ||M1 =
P0 ? P1
2 (e? ? 1)2 /(e? + 1) + O(?3 ) . Compared to [10, Theorem 1], we recover their bound
TV
of 4(e? ? 1)2 kP0 ? P1 k2TV with a smaller constant. We want to note that Duchi et al.?s bound holds
for all values of ? and uses different techniques. However no achievable mechanism is provided. We
instead provide an explicit mechanism that is optimal in high privacy regime.
Similarly, in the high privacy regime, we can show the following converse result.
Corollary 2.8. For any ? ? 0, let Q be any conditional distribution that guarantees ?-local differential privacy. Then, for any pair of distributions P0 and P1 , and any positive ? > 0, there exists a
positive ?? that depends on P0 , P1 , and ? such that for any ? ? ?? , the induced marginals M0 and
M1 satisfy the bound
Dkl M0 ||M1 + Dkl M1 ||M0
? Dkl (P0 ||P1 ) ? (1 ? ?)G(P0 , P1 )e?? .
where G(P0 , P1 ) =
P
x?X (1
? P0 (x)) log(P1 (x)/P0 (x)).
This follows directly from Theorem 2.6 and the fact that under the randomized response mechanism,
Dkl (M0 ||M1 ) = Dkl (P0 ||P1 ) ? G(P0 , P1 )e?? + O(e?2? ) .
Similarly for total variation, we can get the following converse result. This follows from Theorem
2.4 and explicitly computing the total variation achieved by the binary mechanism.
Corollary 2.9. For any ? ? 0, let Q be any conditional distribution that guarantees ?-local differential privacy. Then,
marginals M0 and M1
for any
pair of distributions P0 and
P1 , the induced
satisfy the bound
M0 ? M1
TV ? ((e? ? 1)/(e? + 1))
P0 ? P1
TV , and equality is achieved
by the binary mechanism.
2.4
Connections to hypothesis testing
Under the data collection scenario, there are n individuals each with data Xi sampled from a distribution P? for a fixed ? ? {0, 1}. Let Q be a non-interactive privatization mechanism guaranteeing
?-local differential privacy. The privatized views {Yi }ni=1 , are independently distributed according
to one of the induced marginals M0 or M1 defined in (2).
7
Given the privatized views {Yi }ni=1 , the data analyst wants to test whether they were generated from
M0 or M1 . Let the null hypothesis be H0 : Yi ?s are generated from M0 , and the alternative hypothesis H1 : Yi ?s are generated from M1 . For a choice of rejection region R ? Y n , the probability
of false alarm (type I error) is ? = M0n (R) and the probability of miss detection (type II error) is
? = M1n (Y n \ R). Let ? ? = minR?Y n ,?<?? ? denote the minimum type II error achievable while
keeping type I error rate at most ?? . According to Chernoff-Stein lemma, we know that
?
1
log ? ? = ?Dkl (M0 ||M1 ) .
n?? n
lim
Suppose the analyst knows P0 , P1 , and Q. Then, in order to achieve optimal asymptotic error rate,
one would want to maximize the KL-divergence between the induced marginals over all ?-locally
differentially private mechanisms Q. Theorems 2.3 and 2.6 provide an explicit construction of the
optimal mechanisms in high and low privacy regimes. Further, our converse results in Section 2.3
provides a fundamental limit on the achievable error rates under differential privacy. Precisely,
with data collected from an ?-locally differentially privatization mechanism, one cannot achieve an
asymptotic type II error smaller than
?
1
(1 + ?)(e? ? 1)2
(1 + ?)(e? ? 1)2
2
log ? ? ? ?
kP
?
P
k
?
?
Dkl (P0 ||P1 ) ,
0
1
TV
n?? n
(e? + 1)
2(e? + 1)
lim
whenever ? ? ?? , where ?? is dictated by Theorem 2.3. In the equation above, the second inequality
follows from Pinsker?s inequality. Since (e? ? 1)2 = O(?2 ) for small ?, the effective sample size is
now reduced from n to ?2 n. This is the price of privacy. In the low privacy regime where ? ? ?? ,
for ?? dictated by Theorem 2.6, one cannot achieve an asymptotic type II error smaller than
?
1
log ? ? ? ?Dkl (P0 ||P1 ) + (1 ? ?)G(P0 , P1 )e?? .
n?? n
lim
3
Discussion
In this paper, we have considered f -divergence utility functions and assumed a setting where individuals cannot collaborate (communicate with each other) before releasing their data. It turns out that
the optimality results presented in Section 2 are general and hold for a large class of convex utility
function [22]. In addition, the techniques developed in this work can be generalized to find optimal
privatization mechanisms in a setting where different individuals can collaborate interactively and
each individual can be an analyst [23].
Binary hypothesis testing is a canonical statistical inference problem with a wide range of applications. However, there are a number of nontrivial and interesting extensions to our work. Firstly,
in some scenarios the Xi ?s could be correlated (e.g., when different individuals observe different functions of the same random variable). In this case, the data analyst is interested in inferring whether the data was generated from P0n or P1n , where P?n is one of two possible joint priors on X1 , ..., Xn . This is a challenging problem because knowing Xi reveals information about
Xj , j 6= i. Therefore, the utility maximization problems for different individuals are coupled in
this setting. Secondly, in some cases the data analyst need not have access to P0 and P1 , but
rather two classes of prior distribution P?0 and P?1 for ?0 ? ?0 and ?1 ? ?1 . Such problems
are studied under the rubric of universal hypothesis testing and robust hypothesis testing. One
possible direction is to select the privatization mechanism that maximizes the worst case utility:
Q? = arg maxQ?D? min?0 ??0 ,?1 ??1 Df (M?0 ||M?1 ), where M?? is the induced marginal under
P?? . Finally, the more general problem of private m-ary hypothesis testing is also an interesting but
challenging one. In this setting, the Xi ?s can follow one of m distributions P0 , P1 , ..., Pm?1 , and
therefore the Yi ?s can follow one of m distributions M0 , M1 , ..., Mm?1 . The
Putility can be defined
as the average f -divergence between any two distributions: 1/(m(m ? 1)) i6=j Df (Mi ||Mj ), or
the worst case one: mini6=j Df (Mi ||Mj ).
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4,852 | 5,393 | Reputation-based Worker Filtering in Crowdsourcing
Srikanth Jagabathula1 Lakshminarayanan Subramanian2,3 Ashwin Venkataraman2,3
1
Department of IOMS, NYU Stern School of Business
Department of Computer Science, New York University
3
CTED, New York University Abu Dhabi
[email protected] {lakshmi,ashwin}@cs.nyu.edu
2
Abstract
In this paper, we study the problem of aggregating noisy labels from crowd workers to infer the underlying true labels of binary tasks. Unlike most prior work
which has examined this problem under the random worker paradigm, we consider
a much broader class of adversarial workers with no specific assumptions on their
labeling strategy. Our key contribution is the design of a computationally efficient
reputation algorithm to identify and filter out these adversarial workers in crowdsourcing systems. Our algorithm uses the concept of optimal semi-matchings
in conjunction with worker penalties based on label disagreements, to assign a
reputation score for every worker. We provide strong theoretical guarantees for
deterministic adversarial strategies as well as the extreme case of sophisticated
adversaries where we analyze the worst-case behavior of our algorithm. Finally,
we show that our reputation algorithm can significantly improve the accuracy of
existing label aggregation algorithms in real-world crowdsourcing datasets.
1
Introduction
The growing popularity of online crowdsourcing services (e.g. Amazon Mechanical Turk, CrowdFlower etc.) has made it easy to collect low-cost labels from the crowd to generate training datasets
for machine learning applications. However, these applications remain vulnerable to noisy labels
introduced either unintentionally by unreliable workers or intentionally by spammers and malicious
workers [10, 11]. Recovering the underlying true labels in the face of noisy input in online crowdsourcing environments is challenging due to three key reasons: (a) Workers are often anonymous
and transient and can provide random or even malicious labels (b) The reliabilities or reputations of
the workers are often unknown (c) Each task may receive labels from only a (small) subset of the
workers.
Several existing approaches aim to address the above challenges under the following standard setup.
There is a set T of binary tasks, each with a true label in { 1, 1}. A set of workers W are asked
to label the tasks, and the assignment of the tasks to the workers can be represented by a bipartite
graph with the workers on one side, tasks on the other side, and an edge connecting each worker
to the set of tasks she is assigned. We term this the worker-task assignment graph. Workers are
assumed to generate labels according to a probabilistic model - given a task t, a worker w provides
the true label with probability pw . Note that every worker is assumed to label each task independent
of other tasks. The goal then is to infer the underlying true labels of the tasks by aggregating the
labels provided by the workers. Prior works based on the above model can be broadly classified into
two categories: machine-learning based and linear-algebra based. The machine-learning approaches
are typically based on variants of the EM algorithm [3, 16, 24, 14]. These algorithms perform well
in most scenarios, but they lack any theoretical guarantees. More recently, linear-algebra based
algorithms [9, 6, 2] have been proposed, which provide guarantees on the error in estimating the
true labels of the tasks (under appropriate assumptions), and have also been shown to perform well
on various real-world datasets. While existing work focuses on workers making random errors,
recent work and anecdotal evidence have shown that worker labeling strategies that are common in
practice do not fit the standard random model [19]. Specific examples include vote pollution attacks
1
on Digg [18], malicious behavior in social media [22, 12] and low-precision worker populations in
crowdsourcing experiments [4].
In this paper, we aim to go beyond the standard random model and study the problem of inferring
the true labels of tasks under a much broader class of adversarial worker strategies with no specific
assumptions on their labeling pattern. For instance, deterministic labeling, where the workers always
give the same label, cannot be captured by the standard random model. Also, malicious workers can
employ arbitrary labeling patterns to degrade the accuracy of the inferred labels. Our goal is to
accurately infer the true labels of the tasks without restricting workers? strategies.
Main results. Our main contribution is the design of a reputation algorithm to identify and filter out
adversarial workers in online crowdsourcing systems. Specifically, we propose 2 computationally
efficient algorithms to compute worker reputations using only the labels provided by the workers
(see Algorithms 1 and 2), which are robust to manipulation by adversaries. We compute worker
reputations by assigning penalties to a worker for each task she is assigned. The assigned penalty is
higher for tasks on which there is ?a lot? of disagreement with the other workers. The penalties are
then aggregated in a ?load-balanced? manner using the concept of optimal semi-matchings [7]. The
reputation algorithm is designed to be used in conjunction with any of the existing label aggregation
algorithms that are designed for the standard random worker model: workers with low reputations1
are filtered out and the aggregation algorithm is used on the remaining labels. As a result, our
algorithm can be used to boost the performance of existing label aggregation algorithms.
We demonstrate the effectiveness of our algorithm using a combination of strong theoretical guarantees and empirical results on real-world datasets. Our analysis considers three scenarios. First,
we consider the standard setting in which workers are not adversarial and provide labels according
to the random model. In this setting, we show that when the worker-task assignment graph is (l, r)regular, the reputation scores are proportional to the reliabilities of the workers (see Theorem 1), so
that only unreliable workers are filtered out. As a result, our reputation scores are consistent with
worker reliabilities in the absence of adversarial workers. The analysis becomes significantly complicated for more general graphs (a fact observed in prior works; see [2]); hence, we demonstrate
improvements using simulations and experiments on real world datasets. Second, we evaluate the
performance of our algorithm in the presence of workers who use deterministic labeling strategies
(always label 1 or 1). For these strategies, when the worker-task assignment graph is (l, r)-regular,
we show (see Theorem 2) that the adversarial workers receive lower reputations than their ?honest?
counterparts, provided honest workers have ?high enough? reliabilities ? the exact bound depends
on the prevalence of tasks with true label 1, the fraction of adversarial workers and the average
reliability of the honest workers.
Third, we consider the case of sophisticated adversaries, i.e. worst-case adversarial workers whose
goal is to maximize the number of tasks they affect (i.e. cause to get incorrect labels). Under
this assumption, we provide bounds on the ?damage? they can do: We prove that irrespective of
the label aggregation algorithm (as long as it is agnostic to worker/task identities), there is a nontrivial minimum fraction of tasks whose true label is incorrectly inferred. This bound depends on
the graph structure of the honest worker labeling pattern (see Theorem 3 for details). Our result
is valid across different labeling patterns and a large class of label aggregation algorithms, and
hence provides fundamental limits on the damage achievable by adversaries. Further, we propose a
label aggregation algorithm utilizing the worker reputations computed in Algorithm 2 and prove the
existence of an upper bound on the worst-case accuracy in inferring the true labels (see Theorem 4).
Finally, using several publicly available crowdsourcing datasets (see Section 4), we show that our
reputation algorithm: (a) can help in enhancing the accuracy of state-of-the-art label aggregation
algorithms (b) is able to detect workers in these datasets who exhibit certain ?non-random? strategies.
Additional Related Work: In addition to the references cited above, there have been works which
use gold standard tasks, i.e. tasks whose true label is already known [17, 5, 11] to correct for worker
bias. [8] proposed a way of quantifying worker quality by transforming the observed labels into soft
posterior labels based on the estimated confusion matrix [3]. The authors in [13] propose an empirical Bayesian algorithm to eliminate workers who label randomly without looking at the particular
task (called spammers), and estimate the consensus labels from the remaining workers. Both these
1
As will become evident later, reputations are measures of how adversarial a worker is and are different
from reliabilities of workers.
2
works use the estimated parameters to define ?good workers? whereas we compute the reputation
scores using only the labels provided by the workers. The authors in [20] focus on detecting specific
kinds of spammers and then replace their labels with new workers. We consider all types of adversarial workers, not just spammers and don?t assume access to a pool of workers who can be asked
to label the tasks.
2
Model and reputation algorithms
Notation. Consider a set of tasks T having true labels in {1, 1}. Let yj denotes the true label of a
task tj 2 T and suppose that the tasks are sampled from a population in which the prevalence of the
positive tasks is 2 [0, 1], so that there is a fraction of tasks with true label 1. A set of workers
W provide binary labels to the tasks in T . We let G denote the bipartite worker-task assignment
graph where an edge between worker wi and task tj indicates that wi has labeled tj . Further, let
wi (tj ) denote the label provided by worker wi to task tj , where we set wi (tj ) = 0 if worker wi did
not label task tj . For a task tj , let Wj ? W denote the set of workers who labeled tj and likewise,
for a worker wi , let Ti denote the set of tasks the worker has labeled. Denote by d+
j (resp. dj ) the
|W |?|T |
number of workers labeling task tj as 1 (resp. 1). Finally, let L 2 {1, 0, 1}
denote the
matrix representing the labels assigned by the workers to the tasks, i.e. Lij = wi (tj ). Given L, the
goal is to infer the true labels yj of the tasks.
Worker model. We consider the setting in which workers may be honest or adversarial. That is,
W = H [ A with H \ A = ;. Honest workers are assumed to provide labels according to a
probabilistic model: for task tj with true label yj , worker wi provides label yj with probability pi
and yj with probability 1 pi . Note that the parameter pi doesn?t depend on the particular task that
the worker is labeling, so an honest worker labels each task independently. It is standard to define the
reliability of an honest worker as ?i = 2pi 1, so that we have ?i 2 [ 1, 1]. Further, we assume that
the honest workers are sampled from a population with average reliability ? > 0. Adversaries, on
the other hand, may adopt any arbitrary (deterministic or randomized) labeling strategy that cannot
be described using the above probabilistic process. For instance, the adversary could always label
all tasks as 1, irrespective of the true label. Another example is when the adversary decides her
labels based on existing labels cast by other workers (assuming the adversary has access to such
information). Note however, that adversarial workers need not always provide the incorrect labels.
Essentially, the presence of such workers breaks the assumptions of the model and can adversely
impact the performance of aggregation algorithms. Hence, our focus in this paper is on designing
algorithms to identify and filter out such adversarial workers. Once this is achieved, we can use
existing state-of-the-art label aggregation algorithms on the remaining labels to infer the true labels
of the tasks.
To identify these adversarial workers, we propose an algorithm for computing ?reputation? or ?trust?
scores for each worker. More concretely, we assign penalties (in a suitable way) to every worker and
higher the penalty, worse the reputation of the worker. First note that any task that has all 1 labels
(or 1 labels) does not provide us with any information about the reliabilities of the workers who
labeled the task. Hence, we focus on the tasks that have both 1 and 1 labels and we call this set
the conflict set Tcs . Further, since we have no ?side? information on the identities of workers, any
reputation score computation must be based solely on the labels provided by the workers.
We start with the following basic idea to compute reputation scores: a worker is penalized for every
?conflict? s/he is involved in (a task in the conflict set the worker has labeled on). This idea is
motivated by the fact that in an ideal scenario, where all honest workers have a reliability ?i = 1,
a conflict indicates the presence of an adversary and the reputation score aims to capture a measure
of the number of conflicts each worker is involved in: the higher the number of conflicts, the worse
the reputation score. However, a straightforward aggregation of penalties for each worker may overpenalize (honest) workers who label several tasks.
In order to overcome the issue of over-penalizing (honest) workers, we propose two techniques:
(a) soft and (b) hard assignment of penalties. In the soft assignment of penalties (Algorithm 1),
we assign a penalty of 1/d+
j to all workers who label 1 on task tj and 1/dj to all workers who
label 1 on task tj . Then, for each worker, we aggregate the penalties across all assigned tasks by
taking the average. The above assignment of penalties implicitly rewards agreements by making
the penalty inversely proportional to the number of other workers that agree with a worker. Further,
taking the average normalizes for the number of tasks labeled by the worker. Since we expect the
3
honest workers to agree with the majority more often than not, we expect this technique to assign
lower penalties to honest workers when compared to adversaries. The soft assignment of penalties
can be shown to perform quite well in identifying low reliability and adversarial workers (refer
to Theorems 1 and 2). However, it may still be subject to manipulation by more ?sophisticated?
adversaries who can adapt and modify their labeling strategy to target certain tasks and to inflate
the penalty of specific honest workers. In fact for such worst-case adversaries, we can show that
(Theorem 3) given any honest worker labeling pattern, there exists a lower bound on the number of
tasks whose true label cannot be inferred correctly, by any label aggregation algorithm.
To address the case of these sophisticated adversaries, we propose a hard penalty assignment scheme
(Algorithm 2) where the key idea is not to distribute the penalty evenly across all workers; but to
only choose two workers to penalize per conflict task: one ?representative? worker among those
who labeled 1 and another ?representative? worker among those who labeled 1. While choosing
such workers, the goal is to pick these representative workers in a load-balanced manner to ?spread?
the penalty across all workers, so that it is not concentrated on one/few workers. The final penalty of
each worker is the sum of the accrued penalties across all the (conflict) tasks assigned to the worker.
Intuitively, such hard assignment of penalties will penalize workers with higher degrees and many
conflicts (who are potential ?worst-case? adversaries), limiting their impact.
We use the concept of optimal semi-matchings [7] on bipartite graphs to distribute penalties in a
load balanced manner, which we briefly discuss here. For a bipartite graph B = (U, V, E), a semimatching in B is a set of edges M ? E such that each vertex in V is incident to exactly one edge in
M (note that vertices in U could be incident to multiple edges in M ). A semi-matching generalizes
the notion of matchings on bipartite graphs. To define an optimal semi-matching, we introduce a cost
function for a semi-matching - for each u 2 U , let degM (u) denote the number of edges in M that
PdegM (u)
M (u)+1)
are incident to u and let costM (u) be defined as costM (u)
i = degM (u)(deg
.
2
P= i=1
An optimal semi-matching then, is one which minimizes u2U costM (u). This notion of cost is
motivated by the load balancing problem for scheduling tasks on machines (refer to [7] for further
details). Intuitively, an optimal semi-matching fairly matches the V -vertices across the U -vertices
so that the maximum ?load? on any U -vertex is minimized.
Algorithm 1 SOFT PENALTY
Algorithm 2 HARD PENALTY
1: Input: W , T and L
1: Input: W , T and L
2: For every task tj 2 Tcs , assign penalty sij 2: Create a bipartite graph B cs as follows:
to each worker wi 2 Wj as follows:
sij = d1+ if Lij = 1
j
sij =
if Lij =
1
pen(wi ) =
P
1
dj
3: Output: Penalty of worker wi
3
tj 2Ti \ Tcs
sij
|Ti \ Tcs |
(i) Each worker wi 2 W is represented by
a node on the left (ii) Each task tj 2 Tcs is
represented by two nodes on the right t+
j and
tj (iii) Add the edge (wi , t+
)
if
L
=
1 or
ij
j
edge (wi , tj ) if Lij = 1.
3: Compute an optimal semi-matching OSM on
B cs and let di ( degree of worker wi in OSM
4: Output: Penalty of worker wi pen(wi ) = di
Theoretical Results
Soft penalty. We focus on (l, r)-regular worker-task assignment graphs in which every worker
is assigned l tasks and every object is labeled by r workers. The performance of our reputation
algorithms depend on the reliabilities of the workers as well as the true labels of the tasks. Hence,
we consider the following probabilistic model: for a given (l, r)-regular worker-task assignment
graph G, the reliabilities of the workers and the true labels of tasks are sampled independently (from
distributions described in Section 2). We then analyze the performance of our algorithms as the
task degree r (and hence number of workers |W |) goes to infinity. Specifically, we establish the
following results (the proofs of all theorems are in the supplementary material).
Theorem 1. Suppose there are no adversarial workers, i.e A = ; and that the worker-task assignment graph G is (l, r)-regular. Then, with high probability as r ! 1, for any pair of workers wi
and wi0 , ?i < ?i0 =) pen(wi ) > pen(wi0 ), i.e. higher reliability workers are assigned lower
penalties by Algorithm 1.
4
The probability in the above theorem is according to the model described above. Note that the theorem justifies our definition of the reputation scores by establishing their consistency with worker
reliabilities in the absence of adversarial workers. Next, consider the setting in which adversarial
workers adopt the following uniform strategy: label 1 on all assigned tasks (the 1 case is symmetric).
Theorem 2. Suppose that the worker-task assignment graph G is (l, r)-regular. Let the probability
of an arbitrary worker being honest be q and suppose each adversary adopts the uniform strategy
in which she labels 1 on all the tasks assigned to her. Denote an arbitrary honest worker as hi and
any adversary as a. Then, with high probability as r ! 1, we have
1. If
=
1
2
and ?i = 1, then pen(hi ) < pen(a) if and only if q >
2. If
=
1
2
and q >
1
1+? ,
1
1+?
then pen(hi ) < pen(a) if and only if
?i
(2
q)(1
(2
q q 2 ?2 ) q 2 ?2
q)q + q 2 ?2
The above theorem establishes that when adversaries adopt the uniform strategy, the soft-penalty
algorithm assigns lower penalties to honest workers whose reliability excess a threshold, as long as
the fraction of honest workers is ?large enough?. Although not stated, the result above immediately
extends (with a modified lower bound for ?i ) to the case when > 1/2, which corresponds to
adversaries adopting smart strategies by labeling based on the prevalence of positive tasks.
Sophisticated adversaries. Numerous real-world incidents show evidence of malicious worker behavior in online systems [18, 22, 12]. Moreover, attacks on the training process of ML models
have also been studied [15, 1]. Recent work [21] has also shown the impact of powerful adversarial attacks by administrators of crowdturfing (malicious crowdsourcing) sites. Motivated by these
examples, we consider sophisticated adversaries:
Definition 1. Sophisticated adversaries are computationally unbounded and colluding. Further,
they have knowledge of the labels provided by the honest workers and their goal is to maximize the
number of tasks whose true label is incorrectly identified.
We now raise the following question: In the presence of sophisticated adversaries, does there exist
a fundamental limit on the number of tasks whose true label can be correctly identified, irrespective
of the aggregation algorithm employed to aggregate the worker labels?
In order to answer the above question precisely, we introduce some notation. Let n = |W | and
m
m = |T |. Then, we represent any label aggregation algorithm as a decision rule R : L ! {1, 1} ,
which maps the observed labeling matrix L to a set of output labels for each task. Because of the
absence of any auxiliary information about the workers or the tasks, the class of decision rules, say
C, is invariant to permutations of the identities of workers and/or tasks. More precisely, C denotes
the class of decision rules that satisfy R(P LQ) = R(L)Q, for any n ? n permutation matrix P and
m ? m permutation matrix Q. We say that a task is affected if the decision rule outputs the incorrect
label for the task. We define the quality of a decision rule R(?) as the worst-case number of affected
tasks over all possible true labelings and adversary strategies with a fixed honest worker labeling
pattern. Fixing the honest worker labeling pattern allows isolation of the effect of the adversary
strategy on the accuracy of the decision rule. Considering the worst-case over all possible true
labels makes the metric robust to ground-truth assignments, which are typically application specific.
Next to formally define the quality, let BH denote the honest worker-task assignment graph and
y = (y1 , y2 , . . . , ym ) denote the vector of true labels for the tasks. Note that since the number of
affected tasks also depends on the actual honest worker labels, we further assume that all honest
workers have reliability ?i = 1, i.e they always label correctly. This assumption allows us to
attribute any mis-identification of true labels to the presence of adversaries because, otherwise, in
the absence of any adversaries, the true labels of all the tasks can be trivially identified. Finally, let
Sk denote the strategy space of k adversaries, where each strategy 2 Sk specifies the k ? m label
matrix provided by the adversaries. Since we do not restrict the adversary strategy in any way, it
k?m
follows that Sk = { 1, 0, 1}
. The quality of a decision rule is then defined as
n
o
y,
A?(R, BH , k) =
max
t
2
T
:
R
=
6
y
)
,
j
j
t
j
m
2Sk ,y2{1, 1}
5
where Rty, 2 {1, 1} is the label output by the decision rule R for task t when the true label vector
is y and the adversary strategy is . Note that our notation A?(R, BH , k) makes the dependence of
the quality measure on the honest worker-task assignment graph BH and the number of adversaries k
explicit. We answer the question raised above in the affirmative, i.e. there does exist a fundamental
limit on identification. In the theorem below, PreIm(T 0 ) is the set of honest workers who label
atleast one task in T 0 .
Theorem 3. Suppose that k = |A| and ?h = 1 for all honest workers h 2 H. Then, given
any honest worker-task assignment graph BH , there exists an adversary strategy ? 2 Sk that is
independent of any decision rule R 2 C such that
L ? max m A?(R, ? , y) 8R 2 C, where
y2{ 1,1}
1
max
|T 0 | ,
2 T 0 ?T : |PreIm(T 0 )|?k
and A?(R, ? , y) denotes the number of affected tasks under adversary strategy ? , decision rule
R, and true label vector y (with the assumption that max over an empty set is zero).
L=
We describe the main idea of the proof which proceeds in two steps: (i) we provide an explicit
construction of an adversary strategy ? that depends on BH and y, and (ii) we show the existence of
another true labeling y
? such that R outputs exactly the same labels in both scenarios. The adversary
labeling strategy we construct uses the idea of indistinguishability, which captures the fact that by
carefully choosing their labels, the adversaries can render themselves indistinguishable from honest
workers. Specifically, in the simple case when there is only one honest worker, the adversary simply
labels the opposite of the honest worker on all assigned tasks, so that each task has two labels of
opposite parity. It can be argued that since there is no other information, it is impossible for any
decision rule R 2 C to distinguish the honest worker from the adversary and hence identify the
true label of any task (better than a random guess). We extend this to the general case, where the
adversary ?targets? atmost k honest workers and derives a strategy based on the subgraph of BH
restricted to the targeted workers. The resultant strategy can be shown to result in incorrect labels
for atleast L tasks for some true labeling of the tasks.
Hard penalty. We now analyze the performance of the hard penalty-based reputation algorithm in
the presence of sophisticated adversarial workers. For the purposes of the theorem, we consider a
natural extension of our reputation algorithm to also perform label aggregation (see figure 1).
Theorem 4. Suppose that k = |A| and ?i = 1 for each honest worker, i.e an honest worker always
provides the correct label. Further, let d1
d2
???
d|H| denote the degrees of the honest
workers in the optimal semi-matching on BH . For any true labeling of the tasks and under the
penalty-based label aggregation algorithm (with the convention that di = 0 for i > |H|) :
Pk 1
1. There exists an adversary strategy ? such that the number of tasks affected
i=1 di .
2. No adversary strategy can affect more than U tasks where
Pk
(a) U = i=1 di , when atmost one adversary provides correct labels
P2k
(b) U = i=1 di , in the general case
A few remarks are in order. First, it can be shown [7] that for optimal semi-matchings, the degree
sequence is unique and therefore, the bounds in the theorem above are uniquely defined given BH .
Also, the assumption that ?i = 1 is required for analytical tractability, proving theoretical guarantees in crowd-sourced settings (even without adversaries) for general graph structures is notoriously
hard [2]. Note that the result of Theorem 4 provides both a lower and upper bound for the number
of tasks that can be affected by k adversaries when using the penalty-based aggregation algorithm.
The characterization we obtain is reasonably tight for the case when atmost 1 adversary provides
correct labels; in this case the gap between the upper and lower bounds is dk , which can be ?small?
for k large enough. However, our characterization is loose in the general case when adversaries can
P2k
label arbitrarily; here the gap is i=k di which we attribute to our proof technique and conjecture
Pk
that the upper bound of i=1 di also applies in the more general case.
4
Experiments
In this section, we evaluate the performance of our reputation algorithms on both synthetic and real
datasets. We consider the following popular label aggregation algorithms: (a) simple majority vot6
Random
MV
EM
KOS
KOS +
PRECISION
BEST
Malicious
Uniform
Low
High
Low
High
Low
High
9.9
-1.9
-4.3
-3.9
81.7
7.9
6.3
13.1
7.3
82.1
16.8
-1.6
-8.3
-8.3
92.5
15.6
-49.4
-98.7
-69.6
59.4
26.0
-1.2
-6.5
-6.0
80.8
15.0
-9.1
12.9
10.7
62.4
MV- SOFT MV- HARD MV- SOFT KOS MV- SOFT MV- HARD
PENALTY- BASED AGGREGATION
wt ( worker that task t is mapped
to in OSM in Algorithm 2
Output y(t) = 1 if dwt+ < dwt
y(t) = 1 if dwt+ > dwt and
y(t) = 0 otherwise
(here y refers to the label of the task
and dw refers to the degree of worker
w in OSM)
Figure 1: Left: Percentage decrease in incorrect tasks on synthetic data (negative indicates increase in incorrect
tasks). We implemented both SOFT and HARD and report the best outcome. Also reported is the precision
when removing 15 workers with the highest penalties. The columns specify the three types of adversaries
and High/Low indicates the degree bias of the adversaries. The probability that a worker is honest q was
set to 0.7 and the prevalence of positive tasks was set to 0.5. The numbers reported are an average over
100 experimental runs. The last row lists the combination with the best accuracy in each case. Right: The
penalty-based label aggregation algorithm.
ing MV (b) the EM algorithm [3] (c) the BP-based KOS algorithm [9] and (d) KOS +, a normalized
version of KOS that is amenable for arbitrary graphs (KOS is designed for random regular graphs),
and compare their accuracy in inferring the true labels of the tasks, when implemented in conjunction with our reputation algorithms. We implemented iterative versions of Algorithms 1(SOFT) and
2(HARD), where in each iteration we filtered out the worker with the highest penalty and recomputed
penalties for the remaining workers.
Synthetic Dataset. We analyzed the performance of our soft penalty-based reputation algorithm on
(l, r)-regular graphs in section 3. In many practical scenarios, however, the worker-task assignment
graph forms organically where the workers decide which tasks to label on. To consider this case, we
simulated a setup of 100 workers with a power-law distribution for worker degrees to generate the
bipartite worker-task assignment graph. We assume that an honest worker always labels correctly
(the results are qualitatively similar when honest workers make errors with small probability) and
consider three notions of adversaries: (a) random - who label each task 1 or 1 with prob. 1/2
(b) malicious - who always label incorrectly and (c) uniform - who label 1 on all tasks. Also,
we consider both cases - one where the adversaries are biased to have high degrees and the other
where they have low degrees. Further, we arbitrarily decided to remove 15% of the workers with the
highest penalties and we define precision as the percentage of workers filtered who were adversarial.
Figure 1 shows the performance improvement of the different benchmarks in the presence of our
reputation algorithm.
We make a few observations. First, we are successful in identifying random adversaries as well as
low-degree malicious and uniform adversaries (precision > 80%). This shows that our reputation
algorithms also perform well when worker-task assignment graphs are non-regular, complementing
the theoretical results (Theorems 1 and 2) for regular graphs. Second, our filtering algorithm can
result in significant reduction (upto 26%) in the fraction of incorrect tasks. In fact, in 5 out of 6 cases,
the best performing algorithm incorporates our reputation algorithm. Note that since 15 workers
are filtered out, labels from fewer workers are used to infer the true labels of the tasks. Despite
using fewer labels, we improve performance because the high precision of our algorithms results in
mostly adversaries being filtered out. Third, we note that when the adversaries are malicious and
have high degrees, the removal of 15 workers degrades the performance of the KOS (and KOS +) and
EM algorithms. We attribute this to the fact that while KOS and EM are able to automatically invert
the malicious labels, we discard these labels which hurts performance, since the adversaries have
high degrees. Finally, note that the SOFT (HARD) penalty algorithm tends to perform better when
adversaries are biased towards low (high) degrees, and this insight can be used to aid the choice of
the reputation algorithm to be employed in different scenarios.
Real Datasets. Next, we evaluated our algorithm on some standard datasets: (a) TREC2 : a collection
of topic-document pairs labeled as relevant or non-relevant by workers on AMT. We consider two
versions: stage2 and task2. (b) NLP [17]: consists of annotations by AMT workers for different
NLP tasks (1) rte - which provides binary judgments for textual entailment, i.e. whether one
2
http://sites.google.com/site/treccrowd/home
7
Dataset
rte
temp
bluebird
stage2
task2
MV
EM
KOS +
KOS
Base
Soft
Hard
Base
Soft
Hard
Base
Soft
Hard
Base
Soft
Hard
91.9
93.9
75.9
74.1
64.3
92.1(8)
93.9
75.9
74.1
64.3
92.5(3)
94.3(5)
75.9
81.4(3)
68.4(10)
92.7
94.1
89.8
64.7
66.8
92.7
94.1
89.8
65.3(6)
66.8
93.3(9)
94.1
89.8
78.9(2)
67.3(9)
49.7
56.9
72.2
74.5
57.4
88.8(9)
69.2(4)
75.9(3)
74.5
57.4
91.6(10)
93.7(3)
72.2
75.2(3)
66.7(10)
91.3
93.9
72.2
75.5
59.3
92.7(8)
94.3(7)
75.9(3)
76.6(2)
59.4(4)
92.8(10)
94.3(1)
72.2
77.2(3)
67.9(9)
82.5
81.6 81.7
84.7
62.1 73.2
79.9
78.4 79.8
aggregate 80.0 80.0
80.9
Table 1: Percentage accuracy of benchmark algorithms when combined with our reputation algorithms. For
each benchmark, the best performing combination is shown in bold. The number in the parentheses represents the number of workers filtered by our reputation algorithm (an absence indicates that no performance
improvement was achieved while removing upto 10 workers with the highest penalties).
sentence can be inferred from another (2) temp - which provides binary judgments for temporal
ordering of events. (c) bluebird [23] contains judgments differentiating two kinds of birds in
an image. Table 1 reports the best accuracy achieved when upto 10 workers are filtered using our
iterative reputation algorithms.
The main conclusion we draw is that our reputation algorithms are able to boost the performance
of state-of-the-art aggregation algorithms by a significant amount across the datasets: the average
improvement for MV and KOS + is 2.5%, EM is 3% and for KOS is almost 18%, when using the hard
penalty-based reputation algorithm. Second, we can elevate the performance of simpler algorithms
such as KOS and MV to the levels of the more general (and in some cases, complicated) EM algorithm. The KOS algorithm for instance, is not only easier to implement, but also has tight theoretical
guarantees when the underlying assignment graph is sparse random regular and further is robust to
different initializations [9]. The modified version KOS + can be used in graphs where worker degrees are skewed, but we are still able to enhance its accuracy. Our results provide evidence for the
fact that existing random models are inadequate in capturing the behavior of workers in real-world
datasets, necessitating the need for a more general approach. Third, note that the hard penalty-based
algorithm outperforms the soft version across the datasets. Since the hard penalty algorithm works
well when adversaries have higher degrees (a fact noticed in the simulation results above), this suggests the presence of high-degree adversarial workers in theses datasets. Finally, our algorithm was
successful in identifying the following types of ?adversaries?: (1) uniform workers who always label 1 or 1 (in temp, task2, stage2), (2) malicious workers who mostly label incorrectly (in
bluebird, stage2) and (3) random workers who label each task independent of its true label
(such workers were identified as ?spammers? in [13]). Therefore, our algorithm is able to identify a
broad set of adversary strategies in addition to those detected by existing techniques.
5
Conclusions
This paper analyzes the problem of inferring true labels of tasks in crowdsourcing systems against
a broad class of adversarial labeling strategies. The main contribution is the design of a reputationbased worker filtering algorithm that uses a combination of disagreement-based penalties and optimal semi-matchings to identify adversarial workers. We show that our reputation scores are consistent with the existing notion of worker reliabilities and further can identify adversaries that employ
deterministic labeling strategies. Empirically, we show that our algorithm can be applied in real
crowd-sourced datasets to enhance the accuracy of existing label aggregation algorithms. Further,
we analyze the scenario of worst-case adversaries and establish lower bounds on the minimum
?damage? achievable by the adversaries.
Acknowledgments
We thank the anonymous reviewers for their valuable feedback. Ashwin Venkataraman was supported by the Center for Technology and Economic Development (CTED).
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9
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4,853 | 5,394 | Feedback Detection for Live Predictors
Stefan Wager, Nick Chamandy, Omkar Muralidharan, and Amir Najmi
[email protected], {chamandy, omuralidharan, amir}@google.com
Stanford University and Google, Inc.
Abstract
A predictor that is deployed in a live production system may perturb the features
it uses to make predictions. Such a feedback loop can occur, for example, when a
model that predicts a certain type of behavior ends up causing the behavior it predicts, thus creating a self-fulfilling prophecy. In this paper we analyze predictor
feedback detection as a causal inference problem, and introduce a local randomization scheme that can be used to detect non-linear feedback in real-world problems. We conduct a pilot study for our proposed methodology using a predictive
system currently deployed as a part of a search engine.
1
Introduction
When statistical predictors are deployed in a live production environment, feedback loops can become a concern. Predictive models are usually tuned using training data that has not been influenced
by the predictor itself; thus, most real-world predictors cannot account for the effect they themselves
have on their environment. Consider the following caricatured example: A search engine wants to
train a simple classifier that predicts whether a search result is ?newsy? or not, meaning that the
search result is relevant for people who want to read the news. This classifier is trained on historical data, and learns that high click-through rate (CTR) has a positive association with ?newsiness.?
Problems may arise if the search engine deploys the classifier, and starts featuring search results that
are predicted to be newsy for some queries: promoting the search result may lead to a higher CTR,
which in turn leads to higher newsiness predictions, which makes the result be featured even more.
If we knew beforehand all the channels through which predictor feedback can occur, then detecting
feedback would not be too difficult. For example, in the context of the above example, if we knew
that feedback could only occur through some changes to the search result page that were directly
triggered by our model, then we could estimate feedback by running small experiments where we
turn off these triggering rules. However, in large industrial systems where networks of classifiers all
feed into each other, we can no longer hope to understand a priori all the ways in which feedback
may occur. We need a method that lets us detect feedback from sources we might not have even
known to exist.
This paper proposes a simple method for detecting feedback loops from unknown sources in live
systems. Our method relies on artificially inserting a small amount of noise into the predictions made
by a model, and then measuring the effect of this noise on future predictions made by the model. If
future model predictions change when we add artificial noise, then our system has feedback.
1
To understand how random noise can enable us to detect feedback, suppose that we have a model
with predictions y? in which tomorrow?s prediction y?(t+1) has a linear feedback dependence on today?s prediction y?(t) : if we increase y?(t) by , then y?(t+1) increases by
for some 2 R. Intuitively, we should be able to fit this slope by perturbing y?(t) with a small amount of noise
? ? N 0, ?2 and then regressing the new y?(t+1) against the noise; the reason least squares should
work here is that the noise ? is independent of all other variables by construction. The main contribution of this paper is to turn this simple estimation idea into a general procedure that can be used
to detect feedback in realistic problems where the feedback has non-linearities and jumps.
Counterfactuals and Causal Inference Feedback detection is a problem in causal inference. A
model suffers from feedback if the predictions it makes today affect the predictions it will make tomorrow. We are thus interested in discovering a causal relationship between today?s and tomorrow?s
predictions; simply detecting a correlation is not enough. The distinction between causal and associational inference is acute in the case of feedback: today?s and tomorrow?s predictions are almost
always strongly correlated, but this correlation by no means implies any causal relationship.
In order to discover causal relationships between consecutive predictions, we need to use some form
of randomized experimentation. In our case, we add a small amount of random noise to our predictions. Because the noise is fully artificial, we can reasonably ask counterfactual questions of the
type: ?How would tomorrow?s predictions have changed if we added more/less noise to the predictions today?? The noise acts as an independent instrument that lets us detect feedback. We frame
our analysis in terms of a potential outcomes model that asks how the world would have changed
had we altered a treatment; in our case, the treatment is the random noise we add to our predictions.
This formalism, often called the Rubin causal model [1], is regularly used for understanding causal
inference [2, 3, 4]. Causal models are useful for studying the behavior of live predictive systems on
the internet, as shown by, e.g., the recent work of Bottou et al. [5] and Chan et al. [6].
Outline and Contributions In order to define a rigorous feedback detection procedure, we need
to have a precise notion of what we mean by feedback. Our first contribution is thus to provide such
a model by defining statistical feedback in terms of a potential outcomes model (Section 2). Given
this feedback model, we propose a local noising scheme that can be used to fit feedback functions
with non-linearities and jumps (Section 4). Before presenting general version of our approach, however, we begin by discussing the linear case in Section 3 to elucidate the mathematics of feedback
detection: as we will show, the problem of linear feedback detection using local perturbations reduces to linear regression. Finally, in Section 5 we conduct a pilot study based on a predictive model
currently deployed as a part of a search engine.
2
Feedback Detection for Statistical Predictors
(t)
Suppose that we have a model that makes predictions y?i in time periods t = 1, 2, ... for examples
i = 1, ..., n. The predictive model itself is taken as given; our goal is to understand feedback effects
(t)
(t+1)
between consecutive pairs of predictions y?i and y?i
. We define statistical feedback in terms
(t+1)
(t)
of counterfactual reasoning: we want to know what would have happened to y?i
had y?i been
different. We use potential outcomes notation [e.g., 7] to distinguish between counterfactuals: let
(t+1) (t)
(t)
y?i
[yi ] be the predictions our model would have made at time t + 1 if we had published yi as
(t+1) (t)
(t)
our time-t prediction. In practice we only get to observe y?i
[yi ] for a single yi ; all other values
(t+1) (t)
(t+1)
of y?i
[yi ] are counterfactual. We also consider y
?i
[?], the prediction our model would have
made at time t + 1 if the model never made any of its predictions public and so did not have the
chance to affect its environment. With this notation, we define feedback as
(t)
(t+1)
feedbacki = y?i
2
(t)
[?
yi ]
(t+1)
y?i
[?],
(1)
i.e., the difference between the predictions our model actually made and the predictions it would
have made had it not had the chance to affect its environment by broadcasting predictions in the
past. Thus, statistical feedback is a difference in potential outcomes.
An additive feedback model In order to get a handle on feedback as defined above, we assume
(t+1) (t)
(t+1)
(t)
that feedback enters the model additively: y?i
[yi ] = y
?i
[?] + f (yi ), where f is a feedback
(t)
function, and yi is the prediction published at time t. In other words, we assume that the predictions
made by our model at time t + 1 are the sum of the prediction the model would have made if there
were no feedback, plus a feedback term that only depends on the previous prediction made by the
model. Our goal is to estimate the feedback function f .
(t)
(t+1)
Artificial noising for feedback detection The relationship between y?i and y?i
can be influenced by many things, such as trends, mean reversion, random fluctuations, as well as feedback. In
order to isolate the effect of feedback, we need to add some noise to the system to create a situation
that resembles a randomized experiment. Ideally, we might hope to sometimes turn our predic(t)
tive system off in order to get estimates of y?i [?]. However, predictive models are often deeply
integrated into large software systems, and it may not be clear what the correct system behavior
would be if we turned the predictor off. To side-step this concern, we randomize our system by
(t)
adding artificial noise to predictions: at time t, instead of deploying the prediction y?i , we deploy
(t)
(t)
(t)
(t) iid
y?i = y?i + ?i , where ?i ? N is artificial noise drawn from some distribution N . Because the
(t)
noise ?i is independent from everything else, it puts us in a randomized experimental setup that
(t+1)
(t)
allows us to detect feedback as a causal effect. If the time t + 1 prediction y?i
is affected by ?i ,
(t)
(t+1)
then our system must have feedback because the only way ?i can influence y?i
is through the
interaction between our model predictions and the surrounding environment at time t.
(t)
Local average treatment effect In practice, we want the noise ?i to be small enough that it does
not disturb the regular operation of the predictive model too much. Thus, our experimental setup
(t)
allows us to measure feedback as a local average treatment effect [4], where the artificial noise ?i
acts as a continuous treatment. Provided our additive model holds, we can then piece together these
local treatment effects into a single global feedback function f .
3
Linear Feedback
We begin with an analysis of linear feedback problems; the linear setup allows us to convey the main
insights with less technical overhead. We discuss the non-linear case in Section 4. Suppose that we
have some natural process x(1) , x(2) , ... and a predictive model of the form y? = w ? x. (Suppose for
notational convenience that x includes the constant, and the intercept term is folded into w.) For our
purposes, w is fixed and known; for example, w may have been set by training on historical data.
At some point, we ship a system that starts broadcasting the predictions y? = w ? x, and there is a
concern that the act of broadcasting the y? may perturb the underlying x(t) process. Our goal is to
(t+1) (t)
(t+1) (t)
detect any such feedback. Following earlier notation we write y?i
[?
yi ] = w ? x i
[?
yi ] for the
(t+1)
(t+1)
time t + 1 variables perturbed by feedback, and y?i
[?] = w ? xi
[?] for the counterparts we
would have observed without any feedback.
(t)
(t+1)
(t)
In this setup, any effect of y?i on xi
[?
yi ] is feedback. A simple way to constrain this relationship
(t+1) (t)
(t+1)
(t)
(t+1) (t)
is using a linear model xi
[?
yi ] = x i
[?] + y
?i . In other words, we assume that xi
[?
yi ]
(t)
is perturbed by an amount that scales linearly with y?i . Given this simple model, we find that:
(t+1)
y?i
(t)
[?
yi ]
(t+1)
= y?i
3
[?]
(t)
+ w ? y?i ,
(2)
and so f (y) =
y with
= w ? ; f is the feedback function we want to fit.
(t+1)
We cannot work with (2) directly, because y?i
[?] is not observed. In order to get around this
(t)
(t)
(t)
problem, we add artificial noise to our predictions: at time t, we publish predictions y?i = y?i +?i
(t)
instead of the raw predictions y?i . As argued in Section 2, this method lets us detect feedback
(t+1)
(t)
because y?i
can only depend on ?i through a feedback mechanism, and so any relationship
(t+1)
(t)
between y?i
and ?i must be a symptom of feedback.
(t)
(t+1)
A Simple Regression Approach With the linear feedback model (2), the effect of ?i on y?i
is
(t+1) (t)
(t+1) (t)
(t)
(t)
y?i
[?
yi +?i ] = y
?i
[?
yi ] + ?i . This relationship suggests that we should be able to recover
(t+1)
(t)
by regressing y?i
against the added noise ?i . The following result confirms this intuition.
(t)
Theorem 1. Suppose that (2) holds, and that we add noise ?i to our time t predictions. If we
estimate using linear least squares
h
i
h
i1
0
(t+1) (t)
d y?(t+1) [?yi(t) +?i(t) ], ? (t)
?
Cov
Var y?i
[?
yi ]
p ?
i
i
?=
A , (3)
h
i
, then n ?
) N @0,
2
d ? (t)
?
Var
i
h
i
(t)
where ?2 = Var ?i and n is the number of examples to which we applied our predictor.
Theorem 1 gives us a baseline understanding for the difficulty of the feedback detection problem:
the precision of our feedback estimates scales as the ratio of the artificial noise ?2 to natural noise
(t+1) (t)
Var[?
yi
[?
yi ]]. Note that the proof of Theorem 1 assumes that we only used predictions from a
(t+1) (t)
single time period t + 1 to fit feedback, and that the raw predictions y?i
[?
yi ] are all independent.
If we relax these assumptions we get a regression problem with correlated errors, and need to be
more careful with technical conditions.
(t+1)
(t)
Efficiency and Conditioning The simple regression model (3) treats the term y?i
[?
yi ] as noise.
(t)
(t+1) (t)
This is quite wasteful: if we know y?i we usually have a fairly good idea of what y?i
[?
yi ] should
be, and not using this information needlessly inflates the noise. Suppose that we knew the function1
h
i
(t+1) (t)
(t)
?(y) := E y?i
[?
yi ] y
?i = y .
(4)
Then, we could write our feedback model as
?
? ?
(t+1) (t)
(t)
(t+1) (t)
(t)
y?i
[?
yi +?i ] = ? y
?i
+ y?i
[?
yi ]
?
??
(t)
? y?i
+
(t)
?i ,
(5)
(t)
where ?(?
yi ) is a known offset. Extracting this offset improves the precision of our estimate for ?.
Theorem 2. Under the conditions of Theorem 1 suppose that the function ? from (4) is known and
(t+1)
(t)
that the y?i
are all independent of each other conditional on y?i . Then, given the information
available at time t, the estimate
h
?
?
i
d y?(t+1) [?yi(t) +?i(t) ] ? y?(t) , ? (t)
Cov
i
i
i
?? =
h
i
has asymptotic distribution
(6)
d ? (t)
Var
i
h
h
ii 1
0
(t+1) (t)
(t)
?
?
E
Var
y
?
[?
y
]
y
?
i
p
i
i
A.
(7)
n ??
) N @0,
2
?
1
In practice we do not know ?, but we can estimate it; see Section 4.
4
(t)
(t+1)
Moreover, if the variance of ?i = y?i
linear unbiased estimator of .
(t)
[?
yi ]
(t)
(t)
?(?
yi ) does not depend on y?i , then ?? is the best
Theorem 2 extends the general result from above that the precision with which we can estimate
feedback scales as the ratio of artificial noise to natural noise. The reason why ?? is more efficient
than ? is that we managed to condition away some of the natural noise, and reduced the variance of
our estimate for by
h ?
?i
h
i
h
h
ii
(t)
(t+1) (t)
(t+1) (t)
(t)
Var ? y?i
= Var y?i
[?
yi ]
E Var y?i
[?
yi ] y
?i
.
(8)
In other words, the variance reduction we get from ?? directly matches the amount of variability we
can explain away by conditioning. The estimator (6) is not practical as stated, because it requires
knowledge of the unknown function ? and is restricted to the case of linear feedback. In the next
section, we generalize this estimator into one that does not require prior knowledge of ? and can
handle non-linear feedback.
4
Fitting Non-Linear Feedback
Suppose now that we have the same setup as in the previous section, except that now feedback has
(t+1) (t)
(t+1)
(t)
a non-linear dependence on the prediction: y?i
[?
yi ] = y
?i
[?] + f (?
yi ) for some arbitrary
function f . For example, in the case of a linear predictive model y? = w ? x, this kind of feedback
(t+1) (t)
(t+1)
(t)
could arise if we have feature feedback xi
[?
yi ] = x i
[?] + f(x) (?
yi ); the feedback function
(t)
then becomes f (?) = w ? f(x) (?). When we add noise ?i to the above predictions, we only affect
the feedback term f (?):
?
?
?
?
(t+1) (t)
(t+1) (t)
(t)
(t)
(t)
(t)
y?i
[?
yi +?i ]
y?i
[?
yi ] = f y
?i + ?i
f y?i .
(9)
(t)
Thus, by adding artificial noise ?i , we are able to cancel out the nuisance terms, and isolate the
feedback function f that we want to estimate. We cannot use (9) in practice, though, as we can only
(t+1) (t)
(t+1) (t)
(t)
observe one of y?i
[?
yi +?i ] or y
?i
[?
yi ] in reality; the other one is counterfactual. We can get
(t)
around this problem by conditioning on y?i as in Section 3. Let
h
i
(t+1) (t)
(t)
(t)
? (y) = E y?i
[?
yi +?i ] y
?i = y
(10)
h
i
(t+1)
(t)
= t (y) + 'N ? f (y) , where t (y) = E y?i
[?] y
?i = y
is a term that captures trend effects that are not due to feedback. The ? denotes convolution:
h ?
?
i
(t)
(t)
(t)
(t)
'N ? f (y) = E f y?i + ?i
y?i = y with ?i ? N.
(11)
Using the conditional mean function ? we can write our expression of interest as
?
?
?
?
?
?
(t+1) (t)
(t)
(t)
(t)
(t)
(t)
(t)
y?i
[?
yi +?i ]
? y?i
= f y?i + ?i
'N ? f y?i
+ ?i ,
(12)
?
?
(t)
(t+1)
(t)
where ?i := y?i
[?]
t y?i . If we have a good idea of what ? is, the left-hand side can be
(t+1)
(t)
(t)
(t)
(t)
measured, as it only depends on y?i
[?
yi +?i ] and y
?i . Meanwhile, conditional on y?i , the first
(t)
(t)
(t)
two terms on the right-hand side only depend on ?i , while ?i is independent of ?i and mean(t)
zero. The upshot is that we can treat (12) as a regression problem where ?i is noise. In practice,
(t+1) (t)
(t)
(t)
we estimate ? from an auxiliary problem where we regress y?i
[?
yi +?i ] against y
?i .
5
A Pragmatic Approach There are many possible approaches to solving the non-parametric system of equations (12) for f [e.g., 8, Chapter 5]. Here, we take a pragmatic approach, and constrain
ourselves to solutions of the form ?
?(y) = ?? ? b? (y) and f?(y) = ?f ? bf (y), where b? : R ! Rp?
pf
and bf : R ! R are predetermined basis expansions. This approach transforms our problem
into an ordinary least-squares problem, and works well in terms of producing reasonable feedback
estimates in real-world problems (see Section 5). If this relation in fact holds for some values ?
and f , the result below shows that we can recover f by least-squares.
Theorem 3. Suppose that ? and f are defined as above, and that we have an unbiased estimator
?? of ? with variance V? = Var[ ?? ]. Then, if we fit f by least squares using (12) as described in
Appendix A, the resulting estimate ?f is unbiased and has variance
h i ?
? 1
?
? 1
Var ?f = Xf| Xf
Xf| VY + X? V? X?| Xf Xf| Xf
,
(13)
where the design matrices X? and Xf are defined as
0
1
0
1
..
..
.
.
B ?
B ?
?C
?
?
?C
B
B
(t) C
(t)
(t)
(t) C
|
X? = Bb|? y?i C and Xf = Bb|f y?i + ?i
('N ? bf ) y?i C
@
A
@
A
..
..
.
.
i
h
(t+1) (t)
(t)
and VY is a diagonal matrix with (VY )ii = Var y?i
[?
yi ] y
?i .
(14)
In the case where our spline model is misspecified, we can obtain a similar result using methods
due to Huber [9] and White [10]. In practice, we can treat ?? as known since fitting ?(?) is usually
easier than fitting f (?): estimating ?(?) is just a smoothing problem whereas estimating f (?) requires
(t)
fitting differences. If we also treat the errors ?i in (12) as roughly homoscedatic, (13) reduces to
h
h
ii
h i E Var y?i(t+1) [?yi(t) ] y?i(t)
?
?
?
?
(t)
(t)
(t)
Var ?f ?
,
where
s
=
b
y
?
+
?
'
?
b
y
?
. (15)
i
f
N
f
i
i
i
n E [ksi k22 ]
This simplified form again shows that the precision of our estimate of f (?) scales roughly as the ratio
(t)
of the variance of the artificial noise ?i to the variance of the natural noise.
Our Method in Practice For convenience, we summarize the steps needed to implement our
(t)
(t) iid
method here: (1) At time t, compute model predictions y?i and draw noise terms ?i ? N for
(t)
(t)
(t)
some noise distribution N . Deploy predictions y?i = y?i +??i ?in the live system. (2) Fit a non(t+1)
(t)
(t)
(t)
parametric least-squares regression of y?i
[?
yi +?i ] ? ? y
?i
to learn the function ? (y) :=
h
i
(t+1) (t)
(t)
(t)
E y?i
[?
yi +?i ] y
?i = y . We use the R formula notation, where a ? g(b) means that we
want to learn a function g(b) that predicts a. (3) Set up the non-parametric least-squares regression
problem
?
?
?
?
?
?
(t+1) (t)
(t)
(t)
(t)
(t)
(t)
y?i
[?
yi +?i ]
? y?i
? f y?i + ?i
'N ? f y?i ,
(16)
(t)
where the goal is to learn f . Here, 'N is the density of ?i , and ? denotes convolution. In Appendix
A we show how to carry out these steps using standard R libraries.
(t)
The resulting function f (y) is our estimate of feedback: If we make a prediction y?i at time t, then
(t)
(t)
(t)
our time t + 1 prediction will be boosted by f (?
yi ). The above equation only depends on y?i , ?i ,
6
(t+1)
(t)
(t)
and y?i
[?
yi +?i ], which are all quantities that can be observed in the context of an experiment
with noised predictions. Note that as we only fit f using the differences in (16), the intercept of f
is not identifiable. We fix the intercept (rather arbitrarily) by setting the average fitted feedback over
all training examples to 0; we do not include an intercept term in the basis bf .
Choice of Noising Distribution Adding noise to deployed predictions often has a cost that may
depend on the shape of the noise distribution N . A good choice of N should reflect this cost. For
example, if the practical cost of adding noise only depends on the largest amount of noise we ever
(t)
add, then it may be a good idea to draw ?i uniformly at random from {?"} for some " > 0. In our
(t)
experiments, we draw noise from a Gaussian distribution ?i ? N (0, ?2 ).
5
A Pilot Study
The original motivation for this research was to develop a methodology for detecting feedback in
real-world systems. Here, we present results from a pilot study, where we added signal to historical
data that we believe should emulate actual feedback. The reason for monitoring feedback on this
system is that our system was about to be more closely integrated with other predictive systems, and
there was a concern that the integration could induce bad feedback loops. Having a reliable method
for detecting feedback would provide us with an early warning system during the integration.
The predictive model in question is a logistic regression classifier. We added feedback to historical
(t)
(t)
data collected from log files according to half a dozen rules of the form ?if ai is high and y?i > 0,
(t+1)
(t)
then increase ai
by a random amount?; here y?i is the time-t prediction deployed by our system
(t)
(in log-odds space) and ai is some feature with a positive coefficient. These feedback generation
rules do not obey the additive assumption. Thus our model is misspecified in the sense that there
(t)
is no function f such that a current prediction y?i increased the log-odds of the next prediction by
(t)
f (?
yi ), and so this example can be taken as a stretch case for our method.
Our dataset had on the order of 100,000 data points, half of which were used for fitting the model
itself and half of which were used for feedback simulation. We generated data for 5 simulated time
periods, adding noise with ? = 0.1 at each step, and fit feedback using a spline basis discussed in
Appendix B. The ?true feedback? curve was obtained by fitting a spline regression to the additive
(t+1)
feedback model by looking at the unobservable y?i
[?]; we used a df = 5 natural spline with
knots evenly spread out on [ 9, 3] in log-odds space plus a jump at 0.
For our classifier of interest, we have fairly strong reasons to believe that the feedback function may
have a jump at zero, but probably shouldn?t have any other big jumps. Assuming that we know a
priori where to look for jumps does not seem to be too big a problem for the practical applications
we have considered. Results for feedback detection are shown in Figure 1. Although the fit is not
perfect, we appear to have successfully detected the shape of feedback. The error bars for estimated
feedback were obtained using a non-parametric bootstrap [11] for which we resampled pairs of
(current, next) predictions.
This simulation suggests that our method can be used to accurately detect feedback on scales that
may affect real-world systems. Knowing that we can detect feedback is reassuring from an engineering point of view. On a practical level, the feedback curve shown in Figure 1 may not be too
big a concern yet: the average feedback is well within the noise level of the classifier. But in largescale systems the ways in which a model interacts with its environment is always changing, and it
is entirely plausible that some innocuous-looking change in the future would increase the amount
of feedback. Our methodology provides us with a way to continuously monitor how feedback is
affected by changes to the system, and can alert us to changes that cause problems. In Appendix B,
we show some simulations with a wider range of effect sizes.
7
0.4
0.2
0.0
0.1
Feedback
0.3
True Feedback
Estimated Feedback
0.2
0.4
0.6
0.8
Prediction
Figure 1: Simulation aiming to replicate realistic feedback in a real-world classifier. The red solid
line is our feedback estimate; the black dashed line is the best additive approximation to the true
feedback. The x-axis shows predictions in probability space; the y axis shows feedback in logodds space. The error bars indicate pointwise confidence intervals obtained using a non-parametric
bootstrap with B = 10 replicates, and stretch 1 SE in each direction. Further experiments are
provided in Appendix B.
6
Conclusion
In this paper, we proposed a randomization scheme that can be used to detect feedback in real-world
predictive systems. Our method involves adding noise to the predictions made by the system; this
noise puts us in a randomized experimental setup that lets us measure feedback as a causal effect.
In general, the scale of the artificial noise required to detect feedback is smaller than the scale of
the natural predictor noise; thus, we can deploy our feedback detection method without disturbing
our system of interest too much. The method does not require us to make hypotheses about the
mechanism through which feedback may propagate, and so it can be used to continuously monitor
predictive systems and alert us if any changes to the system lead to an increase in feedback.
Related Work The interaction between models and the systems they attempt to describe has been
extensively studied across many fields. Models can have different kinds of feedback effects on their
environments. At one extreme of the spectrum, models can become self-fulfilling prophecies: for
example, models that predict economic growth may in fact cause economic growth by instilling
market confidence [12, 13]. At the other end, models may distort the phenomena they seek to
describe and therefore become invalid. A classical example of this is a concern that any metric used
to regulate financial risk may become invalid as soon as it is widely used, because actors in the
financial market may attempt to game the metric to avoid regulation [14]. However, much of the
work on model feedback in fields like finance, education, or macro-economic theory has focused on
negative results: there is an emphasis on understanding when feedback can happen and promoting
awareness about how feedback can interact with policy decisions, but there does not appear to be
much focus on actually fitting feedback. One notable exception is a paper by Akaike [15], who
showed how to fit cross-component feedback in a system with many components; however, he did
not add artificial noise to the system, and so was unable to detect feedback of a single component on
itself.
Acknowledgments The authors are grateful to Alex Blocker, Randall Lewis, and Brad Efron for
helpful suggestions and interesting conversations. S. W. is supported by a B. C. and E. J. Eaves
Stanford Graduate Fellowship.
8
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9
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4,854 | 5,395 | DFacTo: Distributed Factorization of Tensors
S. V. N. Vishwanathan
Statistics and Computer Science
Purdue University
West Lafayette IN 47907
[email protected]
Joon Hee Choi
Electrical and Computer Engineering
Purdue University
West Lafayette IN 47907
[email protected]
Abstract
We present a technique for significantly speeding up Alternating Least Squares
(ALS) and Gradient Descent (GD), two widely used algorithms for tensor factorization. By exploiting properties of the Khatri-Rao product, we show how to
efficiently address a computationally challenging sub-step of both algorithms. Our
algorithm, DFacTo, only requires two sparse matrix-vector products and is easy
to parallelize. DFacTo is not only scalable but also on average 4 to 10 times faster
than competing algorithms on a variety of datasets. For instance, DFacTo only
takes 480 seconds on 4 machines to perform one iteration of the ALS algorithm
and 1,143 seconds to perform one iteration of the GD algorithm on a 6.5 million
? 2.5 million ? 1.5 million dimensional tensor with 1.2 billion non-zero entries.
1
Introduction
Tensor data appears naturally in a number of applications [1, 2]. For instance, consider a social
network evolving over time. One can form a users ? users ? time tensor which contains snapshots
of interactions between members of the social network [3]. As another example consider an online
store such as Amazon.com where users routinely review various products. One can form a users ?
items ? words tensor from the review text [4]. Similarly a tensor can be formed by considering the
various contexts in which a user has interacted with an item [5]. Finally, consider data collected by
the Never Ending Language Learner from the Read the Web project which contains triples of noun
phrases and the context in which they occur, such as, (?George Harrison?, ?plays?, ?guitars?) [6].
While matrix factorization and matrix completion have become standard tools that are routinely used
by practitioners, unfortunately, the same cannot be said about tensor factorization. The reasons are
not very hard to see: There are two popular algorithms for tensor factorization namely Alternating
Least Squares (ALS) (Appendix B), and Gradient Descent (GD) (Appendix C). The key step in
both algorithms is to multiply a matricized tensor and a Khatri-Rao product of two matrices (line 4
of Algorithm 2 and line 4 of Algorithm 3). However, this process leads to a computationallychallenging, intermediate data explosion problem. This problem is exacerbated when the dimensions
of tensor we need to factorize are very large (of the order of hundreds of thousands or millions), or
when sparse tensors contain millions to billions of non-zero entries. For instance, a tensor we formed
using review text from Amazon.com has dimensions of 6.5 million ? 2.5 million ? 1.5 million and
contains approximately 1.2 billion non-zero entries.
Some studies have identified this intermediate data explosion problem and have suggested ways of
addressing it. First, the Tensor Toolbox [7] uses the method of reducing indices of the tensor for
sparse datasets and entrywise multiplication of vectors and matrices for dense datasets. However, it
is not clear how to store data or how to distribute the tensor factorization computation to multiple
machines (see Appendix D). That is, there is a lack of distributable algorithms in existing studies.
Another possible strategy to solve the data explosion problem is to use GigaTensor [8]. Unfortunately, while GigaTensor does address the problem of parallel computation, it is relatively slow. To
1
summarize, existing algorithms for tensor factorization such as the excellent Tensor Toolbox of [7],
or the Map-Reduce based GigaTensor algorithm of [8] often do not scale to large problems.
In this paper, we introduce an efficient, scalable and distributed algorithm, DFacTo, that addresses
the data explosion problem. Since most large-scale real datasets are sparse, we will focus exclusively on sparse tensors. This is well justified because previous studies have shown that designing
specialized algorithms for sparse tensors can yield significant speedups [7]. We show that DFacTo
can be applied to both ALS and GD, and naturally lends itself to a distributed implementation.
Therefore, it can be applied to massive real datasets which cannot be stored and manipulated on a
single machine. For ALS, DFacTo is on average around 5 times faster than GigaTensor and around
10 times faster than the Tensor Toolbox on a variety of datasets. In the case of GD, DFacTo is
on average around 4 times faster than CP-OPT [9] from the Tensor Toolbox. On the Amazon.com
review dataset, DFacTo only takes 480 seconds on 4 machines to perform one iteration of ALS and
1,143 seconds to perform one iteration of GD.
As with any algorithm, there is a trade-off: DFacTo uses 3 times more memory than the Tensor
Toolbox, since it needs to store 3 flattened matrices as opposed to a single tensor. However, in
return, our algorithm only requires two sparse matrix-vector multiplications, making DFacTo easy to
implement using any standard sparse linear algebra library. Therefore, there are two merits of using
our algorithm: 1) computations are distributed in a natural way; and 2) only standard operations are
required.
2
Notation and Preliminaries
Our notation is standard, and closely follows [2]. Also see [1]. Lower case letters such as x denote
scalars, bold lower case letters such as x denote vectors, bold upper case letters such as X represent
matrices, and calligraphic letters such as X denote three-dimensional tensors.
The i-th element of a vector x is written as xi . In a similar vein, the (i, j)-th entry of a matrix
X is denoted as xi,j and the (i, j, k)-th entry of a tensor X is written as xi,j,k . Furthermore, xi,:
(resp. x:,i ) denotes the i-th row (resp. column) of X. We will use X?,: (resp. X:,? ) to denote the
sub-matrix of X which contains the rows (resp. columns) indexed by the set ?. For instance, if
? = {2, 4}, then X?,: is a matrix which contains the second and fourth rows of X. Extending the
above notation to tensors, we will write Xi,:,: , X:,j,: and X:,:,k to respectively denote the horizontal,
lateral and frontal slices of a third-order tensor X. The column, row, and tube fibers of X are given
by x:,j,k , xi,:,k , and xi,j,: respectively.
Sometimes a matrix or tensor may not be fully observed. We will use ?X or ?X respectively to
denote the set of indices corresponding to the observed (or equivalently non-zero) entries in a matrix
X
X or a tensor X. Extending this notation, ?X
i,: (resp. ?:,j ) denotes the set of column (resp. row)
indices corresponding to the observed entries in the i-th row (resp. j-th column) of X. We define
X
X
?X
i,:,: , ?:,j,: , and ?:,:,k analogously as the set of indices corresponding to the observed entries of the
i-th horizontal, j-th lateral, or k-th frontal slices of X. Also, nnzr(X) (resp. nnzc(X)) denotes the
number of rows (resp. columns) of X which contain at least one non-zero element.
X> denotes the transpose, X? denotes the Moore-Penrose pseudo-inverse, and kXk (resp. kXk)
denotes the Frobenius norm of a matrix X (resp. tensor X) [10]. Given a matrix A ? Rn?m , the
linear operator vec(A) yields a vector x ? Rnm , which is obtained by stacking the columns of A.
On the other hand, given a vector x ? Rnm , the operator unvec(n,m) (x) yields a matrix A ? Rn?m .
A ? B denotes the Kronecker product, A B the Khatri-Rao product, and A ? B the Hadamard
product of matrices A and B. The outer product of vectors a and b is written as a ? b (see e.g.,
[11]). Definitions of these standard matrix products can be found in Appendix A.
2.1
Flattening Tensors
Just like the vec(?) operator flattens a matrix, a tensor X may also be unfolded or flattened into a
matrix in three ways namely by stacking the horizontal, lateral, and frontal slices. We use Xn to
denote the n-mode flattening of a third-order tensor X ? RI?J?K ; X1 is of size I ? JK, X2 is of
size J ? KI, and X3 is of size K ? IJ. The following relationships hold between the entries of X
2
and its unfolded versions (see Appendix A.1 for an illustrative example):
xi,j,k = x1i,j+(k?1)J = x2j,k+(i?1)K = x3k,i+(j?1)I .
(1)
We can view X1 as consisting of K stacked frontal slices of X, each of size I ? J. Similarly, X2
consists of I slices of size J ? K and X3 is made up of J slices of size K ? I. If we use Xn,m to
denote the m-th slice in the n-mode flattening of X, then observe that the following holds:
x1i,j+(k?1)J = x1,k
i,j ,
x2j,k+(i?1)K = x2,i
j,k ,
x3k,i+(j?1)I = x3,j
k,i .
(2)
One can state a relationship between the rows and columns of various flattenings of a tensor, which
will be used to derive our distributed tensor factorization algorithm in Section 3. The proof of the
below lemma is in Appendix A.2.
0
Lemma 1 Let (n, n0 ) ? {(2, 1), (3, 2), (1, 3)}, and let Xn and Xn be the n and n0 -mode flattening
0
respectively of a tensor X. Moreover, let Xn,m be the m-th slice in Xn , and xnm,: be the m-th row
0
0
of Xn . Then, vec(Xn,m ) = xnm,: .
3
DFacTo
Recall that the main challenge of implementing ALS or GD for solving tensor factorization lies in
multiplying a matricized tensor and a Khatri-Rao product of two matrices: X1 (C B)1 . If B is
of size J ? R and C is of size K ? R, explicitly forming (C B) requires O(JKR) memory and
is infeasible when J and K are large. This is called the intermediate data explosion problem in the
literature [8]. The lemma below will be used to derive our efficient algorithm, which avoids this
problem. Although the proof can be inferred using results in [2], we give an elementary proof for
completeness.
Lemma 2 The r-th column of X1 (C B) can be computed as
i>
h
>
1
X (C B) :,r = unvec(K,I) X2 b:,r
c:,r
(3)
Proof We need to show that
i>
h
>
1
X (C B) :,r = unvec(K,I) X2 b:,r
c:,r
?
? > 2,1
b:,r X c:,r
?
?
..
=?
?.
.
2,I
c:,r
b>
:,r X
1
2,i
c:,r . Using (13)
Or equivalently it suffices to show that X (C B) i,r = b>
:,r X
2,i
>
2,i
vec b>
c:,r = c>
.
:,r X
:,r ? b:,r vec X
2,i
c:,r is a scalar. Moreover, using Lemma 1 we can write vec X
Observe that b>
:,r X
This allows us to rewrite the above equation as
>
2,i
b>
c:,r = x1i,: (c:,r ? b:,r ) = X1 (C B) i,r ,
:,r X
(4)
2,i
= x1i,: .
which completes the proof.
Unfortunately, a naive computation of X1 (C B) :,r by using (3) does not solve the intermediate
>
data explosion problem. This is because X2 b:,r produces a KI dimensional vector, which is
then reshaped by the unvec(K,I) (?) operator into a K ? I matrix. However, as the next lemma
>
asserts, only a small number of entries of X2 b:,r are non-zero.
For convenience, let a vector produced by (X2 )> b:,r be v:,r and a matrix produced by
>
unvec(K,I) (v:,r ) be Mr .
1
We mainly concentrate on the update to A since the updates to B and C are analogous.
3
Lemma 3 The number of non-zeros in v:,r is at most nnzr((X2 )> ) and nnzc(X2 ).
Proof Multiplying an all-zero row in (X2 )> and b:,r produces zero. Therefore, the number of nonzeros in v:,r is equal to the number of rows in (X2 )> that contain at least one non-zero element.
Also, by definition, nnzr((X2 )> ) is equal to nnzc(X2 ).
As a consequence of the above lemma, we only need to explicitly compute the non-zero entries of
>
v:,r . However, the problem of reshaping v:,r via the unvec(K,I) (?) operator still remains. The
next lemma shows how to overcome this difficulty.
Lemma 4 The location of the non-zero entries of Mr depends on (X2 )> and is independent of b:,r .
Proof The product of the (k + (i ? 1)K)-th row of (X2 )> and b:,r is the (k + (i ? 1)K)-th element
>
of v:,r . And, this element is the (i, k)-th entry of Mr by definition of unvec(K,I) (?) . Therefore,
if all the entries in the (k + (i ? 1)K)-th row of (X2 )> are zero, then the (i, k)-th entry of Mr is
zero regardless of b:,r . Consequently, the location of the non-zero entries of Mr is independent of
b:,r , and is only determined by (X2 )> .
Given X one can compute (X2 )> to know the locations of the non-zero entries of Mr . In other
words, we can infer the non-zero pattern and therefore preallocate memory for Mr . We will show
>
below how this allows us to perform the unvec(K,I) (?) operation for free.
Recall the Compressed Sparse Row (CSR) Format, which stores a sparse matrix as three arrays
namely values, columns, and rows. Here, values represents the non-zero values of the matrix; while
columns stores the column indices of the non-zero values. Also, rows stores the indices of the
columns array where each row starts. For example, if a sparse matrix Mr is
1 0 2
Mr =
,
0 3 4
then the CSR of Mr is
value(Mr ) = [ 1
2
3
4 ]
r
col(M ) = [ 0
2
1
2 ]
r
2
4 ].
row(M ) = [ 0
Different matrices with the same sparsity pattern can be represented by simply changing the entries
of the value array. For our particular case, what this means is that we can pre-compute col(Mr ) and
row(Mr ) and pre-allocate value(Mr ). By writing the non-zero entries of v:,r into value(Mr ) we
can ?reshape? v:,r into Mr .
? 2 )> . Then, Algorithm 1 shows the
Let the matrix with all-zero rows in (X2 )> removed be (X
1
? 2 )> , B, C, and
DFacTo algorithm for computing N := X (C B). Here, the input values are (X
r
2
>
? ) and b:,r directly
M preallocated in CSR format. By storing the results of the product of (X
into value(Mr ), we can obtain Mr because Mr was preallocated in the CSR format. Then, the
product of Mr and c:,r yields the r-th column of N. We obtain the output N by repeating these two
sparse matrix-vector products R times.
Algorithm 1: DFacTo algorithm for Tensor Factorization
? 2 )> , B, C, value(Mr ) col(Mr ), row(Mr )
1 Input: (X
2 Output: N
3 while r=1, 2,. . . , R do
? 2 )> b:,r
4
value(Mr ) ? (X
r
5
n:,r ? M c:,r
6 end
It is immediately obvious that using the above lemmas to compute
N requires no extra memory
other than storing Mr , which contains at most nnzc(X2 ) ? ?X non-zero entries. Therefore, we
4
completely avoid the intermediate data explosion problem. Moreover, the same subroutine can be
used for both ALS and GD (see Appendix E for detailed pseudo-code).
3.1
Distributed Memory Implementation
Our algorithm is easy to parallelize using a master-slave architecture of MPI(Message Passing Interface). At every iteration, the master transmits A, B, and C to the slaves. The slaves hold a fraction
of the rows of X2 using which a fraction of the rows of N is computed. By performing a synchronization step, the slaves can exchange rows of N. In ALS, this N is used to compute A which is
transmitted back to the master. Then, the master updates A, and the iteration proceeds. In GD, the
slaves transmit N back to the master, which computes ?A. Then, the master computes the step size
by a line search algorithm, updates A, and the iteration proceeds.
3.2
Complexity Analysis
A naive computation of N requires JK + ?X R flops; forming C
B requires JKR flops
1
?X R flops. Our algorithm
and performing the matrix-matrix
multiplication
X
(C
B)
requires
requires only nnzc(X2 ) + ?X R flops; ?X R flops for computing v:,r and nnzc(X2 )R flops
for computing Mr c:,r . Note that, typically, nnzc(X2 ) both JK and ?X (see Table 1). In terms
of memory, the naive algorithm requires O(JKR) extra memory, while our algorithm only requires
nnzc(X2 ) extra space to store Mr .
4
Related Work
Two papers that are most closely related to our work are the GigaTensor algorithm proposed by [8]
and the Sparse Tensor Toolbox of [7]. As discussed above, both algorithms attack the problem of
computing N
efficiently. In order to
computes two intermediate
matrices
compute n:,r , GigaTensor
>
>
1
1
N1 := X ? 1I (c:,r ? 1J )
and N2 := bin X ? 1I (1K ? b:,r ) . Next, N3 :=
N1 ? N
by computing N3 1JK . As reported in [8], GigaTensor
2 is computed, and n:,r is obtained
uses 2 ?X extra storage and 5 ?X flops to compute one column of N. The Sparse Tensor Toolbox
stores a tensor as a vector of non-zero values and a matrix of corresponding indices. Entries of B
and C are replicated appropriately to create intermediate vectors. A Hadamard product is computed
between the non-zero entries of the matrix and intermediate
and a selected set of
entries
vectors,
are summed to form columns of N. The algorithm uses 2 ?X extra storage and 5 ?X flops to
compute one column of N. See Appendix D for a detailed illustrative example which shows all the
intermediate calculations performed by our algorithm as well as the algorithm of [8] and [7].
Also, [9] suggests the gradient-based optimization of CANDECOMP/PARAFAC (CP) using the
same method as [7] to compute X1 (C B). [9] refers to this gradient-based optimization algorithm
as CPOPT and the ALS algorithm of CP using the method of [7] as CPALS. Following [9], we use
these names, CPALS and CPOPT.
5
Experimental Evaluation
Our experiments are designed to study the scaling behavior of DFacTo on both publicly available
real-world datasets as well as synthetically generated data. We contrast the performance of DFacTo
(ALS) with GigaTensor [8] as well as with CPALS [7], while the performance of DFacTo (GD) is
compared with CPOPT [9]. We also present results to show the scaling behavior of DFacTo when
data is distributed across multiple machines.
Datasets See Table 1 for a summary of the real-world datasets we used in our experiments. The
NELL-1 and NELL-2 datasets are from [8] and consists of (noun phrase 1, context, noun phrase 2)
triples from the ?Read the Web? project [6]. NELL-2 is a version of NELL-1, which is obtained by
removing entries whose values are below a threshold.
5
The Yelp Phoenix dataset is from the Yelp Data Challenge 2 , while Cellartracker, Ratebeer, Beeradvocate and Amazon.com are from the Stanford Network Analysis Project (SNAP) home page. All
these datasets consist of product or business reviews. We converted them into a users ? items ?
words tensor by first splitting the text into words, removing stop words, using Porter stemming [12],
and then removing user-item pairs which did not have any words associated with them. In addition,
for the Amazon.com dataset we filtered words that appeard less than 5 times or in fewer than 5
documents. Note that the number of dimensions as well as the number of non-zero entries reported
in Table 1 differ from those reported in [4] because of our pre-processing.
?
X
Dataset
I
J
K
? nnzc(X1 ) nnzc(X2 ) nnzc(X3 )
Yelp
45.97K
11.54K
84.52K
9.85M
4.32M
6.11M
229.83K
Cellartracker 36.54K 412.36K 163.46K
25.02M
19.23M
5.88M
1.32M
NELL-2
12.09K
9.18K
28.82K
76.88M
16.56M
21.48M
337.37K
Beeradvocate 33.37K
66.06K 204.08K
78.77M
18.98M
12.05M
1.57M
Ratebeer
29.07K 110.30K 294.04K
77.13M
22.40M
7.84M
2.85M
NELL-1
2.90M
2.14M
25.50M 143.68M
113.30M
119.13M
17.37M
Amazon
6.64M
2.44M
1.68M
1.22B
525.25M
389.64M
29.91M
Table 1: Summary statistics of the datasets used in our experiments.
We also generated the following two kinds of synthetic data for our experiments:
? the number of non-zero entries in the tensor is held fixed but we vary I, J, and K.
? the dimensions I, J, and K are held fixed but the number of non-zeros entries varies.
To simulate power law behavior, both the above datasets were generated using the following preferential attachment model [13]: the probability that a non-zero entry is added at index (i, j, k) is given
by pi ? pj ? pk , where pi (resp. pj and pk ) is proportional to the number of non-zero entries at index
i (resp. j and k).
Implementation and Hardware All experiments were conducted on a computing cluster where
each node has two 2.1 GHz 12-core AMD 6172 processors with 48 GB physical memory per node.
Our algorithms are implemented in C++ using the Eigen library3 and compiled with the Intel Compiler. We downloaded Version 2.5 of the Tensor Toolbox, which is implemented in MATLAB4 .
Since open source code for GigaTensor is not freely available, we developed our own version in
C++ following the description in [8]. Also, we used MPICH25 in order to distribute the tensor factorization computation to multiple machines. All our codes are available for download under an
open source license from http://www.joonheechoi.com/research.
Scaling on Real-World Datasets Both CPALS and our implementation of GigaTensor are uniprocessor codes. Therefore, for this experiment we restricted ourselves to datasets which can fit on a
single machine. When initialized with the same starting point, DFacTo and its competing algorithms
will converge to the same solution. Therefore, we only compare the CPU time per iteration of the
different algorithms. The results are summarized in Table 2. On many datasets DFacTo (ALS) is
around 5 times faster than GigaTensor and 10 times faster than CPALS; the differences are more
pronounced on the larger datasets. Also, DFacTo (GD) is around 4 times faster than CPOPT.
The differences in performance between DFacTo (ALS) and CPALS and between DFacTo (GD)
and CPOPT can partially be explained by the fact that DFacTo (ALS, GD) is implemented in C++
while CPALS and CPOPT use MATLAB. However, it must be borne in mind that both MATLAB
and our implementation use an optimized BLAS library to perform their computationally intensive
numerical linear algebra operations.
Compared to the Map-Reduce version implemented in Java and used for the experiments reported
in [8], our C++ implementation of GigaTensor is significantly faster and more optimized. As per [8],
2
https://www.yelp.com/dataset challenge/dataset
http://eigen.tuxfamily.org
4
http://www.sandia.gov/?tgkolda/TensorToolbox/
5
http://www.mpich.org/static/downloads/
3
6
Dataset
Yelp Phoenix
Cellartracker
NELL-2
Beeradvocate
Ratebeer
NELL-1
DFacTo (ALS)
9.52
23.89
32.59
43.84
44.20
322.45
GigaTensor
26.82
80.65
186.30
224.29
240.80
772.24
CPALS
46.52
118.25
376.10
364.98
396.63
-
DFacTo (GD)
13.57
35.82
80.79
94.85
87.36
742.67
CPOPT
45.9
130.32
386.25
481.06
349.18
-
Table 2: Times per iteration (in seconds) of DFacTo (ALS), GigaTensor, CPALS, DFacTo (GD), and
CPOPT on datasets which can fit in a single machine (R=10).
Machines
1
2
4
8
16
32
DFacTo (ALS)
NELL-1
Amazon
Iter.
CPU
Iter.
CPU
322.45 322.45
205.07 167.29
141.02 101.58 480.21 376.71
86.09
62.19 292.34 204.41
81.24
46.25 179.23
98.07
90.31
34.54 142.69
54.60
DFacTo (GD)
NELL-1
Amazon
Iter.
CPU
Iter.
CPU
742.67 104.23
492.38
55.11
322.65
28.55 1143.7 127.57
232.41
16.24 727.79
62.61
178.92
9.70 560.47
28.61
209.39
7.45 471.91
15.78
Table 3: Total Time and CPU time per iteration (in seconds) as a function of number of machines
for the NELL-1 and Amazon datasets (R=10).
the Java implementation took approximately 10,000 seconds per iteration to handle a tensor with
around 109 non-zero entries, when using 35 machines. In contrast, the C++ version was able to
handle one iteration of the ALS algorithm on the NELL-1 dataset on a single machine in 772 seconds. However, because DFacto (ALS) uses a better algorithm, it is able to handsomely outperform
GigaTensor and only takes 322 seconds per iteration.
Also, the execution time of DFacTo (GD) is longer than that of DFacTo (ALS) because DFacTo
(GD) spends more time on the line search algorithm to obtain an appropriate step size.
Scaling across Machines Our goal is to study scaling behavior of the time per iteration as datasets
are distributed across different machines. Towards this end we worked with two datasets. NELL-1
is a moderate-size dataset which our algorithm can handle on a single machine, while Amazon is a
large dataset which does not fit on a single machine. Table 3 shows that the iteration time decreases
as the number of machines increases on the NELL-1 and Amazon datasets. While the decrease in
iteration time is not completely linear, the computation time excluding both synchronization and
line search time decreases linearly. The Y-axis in Figure 1 indicates T4 /Tn where Tn is the single
iteration time with n machines on the Amazon dataset.
(a) DFacTo(ALS)
(b) DFacTo(GD)
Figure 1: The scalability of DFacTo with respect to the number of machines on the Amazon dataset
7
Synthetic Data Experiments We perform two experiments with synthetically generated tensor
data. In the first experiment we fix the number of non-zero entries to be 106 and let I = J = K
and vary the dimensions of the tensor. For the second experiment we fix the dimensions and let
I = J = K and the number of non-zero entries is set to be 2I. The scaling behavior of the three
algorithms on these two datasets is summarized in Table 4. Since we used a preferential attachment
model to generate the datasets, the non-zero indices exhibit a power law behavior. Consequently,
the number of columns with non-zero elements (nnzc(?)) for X1 , X2 and X3 is very close to the
total number of non-zero entries in the tensor. Therefore, as predicted by theory, DFacTo (ALS, GD)
does not enjoy significant speedups when compared to GigaTensor, CPALS and CPOPT. However,
it must be noted that DFacto (ALS) is faster than either GigaTensor or CPALS in all but one case and
DFacTo (GD) is faster than CPOPT in all cases. We attribute this to better memory locality which
arises as a consequence of reusing the memory for N as discussed in Section 3.
I=J =K
104
105
106
107
104
105
106
107
Non-zeros
106
106
106
106
2 ? 104
2 ? 105
2 ? 106
2 ? 107
DFacTo (ALS)
1.14
2.72
7.26
41.64
0.05
0.92
12.06
144.48
GigaTensor
2.80
6.71
11.86
38.19
0.09
1.61
22.08
251.89
CPALS
5.10
6.11
16.54
175.57
0.52
1.50
15.84
214.37
DFacTo (GD)
2.32
5.87
16.51
121.30
0.09
1.81
21.74
275.19
CPOPT
5.21
11.70
29.13
202.71
0.57
2.98
26.04
324.2
Table 4: Time per iteration (in seconds) on synthetic datasets (non-zeros = 106 or 2I, R=10)
Rank Variation Experiments Table 5 shows the time per iteration on various ranks (R) with the
NELL-2 dataset. We see that the computation time of our algorithm increases lineraly in R like the
time complexity analyzed in Section 3.2.
R
NELL-2
5
15.84
10
31.92
20
58.71
50
141.43
100
298.89
200
574.63
500
1498.68
Table 5: Time per iteration (in seconds) on various R
6
Discussion and Conclusion
We presented a technique for significantly speeding up the Alternating Least Squares (ALS) and the
Gradient Descent (GD) algorithm for tensor factorization by exploiting properties of the Khatri-Rao
product. Not only is our algorithm, DFacto, computationally attractive, but it is also more memory
efficient compared to existing algorithms. Furthermore, we presented a strategy for distributing the
computations across multiple machines.
We hope that the availability of a scalable tensor factorization algorithm will enable practitioners
to work on more challenging tensor datasets, and therefore lead to advances in the analysis and
understanding of tensor data. Towards this end we intend to make our code freely available for
download under a permissive open source license.
Although we mainly focused on tensor factorization using ALS and GD, it is worth noting that one
can extend the basic ideas behind DFacTo to other related problems such as joint matrix completion
and tensor factorization. We present such a model in Appendix F. In fact, we believe that this joint
matrix completion and tensor factorization model by itself is somewhat new and interesting in its
own right, despite its resemblance to other joint models including tensor factorization such as [14].
In our joint model, we are given a user ? item ratings matrix Y, and some side information such as
a user ? item ? words tensor X. Preliminary experimental results suggest that jointly factorizing
Y and X outperforms vanilla matrix completion. Please see Appendix F for details of the algorithm
and some experimental results.
8
References
[1] Age Smilde, Rasmus Bro, and Paul Geladi. Multi-way Analysis with Applications in the Chemical Sciences. John Wiley and Sons, Ltd, 2004.
[2] Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review,
51(3):455?500, 2009.
[3] Jure Leskovec, Jon M. Kleinberg, and Christos Faloutsos. Graphs over time: densification
laws, shrinking diameters and possible explanations. In KDD, pages 177?187, 2005.
[4] J. McAuley and J. Leskovec. Hidden Factors and Hidden Topics: Understanding Rating Dimensions with Review Text. In Proceedings of the 7th ACM Conference on Recommender
Systems, pages 165?172, 2013.
[5] Alexandros Karatzoglou, Xavier Amatriain, Linas Baltrunas, and Nuria Oliver. Multiverse
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[6] A. Carlson, J. Betteridge, B. Kisiel, B. Settles, E.R. Hruschka Jr., and T.M. Mitchell. Toward
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[7] Brett W. Bader and Tamara G. Kolda. Efficient matlab computations with sparse and factored
tensors. SIAM Journal on Scientific Computing, 30(1):205?231, 2007.
[8] U. Kang, Evangelos E. Papalexakis, Abhay Harpale, and Christos Faloutsos. Gigatensor: scaling tensor analysis up by 100 times - algorithms and discoveries. In Conference on Knowledge
Discovery and Data Mining, pages 316?324, 2012.
[9] Evrim Acar, Daniel M. Dunlavy, and Tamara G. Kolda. A scalable optimization approach
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2011.
[10] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge Univ Press, 1990.
[11] Dennis S. Bernstein. Matrix Mathematics. Princeton University Press, 2005.
[12] M. Porter. An algorithm for suffix stripping. Program, 14(3):130?137, 1980.
[13] A. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286:509?512,
1999.
[14] Evrim Acar, Tamara G. Kolda, and Daniel M. Dunlavy. All-at-once optimization for coupled matrix and tensor factorizations. In MLG?11: Proceedings of Mining and Learning with
Graphs, August 2011.
9
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4,855 | 5,396 | Distributed Power-law Graph Computing:
Theoretical and Empirical Analysis
Ling Yan
Dept. of Comp. Sci. and Eng.
Shanghai Jiao Tong University
800 Dongchuan Road
Shanghai 200240, China
[email protected]
Cong Xie
Dept. of Comp. Sci. and Eng.
Shanghai Jiao Tong University
800 Dongchuan Road
Shanghai 200240, China
[email protected]
Wu-Jun Li
National Key Lab. for Novel Software Tech.
Dept. of Comp. Sci. and Tech.
Nanjing University
Nanjing 210023, China
[email protected]
Zhihua Zhang
Dept. of Comp. Sci. and Eng.
Shanghai Jiao Tong University
800 Dongchuan Road
Shanghai 200240, China
[email protected]
Abstract
With the emergence of big graphs in a variety of real applications like social
networks, machine learning based on distributed graph-computing (DGC) frameworks has attracted much attention from big data machine learning community.
In DGC frameworks, the graph partitioning (GP) strategy plays a key role to affect the performance, including the workload balance and communication cost.
Typically, the degree distributions of natural graphs from real applications follow
skewed power laws, which makes GP a challenging task. Recently, many methods
have been proposed to solve the GP problem. However, the existing GP methods
cannot achieve satisfactory performance for applications with power-law graphs.
In this paper, we propose a novel vertex-cut method, called degree-based hashing (DBH), for GP. DBH makes effective use of the skewed degree distributions
for GP. We theoretically prove that DBH can achieve lower communication cost
than existing methods and can simultaneously guarantee good workload balance.
Furthermore, empirical results on several large power-law graphs also show that
DBH can outperform the state of the art.
1
Introduction
Recent years have witnessed the emergence of big graphs in a large variety of real applications,
such as the web and social network services. Furthermore, many machine learning and data mining
algorithms can also be modeled with graphs [13]. Hence, machine learning based on distributed
graph-computing (DGC) frameworks has attracted much attention from big data machine learning
community [13, 15, 14, 6, 11, 7]. To perform distributed (parallel) graph-computing on clusters with
several machines (servers), one has to partition the whole graph across the machines in a cluster.
Graph partitioning (GP) can dramatically affect the performance of DGC frameworks in terms of
workload balance and communication cost. Hence, the GP strategy typically plays a key role in
DGC frameworks. The ideal GP method should minimize the cross-machine communication cost,
and simultaneously keep the workload in every machine approximately balanced.
1
Existing GP methods can be divided into two main categories: edge-cut and vertex-cut methods.
Edge-cut tries to evenly assign the vertices to machines by cutting the edges. In contrast, vertex-cut
tries to evenly assign the edges to machines by cutting the vertices. Figure 1 illustrates the edgecut and vertex-cut partitioning results of an example graph. In Figure 1 (a), the edges (A,C) and
(A,E) are cut, and the two machines store the vertex sets {A,B,D} and {C,E}, respectively. In
Figure 1 (b), the vertex A is cut, and the two machines store the edge sets {(A,B), (A,D), (B,D)}
and {(A,C), (A,E), (C,E)}, respectively. In edge-cut, both machines of a cut edge should maintain
a ghost (local replica) of the vertex and the edge data. In vertex-cut, all the machines associated
with a cut vertex should maintain a mirror (local replica) of the vertex. The ghosts and mirrors are
shown in shaded vertices in Figure 1. In edge-cut, the workload of a machine is determined by
the number of vertices located in that machine, and the communication cost of the whole graph is
determined by the number of edges spanning different machines. In vertex-cut, the workload of a
machine is determined by the number of edges located in that machine, and the communication cost
of the whole graph is determined by the number of mirrors of the vertices.
(a) Edge-Cut
(b) Vertex-Cut
Figure 1: Two strategies for graph partitioning. Shaded vertices are ghosts and mirrors, respectively.
Most traditional DGC frameworks, such as GraphLab [13] and Pregel [15], use edge-cut methods [9, 18, 19, 20] for GP. Very recently, the authors of PowerGraph [6] find that the vertex-cut
methods can achieve better performance than edge-cut methods, especially for power-law graphs. Hence, vertex-cut has attracted more and more attention from DGC research community. For
example, PowerGraph [6] adopts a random vertex-cut method and two greedy variants for GP.
GraphBuilder [8] provides some heuristics, such as the grid-based constrained solution, to improve
the random vertex-cut method.
Large natural graphs usually follow skewed degree distributions like power-law distributions, which
makes GP challenging. Different vertex-cut methods can result in different performance for powerlaw graphs. For example, Figure 2 (a) shows a toy power-law graph with only one vertex having
much higher degree than the others. Figure 2 (b) shows a partitioning strategy by cutting the vertices
{E, F, A, C, D}, and Figure 2 (c) shows a partitioning strategy by cutting the vertices {A, E}. We
can find that the partitioning strategy in Figure 2 (c) is better than that in Figure 2 (b) because the
number of mirrors in Figure 2 (c) is smaller which means less communication cost. The intuition
underlying this example is that cutting higher-degree vertices can result in fewer mirror vertices.
Hence, the power-law degree distribution can be used to facilitate GP. Unfortunately, existing vertexcut methods, including those in PowerGraph and GraphBuilder, make rarely use of the power-law
degree distribution for GP. Hence, they cannot achieve satisfactory performance in natural powerlaw graphs. PowerLyra [4] tries to combine both edge-cut and vertex-cut together by using the
power-law degree distribution. However, it is lack of theoretical guarantee.
(b)
(a)
(c)
Sample
Bad partitioning
Good partitioning
Figure 2: Partition a sample graph with vertex-cut.
2
In this paper, we propose a novel vertex-cut GP method, called degree-based hashing (DBH), for
distributed power-law graph computing. The main contributions of DBH are briefly outlined as
follows:
? DBH can effectively exploit the power-law degree distributions in natural graphs for vertexcut GP.
? Theoretical bounds on the communication cost and workload balance for DBH can be derived, which show that DBH can achieve lower communication cost than existing methods
and can simultaneously guarantee good workload balance.
? DBH can be implemented as an execution engine for PowerGraph [6], and hence all
PowerGraph applications can be seamlessly supported by DBH.
? Empirical results on several large real graphs and synthetic graphs show that DBH can
outperform the state-of-the-art methods.
2
Problem Formulation
Let G = (V, E) denote a graph, where V = {v1 , v2 , . . . , vn } is the set of vertices and E ? V ? V
is the set of edges in G. Let |V | denote the cardinality of the set V , and hence |V | = n. vi and vj are
called neighbors if (vi , vj ) ? E. The degree of vi is denoted as di , which measures the number of
neighbors of vi . Please note that we only need to consider the GP task for undirected graphs because
the workload mainly depends on the number of edges no matter directed or undirected graphs the
computation is based on. Even if the computation is based on directed graphs, we can also use the
undirected counterparts of the directed graphs to get the partitioning results.
Assume we have a cluster of p machines. Vertex-cut GP is to assign each edge with the two corresponding vertices to one of the p machines in the cluster. The assignment of an edge is unique, while
vertices may have replicas across different machines. For DGC frameworks based on vertex-cut GP,
the workload (amount of computation) of a machine is roughly linear in the number of edges located
in that machine, and the replicas of the vertices incur communication for synchronization. So the
goal of vertex-cut GP is to minimize the number of replicas and simultaneously balance the number
of edges on each machine.
Let M (e) ? {1, . . . , p} be the machine edge e ? E is assigned to, and A(v) ? {1, . . . , p} be
the span of vertex v over different machines. Hence, |A(v)| is the number of replicas of v among
different machines. Similar to PowerGraph [6], one of the replicas of a vertex is chosen as the master
and the others are treated as the mirrors of the master. We let M aster(v) denote the machine in
which the master of v is located. Hence, the goal of vertex-cut GP can be formulated as follows:
n
min
A
1X
|A(vi )|
n i=1
s.t. max |{e ? E | M (e) = m}| < ?
m
|E|
n
, and max |{v ? V | M aster(v) = m}| < ? ,
m
p
p
where m ? {1, . . . , p} denotes a machine, ? ? 1 and ? ? 1 are imbalance factors. We den
P
p
fine n1
|A(vi )| as replication factor, |E|
max |{e ? E | M (e) = m}| as edge-imbalance, and
p
n
i=1
m
max |{v ? V | M aster(v) = m}| as vertex-imbalance. To get a good balance of workload, ?
m
and ? should be as small as possible.
The degrees of natural graphs usually follow skewed power-law distributions [3, 1]:
Pr(d) ? d?? ,
where Pr(d) is the probability that a vertex has degree d and the power parameter ? is a positive
constant. The lower the ? is, the more skewed a graph will be. This power-law degree distribution makes GP challenging [6]. Although vertex-cut methods can achieve better performance than
edge-cut methods for power-law graphs [6], existing vertex-cut methods, such as random method in
PowerGraph and grid-based method in GraphBuilder [8], cannot make effective use of the powerlaw distribution to achieve satisfactory performance.
3
3
Degree-Based Hashing for GP
In this section, we propose a novel vertex-cut method, called degree-based hashing (DBH), to effectively exploit the power-law distribution for GP.
3.1
Hashing Model
We refer to a certain machine by its index idx, and the idxth machine is denoted as Pidx . We first define two kinds of hash functions: vertex-hash function idx = vertex hash(v) which hashes vertex
v to the machine Pidx , and edge-hash function idx = edge hash(e) or idx = edge hash(vi , vj )
which hashes edge e = (vi , vj ) to the machine Pidx .
Our hashing model includes two main components:
? Master-vertex assignment: The master replica of vi is uniquely assigned to one of the
p machines with equal probability for each machine by some randomized hash function
vertex hash(vi ).
? Edge assignment: Each edge e = (vi , vj ) is assigned to one of the p machines by some
hash function edge hash(vi , vj ).
It is easy to find that the above hashing model is a vertex-cut GP method. The master-vertex assignment can be easily implemented, which can also be expected to achieve a low vertex-imbalance
score. On the contrary, the edge assignment is much more complicated. Different edge-hash functions can achieve different replication factors and different edge-imbalance scores. Please note that
replication factor reflects communication cost, and edge-imbalance reflects workload-imbalance.
Hence, the key of our hashing model lies in the edge-hash function edge hash(vi , vj ).
3.2
Degree-Based Hashing
From the example in Figure 2, we observe that in power-law graphs the replication factor, which is
defined as the total number of replicas divided by the total number of vertices, will be smaller if we
cut vertices with relatively higher degrees. Based on this intuition, we define the edge hash(vi , vj )
as follows:
vertex hash(vi ) if di < dj ,
edge hash(vi , vj ) =
(1)
vertex hash(vj ) otherwise.
It means that we use the vertex-hash function to define the edge-hash function. Furthermore, the
edge-hash function value of an edge is determined by the degrees of the two associated vertices.
More specifically, the edge-hash function value of an edge is defined by the vertex-hash function
value of the associated vertex with a smaller degree. Hence, our method is called degree-based
hashing (DBH). DBH can effectively capture the intuition that cutting vertices with higher degrees
will get better performance.
Our DBH method for vertex-cut GP is briefly summarized in Algorithm 1, where [n] = {1, . . . , n}.
Algorithm 1 Degree-based hashing (DBH) for vertex-cut GP
Input: The set of edges E; the set of vertices V ; the number of machines p.
Output: The assignment M (e) ? [p] for each edge e.
1: Initialization: count the degree di for each i ? [n] in parallel
2: for all e = (vi , vj ) ? E do
3:
Hash each edge in parallel:
4:
if di < dj then
5:
M (e) ? vertex hash(vi )
6:
else
7:
M (e) ? vertex hash(vj )
8:
end if
9: end for
4
4
Theoretical Analysis
In this section, we present theoretical analysis for our DBH method. For comparison, the random vertex-cut method (called Random) of PowerGraph [6] and the grid-based constrained solution (called Grid) of GraphBuilder [8] are adopted as baselines. Our analysis is based on randomization. Moreover, we assume that the graph is undirected and there are no duplicated edges in the
graph. We mainly study the performance in terms of replication factor, edge-imbalance and verteximbalance defined in Section 2. Due to space limitation, we put the proofs of all theoretical results
into the supplementary material.
4.1
Partitioning Degree-fixed Graphs
Firstly, we assume that the degree sequence {di }ni=1 is fixed. Then we can get the following expected
replication factor produced by different methods.
Random assigns each edge evenly to the p machines via a randomized hash function. The result can
be directly got from PowerGraph [6].
Lemma 1. Assume that we have a sequence of n vertices {vi }ni=1 and the corresponding degree
sequence D = {di }ni=1 . A simple randomized vertex-cut on p machines has the expected replication
factor:
" n
#
n
1X
pX
1 di
E
|A(vi )| D =
1? 1?
.
n i=1
n i=1
p
?
By using the Grid hash function, each vertex
? has p rather than p candidate machines compared to
Random. Thus we simply replace p with p to get the following corollary.
Corollary 1. By using Grid for hashing, the expected replication factor on p machines is:
" n
# ? X
n
p
1X
1 di
E
|A(vi )| D =
1? 1? ?
.
n i=1
n i=1
p
Using DBH method in Section 3.2, we obtain the following result by fixing the sequence {hi }ni=1 ,
where hi is defined as the number of vi ?s adjacent edges which are hashed by the neighbors of vi
according to the edge-hash function defined in (1).
Theorem 1. Assume that we have a sequence of n vertices {vi }ni=1 and the corresponding degree
sequence D = {di }ni=1 . For each vi , di ? hi adjacent edges of it are hashed by vi itself. Define
H = {hi }ni=1 . Our DBH method on p machines has the expected replication factor:
#
" n
n
n
pX
1 hi +1
1 di
1X
pX
|A(vi )| H, D =
1? 1?
1? 1?
,
E
?
n i=1
n i=1
p
n i=1
p
where hi ? di ? 1 for any vi .
This theorem says that our DBH method has smaller expected replication factor than Random of
PowerGraph [6].
Next we turn to the analysis of the balance constraints. We still fix the degree sequence and have the
following result for our DBH method.
Theorem 2. Our DBH method on p machines with the sequences {vi }ni=1 , {di }ni=1 and {hi }ni=1
defined in Theorem 1 has the edge-imbalance:
n
P
P
hi
+
max
(di ? hi )
p
max |{e ? E | M (e) = m}|
j?[p] vi ?Pj
i=1
m
=
.
|E|/p
2|E|/p
Although the master vertices are evenly assigned to each machine, we want to show how the randomized assignment is close to the perfect balance. This problem is well studied in the model of
uniformly throwing n balls into p bins when n p(ln p)3 [17].
5
Lemma 2. The maximum number of master vertices for each machine is bounded as follows:
Pr[M axLoad > ka ] = o(1)
if a > 1,
Pr[M axLoad > ka ] = 1 ? o(1) if 0 < a < 1.
r
ln ln p
n
Here M axLoad = max |{v ? V | M aster(v) = m}|, and ka = p + 2npln p 1 ? 2a
ln p .
m
4.2
Partitioning Power-law Graphs
Now we change the sequence of fixed degrees into a sequence of random samples generated from
the power-law distribution. As a result, upper-bounds can be provided for the above three methods,
which are Random, Grid and DBH.
Theorem 3. Let the minimal degree be dmin and each d ? {di }ni=1 be sampled from a power-law
degree distribution with parameter ? ? (2, 3). The expected replication factor of Random on p
machines can be approximately bounded by:
" n
#
pX
1 ??
1 di
?p 1? 1?
,
1? 1?
ED
n i=1
p
p
? = dmin ?
where ?
??1
??2 .
This theorem says that when the degree sequence is under power-law distribution, the upper bound
of the expected replication factor increases as ? decreases. This implies that Random yields a worse
partitioning when the power-law graph is more skewed.
?
Like Corollary 1, we replace p with p to get the similar result for Grid.
Corollary 2. By using Grid method, the expected replication factor on p machines can be approximately bounded by:
"? n
#
pX
1 d i
1 ??
?
1? 1? ?
? p 1? 1? ?
,
ED
n i=1
p
p
? = dmin ?
where ?
Note that
?
??1
??2 .
p 1? 1?
?1
p
??
??
? p 1 ? 1 ? p1
. So Corollary 2 tells us that Grid can reduce
the replication factor but it is not motivated by the skewness of the degree distribution.
Theorem 4. Assume each edge is hashed by our DBH method and hi ? di ? 1 for any vi . The
expected replication factor of DBH on p machines can be approximately bounded by:
" n
#
pX
1 hi +1
1 ??0
EH,D
1? 1?
?p 1? 1?
,
n i=1
p
p
where ??0 = dmin ?
??1
??2
? dmin ?
??1
2??3
+ 12 .
Note that
1 ??
1 ??0
<p 1? 1?
.
p 1? 1?
p
p
??1
Therefore, our DBH method can expectedly reduce the replication factor. The term 2??3
increases
as ? decreases, which means our DBH reduces more replication factor when the power-law graph
is more skewed. Note that Grid and our DBH method actually use two different ways to reduce the
replication factor. Grid reduces more replication factor when p grows. These two approaches can
be combined to obtain further improvement, which is not the focus of this paper.
Finally, we show that our DBH methd also guarantees good edge-balance (workload balance) under
power-law distributions.
6
Theorem 5. Assume each edge is hashed by the DBH method with dmin , {vi }ni=1 , {di }ni=1 and
{hi }ni=1 defined
above. The vertices are evenly assigned. By taking the constant 2|E|/p =
n
P
ED
di = nED [d] /p, there exists ? (0, 1) such that the expected edge-imbalance of DBH
i=1
on p machines can be bounded w.h.p (with high probability). That is,
?
?
n
X
X
h
2|E|
i
EH,D ?
+ max
(di ? hi )? ? (1 + )
.
p
p
j?[p]
i=1
vi ?Pj
Note that any that satisfies 1/ n/p could work for this theorem, which results in a tighter
bound for large n. Therefore, together with Theorem 4, this theorem shows that our DBH method
can reduce the replication factor and simultaneously guarantee good workload balance.
5
Empirical Evaluation
In this section, empirical evaluation on real and synthetic graphs is used to verify the effectiveness
of our DBH method. The cluster for experiment contains 64 machines connected via 1 GB Ethernet.
Each machine has 24 Intel Xeon cores and 96GB of RAM.
5.1
Datasets
The graph datasets used in our experiments include both synthetic and real-world power-law graphs.
Each synthetic power-law graph is generated by a combination of two synthetic directed graphs. The
in-degree and out-degree of the two directed graphs are sampled from the power-law degree distributions with different power parameters ? and ?, respectively. Such a collection of synthetic graphs is
separated into two subsets: one subset with parameter ? ? ? which is shown in Table 1(a), and the
other subset with parameter ? < ? which is shown in Table 1(b). The real-world graphs are shown
in Table 1(c). Some of the real-world graphs are the same as those in the experiment of PowerGraph.
And some additional real-world graphs are from the UF Sparse Matrices Collection [5].
Table 1: Datasets
(a) Synthetic graphs: ? ? ? (b) Synthetic graphs: ? < ?
Alias
S1
S2
S3
S4
S5
S6
S7
S8
S9
5.2
?
2.2
2.2
2.2
2.2
2.1
2.1
2.1
2.0
2.0
?
2.2
2.1
2.0
1.9
2.1
2.0
1.9
2.0
1.9
|E|
71,334,974
88,305,754
134,881,233
273,569,812
103,838,645
164,602,848
280,516,909
208,555,632
310,763,862
Alias
S10
S11
S12
S13
S14
S15
?
2.1
2.0
2.0
1.9
1.9
1.9
|E|
88,617,300
135,998,503
145,307,486
280,090,594
289,002,621
327,718,498
?
2.2
2.2
2.1
2.2
2.1
2.0
(c) Real-world graphs
Alias
Tw
Arab
Wiki
LJ
WG
Graph
Twitter [10]
Arabic-2005 [5]
Wiki [2]
LiveJournal [16]
WebGoogle [12]
|V |
42M
22M
5.7M
5.4M
0.9M
|E|
1.47B
0.6B
130M
79M
5.1M
Baselines and Evaluation Metric
In our experiment, we adopt the Random of PowerGraph [6] and the Grid of GraphBuilder [8]1
as baselines for empirical comparison. The method Hybrid of PowerLyra [4] is not adopted for
comparison because it combines both edge-cut and vertex-cut which is not a pure vertex-cut method.
One important metric is the replication factor, which reflects the communication cost. To test the
speedup for real applications, we use the total execution time for PageRank which is forced to take
100 iterations. The speedup is defined as: speedup = 100% ? (?Alg ? ?DBH )/?Alg , where ?Alg is
the execution time of PageRank with the method Alg. Here, Alg can be Random or Grid. Because
all the methods can achieve good workload balance in our experiments, we do not report it here.
1
GraphLab 2.2 released in July 2013 has used PowerGraph as its engine, and the Grid GP method has been
adopted by GraphLab 2.2 to replace the original Random GP method. Detailed information can be found at:
http://graphlab.org/projects/index.html
7
5.3
Results
Figure 3 shows the replication factor on two subsets of synthetic graphs. We can find that our DBH
method achieves much lower replication factor than Random and Grid. The replication factor of
DBH is reduced by up to 80% compared to Random and 60% compared to Grid.
30
30
Random
Grid
DBH
Random
Grid
DBH
25
Replication Factor
Replication Factor
25
20
15
10
5
20
15
10
5
0
S1
S2
S3
S4
S5
S6
S7
S8
1+10?12
S9
0
S10
S11
S12
S13
S14
1+10?12
S15
(a) Replication Factor
(b) Replication Factor
Figure 3: Experiments on two subsets of synthetic graphs. The X-axis denotes different datasets in Table 1(a)
and Table 1(b). The number of machines is 48.
18
70
Random
Grid
DBH
16
60.6%
Figure 4 (a) shows the replication factor on the real-world graphs. We can also find that DBH
achieves the best performance. Figure 4 (b) shows that the relative speedup of DBH is up to 60%
over Random and 25% over Grid on the PageRank computation.
Random
Grid
60
10
0
WG
LJ
Wiki
Arab
1+10?12
Tw
0
(a) Replication Factor
WG
LJ
Wiki
25%
13.3%
4.28%
8.42%
20
2
6.06%
30
6
4
23.6%
40
26.5%
8
21.2%
10
31.5%
50
12
Speedup(%)
Replication Factor
14
Arab
1+10?12
Tw
(b) Execution Speedup
Figure 4: Experiments on real-world graphs. The number of machines is 48.
Figure 5 shows the replication factor and execution time for PageRank on Twitter graph when the
number of machines ranges from 8 to 64. We can find our DBH achieves the best performance for
all cases.
20
2000
Random
Grid
DBH
18
Random
Grid
DBH
1800
1600
Execution Time (Sec)
Replication Factor
16
14
12
10
8
6
4
1400
1200
1000
800
600
400
2
200
0
8
16
24
Number of Machines
48
64
1+10?12
(a) Replication Factor
8
16
24
Number of Machines
48
64
1+10?12
(b) Execution Time
Figure 5: Experiments on Twitter graph. The number of machines ranges from 8 to 64.
6
Conclusion
In this paper, we have proposed a new vertex-cut graph partitioning method called degree-based
hashing (DBH) for distributed graph-computing frameworks. DBH can effectively exploit the
power-law degree distributions in natural graphs to achieve promising performance. Both theoretical and empirical results show that DBH can outperform the state-of-the-art methods. In our
future work, we will apply DBH to more big data machine learning tasks.
7
Acknowledgements
This work is supported by the NSFC (No. 61100125, No. 61472182), the 863 Program of China
(No. 2012AA011003), and the Fundamental Research Funds for the Central Universities.
8
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9
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4,856 | 5,397 | Scalable Nonlinear Learning with
Adaptive Polynomial Expansions
Alina Beygelzimer
Yahoo! Labs
[email protected]
Alekh Agarwal
Microsoft Research
[email protected]
Daniel Hsu
Columbia University
[email protected]
John Langford
Microsoft Research
[email protected]
Matus Telgarsky?
Rutgers University
[email protected]
Abstract
Can we effectively learn a nonlinear representation in time comparable to linear
learning? We describe a new algorithm that explicitly and adaptively expands
higher-order interaction features over base linear representations. The algorithm
is designed for extreme computational efficiency, and an extensive experimental
study shows that its computation/prediction tradeoff ability compares very favorably against strong baselines.
1
Introduction
When faced with large datasets, it is commonly observed that using all the data with a simpler
algorithm is superior to using a small fraction of the data with a more computationally intense but
possibly more effective algorithm. The question becomes: What is the most sophisticated algorithm
that can be executed given a computational constraint?
At the largest scales, Na?ve Bayes approaches offer a simple, easily distributed single-pass algorithm. A more computationally difficult, but commonly better-performing approach is large scale
linear regression, which has been effectively parallelized in several ways on real-world large scale
datasets [1, 2]. Is there a modestly more computationally difficult approach that allows us to commonly achieve superior statistical performance?
The approach developed here starts with a fast parallelized online learning algorithm for linear models, and explicitly and adaptively adds higher-order interaction features over the course of training,
using the learned weights as a guide. The resulting space of polynomial functions increases the
approximation power over the base linear representation at a modest increase in computational cost.
Several natural folklore baselines exist. For example, it is common to enrich feature spaces with ngrams or low-order interactions. These approaches are naturally computationally appealing, because
these nonlinear features can be computed on-the-fly avoiding I/O bottlenecks. With I/O bottlenecked
datasets, this can sometimes even be done so efficiently that the additional computational complexity
is negligible, so improving over this baseline is quite challenging.
The design of our algorithm is heavily influenced by considerations for computational efficiency, as
discussed further in Section 2. Several alternative designs are plausible but fail to provide adequate
computation/prediction tradeoffs or even outperform the aforementioned folklore baselines. An
extensive experimental study in Section 3 compares efficient implementations of these baselines with
?
This work was performed while MT was visiting Microsoft Research, NYC.
1
Relative error vs time tradeoff
1.0
linear
quadratic
cubic
apple(0.125)
apple(0.25)
apple(0.5)
apple(0.75)
apple(1.0)
relative error
0.8
0.6
0.4
0.2
0.0
?0.2
0
10
1
10
relative time
2
10
Figure 1: Computation/prediction tradeoff points using non-adaptive polynomial expansions and
adaptive polynomial expansions (apple). The markers are positioned at the coordinate-wise median of (relative error, relative time) over 30 datasets, with bars extending to 25th and 75th
percentiles. See Section 3 for definition of relative error and relative time used here.
the proposed mechanism and gives strong evidence of the latter?s dominant computation/prediction
tradeoff ability (see Figure 1 for an illustrative summary).
Although it is notoriously difficult to analyze nonlinear algorithms, it turns out that two aspects
of this algorithm are amenable to analysis. First, we prove a regret bound showing that we can
effectively compete with a growing feature set. Second, we exhibit simple problems where this
algorithm is effective, and discuss a worst-case consistent variant. We point the reader to the full
version [3] for more details.
Related work. This work considers methods for enabling nonlinear learning directly in a highlyscalable learning algorithm. Starting with a fast algorithm is desirable because it more naturally
allows one to improve statistical power by spending more computational resources until a computational budget is exhausted. In contrast, many existing techniques start with a (comparably) slow
method (e.g., kernel SVM [4], batch PCA [5], batch least-squares regression [5]), and speed it up by
sacrificing statistical power, often just to allow the algorithm to run at all on massive data sets.
A standard alternative to explicit polynomial expansions is to employ polynomial kernels with the
kernel trick [6]. While kernel methods generally have computation scaling at least quadratically
with the number of training examples, a number of approximations schemes have been developed to
enable a better tradeoff. The Nystr?m method (and related techniques) can be used to approximate
the kernel matrix while permitting faster training [4]. However, these methods still suffer from the
drawback that the model size after n examples is typically O(n). As a result, even single pass online
implementations [7] typically suffer from O(n2 ) training and O(n) testing time complexity.
Another class of approximation schemes for kernel methods involves random embeddings into a
high (but finite) dimensional Euclidean space such that the standard inner product there approximates the kernel function [8?11]. Recently, such schemes have been developed for polynomial kernels [9?11] with computational scaling roughly linear in the polynomial degree. However, for many
sparse, high-dimensional datasets (such as text data), the embedding of [10] creates dense, high dimensional examples, which leads to a substantial increase in computational complexity. Moreover,
neither of the embeddings from [9, 10] exhibits good statistical performance unless combined with
dense linear dimension reduction [11], which again results in dense vector computations. Such feature construction schemes are also typically unsupervised, while the method proposed here makes
use of label information.
Among methods proposed for efficiently learning polynomial functions [12?16], all but [13] are
batch algorithms. The method of [13] uses online optimization together with an adaptive rule for
creating interaction features. A variant of this is discussed in Section 2 and is used in the experimental study in Section 3 as a baseline.
2
Algorithm 1 Adaptive Polynomial Expansion (apple)
input Initial features S1 = {x1 , . . . , xd }, expansion sizes (sk ), epoch schedule (?k ), stepsizes (?t ).
1: Initial weights w 1 := 0, initial epoch k := 1, parent set P1 := ?.
2: for t = 1, 2, . . . : do
3:
Receive stochastic gradient g t .
4:
Update weights: wt+1 := wt ? ?t [g t ]Sk ,
where [?]Sk denotes restriction to monomials in the feature set Sk .
5:
if t = ?k then
6:
Let Mk ? Sk be the top sk monomials m(x) ? Sk such that m(x) ?
/ Pk , ordered from
highest-to-lowest by the weight magnitude in wt+1 .
7:
Expand feature set: Sk+1 := Sk ? {xi ? m(x) : i ? [d], m(x) ? Mk }, and
Pk+1 := Pk ? {m(x) : m(x) ? Mk }.
8:
k := k + 1.
9:
end if
10: end for
2
Adaptive polynomial expansions
This section describes our new learning algorithm, apple.
2.1
Algorithm description
The pseudocode is given in Algorithm 1. The algorithm proceeds as stochastic gradient descent over
the current feature set to update a weight vector. At specified times ?k , the feature set Sk is expanded
to Sk+1 by taking the top monomials in the current feature set, ordered by weight magnitude in the
current weight vector, and creating interaction features between these monomials and x. Care is
exercised to not repeatedly pick the same monomial for creating higher order monomial by tracking
a parent set Pk , the set of all monomials for which higher degree terms have been expanded. We
provide more intuition for our choice of this feature growing heuristic in Section 2.3.
There are two benefits to this staged process. Computationally, the stages allow us to amortize the
cost of the adding of monomials?which is implemented as an expensive dense operation?over
several other (possibly sparse) operations. Statistically, using stages guarantees that the monomials
added in the previous stage have an opportunity to have their corresponding parameters converge.
We have found it empirically effective to set sk := average k[g t ]S1 k0 , and to update the feature set
at a constant number of equally-spaced times over the entire course of learning. In this case, the
number of updates (plus one) bounds the maximum degree of any monomial in the final feature set.
2.2
Shifting comparators and a regret bound for regularized objectives
Standard regret bounds compare the cumulative loss of an online learner to the cumulative loss of a
single predictor (comparator) from a fixed comparison class. Shifting regret is a more general notion
of regret, where the learner is compared to a sequence of comparators u1 , u2 , . . . , uT .
Existing shifting regret bounds can be used to loosely justify the use of online gradient descent
PT
methods over
feature spaces [17]. These bounds are roughly of the form t=1 ft (wt ) ?
p expanding
P
ft (ut ) . T t<T kut ? ut+1 k, where ut is allowed to use the same features available to wt ,
and ft is the convex cost function in step t. This suggests a relatively high cost for a substantial
total change in the comparator, and thus in the feature space. Given a budget, one could either do a
liberal expansion a small number of times, or opt for including a small number of carefully chosen
monomials more frequently. We have found that the computational cost of carefully picking a small
number of high quality monomials is often quite high. With computational considerations at the
forefront, we will prefer a more liberal but infrequent expansion. This also effectively exposes the
learning algorithm to a large number of nonlinearities quickly, allowing their parameters to jointly
converge between the stages.
It is natural to ask if better guarantees are possible under some structure on the learning problem.
Here, we consider the stochastic setting (rather than the harsher adversarial setting of [17]), and
3
further assume that our objective takes the form
f (w) := E[`(hw, xyi)] + ?kwk2 /2,
(1)
where the expectation is under the (unknown) data generating distribution D over (x, y) ? S ? R,
and ` is some convex loss function on which suitable restrictions will be placed. Here S is such
that S1 ? S2 ? . . . ? S, based on the largest degree monomials we intend to expand. We assume
that in round t, we observe a stochastic gradient of the objective f , which is typically done by first
sampling (xt , yt ) ? D and then evaluating the gradient of the regularized objective on this sample.
This setting has some interesting structural implications over the general setting of online learning
with shifting comparators. First, the fixed objective f gives us a more direct way of tracking the
change in comparator through f (ut ) ? f (ut+1 ), which might often be milder than kut ? ut+1 k.
In particular, if ut = arg minu?Sk f (u) in epoch k, for a nested subspace sequence Sk , then we
immediately obtain f (ut+1 ) ? f (ut ). Second, the strong convexity of the regularized objective
enables the possibility of faster O(1/T ) rates than prior work [17]. Indeed, in this setting, we obtain
the following stronger result. We use the shorthand Et [?] to denote the conditional expectation at
time t, conditioning over the data from rounds 1, . . . , t ? 1.
Theorem 1. Let a distribution over (x, y), twice differentiable convex loss ` with ` ? 0 and
max{`0 , `00 } ? 1, and a regularization parameter ? > 0 be given. Recall the definition (1) of
the objective f . Let (wt , g t )t?1 be as specified by apple with step size ?t := 1/(?(t + 1)), where
Et ([g t ]S(t) ) = [?f (wt )]S(t) and S(t) is the support set corresponding to epoch kt at time t in
apple. Then for any comparator sequence (ut )?
t=1 satisfying ut ? S(t) , for any fixed T ? 1,
!
PT
2
(X + ?)(X + ?D)2
1
t=1 (t + 2)f (ut )
E f (wT +1 ) ?
,
?
PT
T +1
2?2
t=1 (t + 2)
where X ? maxt kxt yt k and D ? maxt max{kwt k, kut k}.
Quite remarkably, the result exhibits no dependence on the cumulative shifting of the comparators
unlike existing bounds [17]. This is the first result of this sort amongst shifting bounds to the best of
our knowledge, and the only one that yields 1/T rates of convergence even with strong convexity. Of
course, we limit ourselves to the stochastic setting, and prove expected regret guarantees on the final
PT
predictor wT as opposed to a bound on t=1 f (wt )/T . A curious distinction is our comparator,
which is a weighted average of f (ut ) as opposed to the more standard uniform average. Recalling
that f (ut+1 ) ? f (ut ) in our setting, this is a strictly harder benchmark than an unweighted average
and overemphasizes the later comparator terms which are based on larger support sets. Indeed, this
is a nice compromise between competing against uT , which is the hardest yardstick, and u1 , which
is what a standard non-shifting analysis compares to. Indeed our improvement can be partially
attributed to the stability of the averaged f values as opposed to just f (uT ) (more details in [3]).
Overall, this result demonstrates that in our setting, while there is generally a cost to be paid for
shifting the comparator too much, it can still be effectively controlled in favorable cases. One
problem for future work is to establish these fast 1/T rates also with high probability.
Note that the regret bound offers no guidance on how or when to select new monomials to add.
2.3
Feature expansion heuristics
Previous work on learning sparse polynomials [13] suggests that it is possible to anticipate the utility
of interaction features before even evaluating them. For instance, one of the algorithms from [13]
orders monomials m(x) by an estimate of E[r(x)2 m(x)2 ]/E[m(x)2 ], where r(x) = E[y|x]? f?(x)
is the residual of the current predictor f? (for least-squares prediction of the label y). Such an index
is shown to be related to the potential error reduction by polynomials with m(x) as a factor. We call
this the SSM heuristic (after the authors of [13], though it differs from their original algorithm).
Another plausible heuristic, which we use in Algorithm 1, simply orders the monomials in Sk by
their weight magnitude in the current weight vector. We can justify this weight heuristic in the
following Q
simple example. Suppose a target function E[y|x] is just a single monomial in x, say,
m(x) := i?M xi for some M ? [d], and that x has a product distribution over {0, 1}d with 0 <
E[xi ] =: p ? 1/2 for all i ? [d]. Suppose we repeatedly perform 1-sparse regression with the current
4
feature set Sk , and pick the top weight magnitude monomial for inclusion in the parent set Pk+1 . It
is easy to show that the weight on a degree ` sub-monomial of m(x) in this regression is p|M |?` , and
the weight is strictly smaller for any term which is not a proper sub-monomial of m(x). Thus we
repeatedly pick the largest available sub-monomial of m(x) and expand it, eventually discovering
m(x). After k stages of the algorithm, we have at most kd features in our regression here, and
hence we find m(x) with a total of d|M | variables in our regression, as opposed to d|M | which
typical feature selection approaches would need. This intuition can be extended more generally to
scenarios where we do not necessarily do a sparse regression and beyond product distributions, but
we find that even this simplest example illustrates the basic motivations underlying our choice?we
want to parsimoniously expand on top of a base feature set, while still making progress towards a
good polynomial for our data.
2.4
Fall-back risk-consistency
Neither the SSM heuristic nor the weight heuristic is rigorously analyzed (in any generality). Despite
this, the basic algorithm apple can be easily modified to guarantee a form of risk consistency,
regardless of which feature expansion heuristic is used. Consider the following variant of the support
update rule in the algorithm apple. Given the current feature budget sk , we add sk ? 1 monomials
ordered by weight magnitudes as in Step 7. We also pick a monomial m(x) of the smallest degree
such that m(x) ?
/ Pk . Intuitively, this ensures that all degree 1 terms are in Pk after d stages,
all degree 2 terms are in Pk after k = O(d2 ) stages and so on. In general, it is easily seen that
k = O(d`?1 ) ensures that all degree ` ? 1 monomials are in Pk and hence all degree ` monomials
are in Sk . For ease of exposition, let us assume that sk is set to be a constant s independent of k.
Then the total number of monomials in Pk when k = O(d`?1 ) is O(sd`?1 ), which means the total
number of features in Sk is O(sd` ).
Suppose we were interested in competing with all ?-sparse polynomials of degree `. The most direct
approach would be to consider the explicit enumeration of all monomials of degree up to `, and then
perform `1 -regularized regression [18] or a greedy variable selection method such as OMP [19] as
means of enforcing sparsity. This ensures consistent estimation with n = O(? log d` ) = O(?` log d)
examples. In contrast, we might need n = O(?(` log d + log s)) examples in the worst case using
this fall back rule, a minor overhead at best. However, in favorable cases, we stand to gain a lot when
the heuristic succeeds in finding good monomials rapidly. Since this is really an empirical question,
we will address it with our empirical evaluation.
3
Experimental study
We now describe of our empirical evaluation of apple.
3.1
Implementation, experimental setup, and performance metrics
In order to assess the effectiveness of our algorithm, it is critical to build on top of an efficient
learning framework that can handle large, high-dimensional datasets. To this end, we implemented
apple in the Vowpal Wabbit (henceforth VW) open source machine learning software1 . VW is a
good framework for us, since it also natively supports quadratic and cubic expansions on top of the
base features. These expansions are done dynamically at run-time, rather than being stored and read
from disk in the expanded form for computational considerations. To deal with these dynamically
enumerated features, VW uses hashing to associate features with indices, mapping each feature to a
b-bit index, where b is a parameter. The core learning algorithm is an online algorithm as assumed
in apple, but uses refinements of the basic stochastic gradient descent update (e.g., [20?23]).
We implemented apple such that the total number of epochs was always 6 (meaning 5 rounds of
adding new features). At the end of each epoch, the non-parent monomials with largest magnitude
weights were marked as parents. Recall that the number of parents is modulated at s? for some
? > 0, with s being the average number of non-zero features per example in the dataset so far. We
will present experimental results with different choices of ?, and we found ? = 1 to be a reliable
1
Please see https://github.com/JohnLangford/vowpal_wabbit and the associated git
repository, where -stage_poly and related command line options execute apple.
5
30
linear
quadratic
cubic
apple
apple-best
ssm
ssm-best
25
20
15
number of datasets (cumulative)
number of datasets (cumulative)
30
10
5
0
?1.5
?1.0
?0.5
0.0
0.5
1.0
1.5
relative error
25
20
linear
quadratic
cubic
apple
apple-best
ssm
ssm-best
15
10
5
0
1
10
100
relative time
(a)
(b)
Figure 2: Dataset CDFs across all 30 datasets: (a) relative test error, (b) relative training time (log
scale). {apple, ssm} refer to the ? = 1 default; {apple, ssm}-best picks best ? per dataset.
default. Upon seeing an example, the features are enumerated on-the-fly by recursively expanding
the marked parents, taking products with base monomials. These operations are done in a way to
respect the sparsity (in terms of base features) of examples which many of our datasets exhibit.
Since the benefits of nonlinear learning over linear learning themselves are very dataset dependent,
and furthermore can vary greatly for different heuristics based on the problem at hand, we found it
important to experiment with a large testbed consisting of a diverse collection of medium and largescale datasets. To this end, we compiled a collection of 30 publicly available datasets, across a number of KDDCup challenges, UCI repository and other common resources (detailed in the appendix).
For all the datasets, we tuned the learning rate for each learning algorithm based on the progressive
validation error (which is typically a reliable bound on test error) [24]. The number of bits in hashing
was set to 18 for all algorithms, apart from cubic polynomials, where using 24 bits for hashing was
found to be important for good statistical performance. For each dataset, we performed a random
split with 80% of the data used for training and the remainder for testing. For all datasets, we used
squared-loss to train, and 0-1/squared-loss for evaluation in classification/regression problems. We
also experimented with `1 and `2 regularization, but these did not help much. The remaining settings
were left to their VW defaults.
For aggregating performance across 30 diverse datasets, it was important to use error and running
time measures on a scale independent of the dataset. Let `, q and c refer to the test errors of linear,
quadratic and cubic baselines respectively (with lin, quad, and cubic used to denote the baseline
algorithms themselves). For an algorithm alg, we compute the relative (test) error:
rel err(alg) =
err(alg) ? min(`, q, c)
,
max(`, q, c) ? min(`, q, c)
(2)
where min(`, q, c) is the smallest error among the three baselines on the dataset, and max(`, q, c)
is similarly defined. We also define the relative (training) time as the ratio to running time of lin:
rel time(alg) = time(alg)/time(lin). With these definitions, the aggregated plots of relative
errors and relative times for the various baselines and our methods are shown in Figure 2. For each
method, the plots show a cumulative distribution function (CDF) across datasets: an entry (a, b)
on the left plot indicates that the relative error for b datasets was at most a. The plots include the
baselines lin, quad, cubic, as well as a variant of apple (called ssm) that replaces the weight
heuristic with the SSM heuristic, as described in Section 2.3. For apple and ssm, the plot shows
the results with the fixed setting of ? = 1, as well as the best setting chosen per dataset from
? ? {0.125, 0.25, 0.5, 0.75, 1} (referred to as apple-best and ssm-best).
3.2
Results
In this section, we present some aggregate results. Detailed results with full plots and tables are
presented in the appendix. In the Figure 2(a), the relative error for all of lin, quad and cubic is
6
10
8
linear
quadratic
cubic
apple
apple-best
number of datasets (cumulative)
number of datasets (cumulative)
12
6
4
2
0
?1.5
?1.0
?0.5
0.0
0.5
1.0
1.5
relative error
(a)
12
10
linear
quadratic
cubic
apple
apple-best
8
6
4
2
0
1
10
100
relative time
(b)
Figure 3: Dataset CDFs across 13 datasets where time(quad) ? 2time(lin): (a) relative test error,
(b) relative training time (log scale).
always to the right of 0 (due to the definition of rel err). In this plot, a curve enclosing a larger area
indicates, in some sense, that one method uniformly dominates another. Since apple uniformly
dominates ssm statistically (with only slightly longer running times), we restrict the remainder of
our study to comparing apple to the baselines lin, quad and cubic. We found that on 12 of the
30 datasets, the relative error was negative, meaning that apple beats all the baselines. A relative
error of 0.5 indicates that we cover at least half the gap between min(`, q, c) and max(`, q, c). We
find that we are below 0.5 on 27 out of 30 datasets for apple-best, and 26 out of the 30 datasets for
the setting ? = 1. This is particularly striking since the error min(`, q, c) is attained by cubic on
a majority of the datasets (17 out of 30), where the relative error of cubic is 0. Hence, statistically
apple often outperforms even cubic, while typically using a much smaller number of features. To
support this claim, we include in the appendix a plot of the average number of features per example
generated by each method, for all datasets. Overall, we find the statistical performance of apple
from Figure 2 to be quite encouraging across this large collection of diverse datasets.
The running time performance of apple is also extremely good. Figure 2(b) shows that the running
time of apple is within a factor of 10 of lin for almost all datasets, which is quite impressive
considering that we generate a potentially much larger number of features. The gap between lin
and apple is particularly small for several large datasets, where the examples are sparse and highdimensional. In these cases, all algorithms are typically I/O-bottlenecked, which is the same for all
algorithms due to the dynamic feature expansions used. It is easily seen that the statistically efficient
baseline of cubic is typically computationally infeasible, with the relative time often being as large
as 102 and 105 on the biggest dataset. Overall, the statistical performance of apple is competitive
with and often better than min(`, q, c), and offers a nice intermediate in computational complexity.
A surprise in Figure 2(b) is that quad appears to computationally outperform apple for a relatively
large number of datasets, at least in aggregate. This is due to the extremely efficient implementation
of quad in VW: on 17 of 30 datasets, the running time of quad is less than twice that of lin. While
we often statistically outperform quad on many of these smaller datasets, we are primarily interested
in the larger datasets where the relative cost of nonlinear expansions (as in quad) is high.
In Figure 3, we restrict attention to the 13 datasets where time(quad)/time(lin) ? 2. On these
larger datasets, our statistical performance seems to dominate all the baselines (at least in terms
of the CDFs, more on individual datasets will be said later). In terms of computational time, we
see that we are often much better than quad, and cubic is essentially infeasible on most of these
datasets. This demonstrates our key intuition that such adaptively chosen monomials are key to
effective nonlinear learning in large, high-dimensional datasets.
We also experimented with picky algorithms of the sort mentioned in Section 2.2. We tried the
original algorithm from [13], which tests a candidate monomial before adding it to the feature set Sk ,
rather than just testing candidate parent monomials for inclusion in Pk ; and also a picky algorithm
based on our weight heuristic. Both algorithms were extremely computationally expensive, even
when implemented using VW as a base: the explicit testing for inclusion in Sk (on a per-example
7
5
Relative error, ordered by average nonzero features per example 103 Relative time, ordered by average nonzero features per example
linear
quadratic
cubic
apple(0.125)
apple(0.25)
apple(0.5)
apple(0.75)
apple(1.0)
4
3
2
2
10
1
10
1
0
0
rcv1
nomao
year
20news
slice
cup98
10
rcv1
nomao
(a)
year
20news
slice
cup98
(b)
Figure 4: Comparison of different methods on the top 6 datasets by non-zero features per example:
(a) relative test errors, (b) relative training times.
Test AUC
Training time (in s)
lin
0.81664
1282
lin + apple
0.81712
2727
bigram
0.81757
2755
bigram + apple
0.81796
7378
Table 1: Test error and training times for different methods in a large-scale distributed setting. For
{lin, bigram} + apple, we used ? = 0.25.
basis) caused too much overhead. We ruled out other baselines such as polynomial kernels for
similar computational reasons.
To provide more intuition, we also show individual results for the top 6 datasets with the highest
average number of non-zero features per example?a key factor determining the computational cost
of all approaches. In Figure 4, we show the performance of the lin, quad, cubic baselines, as well
as apple with 5 different parameter settings in terms of relative error (Figure 4(a)) and relative time
(Figure 4(b)). The results are overall quite positive. We see that on 3 of the datasets, we improve
upon all the baselines statistically, and even on other 3 the performance is quite close to the best of
the baselines with the exception of the cup98 dataset. In terms of running time, we find cubic to be
extremely expensive in all the cases. We are typically faster than quad, and in the few cases where
we take longer, we also obtain a statistical improvement for the slight increase in computational
cost. In conclusion, on larger datasets, the performance of our method is quite desirable.
Finally, we also implemented a parallel version of our algorithm, building on the repeated averaging
approach [2, 25], using the built-in AllReduce communication mechanism of VW, and ran an experiment using an internal advertising dataset consisting of approximately 690M training examples,
with roughly 318 non-zero features per example. The task is the prediction of click/no-click
events. The data was stored in a large Hadoop cluster, split over 100 partitions. We implemented the
lin baseline, using 5 passes of online learning with repeated averaging on this dataset, but could
not run full quad or cubic baselines due to the prohibitive computational cost. As an intermediate,
we generated bigram features, which only doubles the number of non-zero features per example.
We parallelized apple as follows. In the first pass over the data, each one of the 100 nodes locally
selects the promising features over 6 epochs, as in our single-machine setting. We then take the
union of all the parents locally found across all nodes, and freeze that to be the parent set for the rest
of training. The remaining 4 passes are now done with this fixed feature set, repeatedly averaging
local weights. We then ran apple, on top of both lin as well as bigram as the base features to
obtain maximally expressive features. The test error was measured in terms of the area under ROC
curve (AUC), since this is a highly imbalanced dataset. The error and time results, reported in Table 1, show that using nonlinear features does lead to non-trivial improvements in AUC, albeit at
an increased computational cost. Once again, this should be put in perspective with the full quad
baseline, which did not finish in over a day on this dataset.
Acknowledgements: We thank Leon Bottou, Rob Schapire and Dean Foster for helpful discussions.
8
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4,857 | 5,398 | Orbit Regularization
Andr?e F. T. Martins?
Instituto de Telecomunicac?o? es
Instituto Superior T?ecnico
1049?001 Lisboa, Portugal
[email protected]
Renato Negrinho
Instituto de Telecomunicac?o? es
Instituto Superior T?ecnico
1049?001 Lisboa, Portugal
[email protected]
Abstract
We propose a general framework for regularization based on group-induced majorization. In this framework, a group is defined to act on the parameter space and
an orbit is fixed; to control complexity, the model parameters are confined to the
convex hull of this orbit (the orbitope). We recover several well-known regularizers as particular cases, and reveal a connection between the hyperoctahedral group
and the recently proposed sorted `1 -norm. We derive the properties a group must
satisfy for being amenable to optimization with conditional and projected gradient algorithms. Finally, we suggest a continuation strategy for orbit exploration,
presenting simulation results for the symmetric and hyperoctahedral groups.
1
Introduction
The main motivation behind current sparse estimation methods and regularized empirical risk minimization is the principle of parsimony, which states that simple explanations should be preferred
over complex ones. Traditionally, this has been done by defining a function ? : V ? R that evaluates the complexity of a model w ? V and trades off this quantity with a data-dependent term. The
penalty function ? is often designed to be a convex surrogate of an otherwise non-tractable quantity,
a strategy which has led to important achievements in sparse regression [1], compressed sensing
[2], and matrix completion [3], allowing to successfully recover parameters from highly incomplete
information. Prior knowledge about the structure of the variables and the intended sparsity pattern,
when available, can be taken into account when designing ? via sparsity-inducing norms [4]. Performance bounds under different regimes have been established theoretically [5, 6], contributing to
a better understanding of the success and failure modes of these techniques.
In this paper, we introduce a new way to characterize the complexity of a model via the concept
of group-induced majorization. Rather than regarding complexity in an absolute manner via ?, we
define it relative to a prototype model v ? V , by requiring that the estimated model w satisfies
w G v,
(1)
where G is an ordering relation on V induced by a group G. This idea is rooted in majorization
theory, a well-established field [7, 8] which, to the best of our knowledge, has never been applied to
machine learning. We therefore review these concepts in ?2, where we show that this formulation
subsumes several well-known regularizers and motivates new ones. Then, in ?3, we introduce two
important properties of groups that serve as building blocks for the rest of the paper: the notions
of matching function and region cones. In ?4, we apply these tools to the permutation and signed
permutation groups, unveiling connections with the recent sorted `1 -norm [9] as a byproduct. In ?5
we turn to algorithmic considerations, pinpointing the group-specific operations that make a group
amenable to optimization with conditional and projected gradient algorithms.
?
Also at Priberam Labs, Alameda D. Afonso Henriques, 41 - 2? , 1000?123, Lisboa, Portugal.
1
Figure 1: Examples of orbitopes for the orthogonal group O(d) (left) and the hyperoctahedral
group P? (right). Shown are also the corresponding region cones, which in the case of O(d) degenerates into a ray.
A key aspect of our framework is a decoupling in which the group G captures the invariances of
the regularizer, while the data-dependent term is optimized in the group orbitopes. In ?6, we build
on this intuition to propose a simple continuation algorithm for orbit exploration. Finally, ?7 shows
some simulation results, and we conclude in ?8.
2
2.1
Orbitopes and Majorization
Vector Spaces and Groups
Let V be a vector space with an inner product h?, ?i. We will be mostly concerned with the case where
V = Rd , i.e., the d-dimensional real Euclidean space, but some of the concepts introduced here
generalize to arbitrary Hilbert spaces. A group is a set G endowed with an operation ? : G ? G ? G
satisfying closure (g ? h ? G, ?g, h ? G), associativity ((f ? g) ? h = f ? (g ? h), ?f, g, h ? G),
existence of identity (?1G ? G such that 1G ? g = g ? 1G = g, ?g ? G), and existence of inverses
(each g ? G has an inverse g ?1 ? G such that g ? g ?1 = g ?1 ? g = 1G ). Throughout, we use
boldface letters u, v, w, . . . for vectors, and g, h, . . . for group elements. We also omit the group
operation symbol, writing gh instead of g ? h.
2.2
Group Actions, Orbits, and Orbitopes
A (left) group action of G on V [10] is a function ? : G ? V ? V satisfying ?(g, ?(h, v)) =
?(g ? h, v) and ?(1G , v) = v for all g, h ? G and v ? V . When the action is clear from the context,
we omit the letter ?, writing simply gv for the action of the group element g on v, instead of ?(g, v).
In this paper, we always assume our actions are linear, i.e., g(c1 v 1 + c2 v 2 ) = c1 gv 1 + c2 gv 2 for
scalars c1 and c2 and vectors v 1 and v 2 . In some cases, we also assume they are norm-preserving,
i.e., kgvk = kvk for any g ? G and v ? V . When V = Rd , we may regard the groups underlying
these actions as subgroups of the general linear group GL(d) and of the orthogonal group O(d),
respectively. GL(d) is the set of d-by-d invertible matrices, and O(d) the set of d-by-d orthogonal
matrices {U ? Rd?d | U > U = U U > = Id }, where Id denotes the d-dimensional identity matrix.
A group action defines an equivalence relation on V , namely w ? v iff there is g ? G such that
w = gv. The orbit of a vector v ? V under the action of G is the set Gv := {gv | g ? G}, i.e., the
vectors that result from acting on v with some element of G. Its convex hull is called the orbitope:
OG (v) := conv(Gv).
(2)
Fig. 1 (left) illustrates this concept for the orthogonal group in R2 . An important concept associated
with group actions and orbitopes is that of G-majorization [7]:
Definition 1 Let v, w ? V . We say that w is G-majorized by v, denoted w G v, if w ? OG (v).
Proposition 2 If the group action is linear, then G is reflexive and transitive, i.e., it is a pre-order.
Proof: See supplemental material.
Group majorization plays an important role in the area of multivariate inequalities in statistics [11].
In this paper, we use this concept for representing model complexity, as described next.
2.3
Orbit Regularization
We formulate our learning problem as follows:
minimize L(w) s.t. w G v,
(3)
where L : V ? R is a loss function, G is a given group, and v ? V is a seed vector. This
formulation subsumes several well-known cases, outlined below.
2
? `2 -regularization. If G := O(d) is the orthogonal group acting by multiplication, we recover `2
regularization. Indeed, we have Gv = {U v ? Rd | U ? O(d)} = {w ? Rd | kwk2 = kvk2 }, for
any seed v ? Rd . That is, the orbitope OG (v) = conv(Gv) becomes the `2 -ball with radius kvk2 .
The only property of the seed that matters in this case is its `2 -norm.
? Permutahedron. Let P be the symmetric group (also called the permutation group), which can
be represented as the set of d-by-d permutation matrices. Given v ? Rd , the orbitope induced by v
under P is the convex hull of all the permutations of v, which can be equivalently described as the
vectors that are transformations of v through a doubly stochastic matrix:
OP (v) = conv{P v | P ? P} = {M v | M 1 = 1, M > 1 = 1, M ? 0}.
(4)
This set is called the permutahedron [12]. We will revisit this case in ?4.
? Signed permutahedron. Let P? be the hyperoctahedral group (also called the signed permutation group), i.e., the d-by-d matrices with entries in {0, ?1} such that the sum of the absolute values
in each row and column is 1. The action of P? on Rd permutes the entries of vectors and arbitrarily
switches signs. Given v ? Rd , the orbitope induced by v under P? is:
OP? (v) = conv{Diag(s)P v | P ? P, s ? {?1}d },
(5)
where Diag(s) denotes a diagonal matrix formed by the entries of s. We call this set the signed
permutahedron; it is depicted in Fig. 1 and will also be revisited in ?4.
? `1 and `? -regularization. As a particular case of the signed permutahedron, we recover `1 and
`? balls by choosing seeds of the form v = ?e1 (a scaled canonical basis vector) and v = ?1 (a
constant vector), respectively, where ? is a scalar. In the first case, we obtain the `1 -ball, OG (v) =
? conv({e1 , . . . , ed }) and in the second case, we get the `? -ball OG (v) = ? conv({?1}d ).
? Symmetric matrices with majorized eigenvalues. Let G := O(d) be again the orthogonal
group, but now acting by conjugation on the vector space of d-by-d symmetric matrices, V = Sd .
Given a seed v ? A ? Sd , its orbit is Gv = {U AU > | U ? O(d)} = {U Diag(?(A))U > | U ?
O(d)}, where ?(A) denotes a vector containing the eigenvalues of A in decreasing order (so we
may assume without loss of generality that the seed is diagonal). The orbitope OG (v) becomes:
OG (v) := {B ? Sd | ?(B) P ?(A)},
(6)
which is the set of matrices whose eigenvalues are in the permutahedron OP (?(A)) (see example
above). This is called the Schur-Horn orbitope in the literature [8].
? Squared matrices with majorized singular values. Let G := O(d) ? O(d) act on Rd?d (the
space of squared matrices, not necessarily symmetric) as gU,V A := U AV > . Given a seed v ? A,
its orbit is Gv = {U AV > | U, V ? O(d)} = {U Diag(?(A))V > | U, V ? O(d)}, where ?(A)
contains the singular values of A in decreasing order (so we may assume without loss of generality
that the seed is diagonal and non-negative). The orbitope OG (v) becomes:
OG (v) := {B ? Rd?d | ?(B) P ?(A)},
(7)
which is the set of matrices whose singular values are in the permutahedron OP (?(A)).
? Spectral and nuclear norm regularization. The previous case subsumes spectral and nuclear
norm balls: indeed, for a seed A = ?Id , the orbitope becomes the convex hull of orthogonal matrices, which is the spectral norm ball {A ? Rd?d | kAk2 := ?1 (A) ? ?}; while for a seed
A = ? Diag(e1 ), the orbitope becomes the convex hull ofPrank-1 matrices with norm bounded by
?, which is the nuclear norm ball {A ? Rd?d | kAk? := i ?i ? ?}. This norm has been widely
used for low-rank matrix factorization and matrix completion [3].
Besides these examples, other regularization strategies, such as non-overlapping `2,1 and `?,1 norms
[13, 4] can be obtained by considering products of the groups above. We omit details for space.
2.4
Relation with Atomic Norms
Atomic norms have been recently proposed as a toolbox for structured sparsity [6]. Let A ? V
be a centrally symmetric set of atoms, i.e., v ? A iff ?v ? A. The atomic norm induced by A
is defined as kwkA := inf{t > 0 | w ? t conv(A)}. The corresponding atomic ball is the set
{w | kwkA ? 1} = conv(A). Not surprisingly, orbitopes are often atomic norm balls.
3
Proposition 3 (Atomic norms) If G is a subgroup of the general linear group GL(d) and satisfies
?v ? Gv, then the set OG (v) is the ball of an atomic norm.
Proof: Under the given assumption, the set Gv is centrally symmetric, i.e., it satisfies w ? Gv iff
?w ? Gv (indeed, the left hand side implies that w = gv for some g ? G, and ?v ? Gv implies
that ?v = hv for some h ? G, therefore, ?w = ?gh?1 (?v) = gh?1 v ? Gv). As shown by
Chandrasekaran et al. [6], this guarantees that k.kGv satisfies the axioms of a norm.
Corollary 4 For any choice of seed, the signed permutahedron OP? (v) and the orbitope formed
by the squared matrices with majorized singular values are both atomic norm balls. If d is even and
d/2
v is of the form v = (v + , ?v + ), with v + ? R+ , then the permutahedron OP (v) and the orbitope
formed by the symmetric matrices with eigenvalues majorized by ?(v) are both atomic norm balls.
3
Matching Function and Region Cones
We now construct a unifying perspective that highlights the role of the group G. Two key concepts
that play a crucial role in our analysis are that of matching function and region cone. In the sequel,
these will work as building blocks for important algorithmic and geometric characterizations.
Definition 5 (Matching function) The matching function of G, mG : V ? V ? R, is defined as:
mG (u, v) := sup{hu, wi | w ? Gv}.
(8)
Intuitively, mG (u, v) ?aligns? the orbits of u and v before taking the inner product. Note also that
mG (u, v) = sup{hu, wi | w ? OG (v)}, since we may equivalently maximize the linear objective
over OG (v), which is the convex hull of Gv. We therefore have the following
Proposition 6 (Duality) Fix v ? V , and define the indicator function of the orbitope, IOG (v) (w) =
0 if w ? OG (v), and ?? otherwise. The Fenchel dual of IOG (v) is mG (., v). As a consequence,
letting L? : V ? R is the Fenchel dual of the loss L, the dual problem of Eq. 3 is:
maximize ? L? (?u) ? mG (u, v) w.r.t. u ? V.
(9)
Note that if k.kGv is a norm (e.g., if the conditions of Prop. 3 are satisfied), then the statement above
means that mG (., v) = k.k?Gv is its dual norm. We will revisit this dual formulation in ?4.
The following properties have been established in [14, 15].
Proposition 7 For any u, v ? V , we have: (i) mG (c1 u, c2 v) = c1 c2 mG (u, v) for c1 , c2 ? 0;
(ii) mG (g1 u, g2 v) = mG (u, v) for g1 , g2 ? G; (iii) mG (u, v) = mG (v, u). Furthermore, the
following three statements are equivalent: (i) w G v, (ii) f (w) ? f (v) for all G-invariant convex
functions f : V ? R, (iii) mG (u, w) ? mG (u, v) for all u ? V .
In the sequel, we always assume that G is a subgroup of the orthogonal group O(d). This implies
that the orbitope OG (v) is compact for any v ? V (and therefore the sup in Eq. 8 can be replaced by
a max), and that kgvk = kvk for any v ? V . Another important concept is that of the normal cone
of a point w ? V with respect to the orbitope OG (v), denoted as NGv (w) and defined as follows:
NGv (w) := {u ? V | hu, w0 ? wi ? 0 ?w0 G v}.
(10)
Normal cones plays an important role in convex analysis [16]. The particular case of the normal
cone at the seed v (illustrated in Fig. 1) is of great importance, as will be seen below.
Definition 8 (Region cone) Given v ? V , the region cone at v is KG (v) := NGv (v). It is the set
of points that are ?maximally aligned? with v in terms of the matching function:
KG (v) = {u ? V | mG (u, v) = hu, vi}.
(11)
4
Permutahedra and Sorted `1 -Norms
In this section, we focus on the permutahedra introduced in ?2. Below, given a vector w ? Rd , we
denote by w(k) its kth order statistic, i.e., we will ?sort? w so that w(1) ? w(2) ? . . . ? w(d) . We
also consider the order statistics of the magnitudes |w|(k) by sorting the absolute values.
4
4.1
Signed Permutahedron
We start by defining the ?sorted `1 -norm,? proposed by Bogdan et al. [9] in their recent SLOPE
method as a means to control the false discovery rate, and studied by Zeng and Figueiredo [17].
Definition 9 (Sorted `1 -norm) Let v, w ? Rd , with v1 ? v2 ? . . . ? vd ? 0 and v1 > 0. The
Pd
sorted `1 -norm of w (weighted by v) is defined as: kwkSLOPE,v := j=1 vj |w|(j) .
In [9] it is shown that k.kSLOPE,v satisfies the axioms of a norm. The rationale is that larger components of w are penalized more than smaller ones, in a way controlled by the prescribed v. For
v = 1, we recover the standard `1 -norm, while the `? -norm corresponds toP
v = e1 . Another special case is the OSCAR regularizer [18, 19], kwkOSCAR,?1 ,?2 := ?1 kwk1 + ?2 i<j max{|wi |, |wj |},
corresponding to a linearly spaced v, vj = (?1 + ?2 (d ? j)) for j = 1, . . . , d. The next proposition
reveals a connection between SLOPE and the atomic norm induced by the signed permutahedron.
Proposition 10 Let v ? Rd+ be as in Def. 9. The sorted `1 -norm weighted by v and the atomic norm
induced by the P ? -orbitope seeded at v are dual to each other: k.k?P? v = k.kSLOPE,v .
Proof: From Prop. 6, we have kwk?P? v = mP? (w, v). Let P be a signed permutation matrix s.t.
? := P w has its components sorted by decreasing magnitude, |w|
w
? 1 ? . . . ? |w|
? d . From Prop. 7,
? v) = h|w|,
? vi = kwkSLOPE,v .
we have mP? (w, v) = m(w,
The next proposition [7, 14] provides a characterization of the P? -orbitope in terms of inequalities
about the cumulative distribution of the order statistics.
Proposition 11 (Submajorization ordering) The orbitope OP? (v) can be characterized as:
n
o
P
P
OP? (v) = w ? Rd
|w|
?
|v|
,
?i
=
1,
.
.
.
,
d
.
(12)
(j)
(j)
j?i
j?i
Prop. 11 leads to a precise characterization
Pof the atomic
P norm kwkP? v , and therefore of the dual
norm of SLOPE: kwkP? v = maxi=1,...,d j?i |w|(j) / j?i |v|(j) .
4.2
Permutahedron
The unsigned counterpart of Prop. 11 goes back to Hardy et al. [20].
Proposition 12 (Majorization ordering) The P-orbitope seeded at v can be characterized as:
n
o
P
P
OP (v) = w ? Rd 1> w = 1> v ?
w
?
v
,
?i
=
1,
.
.
.
,
d
?
1
.
(13)
j?i (j)
j?i (j)
As seen in Corollary 4, if d is even and v = (v + , ?v + ), with v ? 0, then kwkPv qualifies as a
Pd
norm (we need to confine to the linear subspace V := {w ? Rd |
w = 0}). From Prop. 12,
Pj=1 j P
we have that this norm can be written as: kwkPv = maxi=1,...,d?1 j?i w(j) / j?i v(j) .
Proposition 13 Assume the conditions above hold and that v1 ? v2 ? . . . ? vd/2 ? 0 and v1 > 0.
Pd/2
The dual norm of k.kPv is kwk?Pv = j=1 vj (w(j) ? w(d?j+1) ).
Proof: Similar to the proof of Prop. 11.
5
Conditional and Projected Gradient Algorithms
Two important classes of algorithms in sparse modeling are the conditional gradient method [21, 22]
and the proximal gradient method [23, 24]. Under Ivanov regularization as in Eq. 3, the latter reduces
to the projected gradient method. In this section, we show that both algorithms are a good fit for
solving Eq. 3 for arbitrary groups, as long as the two building blocks mentioned in ?3 are available:
(i) a procedure for evaluating the matching function (necessary for conditional gradient methods)
and (ii) a procedure for projecting onto the region cone (necessary for projected gradient).
5
1: Initialize w 1 = 0
2: for t = 1, 2, . . . do
3:
Choose a stepsize ?t
4:
a = wt ? ?t ?L(wt )
5:
wt+1 = arg minwG v kw ? ak
6: end for
1: Initialize w 1 = 0
2: for t = 1, 2, . . . do
3:
ut = arg maxuG v h??L(wt ), ui
4:
?t = 2/(t + 2)
5:
wt+1 = (1 ? ?t )wt + ?t ut
6: end for
Figure 2: Conditional gradient (left) and projected gradient (right) algorithms.
5.1
Conditional Gradient
The conditional gradient method is shown in Fig. 2 (left). We assume that a procedure is available
for computing the gradient of the loss. The relevant part is the maximization in line 3, which
corresponds precisely to an evaluation of the matching function m(s, v), with s = ??L(wt )
(cf. Eq. 8). Fortunately, this step is efficient for a variety of cases:
Permutations. If G = P, the matching function can be evaluated in time O(d log d) with a simple sort operation. Without losing generality, we assume the seed v is sorted in descending order
(otherwise, pre-sort it before the main loop starts). Then, each time we need to evaluate m(s, v),
we compute a permutation P such that P s is also sorted. The minimizer in line 3 will equal P ?1 v.
Signed permutations. If G = P? , a similar procedure with the same O(d log d) runtime also
works, except that now we sort the absolute values, and set the signs of P ?1 v to match those of s.
Symmetric matrices with majorized eigenvalues. Let A = UA ?(A)UA> ? Sd and B =
UB ?(B)UB> ? Sd , where the eigenvalues ?(A) and ?(B) are sorted in decreasing order.
In this case, the matching function becomes mG (A, B) = maxV ?O(d) trace(A> V BV > ) =
h?(A), ?(B)i due to von Neumann?s trace inequality [25], the maximizer being V = UA UB> .
Therefore, we need only to make an eigen-decomposition and set B 0 = UA ?(B)UA> .
Squared matrices with majorized singular values. Let A = UA ?(A)VA> ? Rd?d and
B = UB ?(B)VB> ? Rd?d , where the singular values are sorted. We have mG (A, B) =
maxU,V ?O(d) trace(A> U BV > ) = h?(A), ?(B)i also from von Neumann?s inequality [25]. To
evaluate the matching function, we need only to make an SVD and set B 0 = UA ?(B)VA> .
5.2
Projected Gradient
The projected gradient algorithm is illustrated in Fig. 2 (right); the relevant part is line 5, which
involves a projection onto the orbitope OG (v). This projection may be hard to compute directly,
since the orbitope may lack a concise half-space representation. However, we next transform this
problem into a projection onto the region cone KG (v) (the proof is in the supplemental material).
Proposition 14 Assume G is a subgroup of O(d). Let g ? G be such that ha, gvi = mG (a, v).
Then, the solution of the problem in line 5 is w? = a ? ?KG (gv) (a ? gv).
Thus, all is necessary is computing the arg-max associated with the matching function, and a black
box that projects onto the region cone KG (v). Again, this step is efficient in several cases:
Permutations. If G = P, the region cone of a point v is the set of points w satisfying vi > vj ?
wi ? wj , for all i, j ? 1, . . . , d. Projecting onto this cone is a well-studied problem in isotonic
regression [26, 27], with existing O(d) algorithms.
Signed permutations. If G = P? , this problem is precisely the evaluation of the proximity operator of the sorted `1 -norm, also solvable in O(d) runtime with a stack-based algorithm [9].
6
Continuation Algorithm
Finally, we present a general continuation procedure for exploring regularization paths when L
is a convex loss function (not necessarily differentiable) and the seed v is not prescribed. The
6
Require: Factor > 0, interpolation parameter ? ? [0, 1]
1: Initialize seed v 0 randomly and set kv 0 k =
2: Set t = 0
3: repeat
4:
Solve wt = arg minwG vt L(w)
5:
Pick v 0t ? Gv t ? KG (wt )
6:
Set next seed v t+1 = (1 + )(?v 0t + (1 ? ?)wt )
7:
t?t+1
8: until kw t kGvt < 1.
b ? {w1 , w2 , . . .}
9: Use cross-validation to choose the best w
Figure 3: Left: Continuation algorithm. Right: Reachable region WG for the hyperoctahedral group, with a
reconstruction loss L(w) = kw ? ak2 . Only points v s.t. ??L(v) = a ? v ? KG (v) belong to this set.
Different initializations of v 0 lead to different paths along WG , all ending in a.
procedure?outlined in Fig. 3?solves instances of Eq. 3 for a sequence of seeds v 1 , v 2 , . . ., using
a simple heuristic for choosing the next seed given the previous one and the current solution.
The basic principle behind this procedure is the same as in other homotopy continuation methods
[28, 29, 30, 31]: we start with very strong regularization (using a small norm ball), and then gradually
weaken the regularization (increasing the ball) while ?tracking? the solution. The process stops
when the solution is found to be in the interior of the ball (the condition in line 8), which means the
regularization constraint is no longer active. The main difference with respect to classical homotopy
methods is that we do not just scale the ball (in our case, the G-orbitope); we also generate new
seeds that shape the ball along the way. To do so, we adopt a simple heuristic (line 6) to make
the seed move toward the current solution wt before scaling the orbitope. This procedure depends
on the initialization (see Fig. 3 for an illustration), which drives the search into different regions.
Reasoning in terms of groups, line 4 makes us move inside the orbits, while line 6 is an heuristic
to jump to a nearby orbit. For any choice of > 0 and ? ? [0, 1], the algorithm is convergent and
produces a strictly decreasing sequence L(w1 ) > L(w2 ) > ? ? ? before it terminates (a proof is
provided as supplementary material). We expect that, eventually, a seed v will be generated that is
b Although it may not be obvious at first sight why would it be desirable
close to the true model w.
b we provide a simple result below (Prop. 15) that sheds some light on this matter, by
that v ? w,
characterizing the set of points in V that are ?reachable? by optimizing Eq. 3.
From the optimality conditions of convex programming [32, p. 257], we have that w? is a solution
of the optimization problem in Eq. 3 if and only if 0 ? ?L(w? ) + NGv (w? ), where ?L(w) denotes
the subdifferential of L at w, and NGv (w) is the normal cone to OG (v) at w, defined in ?3. For
certain seeds v ? V , it may happen that the optimal solution w? of Eq. 3 is the seed itself. Let WG
be the set of seeds with this property:
WG := {v ? V | L(v) ? L(w), ?w G v} = {v ? V | 0 ? ?L(v) + KG (v)},
(14)
where KG (v) is the region cone and the right hand side follows from the optimality conditions. We
next show that this set is all we need to care about.
c
Proposition
15 Consider ?the set of points that are solutions of Eq. 3 for some seed v ? V , WG :=
?
c
w ? V ?v ? V : w ? arg minwG v L(w) . We have WG = WG .
cG . For the reverse direction, suppose that w? ? W
cG ,
Proof: Obviously, v ? WG ? v ? W
?
?
in which case there is some v ? V such that w G v and L(w ) ? L(w) for any w G v.
Since G is a pre-order, it must hold in particular that L(w? ) ? L(w) for any w G w? G v.
Therefore, we also have that w? ? arg minwG w? L(w), i.e., w? ? WG .
7
Simulation Results
We describe the results of numerical experiments when regularizing with the permutahedron (symmetric group) and the signed permutahedron (hyperoctahedral group). All problems were solved
b ? Rd
using the conditional gradient algorithm, as described in ?5. We generated the true model w
7
Figure 4: Learning curves for the permutahedron and signed permutahedron regularizers with a perfect seed.
Shown are averages and standard deviations over 10 trials. The baselines are `1 (three leftmost plots, resp. with
k = 150, 250, 400), and `2 (last plot, with k = 500).
Figure 5: Mean squared errors in the training set (left) and the test set (right) along the regularization path.
For the permutahedra regularizers, this path was traced with the continuation algorithm. The baseline is `1
regularization. The horizontal lines in the right plot show the solutions found with validation in a held-out set.
by sampling the entries from a uniform distribution in [0, 1] and subtracted the mean, keeping k ? d
b was normalized to have unit `2 -norm. Then, we sampled a random nnonzeros; after which w
by-d matrix X with i.i.d. Gaussian entries and variance ? 2 = 1/d, and simulated measurements
b + n, where n ? N (0, ?n2 ) is Gaussian noise. We set d = 500 and ?n = 0.3?.
y = Xw
For the first set of experiments (Fig. 4), we set k ? {150, 250, 400, 500} and varied the number
of measurements n. To assess the advantage of knowing the true parameters up to a group transb up to a constant
formation, we used for the orbitope regularizers a seed in the orbit of the true w,
factor (this constant, and the regularization constants for `1 and `2 , were all chosen with validation in a held-out set). As expected, this information was beneficial, and no significant difference
was observed between the permutahedron and the signed permutahedron. For the second set of
experiments (Fig. 5), where the aim is to assess the performance of the continuation method, no
information about the true model was given. Here, we fixed n = 250 and k = 300 and ran the
continuation algorithm with = 0.1 and ? = 0.0, for 5 different initializations of v 0 . We observe
that this procedure was effective at exploring the orbits, eventually finding a slightly better model
than the one found with `1 and `2 regularizers.
8
Conclusions and Future Work
In this paper, we proposed a group-based regularization scheme using the notion of orbitopes. Simple choices of groups recover commonly used regularizers such as `1 , `2 , `? , spectral and nuclear
matrix norms; as well as some new ones, such as the permutahedron and signed permutahedron.
As a byproduct, we revealed a connection between the permutahedra and the recently proposed
sorted `1 -norm. We derived procedures for learning with these orbit regularizers via conditional and
projected gradient algorithms, and a continuation strategy for orbit exploration.
There are several avenues for future research. For example, certain classes of groups, such as reflection groups [33], have additional properties that may be exploited algorithmically. Our work should
be regarded as a first step toward group-based regularization?we believe that the regularizers studied here are just the tip of the iceberg. Groups and their representations are well studied in other
disciplines [10], and chances are high that this framework can lead to new regularizers that are a
good fit to specific machine learning problems.
Acknowledgments
We thank all reviewers for their valuable comments. This work was partially supported by FCT
grants PTDC/EEI-SII/2312/2012 and PEst-OE/EEI/LA0008/2011, and by the EU/FEDER programme, QREN/POR Lisboa (Portugal), under the Intelligo project (contract 2012/24803).
8
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4,858 | 5,399 | Covariance shrinkage for autocorrelated data
Daniel Bartz
Department of Computer Science
TU Berlin, Berlin, Germany
[email protected]
?
Klaus-Robert Muller
TU Berlin, Berlin, Germany
Korea University, Korea, Seoul
[email protected]
Abstract
The accurate estimation of covariance matrices is essential for many signal processing and machine learning algorithms. In high dimensional settings the sample
covariance is known to perform poorly, hence regularization strategies such as
analytic shrinkage of Ledoit/Wolf are applied. In the standard setting, i.i.d. data
is assumed, however, in practice, time series typically exhibit strong autocorrelation structure, which introduces a pronounced estimation bias. Recent work by
Sancetta has extended the shrinkage framework beyond i.i.d. data. We contribute
in this work by showing that the Sancetta estimator, while being consistent in the
high-dimensional limit, suffers from a high bias in finite sample sizes. We propose
an alternative estimator, which is (1) unbiased, (2) less sensitive to hyperparameter choice and (3) yields superior performance in simulations on toy data and on a
real world data set from an EEG-based Brain-Computer-Interfacing experiment.
1
Introduction and Motivation
Covariance matrices are a key ingredient in many algorithms in signal processing, machine learning
and statistics. The standard estimator, the sample covariance matrix S, has appealing properties in
the limit of large sample sizes n: its entries are unbiased and consistent [HTF08]. On the other hand,
for sample sizes of the order of the dimensionality p or even smaller, its entries have a high variance
and the spectrum has a large systematic error. In particular, large eigenvalues are overestimated and
small eigenvalues underestimated, the condition number is large and the matrix difficult to invert
[MP67, ER05, BS10]. One way to counteract this issue is to shrink S towards a biased estimator T
(the shrinkage target) with lower variance [Ste56],
Csh := (1 ? ?)S + ?T,
the default choice being T = p?1 trace(S)I, the identity multiplied by the average eigenvalue.
For the optimal shrinkage intensity ?? , a reduction of the expected mean squared error is guaranteed [LW04]. Model selection for ? can be done by cross-validation (CV) with the known drawbacks: for (i) problems with many hyperparameters, (ii) very high-dimensional data sets, or (iii)
online settings which need fast responses, CV can become unfeasible and a faster model selection
method is required. A popular alternative to CV is Ledoit and Wolf?s analytic shrinkage procedure [LW04] and more recent variants [CWEH10, BM13]. Analytic shrinkage directly estimates the
shrinkage intensity which minimizes the expected mean squared error of the convex combination
with a negligible computational cost, especially for applications which rely on expensive matrix
inversions or eigendecompositions in high dimensions.
All of the above algorithms assume i.i.d. data. Real world time series, however, are often non-i.i.d. as
they possess pronounced autocorrelation (AC). This makes covariance estimation in high dimensions
even harder: the data dependence lowers the effective sample size available for constructing the
estimator [TZ84]. Thus, stronger regularization ? will be needed. In Figure 1 the simple case of an
autoregressive model serves as an example for an arbitrary generative model with autocorrelation.
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population
sample
30
30
variance
no AC (AR?coeff. = 0)
low AC (AR?coeff. = 0.7)
high AC (AR?coeff. = 0.95)
eigenvalue
autocorrelation (AC)
1
20
10
0
150
20
10
160
170
180
190
200
0
150
#eigenvalue
160
170
180
190
200
#sample eigendirection
Figure 1: Dependency of the eigendecomposition on autocorrelation. p = 200, n = 250.
The Figure shows, for three levels of autocorrelation (left), the population and sample eigenvalues
(middle): with increasing autocorrelation the sample eigenvalues become more biased. This bias
is an optimistic measure for the quality of the covariance estimator: it neglects that population and
sample eigenbasis also differ [LW12]. Comparing sample eigenvalues to the population variance in
the sample eigenbasis, the bias is even larger (right).
In practice, violations of the i.i.d. assumption are often ignored [LG11, SBMK13, GLL+ 14], although Sancetta proposed a consistent shrinkage estimator under autocorrelation [San08]. In this
paper, we contribute by showing in theory, simulations and on real world data, that (i) ignoring autocorrelations for shrinkage leads to large estimation errors and (ii) for finite samples Sancetta?s estimator is still substantially biased and highly sensitive to the number of incorporated time lags. We
propose a new bias-corrected estimator which (iii) outperforms standard shrinkage and Sancetta?s
method under the presence of autocorrelation and (iv) is robust to the choice of the lag parameter.
2
Shrinkage for autocorrelated data
Ledoit and Wolf derived a formula for the optimal shrinkage intensity [LW04, SS05]:
P
ij Var Sij
?
h
? =P
2 i .
E
S
?
T
ij
ij
ij
(1)
? is obtained by replacing expectations with sample estimates:
The analytic shrinkage estimator ?
n
n
2
X
1X
d Sij = 1
x
x
x
x
?
(2)
Var Sij
?? Var
it jt
is js
n2 s=1
n t=1
h
2 i
b (Sij ? Tij )2 = (Sij ? Tij )2 ,
E Sij ? Tij
?? E
(3)
where xit is the tth observation of variable i. While the estimator eq. (3) is unbiased even under a
violation of the i.i.d. assumption, the estimator eq. (2) is based on
!
n
1X
i.i.d. 1
Var
xit xjt = Var (xit xjt ) .
n t=1
n
If the data are autocorrelated, cross terms cannot be ignored and we obtain
!
n
n
1X
1 X
Var
xit xjt = 2
Cov(xit xjt , xis xjs )
n t=1
n s,t=1
=
=:
n?1
1
2 X n?s
Cov(xit xjt , xit xjt ) +
Cov(xit xjt , xi,t+s xj,t+s )
n
n s=1 n
n?1
1
2X
?ij (0) +
?ij (s)
n
n s=1
(4)
Figure 2 illustrates the effect of ignoring the cross terms for increasing autocorrelation (larger ARcoefficients, see section 3 for details on the simulation). It compares standard shrinkage to an oracle
shrinkage based on the population variance of the sample covariance1 . The population variance of S
1
calculated by resampling.
2
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1
population
sample
pop. var(S) shrinkage
standard shrinkage
15
0.8
variance
0.06
impr. over sample cov.
shrinkage intensity ?
norm. ?ij var(Sij)
0.1
0.08
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
AR?coefficients
1
10
AR?coeff. = 0.7
5
0
0
50
100
150
200
#sample eigendirection
Figure 2: Dependency of shrinkage on autocorrelation. p = 200, n = 250.
increases because the effective sample size is reduced [TZ84], yet the standard shrinkage variance
estimator eq. (2) does not increase (outer left). As a consequence, for oracle shrinkage the shrinkage
intensity increases, for the standard shrinkage estimator it even decreases because the denominator
in eq. (1) grows (middle left). With increasing autocorrelation, the sample covariance becomes
a less precise estimator: for optimal (stronger) shrinkage more improvement becomes possible,
yet standard shrinkage does not improve (middle right). Looking at the variance estimates in the
sample eigendirections for AR-coefficients of 0.7, we see that the bias of standard shrinkage is only
marginally smaller than the bias of the sample covariance, while oracle shrinkage yields a substantial
bias reduction (outer right).
Sancetta-estimator An estimator for eq. (4) was proposed by [San08]:
n?s
X
? San (s) := 1
?
(xit xjt ? Sij ) (xi,t+s xj,t+s ? Sij ) ,
ij
n t=1
!
n?1
X
San,b
1 ? San
San
d
?
:=
Var Sij
?ij (0) + 2
?(s/b)?ij (s) , b > 0,
n
s=1
(5)
where ? is a kernel which has to fulfill Assumption B in [And91]. We will restrict our analysis to
the truncated kernel ?TR (x) = {1 for |x| ? 1, 0 otherwise} to obtain less cluttered formulas2 . The
kernel parameter b describes how many time lags are taken into account.
The Sancetta estimator behaves well in the high dimensional limit: the main theoretical result states
that for (i) a fixed decay of the autocorrelation, (ii) b, n ? ? and (iii) b2 increasing at a lower rate
than n, the estimator is consistent independently of the rate of p (for details, see [San08]). This is
in line with the results in [LW04, CWEH10, BM13]: as long as n increases, all of these shrinkage
estimators are consistent.
Bias of the Sancetta-estimator In the following we will show that the Sancetta-estimator is suboptimal in finite samples: it has a non-negligible bias. To understand this, consider a lag s large
enough to have ?ij (s) ? 0. If we approximate the expectation of the Sancetta-estimator, we see that
it is biased downwards:
" n?s
#
h
i
X
1
2
? San
E ?
xit xjt xi,t+s xj,t+s ? Sij
.
ij (s) ? E
n t=1
2
n?s 2
n?s
?
E [Sij ] ? E Sij
=?
Var (Sij ) < 0.
n
n
Bias-corrected (BC) estimator We propose a bias-corrected estimator for the variance of the
entries in the sample covariance matrix:
n?s
1X
2
? BC
:=
?
(s)
xit xjt xi,t+s xj,t+s ? Sij
,
ij
n t=1
d Sij
Var
2
BC,b
1
:=
n ? 1 ? 2b + b(b + 1)/n
? BC
?
ij (0) + 2
(6)
n?1
X
!
BC
?
?TR (s/b)?ij (s) ,
b > 0.
s=1
in his simulations, Sancetta uses the Bartlett kernel. For fixed b, this increases the truncation bias.
3
? BC (s) is very similar to ?
? San (s), but slightly easier to compute. The main difference
The estimator ?
ij
ij
BC,b
d Sij
is the denominator in Var
: it is smaller than n and thus corrects the downwards bias.
2.1
Theoretical results
It is straightforward to extend the theoretical results on the Sancetta estimator ([San08], see summary above) to our proposed estimator. In the following, to better understand the limitations of the
Sancetta estimator, we will provide a complementary theoretical analysis on the behaviour of the
estimator for finite n.
Our theoretical results are based on the analysis of a sequence of statistical models indexed by p. Xp
denotes a p ? n matrix of n observations of p variables with mean zero and covariance matrix Cp .
Y p = R>
p Xp denotes the same observations rotated in their eigenbasis, having diagonal covariance
p
p
3
?p = R>
p Cp Rp . Lower case letters xit and yit denote the entries of Xp and Yp , respectively . The
analysis is based on the following assumptions:
Assumption 1 (A1, bound on average eighth moment). There exists a constant K1 independent of
p such that
p
1X
E[(xpi1 )8 ] ? K1 .
p i=1
Assumption 2 (A2, uncorrelatedness of higher moments). Let Q denote the set of quadruples
{i,j,k,l} of distinct integers.
P
p
p
2 p p
i,j,kl,l?Qp Cov [yi1 yj1 , yk,1+s yl,1+s ]
= O p?1 ,
|Qp |
and
h
i
p p 2
p
p
2
Cov
(y
y
)
,
(y
y
)
i1 j1
i,j,kl,l?Qp
k,1+s l,1+s
P
?s :
|Qp |
= O p?1 ,
hold.
Assumption 3 (A3, non-degeneracy). There exists a constant K2 such that
p
1X
E[(xpi1 )2 ] ? K2 .
p i=1
Assumption 4 (A4, moment relation). There exist constants ?4 , ?8 , ?4 and ?8 such that
E[yi8 ]
?
(1 + ?8 )E2 [yi4 ],
E[yi4 ]
?
(1 + ?4 )E2 [yi2 ],
E[yi8 ]
?
(1 + ?8 )E2 [yi4 ],
E[yi4 ]
?
(1 + ?4 )E2 [yi2 ].
Remarks on the assumptions A restriction on the eighth moment (assumption A1) is necessary
because the estimators eq. (2), (3), (5) and (6) contain fourth moments, their variances therefore
contain eighths moments. Note that, contrary to the similar assumption in the eigenbasis in [LW04],
A1 poses no restriction on the covariance structure [BM13]. To quantify the effect of averaging
over dimensions, assumption A2 restricts the correlations of higher moments in the eigenbasis. This
assumption is trivially fulfilled for Gaussian data, but much weaker (see [LW04]). Assumption A3
rules out the degenerate case of adding observation channels without any variance and assumption
A4 excludes distributions with arbitrarily heavy tails.
Based on these assumptions, we can analyse the difference between the Sancetta-estimator and our
proposed estimator for large p:
Theorem 1 (consistency under ?fixed n?-asympotics). Let A1, A2, A3, A4 hold. We then have
1 X
Var (Sij ) = ?(1)
p2 ij
3
We shall often drop the sequence index p and the observation index t to improve readability of formulas.
4
no AC (b = 10)
?3
?ij var(Sij)/p2
9.5
x 10
low AC (b = 20)
high AC (b = 90)
0.09
30
25
9
0.08
8.5
0.07
8
0.06
7.5
0.05
7
0.04
6.5
0.03
population
shrinkage
Sancetta
bias?corr
20
15
6
0
100
200
300
400
500
10
5
0.02
0
100
200
300
400
500
0
0
dimensionality
dimensionality
100
200
300
400
500
dimensionality
Figure 3: Dependence of the variance estimates on the dimensionality. Averaged over R = 50
models. n = 250.
2
P 2 !
1 X
2
San,b
j ?j
San,b
San,b
d
+O P
E
+ BiasTR
Var
(Sij ) ? Var (Sij )
p2
= Bias
( j ?j )2
ij
2
P 2 !
1 X
BC,b
j ?j
BC,b 2
d
E
2
Var
(Sij ) ? Var (Sij )
= BiasTR
+O P
( j ?j ) 2
p ij
where the ?i denote the eigenvalues of C and
)
(
n
n
b
4 X X X
1 X 1 + 2b ? b(b + 1)/n
San,b
:= ? 2
Cov [xit xjt , xiu xju ]
Var (Sij ) ? 3
Bias
p ij
n
n s=1 t=n?s u=1
:= ?
BiasSan,b
TR
n
1 2 X X n?s
Cov [xit xjt , xi,t+s xj,t+s ]
p2 n ij
n
s=b+1
BiasBC,b
TR := ?
1
2
2
p n ? 1 ? 2b +
X n?1
X
b(b+1)
n
Cov [xit xjt , xi,t+s xj,t+s ]
ij s=b+1
Proof. see the supplemental material.
Remarks on Theorem 1 (i) The mean squared error of both estimators consists of a bias and a
variance term. Both estimators have a truncation bias which is a consequence of including only a
limited number of time lags into the variance estimation. When b is chosen sufficiently high, this
term gets close to zero. (ii) The Sancetta-estimator has an additional bias term which is smaller than
zero in each dimension and therefore does not average out. Simulations will show that, as a consequence, the Sancetta-estimator has a strong bias which gets larger with increasing lag parameter b.
P
P
2
(iii) The variance of both estimators behaves as O( i ?i2 / ( i ?i ) ): the more the variance of the
data is spread over the eigendirections, the smaller the variance of the estimators. This bound is
minimal if the eigenvalues are identical. (iv) Theorem 1 does not make a statement on the relative
sizes of the variances of the estimators. Note that the BC estimator mainly differs by a multiplicative
factor > 1, hence the variance is larger, but not relative to the expectation of the estimator.
3
Simulations
Our simulations are based on those in [San08]: We average over R = 50 multivariate Gaussian
AR(1) models
~xt = A~xt?1 + ~t ,
4
with parameter matrix A = ?AC ? I , with ?no AC = 0, ?low AC = 0.7, and ?high AC = 0.95 (see Figure 1). The innovations it are Gaussian with variances ?i2 drawn from a log-normal distribution
4
more complex parameter matrices or a different generative model do not pose a problem for the biascorrected estimator. The simple model was chosen for clarity of presentation.
5
no AC
?3
?ij var(Sij)/p2
8
x 10
shrinkage intensity ?
high AC
20
0.08
15
0.06
10
0.04
5
6
4
2
0
PRIAL
low AC
0.1
25
50
75
100
0.5
0.4
0.02
0
25
50
75
100
0
0
0.6
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
25
50
75
100
25
50
75
100
0.3
0.2
0.1
0
25
50
75
100
0.1
0
25
50
75
100
0
0
0.4
0.7
1
0.35
0.6
0.8
0.3
0.5
0.6
0.25
0.4
0.4
0.2
0.3
0.2
0
25
50
75
100
0.2
0
25
50
75
100
0
0
pop. var(S)
shrinkage
Sancetta
bias?corr
25
50
75
100
lag parameter b
Figure 4: Robustness to the choice of lag parameter b. Variance estimates (upper row), shrinkage
intensities (middle row) and improvement over sample covariance (lower row). Averaged over R =
50 models. p = 200, n = 250.
with mean ? = 1 and scale parameter ? = 0.5. For each model, we generate K = P
50 data sets to
calculate the std. deviations of the estimators and to obtain an approximation of p?2 ij Var (Sij ).
Simulation 1 analyses the dependence of the estimators on the dimensionality of the data. The number of observations is fixed at n = 250 and the lag parameter b chosen by hand such that the whole
autocorrelation is covered5 : bno AC = 10, blow AC = 20 and bhigh AC = 90. Figure 3 shows that the
standard shrinkage estimator is unbiased and has low variance in the no AC-setting, but under the
presence of autocorrelation it strongly underestimates the variance. As predicted by Theorem 1,
the Sancetta estimator is also biased; its remains stays constant for increasing dimensionality. Our
proposed estimator has no visible bias. For increasing dimensionality the variances of all estimators decrease. Relative to the average estimate, there is no visible difference between the standard
deviations of the Sancetta and the BC estimator.
Simulation 2 analyses the dependency on the lag parameter b for fixed dimensionality p = 200
and number of observations n = 250. In addition to variance estimates, Figure 4 reports shrinkage
intensities and the percentage improvement in absolute loss (PRIAL) over the sample covariance
matrix:
EkS ? Ck ? EkC{pop., shr, San., BC} ? Ck
.
PRIAL C{pop., shr, San., BC} =
EkS ? Ck
The three quantities show very similar behaviour. Standard shrinkage performs well in the no ACcase, but is strongly biased in the autocorrelated settings. The Sancetta estimator is very sensitive to
the choice of the lag parameter b. For low AC, the bias at the optimal b is small: only a small number
of biased terms are included. For high AC the optimal b is larger, the higher number of biased terms
causes a larger bias. The BC-estimator is very robust: it performs well for all b large enough to
capture the autocorrelation. For very large b its variance increases slightly, but this has practically
5
for b < 1, optimal in the no AC-setting, Sancetta and BC estimator are equivalent to standard shrinkage.
6
0.8
0.2
0
0.7
0.65
sample cov
standard shrinkage
sancetta
bias?corr
cross?val.
0.6
25
50
0.55
0
75
time lag
5
10
15
0.02
0.01
0
0
20
accuracy
AC
0
200
time lag
300
0.05
10
15
20
0
5
10
15
20
number of trials per class
0.2
0.7
0.65
sample cov
standard shrinkage
sancetta
bias?corr
cross?val.
0.6
100
0.1
0
5
0.8
0.5
0.15
number of trials per class
0.75
b = 300
0.03
0.55
0
5
10
15
20
number of trials per class
0.05
shrinkage intensity
1
?0.5
0
0.04
number of trials per class
accuracy ? acc(sample cov)
?0.5
0
0.05
shrinkage intensity
accuracy
AC
b = 75
0.75
0.5
accuracy ? acc(sample cov)
1
0.04
0.03
0.02
0.01
0
0
0.15
0.1
0.05
0
5
10
15
number of trials per class
20
0
5
10
15
20
number of trials per class
Figure 5: BCI motor imagery data for lag parameter b = 75 (upper row) and b = 300 (lower row).
Averaged over subjects and K = 100 runs.
no effect on the PRIAL. An interesting aspect is that our proposed estimator even outperforms
shrinkage based on the the population Var (Sij ) (calculated by resampling). This results from the
d Sij BC,b with the sample estimate eq. (3) of the denominator in
correlation of the estimator Var
eq. (1).
4
Real World Data: Brain Computer Interface based on Motor Imagery
As an example of autocorrelated data we reanalyzed a data set from a motor imagery experiment. In
the experiment, brain activity for two different imagined movements was measured via EEG (p = 55
channels, 80 subjects, 150 trials per subject, each trial with ntrial = 390 measurements [BSH+ 10]).
The frequency band was optimized for each subject and from the class-wise covariance matrices, 1-3
filters per class were extracted by Common Spatial Patterns (CSP), adaptively chosen by a heuristic
(see [BTL+ 08]). We trained Linear Discriminant Analysis on log-variance features.
To improve the estimate of the class covariances on these highly autocorrelated data, standard shrinkage, Sancetta shrinkage, cross-validation and and our proposed BC shrinkage estimator were applied. The covariance structure is far from diagonal, therefore, for each subject, we used the average
of the class covariances of the other subjects as shrinkage target [BLT+ 11]. Shrinkage is dominated by the influence of high-variance directions [BM13], which are pronounced in this data set.
To reduce this effect, we rescaled, only for the calculation of the shrinkage intensities, the first five
principal components to have the same variance as the sixth principal component.
We analyse the dependency of the four algorithms on the number of supplied training trials. Figure 5
(upper row) shows results for an optimized time lag (b = 75) which captures well the autocorrelation of the data (outer left). Taking the autocorrelation into account makes a clear difference
(middle left/right): while standard shrinkage outperforms the sample covariance, it is clearly outperformed by the autocorrelation-adjusted approaches. The Sancetta-estimator is slightly worse
than our proposed estimator. The shrinkage intensities (outer right) are extremely low for standard
shrinkage and the negative bias of the Sancetta-estimator shows clearly for small numbers of training trials. Figure 5 (lower row) shows results for a too large time lag (b = 300). The performance
of the Sancetta-estimator strongly degrades as its shrinkage intensities get smaller, while our proposed estimator is robust to the choice of b, only for the smallest number of trials we observe a
small degradation in performance. Figure 6 (left) compares our bias-corrected estimator to the four
other approaches for 10 training trials: it significantly outperforms standard shrinkage and Sancetta
shrinkage for both the larger (b = 300, p ? 0.01) and the smaller time lag (b = 75, p ? 0.05).
7
time demand
bias?corr
0.9
1
**90%
**90%
0.9
1
**77.50%
**78.75%
0.9
1
51.25%
53.75%
0.9
*60%
**60%
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
b = 75
b = 300
0.6
*38.75%
**38.75%
0.6
**8.75%
**8.75%
0.5
0.5
0.6
0.7
0.8
0.9
sample covariance
0.6
**21.25%
**20%
0.5
0.5
0.6
0.7
0.8
0.9
0.6
0.5
0.5
47.50%
45.00%
0.6
0.7
standard shrinkage
0.8
CV
0.9
0.5
0.5
0.6
0.7
0.8
0.9
normalized runtime
subject?wise classification accuracies
1
120
100
80
60
40
20
0
SC Shr San BC
CV
Sancetta estimator
Figure 6: Subject-wise BCI classification accuracies for 10 training trials (left) and time demands
(right). ?? /? := significant at p ? 0.01 or p ? 0.05, respectively.
Analytic shrinkage procedures optimize only the mean squared error of the covariance matrix, while
cross-validation directly optimizes the classification performance. Yet, Figure 5 (middle) shows that
for small numbers of training trials our proposed estimator outperforms CV, although the difference
is not significant (see Fig. 6). For larger numbers of training trials CV performs better. This shows
that the MSE is not a very good proxy for classification accuracies in the context of CSP: for optimal MSE, shrinkage intensities decrease with increasing number of observations. CV shrinkage
intensities instead stay on a constant level between 0.1 and 0.15. Figure 6 (right) shows that the
three shrinkage approaches (b = 300) have a huge performance advantage over cross-validation (10
folds/10 parameter candidates) with respect to runtime.
5
Discussion
Analytic Shrinkage estimators are highly useful tools for covariance matrix estimation in timecritical or computationally expensive applications: no time-consuming cross-validation procedure
is required. In addition, it has been observed that in some applications, cross-validation is not a
good predictor for out-of-sample performance [LG11, BKT+ 07]. Its speed and good performance
has made analytic shrinkage widely used: it is, for example, state-of-the-art in ERP experiments
[BLT+ 11]. While standard shrinkage assumes i.i.d. data, many real world data sets, for example
from video, audio, finance, biomedical engineering or energy systems clearly violate this assumption as strong autocorrelation is present. Intuitively this means that the information content per data
point becomes lower, and thus the covariance estimation problem becomes harder: the dimensionality remains unchanged but the effective number of samples available decreases. Thus stronger
regularization is required and standard analytic shrinkage [LW04] needs to be corrected.
Sancetta already moved the first step into this important direction by providing a consistent estimator
under i.i.d. violations [San08]. In this work we analysed finite sample sizes and showed that (i) even
apart from truncation bias ?which results from including a limited number of time lags? Sancetta?s
estimator is biased, (ii) this bias is only negligible if the autocorrelation decays fast compared to
the length of the time series and (iii) the Sancetta estimator is very sensitive to the choice of lag
parameter.
We proposed an alternative estimator which is (i) both consistent and ?apart from truncation bias?
unbiased and (ii) highly robust to the choice of lag parameter: In simulations on toy and real world
data we showed that the proposed estimator yields large improvements for small samples and/or suboptimal lag parameter. Even for optimal lag parameter there is a slight but significant improvement.
Analysing data from BCI motor imagery experiments we see that (i) the BCI data set possesses
significant autocorrelation, that (ii) this adversely affects CSP based on the sample covariance and
standard shrinkage (iii) this effect can be alleviated using our novel estimator, which is shown to
(iv) compare favorably to Sancetta?s estimator.
Acknowledgments
This research was also supported by the National Research Foundation grant (No. 2012-005741)
funded by the Korean government. We thank Johannes H?ohne, Sebastian Bach and Duncan Blythe
for valuable discussions and comments.
8
References
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9
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4,859 | 54 | 860
A METHOD FOR THE DESIGN OF STABLE LATERAL INHIBITION
NETWORKS THAT IS ROBUST IN THE PRESENCE
OF CIRCUIT PARASITICS
J.L. WYATT, Jr and D.L. STANDLEY
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
In the analog VLSI implementation of neural systems, it is
sometimes convenient to build lateral inhibition networks by using
a locally connected on-chip resistive grid. A serious problem
of unwanted spontaneous oscillation often arises with these
circuits and renders them unusable in practice. This paper reports
a design approach that guarantees such a system will be stable,
even though the values of designed elements and parasitic elements
in the resistive grid may be unknown. The method is based on a
rigorous, somewhat novel mathematical analysis using Tellegen's
theorem and the idea of Popov multipliers from control theory. It
is thoroughly practical because the criteria are local in the sense
that no overall analysis of the interconnected system is required,
empirical in the sense that they involve only measurable frequency
response data on the individual cells, and robust in the sense that
unmodelled parasitic resistances and capacitances in the interconnection network cannot affect the analysis.
I.
INTRODUCTION
The term "lateral inhibition" first arose in neurophysiology to
describe a common form of neural circuitry in which the output of
each neuron in some population is used to inhibit the response of
each of its neighbors. Perhaps the best understood example is the
horizontal cell layer in the vertebrate retina, in which lateral
inhibition simultaneously enhances intensity edges and acts as an
automatic lain control to extend the dynamic range of the retina
as a whole. The principle has been used in the design of artificial
neural system algorithms by Kohonen 2 and others and in the electronic
design of neural chips by Carver Mead et. al. 3 ,4.
In the VLSI implementation of neural systems, it is convenient
to build lateral inhibition networks by using a locally connected
on-chip resistive grid. Linear resistors fabricated in, e.g.,
polysilicon, yield a very compact realization, and nonlinear
resistive grids, made from MOS transistors, have been found useful
for image segmentation. 4 ,5 Networks of this type can be divided into
two classes: feedback systems and feedforward-only systems. In the
feedforward case one set of amplifiers imposes signal voltages or
? American Institute of Physics 1988
861
currents on the grid and another set reads out the resulting response
for subsequent processing, while the same amplifiers both "write" to
the grid and "read" from it in a feedback arrangement. Feedforward
networks of this type are inherently stable, but feedback networks
need not be.
A practical example is one of Carver Meadls retina chips3 that
achieves edge enhancement by means of lateral inhibition through a
resistive grid. Figure 1 shows a single cell in a continuous-time
version of this chip. Note that the capacitor voltage is affected
both by the local light intensity incident on that cell and by the
capacitor voltages on neighboring cells of identical design. Any
cell drives its neighbors, which drive both their distant neighbors
and the original cell in turn. Thus the necessary ingredients for
instability--active elements and signal feedback--are both present
in this system, and in fact the continuous-time version oscillates
so badly that the original design is scarcely usable in practice
with the lateral inhibition paths enabled. 6 Such oscillations can
I
incident
light
v
out
Figure 1. This photoreceptor and signal processor Circuit, using two
MOS transconductance amplifiers, realizes lateral inhibition by
communicating with similar units through a resistive grid.
readily occur in any resistive grid circuit with active elements and
feedback,even when each individual cell is quite stable. Analysis
of the conditions of instability by straightforward methods appears
hopeless, since any repeated array contains many cells, each of
which influences many others directly or indirectly and is influenced
by them in turn, so that the number of simultaneously active feedback loops is enormous.
This paper reports a practical design approach that rigorously
guarantees such a system will be stable. The very simplest version
of the idea is intuitively obvious: design each individual cell so
that, although internally active, it acts like a passive system as
seen from the resistive grid. In circuit theory language, the
design goal here is that each cellis output impedance should be a
positive-real? function. This is sometimes not too difficult in
practice; we will show that the original network in Fig. 1 satisfies
this condition in the absence of certain parasitic elements. More
important, perhaps, it is a condition one can verify experimentally
862
by frequency-response measurements.
It is physically apparent that a collection of cells that
appear passive at their terminals will form a stable system when
interconnected through a passive medium such as a resistive grid.
The research contributions, reported here in summary form, are
i) a demonstration that this passivity or positive-real condition
is much stronger than we actually need and that weaker conditions,
more easily achieved in practice, suffice to guarantee stability of
the linear network model, and ii) an extension of i) to the nonlinear
domain that furthermore rules out large-signal oscillations under
certain conditions.
II.
FIRST-ORDER LINEAR ANALYSIS OF A SINGLE CELL
We begin with a linear analysis of an elementary model for the
circuit in Fig. 1. For an initial approximation to the output
admittance of the cell we simplify the topology (without loss of
relevant information) and use a naive'model for the transconductance
amplifiers, as shown in Fig. 2.
e
+
Figure 2. Simplified network topology and transconductance amplifier
model for the circuit in Fig. 1. The capacitor in Fig. 1 has been
absorbed into CO2 ?
Straightforward calculations show that the output admittance is
given by
yes)
(1)
This is a positive-real, i.e., passive, admittance since it can always
be realized by a network of the form shown in Fig. 3, where
= (gm2+
-1 -1
-1
Ro2 ) , R2= (gmlgm2Rol)
, and L = COI/gmlgm2?
Although the original circuit contains no inductors, the
realization has both capacitors and inductors and thus is capable
of damped oscillations. Nonetheless, i f the transamp model in
Fig. 2 were perfectly accurate, no network created by interconnecting
such cells through a resistive grid (with parasitic capacitances)
could exhibit sustained oscillations. For element values that may
be typical in practice, the model in Fig. 3 has a lightly damped
resonance around I KHz with a Q ~ 10. This disturbingly high Q
suggests that the cell will be highly sensitive to parasitic elements
not captured by the simple models in Fig. 2. Our preliminary
Rl
863
yes)
Figure 3. Passive network realization of the output admittance (eq.
(1) of the circuit in Fig. 2.
analysis of a much more complex model extracted from a physical
circuit layout created in Carver Mead's laboratory indicates that
the output impedance will not be passive for all values of the transamp bias currents. But a definite explanation of the instability
awaits a more careful circuit modelling effort and perhaps the design
of an on-chip impedance measuring instrument.
III.
POSITIVE-REAL FUNCTIONS, e-POSITlVE FUNCTIONS, AND
STABILITY OF LINEAR NETWORK MODELS
In the following discussion s = cr+jw is a complex variable,
H(s) is a rational function (ratio of polynomials) in s with real
coefficients, and we assume for simplicity that H(s) has no pure
imaginary poles. The term closed right halE plane refers to the set
of complex numbers s with Re{s} > o.
Def. I
The function H(s) is said to be positive-real if a) it has no
poles in the right half plane and b) Re{H(jw)} ~ 0 for all w.
If we know at the outset that H(s) has no right half plane poles,
then Def. I reduces to a simple graphical criterion: H1s} is positivereal if and only if the Nyquist diagram of H(s) (i.e. the plot of
H(jW) for w ~ 0, as in Fig. 4) lies entirely in the closed right half
plane.
Note that positive-real functions are necessarily stable since
they have no right half plane poles, but stable functions are not
necessarily positive-real, as Example 1 will show.
A deep link between positive real functions, physical networks
and passivity is established by the classical result 7 in linear
circuit theory which states that H(s) is positive-real if and only if
it is possible to synthesize a 2-terminal network of positive linear
resistors, capacitors, inductors and ideal transformers that has H(s)
as its driving-point impedance or admittance.
864
Oef. 2
The function H(s) is said to be a-positive for a particular value
of e(e ~ 0, e ~ ~), if a) H{s) has no poles in the right half plane,
and b) the Nyquist plot of H(s) lies strictly to the right of the
straight line passing through the origin at an angle a to the real
positive axis.
Note that every a-positive function is stable and any function
that is e-positive with e = ~/2 is necessarily positive-real.
I {G(jw)}
m
Re{G(jw) }
Figure 4. Nyquist diagram for a fUnction that is a-positive but
not positive-real.
Example 1
The function
G (s)
=
(s+l) (s+40)
(s+5) (s+6) (s+7)
(2)
is a-positive (for any e between about 18? and 68?) and stable, but it
is not positive-real since its Nyquist diagram, shown in Fig. 4,
crosses into the left half plane.
The importance of e-positive functions lies in the following
observations: 1) an interconnection of passive linear resistors and
capacitors and cells with stable linear impedances can result in an
unstable network, b) such an instability cannot result if the
impedances are also positive-real, c) a-positive impedances form a
larger class than positive-real ones and hence a-positivity is a less
demanding synthesis goal, and d) Theorem 1 below shows that such an
instability cannot result if the impedances are a-positive, even if
they are not positive-real.
Theorem 1
Consider a linear network of arbitrary topology, consisting of
any number of passive 2-terminal resistors and capacitors of arbitrary
value driven by any number of active cells. If the output impedances
865
'II" ,
of all the active cells are a-positive for some common a, 0<a 22
then the network is stable.
The proof of Theorem 1 relies on Lemma 1 below.
Lemma 1
If H(s) is a-positive for some fixed a, then for all So in the
closed first quadrant of the complex plane, H(so) lies strictly to
the right of the straight line passing through the origin at an angle
a to the real positive axis, i.e., Re{so} ~ 0 and Im{so} ~ 0 ~
a-'II" < L H (so) < a.
Proof of Lemma 1 (Outline)
Let d be the function that assigns to each s in the closed right
half plane the perpendicular distance des) from H(s) to the line
defined in Def. 2. Note that des) is harmonic in the closed right
half plane, since H is analytic there. It then follows, by application
of the maximum modulus principle8 for harmonic functions, that d takes
its minimum value on the boundary of its domain, which is the
imaginary axis. This establishes Lemma 1.
Proof of Theorem 1 (OUtline)
The network is unstable or marginally stable if and only if it
has a natural frequency in the closed right half plane, and So is a
natural frequency if and only if the network equations have a nonzero
solution at so. Let {Ik} denote the complex branch currents Of such
a solution. By Tellegen I s theorern9 the sum of the complex powers
absorbed by the circuit elements must vanish at such a solution, i.e.,
~
IIk12/s0Ck +
capac~tances
L
cell
terminal pairs
(3)
where the second term is deleted in the special case so=O, since the
complex power into capacitors vanishes at so=O.
If the network has a natural frequency in the closed right half
plane, it must have one in the closed first quadrant since natural
frequencies are either real or else occur in complex conjugate pairs.
But (3) cannot be satisfied for any So in the closed first quadrant,
as we can see by dividing both sides of (3) by
IIkI2, where the
k
sum is taken over all network branches. After this division, (3)
asserts that zero is a convex combination of terms of the form Rk,
terms of the form (CkSo)-I, and terms of the form Zk(So).
Visualize where these terms lie in the complex plane: the first set lies
on the real positive axis, the second set lies in the closed 4-th
~adrant since So lies in the closed 1st quadrant by assumption, and
the third set lies to the right of a line passing through the origin
at an angle a by Lemma 1. Thus all these terms lie strictly to the
right of this line, which implies that no convex combination of them
can equal zero. Hence the network is stable!
866
IV.
STABILITY RESULT FOR NETWORKS WITH NONLINEAR
RESISTORS AND CAPACITORS
The previous result for linear networks can afford some limited
insight into the behavior of nonlinear networks. First the nonlinear
equations are linearized about an equilibrium point and Theorem 1 is
applied to the linear model. If the linearized model is stable, then
the equilibrium point of the original nonlinear network is locally
stable, i.e., the network will return to that equilibrium point if
the initial condition is sufficiently near it. But the result in this
section, in contrast, applies to the full nonlinear circuit model and
allows one to conclude that in certain circumstances the network
cannot oscillate even if the initial state is arbitrarily far from
the equilibrium point.
Def. 3
A function H(s) as described in Section III is said tc satisfy
the Popov criterion lO if there exists a real number r>O such that
Re{(l+jwr) H(jw)} ~ 0 for all w.
Note that positive real functions satisfy the Popov criterion
with r=O. And the reader can easily verify that G(s) in Exam~le I
satisfies the Popov criterion for a range of values of r. The important
effect of the term (l+jwr) in Def. 3 is to rotate the Nyquist plot
counterclockwise by progressively greater amounts up to 90? as w
increases.
Theorem 2
Consider a network consisting of nonlinear 2-terminal resistors
and capacitors, and cells with linear output impedances ~(s). Suppose
i) the resistor curves are characterized by continuously
diffefentiable functions i k = gk(vk ) where gk(O) = 0 and
o < gk(vk ) < G < 00 for all values of k and vk'
ii) the capacitors are characterized by i k = Ck(Vk)~k with
< CI < Ck(v k ) < C2 < 00 for all values of k and vk'
o
iii) the impedances Zk(s) have no poles in the closed right
half plane and all satisfy the Popov criterion for some common
value of r.
If these conditions are satisfied, then the network is stable in the
sense that, for any initial condition,
f oo(
o
I
all branches
i~(t) )
dt
<
00
?
The proof, based on Tellegen's theorem, is rather involved.
will be omitted here and will appear elsewhere.
(4)
It
867
ACKNOWLEDGEMENT
We sincerely thank Professor Carver Mead of Cal Tech for
enthusiastically supporting this work and for making it possible for
us to present an early report on it in this conference proceedings.
This work was supportedJ::? Defense Advanced Research Projects Agency
(DoD), through the Office of Naval Research under ARPA Order No.
3872, Contract No. N00014-80-C-0622 and Defense Advanced Research
Projects Agency (DARPA) Contract No. N00014-87-R-0825.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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C.A. Mead, Analog VLSI and Neural Systems, Addison-Wesley, to
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J. Hutchinson, C. Koch, J. Luo and C. Mead, "Computing Motion
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B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and
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and Electrical Networks, MIT Press, Cambridge, MA,1970.
M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall,
Englewood Cliffs, NJ, 1970, pp. 211-217.
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4,860 | 540 | Learning to Segment Images
Using Dynamic Feature Binding
Michael C. Moser
Dept. of Compo Science &
Inst. of Cognitive Science
University of Colorado
Boulder, CO 80309-0430
Richard S. Zemel
Dept. of Compo Science
University of Toronto
Toronto, Ontario
Canada M5S lA4
Marlene Behrmann
Dept. of Psychology &
Faculty of Medicine
University of Toronto
Toronto, Ontario
Canada M5S lAl
Abstract
Despite the fact that complex visual scenes contain multiple, overlapping
objects, people perform object recognition with ease and accuracy. One
operation that facilitates recognition is an early segmentation process in
which features of objects are grouped and labeled according to which object they belong. Current computational systems that perform this operation are based on predefined grouping heuristics. We describe a system
called MAGIC that learn. how to group features based on a set of presegmented examples. In many cases, MAGIC discovers grouping heuristics
similar to those previously proposed, but it also has the capability of finding nonintuitive structural regularities in images. Grouping is performed
by a relaxation network that aUempts to dynamically bind related features. Features transmit a complex-valued signal (amplitude and phase)
to one another; binding can thus be represented by phase locking related
features. MAGIC'S training procedure is a generalization of recurrent back
propagation to complex-valued units.
When a visual image contains multiple, overlapping objects, recognition is difficult
because features in the image are not grouped according to which object they belong.
Without the capability to form such groupings, it would be necessary to undergo a
massive search through all subsets of image features. For this reason, most machine
vision recognition systems include a component that performs feature grouping or
image .egmentation (e.g., Guzman, 1968; Lowe, 1985; Marr, 1982).
436
Learning to Segment Images Using Dynamic Feature Binding
A multitude of heuristics have been proposed for segmenting images. Gestalt psychologists have explored how people group elements of a display and have suggested
a range of grouping principles that govern human perception (Rock &z: Palmer, 1990).
Computer vision researchers have studied the problem from a more computational perspective. They have investigated methods of grouping elements of an image
based on nonaccidental regularitie..-feature combinations that are unlikely to occur
by chance when several objects are juxtaposed, and are thus indicative of a single
object (Kanade, 1981; Lowe &z: Binford, 1982).
In these earlier approaches, the researchers have hypothesized a set of grouping
heuristics and then tested their psychological validity or computational utility. In
our work, we have taken an adaptive approach to the problem of image segmentation in which a system learns how to group features based on a set of examples.
We call the system MAGIC, an acronym for multiple-object !daptive grouping of
image ~omponents. In many cases MAGIC discovers grouping heuristics similar to
those proposed in earlier work, but it also has the capability offinding nonintuitive
structural regularities in images.
is trained on a set of presegmented images containing multiple objects. By
"presegmented," we mean that each image feature is labeled as to which object it
belongs. MAGIC learns to detect configurations of the image features that have a
consistent labeling in relation to one another across the training examples. Identifying these configurations allows MAGIC to then label features in novel, unsegmented
images in a manner consistent with the training examples.
MAGIC
1
REPRESENTING FEATURE LABELINGS
Before describing MAGIC, we must first discuss a representation that allows for
the labeling of features. Von der Malsburg (1981), von der Malsburg &z: Schneider
(1986), Gray et al. (1989), and Eckhorn et al. (1988), among others, have suggested
a biologically plausible mechanism of labeling through temporal correlations among
neural signals, either the relative timing of neuronal spikes or the synchronization of
oscillatory activities in the nervous system. The key idea here is that each processing
unit conveys not just an activation value-average firing frequency in neural termsbut also a second, independent value which represents the relative phcue of firing.
The dynamic grouping or binding of a set of features is accomplished by aligning
the phases of the features. Recent work (Goebel, 1991; Hummel &z: Biederman, in
press) has used this notion of dynamic binding for grouping image features, but has
been based on relatively simple, predetermined grouping heuristics.
2
THE DOMAIN
Our initial work has been conducted in the domain of two-dimensional geometric
contours, including rectangles, diamonds, crosses, triangles, hexagons, and octagons. The contours are constructed from four primitive feature types-oriented
line segments at 0?, 45?, 90?, and 135?-and are laid out on a 15 X 20 grid. At
each location on the grid are units, called feature unib, that detect each of the four
primitive feature types. In our present experiments, images contain two contours.
Contours are not permitted to overlap in their activation of the same feature unit.
437
438
Mozer, Zemel , and Behrmann
hidden
layer
_ _r
Figure 1: The architedure of MAGIC. The lower layer contains the feature units; the
upper layer contains the hidden units. Each layer is arranged in a spatiotopic array
with a number of different feature types at each position in the array. Each plane in
the feature layer corresponds to a different feature type. The grayed hidden units
are reciprocally conneded to all features in the corresponding grayed region of the
feature layer. The lines between layers represent projections in both directions.
3
THE ARCHITECTURE
The input to MAGIC is a paUern of activity over the feature units indicating which
features are present in an image. The initial phases ofthe units are random. MAGIC'S
task is to assign appropriate phase values to the units. Thus, the network performs
a type of paUern completion.
The network architedure consists of two layers of units, as shown in Figure 1. The
lower (input) layer contains the feature units, arranged in spatiotopic arrays with
one array per feature type. The upper layer contains hidden units that help to align
the phases of the feature units; their response properties are determined by training.
Each hidden unit is reciprocally conneded to the units in a local spatial region of
all feature arrays. We refer to this region as a patch; in our current simulations, the
patch has dimensions 4 x 4. For each patch there is a corresponding fixed-size pool
of hidden units. To achieve uniformity of response across the image, the pools are
arranged in a spatiotopic array in which neighboring pools respond to neighboring
patches and the weights of all pools are consbained to be the same.
The feature units activate the hidden units, which in turn feed back to the feature
units. Through a relaxation process, the system settles on an assignment of phases
to the features.
Learning to Segment Images Using Dynamic Feature Binding
4
NETWORK DYNAMICS
Formally, the response of each feature unit i, ~i, is a complex value in polar form,
(<<li, pil, where ?li is the amplitude or activation and Pi is the phase. Similarly, the
response of each hidden unit ;, 11;, has components (b;, q;). The weight connecting
unit i to unit ;, wiiJ is also complex valued, having components (Pii,8ii ). The
activation rule we propose is a generalization of the dot product to the complex
domain:
neti
x?wi
Ei~iWii
([(Ei?lip;i cos(Pi - 8;i?2
t
-1
an
+ (Ei?liPii sin(pi -
8ii ?2] ! ,
[Ei?lip;iSin(Pi - 8ii )])
Ei?liP;i COS(pi - 8;i)
where net; is the net input to hidden unit;. The net input is passed through
a squashing nonlinearity that maps the amplitude of the response from the range
o -+ 00 to 0 -+ 1 but leaves the phase unaffected:
1Ii
neti (1 _ e-Inetjl:l) .
Inet;1
The :Bow of activation from the hidden layer to the feature layer follows the same
dynamics, although in the current implementation the amplitudes of the features
are clamped, hence the top-down How affects only the phases. One could imagine a
more general architecture in which the relaxation process determined not only the
phase values, but cleaned up noise in the feature amplitudes as well.
The intuition underlying the activation rule is as follows. The activity of a hidden
unit, b;, should be monotonically related to how well the feature response pattern
matches the hidden unit weight vector, just as in the standard real-valued activation
rule. Indeed, one can readily see that if the feature and weight phases are equal
(Pi
8;i), the rule for bi reduces to the real-valued case. Even if the feature
and weight phases differ by a constant (Pi
8i i + e), b; is unaffected. This is
a critical property of the activation rule: Because ab.olute phase values have no
inhinsic meaning, the response of a unit should depend only on the relative phases.
The activation rule achieves this by essentially ignoring the average difference in
phase between the feature units and the weights. The hidden phase, q;, reHects this
average difference.
=
5
=
LEARNING ALGORITHM
During training, we would like the hidden units to learn to detect configurations
of features that reliably indicate phase relationships among the features. We have
experimented with a variety of training algorithms. The one with which we have
had greatest success involves running the network for a fixed number of iterations
and, after each iteration, attempting to adjust the weights so that the feature phase
pattern will match a target phase pattern. Each training hial proceeds as follows:
439
440
Mozer, Zemel, and Behrmann
1. A training example is generated at random. This involves selecting two contours and instantiating them in an image. The features of one contour have
target phase 0? and the features of the other contour have target phase 180?.
2. The training example is presented to MAGIC by clamping the amplitude of a
feature unit to 1.0 ifits corresponding image feature is present, or 0.0 otherwise.
The phases ofthe feature units are set to random values in the range 0? to 360?.
3. Activity is allowed to :flow from the feature units to the hidden units and back
to the feature units. Because the feature amplitudes are clamped, they are
unaffected.
4. The new phase pattern over the feature units is compared to the target phase
pattern (see step I), and an error measure is computed:
E = -(Et(l( cos(Pi - Pi))2 - (Eta. sin(Pi - Pi))2,
where p is the target phase pattern. This error ignores the absolute difference
between the target and actual phases. That is, E is minimized when Pi - Pi is
a constant for all i, regardless of the value of Pi - Pi.
5. Using a generalization of back propagation to complex valued units, error gradients are computed for the feature-to-hidden and hidden-to-feature weights.
6. Steps 3-5 are repeated for a maximum of 30 iterations. The trial is terminated
if the error increases on five consecutive iterations.
7. Weights are updated by an amount proportional to the average error gradient
over iterations.
Learning is more robust when the feature-to-hidden weights are constrained to be
symmetric with the hidden-to-feature weights. For complex weights, symmetry
means that the weight from feature unit i to hidden unit j is the complex conjugate of the weight from hidden unit j to feature unit i. Weight symmetry ensures
that MAGIC will converge to a fixed point. (The proof is based on discrete-time
update and a two-layer architecture with sequential layer updates and no intralayer
connections. )
Simulations reported below use a learning rate of .005 for the amplitudes and 0.02
for the phases. About 10,000 learning trials are required for stable performance,
although MAGIC rapidly picks up on the most salient aspects of the domain.
6
SIMULATION RESULTS
We trained a network with 20 hidden units per pool on images containing either
two rectangles, two diamonds, or a rectangle and a diamond. The shapes were of
varying size and appeared in various locations. A subset of the resulting weights are
shown in Figure 2. Each hidden unit attempts to detect and reinstantiate activity
patterns that match its weights. One clear and prevalent pattern in the weights is
the collinear arrangement of segments of a given orientation, all having the same
phase value. When a hidden unit having weights of this form responds to a patch of
the feature array, it tries align the phases of the patch with the phases of its weight
vector. By synchronizing the phases of features, it acts to group the features. Thus,
one can interpret the weight vectors as the rules by which features are grouped.
Learning to Segmem Images Using Dynamic Feature Binding
G
:'" "::OO
,',' 'Q QG)
': : ,',' 'G J::O :O
:' ":'8' ,','
.
..::. . -::.
':-.
..;:.
.:;"
.;..
.;::-
.;:.-
' ~G
:
V
Phase Spectrum
Figure 2: Sample of feature-to-hidden weights learned by MAGIC. The area of a
circle represents the amplitude of a weight, the orientation of the internal tick mark
represents the phase angle. The weights are arranged such that the connections
into each hidden unit are presented on a light gray background. Each hidden unit
has a total of 64 incoming weights--t x 4 locations in its receptive field and four
feature types at each location. The weights are further grouped by feature type,
and for each feature type they are arranged in a 4 X 4 pattern homologous to the
image patch itself.
Whereas traditional grouping principles indicate the conditions under which features
should be bound together as part of the same object, the grouping principles learned
by MAGIC also indicate when features should be segregated into different objects.
For example, the weights of the vertical and horizontal segments are generally 180 0
out of phase with the diagonal segments. This allows MAGIC to segregate the vertical
and horizontal features of a rectangle from the diagonal features of a diamond. We
had anticipated that the weights to each hidden unit would contain two phase
values at most because each image patch contains at most two objects. However,
some units make use of three or more phases, suggesting that the hidden unit is
performing several distinct functions. As is the usual case with hidden unit weights,
these patterns are difficult to interpret.
Figure 3 presents an example of the network segmenting an image. The image
contains two diamonds. The top left panel shows the features of the diamonds and
their initial random phases. The succeeding panels show the network's response
during the relaxation process. The lower right panel shows the network response at
equilibrium. Features of each object have been assigned a uniform phase, and the
two objects are 1800 out of phase. The task here may appear simple, but it is quite
challenging due to the illusory diamond generated by the overlapping diamonds.
441
442
Mozer, Zemel, and Behrmann
"'",
''''?''''tiIi'?''''
<~:~::-::4$s.,'
.,.."
...
..... ,.. "
'"
.
"
:;::.
.~
" ?..... # ",
?" ... ,;..,; ,'
, ,;? "
.;::'
..?
..-?
:.;.
.:::.
~
;::::
~
~~
.:::.
?
.
'::!?
~
Iteration 0
Iteration 2
Iteration 4
Iteration 6
Iteration 10
Iteration 25
Figure 3: An example of MAGIC segmenting an image. The "iteration" refers to
the number of times activity has flowed from the feature units to the hidden units
and back. The phase value of a feature is represented by a gray level. The periodic
phase continuum can only be approximated by the linear gray level continuum, but
the basic information is conveyed nonetheless.
7
CURRENT DIRECTIONS
We are currently extending
MAGIC
in several diredions, which we outline here.
? A natural principle for the hierarchical decomposition of objects emerges from
the relative frequency of feature configurations during training. More frequent
configurations result in a robust hidden representation, and hence the features
forming these configurations will be tightly coupled. A coarse quantization of
phases will lead to parses of the image in which only the highest frequency
configurations are considered as "objeds." Finer quantizations will lead to a
further decomposition of the image. Thus, the continuous phase representation
allows for the construdion of hierarchical descriptions of objeds.
? Spatially local grouping principles are unlikely to be sufficient for the image
segmentation task. Indeed, we have encountered incorred solutions produced
by MAGIC that are locally consistent but globally inconsistent. To solve this
problem, we are investigating an architecture in which the image is processed
at several spatial scales simultaneously.
? Simulations are also underway to examine MAGIC'S performance on real-world
images-overlapping handwriUen leUers and digits-where it is somewhat less
clear to which types of paUerns the hidden units should respond.
? Zemel, Williams, and Mozer (to appear) have proposed a mathematical framework that-with slight modifications to the model-allow it to be interpreted
Learning
to
Segment Images Using Dynamic Feature Binding
as a mean-field approximation to a stochastic phase model .
? Behrmann, Zemel, and Mozer (to appear) are conducting psychological experiments to examine whether limitations of the model match human limitations.
Acknowledgements
This research was supported by NSF Presidential Young Investigator award ffiI-9058450,
grant 90-21 from the James S. McDonnell Foundation, and DEC external research grant
1250 to MM, and by a National Sciences and Engineering Research Council Postgraduate
Scholarship to RZ. Our thanks to Paul Smolensky, Chris Williams, Geoffrey Hinton, and
Jiirgen Schmidhuber for helpful comments regarding this work.
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Guzman, A. (1968). Decomposition of a visual scene into three-dimensional bodies. AFIPS
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4,861 | 5,400 | A Dual Algorithm for Olfactory Computation in the
Locust Brain
Sina Tootoonian
[email protected]
M?at?e Lengyel
[email protected]
Computational & Biological Learning Laboratory
Department of Engineering, University of Cambridge
Trumpington Street, Cambridge CB2 1PZ, United Kingdom
Abstract
We study the early locust olfactory system in an attempt to explain its wellcharacterized structure and dynamics. We first propose its computational function
as recovery of high-dimensional sparse olfactory signals from a small number
of measurements. Detailed experimental knowledge about this system rules out
standard algorithmic solutions to this problem. Instead, we show that solving a
dual formulation of the corresponding optimisation problem yields structure and
dynamics in good agreement with biological data. Further biological constraints
lead us to a reduced form of this dual formulation in which the system uses independent component analysis to continuously adapt to its olfactory environment
to allow accurate sparse recovery. Our work demonstrates the challenges and rewards of attempting detailed understanding of experimentally well-characterized
systems.
1
Introduction
Olfaction is perhaps the most widespread sensory modality in the animal kingdom, often crucial for
basic survival behaviours such as foraging, navigation, kin recognition, and mating. Remarkably,
the neural architecture of olfactory systems across phyla is largely conserved [1]. Such convergent
evolution suggests that what we learn studying the problem in small model systems will generalize
to larger ones. Here we study the olfactory system of the locust Schistocerca americana. While
we focus on this system because it is experimentally well-characterized (Section 2), we expect our
results to extend to other olfactory systems with similar architectures. We begin by observing that
although most odors are mixtures of hundreds of molecular species, with typically only a few of
these dominating in concentration ? i.e. odors are sparse in the space of molecular concentrations
(Fig. 1A). We introduce a simple generative model of odors and their effects on odorant receptors
that reflects this sparsity (Section 3). Inspired by recent experimental findings [2], we then propose
that the function of the early olfactory system is maximum a posteriori (MAP) inference of these
concentration vectors from receptor inputs (Section 4). This is basically a sparse signal recovery
problem, but the wealth of biological evidence available about the system rules out standard solutions. We are then led by these constraints to propose a novel solution to this problem in term of
its dual formulation (Section 5), and further to a reduced form of this solution (Section 6) in which
the circuitry uses ICA to continuously adapt itself to the local olfactory environment (Section 7).
We close by discussing predictions of our theory that are amenable to testing in future experiments,
and future extensions of the model to deal with readout and learning simultaneously, and to provide
robustness against noise corrupting sensory signals (Section 8).
1
A
B
C
Odors
~1000
glomeruli
~1000
PNs
~100
bLNs
Behaviour
~300 LNs
Relative concentration
D
50,000
KCs
antennal lobe (AL)
90,000
ORNs
Molecules
E
0
1
Time (s)
2
antenna
mushroom body
(MB) 1 GGN
0
1
Time (s)
2
Figure 1: Odors and the olfactory circuit. (A) Relative concentrations of ? 70 molecules in the
odor of the Festival strawberry cultivar, demonstrating sparseness of odor vectors. (B,C) Diagram
and schematic of the locust olfactory circuit. Inputs from 90,000 ORNs converge onto ? 1000
glomeruli, are processed by the ? 1000 cells (projection neurons, PN, and local internuerons, LNs)
of the antennal lobe, and read out in a feedforward manner by the 50,000 Kenyon cells (KC) of the
mushroom body, whose activity ultimately is read out to produce behavior. (D,E) Odor response
of a PN (D) and a KC (E) to 7 trials of 44 mixtures of 8 monomolecular components (colors)
demonstrating cell- and odor-specific responses. The odor presentation window is in gray. PN
responses are dense and temporally patterned. KC responses are sparse and are often sensitive to
single molecules in a mixture. Panel A is reproduced from [8], B from [6], and D-E from the dataset
in [2].
2
Biological background
A schematic of the locust olfactory system is shown in Figure 1B-C. Axons from ? 90, 000 olfactory
receptor neurons (ORNs) each thought to express one type of olfactory receptor (OR) converge onto
approximately 1000 spherical neuropilar structures called ?glomeruli?, presumably by the ?1-OR-to1-glomerulus? rule observed in flies and mice. The functional role of this convergence is thought to
be noise reduction through averaging.
The glomeruli are sampled by the approximately 800 excitatory projection neurons (PNs) and 300
inhibitory local interneurons (LNs) of the antennal lobe (AL). LNs are densely connected to other
LNs and to the PNs; PNs are connected to each-other only indirectly via their dense connections
to LNs [3]. In response to odors, the AL exhibits 20 Hz local field potential oscillations and odorand cell-specific activity patterns in its PNs and LNs (Fig. 1D). The PNs form the only output of
the AL and project densely [4] to the 50,000 Kenyon cells (KCs) of the mushroom body (MB).
The KCs decode the PNs in a memoryless fashion every oscillation cycle, converting the dense
and promiscuous PN odor code into a very sparse and selective KC code [5], often sensitive to
a single component in a complex odor mixture [2] (Fig. 1E). KCs make axo-axonal connections
with neighbouring KCs [6] but otherwise only communicate with one-another indirectly via global
inhibition mediated by the giant GABA-ergic neuron [7]. Thus, while the AL has rich recurrency,
there is no feedback from the KCs back to the AL: the PN to KC circuit is strictly feedforward. As
we shall see below, this presents a fundamental challenge to theories of AL-MB computation.
2
3
Generative model
Natural odors are mixtures of hundreds of different types of molecules at various concentrations (e.g.
[8]), and can be represented as points in RN
+ , where each dimension represents the concentration
of one of the N molecular species in ?odor space?. Often a few of these will be at a much higher
concentration than the others, i.e. natural odors are sparse. Because the AL responds similarly across
concentrations [9] , we will ignore concentration in our odor model and consider odors as binary
vectors x ? {0, 1}N . We will also assume that molecules appear in odor vectors independently of
one-another with probability k/N , where k is the average complexity of odors (# of molecules/odor,
equivalently the Hamming weight of x) in odor space.
We assume a linear noise-free observation model y = Ax for the M -dimensional glomerular activity vector (we discuss observation noise in Section 7). A is an M ? N affinity matrix representing
the response of each of the M glomeruli to each of the N molecular odor components and has elements drawn iid. from a zero-mean Gaussian with variance 1/M . Our generative model for odors
and observations is summarized as
x = {x1 , . . . , xN }, xi ? Bernoulli(k/N ),
4
y = Ax,
Aij ? N (0, M ?1 )
(1)
Basic MAP inference
Inspired by the sensitivity of KCs to monomolecular odors [2], we propose that the locust olfactory
system acts as a spectrum analyzer which uses MAP inference to recover the sparse N -dimensional
odor vector x responsible for the dense M -dimensional glomerular observations y, with M
N e.g. O(1000) vs. O(10000) in the locust. Thus, the computational problem is akin to one in
compressed sensing [10], which we will exploit in Section 5. We posit that each KC encodes the
presence of a single molecular species in the odor, so that the overall KC activity vector represents
the system?s estimate of the odor that produced the observations y.
To perform MAP inference on binary x from y given A, a standard approach is to relax x to the
positive orthant RN
+ [11], smoothen the observation model with isotropic Gaussian noise of variance
? 2 and perform gradient descent on the log posterior
log p(x|y, A, k) = C ? ?kxk1 ?
where ? = log((1 ? q)/q), q = k/N , kxk1 =
of the posterior determines the x dynamics:
PM
i=1
1
ky ? Axk22
2? 2
(2)
xi for x 0, and C is a constant. The gradient
1
AT (y ? Ax)
(3)
2? 2
Given our assumed 1-to-1 mapping of KCs to (decoded) elements of x, these dynamics fundamentally violate the known biology for two reasons. First, they stipulate KC dynamics where there are
none. Second, they require all-to-all connectivity of KCs via AT A where none exist. In reality, the
dynamics in the circuit occur in the lower (? M ) dimensional measurement space of the antennal
lobe, and hence we need a way of solving the inference problem there rather than directly in the high
(N ) dimensional space of KC activites.
x? ? ?x log p = ?? sgn(x) +
5
Low dimensional dynamics from duality
To compute the MAP solution using lower-dimensional dynamics, we consider the following compressed sensing (CS) problem:
minimize kxk1 ,
subject to ky ? Axk22 = 0
(4)
whose Lagrangian has the form
L(x, ?) = kxk1 + ?ky ? Axk22
(5)
where ? is a scalar Lagrange multiplier. This is exactly the equation for our (negative) log posterior
(Eq. 2) with the constants absorbed by ?. We will assume that because x is binary, the two systems
will have the same solution, and will henceforth work with the CS problem.
3
To derive low dimensional dynamics, we first reformulate the constraint and solve
minimize kxk1 ,
subject to y = Ax
(6)
with Lagrangian
L(x, ?) = kxk1 + ?T (y ? Ax)
(7)
where now ? is a vector of Lagrange multipliers. Note that we are still solving an N -dimensional
minimization problem with M N constraints, while we need M -dimensional dynamics. Therefore, we consider the dual optimization problem of maximizing g(?) where g(?) = inf x L(x, ?)
is the dual Lagrangian of the problem. If strong duality holds, the primal and dual objectives have
the same value at the solution, and the primal solution can be found by minimizing the Lagrangian
at the optimal value of ? [11]. Were x ? RN , strong duality would hold for our problem by Slater?s
sufficiency condition [11]. The binary nature of x robs our problem of the convexity required for
this sufficiency condition to be applicable. Nevertheless we proceed assuming strong duality holds.
The dual Lagrangian has a closed-form expression for our problem. To see this, let b = AT ?.
Then, exploiting the form of the 1-norm and x being binary, we obtain the following:
g(?)??T y = inf kxk1 ?bT x = inf
x
x
T
M
X
(|xi |?bi xi ) =
i=1
M
X
i=1
inf (|xi |?bi xi ) = ?
xi
M
X
[bi ?1]+ (8)
i=1
T
or, in vector form, g(?) = ? y ? 1 [b ? 1]+ , where [?]+ is the positive rectifying function.
Maximizing g(?) by gradient descent yields M dimensional dynamics in ?:
?? ? ?? g = y ? A ?(AT ? ? 1)
(9)
where ?(?) is the Heaviside function. The solution to the CS problem ? the odor vector that produced
the measurements y ? is then read out at the convergence of these dynamics to ?? as
x? = argminx L(x, ?? ) = ?(AT ?? ? 1)
(10)
A natural mapping of equations 9 and 10 to antennal lobe dynamics is for the output of the M
glomeruli to represent y, the PNs to represent ?, and the KCs to represent (the output of) ?, and
hence eventually x? . Note that this would still require the connectivity between PNs and KCs to
be negative reciprocal (and determined by the affinity matrix A). We term the circuit under this
mapping the full dual circuit (Fig. 2B). These dynamics allow neuronal firing rates to be both
positive and negative, hence they can be implemented in real neurons as e.g. deviations relative to a
baseline rate [12], which is subtracted out at readout.
We measured the performance of a full dual network of M = 100 PNs in recovering binary odor
vectors containing an average of k = 1 to 10 components out of a possible N = 1000. The
results in Figure 2E (blue) show that the dynamics exhibit perfect recovery.1 For comparison, we
have included the performance of the purely feedforward circuit (Fig. 2A), in which the glomerular
vector y is merely scaled by the k-specific amount that yields minimum error before being read
out by the KCs (Fig. 2E, black). In principle, no recurrent circuit should perform worse than
this feedfoward network, otherwise we have added substantial (energetic and time) costs without
computational benefits.
6
The reduced dual circuit
The full dual antennal lobe circuit described by Equations 9 and 10 is in better agreement with the
known biology of the locust olfactory system than 2 for a number of reasons:
1. Dynamics are in the lower dimensional space of the antennal lobe PNs (?) rather than the
mushroom body KCs (x).
2. Each PN ?i receives private glomerular input yi
3. There are no direct connections between PNs; their only interaction with other PNs is
indirect via inhibition provided by ?.
1
See the the Supplementary Material for considerations when simulating the piecewise linear dynamics of
9.
4
Full Dual
PNs
KCs
D
PN activation
glom.
Odor
B
Feedforward Circuit
0.04
0.02
0
LN activation
Odor
A
1
0.8
0.6
0.4
0.2
0
E
Time (a.u.)
8
Feedforward
Full Dual
Reduced Dual
7
6
Distance
Reduced Dual
Odor
C
5
4
3
2
1
LNs
0
1
2
3
4
5
k
6
7
8
9
10
Figure 2: Performance of the feedforward and the dual circuits. (A-C) Circuit schematics. Arrows
(circles) indicate excitatory (inhibitory) connections. (D) Example PN and LN odor-evoked dynamics for the reduced dual circuit. Top: PNs receive cell-specific excitation or inhibition whose strength
is changed as different LNs are activated, yielding cell-specific temporal patterning. Bottom: The
LNs whose corresponding KCs encode the odor (red) are strongly excited and eventually breach
the threshold (dashed line), causing changes to the dynamics (time points marked with dots). The
excitation of the other LNs (pink) remains subthreshold. (E) Hamming distance between recovered
and true odor vector as a function of odor density k. The dual circuits generally outperform the
feedforward system over the entire range tested. Points are means, bars are s.e.m., computed for 200
trials (feedforward) and all trials from 200 attempts in which the steady-state solution was found
(dual circuits, greater than 90%).
4. The KCs serve merely as a readout stage and are not interconnected.2
However, there is also a crucial disagreement of the full dual dynamics with biology: the requirement
for feedback from the KCs to the PNs. The mapping of ? to PNs and ? to the KCs in Equation 9
implies negative reciprocal connectivity of PNs and KCs, i.e. a feedforward connection of Aij from
PN i to KC j, and a feedback connection of ?Aij from KC j to PN i. This latter connection from
KCs to PNs violates biological fact ? no such direct and specific connectivity from KCs to PNs exists
in the locust system, and even if it did, it would most likely be excitatory rather than inhibitory, as
KCs are excitatory.
Although KCs are not inhibitory, antennal lobe LNs are and connect densely to the PNs. Hence they
could provide the feedback required to guide PN dynamics. Unfortunately, the number of LNs is on
the order of that of the PNs, i.e. much fewer than the number of the KCs, making it a priori unlikely
that they could replace the KCs in providing the detailed pattern of feedback that the PNs require
under the full dual dynamics.
To circumvent this problem, we make two assumptions about the odor environment. The first is
that any given environment contains a small fraction of the set of all possible molecules in odor
space. This implies the potential activation of only a small number of KCs, whose feedback patterns
(columns of A) could then be provided by the LNs. The second assumption is that the environment
changes sufficiently slowly that the animal has time to learn it, i.e. that the LNs can update their
feedback patterns to match the change in required KC activations.
This yields the reduced dual circuit, in which the reciprocal interaction of the PNs with the KCs via
the matrix A is replaced with interaction with the M LNs via the square matrix B. The activity of
the LNs represents the activity of the KCs encoding the molecules in the current odor environment,
2
Although axo-axonal connections between neighbouring KC axons in the mushroom body peduncle are
known to exist [6], see also Section 2.
5
and the columns of B are the corresponding columns of the full A matrix:
?? ? y ? B ?(BT ? ? 1),
x = ?(AT ? ? 1)
(11)
Note that instantaneous readout of the PNs is still performed by the KCs as in the full dual. The
performance of the reduced dual is shown in red in Figure 2E, demonstrating better performance
than the feedforward circuit, though not the perfect recovery of the full dual. This is because the
solution sets of the two equations are not the same: Suppose that B = A:,1:M , and that y =
Pk
T
i=1 A:,i . The corresponding solution set for reduced dual is ?1 (y) = {? : (B:,1:k ) ? > 1 ?
T
T
T
(B:,k+1:M ) ? < 1}, equivalently ?1 (y) = {? : (A:,1:k ) ? > 1 ? (A:,k+1:M ) ? < 1}. On the
other hand, the solution set for the full dual is ?0 (y) = {? : (A:,1:k )T ? > 1 ? (A:,k+1:M )T ? <
1 ? (A:,M +1:N )T ? < 1}. Note the additional requirement that the projection of ? onto columns
M + 1 to N of A must also be less than 1. Hence any solution to the full dual is a solution to the
reduced dual , but not necessarily vise-versa: ?0 (y) ? ?1 (y). Since only the former are solutions to
the full problem, not all solutions to the reduced dual will solve it, leading to the reduced peformance
observed. This analysis also implies that increasing (or decreasing) the number of columns in B, so
that it is no longer square, will improve (worsen) the performance of the reduced dual, by making
its solution-set a smaller (larger) superset of ?0 (y).
7
Learning via ICA
Figure 2 demonstrates that the reduced dual has reasonable performance when the B matrix is
correct, i.e. it contains the columns of A for the KCs that would be active in the current odor
environment. How would this matrix be learned before birth, when presumably little is known about
the local environment, or as the animal moves from one odor environment to another?
Recall that, according to our generative model (Section 2) and the additional assumptions made for
deriving the reduced dual circuit (Section 6), molecules appear independently at random in odors
of a given odor environment and the mapping from odors x to glomerular responses y is linear
in x via the square mixing matrix B. Hence, our problem of learning B is precisely that of ICA
(or more precisely, sparse coding, as the observation noise variance is assumed to be ? 2 > 0 for
inference), with binary latent variables x. We solve this problem using MAP inference via EM
with a mean-field variational approximation q(x) to the posterior p(x|y, B) [13], where q(x) ,
QM
QM xi
1?xi
. The E-step, after observing that for binary x,
i=1 Bernoulli(xi ; qi ) =
i=1 qi (1 ? qi )
q
1
2
T
x = x, is ?q ? ?? ? log 1?q + ?2 B y ? ?12 Cq, with ? = ?1 + 2?1 2 c, ? = log((1 ? q0 )/q0 ),
q0 = k/M , the vector c = diag(BT B), and C = BT B ? diag(c), i.e. C is BT B with the
diagonal elements set to zero. To yield more plausible neural dynamics, we change variables to
? As vi is monotonically increasing
v = log(q/(1 ? q)). By the chain rule v? = diag(?vi /?qi )q.
in qi , and so the corresponding partial derivatives are all positive, and the resulting diagonal matrix
is positive definite, we can ignore it in performing gradient descent and still minimize the same
objective. Hence we have
?v ? ?? ? v +
1 T
1
B y ? 2 Cq(v),
?2
?
q(v) =
1
,
1 + exp(?v)
(12)
with the obvious mapping of v to LN membrane potentials, and q as the sigmoidal output function
representing graded voltage-dependent transmitter release observed in locust LNs.
The M-step update is made by changing B to increase log p(B) + Eq log p(x, y|B), yielding
1
1
B + 2 (rqT + B diag(q(1 ? q))),
M
?
Note that this update rule takes the form of a local learning rule.
?B ? ?
r , y ? Bq.
(13)
Empirically, we observed convergence within around 10,000 iterations using a fixed step size of
dt ? 10?2 , and ? ? 0.2 for M in the range of 20?100 and k in the range of 1?5. In cases when
the algorithm did not converge, lowering ? slightly typically solved the problem. The performance
of the algorithm is shown in figure 3. Although the B matrix is learned to high accuracy, it is
not learned exactly. The resulting algorithmic noise renders the performance of the dual shown in
Fig. 2E an upper bound, since there the exact B matrix was used.
6
?2
MSE
10
?4
10
Column of Btrue
?6
10
C
0
2000
4000
6000
Iteration
8000
10000
1
Coefficient of Blearned
B
0
10
Coefficient of Binitial
A
0
Column of Btrue
-1
Figure 3: ICA performance for M = 40, k = 1, dt = 10?2 . (A) Time course of mean squared
error between the elements of the estimate B and their true values for 10 different random seeds.
? = 0.162 for six of the seeds, 0.15 for three, and 0.14 for one. (B,C) Projection of the columns of
Btrue into the basis of the columns of B before (B) and after learning (C), for one of the random
seeds. Plotted values before learning are clipped to the -1?1 range.
8
8.1
Discussion
Biological evidence and predictions
Our work is consistent with much of the known anatomy of the locust olfactory system, e.g. the
lack of connectivity between PNs and dense connectivity between LNs, and between LNs and PNs
[3]; direct ORN inputs to LNs (observed in flies [14]; unknown in locust); dense connectivity from
PNs to KCs [4]; odor-evoked dynamics in the antennal lobe [2], vs. memoryless readout in the KCs
[5]. In addition, we require gradient descent PN dynamics (untested directly, but consistent with PN
dynamics reaching fixed-points upon prolonged odor presentation [15]), and short-term plasticity in
the antennal lobe for ICA (a direct search for ICA has not been performed, but short-term plasticity
is present in trial-to-trial dynamics [16]).
Our model also makes detailed predictions about circuit connectivity. First, it predicts a specific
structure for the PN-to-KC connectivity matrix, namely AT , the transpose of the affinity matrix.
This is superficially at odds with recent work in flies suggesting random connectivity between PNs
and KCs (detailed connectivity information is not present in the locust). Murthy and colleagues
[17] examined a small population of genetically identifiable KCs and found no evidence of response
stereotypy across flies, unlike that present at earlier stages in the system. Our model is agnostic
to permutations of the output vector as these reassign the mapping between KCs and molecules
and affect neither information content nor its format, so our results would be consistent with [17]
under animal-specific permutations. Caron and co-workers [18] analysed the structural connectivity of single KCs to glomeruli and found it consistent with random connectivity conditioned on a
glomerulus-specific connection probability. This is also consistent with our model, with the observed randomness reflecting that of the affinity matrix itself. Our model would predict (a) the
observation of repeated connectivity motifs if enough KCs (across animals) were observed, and that
(b) each connectivity motif corresponds to the (binarized) glomerular response vector evoked by a
particular molecule. In addition we predict symmetric inhibitory connectivity between LNs (BT B),
and negative reciprocal connectivity between PNs and LNs (Bij from PN i to LN j and ?Bij from
LN to PN).
8.2
Combining learning and readout
We have presented two mechanisms above ? the reduced dual for readout and and ICA for learning
? both of which need to be at play to guarantee high performance. In fact, these two mechanisms
must be active simultaneously in the animal. Here we sketch a possible mechanism for combining
them. The key is equation 12, which we repeat below, augmented with an additional term from the
PNs:
1 T
1
?v ? ?v + ?? + 2 B y ? 2 C q(v) + BT ? ? 1 = ?v + Ilearning + Ireadout .
?
?
7
A8
B
Feedforward
Full Dual
Reduced Dual
7
?0.5
6
Distance
5
4
0
3
2
1
0
1
2
3
4
5
k
6
7
8
9
0.5
?0.5
10
0
0.5
Figure 4: Effects of noise. (A) As in Figure 2E but with a small amount of additive noise in the
observations. The full dual still outperforms the feedforward circuit which in turn outperforms the
reduced dual over nearly half the tested range. (B) The feedback surface hinting at noise sensitivity.
PN phase space is colored according to activation of each of the KCs and a 2D projection around
the origin is shown. The average size of a zone with a uniform color is quite small, suggesting that
small perturbations would change the configuration of KCs activated by a PN, and hence the readout
performance.
Suppose (a) the two input channels were segregated e.g. on separate dendritic compartments, and
such that (b) the readout component was fast but weak, while (c) the learning component was slow
but strong, and (d) the v time constant was faster than both. Early after odor presentation, the main
input to the LN would be from the readout circuit, driving the PNs to their fixed point. The input
from the learning circuit would eventually catch up and dominate that of the readout circuit, driving
the LN dynamics for learning. Importantly, if B has already been learned, then the output of the
LNs, q(v), would remain essentially unchanged throughout, as both the learning and readout circuits
would produce the same (steady-state) activation vector in the LNs. If the matrix is incorrect, then
the readout is likely to be incorrect already, and so the important aspect is the learning update which
would eventually dominate. This is just one possibility for combining learning and readout. Indeed,
even the ICA updates themselves are non-trivial to implement. We leave the details of both to future
work.
8.3
Noise sensitivity
Although our derivations for serving inference and learning rules assumed observation noise, the
data that we provided to the models contained none. Adding a small amount of noise reduces
the performance of the dual circuits, particularly that of the reduced dual, as shown in Figure 4A.
Though this may partially be attributed to numerical integration issues (Supplementary Material),
there is likely a fundamental theoretical cause underlying it. This is hinted at by the plot in figure
4B of a 2D projection in PN space of the overlayed halfspaces defined by the activation of each of
? As ? crosses into
the N KCs. In the central void no KC is active and ? can change freely along ?.
?
a halfspace, the corresponding KC is activated, changing ? and the trajectory of ?. The different
colored zones indicate different patterns of KC activation and correspondingly different changes to
? The small size of these zones suggests that small changes in the trajectory of ? caused e.g. by
?.
noise could result in very different patterns of KC activation. For the reduced dual, most of these
halfspaces are absent for the dynamics since B has only a small subset of the columns of A, but
are present during readout, exacerbating the problem. How the biological system overcomes this
apparently fundamental sensitivity is an important question for future work.
Acknowledgements This work was supported by the Wellcome Trust (ST, ML).
8
References
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9
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4,862 | 5,401 | Online Optimization for Max-Norm Regularization
Jie Shen
Dept. of Computer Science
Rutgers University
Piscataway, NJ 08854
Huan Xu
Dept. of Mech. Engineering
National Univ. of Singapore
Singapore 117575
Ping Li
Dept. of Statistics
Dept. of Computer Science
Rutgers University
[email protected]
[email protected]
[email protected]
Abstract
Max-norm regularizer has been extensively studied in the last decade as it promotes an effective low rank estimation of the underlying data. However, maxnorm regularized problems are typically formulated and solved in a batch manner,
which prevents it from processing big data due to possible memory bottleneck.
In this paper, we propose an online algorithm for solving max-norm regularized
problems that is scalable to large problems. Particularly, we consider the matrix
decomposition problem as an example, although our analysis can also be applied
in other problems such as matrix completion. The key technique in our algorithm
is to reformulate the max-norm into a matrix factorization form, consisting of a
basis component and a coefficients one. In this way, we can solve the optimal
basis and coefficients alternatively. We prove that the basis produced by our algorithm converges to a stationary point asymptotically. Experiments demonstrate
encouraging results for the effectiveness and robustness of our algorithm.
See the full paper at arXiv:1406.3190.
1 Introduction
In the last decade, estimating low rank matrices has attracted increasing attention in the machine
learning community owing to its successful applications in a wide range of domains including subspace clustering [13], collaborative filtering [9] and visual texture analysis [25], to name a few.
Suppose that we are given an observed data matrix Z of size p ? n, i.e., n observations in p ambient
dimensions, with each observation being i.i.d. sampled from some unknown distribution, we aim
to learn a prediction matrix X with a low rank structure to approximate Z. This problem, together
with its many variants, typically involves minimizing a weighted combination of the residual error
and matrix rank regularization term.
Generally speaking, it is intractable to optimize a matrix rank [15]. To tackle this challenge, researchers suggest alternative convex relaxations to the matrix rank. The two most widely used convex surrogates are the nuclear norm 1 [15] and the max-norm 2 [19]. In the work of [6], Cand`es et al.
proved that under mild conditions, solving a convex optimization problem consisting of a nuclear
norm regularization and a weighted ?1 norm penalty can exactly recover the low-rank component of
the underlying data even if a constant fraction of the entries are arbitrarily corrupted. In [20], Srebro
and Shraibman studied collaborative filtering and proved that the max-norm regularization formulation enjoyed a lower generalization error than the nuclear norm. Moreover, the max-norm was
shown to empirically outperform the nuclear norm in certain practical applications as well [11, 12].
To optimize a max-norm regularized problem, however, algorithms proposed in prior work [12, 16,
19] require to access all the data. In a large scale setting, the applicability of such batch optimiza1
2
Also known as the trace norm, the Ky-Fan n-norm and the Schatten 1-norm.
Also known as the ?2 -norm.
1
tion methods will be hindered by the memory bottleneck. In this paper, by utilizing the matrix
factorization form of the max-norm, we propose an online algorithm to solve max-norm regularized
problems. The main advantage of online algorithms is that the memory cost is independent from the
sample size, which makes online algorithms a good fit for the big data era [14, 18].
Specifically, we are interested in the max-norm regularized matrix decomposition (MRMD) problem. Assume that the observed data matrix Z can be decomposed into a low rank component X and
a sparse one E, we aim to simultaneously and accurately estimate the two components, by solving
the following convex program:
min
X,E
?1
1
2
?Z ? X ? E?F + ?X?2max + ?2 ?E?1,1 .
2
2
(1.1)
Here ???F denotes the Frobenius norm, ???max is the max-norm (which promotes low rank), ???1,1
is the ?1 norm of a matrix seen as a vector, and ?1 and ?2 are two non-negative parameters.
Our main contributions are two-folds: 1) We develop an online method to solve this MRMD
problems, making it scalable to big data. 2) We prove that the solutions produced by our algorithm
converge to a stationary point asymptotically.
1.1 Connection to Matrix Completion
While we mainly focus on the matrix decomposition problem, our method can be extended to the
matrix completion (MC) problem [4, 7] with max-norm regularization [5], which is another popular
topic in machine learning and signal processing. The MC problem can be described as follows:
min
X
1
?
2
2
?P? (Z ? X)?F + ?X?max ,
2
2
where ? is the set of indices of observed entries in Z and P? (M ) is the orthogonal projector onto
the span of matrices vanishing outside of ? so that the (i, j)-th entry of P? (M ) is equal to Mij if
(i, j) ? ? and zero otherwise. Interestingly, the max-norm regularized MC problem can be cast into
our framework. To see this, let us introduce an auxiliary matrix M , with Mij = C > 0 if (i, j) ? ?
and Mij = C1 otherwise. The following reformulated MC problem,
min
X,E
1
?
2
2
?Z ? X ? E?F + ?X?max + ?M ? E?1,1 ,
2
2
where ??? denotes the entry-wise product, is equivalent to our MRMD formulation (1.1). Furthermore, when C tends to infinity, the reformulated problem converges to the original MC problem.
1.2 Related Work
Here we discuss some relevant work in the literature. Most previous works on max-norm focused
on showing that the max-norm was empirically superior to the nuclear norm in a wide range of applications, such as collaborative filtering [19] and clustering [11]. Moreover, in [17], Salakhutdinov
and Srebro studied the influence of data distribution for the max-norm regularization and observed
good performance even when the data were sampled non-uniformly.
There are also studies which investigated the connection between the max-norm and the nuclear
norm. A comprehensive study on this problem, in the context of collaborative filtering, can be found
in [20], which established and compared the generalization bounds for the nuclear norm regularization and max-norm regularization, and showed that the generalization bound of the max-norm
regularization scheme is superior. More recently, Foygel et al. [9] attempted to unify the nuclear
norm and max-norm for gaining further insights on these two important regularization schemes.
There are few works to develop efficient algorithms for solving max-norm regularized problems, particularly large scale ones. Rennie and Srebro [16] devised a gradient-based optimization method and
empirically showed promising results on large collaborative filtering datasets. In [12], the authors
presented large scale optimization methods for max-norm constrained and max-norm regularized
problems with a theoretical guarantee to a stationary point. Nevertheless, all those methods were
formulated in a batch manner, which can be hindered by the memory bottleneck.
2
From a high level, the goal of this paper is similar to that of [8]. Motivated by the celebrated Robust
Principal Component Analysis (RPCA) problem [6, 23, 24], the authors of [8] developed an online
implementation for the nuclear-norm regularized matrix decomposition. Yet, since the max-norm
is a much more complicated mathematical entity (e.g., even the subgradient of the max-norm is not
completely characterized to the best of our knowledge), new techniques and insights are needed in
order to develop online methods for the max-norm regularization. For example, after taking the
max-norm with its matrix factorization form, the data are still coupled and we propose to convert
the problem to a constrained one for stochastic optimization.
The main technical contribution of this paper is to convert max-norm regularization to an appropriate
matrix factorization problem amenable to online implementation. Part of our proof ideas are inspired
by [14], which also studied online matrix factorization. In contrast to [14], our formulation contains
an additive sparse noise matrix, which enjoys the benefit of robustness to sparse contamination. Our
proof techniques are also different. For example, to prove the convergence of the dictionary and
to well define their problem, [14] needs to assume that the magnitude of the learned dictionary is
constrained. In contrast, in our setup we prove that the optimal basis is uniformly bounded, and
hence our problem is naturally well defined.
2
Problem Setup
We first introduce our notations. We use bold letters to denote vectors. The i-th row and j-th column
of a matrix M are denoted by m(i) and mj , respectively. The ?1 norm and ?2 norm of a vector v
are denoted by ?v?1 and ?v?2 , respectively. The ?2,? norm of a matrix is defined as the maximum
?2 row norm. Finally, the trace of a square matrix M is denoted as Tr(M ).
We are interested in developing an online algorithm for the MRMD Problem (1.1). By taking the
matrix factorization form of the max-norm [19]:
?X?max , min{?L?2,? ? ?R?2,? : X = LR? , L ? Rp?d , R ? Rn?d },
L,R
(2.1)
where d is the intrinsic dimension of the underlying data, we can rewrite Problem (1.1) into the
following equivalent form:
min
L,R,E
1
?1
?Z ? LRT ? E?2F + ?L?22,? ?R?22,? + ?2 ?E?1,1 .
2
2
(2.2)
Intuitively, the variable L corresponds to a basis and the variable R is a coefficients matrix with
each row corresponding to the coefficients. At a first sight, the problem can only be optimized
in a batch manner as the term ?R?22,? couples all the samples. In other words, to compute the
optimal coefficients of the i-th sample, we are required to compute the subgradient of ?R?2,? ,
which needs to access all the data. Fortunately, we have the following proposition that alleviates the
inter-dependency among samples.
Proposition 2.1. Problem (2.2) is equivalent to the following constrained program:
minimize
L,R,E
subject to
1
?1
?Z ? LRT ? E?2F + ?L?22,? + ?2 ?E?1,1 ,
2
2
?R?22,? = 1.
(2.3)
Proposition 2.1 states that our primal MRMD problem can be transformed to an equivalent constrained one. In the new formulation (2.3), the coefficients of each individual sample (i.e., a row of
the matrix R) is uniformly constrained. Thus, the samples are decoupled. Consequently, we can,
equipped with Proposition 2.1, rewrite the original problem in an online fashion, with each sample
being separately processed:
minimize
L,R,E
n
n
?
1?
?1
?ei ?1 ,
?zi ? Lri ? ei ?22 + ?L?22,? + ?2
2 i=1
2
i=1
subject to ?i ? 1, 2, . . . , n, ?ri ?22 ? 1,
3
where zi is the i-th observed sample, ri is the coefficients and ei is the sparse error. Combining the
first and third terms in the above equation, we have
n
?
? i , L, ri , ei ) + ?1 ?L?2 ,
minimize
?(z
2,?
L,R,E
2
(2.4)
i=1
subject to
?i ? 1, 2, . . . , n, ?ri ?22 ? 1,
where
? L, r, e) , 1 ?z ? Lr ? e?2 + ?2 ?e?1 .
?(z,
2
2
This is indeed equivalent to optimizing (i.e., minimizing) the empirical loss function:
n
1?
?1
fn (L) ,
?L?22,? ,
?(zi , L) +
n i=1
2n
where
?(z, L) =
min
r,e,?r?22 ?1
? L, r, e).
?(z,
(2.5)
(2.6)
(2.7)
When n goes to infinity, the empirical loss converges to the expected loss, defined as follows
f (L) = lim fn (L) = Ez [?(z, L)].
n?+?
3
(2.8)
Algorithm
We now present our online implementation to solve the MRMD problem. The detailed algorithm
is listed in Algorithm 1. Here we first briefly explain the underlying intuition: We optimize the
coefficients r, the sparse noise e and the basis L in an alternating manner, which is known to be a
successful strategy [8, 10, 14]. At the t-th iteration, given the basis Lt?1 , we can optimize over r
and e by examining the Karush Kuhn Tucker (KKT) conditions. To update the basis Lt , we then
optimize the following objective function:
1??
?1
?(zi , L, ri , ei ) + ?L?22,? ,
t i=1
2t
t
gt (L) ,
(3.1)
where {ri }ti=1 and {ei }ti=1 have been computed in previous iterations. It is easy to verify that
Eq. (3.1) is a surrogate function of the empirical cost function ft (L) defined in Eq. (2.6). The basis
Lt can be optimized by block coordinate decent, with Lt?1 being a warm start for efficiency.
4
Main Theoretical Results and Proof Outline
In this section we present our main theoretic result regarding the validity of the proposed algorithm.
We first discuss some necessary assumptions.
4.1
Assumptions
1. The observed data are i.i.d. generated from a distribution with compact support Z.
2. The surrogate functions gt (L) in Eq. (3.1) are strongly convex. Particularly, we assume that
the smallest eigenvalue of the positive semi-definite matrix 1t At defined in Algorithm 1 is
not smaller than some positive constant ?1 . Note that we can easily enforce this assumption
by adding a term ?21 ?L?2F to gt (L).
? L, r, e) is strongly convex
3. The minimizer for Problem (2.7) is unique. Notice that ?(z,
w.r.t. e and convex w.r.t. r. Hence, we can easily enforce this assumption by adding a term
??r?22 , where ? is a small positive constant.
4.2
Main Theorem
The following theorem is the main theoretical result of this work. It states that when t tends to
infinity, the basis Lt produced by Algorithm 1 converges to a stationary point.
Theorem 4.1 (Convergence to a stationary point of Lt ). Assume 1, 2 and 3. Given that the intrinsic
dimension of the underlying data is d, the optimal basis Lt produced by Algorithm 1 asymptotically
converges to a stationary point of Problem (2.8) when t tends to infinity.
4
Algorithm 1 Online Max-Norm Regularized Matrix Decomposition
Input: Z ? Rp?n (observed samples), parameters ?1 and ?2 , L0 ? Rp?d (initial basis), zero
matrices A0 ? Rd?d and B0 ? Rp?d
Output: optimal basis Lt
1: for t = 1 to n do
2:
Access the t-th sample zt .
3:
Compute the coefficient and noise:
? t , Lt?1 , r, e).
{rt , et } = arg min ?(z
(3.2)
r,e,?r?22 ?1
4:
5:
Compute the accumulation matrices At and Bt :
At
?
At?1 + rt r?
t ,
Bt
?
Bt?1 + (zt ? et ) r?
t .
Compute the basis Lt by optimizing the surrogate function (3.1):
1??
?1
?(zi , L, ri , ei ) + ?L?22,?
t i=1
2t
L
(
)
)
( ? )
?1
1 1 ( ?
= arg min
Tr L LAt ? Tr L Bt + ?L?22,? .
t
2
2t
L
t
Lt = arg min
(3.3)
6: end for
4.3
Proof Outline for Theorem 4.1
The essential tools for our analysis are from stochastic approximation [3] and asymptotic statistics [21]. There are three main steps in our proof:
(I) We show that the positive stochastic process gt (Lt ) defined in Eq. (3.1) converges almost surely.
(II) Then we prove that the empirical loss function, ft (Lt ) defined in Eq. (2.6) converges almost
surely to the same limit of its surrogate gt (Lt ). According to the central limit theorem, we can
expect that ft (Lt ) also converges almost surely to the expected loss f (Lt ) defined in Eq. (2.8),
implying that gt (Lt ) and f (Lt ) converge to the same limit.
(III) Finally, by taking a simple Taylor expansion, it justifies that the gradient of f (L) taking at Lt
vanishes as t tends to infinity, which concludes Theorem 4.1.
Theorem 4.2 (Convergence of the surrogate function gt (Lt )). The surrogate function gt (Lt ) we
defined in Eq. (3.1) converges almost surely, where Lt is the solution produced by Algorithm 1.
To establish the convergence of gt (Lt ), we verify that gt (Lt ) is a quasi-martingale [3] that converges
almost surely. To this end, we show that the expectation of the difference of gt+1 (Lt+1 ) and gt (Lt )
can be upper bounded by a family of functions ?(?, L) indexed by L ? L, where L is a compact
set. Then we show that the family of functions satisfy the hypotheses in the corollary of Donsker
Theorem [21] and thus can be uniformly upper bounded. Therefore, we conclude that gt (Lt ) is a
quasi-martingale and converges almost surely.
Now let us verify the hypotheses in the corollary of Donsker Theorem. First we prove that the index
set L is uniformly bounded.
Proposition 4.3. Let rt , et and Lt be the optimal solutions produced by Algorithm 1. Then,
1. The optimal solutions rt and et are uniformly bounded.
2. The matrices 1t At and 1t Bt are uniformly bounded.
5
3. There exists a compact set L, such that for all Lt produced by Algorithm 1, Lt ? L. That
is, there exists a positive constant Lmax that is uniform over t, such that for all t > 0,
?Lt ? ? Lmax .
To prove the third claim (which is required for our proof of convergence of gt (Lt )), we should prove
that for all t > 0, rt , et , 1t At and 1t Bt can be uniformly bounded, which can easily be verified.
Then, by utilizing the first order optimal condition of Problem (3.3), we can build an equation that
connects Lt with the four items we mentioned in the first and second claim. From Assumption 2, we
know that the nuclear norm of 1t At can be uniformly lower bounded. This property provides us the
way to show that Lt can be uniformly upper bounded. Note that in [8, 14], both papers assumed that
the dictionary (or basis) is uniformly bounded. In contrast, here in the third claim of Proposition 4.3,
we prove that such condition naturally holds in our problem.
Next, we show that the family of functions ?(z, L) is uniformly Lipschitz w.r.t. L.
Proposition 4.4. Let L ? L and denote the minimizer of ?(z, L, r, e) defined in (2.7) as:
1
{r? , e? } = arg min ?z ? Lr ? e?22 + ?2 ?e?1 .
r,e,?r?2 ?1 2
Then, the function ?(z, L) defined in Problem (2.7) is continuously differentiable and
?L ?(z, L) = (Lr? + e? ? z)r?T .
Furthermore, ?(z, ?) is uniformly Lipschitz and bounded.
By utilizing the corollary of Theorem 4.1 from [2], we can verify the differentiability of ?(z, L) and
the form of its gradient. As all of the items in the gradient are uniformly bounded (Assumption 1
and Proposition 4.3), we show that ?(z, L) is uniformly Lipschitz and bounded.
Based on Proposition 4.3 and 4.4, we verify that all the hypotheses in the corollary of Donsker
Theorem [21] are satisfied. This implies the convergence of gt (Lt ). We now move to step (II).
Theorem 4.5 (Convergence of f (Lt )). Let f (Lt ) be the expected loss function defined in Eq. (2.8)
and Lt is the solution produced by the Algorithm 1. Then,
1. gt (Lt ) ? ft (Lt ) converges almost surely to 0.
2. ft (Lt ) defined in Eq. (2.6) converges almost surely.
3. f (Lt ) converges almost surely to the same limit of ft (Lt ).
We apply Lemma 8 from [14] to prove the first claim. We denote the difference of gt (Lt ) and ft (Lt )
by bt . First we show that bt is uniformly Lipschitz. Then we show that the difference between Lt+1
and Lt is O( 1t ), making bt+1 ? bt be uniformly upper bounded by O( 1t ). Finally, we verify the
convergence of the summation of the serial { 1t bt }?
t=1 . Thus, Lemma 8 from [14] applies.
Proposition 4.6. Let {Lt } be the basis sequence produced by the Algorithm 1. Then,
1
?Lt+1 ? Lt ?F = O( ).
t
(4.1)
Proposition 4.6 can be proved by combining the strong convexity of gt (L) (Assumption 2 in Section 4.1) and the Lipschitz of gt (L); see the full paper for details.
Equipped with Proposition 4.6, we can verify that the difference of the sequence bt = gt (Lt ) ?
ft (Lt ) can be upper bounded by O( 1t ). The convergence of the summation of the serial { 1t bt }?
t=1
can be examined by the expectation convergence property of quasi-martingale gt (Lt ), stated in [3].
Applying the Lemma 8 from [14], we conclude that gt (Lt ) ? ft (Lt ) converges to zero a.s..
After the first claim of Theorem 4.5 being proved, the second claim follows immediately, as gt (Lt )
converges a.s. (Theorem 4.2). By the central limit theorem, the third claim can be verified.
According to Theorem 4.5, we can see that gt (Lt ) and f (Lt ) converge to the same limit a.s. Let
t tends to infinity, as Lt is uniformly bounded (Proposition 4.3), the term ?2t1 ?Lt ?22,? in gt (Lt )
vanishes. Thus gt (Lt ) becomes differentiable. On the other hand, we have the following proposition
about the gradient of f (L).
6
Proposition 4.7 (Gradient of f (L)). Let f (L) be the expected loss function defined in Eq. (2.8).
Then, f (L) is continuously differentiable and ?f (L) = Ez [?L ?(z, L)]. Moreover, ?f (L) is uniformly Lipschitz on L.
Thus, taking a first order Taylor expansion for f (Lt ) and gt (Lt ), we can show that the gradient of
f (Lt ) equals to that of gt (Lt ) when t tends to infinity. Since Lt is the minimizer for gt (L), we know
that the gradient of f (Lt ) vanishes. Therefore, we have proved Theorem 4.1.
5
Experiments
In this section, we report some simulation results on synthetic data to demonstrate the effectiveness
and robustness of our online max-norm regularized matrix decomposition (OMRMD) algorithm.
Data Generation. The simulation data are generated by following a similar procedure in [6]. The
clean data matrix X is produced by X = U V T , where U ? Rp?d and V ? Rn?d . The entries
of U and V are i.i.d. sampled from the Gaussian distribution N (0, 1). We introduce a parameter ?
to control the sparsity of the corruption matrix E, i.e., a ?-fraction of the entries are non-zero and
following an i.i.d. uniform distribution over [?1000, 1000]. Finally, the observation matrix Z is
produced by Z = X + E.
Evaluation Metric. Our goal is to estimate the correct subspace for the underlying data. Here, we
evaluate the fitness of our estimated subspace basis L and the ground truth basis U by the Expressed
Variance (EV) [22]:
Tr(LT U U T L)
EV(U, L) ,
.
Tr(U U T )
The values of EV range in [0, 1] and a higher EV value indicates a more accurate subspace recovery.
Other Settings. Through the experiments, we set the ambient dimension p = 400 and the total
number
? of samples n = 5000 unless otherwise specified. We fix the tunable parameter ?1 = ?2 =
1/ p, and use default parameters for all baseline algorithms we compare with. Each experiment is
repeated 10 times and we report the averaged EV as the result.
0.5
fraction of corruption
fraction of corruption
0.5
0.38
0.26
0.14
0.02
0.02
0.38
0.26
0.14
0.02
0.02
0.14
0.26
0.38
0.5
rank / ambient dimension
(a) OMRMD
0.14
0.26
0.38
0.5
rank / ambient dimension
(b) OR-PCA
Figure 1: Performance of subspace recovery under different rank and corruption fraction. Brighter
color means better performance.
We first study the effectiveness of the algorithm, measured by the EV value of its output after the last
sample, and compare it to the nuclear norm based online RPCA (OR-PCA) algorithm [8]. Specifically, we vary the intrinsic dimension d from 0.02p to 0.5p, with a step size 0.04p, and the corruption
fraction ? from 0.02 to 0.5, with a step size 0.04. The results are reported in Figure 1 where brighter
color means higher EV (hence better performance). We observe that for easier tasks (i.e., when
corruption and rank are low), both algorithms perform comparably. On the other hand, for more
difficult cases, OMRMD outperforms OR-PCA. This is possibly because the max-norm is a tighter
approximation to the matrix rank.
We next study the convergence of OMRMD by plotting the EV curve against the number of samples. Besides OR-PCA, we also add Principal Component Pursuit (PCP) [6] and an online PCA
7
1
0.8
0.8
0.6
OMRMD
OR?PCA
PCP
Online PCA
0.4
EV
EV
1
0.2
0
1
1000 2000 3000 4000 5000
Number of Samples
(a) ? = 0.01
1
0.8
0.8
0.6
0.6
EV
EV
1000 2000 3000 4000 5000
Number of Samples
(b) ? = 0.3
1
0.4
0.4
OMRMD
OR?PCA
PCP
Online PCA
0.2
0
1
OMRMD
OR?PCA
PCP
Online PCA
0.4
0.2
0
1
0.6
0.2
0
1000 2000 3000 4000 5000
Number of Samples
(c) ? = 0.5
OMRMD
OR?PCA
PCP
2
4
6
8
10
Number of Samples x 104
(d) p = 3000, d = 300, ? = 0.3
Figure 2: EV value against number of samples. p = 400 and d = 80 in (a) to (c).
algorithm [1] as baseline algorithms to compare with. The results are reported in Figure 2. As expected, PCP achieves the best performance since it is a batch method and needs to access all the data
throughout the algorithm. Online PCA degrades significantly even with low corruption (Figure 2a).
OMRMD is comparable to OR-PCA when the corruption is low (Figure 2a), and converges significantly faster when the corruption is high (Figure 2b and 2c). Indeed, this is true even with high
dimension and as many as 100, 000 samples (Figure 2d). This observation agrees with Figure 1,
and again suggests that for large corruption, max-norm may be a better fit than the nuclear norm.
Additional experimental results are available in the full paper.
6
Conclusion
In this paper, we developed an online algorithm for max-norm regularized matrix decomposition
problem. Using the matrix factorization form of the max-norm, we convert the original problem
to a constrained one which facilitates an online implementation for solving the original problem.
We established theoretical guarantees that the solutions will converge to a stationary point asymptotically. Moreover, we empirically compared our proposed algorithm with OR-PCA, which is a
recently proposed online algorithm for nuclear-norm based matrix decomposition. The simulation
results suggest that the proposed algorithm outperforms OR-PCA, in particular for harder task (i.e.,
when a large fraction of entries are corrupted). Our experiments, to an extent, empirically suggest
that the max-norm might be a tighter relaxation of the rank function compared to the nuclear norm.
Acknowledgments
The research of Jie Shen and Ping Li is partially supported by NSF-DMS-1444124, NSF-III1360971, NSF-Bigdata-1419210, ONR-N00014-13-1-0764, and AFOSR-FA9550-13-1-0137. Part
of the work of Jie Shen was conducted at Shanghai Jiao Tong University. The work of Huan Xu is
partially supported by the Ministry of Education of Singapore through AcRF Tier Two grant R-265000-443-112.
8
References
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9
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4,863 | 5,402 | Finding a sparse vector in a subspace:
Linear sparsity using alternating directions
Qing Qu, Ju Sun, and John Wright
{qq2105, js4038, jw2966}@columbia.edu
Dept. of Electrical Engineering, Columbia University, New York City, NY, USA, 10027
Abstract
We consider the problem of recovering the sparsest vector in a subspace S ? Rp
with dim (S) = n. This problem can be considered a homogeneous variant of
the sparse recovery problem, and finds applications in sparse dictionary learning,
sparse PCA, and other problems in signal processing and machine learning. Simple
convex heuristics for this problem provably break down when ?
the fraction of
nonzero entries in the target sparse vector substantially exceeds 1/ n. In contrast,
we exhibit a relatively simple nonconvex approach based on alternating directions,
which provably succeeds even when the fraction of nonzero entries is ?(1). To
our knowledge, this is the first practical algorithm to achieve this linear scaling.
This result assumes a planted sparse model, in which the target sparse vector is
embedded in an otherwise random subspace. Empirically, our proposed algorithm
also succeeds in more challenging data models arising, e.g., from sparse dictionary
learning.
1
Introduction
Suppose we are given a linear subspace S of a high-dimensional space Rp , which contains a sparse
vector x0 6= 0. Given arbitrary basis of S, can we efficiently recover x0 ? Equivalently, provided a
matrix A ? R(p?n)?p , can we efficiently find a nonzero sparse vector x such that Ax = 0? In the
language of sparse approximation, can we solve
min kxk0 s.t. Ax = 0, x 6= 0
?
(1)
x
Variants of this problem have been studied in the context of applications to numerical linear algebra
[15], graphical model learning [27], nonrigid structure from motion [16], spectral estimation and
Prony?s problem [11], sparse PCA [29], blind source separation [28], dictionary learning [24],
graphical model learning [3], and sparse coding on manifolds [21].
However, in contrast to the standard sparse regression problem (Ax = b, b 6= 0), for which convex
relaxations perform nearly optimally for broad classes of designs A [14, 18], the computational
properties of problem (1) are not nearly as well understood. It has been known for several decades
that the basic formulation
min kxk0 , s.t. x ? S \ {0},
(2)
x
is NP-hard [15]. However, it is only recently that efficient computational surrogates with nontrivial
recovery guarantees have been discovered. In the context of sparse dictionary learning, Spielman et
al. [24] introduced a relaxation which replaces the nonconvex problem (2) with a sequence of linear
programs:
min kxk1 , s.t. xi = 1, x ? S, 1 ? i ? p,
(3)
x
and proved that when S is generated as a span of n random sparse vectors, with high probability
the relaxation
? recovers these vectors, provided the probability of an entry being nonzero is at most
? ? O (1/ n).
1
In a planted sparse model, in which S consists of a single sparse vector x0 embedded in a ?generic?
subspace, Hand et al. proved
that (3) also correctly recovers x0 , provided the fraction of nonzeros in
?
x0 scales as ? ? O (1/ n) [19].
?
Unfortunately, the results of [24, 19] are essentially sharp: when ? substantially exceeds 1/ n,
in both models the relaxation (3) provably breaks down. Moreover, the most natural semidefinite
programming relaxation of (1),
>
min kXk1 , s.t.
A A, X = 0, trace[X] = 1, X 0.
(4)
X
?
also breaks down at exactly the same threshold of ? ? 1/ n.1
?
One might naturally conjecture that this 1/ n threshold is simply an intrinsic price we must pay for
having an efficient algorithm, even in these random models. Some evidence towards this conjecture
might be borrowed from the surface similarity of (2)-(4) and sparse PCA [29]. In sparse PCA, there is
a substantial gap between what can be achieved with efficient algorithms and the information theoretic
?
optimum [10]. Is this also the case for recovering a sparse vector in a subspace? Is ? ? O (1/ n)
simply the best we can do with efficient, guaranteed algorithms?
Remarkably, this is not the case. Recently, Barak et al. introduced a new rounding technique for
sum-of-squares relaxations, and showed that the sparse vector x0 in the planted sparse model can be
recovered when p ? ? n2 and ? ? ?(1) [8]. It is perhaps surprising that this is possible at all with
a polynomial time algorithm. Unfortunately, the runtime of this approach is a high-degree polynomial
in p, and so for machine learning problems in which p is either a feature dimension or sample size,
this algorithm is of theoretical interest only. However, it raises an interesting algorithmic
question: Is
?
there a practical algorithm that provably recovers a sparse vector with ? 1/ n nonzeros from a
generic subspace S?
In this paper, we address this problem, under the following hypotheses: we assume the planted
sparse model, in which a target sparse vector x0 is embedded in an otherwise random n-dimensional
subspace of Rp . We allow x0 to have up to ?0 p nonzero entries, where ?0 is a constant. We provide
a relatively simple
algorithm which, with very high probability, exactly recovers x0 , provided that
p ? ? n4 log2 n .
Our algorithm is based on alternating directions, with two special twists. First, we introduce a
special data driven initialization, which seems to be important for achieving ? = ?(1). Second, our
theoretical results require a second, linear programming based rounding phase, which is similar to
[24]. Our core algorithm has very simple iterations, of linear complexity in the size of the data, and
hence should be scalable to moderate-to-large scale problems.
In addition to enjoying theoretical guarantees in a regime (? = ?(1)) that is out of the reach
of previous practical algorithms, it performs well in simulations ? succeeding empirically with
p ? ? (n log n). It also performs well empirically on more challenging data models, such as the
dictionary learning model, in?which the subspace of interest contains not one, but n target sparse
vectors. Breaking the O(1/ n) sparsity barrier with a practical algorithm is an important open
problem in the nascent literature on algorithmic guarantees for dictionary learning [5, 4, 2, 1]. We are
optimistic that the techniques introduced here will be applicable in this direction.
2
Problem Formulation and Global Optimality
We study the problem of recovering a sparse vector x0 6= 0 (up to scale), which is an element of a
known subspace S ? Rp of dimension n, provided an arbitrary orthonormal basis Y ? Rp?n for S.
Our starting point is the nonconvex formulation (2). Both the objective and constraint are nonconvex,
and hence not easy to optimize over. We relax (2) by replacing the `0 norm with the `1 norm. For the
constraint x 6= 0, which is necessary to avoid a trivial solution, we force x to live on the unit sphere
kxk2 = 1, giving
min kxk1 , s.t. x ? S, kxk2 = 1.
(5)
x
1
This breakdown behavior is again in sharp contrast to the standard sparse approximation problem (with
b 6= 0), in which it is possible to handle very large fractions of nonzeros (say, ? = ?(1/ log n), or even
? = ?(1)) using a very simple `1 relaxation [14, 18]
2
This formulation is still nonconvex, and so we should not expect to obtain an efficient algorithm
that can solve it globally for general inputs S. Nevertheless, the geometry of the sphere is benign
enough that for well-structured inputs it actually will be possible to give algorithms that find the
global optimum of this problem.
The formulation (5) can be contrasted with (3), in which we optimize the `1 norm subject to the
constraint kxk? = 1. Because k?k? is polyhedral, that formulation immediately yields a sequence
of linear programs. This is very convenient for computation
and analysis, but suffers from the
?
aforementioned breakdown behavior around kx0 k0 ? p/ n.
In contrast, the sphere kxk2 = 1 is a more complicated geometric constraint, but will allow much
larger numbers of nonzeros in x0 . For example, if we consider the global optimizer of a variant of
(5):
min kYqk1 ,
q?Rn
s.t.
kqk2 = 1,
(6)
under the planted sparse model (detailed below), e1 is the unique to (6) with very high probability:
?
Theorem 2.1 (`1 /`2 recovery, planted sparse model). There exists a constant ?0 ? (1/ n, 1/2)
such that if the subspace S follows the planted sparse model
S = span (x0 , g1 , . . . , gn?1 ) ? Rp ,
(7)
x0 ?i.i.d. ?1?p Ber(?),
with gi ?i.i.d. N (0, 1/p), and
with x0 , g1 , . . . , gn?1 mutually independent and
?
1/ n < ? < ?0 , then ?e0 are the only global minimizers to (6) if Y = [x0 , g1 , . . . , gn?1 ], provided
p ? ? (n log n).
Hence, if we could find the global optimizer of (6), we would be able to recover x0 whose number of
nonzero entries is quite large ? even linear in the dimension p (? = ?(1)). On the other hand, it is
not obvious that this should be possible: (6) is nonconvex. In the next section, we will describe a
simple heuristic algorithm for (a near approximation of) the `1 /`2 problem (6), which guarantees
to find a stationary point. More surprisingly, we will then prove that for a class of random problem
instances, this algorithm, plus an auxiliary rounding technique, actually recovers the global optimum
? the target sparse vector x0 . The proof requires a detailed probabilistic analysis, which is sketched in
Section 4.2.
Before continuing, it is worth noting that the formulation (5) is in no way novel ? see, e.g., the work
of [28] in blind source separation for precedent. However, the novelty originates from our algorithms
and subsequent analysis.
3
Algorithm based on Alternating Direction Method (ADM)
To develop an algorithm for solving (6), we work with the orthonormal basis Y ? Rp?n for S. For
numerical purposes, and also for coping with noise in practical application, it is useful to consider a
slight relaxation of (6), in which we introduce an auxiliary variable x ? Yq:
1
2
(8)
min kYq ? xk2 + ? kxk1 , s.t. kqk2 = 1,
q,x 2
Here, ? > 0 is a penalty parameter. It is not difficult to see that this problem is equivalent to
minimizing the Huber m-estimator over Yq. This relaxation makes it possible to apply alternating
direction method to this problem, which, starting from some initial point q(0) , alternates between
optimizing with respect to x and optimizing with respect to q:
2
1
x(k+1) = arg min
Yq(k) ? x
+ ? kxk1 ,
(9)
2
2
x
2
1
(10)
q(k+1) = arg min
Yq ? x(k+1)
s.t. kqk2 = 1.
2
2
q
Both (9) and (10) have simple closed form solutions:
x(k+1) = S? [Yq(k) ],
Y> x(k+1)
q(k+1) =
Y> x(k+1)
,
2
3
(11)
Algorithm 1 Nonconvex ADM
Input: A matrix Y ? Rp?n with Y> Y = I, initialization q(0) , threshold ? > 0.
? 0 = Yq(k)
Output: The recovered sparse vector x
1: Set k = 0,
2: while not converged do
3:
x(k+1) = S? [Yq(k) ],
> (k+1)
4:
q(k+1) = YY> xx(k+1) ,
k
k2
5:
Set k = k + 1.
6: end while
where S? [x] = sign(x) max {|x| ? ?, 0} is the soft-thresholding operator. The proposed ADM
algorithm is summarized in Algorithm 1.
For general input Y and initialization q(0) , Algorithm 1 is guaranteed to produce a stationary point
of problem (8). This is a consequence of recent general analyses of alternating direction methods
for nonsmooth and nonconvex problems ? see [6, 7]. However, if our goal is to recover the sparsest
vector x0 , some additional tricks are needed.
Initialization. Because the problem (6) is nonconvex, an arbitrary or random initialization is
unlikely to produce a global minimizer.2 Therefore, good initializations are critical for the proposed
ADM algorithm to succeed. For this purpose, we suggest to use every normalized row of Y as
initializations for q, and solve a sequence of p nonconvex programs (6) by the ADM algorithm.
To get an intuition of why our initialization works, recall the planted sparse model: S =
p?n
?? | g
. Suppose we take a row zi of Z,
span(x0 , g1 , . . . , gn?1 ). Write Z = [x0 | g1 | ??
n?1 ] ? R
in which x0 (i) is nonzero, then x0 (i) = ? 1/ ?p . Meanwhile, the entries of g1 (i), . . . gn?1 (i)
?
are all N (0, 1/p), and so have size about 1/ p. Hence, when ? is not too large, x0 (i) will be
somewhat bigger than most of the other entries in zi . Put another way, zi is biased towards the first
standard basis vector e1 .
Now, under our probabilistic assumptions, Z is very well conditioned: Z> Z ? I.3 Using, e.g.,
? for S of the form
Gram-Schmidt, we can find a basis Y
? = ZR,
Y
(12)
where R is upper triangular, and R is itself well-conditioned: R ? I. Since the i-th row of Z is
? i is also biased in the direction
biased in the direction of e1 and R is well-conditioned, the i-th row y
of e1 .
? ? = x0 . Since Ze1 = x0 , we have
We know that the global optimizer q? should satisfy Yq
q? = R?1 e1 ? e1 . Here, the approximation comes from R ? I. Hence, for this particular choice
of Y, described in (12), the i-th row is biased in the direction of the global optimizer. This is what
makes the rows of Y a particularly effective choice for initialization.
?
What if we are handed some other basis Y = YU,
where U is an orthogonal matrix? Suppose
? then it is easy to check that, with input matrix
q? is a global optimizer to (6) with input matrix Y,
Y, U> q? is also a global optimizer to (6), which implies that our initialization is invariant to any
rotation of the basis. Hence, even if we are handed an arbitrary basis for S, the i-th row is still biased
in the direction of the global optimizer.
? denote the output of Algorithm 1. We will prove that with our particular initializaRounding. Let q
tion and an appropriate choice of ?, the solution of our ADM algorithm falls within a certain radius
of the globally optimal solution q? to (6). To recover q? , or equivalently to recover the sparse vector
x0 = Yq? , we solve the linear program
min kYqk1
q
s.t.
hr, qi = 1,
(13)
2
More precisely, in our models, random initialization does work, but only when the subspace dimension n is
extremely low compared to the ambient dimension p.
3
This is the common heuristic that ?tall random matrices are well conditioned? [25].
4
? . We will prove that if r is close enough to q? , then this relaxation exactly recovers q? ,
with r = q
and hence x0 .
4
4.1
Analysis
Main Results
In this section, we describe our main theoretical result, which shows that with high probability, the
algorithm described in the previous section succeeds.
Theorem 4.1. Suppose that S satisfies the planted sparse model, and let Y be an arbitrary basis?
for
S. Let y1 . . . yp ? Rn denote the (transposes of) the rows of Y. Apply Algorithm 1 with ? = 1/ p,
?1, . . . , q
? p . Solve the linear program
using initializations q(0) = y1 , . . . , yp , to produce outputs q
?1, . . . , q
? p , to produce q
?1, . . . , q
? p . Set i? ? arg mini kY?
(13) with r = q
qi k0 . Then
Y?
qi? = ?x0 ,
(14)
for some ? 6= 0, with overwhelming probability, provided
p > Cn4 log2 n,
and
1
? ? ? ? ?0 .
4 n
(15)
Here, C and ?0 > 0 are universal constants.
We can see that the result in Theorem 4.1 is suboptimal compared to the global optimality condition
and Barak et al.?s result in the sense of the sampling complexity that we require p ? Cn4 log2 n.
While for the global optimality condition, we only need p > Cn to guarantee a global optimal
solution exists with high probability. For Barak et al.?s result, we need p > Cn2 . Nonetheless,
compared to Barak et al., we believe this is the first practical and efficient method that is guaranteed
to achieve ? ? O(1) rate. The lower bound on ? in Theorem 4.1 is mostly for convenience in the
proof; in fact,
? the LP rounding stage of our algorithm already succeeds with high probability when
? ? O (1/ n).
4.2
A Sketch of Analysis
The proof of our main result requires rather detailed technical analysis of the iteration-by-iteration
properties of Algorithm 1. In this subsection, we briefly sketch the main ideas. For detailed proofs,
please see the technical supplement to this paper.
As noted in Section 3, the ADM algorithm is invariant to change of basis. So, we can assume without
? = ZR defined in that section. In
loss of generality that we are working with the particular basis Y
order to further streamline the presentation, we are going to sketch the proof under the assumption
that
Y = [x0 | g1 | ? ? ? | gn?1 ],
(16)
? This may seem plausible, but when p is large Y is already
rather than the orthogonalized version Y.
? In fact, in our proof, we simply carry through the
nearly orthogonal, and hence Y is very close to Y.
?
argument for Y, and then note that Y and Y are close enough that all steps of the proof still hold
? With that noted, let y1 , . . . , yp ? Rn denote the transposes of the rows of Y,
with Y replaced by Y.
and note that these are independent random vectors. From (11), we can see one step of the ADM
algorithm takes the form:
Pp
1
i
i > (k)
q ]
i=1 y S? [ y
p
(k+1)
.
q
=
(17)
P
1 p
>
p i=1 yi S? [(yi ) q(k) ]
2
This is a very favorable form for analysis: if q is viewed as fixed, the term in the numerator is a sum
of p independent random vectors. To this end, we define a vector valued random process Q(q) on
q ? Sn?1 , via
p
>
1X i
Q(q) =
y S? [ yi q].
(18)
p i=1
5
We study the behavior of the iteration (17) through the random process Q(q). We wish to show
that w.h.p. in our choice of Y, q(k) converges to (?e1 ), so that the algorithm successfully retrieves
the sparse vector x0 = Ye1 . Thus, we hope that in general,
Q(q)
is more concentrated on the first
q1
coordinate than q. Let us partition the vector q as q =
, with q1 ? R and q2 ? Rn?1 , and
q2
Q1 (q)
correspondingly partition Q(q) =
, where
Q2 (q)
p
Q1 (q) =
h > i
1X
x0i S? yi q
p i=1
p
Q2 (q) =
and
1 X i h i > i
g S? y
q .
p i=1
(19)
The inner product of Q(q)/ kQ(q)k2 and e1 is strictly larger than the inner product of q and e1 if
and only if
kQ2 (q)k2
|Q1 (q)|
>
.
(20)
|q1 |
kq2 k2
In the appendix, we show that with high probability, this inequality holds uniformly over a significant
portion of the sphere, so the algorithm moves in the correct direction. To complete the proof of
Theorem 4.1, we combine the following observations:
1. Algorithm 1 converges.
?
2. Rounding succeeds when |r1 | > 2 ?. With high probability, the linear programming based
rounding (13) will produce ?x0 , up
? to scale, whenever it is provided with an input r whose first
coordinate has magnitude at least 2 ?.
?
3. No jumps away from the caps. With high probability, for all q such that |q|1 > C? ?,
?
|Q1 (q)|
q
? 2 ?.
(21)
2
|Q1 (q)2 | + kQ2 (q)k2
4. Uniform progress away from the equator. With high probability, for every q such that
?
|q1 | ? C? ?, the bound
c
|Q1 (q)| kQ2 (q)k2
?
>
|q1 |
kqk2
np
(k)
holds. This implies that if at any iteration k of the algorithm, |q1 | >
?
0
(k0 )
eventually obtain a point q(k ) , k 0 > k, for which |q1 | > C? ?.4
?1 ,
2 ?n
?1
2 ?n
?
(22)
the algorithm will
5. Location of stationary points. Steps 1, 3 and 4 above imply that if Algorithm 1 ever obtains a
?
?
(k)
point q(k) with |q1 | > 2?1?n , it will converge to a point q? with q?1 > C? ?, provided 2?1?n < 2 ?
(i.e., ? > 4?1 n ).
(0)
6. Good initializers. With high probability, at least one of the initializers q(0) satisfies |q1 | >
?1 .
2 ?n
Taken together, these claims imply that from at least one of the initializers q(0) , the ADM algorithm
? which is accurate enough for LP rounding to exactly return x0 , up to scale.
will produce an output q
As x0 is the sparsest nonzero vector in the subspace S with overwhelming probability, it will be
selected as Yqi? , and hence produced by the algorithm.
5
Experimental Results
In this section, we show the performance of the proposed ADM algorithm on both synthetic and real
datasets. On the synthetic dataset, we show the phase transition of our algorithm on both the planted
sparse vector and dictionary learning models; for the real dataset, we demonstrate how seeking sparse
vectors can help discover interesting patterns.
4
In fact, the rate of progress guaranteed in (22) can be used to bound the complexity of the algorithm; we do
not dwell on this here.
6
5.1
Phase Transition on Synthetic Data
For the planted sparse model, for each pair of (k, p), we generate the n dimensional subspace
S ? Rp by a k sparse vector x0 with nonzero entries equal to 1 and a random Gaussian matrix
i.i.d.
G ? Rp?(n?1) with Gij ? N (0, 1/p), so that the basis Yof the subspace S can be constructed
by Y = GS ([x0 , G]) U, where GS (?) denotes the Gram-Schmidt orthonormalization operator and
U ? Rn?n is an arbitrary orthogonal matrix. We fix the?relationship between n and p as p = 5n log n,
and set the regularization parameter in (8) as ? = 1/ p. We use all the normalized rows of Y as
initializations of q for the proposed ADM algorithm, and
for 5000 iterations. We
run every program
x0
assume the proposed method to be success whenever
kx0 k ? Yq
? for at least one of the p
2
2
programs, for some error tolerance = 10?3 . For each pair of (k, p), we repeat the simulation for 5
times.
Figure 1: Phase transition for the planted sparse model (left) and dictionary learning (right) using the ADM
algorithm, with fixed relationship between p and n: p = 5n log n. White indicates success and black indicates
failure.
Second, we consider the same dictionary learning model as in [24]. Specifically, the observation is
assumed to be Y = A0 X0 where A0 is a square, invertible matrix, and X0 a n ? p sparse matrix.
Since A0 is invertible, the row space of Y is the same as that of X0 . For each pair of (k, n), we
>
generate X0 = [x1 , ? ? ? , xn ] , where each vector xi ? Rp is k-sparse with every nonzero
>entry
following i.i.d. Gaussian distribution, and construct the observation by Y> = GS X>
0 U . We
repeat the same experiment as for the planted sparse model presented above. The only difference is
that we assume the proposed method to be success as long as one sparse row of X0 is recovered by
those p programs.
Fig. 1 shows the phase transition between the sparsity level k = ?p and p for both models. It seems
clear for both problems our algorithm can work well into (beyond) the linear regime in sparsity level.
Hence for the planted sparse model, to close the gap between our algorithm and practice is one future
direction. Also, how to extend our analysis for dictionary learning is another interesting direction.
5.2
Exploratory Experiments on Faces
It is well known in computer vision convex objects only subject to illumination changes produce
image collection that can be well approximated by low-dimensional space in raw-pixel space [9].
We will play with face subspaces here. First, we extract face images of one person (65 images)
under different illumination conditions. Then we apply robust principal component analysis [12]
to the data and get a low dimensional subspace of dimension 10, i.e., the basis Y ? R32256?10 . We
apply the ADM algorithm to find the sparsest element in such a subspace, by randomly selecting
10% rows as initializations for q. We judge the sparsity in a `1 /`2 sense, that is, the sparsest vector
? 0 = Yq? should produce the smallest kYqk1 / kYqk2 among all results. Once some sparse vectors
x
are found, we project the subspace onto orthogonal complement of the sparse vectors already found,
and continue the seeking process in the projected subspace. Fig. 2 shows the first four sparse vectors
we get from the data. We can see they correspond well to different extreme illumination conditions.
Second, we manually select ten different persons? faces under the normal lighting condition. Again,
the dimension of the subspace is 10 and Y ? R32256?10 . We repeat the same experiment as stated
above. Fig. 3 shows four sparse vectors we get from the data. Interestingly, the sparse vectors roughly
7
Figure 2: Four sparse vectors extracted by the ADM algorithm for one person in the Yale B database under
different illuminations.
correspond to differences of face images concentrated around facial parts that different people tend to
differ from each other.
Figure 3: Four sparse vectors extracted by the ADM algorithm for 10 persons in the Yale B database under
normal illuminations.
In sum, our algorithm seems to find useful sparse vectors for potential applications, like peculiar
discovery in first setting, and locating differences in second setting. Netherless, the main goal of this
experiment is to invite readers to think about similar pattern discovery problems that might be cast as
searching for a sparse vector in a subspace. The experiment also demonstrates in a concrete way the
practicality of our algorithm, both in handling data sets of realistic size and in producing attractive
results even outside of the (idealized) planted sparse model that we adopt for analysis.
6
Discussion
The random models we assume for the subspace can be easily extended to other random models,
particularly for dictionary learning. Moreover we believe the algorithm paradigm works far beyond
the idealized models, as our preliminary experiments on face data have clearly shown. For the
particular planted sparse model, the performance gap in terms of (p, n, ?) between the empirical
simulation and our result is likely due to analysis itself. Advanced techniques to bound the empirical
process, such as decoupling [17] techniques, can be deployed in place of our crude union bound
to cover all iterates. Our algorithmic paradigm as a whole sits well in the recent surge of research
endeavors in provable and practical nonconvex approaches towards many problems of interest, often
in large-scale setting [13, 22, 20, 23, 26]. We believe this line of research will become increasingly
important in theory and practice. On the application side, the potential of seeking sparse/structured
element in a subspace seems largely unexplored, despite the cases we mentioned at the start. We
hope this work can invite more application ideas.
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9
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4,864 | 5,403 | Compressive Sensing of Signals from a GMM with
Sparse Precision Matrices
1
1
2
1
Jianbo Yang
Xuejun Liao
Minhua Chen
Lawrence Carin
1
Department of Electrical and Computer Engineering, Duke University
2
Department of Statistics & Department of Computer Science, University of Chicago
{jianbo.yang;xjliao;lcarin@[email protected]},{[email protected]}
Abstract
This paper is concerned with compressive sensing of signals drawn from a Gaussian mixture model (GMM) with sparse precision matrices. Previous work has
shown: (i) a signal drawn from a given GMM can be perfectly reconstructed from
r noise-free measurements if the (dominant) rank of each covariance matrix is
less than r; (ii) a sparse Gaussian graphical model can be efficiently estimated
from fully-observed training signals using graphical lasso. This paper addresses a
problem more challenging than both (i) and (ii), by assuming that the GMM is unknown and each signal is only observed through incomplete linear measurements.
Under these challenging assumptions, we develop a hierarchical Bayesian method
to simultaneously estimate the GMM and recover the signals using solely the incomplete measurements and a Bayesian shrinkage prior that promotes sparsity of
the Gaussian precision matrices. In addition, we provide theoretical performance
bounds to relate the reconstruction error to the number of signals for which measurements are available, the sparsity level of precision matrices, and the ?incompleteness? of measurements. The proposed method is demonstrated extensively
on compressive sensing of imagery and video, and the results with simulated and
hardware-acquired real measurements show significant performance improvement
over state-of-the-art methods.
1
Introduction
Gaussian mixture models (GMMs) [1, 2, 3] have become a popular signal model for compressive
sensing [4, 5] of imagery and video, partly because the information domain in these problems can
be decomposed into subdomains known as pixel/voxel patches [3, 6]. A GMM employs a Gaussian
precision matrix to capture the statistical relations between local pixels/voxels within a patch, and
meanwhile captures the global statistics between patches using its clustering mechanism.
Compressive sensing (CS) of signals drawn from a GMM admits closed-form minimum mean
squared error (MMSE) reconstruction from linear measurements. Recent theoretical analysis in
[7] shows that, given a sensing matrix with entries i.i.d. drawn from a zero-mean, fixed-variance,
Gaussian distribution or Bernoulli distribution with parameter 0.5, if the GMM is known and the
(dominant) rank of each covariance matrix is less than r, each signal can be perfectly reconstructed
from r noise-free measurements. Though this is a much less stringent reconstruction condition than
that prescribed by standard restricted-isometry-property (RIP) bounds, it relies on the assumption
of knowing the exact GMM. If a sufficient number of fully observed signals are available beforehand, one can use maximum likelihood (ML) estimators to train a GMM [8, 9, 7, 1, 10] for use in
reconstructing the signals in question. Unfortunately, finding an accurate GMM a priori is usually a
challenge in practice, because it is difficult to obtain training signals that match the statistics of the
interrogated signals.
1
Recent work [2] on GMM-based methods proposes to solve this problem by estimating the Gaussian components, based on measurements of the signals under interrogation, without resorting to
any fully-observed signals to train a model in advance. The method of [2] has two drawbacks: (i)
it estimates full dense Gaussian covariance matrices, with the number of free parameters to be estimated growing quadratically fast with the signal dimensionality n; (ii) it does not have performance
guarantees, because all previous theoretical results, including those in [7], assume the GMM is given
and thus are no longer applicable to the method of [2]. This paper addresses these two issues.
First, we effectively reduce the number of GMM parameters by restricting the GMM to have sparse
precision matrices with group sparsity patterns, making the GMM a mixture of group-sparse Gaussian graphical models. The group sparsity is motivated by the Markov random field (MRF) property
of natural images and video [11, 12, 13]. Instead of having n2 parameters for each Gaussian component as in [2], we have only n + s parameters, where s is the number of nonzero off-diagonals
of the precision matrix. We develop a variational maximum-marginal-likelihood estimator (variational MMLE) to simultaneously estimate the GMM and reconstruct the signals, with a Bayesian
shrinkage prior used to promote sparsity of the Gaussian precision matrices. Our variational MMLE maximizes the marginal likelihood of the GMM given only the linear measurements, with the
unknown signals treated as random variables and integrated out of the likelihood. A key step of
the variational MMLE is using Bayesian graphical lasso to reestimate the sparse Gaussian precision
matrices based on a posteriori signal samples conditional on the linear measurements.
Second, we provide theoretical performance bounds under the assumption that the GMM is not
exactly known. Assuming the GMM has sparse precision matrices, our theoretical results relate
the signal reconstruction error to the number of signals for which measurements are available, the
sparsity level of the precision matrices, and the ?incompleteness? of measurements, where the last
is defined as the uncertainty (variance) of a signal given its linear measurements.
In the experiments, we present reconstruction results of the proposed method on both simulated
measurements and real measurements acquired by actual hardware [6]. The proposed method outperforms the state-of-art CS reconstruction algorithms by significant margins.
Notations. Let N (x|?, ??1 ) denote a Gaussian density of x with mean ? and precision matrix ?,
kM kF denote the Frobenius matrix norm of matrix M , kM kmax denote the largest entry of M in
terms of magnitude, tr(M ) denote the trace of M , ?0 = ??1
0 denote the true precision matrix (i.e.,
the inverse of true covariance matrix ?0 ), ?? denote the estimate of ?0 by the proposed model.
Herein, the eigenvalues of ?0 are assumed to be bounded in a constant interval [?1 , ?2 ] ? (0, ?), to
guarantee the existence of ?0 . For functions f (x) and g(x), we write f (x) g(x) when f (x) =
O(g(x)) and g(x) = O(f (x)) hold simultaneously.
2
2.1
Learning a GMM of Unknown Signals from Linear Measurements
Signal Reconstruction with a Given GMM
The linear measurement of an unknown signal x ? Rn can be written as y = ?x + , where
? ? Rm?n is a sensing matrix, and ? Rm denote measurement noises (we are interested in
m < n). Assuming ? N (|0, R), one has p(y|x) = N (y|?x, R). We further assume R to be a
scaled identity matrix, R = ??1 I, and thus the noise is white Gaussian.
PK
?1
If x is governed by a GMM, i.e., p(x) = z=1 ? (z) N (x|?(z) , ?(z) ), one may obtain
p(y, x, z) = ? (z) N (y|?x, R)N (x|?(z) , ?(z)
p(y) =
K
X
? (z) N (y|??(z) , R + ??(z)
?1
?0 ),
?1
),
p(x, z|y) = ?(z) N (x|? (z) , (C(z) )?1 ),
(1)
z=1
where
C(z) =
?0 R?1 ? + ?(z)
?1
? (z) = ?z + C(z) ?0 R?1 (y ? ??z ),
,
?1
? (z) N (y|??(z) , R + ??(z) ?0 )
?(z) = PK
?1 0 .
(l) N (y|??(l) , R + ??(l)
?)
l=1 ?
(2)
When the GMM is exactly known, the signal is reconstructed analytically as the conditional mean,
b , E(x|y) =
x
PK
2
z=1 ?
(z)
? (z) .
(3)
It has been shown in [7] that, if the (dominant) rank of each Gaussian covariance matrix is less than
r, the signal can be perfectly reconstructed from only r measurements in the low-noise regime.
2.2 Restriction of the GMM to a mixture of Gaussian Markov Random Fields
A Markov random field (MRF), also known as an undirected graphical model, provides a graphical
representation of the joint probability distribution over multiple random variables, by considering
the conditional dependences among the variables [11, 12, 13]. In image analysis, each node of
an MRF corresponds to a pixel of the image in question, and an edge between two nodes is often
modeled by a potential function to characterize the conditional dependence between the associated
pixels. Because of the local smoothness structure of images, the edges of an MRF are usually
chosen based on a pairwise neighborhood structure: each pixel only has edge connections with
its neighbors. The widely used scheme is that each pixel only has edge connections with its four
immediate neighboring pixels to the left, right, top and bottom [11]. Therefore, an MRF for image
representation is an undirected graph with only a limited number of edges between its nodes.
Generally, learning and inference of an MRF are nontrivial, due to the nonlinearity and nonconvexity of the potential functions [14]. A popular special case of MRF is the Gaussian Markov
random field (GMRF) which is an MRF with a multivariate Gaussian distribution over node variables. The best-known advantage of a GMRF is its simplicity of learning and inference, because
of the nice properties of a multivariate Gaussian distribution. According to Hammersley-Clifford?s
theorem [15], the conditional dependence of the node variables in a GMRF is encoded in the precision matrix. As mentioned before, an MRF is sparse for image analysis problems, on account of
the neighborhood structure in the pixel domain. Therefore, the multivariate Gaussian distribution
associated with a GMRF has a sparse precision matrix. This property of a GMRF in image analysis
is demonstrated in Section 1 of the Supplementary Material.
Inspired by the GMRF interpretation, we place a shrinkage prior on each precision matrix to promote
sparsity when estimating the GMM. The Laplacian shrinkage prior used in [16] is chosen, but other
shrinkage priors [17] could also be used. Specifically, we impose a Laplacian shrinkage prior on the
off-diagonal elements of each of K precision matrices,
p(?
(k)
)=
n Y
Y
q
(k)
? (k) ?ij
2
i=1 j<i
(k)
q
(k)
(k)
exp(? ? (k) ?ij |?ij |), ?k = 1, . . . , K,
(4)
(k)
with the symmetry constraints ?ij = ?ji . In (4), ? (k) > 0 is a ?global? scaling parameter for all
(k)
(k)
the elements of {?ij |i = 1, ..., n, j < i} and generally fixed to be one [18], and ?ij is a ?local?
(k)
weight for the element ?ij . With the Laplacian prior (4), many off-diagonal elements of ?(k) are
encouraged to be close to zero. However, in the inference procedure, the above Laplacian shrinkage
prior (4) is inconvenient due to the lack of analytic updating expressions. This issue is overcome by
using an equivalent scale mixture of normals representation [16] of (4) as shown below:
q
(k)
? (k) ?ij
2
Z
(k)
q
?ij
?1 (k) ?1
(k)
(k)
(k)
(k)
(k)
(k)
exp(? ? ?ij |?ij |) = N (?ij |0, ? (k) ?ij
)InvGa(?ij |1,
)d?ij
2
(5)
(k)
where ?ij is an augmented variable drawn from an inverse gamma distribution. Further, one may
(k)
place a gamma prior on ?ij . Then, a draw of the precision matrix may be represented by
?(k) ?
n Y
Y
(k)
N (?ij |0, ? (k)
?1
(k) ?1
?ij
(k)
(k)
(k)
), ?ij ? InvGa(?ij |1,
i=1 j<i
?ij
(k)
(k)
), ?ij ? Ga(?ij |a0 , b0 )
2
(6)
where a0 , b0 are the hyperparameters.
?1
Suppose {xi }N
are samples drawn from N (x|0, ?(k) ) and S denotes the empirical covariance
PNi=1
(k)
1
matrix N i=1 (xi ? x)(xi ? x)0 where x is the empirical mean of {xi }N
i=1 . If the elements ?
are drawn as in (6), the logarithm of the joint likelihood can be expressed as
(k)
log p({xi }N
)
i=1 , ?
N
?
2
log det(?
(k)
) ? tr(S?
(k)
!
q
n X
X
2
(k)
(k)
(k)
)?
? ?ij |?ij | .
N
i=1 j<i
(7)
From the optimization perspective, the maximum a posterior (MAP) estimations of ?(k) in (7) is
known as the adaptive graphical lasso problem [18].
3
2.3 Group sparsity based on banding patterns
The Bayesian adaptive graphical lasso described above assumes the precision matrix is sparse, and
the same Laplacian prior is imposed on all off-diagonal elements of the precision matrix without any
discrimination. However, the aforementioned neighborhood structure of image pixels implies that
the entries of the precision matrix corresponding to the pairs between neighboring pixels tend to have
significant values. This is consistent with the observations as seen from the demonstration in Section
1 of the Supplementary Material: (i) the bands scattered along a few lines above or below the main
diagonal are constituted by the entries with significant values in the precision matrix; (ii) the entries
in the bands correspond to the pairwise neighborhood structure of the graph, since vectorization of
an image patch is constituted by stacking all columns of pixels in a patch on the top of each other;
(iii) the existence of multiple bands in some Gaussian components reveals that, besides the four
immediate neighboring pixels, other indirected neighboring pixels may also lead to nonnegligible
conditional dependence, though the entries in the associated bands have relatively smaller values.
Inspired by the banding patterns mentioned above, we categorize the elements in the set
(k)
(k)
(k)
{?ij }ni=1,j<i into two groups {?ij |(i, j) ? L1 } and {?ij |(i, j) ? L2 }, where L1 denotes the
set of indices corresponding to the elements in the bands and L2 represents the set of indices for the
(k)
elements not in the bands. For the elements in the group {?ij |(i, j) ? L2 }, the Laplacian prior is
(k)
used to encourage a sparse precision matrix. For the elements in the group {?ij |(i, j) ? L1 } , the
sparsity is not desired so a normal prior with Gamma hyperparameters is used instead. Accordingly,
the expressions in (6) can be replaced by
?(k) ?
n Y
Y
(k)
N (?ij |0, ? (k)
?1
(k) ?1
?ij
)
i=1 i<j
(
(k)
?ij
?
(k)
Ga(?ij |c0 , d0 ),
if (i, j) ? L1
(k)
(k)
InvGa(?ij |1,
?ij
2
(k)
(k)
), ?ij ? Ga(?ij |a0 , b0 ),
(8)
.
if (i, j) ? L2
With the prior distribution of ?(k) in (6) replaced with that in (8), the joint log-likelihood in (7)
changes to
(k)
log p({xi }N
)
i=1 , ?
?
N
? ?log det(?(k) ) ? tr(S?(k) ) ?
2
X
(i,j)?L1
2 (k) (k) (k) 2
? ?ij k?ij k ?
N
X
(i,j)?L2
2
N
?
q
(k)
(k)
? (k) ?ij |?ij |? .
(9)
To the best of our knowledge, the maximum a posterior (MAP) estimations of ?(k) in (9) has not
been studied in the family of graphical lasso or its variants, from the optimization perspective.
2.4 Hierarchical Bayesian model and inference
We consider the collective compressive sensing of the signals X = {xi ? Rn }N
i=1 that are drawn
from an unknown GMM. The noisy linear measurements of X are given by Y = {y i ? Rm : y i =
?i xi + i }N
i=1 . We assume the sensing matrices to be signal-dependent to account for generality
(i.e., ?i depends on the signal index i).
The unification of signal reconstruction with a given GMM (presented in Section 2.1) and GMRF learning with fully-observed training signals (presented in Section 2.2) leads to the following
Bayesian model,
y i |xi ? N (y i |?i xi , ??1 I), xi ?
K
X
? (z) N (xi |?(z) , ?(z)
?1
), ? ? Ga(?|e0 , f0 )
(10)
z=1
?(k) ?
n Y
Y
(k)
N (?ij |0, ? (k)
?1
(k) ?1
?ij
(k)
(k)
(k)
), ?ij ? InvGa(?ij |1,
i=1 i<j
?ij
(k)
(k)
), ?ij ? Ga(?ij |a0 , b0 ), (11)
2
The expression in (11) could be replaced by (8) if the group sparsity is considered in the precision
matrix. In addition to the precision matrices, we further add the following standard priors on the
other parameters of the GMM to make the proposed model a full hierarchical Bayesian model,
?(k) ? N (?(k) |m0 , (?0 ?(k) )?1 ), ? ? Dirichlet(? (1) , . . . , ? (K) |a0 ),
4
(12)
where m0 , a0 and ?0 are hyperparameters.
We develop the inference procedure for the proposed Bayesian hierarchical model. Let the symbols
Z, ?, ?, ?, ?, ? denote the sets {zi }, {?(k) }, {?(k) }, {? (k) }, {?(k) }, {? (k) } respectively. The
marginalized likelihood function is written as
Z
L(?) = ln
p(Y, ?, ?)d?
where ? , {X, Z, ?, ?} and ? , {?, ?, ?, ?} denote the set of the latent variables and parameters of the model, respectively. An expectation-maximization (EM) algorithm [19] could be used to
find the optimal ? by alternating the following two steps
? E-step: Find p(?|Y, ?? ) with ?? computed at the M-step, and obtain the expected complete log-likelihood E? (ln p(Y, ?, ?? )).
? M-step: Find an improved estimate of ?? by maximizing the expected complete loglikelihood given at the E-step.
However, it is intractable to compute the exact posterior p (?|Y, ?) at the E step. We develop a
variational inference approach to overcome the intractability. Based on the mean field theory [20],
we approximate the posterior distribution p (?|Y, ?) by a proposal distribution q(?) that factorizes
over the variables as follows
q(?) = q(X, Z, ?, ?) = q(X, Z)q(?)q(?).
(13)
Then, we find an optimal distribution q(?) that minimizes the Kullback-Leibler (KL) divergence
R
q(?)
KL(q(?)||p(?|Y, ?)) = q(?) ln p(?|Y,?)
d?, or equivalently, maximizes the evidence lower
bound (ELBO) of the log-marginal data likelihood [21], denoted by F(q(?), ?),
Z
ln p(Y, ?) = ln
q(?)
p (Y, ?, ?)
d? ?
q(?)
Z
q(?) ln
p (Y, ?, ?)
d? , F(q(?), ?)
q(?)
(14)
where the inequality is held based on the Jensen?s inequality.
With the above approximation, the entire algorithm becomes a variational EM algorithm and it
iterates between the following VE-step and VM-step until convergence:
? VE-step: Find the optimal posterior distribution q ? (?) that maximizes F(q(?), ?? ) with
?? computed at the VM-step.
? VM-step: Find the optimal ?? that maximizes F(q ? (?), ?) with q ? (?) computed at the
VE-step.
The full update equations of the variational EM algorithm are given in Section 2 of the Supplementary Material.
3
Theoretical Analysis
The proposed hierarchical Bayesian model unifies the task of signal recovery and the task of estimating the mixture of GMRF, with a common goal of maximizing the ELBO of the log-marginal
likelihood of the measurements. This section provides a theoretical analysis to further reveal the
mutual influence between these two tasks (Theorem 1 and Theorem 2), and establish a theoretical
performance bound (Theorem 3) to relate the reconstruction error to the number of signals being
measured, the sparsity level of precision matrices, and the ?incompleteness? of measurements. The
proofs of these theorems are presented in Sections 3-5 of the Supplementary Material. For convenience, we consider the single Gaussian case, so the superscript (k) is omitted in the sequel. We
begin with the definitions and assumptions used in the theorems.
e i and x
b i be the signals estimated from measurement y i , using the true precision
Definition 3.1 Let x
matrix ?0 and the estimated precision matrix ?? respectively, according to (3),
b i =? + ?0 + ?0i R?1 ?i
x
?1
?0i R?1 (yi ? ?i ?) = ? + Ci ?0i R?1 (yi ? ?i ?)
?1 0 ?1
?1 0 ?1
ei =? + ?0 + ? + ?0i R?1 ?i
x
?i R (yi ? ?i ?) = ? + C?1
+?
?i R (yi ? ?i ?) .
i
b i as the
Assuming yi ? Rr is noise-free and the (dominant) rank of ?0 is less than r, one obtains x
b i = xi . Then the reconstruction error of x
e i is k? i k2 , where ? i = x
ei ? x
bi.
true signal xi [7], i.e., x
5
Definition 3.2 The estimation error of ?? is defined as k?kF where ? = ?? ? ?0 .
At each VM-step of the variational EM algorithm developed in Section 2.4, ?? is updated based on
the empirical covariance matrix ?em computed from {e
xi }, i.e.,
?em =
N
N
N
N
1 X
1 X
1 X
1 X
eix
e 0i +
bix
b 0i +
x
Ci =
x
(2b
xi ?i0 + ?i ?i0 + Ci ),
N i=1
N i=1
N i=1
N i=1
{z
} |
{z
}
|
?0
em
(15)
?de
where {b
xi } and {e
xi } are considered to both have zero mean, as one can always center the signals
with respect to their means [2].
Definition 3.3 The deviation of empirical matrix ?0em is defined as ?de = ?em ??0em according to
(15), and we use ?
?de , k?de kmax to measure this deviation. Considering
q the developed variational
EM algorithm can converge to a local minimum, we assume ?
?de ? c
log n
N
for a constant c > 01 .
3.1 Theoretical results
Theorem 1 Assuming kCi kF k?kF < 1, the reconstruction error of the i-th signal is upper boundkCi kF k?kF
kb
xi k2 .
ed as k? i k2 ? 1?kC
i k k?k
F
F
Theorem 1 establishes the error bound of signal recovery in terms of ?. In this theorem, ?? can be
obtained by any GMRF estimation methods, including [1, 2] and the proposed method.
?
??
?
??
Let ? = min(i,j)?S c N ij , ? = max(i,j)?S N ij , S = {(i, j) : ?ij 6= 0, i 6= j}, S c = {(i, j) :
?ij = 0, i 6= j} and the cardinality of S be s. The following theorem establishes an upper bound of
k?kF on account of ?de .
q
log n
Theorem 2 Given the empirical covariance matrix ?em , if ?, ?
?de , then we have
N + ?
p
?
?de }.
k?kF = Op { (n + s) log n/N + n + s?
Note that the standard graphical lasso and its variants [18, 23] assume the true signal samples {xi }
are fully observed when estimating ?? , so they correspond to the simple case that ?
?de = 0. Loh
and Wainwright [22, Corollary 5] also provides an upper bound of k?kF taking ?de into account.
However, they assume ?0em is attainable and the proof of their corollary relies on their proposed
GMRF estimation algorithm, so the theoretical result in [22] cannot be used here.
PN
PN
b max = supi kb
Let 0 = N1 i=1 kb
xi ? ?k2 , ? = N1 i=1 tr(Ci ), ?max = supi k?i k2 , x
xi k2 and
? = maxi kCi kF . A combination of Theorem 1 and 2 leads to the following theorem which relates the error bound of signal reconstruction to the number of partially-observed signals (observed
through incomplete linear measurements), the sparsity level of precision matrices, and the uncertainty of signal reconstruction (i.e., ? and ?) which represent the ?incompleteness? of the measurements.
q
log n
?de , ?k?kF < ?
Theorem 3 Given the empirical covariance matrix ?em , if ?, ?
N + ?
?
where ? is a constant and (1 ?p
?)/ n + s > M 0 (?max + 2b
xmax )? with M being an appropriate
?
constant to make k?kF ? M (n + s) log n/N
+
M
n
+
s?
?de hold with high probability, then
?
PN
(log n)/N +?
1
bi k2 ? (1??)/?n+s?M (? +2bx )? M 0 ?.
we obtain that N i=1 ke
xi ? x
0
max
max
From Theorem 3, we find that when the number of partially-observed signals N tends to infinity and
the uncertainty of signal reconstruction tr(Ci ) tends to zero ? i, the average reconstruction error
PN
1
bi k2 is close to zero with high probability.
xi ? x
i=1 ke
N
4
Experiments
The performance of the proposed methods is evaluated on the problems of compressive
sensing (CS) of imagery and high-speed video2 . For convenience, the proposed method
is termed as Sparse-GMM when using the non-group sparsity described in Section 2.2,
1
2
A similar assumption is made in expression (3.13) of [22].
The complete results can be found at the website: https://sites.google.com/site/nipssgmm/.
6
and is termed Sparse-GMM(G) when using the group sparsity described in Section 2.3.
For Sparse-GMM(G), we construct the two groups L1 and L2 as follows : L1 =
{(i, j) : pixel i is one of four immediate neighbors, in the spatial domain, of pixel j, i 6= j} and
L2 = {(i, j) : i, j = 1, 2, ? ? ? , n, i 6= j} \ L1 . The proposed methods are compared with state-ofthe-art methods, including: a GMM pre-trained from training patches (GMM-TP) [7, 8], a piecewise
linear estimator (PLE) [2], generalized alternating projection (GAP) [24], Two-step Iterative Shrinkage/Thresholding (TwIST) [25], KSVD-OMP [26].
For
of the scaled mixture of Gaussians are set as
p the proposed methods, the hyperparameters
a0 /b0 /N ? 300, c0 = d0 = 10?6 , the hyperparameter of Dirichlet prior ?0 is set as a vector with all elements being one, the hyperparameters of the mean of each Gaussian component are
?6
set as ?0 = 1, and m0 is set to the mean of the initialization of {b
xi }N
for
i=1 . We fixed ? = 10
the proposed methods, GMM-TP and PLE. The number of dictionary elements in KSVD is set to
the best in {64, 128, 256, 512}. The TwIST adopts the total-variation (TV) norm, and the results of
TwIST reported here represented the best among the different settings of regularization parameter in
the range of [10?4 , 1]. In GAP, the spatial transform is chosen between DCT and waveletes and the
one with the best result is reported, and the temporal transform for video is fixed to be DCT.
4.1 Simulated measurements
Compressive sensing of still images. Following the single pixel camera [27], an image xi is projected onto the rows of a random sensing matrix ?i ? Rm?n to obtain the compressive measurements y i for i = 1, . . . , N . Each sensing matrix ?i is constituted by the elements drawn
from a uniform distribution in [0, 1]. The USPS handwritten digits dataset 3 and the
face dataset [28] are used in this experiment. In each dataset, we randomly select 300 images
and each image is resized to the scale of 12 ? 12. Eight settings of CS ratios are adopted with
m
n ? {0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40}. Since signal xi in the single pixel camera represents an entire image which generally has unique statistics, it is infeasible to find suitable training
data in practice. Therefore, GMM-TP and KSVD-OMP are not compared to in this experiment4 . For
PLE, Sparse-GMM and Sparse-GMM(G), the minimum-norm estimates from the measurements,
b i = arg minx {kxk22 : ?i x = y i } = ?0i (?i ?0i )?1 y i , i = 1, . . . , N , are used to initialize the
x
GMM. The number of GMM components K in PLE, Sparse-GMM, and Sparse-GMM(G) is tuned
among 2 ? 10 based on Bayesian information criterion (BIC).
GAP
TwIST
PLE
Sparse-GMM
Sparse-GMM(G)
14
12
32
GAP
TwIST
PLE
Sparse-GMM
Sparse-GMM(G)
25
20
PSRN (dB)
PSRN (dB)
16
15
GAP (23.72)
TwIST (24.81)
GMM-TP (24.47)
KSVD-OMP (22.37)
PLE (25.35)
Sparse-GMM (27.3)
Sparse-GMM(G) (28.05)
30
28
PSRN (dB)
18
26
24
10
10
22
8
0.05
0.1
0.15
0.2
0.25
0.3
CS measurements fraction in a patch
0.35
0.4
5
0.05
0.1
0.15
0.2
0.25
0.3
CS measurements fraction in a patch
0.35
0.4
20
5
10
15
20
Frames
25
30
Figure 1: A comparison of reconstruction performances, in terms of PSNR, among different methods
for CS of imagery on USPS handwritten digits (left) and face datasets (middle), and CS
of video on NBA game dataset (right), with the average PSNR over frames shown in the brackets.
Compressive sensing of high-speed video. Following the Coded Aperture Compressive Temporal
Imaging (CACTI) system [6], each frame of video to be reconstructed is encoded with a shifted
binary mask which is designed by randomly drawing values from {0, 1} at every pixel location,
with a 0.5 probability of drawing 1. Each signal xi represents the vectorization of T consecutive
spatial frames, obtained by first vectorizing each frame into a column and then stacking the resulting
T columns on top of each other. The measurement y i is constituted by y i = ?i xi where ?i =
[?i,1 , . . . , ?i,T ] and ?i,t is a diagonal matrix with its diagonal being the mask that is applied to
the t-th frame. A video containing NBA game scenes is used in the experiment. It has 32 frames,
each of size 256 ? 256, and T is set to be 8. For GMM-TP, KSVD-OMP, PLE, Sparse-GMM and
Sparse-GMM(G), we partition each 256 ? 256 measurement frame into a set of 64 ? 64 blocks,
and each block is considered as if it were a small frame and is processed independently of other
blocks.5 The patch is of size 4 ? 4 ? T . Since each block is only 64 ? 64, a small number of GMM
components are sufficient to capture its statistics, and we find the results are robust to K as long as
2 ? K ? 5 for PLE, Sparse-GMM and Sparse-GMM(G). Following [8, 26], we use the patches
3
It is downloaded from http://cs.nyu.edu/?roweis/data.html.
The results of other settings can be found at https://sites.google.com/site/nipssgmm/.
5
This subimage processing strategy has also been used in [2].
4
7
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GMM-TP
#2
#3
#5
#1
#6
#11
#7
#12
#8
#9
#13
#10
#14
#2
#3
#4
#5
#7
#8
#9
#10
#11
#12
#13
#14
#6
#11
#11
#6
#11
5
#2
#3
#4
#5
#7
#8
#9
#10
#12
#13
#14
Sparse-GMM
TwIST
#2
#3
#4
#5
#7
#8
#9
#10
#12
#13
#14
#1
#6
#11
#2
#7
#12
#3
#8
#13
#4
#9
#5
#1
#6
#10
#11
#14
Sparse-GMM(G)
#1
#1
#6
#6
GAP
#1
PLE
KSVD-OMP
#4
#2
#3
#4
#5
#7
#8
#9
#10
#12
#13
#14
Raw measurement
(Coded image)
#2
#3
#4
#5
#7
#8
#9
#10
#12
#13
#14
GMM-TP
GAP
TwIST
Sparse-GMM
Sparse-GMM(G)
Conclusions
The success of compressive sensing of signals from a GMM highly depends on the quality of the
estimator of the unknown GMM. In this paper, we have developed a hierarchical Bayesian method
to simultaneously estimate the GMM and recover the signals, all based on using only incomplete
linear measurements and a Bayesian shrinkage prior for promoting sparsity of the Gaussian precision matrices. In addition, we have obtained theoretical results under the challenging assumption
that the underlying GMM is unknown and has to be estimated from measurements that contain only
incomplete information about the signals. Our results extend substantially from previous theoretical
results in [7] which assume the GMM is exactly known. The experimental results with both simulated and hardware-acquired measurements show the proposed method significantly outperforms
state-of-the-art methods.
Acknowledgement
The research reported here was funded in part by ARO, DARPA, DOE, NGA and ONR.
6
The results of the training videos containing general scenes can be found at the aforementioned website.
8
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4.2 Real measurements
We demonstrate the efficacy of
the proposed methods on the CS
of video, with the measurements
acquired by the actual hardware
of CACTI camera [6]. A letter is
placed on the blades of a chopper wheel that rotates at an angular velocity of 15 blades per second. The training data are obtained from the videos of a chopper wheel rotating at several orientations, positions and velocities. These training videos are
captured by a regular camcorder
at frame-rates that are different from the high-speed frame rate Figure 3: Reconstructed images 256 ? 256 ? T by differenachieved by CACTI reconstruc- t methods from the ?raw measurement? acquired from CACTI
tion. Other settings of the meth- with T = 14. The region in the red boxes are enlarged and
ods are the same as in the experi- shown at the right bottom part for better comparison.
ments on simulated data. The reconstruction results are shown in Figure 3, which shows that SparseGMM(G) generally yields sharper reconstructed frames with less ghost effects than other methods.
#1
0.8
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20
Results. From the results shown in
Figure 1, we observe that the proposed
methods, especially Sparse-GMM(G),
outperforms other methods with significant margins in all considered settings. The better performance of SparseGMM(G) over Sparse-GMM validates Figure 2: Plots of an example precision matrix (in magthe advantage of considering group s- nitude) learned by different GMM methods on the Face
parsity in the model. Figure 2 shows the dataset with m/n = 0.4. It is preferred to view the figure
an example precision matrix of one of K electronically. The magnitudes in each precision matrix
Gaussian components that are learned are scaled to the range of [0, 1].
by the methods of PLE, Sparse-GMM, and Sparse-GMM(G) on the face dataset. From this figure,
we can see that Sparse-GMM and Sparse-GMM(G) show much clearer groups sparsity than PLE,
demonstrating the benifits of using group sparsity constructed from the banding patterns.
20
Sparse-GMM
1
0.8
20
0.8
of a randomly-selected video containing traffic scenes6 , which are irrelevant to the NBA game, as
training data to learn a GMM for GMM-TP with 20 components, and we use it to initialize PLE,
Sparse-GMM, and Sparse-GMM(G). The same training data are used to learn the dictionaries for
KSVD-OMP.
PLE
Sparse-GMM
Sparse-GMM(G)
140
0
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4,865 | 5,404 | On the Relationship Between LFP & Spiking Data
David E. Carlson1 , Jana Schaich Borg2 , Kafui Dzirasa2 , and Lawrence Carin1
1
Department of Electrical and Computer Engineering
2
Department of Psychiatry and Behavioral Sciences
Duke University
Duham, NC 27701
{david.carlson, jana.borg, kafui.dzirasa, lcarin}@duke.edu
Abstract
One of the goals of neuroscience is to identify neural networks that correlate with
important behaviors, environments, or genotypes. This work proposes a strategy
for identifying neural networks characterized by time- and frequency-dependent
connectivity patterns, using convolutional dictionary learning that links spike-train
data to local field potentials (LFPs) across multiple areas of the brain. Analytical
contributions are: (i) modeling dynamic relationships between LFPs and spikes;
(ii) describing the relationships between spikes and LFPs, by analyzing the ability
to predict LFP data from one region based on spiking information from across the
brain; and (iii) development of a clustering methodology that allows inference
of similarities in neurons from multiple regions. Results are based on data sets in
which spike and LFP data are recorded simultaneously from up to 16 brain regions
in a mouse.
1
Introduction
One of the most fundamental challenges in neuroscience is the ?large-scale integration problem?:
how does distributed neural activity lead to precise, unified cognitive moments [1]. This paper seeks
to examine this challenge from the perspective of extracellular electrodes inserted into the brain. An
extracellular electrode inserted into the brain picks up two types of signals: (1) the local field potential (LFP), which represents local oscillations in frequencies below 200 Hz; and (2) single neuron
action potentials (also known as ?spikes?), which typically occur in frequencies of 0.5 kHz. LFPs
represent network activity summed over long distances, whereas action potentials represent the precise activity of cells near the tip of an electrode. Although action potentials are often treated as the
?currency? of information transfer in the brain, relationships between behaviors and LFP activity
can be equally precise, and sometimes even more precise, than those with the activity of individual
neurons [2, 3]. Further, LFP network disruptions are highly implicated in many forms of psychiatric
disease [4]. This has led to much interest in understanding the mechanisms of how LFPs and action
potentials interact to create specific types of behaviors. New multisite recording techniques that
allow simultaneous recordings from a large number of brain regions provide unprecedented opportunities to study these interactions. However, this type of multi-dimensional data poses significant
challenges that require new analysis techniques.
Three of the most challenging characteristics of multisite recordings are that: 1) the networks they
represent are dynamic in space and time, 2) subpopulations of neurons within a local area can have
different functions and may therefore relate to LFP oscillations in specific ways, and 3) different frequencies of LFP oscillations often relate to single neurons in specific ways [5]. Here new models are
proposed to examine the relationship between neurons and neural networks that accommodate these
characteristics. First, each LFP in a brain region is modeled as convolutions between a bounded-time
dictionary element and the observed spike trains. Critically, the convolutional factors are allowed
to be dynamic, by binning the LFP and spike time series, and modeling the dictionary element for
1
each bin of the time series. Next, a clustering model is proposed making each neuron?s dictionary
element a scaled version of an autoregressive template shared among all neurons in a cluster. This
allows one to identify sub-populations of neurons that have similar dynamics over their functional
connectivity to a brain region. Finally, we provide a strategy for exploring which frequency bands
characterize spike-to-LFP functional connectivity. We show, using two novel multi-region electrophysiology datasets from mice, how these models can be used to identify coordinated interactions
within and between different neuronal subsystems, defined jointly by the activity of single cells and
LFPs. These methods may lead to better understanding of the relationship between brain activity
and behavior, as well as the pathology underlying brain diseases.
2
Model
2.1 Data and notation
The data used here consists of multiple LFP and spike-train time series, measured simultaneously
from M regions of a mouse brain. Spike sorting is performed on the spiking data by a VB implementation of [6], from which J single units are assumed detected from across the multiple regions
(henceforth we refer to single units as ?neurons?); the number of observed neurons J depends on
the data considered, and is inferred as discussed in [6]. Since multiple microwires are inserted into
single brain regions in our experiments (described in [7]), we typically detect between 4-50 neurons
for each of the M regions in which the microwires are inserted (discussed further when presenting
results). The analysis objective is to examine the degree to which one may relate (predict) the LFP
data from one brain region using the J-neuron spiking data from all brain regions. This analysis allows the identification of multi-site neural networks through the examination of the degree to which
neurons in one region are predictive of LFPs in another.
Let x ? RT represent a time series of LFP data measured from a particular brain region. The T
samples are recorded on a regular grid, with temporal interval ?. The spike trains from J different neurons (after sorting) are represented by the set of vectors {y1 , . . . , yJ }, binned in the same
manner temporally as the LFP data. Each yj ? ZT+ is reflective of the number of times neuron
j ? {1, . . . , J} fired within each of the T time bins, where Z+ represents nonnegative integers.
In the proposed model LFP data x are represented as a superposition of signals associated with each
neuron yj , plus a residual that captures LFP signal unrelated to the spiking data. The contribution
to x from information in yj is assumed generated by the convolution of yj with a bounded-time
dictionary element dj (residing within the interval -L to L, with L T ). This model is related to
convolutional dictionary learning [8], where the observed (after spike sorting) signal yj represents
the signal we convolve the learned dictionary dj against.
We model dj as time evolving, motivated by the expectation that neuron j may contribute differently
to specified LFP data, based upon the latent state of the brain (which will be related to observed
animal activity). The time series x is binned into a set of B equal-size contiguous windows, where
x = vec([x1 , . . . , xB ]), and likewise y = vec([yj1 , . . . , yjB ]). The dictionary element for neuron
j is similarly binned as {dj1 , . . . , djB }, and the contribution of neuron j to xb is represented as a
convolution of djb and yjb . This bin size is a trade-off between how finely time is discretized and
the computational costs.
In the experiments, in one example the bins are chosen to be 30 seconds wide (novel-environment
data) and in the other 1 minute (sleep-cycle data), and these are principally chosen for computational
convenience (the second data set is nine times longer). Similar results were found with windows as
narrow as 10 second, or as wide as 2 minutes.
2.2 Modeling the LFP contribution of multiple neurons jointly
Given {y1 , . . . , yJ }, the LFP voltage at time window b is represented as
xb =
J
X
yjb ? djb + b
(1)
j=1
where ? represents the convolution operator. Let Dj = [dj1 , . . . , djB ] ? R(2L+1)?B represent
the sequence of dictionary elements used to represent the LFP data over the B windows, from the
perspective of neuron j. We impose the clustering prior
Dj = ?j Aj , Aj ? G, G ? DP(?, G0 )
2
(2)
where G is a draw from a Dirichlet process (DP) [9, 10], with scale parameter ? > 0 and base
probability measure G0 . Note that we cluster the shape of the dictionary elements, and each neuron
has its own scaling ? ? R. Concerning the base measure, we impose an autoregressive prior on the
temporal dynamics, and therefore G0 is defined by an AR(?, ?) process
ab = ?ab?1 + ?t , ?t ? N (0, ? ?1 I)
(3)
where I is the identity matrix. ThisP
AR prior is used to constitute the B columns of the DP ?atoms?
?
A?h = (a?h1 , . . . , a?hB ), with G = k=1 ?k ?A?k . The elements
of the vector ? = (?1 , ?2 , . . . ) are
Q
drawn from the ?stick-breaking? [9] process ?h = Vh i<h (1 ? Vi ) with Vh ? Beta(1, ?). We
place the prior Gamma(a? , b? ) on ?, and priors Uniform(0,1) and Gamma(a? , b? ) respectively on
? and ?. To complete the model, we place the prior N (0, ? ?1 I) on b , and ?j ? N (0, 1).
In the implementation, a truncated stick-breaking representation is employed for G, using K ?sticks?
(VK = 1), which simplifies the implementation and has been shown to be effective in practice [9] if
K is made large enough, and the size of K is inferred during the inference algorithm.
Special cases of this model are clear. For example, if the Aj are simply drawn i.i.d. from G0 , rather
than from the DP, each neuron is allowed to contribute its own unique dictionary shape to represent
xb , called a non-clustering model in the results. In [11] the authors considered a similar model, but
the time evolution of dj was not considered (each neuron was assumed to contribute in the same way
to represent the LFP, independent of time). Further, in [11] only a single neuron was considered, and
therefore no clustering was considered. A multi-neuron version of this model is inferred by setting
B = 1.
3
3.1
Inference
Mean-field Variational Inference
Letting ? = {z, ?, A1,...,K , V1,...,K , ?, ?, ?}, the full likelihood of the clustering model
p(x, ?)
=
B
Y
[p(xb |?)]
J
Y
[p(zj |?)p(?j )]
j=1
b=1
K
Y
[p(A?k |?, ?)p(Vk |?)] p(?, ?, ?)
(4)
k=1
The non-clustering model can be recovered by setting zj = ?j and the truncation level in the stickbreaking process K to J. The time-invariant model is recovered by setting the number of bins B
to 1, with or without clustering. The model of [11] is recovered by using a single bin and a single
neuron.
Many recent methods [12, 13] have been proposed to provide quick approximations to the Dirichlet
process mixture model. Critically, in these models the latent assignment variables are conditionally
independent when the DP parameters are given. However, in the proposed model this assumption
does not hold because the observation x is the superposition of the convolved draws from the Dirichlet process.
A factorized variational distribution q is proposed to approximate the posterior distribution, and
the non-clustering model arises as a special case of the clustering model. The inference to fit the
distribution q is based on Bayesian Hierarchical Clustering [13] and the VB Dirichlet Process SplitMerge method [12]. The proposed model does not fit in either of these frameworks, so a method
to learn K by merging clusters by adapting [12, 13] is presented in Section 3.1.1. The factorized
distribution q takes the form: "
#
Y
Y
Y
q(?) =
q(zj )
q(?jk ) q(?)q(?)q(?)
[q(A?k )q(Vk )]
(5)
j
k
k
Standard forms on these distributions are assumed, with q(zj ) = Categorical(rj ), q(?) =
? kB ), ??1
Gamma(a0? , b0? ), q(?) = N(0,1) (?
? , ???1 ), q(Ak ) = N (vec(Ak ); vec(?
ak1 , . . . , a
k ), ?k =
0
0
?k , and q(?) = Gamma(a? , b? ). To facilitate inference, the distribution on ?j is split into
?1
q(?jk ) = N (?jk , ?jk
), the variational distribution for ? on the j th spike train given that it is in
cluster k. The non-clustering model can be represented as a special case of the clustering model
where q(?jk ) = ?1 , and q(zj ) = ?j . As noted in [12], this factorized posterior has the property that
a q with K 0 clusters is nested in a representation of q for K clusters for K ? K 0 , so any number of
clusters up to K 0 is represented.
3
Variational algorithms find a q that minimizes the KL divergence from the true, intractable posterior
[14], finding a q that locally maximizes the evidence lower bound (ELBO) objective:
log p(x|?) ? L(q) = Eq [log p(x, z, ?, A?1,...,K , ?, ?, ?|?) ? log q(z, ?, A?1,...,K , ?, ?, ?)] (6)
To facilitate inference, approximations to p(y|?) are developed. Let Tb be the number of time
PTb
points in bin b, and define Rjib ? R(2L+1)?(2L+1) with entries Rjib,ik = T1b t=1
yjb,t yib,t+k?i ;
P
P
?j
? kb ), or
yjb,t is yj at time point t in window/bin b. Let xb = xb ? j 0 6=j yb ? ( k rjk ?jk a
th
the residual after all but the contribution from the j neuron have been removed, and define let
PTb
?j
j
?jb
? R2L+1 with entries ?ji
= T1b t=1
yjb,t xb,t+i for i ? {?L, . . . , L}. Both Rjb and ?jb
can be efficiently estimated with the FFT. For each time bin b, we can write: log p(x?j
b |yjb , djb ) =
PL
?j
?j T
? Tb
?
2
T
const ? 2 (xb,t ? `=?L yj,b,t+` dj,b,?` ) ' const ? 2 (djb Rjjb d ? 2(?jb ) djb )
P
P
0
0 ?
To define the key updates, let ykb
= j rjk ?jk yjb , and x?k
kb . ?kbb0 denotes
j 0 6=j yk ? a
b = xb ?
0
the block in ?k indexing the b and the b bins, which is efficiently calculated because ??1
k is a block
tri-diagonal matrix from the first-order autoregressive process, and explicit equations exist. Letting
?k = P rjk , then q(Vk ) is updated by are ak = 1 + N
? , bk = ?? + PK0
?0
N
j
k =k+1 Nk . For q(?jk ), the
P
?j
?1 P
T
? kb + ?kbb )), and ?jk = ?jk
? Tkb Rjb ?jb
parameters are updated ?jk = 1 + b trace(Rjb (?
akb a
.
ba
The clustering latent variables are updated sequentially by:
?X
?1
T
? kb a
? Tkb ))?2?jk (x?j
? kb ))]+Eq [q(?)]
[(?jk +?jk
)tr(Rjb (Tb ?kbb + a
log(rjk ) ? ?
b ) (yb Rjbb a
2
b
x?k
b
yb?k
and
can be used to calculate q(A?k ). The mean of the distribution q(Ak ) is evaluated
using the forward filtering-backward smoothing algorithm, and ??1
k is a block tridiagonal matrix,
enabling efficient computations. Further details on updating q(A?k ) are found in Section A of the
Supplemental Material. Approximating distributions q(?), q(?) and q(?) are standard [14, 15].
3.1.1 Merge steps
The model is initialized to K = J clusters and the algorithm first finds q for the non-clustering
model. This initialization is important because of the superposition measurement model. The algorithm proceeds to merge down to K 0 , where K 0 is a local mode of the VB algorithm. The procedure
is as follows: (i) Randomly choose two clusters k and k 0 to merge. (ii) Propose a new variational distribution q? with K ? 1 clusters. (iii) Calculate the change in the variational lower bound,
L(?
q ) ? L(q), and accept the merge if the variational lower bound increases. As in [12], intelligent
sampling of k and k 0 significantly improves performance. Here, we sample k and k 0 with weight
proportional to exp(?K(Ak , Ak0 ; c0 )), where K(?, ?; c0 ) is the radial basis function. In [13] all
pairwise clusterings were considered, but that is computationally infeasible in this problem. This
approach for merging clusters is similar to that developed in [12].
This algorithm requires efficient estimation of the difference in the lower bound. For a proposed k
and k 0 , a new variational distribution q? is proposed, with q?(zj = k) = q(zj = k) + q(zj = k 0 )
?
6=k0 ?
?k + N
?k0 , b0 + PK,k
?
0
and q?(zj = k 0 ) = 0, q?(?k ) = Beta(a0P+ P
N
k? =k+1 Nk ), q(?k ) = ?0 , and
q(Ak ) is calculated. Letting H(q) = ? j k rjk log rjk , the difference in the lower bound can
be calculated:
p(Ak |?, ?) p(?k ))
? H(?
q ) + H(?
p)
(7)
L(?
q ) ? L(q) = Eq? log p(y|A1,...,K , ?, ? )
q?(Ak ) q(?k )
p(Ak |?, ?)p(A0k |?, ?) p(?k )p(?k0 )
? Eq log p(y|A1,...,K , ?, ? )
+ H(q) ? H(p)
q(Ak )q(A0k )
q(?k )q(?k0 )
Explicit details on the calculations of these variables are found in Section A of the Supplementary
Material, and the block tridiagonal nature of ?k allows the complete calculation of this value in
?k + N
?k0 ) + L3 )). This is linear in the amount of data used in the model. The algorithm
O(BTb ((N
is stopped after 10 merges in a row are rejected.
3.2 Integrated Nested Laplacian Approximation for the Non-Clustering Model
The VB inference method assumes a separable posterior. In the non-clustering model, Integrated
Nested Laplacian Approximation (INLA) [16] was used to estimate of the joint posterior, without
4
Animal
1
2
3
4
5
6
Invariant
0.1394
0.1465
0.2251
0.0867
0.1238
0.0675
Non-Cluster
0.1968
0.2382
0.3050
0.1433
0.1867
0.1407
Clustering
0.2094
0.2340
0.3414
0.1434
0.1882
0.1351
Animal
7
8
9
10
11
Invariant
0.1385
0.0902
0.1597
0.0311
0.675
Non-Cluster
0.2567
0.3440
0.1881
0.0803
0.1064
Clustering
0.2442
0.3182
0.2362
0.0865
0.1161
Table 1: Mean held-out RFE of the multi-cell models predicting the Hippocampus LFP. ?Invariant? denotes the
time-invariant model, ?Non-cluster? and ?clustering? denote the dynamic model without and with clustering.
0.1
5 Min
15 Min
38 Min
Dynamic
0.02
0.01
0.05
0.5
Invar iant
Non-Clus ter
Clus ter ing
0.4
Hold-out RFE
Amplitude, a.u.
0.04
0.03
Joint Model Prediction in HP
Dictionary Element of a VTA Cell
Single Neuron Hold-out RFE
0.3
0
0.2
?0.05
0.1
0
0
0.01
0.02
0.03
Time-Invariant
0.04
?0.1
?0.5
0
Time, seconds
0.5
0
0
10
20
30
40
Experiment Time, Minutes
Figure 1: (Left) Mean single-cell holdout RFE predicting mouse 3?s Nucleus Accumbens LFP comparing the
dynamic and time-invariant model. Each point is a single neuron. (Middle) Convolutional dictionary for a
VTA cell predicting mouse 3?s Nucleus Accumbens LFP at 5 minutes, 15 minutes, and 38 minutes after the
experiment start. (Right) Hold-out RFE over experiment time with the time-invariant, non-clustering, and the
clustering model to predict mouse 3?s Hippocampus LFP.
assuming separability. Comparisons to INLA constitute an independent validation of VB, for inference in the non-clustering version of the model. The INLA inference procedure is detailed in
Supplemental Section B. INLA inference was found to be significantly slower than the VB approximation, so experimental results below are shown for VB. The INLA and VB predictive performance
were quantitatively similar for the non-clustering model, providing confidence in the VB results.
4
Experiments
4.1 Results on Mice Introduced to a Novel Environment
This data set is from a group of 12 mice consisting of male Clock-?19 (mouse numbers 7-12)
and male wild-type littermate controls (mouse numbers 1-6) (further described in [7]). For each
animal, 32-48 total microwires were implanted, with 6-16 wires in each of the Nucleus Accumbens,
Hippocampus (HP), Prelimbic Cortex (PrL), Thalamus, and the Ventral Tegmental Area (VTA).
LFPs were averaged over all electrodes in an area and filtered from 3-50Hz and sampled at 125
Hz. Neuronal activity was recorded using a Multi-Neuron Acquisition Processor (Plexon). 99-192
individual spike trains (single units) were detected per animal. In this dataset animals begin in their
home cage, and after 10 minutes are placed in a novel environment for 30 minutes. For analysis, this
40 minute data sequence was binned into 30 second chunks, giving 80 bins. For all experiments we
choose L such that the dictionary element covered 0.5 seconds before and after each spike event.
Cross-validation was performed using leave-one-out analysis over time bins, using the error metric
of reduction in fractional error (RFE), 1 ? ||xb ? x?b ||22 /||xb ||22 . Figure 1(left) shows the average
hold-out RFE for the time-invariant model and the dynamic model for single spike train predicting
mouse 3?s Nucleus Accumbens, showing that the dynamic model can give strong improvements
on the scale of a single cell (these results are typical). The dynamic model has a higher hold-out
RFE on 98.4% of detected cells across all animals and all regions, indicating that the dynamic
model generally outperforms the time-invariant model. A dynamic dictionary element from a VTA
cell predicting mouse 3?s Nucleus Accumbens is shown in Figure 1(middle). At the beginning
of the experiment, this cell is linked with a slow, high-amplitude oscillation. After the animal is
initially placed into a new environment (illustrated by the 15-minute data point), the amplitude of the
dictionary element drops close to zero. Once the animal becomes accustomed to its new environment
(illustrated by the 38-minute data point), the cell?s original periodic dictionary element begins to
appear again. This example shows how cells and LFPs clearly have time-evolving relationships.
The leave-one-out performance of the time-invariant, non-clustering, and clustering models predicting animal 3?s Hippocampus LFP with 182 neurons is shown in Figure 1(right). These results show
5
10
5
0
Accumbens HP
PrL Thalamus VTA
0
?0.02
10
20
30
Experiment Time, Minutes
Cluster?s Cell Locations
Number of Cells
40
0.02
40
6
4
2
0
Accumbens HP
Dictionary, Seconds
?0.02
Cluster Factor Evolution
?0.4
?0.2
0
0.2
0.4
Cluster Factor Evolution
PrL Thalamus VTA
0.05
?0.4
?0.2
0
0.2
0.4
Number of Cells
0
Dictionary, Seconds
0.02
10
20
30
Experiment Time, Minutes
Cluster?s Cell Locations
Number of Cells
Dictionary, Seconds
Cluster Factor Evolution
?0.4
?0.2
0
0.2
0.4
0
10
20
30
Experiment Time, Minutes
Cluster?s Cell Locations
40
?0.05
20
10
0
Accumbens HP
PrL Thalamus VTA
Figure 2: Example clusters predicting mouse 3?s Hippocampus LFP. The top part shows the convolutional factor throughout the duration of the experiment, and the bottom part shows the location of the cells in the cluster.
Some of the clusters are dynamic whereas others were consistent through the duration of the experiment.
Hippocampus Cells Predicting Thalamus LFP 25-35Hz
0.6
8
13
0.5
18
0.4
Cluster Contribution
Raw Energy
Residual
500
8
28
0.2
33
300
200
18
0.15
23
0.1
28
33
0.1
38
43
Frequency, Hz
0.3
RFE
23
0.2
13
400
Energy, a.u.
Frequency, Hz
600
RFE
Hipp ocampus Cells Predicting Thalamus LFP
10
20
30
Exp erimental Time, min
40
100
0
0
0.05
38
43
10
20
30
Experimental Time, min
40
10
20
30
Exp erimental Time, min
40
Figure 3: (Left) RFE as a function of time bin and frequency bin for all Hippocampus cells predicting the
Thalamus LFP. There is a change in the predictive properties around 10 minutes. (Middle) Total energy versus
the unexplained residual for the Hippocampus cells predicting the Thalamus LFP for the frequency band 25-35
Hz. (Right) RFE using only the cluster of cells shown in Figure 2(right).
that predictability changes over time, and indicate that there is a strong increase in LFP predictability
when the mouse is placed in the novel environment. Using dynamics improves the results dramatically, and the clustering hold-out results showed further improvements in hold-out performance.
The mean hold-out RFE results for the Hippocampus for 11 animals are shown in Table 1 (1 animal
was missing this region recording). Results for other regions are shown in Supplemental Tables 1,
2, 3, and 4, and show similar results.
In this dataset, there is little quantitative difference between the clustering and non-clustering models; however, the clustering result is much better for interpretation. One reason for this is that
spike-sorting procedures are notoriously imprecise, and often under- or over-cluster. A clustering
model with equivalent performance is evidence that many neurons have the same shapes and dynamics, and repeated dynamic patterns reduces concerns that dynamics are the result of failure to
distinguish distinct neurons. Similarly, clustering of neuron shapes in a single electrode could be
the result of over-clustering from the spike-sorting algorithm, but clustering across electrodes gives
strong evidence that truly different neurons are clustering together. Additionally, neural action potential shapes drift over time [6, 17], but since cells in a cluster come from different electrodes and
regions, this is strong evidence that the dynamics are not due to over-sorting drifting neurons.
Each cluster has both a dynamic shape result as well as well as a neural distribution over regions.
Example clustering shapes and histogram cell locations for clusters predicting mouse 3?s Thalamus
LFP are shown in Figure 2. The top part of this figure shows the base dictionary element evolution
over the duration of the experiment. Note that both the (left) and (middle) plots show a dynamic
effect around 10 minutes, and the cells primarily come from the Ventral Tegmental Area. The (right)
plot shows a fairly stable factor, and its cells are mostly in the Hippocampus region.
The ability to predict the LFP constitutes functional connectivity between a neuron and the neuronal
circuit around the electrode for the LFP [18]. Neural circuits have been shown to transfer information
through specific frequencies of oscillations, so it is of scientific interest to know the functional
connectivity of a group of neurons as a function of frequency [5]. Frequency relationships were
explored by filtering the LFP signal after the predicted signal has been removed, using a notch filter
at 1 Hz intervals with a 1 Hz bandwidth, and the RFE was calculated for each held-out time bin and
frequency bin.
All cells in the Thalamus were used to predict each frequency band in mouse 3?s Hippocampus LFP,
and this result is shown in Figure 3(left). This figure shows an increase in RFE of the 25-35 Hz band
after the animal has been moved to a new location. The RFE on the band from 25-35 Hz is shown
6
Region
Time-Invariant
Non-Clustering
Clustering
PrLCx
0.1055
0.1686
0.1749
MOFCCx
0.1304
0.1994
0.2029
NAcShell
0.0904
0.1599
0.1609
NAcCore
0.1076
0.1796
0.1814
Amyg
0.0883
0.1422
0.1390
Hipp
0.2091
0.2662
0.2798
V1
0.1366
0.1972
0.2020
VTA
0.1317
0.1907
0.1923
Region
Time-Invariant
Non-Clustering
Clustering
Subnigra
0.1309
0.1939
0.1950
Thal
0.1550
0.2188
0.2204
LHb
0.1240
0.1801
0.1813
DLS
0.1237
0.1973
0.2012
DMS
0.1518
0.2363
0.2378
M1
0.1350
0.2034
0.2080
OFC
0.1878
0.2695
0.2723
FrA
0.1164
0.1894
0.1912
Table 2: Mean held-out RFE of the animal going through sleep cycles in each region.
Mean Factors for Cell in V1
Mean Factors for Cell in NAcShell
0.04
0.03
0.1
0.02
0.02
0.05
0
?0.05
?0.1
V1
HP
MDThal
VTA
?0.15
?0.2
?0.5
0
Time, Seconds
0.5
0
?0.02
?0.04
V1
HP
MDThal
VTA
?0.06
?0.08
?0.5
0
Time, Seconds
0.5
Amplitude, a.u.
0.15
Amplitude, a.u.
Amplitude, a.u.
Mean Factors for Cell in HP
0.01
0
?0.01
?0.02
V1
HP
MDThal
VTA
?0.03
?0.04
?0.5
0
Time, Seconds
0.5
Figure 4: The predictive patterns of individual neurons predicting multiple regions. (Left) A Hippocampus
cell is the best single cell predictor of the V1 LFP (Middle) A V1 cell with a relationship only to the V1 LFP.
(Right) A Nucleus Accumbens Shell cell that is equivalent in predictive ability to the best V1 cell.
in Figure 3(middle), and shows that while the raw energy in this frequency band is much higher
after the move to the novel environment, the cells from the Hippocampus can explain much of the
additional energy in this band. In Figure 3(right), we show the same result using only the cluster in
Figure 2. Note that there is a change around 10 minutes that is due to both a slight change in the
convolutional dictionary and a change in the neural firing patterns.
4.2 Results on Sleep Data Set
The second data set was recorded from one mouse going through different sleep cycles over 6 hours.
64 microwires were implanted in 16 different regions of the brain, using the Prelimbix Cortex (PrL),
Medial Orbital Frontal Cortex (MOFCCx), the core and shell of the Nucleus Accumbens (NAc),
Basal Amygdala (Amy), Hippocampus (Hipp), V1, Ventral Tegmental Area (VTA), Substantia nigra (Subnigra), Medial Dorsal Thalamus (MDThal), Lateral Habenula (LHb), Dorsolateral Striatum (DLS), Dorsomedial Striatum (DMS), Motor Cortex (M1), Orbital Frontal Cortex (OFC), and
Frontal Association Cortex (FrA). LFPs were averaged over all electrodes in an area and filtered
from 3-50Hz and sampled at 125Hz, and L was set to 0.5 seconds. 163 total neurons (single units)
were detected using spike sorting, and the data were split into 360 1-minute time bins. The leaveone-out predictive performance was higher for the dynamic single cell model on 159 out of 163
neurons predicting the Hippocampus LFP. The mean hold-out RFEs for all recorded regions of the
brain are shown in Table 2 for all models, and the clustering model is the best performing model in
15 of the 16 regions.
Previously published work looked at the predictability of the V1 LFP signal from individual V1
neurons [11,18,19]. Our experiments find that the dictionary elements for all V1 cells (4 electrodes,
4 cells in this dataset) are time-invariant and match the single-cell time-invariant dictionary shape
of [11]. The dictionary elements for a single V1 cell predicting multiple regions are shown in Figure
4(middle; for simplicity, only a subset of brain regions recorded from are shown). This suggests that
the V1 cell has a connection to the V1 region, but no other brain region that was recorded from in
this model. However, cells in other brain regions showed functional connectivity to V1. The best
individual predictor is a cell in the Hippocampus shown in Figure 4(left). An additional example
cell is a cell in the Nucleus Accumbens shell that has the same RFE as the best V1 cell, and its shape
is shown in Figure 4(right).
Sleep states are typically defined by dynamic changes in functional connectivity across brain regions
as measured by EEG (LFPs recorded from the scalp) [20], but little is known about how single neurons contribute to, or interact with, these network changes. To get sleep covariates, each second of
data was scored into ?awake? or ?sleep? states using the methods in [21], and the sleep state was
averaged over the time bin. We defined a time bin to be a sleep state if ? 95% of the individual sec7
0
?0.02
?0.2
0
0.2
Dictionary Element, s
0.4
6
4
Su
rL
A
P
M
O
D
Fr
0
C
2
F
Numb er of Cells
P
H
rL
0.02
?0.04
?0.4
0.4
5
P
TA
yg
1
m
V
0
0.2
Dictionary Element, s
10
0
V
M
A
T
ha
FC
0
l
2
?0.2
Awake
Sleep
LS
?0.05
?0.4
0.4
0.04
bn
0.2
Numb er of Cells
0
Time, s
4
O
Number of Cells
?0.2
0
ra
?0.1
Awake
Sleep
ig
?0.05
0.05
Amplitude, a.u.
Amplitude, a.u.
Amplitude, a.u.
0
?0.4
Anti-Sleep Cluster
Pro-Sleep Cluster
Cluster predicting V1 Region
0.05
Figure 5: (Left) The cluster predicting the V1 region of the brain, matching known pattern for individual V1
cells [11, 18]. (Middle,Right) Clusters predicting the motor cortex that show positive (pro) and negative (anti)
relationships between amplitude and sleep.
Sleep-Increased Cluster RFE by Frequency
Sleep-Neutral Cluster RFE by Frequency
0.025
Awake
Sleep
0.01
0.03
0.1
0.02
0.05
0.005
10
20
30
Frequency, Hz
40
50
Awake
Sleep
Mean RFE
0.015
Sleep-Decreased Cluster RFE by Frequency
0.05
0.04
0.15
Mean RFE
Mean RFE
0.02
0
0
0.2
Awake
Sleep
0
0
0.01
10
20
30
Frequency, Hz
40
50
0
0
10
20
30
Frequency, Hz
40
50
Figure 6: Mean RFE when the animal is awake and when it is asleep. (Left) Cluster?s convolution factor is
stable, and shows only minor differences between sleep and awake prediction. (Middle and Right) Clusters
shown in Figure 5 (left and right), depicting varying patterns with the mouse?s sleep state
onds are scored as a sleep state, and the animal is awake if ? 5% of the individual seconds are scored
as a sleep state. In Figure 5(middle) we show a cluster that is most strongly positively correlated
with sleep (pro-sleep), and in Figure 5(right) we show a cluster that is most negatively correlated
with sleep (pro-awake). Both figures show the neuron locations as well as the mean waveform shape
during sleep and wake. In this case, the pro-sleep cluster is dominantly Hippocampus cells and the
anti-sleep cluster comes from many different regions. There may be concern that because these are
the maximally correlated clusters, that these results may be atypical. To address this concern, the
p-value for finding a cluster this strongly correlated has a p-value 4 ? 10?6 for Pearson correlation
with the Bonferroni correction for multiple tests. Furthermore, 4 of the 25 clusters detected showed
correlation above .4 between amplitude and sleep state, so this is not an isolated phenomena.
The RFE changes as both a function of frequency and sleep state for some clusters of neurons. Using
1Hz bandwidth frequency bins, in Figure 6 (middle and right) we show the mean RFE using only
the clusters in Figure 5 (middle and right). The cluster associated positively with sleeping shifts
its frequency peak and increases its ability to predict when the animal is sleeping. Likewise, the
sleep-decreased cluster performs worst at predicting when the animal is asleep. For comparison, in
Figure 6 (left) we include the frequency results for cluster with a stable dictionary element. The
total RFE is comparable and there is a not a dramatic shift in the peak frequency between the sleep
and awake states.
5
Conclusions
Novel models and methods are developed here to account for time-varying relationships between
neurons and LFPs. Within the context of our experiments, significantly improved predictive performance is realized when one accounts for temporal dynamics in the neuron-LFP interrelationship.
Further, the clustering model reveals which neurons have similar relationships to a specific brain region, and the frequencies that are predictable in the LFP change with known dynamics of the animal
state. In future work, these ideas can be incorporated with attempts to learn network structure, and
LFPs can be considered a common input when exploring networks of neurons [19, 22, 23]. Moreover, future experiments are being designed to place additional electrodes in a single brain region,
with the goal of detecting 100 neurons in a single brain region while recording LFPs in up to 20
regions. The methods proposed here will facilitate exploration of both the diversity of neurons and
the differences in functional connectivity on an individual neuron scale.
Acknowlegements The research reported here was funded in part by ARO, DARPA, DOE, NGA
and ONR. We thank the reviewers for their helpful comments.
8
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4,866 | 5,405 | A Synaptical Story of Persistent Activity with
Graded Lifetime in a Neural System
Yuanyuan Mi,
Luozheng Li
State Key Laboratory of Cognitive Neuroscience & Learning,
Beijing Normal University, Beijing 100875, China
[email protected], [email protected]
Dahui Wang
State Key Laboratory of Cognitive Neuroscience & Learning,
School of System Science, Beijing Normal University,Beijing 100875, China
[email protected]
Si Wu
State Key Laboratory of Cognitive Neuroscience & Learning,
IDG/McGovern Institute for Brain Research,
Beijing Normal University ,Beijing 100875, China
[email protected]
Abstract
Persistent activity refers to the phenomenon that cortical neurons keep firing even
after the stimulus triggering the initial neuronal responses is moved. Persistent
activity is widely believed to be the substrate for a neural system retaining a memory trace of the stimulus information. In a conventional view, persistent activity is
regarded as an attractor of the network dynamics, but it faces a challenge of how
to be closed properly. Here, in contrast to the view of attractor, we consider that
the stimulus information is encoded in a marginally unstable state of the network
which decays very slowly and exhibits persistent firing for a prolonged duration.
We propose a simple yet effective mechanism to achieve this goal, which utilizes
the property of short-term plasticity (STP) of neuronal synapses. STP has two
forms, short-term depression (STD) and short-term facilitation (STF), which have
opposite effects on retaining neuronal responses. We find that by properly combining STF and STD, a neural system can hold persistent activity of graded lifetime,
and that persistent activity fades away naturally without relying on an external
drive. The implications of these results on neural information representation are
discussed.
1
Introduction
Stimulus information is encoded in neuronal responses. Persistent activity refers to the phenomenon
that cortical neurons keep firing even after the stimulus triggering the initial neural responses is
removed [1, 2, 3]. It has been widely suggested that persistent activity is the substrate for a neural system to retain a memory trace of the stimulus information [4]. For instance, in the classical
delayed-response task where an animal needs to memorize the stimulus location for a given period of time before taking an action, it was found that neurons in the prefrontal cortex retained
high-frequency firing during this waiting period, indicating that persistent activity may serve as the
1
neural substrate of working memory [2]. Understanding the mechanism of how persistent activity is
generated in neural systems has been at the core of theoretical neuroscience for decades [5, 6, 7].
In a conventional view, persistent activity is regarded as an emergent property of network dynamics: neurons in a network are reciprocally connected with each other via excitatory synapses, which
form a positive feedback loop to maintain neural responses in the absence of an external drive; and
meanwhile a matched inhibition process suppresses otherwise explosive neural activities. Mathematically, this view is expressed as the dynamics of an attractor network, in which persistent activity
corresponds to a stationary state (i.e., an attractor) of the network. The notion of attractor dynamics
is appealing, which qualitatively describes a number of brain functions, but its detailed implementation in neural systems remains to be carefully evaluated.
A long-standing debate on the feasibility of attractor dynamics is on how to properly close the attractor states in a network: once a neural system is evolved into a self-sustained active state, it will
stay there forever until an external force pulls it out. Solutions including applying a strong global inhibitory input to shut-down all neurons simultaneously, or applying a strong global excitatory input
to excite all neurons and force them to fall into the refractory period simultaneously, were suggested [9], but none of them appears to be natural or feasible in all conditions. From the computational
point of view, it is also unnecessary for a neural system to hold a mathematically perfect attractor
state lasting forever. In reality, the brain only needs to hold the stimulus information for a finite
amount of time necessary for the task. For instance, in the delayed-response task, the animal only
needed to memorize the stimulus location for the waiting period [1].
To address the above issues, here we propose a novel mechanism to retain persistent activity in neural systems, which gives up the concept of prefect attractor, but rather consider that a neural system
is in a marginally unstable state which decays very slowly and exhibits persistent firing for a prolonged period. The proposed mechanism utilizes a general feature of neuronal interaction, i.e., the
short-term plasticity (STP) of synapses [10, 11]. STP has two forms: short-term depression (STD)
and short-term facilitation (STF). The former is due to depletion of neurotransmitters after neural
firing, and the latter is due to elevation of calcium level after neural firing which increases the release probability of neurotransmitters. STD and STP have opposite effects on retaining prolonged
neuronal responses: the former weakens neuronal interaction and hence tends to suppress neuronal
activities; whereas, the latter strengthens neuronal interaction and tends to enhance neuronal activities. Interestingly, we find that the interplay between the two processes endows a neural system
with the capacity of holding persistent activity with desirable properties, including: 1) the lifetime
of persistent activity can be arbitrarily long depending on the parameters; and 2) persistent activity
fades away naturally in a network without relying on an external force. The implications of these
results on neural information representation are discussed.
2
The Model
Without loss of generality, we consider a homogeneous network in which neurons are randomly and
sparsely connected with each other with a small probability p. The dynamics of a single neuron is
described by an integrate-and-fire process, which is given by
?
dvi
= ?(vi ? VL ) + Rm hi ,
dt
for i = 1 . . . N,
(1)
where vi is the membrane potential of the ith neuron and ? the membrane time constant. VL is the
resting potential. hi is the synaptic current and Rm the membrane resistance. A neuron fires when
its potential exceeds the threshold, i.e., vi > Vth , and after that vi is reset to be VL . N the number
of neurons.
The dynamics of the synaptic current is given by
1 ?
dhi
sp
?
ext
= ?hi +
?s
Jij u+
?(t ? text
i ),
j xj ?(t ? tj ) + I
dt
Np j
(2)
where ?s is the synaptic time constant, which is about 2 ? 5ms. Jij is the absolute synaptic efficacy
from neurons j to i. Jij = J0 if there is a connection from the neurons j to i, and Jij = 0 otherwise.
tsp
j denotes the spiking moment of the neuron j. All neurons in the network receive an external input
2
in the form of Poisson spike train. I ext represents the external input strength and text
the moment
i
of the Poisson spike train the neuron i receives.
The variables uj and xj measure, respectively, the STF and STD effects on the synapses of the jth
neuron, whose dynamics are given by [12, 13]
duj
dt
dxj
?d
dt
?f
sp
= ?uj + ?f U (1 ? u?
j )?(t ? tj ),
(3)
sp
?
= 1 ? xj ? ?d u+
j xj ?(t ? tj ),
(4)
?
where uj is the release probability of neurotransmitters, with u+
j and uj denoting, respectively,
the values of uj just after and just before the arrival of a spike. ?f is the time constant of STF.
?
U controls the increment of uj produced by a spike. Upon the arrival of a spike, u+
j = uj +
+
?
U (1 ? u?
j ). xj represents the fraction of available neurotransmitters, with xj and xj denoting,
respectively, the values of xj just after and just before the arrival of a spike. ?d is the recover time
?
+ ?
of neurotransmitters. Upon the arrival of a spike, x+
j = xj ? uj xj . The time constants ?f and ?d
are typically in the time order of hundreds to thousands of milliseconds, much larger than ? and ?s ,
that is, STP is a slow process compared to neural firing.
2.1
Mean-field approximation
As to be confirmed by simulation, neuronal firings in the state of persistent activity are irregular and
largely independent to each other. Therefore, we can assume that the responses of individual neurons
are statistically equivalent in the state of persistent activity. Under this mean-field approximation,
the dynamics of a single neuron, and so does the mean activity of the network, can be written as [7]
dh
(5)
= ?h + J0 uxR + I,
dt
du
?f
= ?u + ?f U (1 ? u)R,
(6)
dt
dx
?d
= 1 ? x ? ?d uxR,
(7)
dt
where the state variables are the same for all neurons. R is the firing rate of a neuron, which is also
the mean activity of the neuron ensemble. I = I ext ? denotes the external input with ? the rate of
the Poisson spike train. The exact relationship between the firing rate R and the synaptic input h is
difficult to obtain. Here, we assume it to be of the form,
?s
R = max(?h, 0),
(8)
with ? a positive constant.
3
The Mechanism
By using the mean-field model, we first elucidate the working mechanism underlying the generation
of persistent activity of finite lifetime. Later we carry out simulation to confirm the theoretical
analysis.
3.1
How to generate persistent activity of finite lifetime
For the illustration purpose, we only study the dynamics of the firing rate R and assume that the
variables u and x reach to their steady values instantly. This approximation is in general inaccurate,
since u and x are slow variables compared to R. Nevertheless, it gives us insight into understanding
the network dynamics.
By setting du/dt = 0 and dx/dt = 0 in Eqs.(6,7) and substituting them into Eqs.(5,8), we get that,
for I = 0 and R ? 0,
?s
dR
J0 ??f U R2
= ?R +
? F (R).
dt
1 + ?f U R + ?d ?f U R2
3
(9)
J 0<Jc
J >Jc
0
J =Jc
F(R)
0
R*
0
R
Figure 1: The steady states of the network, i.e., the solutions of Eq.(9), have three forms depending
on the parameter values. The three lines correspond to the different neuronal connection strenghths,
which are J0 = 4, 4.38, 5, respectively. The other parameters are: ?s = 5ms, ?d = 100ms, ?f =
700ms, ? = 1, U = 0.05 and Jc = 4.38.
(
)
?
Define a critical connection strength Jc ? 1 + 2 ?d /(?f U ) /?, which is the point the network
dynamics experiences saddle-node bifurcation (see Figure 1). Depending on the parameters, the
steady states of the network have three forms
? When J0 < Jc , F (R) = 0 has only one solution at R = 0, i.e., the network is only stable
at the silent state;
? When J0 > Jc , F (R) = 0 has three solutions, and the network can be stable at the silent
state and an active state;
? When J0 = Jc , F (R) = 0 has two solutions, one is the stable silent state, and the other is
a neutral stable state, referred to as R? .
The interesting behavior occurs at J0 = Jc? , i.e., J0 is slightly smaller than the critical connection
strength Jc . In this case, the network is only stable at the silent state. However, since near to the
state R? , F (R) is very close to zero (and so does |dR/dt|), the decay of the network activity is very
slow in this region (Figure 2A). Suppose that the network is initially at a state R > R? , under the
network dynamics, the system will take a considerable amount of time to pass through the state R?
before reaching to silence. This is manifested by that the decay of the network activity exhibits a
long plateau around R? before dropping to silence rapidly (Figure 2B). Thus, persistent activity of
finite lifetime is achieved.
The lifetime of persistent activity, which is dominated by the time of the network state passing
through the point R? , is calculated to be (see Appendix A),
T ??
2?s
F (R? )F ?? (R? )
,
(10)
where F ?? (R? ) = d2 F (R)/d2 R|R? . By varying the STP effects, such as ?d and ?f , the value of
F (R? )F ?? (R? ) is changed, and the lifetime of persistent activity can be adjusted.
3.2
Persistent activity of graded lifetime
We formally analyze the condition for the network holding persistent activity of finite lifetime.
Inspired by the result in the proceeding section, we focus on the parameter regime of J0 = Jc , i.e.,
the situation when the network has the stable silent state and a neutral stable active state.
Denote (R? , u? , x? ) to be the neutral stable state of the network at J0 = Jc . Linearizing the network
dynamics at this point, we obtain
(
)
(
)
R ? R?
R ? R?
d
u ? u?
? A u ? u?
,
(11)
dt
x ? x?
x ? x?
4
A
B
0
R(Hz)
F(R)
R*
R*
0
R
0
2
4
6
8
t(s)
Figure 2: Persistent activity of finite lifetime. Obtained by solving Eqs.(5-8). (A) When J0 = Jc? ,
the function F (R), and so does dR/dt, is very close to zero at the state R? . Around this point, the
network activity decays very slowly. The inset shows the fine structure in the vicinity of R? . (B) An
external input (indicated by the red bar) triggers the network response. After removing the external
input, the network activity first decays quickly, and then experiences a long plateau before dropping
to silence rapidly. The parameters are: ?s = 5ms, ?d = 10ms, ?f = 800ms, ? = 1, U = 0.5,
I = 10, Jc = 1.316 and J0 = 1.315.
where A is the Jacobian matrix (see Appendix B).
It turns out that the matrix A always has one eigenvector with vanishing eigenvalue, a property due to
that (R? , u? , x? ) is the neutral stable state of the network dynamics. As demonstrated in Sec.3.1, by
choosing J0 = Jc? , we expect that the network state will decay very slowly along the eigenvector of
vanishing eigenvalue, which we call the decay-direction. To ensure this always happens, it requires
that the real parts of the other two eigenvalues of A are negative, so that any perturbation of the
network state away from the decay-direction will be pulled back; otherwise, the network state may
approach to silence rapidly via other routes avoiding the state (R? , u? , x? ). This idea is illustrated
in Fig.3.
The condition for the real parts of the other two eigenvalues of A being smaller than zero is calculated to be (see Appendix B):
?
2
1
U
1
1
1
?
+
+
?
> 0.
(12)
?f ?d
?d ?f ?d
?d ?s 1 + ?f U
?f ?s
?d
This inequality together with J0 =
of finite lifetime.
Jc?
form the condition for the network holding persistent activity
R* R
3-D view
Decay-direction
t
Figure 3: Illustration of the slow-decaying process of the network activity. The network dynamics
experiences a long plateau before dropping to silence quickly. The inset presents a 3-D view of the
local dynamics in the plateau region, where the network state is attracted to the decay-direction to
ensure slow-decaying.
By solving the network dynamics Eqs.(5-8), we calculate how the lifetime of persistent activity
changes with the STP effect. Fig.4A presents the results of fixing U and J0 and varying ?d and
5
?f , We see that below the critical line J0 = Jc , which is the region for J0 > Jc , the network has
prefect attractor states never decaying; and above the critical line, the network has only the stable
silent state. Close to the critical line, the network activity decays slowly and displays persistent
activity of finite lifetime. Fig.4B shows a case that when the STF strength (?f ) is fixed, the lifetime
of persistent activity decreases with the STD strength (?d ). This is understandable, since STD tends
to suppress neuronal responses. Fig.4C shows a case that when ?d is fixed, the lifetime of persistent
activity increases with ?f , due to that STF enhances neuronal responses. These results demonstrate
that by regulating the effects of STF and STD, the lifetime of persistent activity can be adjusted.
A
Decay time (s)
B
Decay time (s)
C
attractor
10
5
0
0
0.5
1
1.5
1
1.5
?d (s)
10
5
0
0
0.5
?f (s)
Figure 4: (A). The lifetimes of the network states with respect to ?f and ?d when U and J0 are fixed.
We use an external input to trigger a strong response of the network and then remove the input. The
lifetime of a network state is measured from the offset of the external input to the moment when the
network returns to silence. The white line corresponds to the condition of J0 = Jc , below which
the network has attractors lasting forever; and above which, the lifetime of a network state gradually
decreases (coded by colour). (B) When ?f = 1250ms is fixed, the lifetime of persistent activity
decreases with ?d (the vertical dashed line in A). (C) When ?d = 260ms is fixed, the lifetime of
persistent activity increases with ?f (the horizontal dashed line in A). The other parameters are:
?s = 5ms, ? = 1, U = 0.05 and J0 = 5.
4
Simulation Results
We carry out simulation with the spiking neuron network model given by Eqs.(1-4) to further confirm
the above theoretical analysis. A homogenous network with N = 1000 neurons is used, and in the
network neurons are randomly and sparsely connected with each other with a probability p = 0.1.
At the state of persistent activity, neurons fire irregularly (the mean value of Coefficient of Variation
is 1.29)and largely independent to each other(the mean correlation of all spike train pairs is 0.30)
with each other (Fig.5A). Fig.5 present the examples of the network holding persistent activity with
varied lifetimes, through different combinations of STF and STD satisfying the condition Eq.(12).
5
Conclusions
In the present study, we have proposed a simple yet effective mechanism to generate persistent
activity of graded lifetime in a neural system. The proposed mechanism utilizes the property of STP,
a general feature of neuronal synapses, and that STF and STD have opposite effects on retaining
neuronal responses. We find that with properly combined STF and STD, a neural system can be in a
marginally unstable state which decays very slowly and exhibits persistent firing for a finite lifetime.
This persistent activity fades away naturally without relying on an external force, and hence avoids
the difficulty of closing an active state faced by the conventional attractor networks.
STP has been widely observed in the cortex and displays large diversity in different regions [14, 15, 16]. Compared to static ones, dynamical synapses with STP greatly enriches the response
patterns and dynamical behaviors of neural networks, which endows neural systems with information processing capacities which are otherwise difficult to implement using purely static synapses.
The research on the computational roles of STP is receiving increasing attention in the field [12]. In
6
A
C
D
B
E
Figure 5: The simulation results of the spiking neural network. (A) A raster plot of the responses of
50 example neurons randomly chosen from the network. The external input is applied for the first
0.5 second. The persistent activity lasts about 1100ms. The parameters are: ?f = 800ms, ?d =
500ms, U = 0.5, J0 = 28.6. (B) The firing rate of the network for the case (A). (C) An example
of persistent activity of negligible lifetime. The parameters are:?f = 800ms, ?d = 1800ms, U =
0.5, J0 = 28.6. (D) An example of persistent activity of around 400ms lifetime. The parameters
are:?f = 600ms, ?d = 500ms, U = 0.5, J0 = 28.6. (E) An example of the network holding an
attractor lasting forever. The parameters are: ?f = 800ms, ?d = 490ms, U = 0.5, J0 = 28.6.
terms of information presentation, a number of appealing functions contributed by STP were proposed. For instances, Mongillo et al. proposed an economical way of using the facilitated synapses
due to STF to realize working memory in the prefrontal cortex without recruiting neural firing [8];
Pfister et al. suggested that STP enables a neuron to estimate the membrane potential information
of the pre-synaptic neuron based on the spike train it receives [17]. Torres et al. found that STD
induces instability of attractor states in a network, which could be useful for memory searching [18];
Fung et al. found that STD enables a continuous attractor network to have a slow-decaying state in
the time order of STD, which could serve for passive sensory memory [19]. Here, our study reveals
that through combining STF and STD properly, a neural system can hold stimulus information for
an arbitrary time, serving for different computational purposes. In particular, STF tends to increase
the lifetime of persistent activity; whereas, STD tends to decrease the lifetime of persistent activity.
This property may justify the diverse distribution of STF and STD in different cortical regions. For
instances, in the prefrontal cortex where the stimulus information often needs to be held for a long
time in order to realize higher cognitive functions, such as working memory, STF is found to be
dominating; whereas, in the sensory cortex where the stimulus information will be forwarded to
higher cortical regions shortly, STD is found to be dominating. Furthermore, our findings suggest
that a neural system may actively regulate the combination of STF and STD, e.g., by applying appropriate neural modulators [10], so that it can hold the stimulus information for a flexible amount
of time depending on the actual computational requirement. Further experimental and theoretical
studies are needed to clarify these interesting issues.
6
Acknowledgments
This work is supported by grants from National Key Basic Research Program of China
(NO.2014CB846101), and National Foundation of Natural Science of China (No.11305112, Y.Y.M.;
No.31261160495, S.W.; No.31271169,D.H.W.), and the Fundamental Research Funds for the central Universities (No.31221003, S.W.), and SRFDP (No.20130003110022, S.W), and Natural Science Foundation of Jiangsu Province BK20130282.
7
Appendix A: The lifetime of persistent activity
Consider the network dynamics Eq.(9). When J0 = Jc , the network has a stable silent state (R = 0)
and an unstable active state, referred to as R? (Fig.1). We consider that J0 = Jc? . In this case,
F (R? ) is slightly smaller than zero (Fig.2A). Starting from a state R > R? , the network will take
a considerable amount of time to cross the point R? , since dR/dt is very small in this region, and
the network exhibits persistent activity for a considerable amount of time. We estimate the time
consuming for the network crossing the point R? .
According to Eq.(9), we have
?
?
?T
?R+
dt
=
?
R?
0
=
?
=
?
?s
dR
F (R)
?R+
?
?
R?
F (R? )
[
2?s
F (R? )F ?? (R? )
2?s
F (R? )F ?? (R? )
?s dR
,
+ (R ? R? )2 F ?? (R? )/2
?
R+
? R?
?
R?
? R?
]
arctg ?
? arctg ?
,
F (R? )/F ?? (R? )
F (R? )/F ?? (R? )
G(R? ),
(13)
?
?
where R+
and R?
denote, respectively, the points slightly larger or smaller than R? , F ? (R? ) =
dF (R)/dR|R? , and F ?? (R? ) = dF ? (R)/dR|R? . To get the above result, we used the second-order
Taylor expansion of F (R) at R? , and the condition F ? (R? ) = 0.
In the limit of F (R? ) ? 0, the value of G(R? ) is bounded. Thus, the lifetime of persistent activity
is in the order of
2?s
T ??
.
(14)
F (R? )F ?? (R? )
Appendix B: The condition for the network holding persistent activity of finite
lifetime
Denote (R? , u? , x? ) to be the neutral stable state of the network when J0 = Jc , which is calculated
to be (by solving Eqs.(5-8)),
?
?f U R?
1 + ?f U R?
?
R? = 1/?f ?d U , u? =
(15)
,
x
=
.
1 + ?f U R ?
1 + ?f U R? + ?f ?d U R? 2
Linearizing the network dynamics at this point, we obtain Eq.(12), in which the Jacobian matrix A
is given by
)
(
(J0 u? x? ? 1)/?s ,
J0 x? R? /?s ,
J0 u? R? /?s
?
?
U (1 ? u ),
?1/?f ? U R ,
0
.
(16)
A=
?u? x? ,
?x? R? ,
?1/?d ? u? R?
The eigenvalues of the Jacobian matrix satisfy the equality |A ? ?I| = 0. Utilizing Eqs.(15), this
equality becomes
?(?2 + b? + c?) = 0,
(17)
where the coefficients b and c are given by
b
=
c
=
1
1
+
+ u? R ? + U R ? ,
?d
?f
?
2
1
U
1
1
1
?
+
+
?
.
?f ?d
?d ?f ?d
?d ?s 1 + ?f U
?f ?s
(18)
(19)
?d
From Eq.(17), we see that the matrix A has three eigenvalues. One eigenvalue, referred to as ?1 , is
always zero. The other two eigenvalues satisfy that ?2 + ?3 = ?b and ?2 ?3 = c. Since b > 0, the
condition for the real parts of ?2 and ?3 being negative is c > 0.
8
References
[1] J. Fuster and G. Alexander. Neuron activity related to short-term memory. Science 173, 652654 (1971).
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monkeys dorsolateral prefrontal cortex. J. Neurophysiol. 61, 331-349 (1989).
[3] R. Romo, C. D. Brody, A. Hernandez. Lemus L. Neuronal correlates of parametric working
memory in the prefrontal cortex. Nature 399, 470-473 (1999).
[4] D.J. Amit. Modelling brain function. New York: Cambridge University Press. (1989)
[5] S. Amari. Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern.
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[6] X.J. Wang. Synaptic basis of cortical persistent activity: the importance of NMDA receptors
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[10] H. Markram and M. Tsodyks. Redistribution of synaptic efficacy between neocortical pyramidal neurons. Nature. 382(6594): 807-810(1996).
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[12] M. Tsodyks and S. Wu. Short-Term Synaptic Plasticity. Scholarpedia, 8(10): 3153(2013).
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[14] H. Markram, Y. Wang and M. Tsodyks. Differential signaling via the same axon of neocortical pyramidal neurons. Proceedings of the National Academy of Sciences. 95(9): 53235328(1998).
[15] J. S. Dittman, A. C. Kreitzer and W. G. Regehr. Interplay between facilitation, depression, and
residual calcium at three presynaptic terminals. J. Neurosci. 20: 1374-1385(2000).
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9
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4,867 | 5,406 | Sparse PCA via Covariance Thresholding
Andrea Montanari
Electrical Engineering and Statistics
Stanford University
[email protected]
Yash Deshpande
Electrical Engineering
Stanford University
[email protected]
Abstract
In sparse principal component analysis we are given noisy observations of a lowrank matrix of dimension n ? p and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here that the principal components
v1 , . . . , vr have at most k1 , ? ? ? , kq non-zero entries respectively, and study the
high-dimensional regime in which p is of the same order as n.
In an influential paper, Johnstone and Lu [JL04] introduced a simple algorithm
that estimates the support of the principal vectors v1 , . . . , vr by the largest entries
in the diagonal of the empirical covariance.
This method can be shown to succeed
p
with highpprobability if kq ? C1 n/ log p, and to fail with high probability if
kq ? C2 n/ log p for two constants 0 < C1 , C2 < ?. Despite a considerable
amount of work over the last ten years, no practical algorithm exists with provably
better support recovery guarantees.
Here we analyze a covariance thresholding algorithm that was recently proposed
by Krauthgamer, Nadler and Vilenchik [KNV13]. We confirm empirical evidence
presented by these authors and rigorously
prove that the algorithm succeeds with
?
high probability for k of order n. Recent conditional lower bounds [BR13]
suggest that it might be impossible to do significantly better.
The key technical component of our analysis develops new bounds on the norm of
kernel random matrices, in regimes that were not considered before.
1
Introduction
In the spiked covariance model proposed by [JL04], we are given data x1 , x2 , . . . , xn with xi ? Rp
of the form1 :
xi =
r
X
p
?q uq,i vq + zi ,
(1)
q=1
Here v1 , . . . , vr ? Rp is a set of orthonormal vectors, that we want to estimate, while uq,i ?
N(0, 1) and zi ? N(0, Ip ) are independent and identically distributed. The quantity ?q ? R>0
quantifies the signal-to-noise ratio. We are interested in the high-dimensional limit n, p ? ? with
limn?? p/n = ? ? (0, ?). In the rest of this introduction we will refer to the rank one case, in
order to simplify the exposition, and drop the subscript q = {1, 2, . . . , r}. Our results and proofs
hold for general bounded rank.
The standard method
Pn of principal component analysis involves computing the sample covariance
matrix G = n?1 i=1 xi xT
i and estimates v = v1 by its principal eigenvector vPC (G). It is a
well-known fact that, in the high dimensional asymptotic p/n ? ? > 0, this yields an inconsistent
1
Throughout the paper, we follow the convention of denoting scalars by lowercase, vectors by lowercase
boldface, and matrices by uppercase boldface letters.
1
estimate [JL09]. Namely kvPC ? vk2 6? 0 in the high-dimensional asymptotic limit, unless ? ? 0
(i.e. p/n ? 0). Even worse, Baik, Ben-Arous
and P?ech?e [BBAP05] and Paul [Pau07] demonstrate
?
a phase transition phenomenon: if ? < ?
? the estimate is asymptotically orthogonal to the signal
hvPC , vi ? 0. On the other hand, for ? > ?, hvPC , vi remains strictly positive as n, p ? ?. This
phase transition phenomenon has attracted considerable attention recently within random matrix
theory [FP07, CDMF09, BGN11, KY13].
These inconsistency results motivated several efforts to exploit additional structural information on
the signal v. In two influential papers, Johnstone and Lu [JL04, JL09] considered the case of a
signal v that is sparse in a suitable basis, e.g. in the wavelet domain. Without loss of generality, we
will assume here that v is sparse in the canonical basis e1 , . . . ep . In a nutshell, [JL09] proposes the
following:
1. Order the diagonal entries of the Gram matrix Gi(1),i(1) ? Gi(2),i(2) ? ? ? ? ? Gi(p),i(p) ,
and let J ? {i(1), i(2), . . . , i(k)} be the set of indices corresponding to the k largest
entries.
2. Set to zero all the entries Gi,j of G unless i, j ? J, and estimate v with the principal
eigenvector of the resulting matrix.
Johnstone and Lu formalized the sparsity assumption by requiring that v belongs to a weak `q -ball
with q ? (0, 1). Instead, here we consider a strict
? sparsity constraint where v has exactly k non-zero
entries, with magnitudes bounded below by ?/ k for some constant ? > 0. The case of ? = 1 was
studied by Amini and Wainwright in [AW09].
Within this model, it was proved that diagonalp
thresholding successfully recovers the support of
v provided v is sparse enough, namely k ? C n/ log p with C = C(?, ?) a constant [AW09].
(Throughout the paper we denote by C constants that can change from point to point.) This result is
a striking improvement over vanilla PCA. While the latter requires a number of samples scaling as
the number of parameters2 n & p, sparse PCA via diagonal thresholding achieves the same objective
with a number of samples scaling as the number of non-zero parameters, n & k 2 log p.
At the same time, this result is not as optimistic as might have been expected. By searching exhaustively over all possible supports of size k (a method that has complexity of order pk ) the correct
support can be identified with high probability as soon as n & k log p. On the other hand, no method
can succeed for much smaller n, because of information theoretic obstructions [AW09].
Over the last ten years, a significant effort has been devoted to developing practical algorithms
that outperform diagonal thresholding, see e.g. [MWA05, ZHT06, dEGJL07, dBG08, WTH09]. In
particular, d?Aspremont et al. [dEGJL07] developed a promising M-estimator based on a semidefinite programming (SDP) relaxation. Amini and Wainwright [AW09] carried out an analysis of this
method and proved that, if (i) k ? C(?) n/ log p, and (ii) if the SDP solution has rank one, then the
SDP relaxation provides a consistent estimator of the support of v.
At first sight, this appears as a satisfactory solution of the original problem. No procedure can
estimate the support of v from less than k log p samples, and the SDP relaxation succeeds in doing it
from ?at most? a constant factor more samples. This picture was upset by a recent, remarkable result
by Krauthgamer, Nadler and Vilenchik [KNV13]
? who showed that the rank-one condition assumed
by Amini and Wainwright does not hold for n . k . (n/ log p). This result is consistent with
recent work of Berthet and Rigollet [BR13]
demonstrating that sparse PCA cannot be performed in
?
polynomial time in the regime k & n, under a certain computational complexity conjecture for
the so-called planted clique problem.
In summary, the problem of support recovery in sparse PCA demonstrates a fascinating interplay
between computational and statistical barriers.
From a statistical perspective, and disregarding computational considerations, the support of v
can be estimated consistently if and only if k . n/ log p. This can be done, for instance,
by exhaustive search over all the kp possible supports of v. (See [VL12, CMW+ 13] for a
minimax analysis.)
2
Throughout the introduction, we write f (n) & g(n) as a shorthand of ?f (n) ? C g(n) for a some constant
C = C(?, ?)?. Further C denotes a constant that may change from point to point.
2
From a computational perspective, the problem appears to be much more difficult. There is rigorous evidence
? [BR13, MW13] that no polynomial algorithm can reconstruct the support
unless k . n. On the positive p
side, a very simple algorithm (Johnstone and Lu?s diagonal
thresholding) succeeds for k . n/ log p.
Of course, several elements are still missing in this emerging picture. In the present paper we address
one of them, providing an answer to the following question:
Is there a polynomial time algorithm
to solve the sparse PCA
p that is guaranteed
?
problem with high probability for n/ log p . k . n?
We answer this question positively by analyzing a covariance thresholding algorithm that proceeds,
briefly, as follows. (A precise, general definition, with some technical changes is given in the next
section.)
1. Form
? the Gram matrix G and set to zero all its entries that are in modulus smaller than
? / n, for ? a suitably chosen constant.
b1 of this thresholded matrix.
2. Compute the principal eigenvector v
b1 .
3. Denote by B ? {1, . . . , p} be the set of indices corresponding to the k largest entries of v
4. Estimate the support of v by ?cleaning? the set B. (Briefly, v is estimated by thresholding
bB obtained by zeroing the entries outside B.)
Gb
vB with v
Such a covariance thresholding approach was proposed in [KNV13], and is in turn related to earlier
work by Bickel and Levina [BL08]. The formulation discussed in the next section presents some
technical differences that have been introduced to simplify the analysis. Notice that, to simplify
proofs, we assume k to be known: This issue is discussed in the next two sections.
The rest of the paper is organized as follows. In the next section we provide a detailed description of
the algorithm and state our main results. Our theoretical results assume full knowledge of problem
parameters for ease of proof. In light of this, in Section 3 we discuss a practical implementation
of the same idea that does not require prior knowledge of problem parameters, and is entirely datadriven. We also illustrate the method through simulations. The complete proofs are available in the
accompanying supplement, in Sections 1, 2 and 3 respectively.
2
Algorithm and main result
For notational convenience, we shall assume hereafter that 2n sample vectors are given (instead of
n): {xi }1?i?2n . These are distributed according to the model (1). The number of spikes r will be
treated as a known parameter in the problem.
We will make the following assumptions:
A1 The number of spikes r and the signal strengths ?1 , . . . , ?r are fixed as n, p ? ?.
Further ?1 > ?2 > . . . ?r are all distinct.
A2 Let Qq and kq denote the support of vq and its size respectively. We let Q = ?q Qqpand
P
k = q kq throughout. Then the non-zero entries of the spikes satisfy |vq,i | ? ?/ kq
for all i ? Qq for some ? > 0. Further, for any q, q 0 we assume |vq,i /vq0 ,i | ? ? for every
i ? Qq ? Qq0 , for some constant ? > 1.
As before, we are interested in the high-dimensional limit of n, p ? ? with p/n ? ?. A more
detailed description of the covariance thresholding algorithm for the general model (1) is given in
Algorithm 1. We describe the basic intuition for the simpler rank-one case (omitting the subscript
q ? {1, 2, . . . , r}), while stating results in generality.
We start by splitting the data into two halves: (xi )1?i?n and (xi )n<i?2n and compute the respective
sample covariance matrices G and G0 respectively. As we will see, the matrix G is used to obtain
a good estimate for the spike v. This estimate, along with the (independent) second part G0 , is then
used to construct a consistent estimator for the supports of v.
3
Let us focus on the first phase of the algorithm, which aims to obtain a good estimate of v. We
b = G ? I. For ? > ??, the principal eigenvector of G, and hence of ?
b is
first compute ?
?
b
positively correlated with v, i.e. limn?? hb
v1 (?), vi > 0. However, for ? < ?, the noise
b dominates and the two vectors become asymptotically orthogonal, i.e. for instance
component in ?
b vi = 0. In order to reduce the noise level, we exploit the sparsity of the spike v.
limn?? hb
v1 (?),
Denoting by X ? Rn?p the matrix with rows x1 , . . . xn , by Z ? Rn?p the matrix with rows z1 ,
. . . zn , and letting u = (u1 , u2 , . . . , un ), the model (1) can be rewritten as
p
X = ? u vT + Z .
(2)
?
Hence, letting ? 0 ? ?kuk2 /n ? ?, and w ? ?ZT u/n
b = ? 0 vvT + v wT + w vT + 1 ZT Z ? Ip , .
?
n
(3)
For a moment, let us neglect the cross terms (vwT + wvT ). The ?signal? component
? 0 vvT is
?
2
sparse with
? k entries of magnitude ?/k, which (in the regime of interest k = n/C)
? is equivalent
to ?
C?/ n. The ?noise? component ZT Z/n ? Ip is dense with entries of order 1/ n. Assuming
k/ n a small enough constant,?it should be possible to remove most of the noise by thresholding
the entries at level of order 1/ n. For technical reasons, we use the soft thresholding function
? : R ? R?0 ? R, ?(z; ? ) = sgn(z)(|z| ? ? )+ . We will omit the second argument wherever it is
clear from context. Classical denoising theory [DJ94, Joh02] provides upper bounds the estimation
error of such a procedure. Note however that these results fall short of our goal. Classical theory
measures estimation error by (element-wise) `p norm, while here we are interested in the resulting
principal eigenvector. This would require bounding, for instance, the error in operator norm.
?
?
Since the soft thresholding function ?(z; ? / n) is affine when z ? / n, we would expect that
the following decomposition holds approximately (for instance, in operator norm):
1 T
0
T
b
?(?) ? ? ? vv + ?
Z Z ? Ip .
(4)
n
The main technical challenge now is to control the operator norm of the perturbation ?(ZT Z/n?Ip ).
It is easy to see that ?(ZT Z/n ? Ip ) has entries of variance ?(? )/n, for ?(? ) ? 0 as ? ? ?. If
entries were independent with mild decay, this would imply ?with high probability?
? 1 ZT Z
. C?(? ),
(5)
n
2
for some constant C. Further, the first component in the decomposition (4) is still approximately
rank one with norm of the order of ? 0 ? ?. Consequently, with standard linear algebra results on
the perturbation of eigenspaces [DK70], we obtain an error bound kb
v ? vk . ?(? )/C 0 ?
Our first theorem formalizes this intuition and provides a bound on the estimation error in the prinb
cipal components of ?(?).
bq denote
Theorem 1. Under the spiked covariance model Eq. (1) satisfying Assumption A1, let v
b using threshold ? . For every ?, (?q )r ? (0, ?), integer r and every
the q th eigenvector of ?(?)
q=1
? > 0 there exist constants, ? = ? (?, ?, (?q )rq=1 , r, ?) and 0 < c? = c? (?, ?, (?q )rq=1 , r, ?) < ?
P
P
?
such that, if q kq = q |supp(vq )| ? c? n), then
n
o
?
P min(kb
vq ? vq k , kb
vq + vq k) ? ? ?q ? {1, . . . , r} ? 1 ? 4 .
(6)
n
Random matrices of the type ?(ZT Z/n ? Ip ) are called inner-product kernel random matrices and
have attracted recent interest within probability theory [EK10a, EK10b, CS12]. In this literature, the
asymptotic eigenvalue distribution of a matrix f (ZT Z/n) is the object of study. Here f : R ? R
is a kernel function and is applied entry-wise to the matrix ZT Z/n, with Z a matrix as above.
Unfortunately, these results cannot be applied to our problem for the following reasons:
? The results [EK10a, EK10b] are perturbative and provide conditions under which the spectrum of f (ZT Z/n) is close to a rescaling of the spectrum of (ZT Z/n) (with rescaling
4
Algorithm 1 Covariance Thresholding
1: Input: Data (xi )1?i?2n , parameters r, (kq )q?r ? N, ?, ?, ? ? R?0 ;
Pn
P2n
0
T
2: Compute the Gram matrices G ? i=1 xi xT
i /n , G ?
i=n+1 xi xi /n;
b = G ? Ip (resp. ?
b 0 = G0 ? Ip );
3: Compute ?
b by soft-thresholding the entries of ?:
b
4: Compute the matrix ?(?)
?
?
?
b
b
? / n,
?
??ij ? ?n if ?ij ?
?
b ij = 0
b ij < ? /?n,
?(?)
if ?? / n < ?
?
??
b ij ? ?? /?n,
b ij + ??
if ?
n
b
5: Let (b
vq )q?r be the first r eigenvectors
of ?(p
?);
6: Define sq ? Rp by sq,i = v
bq,i I( vbq,i ? ?/2 kq );
b = {i ? [p] : ? q s.t. |(?
b 0 sq )i | ? ?}.
7: Output: Q
factors depending on the Taylor expansion of f close to 0). We are interested instead in a
non-perturbative regime in which the spectrum of f (ZT Z/n) is very different from the one
of (ZT Z/n) (and the Taylor expansion is trivial).
? [CS12] consider n-dependent kernels, but their results are asymptotic and concern the weak
limit of the empirical spectral distribution of f (ZT Z/n). This does not yield an upper
bound on the spectral norm3 of f (ZT Z/n).
Our approach to prove Theorem 1 follows instead the so-called ?-net method: we develop high
probability bounds on the maximum Rayleigh quotient:
X h?
zi , ?
zj i ?
T
?
max hy, ?(Z Z/n)yi = max
;?
yi yj ,
n
n
y?S p?1
y?S p?1
i,j
Here S p?1 = {y ? Rp : kyk = 1} is the unit sphere and ?
zi denote the columns of Z. Since
?(ZT Z/n) is not Lipschitz continuous in the underlying Gaussian variables Z, concentration does
not follow immediately from Gaussian isoperimetry. We have to develop more careful (non-uniform)
bounds on the gradient of ?(ZT Z/n) and show that they imply concentration as required.
b is a reasonable estimate of the spike v in `2 distance (up to
While Theorem 1 guarantees that v
a sign flip), it does not provide a consistent estimator of its support. This brings us to the second
b is not even expected to be sparse, it is easy to see that the largest
phase of our algorithm. Although v
b should have significant overlap with supp(v). Steps 6, 7 and 8 of the algorithm exploit
entries of v
b of the support of the spike v. Our second theorem
this property to construct a consistent estimator Q
guarantees that this estimator is indeed consistent.
Theorem 2. Consider the spiked covariance model of Eq. (1) satisfying Assumptions A1, A2. For
any ?, (?q )q?r ? (0, ?), ?, ? P
> 0 and integer r, there ?
exist constants c? , ?, ? dependent on
?, (?q )q?r , ?, ?, r, such that, if q kq = |supp(vq )| ? c? n, the Covariance Thresholding algorithm of Table 1 recovers the joint supports of vq with high probability.
Explicitly, there exists a constant C > 0 such that
n
o
b = ?q supp(vq ) ? 1 ? C .
P Q
n4
(7)
Given the results above, it is useful to pause for a few remarks.
Remark 2.1. We focus on a consistent estimation of the joint supports ?q supp(vq ) of the spikes.
In the rank-one case, this obviously corresponds to the standard support recovery. Once this is
accomplished, estimating the individual supports poses no additional difficulty: indeed, since | ?q
?
b yields estimates for vq
supp(vq ))| = O( n) an extra step with n fresh samples xi restricted to Q
3
Note that [CS12] also provide a finite n bound for the spectral norm of f (ZT Z/n) via the moment method,
but this bound diverges with n and does not give a result of the type of Eq. (5).
5
p
with `2 error of order k/n. This implies consistent estimation of supp(vq ) when vq have entries
bounded below as in Assumption A2.
Remark 2.2. Recovering the signed supports Qq,+ = {i ? [p] : vq,i > 0} and Qq,? = {i ?
[p] : vq,i < 0} is possible using the same technique as recovering the supports supp(vq ) above, and
poses no additional difficulty.
p
Remark 2.3. Assumption A2 requires |vq,i | ? ?/ kq for all i ? Qq . This is a standard requirement
in the support recovery literature [Wai09, MB06]. The second part of assumption A2 guarantees
that when the supports of two spikes overlap, their entries are roughly of the same order. This is
necessary for our proof technique to go through. Avoiding such an assumption altogether remains
an open question.
Our covariance thresholding algorithm assumes knowledge of the correct support sizes kq . Notice
that the same assumption is made in earlier theoretical, e.g. in the analysis of SDP-based reconstruction by Amini and Wainwright [AW09]. While this assumption is not realistic in applications,
it helps to focus our exposition
on the most challenging aspects of the problem. Further, a ballpark
P
estimate of kq (indeed q kq ) is actually sufficient, with which we use the following steps in lieu of
Steps 7, 8 of Algorithm 1.
7: Define s0q ? Rp by
s0q,i =
?
if |b
vq,i | > ?/(2 k0 )
otherwise.
vbq,i
0
(8)
8: Return
b = ?q {i ? [p] : |(?
b 0 s0 )i | ? ?} .
Q
q
(9)
P
The next theorem shows that this procedure is effective even if k0 overestimates q kq by an order
of magnitude. Its proof is deferred to Section 2.
Theorem 3. Consider the spiked covariance model of Eq.
? (0, ?),Plet constants
P (1). For any ?, ? ?
c? , ?, ? be given as in Theorem 2. Further assume k = q |supp(vq )| ? c? n, and q k ? k0 ?
P
20 q kq . Then, the Covariance Thresholding algorithm of Table 1, with the definitions in Eqs. (8)
and (9), recovers the joint supports of vq successfully, i.e.
b = ?q supp(vq ) ? 1 ? C .
(10)
P Q
n4
3
Practical aspects and empirical results
Specializing to the rank one case, Theorems 1 and 2 show that Covariance Thresholding succeeds
with high probability for a number of samples n & k 2 , while Diagonal Thresholding requires n &
k 2 log p. The reader might wonder whether eliminating the log p factor has any practical relevance
or is a purely conceptual improvement. Figure 1 presents simulations on synthetic data under the
strictly sparse model, and the Covariance Thresholding algorithm of Table 1, used in the proof of
Theorem 2. The objective is to check whether the log p factor has an impact at moderate p. We
compare this with Diagonal Thresholding.
?
We plot the empirical success probability as a function of k/ n for several values of p, with p = n.
The empirical success probability was computed by using 100 independent instances of the problem.
A few observations are of interest: (i) Covariance Thresholding appears to have a significantly
larger success probability in the ?difficult? regime where Diagonal Thresholding starts to fail; (ii)
The curves for Diagonal
Thresholding appear to decrease monotonically with p indicating that k
?
proportional to n is not the right scaling for this algorithm (as is known from theory); (iii) In
contrast, the curves for Covariance Thresholding become
steeper for larger p, and, in particular,
?
the success
? probability increases with p for k ? 1.1 n. This indicates a sharp threshold for k =
const ? n, as suggested by our theory.
In terms of practical applicability, our algorithm in Table 1 has the shortcomings of requiring knowledge of problem parameters ?q , r, kq . Furthermore, the thresholds ?, ? suggested by theory need not
6
0.6
0.4
0.2
p = 625
p = 1250
p = 2500
p = 5000
1
0.8
0.6
0.4
0.2
0
0.5
1
?
k/ n
1.5
2
0.5
1
?
k/ n
1.5
2
Fraction of support recovered
0.8
Fraction of support recovered
Fraction of support recovered
p = 625
p = 1250
p = 2500
p = 5000
1
p = 625
p = 1250
p = 2500
p = 5000
1
0.8
0.6
0.4
0.2
0
0.5
1
?
k/ n
1.5
2
Figure 1: The support recovery phase transitions for Diagonal Thresholding (left) and Covariance
Thresholding (center) and the data-driven version of Section 3 (right). For Covariance Threshold?
ing, the fraction of support recovered correctly increases monotonically with p, as long as k ? c n
with c ? 1.1. Further, it appears to converge to one throughout this region. For Diagonal Thresholding,
the fraction of support recovered correctly decreases monotonically with p for all k of order
?
n. This confirms that Covariance Thresholding
(with or without knowledge of the support size k)
?
succeeds with high probability for k ? c n, while Diagonal Thresholding requires a significantly
sparser principal component.
be optimal. We next describe a principled approach to estimating (where possible) the parameters of
interest and running the algorithm in a purely data-dependent manner. Assume the following model,
for i ? [n]
Xp
?q uq,i vq + ?zi ,
xi = ? +
q
p
where ? ? R is a fixed mean vector, ui have mean 0 and variance 1, and zi have mean 0 and covariance Ip . Note that our focus in this section is not on rigorous analysis, but instead to demonstrate
a principled approach to applying covariance thresholding in practice. We proceed as follows:
P
b = ni=1 xi /n be the empirical mean estimate for ?. Further letting
Estimating ?, ?: We let ?
P
X = X ? 1b
?T we see that pn ? ( q kq )n ? pn entries of X are mean 0 and variance
? 2 . We let ?
b = MAD(X)/? where MAD( ? ) denotes the median absolute deviation of
the entries of the matrix in the argument, and ? is a constant scale factor. Guided by the
Gaussian case, we take ? = ??1 (3/4) ? 0.6745.
Choosing ? : Although in the statement of the theorem, our choice of ? depends on the SNR
?/? 2 , we believe this is an artifact of our analysis. Indeed it is reasonable to threshold
?at the noise level?, as follows. The noise component of entry i, j of the sample covariance (ignoring lower order terms) is given by ? 2 hzi , zj i/n. By the central limit theo? d
4
rem, hzi , zj i/ n ? N(0, 1). Consequently, ? 2 hz
? i , zj i/n ? N(0, ? /n), and we need to
choose the (rescaled) threshold proportional to ? 4 = ? 2 . Using previous estimates, we
let ? = ? 0 ? ?
b2 for a constant ? 0 . In simulations, a choice 3 . ? 0 . 4 appears to work well.
b = XT X/n ? ? 2 Ip and soft threshold it to get ?(?)
b using ? as above.
Estimating r: We define ?
b has r eigenvalues that are separated
Our proof of Theorem 1 relies on the fact that ?(?)
from the bulk of the spectrum4 . Hence, we estimate r using rb: the number of eigenvalues
b
separated from the bulk in ?(?).
b Our theoretical analysis indicates that
bq denote the q th eigenvector of ?(?).
Estimating vq : Let v
bq is expected to be close to vq . In order to denoise v
bq , we assume v
bq ? (1 ? ?)vq + ?q ,
v
where ?q is additive random noise. We then threshold vq ?at the noise level? to recover a better estimate of vq . To do this, we estimate the standard deviation of the
?noise? ? by ?
c? = MAD(vq )/?. Here we set ?again guided by the Gaussian heuristic?
? ? 0.6745. Since vq is sparse, this procedure returns a good estimate for the size of the
bq : set
noise deviation. We let ?H (b
vq ) denote the vector obtained by hard thresholding v
4
The support of the bulk spectrum can be computed numerically from the results of [CS12].
7
bq,i if |b
bq? = ?(b
(?H (b
vq ))i = v
vq,i | ? ? 0 ?c
vq )/ k?(b
vq )k and
?q and 0 otherwise. We then let v
bq? as our estimate for vq .
return v
Note that ?while different in several respects? this empirical approach shares the same philosophy
of the algorithm in Table 1. On the other hand, the data-driven algorithm presented in this section is
less straightforward to analyze, a task that we defer to future work.
Figure 1 also shows results of a support recovery experiment using the ?data-driven? version of
this section.
Covariance thresholding in this form also appears to work for supports of size k ?
?
const n. Figure 2 shows the performance of vanilla PCA, Diagonal Thresholding and Covariance
Thresholding on the ?Three Peak? example of Johnstone and Lu [JL04]. This signal is sparse in
the wavelet domain and the simulations employ the data-driven version of covariance thresholding.
A similar experiment with the ?box? example of Johnstone and Lu is provided in the supplement.
These experiments demonstrate that, while for large values of n both Diagonal Thresholding and
Covariance Thresholding perform well, the latter appears superior for smaller values of n.
PCA
DT
CT
0.3
0.1
0.1
0.2
5 ? 10?2
n = 1024
5 ? 10?2
0.1
0
0
0
?5 ? 10?2
0
1,000
2,000
3,000
4,000
0
0.1
0.1
5 ? 10?2
n = 1625
5 ? 10?2
0
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
0.1
5 ? 10?2
0
?5 ? 10?2
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
0.1
0.1
0.1
5 ? 10?2
n = 2580
5 ? 10?2
5 ? 10?2
0
0
?5 ? 10?2
0
1,000
2,000
3,000
0
4,000
0.1
5 ? 10?2
n = 4096
0
1,000
2,000
3,000
4,000
0.1
0.1
5 ? 10?2
5 ? 10?2
0
0
0
0
1,000
2,000
3,000
4,000
0
1,000
2,000
3,000
4,000
Figure 2: The results of Simple PCA, Diagonal Thresholding and Covariance Thresholding (respectively) for the ?Three Peak? example of Johnstone and Lu [JL09] (see Figure 1 of the paper). The
signal is sparse in the ?Symmlet 8? basis. We use ? = 1.4, p = 4096, and the rows correspond to
sample sizes n = 1024, 1625, 2580, 4096 respectively. Parameters for Covariance Thresholding are
chosen as in Section 3, with ? 0 = 4.5. Parameters for Diagonal Thresholding are from [JL09]. On
each curve, we superpose the clean signal (dotted).
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4,868 | 5,407 | Low Rank Approximation Lower Bounds in
Row-Update Streams
David P. Woodruff
IBM Research Almaden
[email protected]
Abstract
We study low-rank approximation in the streaming model in which the rows of
an n ? d matrix A are presented one at a time in an arbitrary order. At the end
of the stream, the streaming algorithm should output a k ? d matrix R so that
kA ? AR? Rk2F ? (1 + )kA ? Ak k2F , where Ak is the best rank-k approximation
to A. A deterministic streaming algorithm of Liberty (KDD, 2013), with an improved analysis of Ghashami and Phillips (SODA, 2014), provides such a streaming algorithm using O(dk/) words of space. A natural question is if smaller
space is possible. We give an almost matching lower bound of ?(dk/) bits of
space, even for randomized algorithms which succeed only with constant probability. Our lower bound matches the upper bound of Ghashami and Phillips up to
the word size, improving on a simple ?(dk) space lower bound.
1
Introduction
In the last decade many algorithms for numerical linear algebra problems have been proposed, often
providing substantial gains over more traditional algorithms based on the singular value decomposition (SVD). Much of this work was influenced by the seminal work of Frieze, Kannan, and Vempala
[8]. These include algorithms for matrix product, low rank approximation, regression, and many
other problems. These algorithms are typically approximate and succeed with high probability.
Moreover, they also generally only require one or a small number of passes over the data.
When the algorithm only makes a single pass over the data and uses a small amount of memory,
it is typically referred to as a streaming algorithm. The memory restriction is especially important
for large-scale data sets, e.g., matrices whose elements arrive online and/or are too large to fit in
main memory. These elements may be in the form of an entry or entire row seen at a time; we
refer to the former as the entry-update model and the latter as the row-update model. The rowupdate model often makes sense when the rows correspond to individual entities. Typically one is
interested in designing robust streaming algorithms which do not need to assume a particular order
of the arriving elements for their correctness. Indeed, if data is collected online, such an assumption
may be unrealistic.
Muthukrishnan asked the question of determining the memory required of data stream algorithms
for numerical linear algebra problems, including best rank-k approximation, matrix product, eigenvalues, determinants, and inverses [18]. This question was posed again by Sarl?os [21]. A number
of exciting streaming algorithms now exist for matrix problems. Sarl?os [21] gave 2-pass algorithms
for matrix product, low rank approximation, and regression, which were sharpened by Clarkson and
Woodruff [5], who also proved lower bounds in the entry-update model for a number of these problems. See also work by Andoni and Nguyen for estimating eigenvalues in a stream [2], and work in
[1, 4, 6] which implicitly provides algorithms for approximate matrix product.
In this work we focus on the low rank approximation problem. In this problem we are given an
n ? d matrix A and would like to compute a matrix B of rank at most k for which kA ? BkF ?
1
(1 + )kA ? Ak kF . Here, for a matrix A, kAkF denotes its Frobenius norm
Ak is the best rank-k approximation to A in this norm given by the SVD.
qP
n
i=1
Pd
j=1
A2i,j and
Clarkson and Woodruff [5] show in the entry-update model, one can compute a factorization B =
L ? U ? R with L ? Rn?k , U ? Rk?k , and R ? Rk?d , with a streaming algorithm using O(k?2 (n +
d/2 ) log(nd)) bits of space. They also show a lower bound of ?(k?1 (n + d) log(nd)) bits of
space. One limitation of these bounds is that they hold only when the algorithm is required to output
a factorization L ? U ? R. In many cases n d, and using memory that grows linearly with n (as
the above lower bounds show is unavoidable) is prohibitive. As observed in previous work [9, 16],
in downstream applications we are often only interested in an approximation to the top k principal
components, i.e., the matrix R above, and so the lower bounds of Clarkson and Woodruff can be
too restrictive. For example, in PCA the goal is to compute the most important directions in the row
space of A.
By reanalyzing an algorithm of Liberty [16], Ghashami and Phillips [9] were able to overcome this
restriction in the row-update model, showing that Liberty?s algorithm is a streaming algorithm which
finds a k ? d matrix R for which kA ? AR? RkF ? (1 + )kA ? Ak kF using only O(dk/) words of
space. Here R? is the Moore-Penrose pseudoinverse of R and R? R denotes the projection onto the
row space of R. Importantly, this space bound no longer depends on n. Moreover, their algorithm
is deterministic and achieves relative error. We note that Liberty?s algorithm itself is similar in spirit
to earlier work on incremental PCA [3, 10, 11, 15, 19], but that work missed the idea of using a
Misra-Gries heavy hitters subroutine [17] which is used to bound the additive error (which was then
improved to relative error by Ghashami and Phillips). It also seems possible to obtain a streaming
algorithm using O(dk(log n)/) words of space, using the coreset approach in an earlier paper by
Feldman et al. [7].
This work is motivated by the following questions: Is the O(dk/) space bound tight or can one
achieve an even smaller amount of space? What if one also allows randomization?
In this work we answer the above questions. Our main theorem is the following.
Theorem 1. Any, possibly randomized, streaming algorithm in the row-update model which outputs
a k ? d matrix R and guarantees that kA ? AR? Rk2F ? (1 + )kA ? Ak k2F with probability at least
2/3, must use ?(kd/) bits of space.
Up to a factor of the word size (which is typically O(log(nd)) bits), our main theorem shows that the
algorithm of Liberty is optimal. It also shows that allowing for randomization and a small probability
of error does not significantly help in reducing the memory required. We note that a simple argument
gives an ?(kd) bit lower bound, see Lemma 2 below, which intuitively follows from the fact that
if A itself is rank-k, then R needs to have the same rowspace of A, and specifying a random kdimensional subspace of Rd requires ?(kd) bits. Hence, the main interest here is improving upon
this lower bound to ?(kd/) bits of space. This extra 1/ factor is significant for small values of ,
e.g., if one wants approximations as close to machine precision as possible with a given amount of
memory.
The only other lower bounds for streaming algorithms for low rank approximation that we know of
are due to Clarkson and Woodruff [5]. As in their work, we use the Index problem in communication
complexity to establish our bounds, which is a communication game between two players Alice and
Bob, holding a string x ? {0, 1}r and an index i ? [r] =: {1, 2, . . . , r}, respectively. In this
game Alice sends a single message to Bob who should output xi with constant probability. It is
known (see, e.g., [13]) that this problem requires Alice?s message to be ?(r) bits long. If Alg is a
streaming algorithm for low rank approximation, and Alice can create a matrix Ax while Bob can
create a matrix Bi (depending on their respective inputs x and i), then if from the output of Alg
on the concatenated matrix [Ax ; Bi ] Bob can output xi with constant probability, then the memory
required of Alg is ?(r) bits, since Alice?s message is the state of Alg after running it on Ax .
The main technical challenges are thus in showing how to choose Ax and Bi , as well as showing
how the output of Alg on [Ax ; Bi ] can be used to solve Index. This is where our work departs
significantly from that of Clarkson and Woodruff [5]. Indeed, a major challenge is that in Theorem
1, we only require the output to be the matrix R, whereas in Clarkson and Woodruff?s work from
the output one can reconstruct AR? R. This causes technical complications, since there is much less
information in the output of the algorithm to use to solve the communication game.
2
The intuition behind the proof of Theorem 1 is that given a 2 ? d matrix A = [1, x; 1, 0d ], where
x is a random unit vector, then if P = R? R is a sufficiently good projection matrix for the low
rank approximation problem on A, then the second row of AP actually reveals a lot of information
about x. This may be counterintuitive at first, since one may think that [1, 0d ; 1, 0d ] is a perfectly
good low rank approximation. However, it turns out that [1, x/2; 1, x/2] is a much better low rank
approximation in Frobenius norm, and even this is not optimal. Therefore, Bob, who has [1, 0d ]
together with the output P , can compute the second row of AP , which necessarily reveals a lot of
information about x (e.g., if AP ? [1, x/2; 1, x/2], its second row would reveal a lot of information
about x), and therefore one could hope to embed an instance of the Index problem into x. Most of
the technical work is about reducing the general problem to this 2 ? d primitive problem.
2
Main Theorem
This section is devoted to proving Theorem 1. We start with a simple lemma showing an ?(kd)
lower bound, which we will refer to. The proof of this lemma is in the full version.
Lemma 2. Any streaming algorithm which, for every input A, with constant probability (over its
internal randomness) succeeds in outputting a matrix R for which kA ? AR? RkF ? (1 + )kA ?
Ak kF must use ?(kd) bits of space.
Returning to the proof of Theorem 1, let c > 0 be a small constant to be determined. We consider
the following two player problem between Alice and Bob: Alice has a ck/ ? d matrix A which
can be written as a block matrix [I, R], where I is the ck/ ? ck/ identity matrix, and R is a
ck/ ? (d ? ck/) matrix in which the entries are in {?1/(d ? ck/)1/2 , +1/(d ? ck/)1/2 }. Here
[I, R] means we append the columns of I to the left of the columns of R. Bob is given a set of k
standard unit vectors ei1 , . . . , eik , for distinct i1 , . . . , ik ? [ck/] = {1, 2, . . . , ck/}. Here we need
c/ > 1, but we can assume is less than a sufficiently small constant, as otherwise we would just
need to prove an ?(kd) lower bound, which is established by Lemma 2.
Let B be the matrix [A; ei1 , . . . , eik ] obtained by stacking A on top of the vectors ei1 , . . . , eik .
The goal is for Bob to output a rank-k projection matrix P ? Rd?d for which kB ? BP kF ?
(1 + )kB ? Bk kF .
Denote this problem by f . We will show the randomized 1-way communication complexity of this
1?way
problem R1/4
(f ), in which Alice sends a single message to Bob and Bob fails with probability
at most 1/4, is ?(kd/) bits. More precisely, let ? be the following product distribution on Alice
and Bob?s inputs: the entries of R are chosen independently and uniformly at random in {?1/(d ?
ck/)1/2 , +1/(d ? ck/)1/2 }, while {i1 , . . . , ik } is a uniformly random set among all sets of k
1?way
1?way
distinct indices in [ck/]. We will show that D?,1/4
(f ) = ?(kd/), where D?,1/4
(f ) denotes
the minimum communication cost over all deterministic 1-way (from Alice to Bob) protocols which
fail with probability at most 1/4 when the inputs are distributed according to ?. By Yao?s minimax
1?way
1?way
principle (see, e.g., [14]), R1/4
(f ) ? D?,1/4
(f ).
1?way
We use the following two-player problem Index in order to lower bound D?,1/4
(f ). In this probr
lem Alice is given a string x ? {0, 1} , while Bob is given an index i ? [r]. Alice sends a single
message to Bob, who needs to output xi with probability at least 2/3. Again by Yao?s minimax prin1?way
1?way
ciple, we have that R1/3
(Index) ? D?,1/3
(Index), where ? is the distribution for which x and
i are chosen independently and uniformly at random from their respective domains. The following
is well-known.
1?way
Fact 3. [13] D?,1/3
(Index) = ?(r).
1?way
Theorem 4. For c a small enough positive constant, and d ? k/, we have D?,1/4
(f ) = ?(dk/).
Proof. We will reduce from the Index problem with r = (ck/)(d ? ck/). Alice, given her string x
to Index, creates the ck/ ? d matrix A = [I, R] as follows. The matrix I is the ck/ ? ck/ identity
matrix, while the matrix R is a ck/?(d?ck/) matrix with entries in {?1/(d?ck/)1/2 , +1/(d?
ck/)1/2 }. For an arbitrary bijection between the coordinates of x and the entries of R, Alice sets a
3
ck/"
B2
ck/"
B1
k
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
1
0
d
0
0
0
0
1
0
0
ck/"
RS
R
Alice
RT
0
Bob
S
R
T
given entry in R to ?1/(d ? ck/)1/2 if the corresponding coordinate of x is 0, otherwise Alice sets
the given entry in R to +1/(d ? ck/)1/2 . In the Index problem, Bob is given an index, which under
the bijection between coordinates of x and entries of R, corresponds to being given a row index i
and an entry j in the i-th row of R that he needs to recover. He sets i` = i for a random ` ? [k],
and chooses k ? 1 distinct and random indices ij ? [ck/] \ {i` }, for j ? [k] \ {`}. Observe that
if (x, i) ? ?, then (R, i1 , . . . , ik ) ? ?. Suppose there is a protocol in which Alice sends a single
message to Bob who solves f with probability at least 3/4 under ?. We show that this can be used
to solve Index with probability at least 2/3 under ?. The theorem will follow by Fact 3. Consider
the matrix B which is the matrix A stacked on top of the rows ei1 , . . . , eik , in that order, so that B
has ck/ + k rows.
We proceed to lower bound kB ? BP k2F in a certain way, which will allow our reduction to Index
to be carried out. We need the following fact:
?
Fact 5. ((2.4) of [20]) Let
? A be an m ? n matrix with i.i.d. entries which are each +1/ n with
probability 1/2 and ?1/ n with probability 1/2, and suppose m/n < 1. Then for all t > 0,
p
0
3/2
Pr[kAk2 > 1 + t + m/n] ? ?e?? nt .
where ?, ?0 > 0 are absolute constants. Here kAk2 is the operator norm supx kAxk/kxk of A.
We apply Fact 5 to the matrix R, which implies,
p
?
0
3/4
Pr[kRk2 > 1 + c + (ck/)/(d ? (ck/))] ? ?e?? (d?(ck/))c ,
and using that d ? k/ and c > 0 is a sufficiently small constant, this implies
?
Pr[kRk2 > 1 + 3 c] ? e??d ,
(1)
where ?
? > 0 is an ?
absolute constant (depending on c). Note
? that for c > 0 sufficiently small,
(1 + 3 c)2 ? 1 + 7 c. Let E be the event that kRk22 ? 1 + 7 c, which we condition on.
We partition the rows of B into B1 and B2 , where B1 contains those rows whose projection onto
the first ck/ coordinates equals ei for some i ?
/ {i1 , . . . , ik }. Note that B1 is (ck/ ? k) ? d and
B2 is 2k ? d. Here, B2 is 2k ? d since it includes the rows in A indexed by i1 , . . . , ik , together with
the rows ei1 , . . . , eik . Let us also partition the rows of R into RT and RS , so that the union of the
rows in RT and in RS is equal to R, where the rows of RT are the rows of R in B1 , and the rows
of RS are the non-zero rows of R in B2 (note that k of the rows are non-zero and k are zero in B2
restricted to the columns in R).
Lemma 6. For any unit vector u, write u = uR + uS + uT , where S = {i1 , . . . , ik }, T = [ck/] \ S,
and R = [d] \ [ck/], and ?
where uA for a set A is 0 on indices j ?
/ A. Then, conditioned on E
occurring, kBuk2 ? (1 + 7 c)(2 ? kuT k2 ? kuR k2 + 2kuS + uT kkuR k).
4
Proof. Let C be the matrix consisting of the top ck/ rows of B, so that C has the form [I, R],
where I is a ck/ ? ck/ identity matrix. By construction of B, kBuk2 = kuS k2 + kCuk2 . Now,
Cu = uS + uT + RuR , and so
kCuk22
and so
= kuS + uT k2 + kRuR k2 + 2(us + uT )T RuR
?
? kuS + uT k2 + (1 + 7 c)kuR k2 + 2kuS + uT kkRuR k
?
?
? (1 + 7 c)(kuS k2 + kuT k2 + kuR k2 ) + (1 + 3 c)2kuS + uT kkuR k
?
? (1 + 7 c)(1 + 2kuS + uT kkuR k),
kBuk2
?
? (1 + 7 c)(1 + kuS k2 + 2kuS + uT kkuR k)
?
= (1 + 7 c)(2 ? kuR k2 ? kuT k2 + 2kuS + UT kkuR k).
We will also make use of the following simple but tedious fact, shown in the full version.
?
Fact
For x ? [0, 1], the function f (x) = 2x 1 ? x2 ? x2 is maximized when x =
q 7. ?
?
?
1/2 ? 5/10. We define ? to be the value of f (x) at its maximum, where ? = 2/ 5 + 5/10 ?
1/2 ? .618.
?
Corollary 8. Conditioned on E occurring, kBk22 ? (1 + 7 c)(2 + ?).
Proof. By Lemma 6, for any unit vector u,
?
kBuk2 ? (1 + 7 c)(2 ? kuT k2 ? kuR k2 + 2kuS + uT kkuR k).
Suppose we replace the vector uS + uT with an arbitrary vector supported on coordinates in S with
the same norm as uS +uT . Then the right hand side of this expression cannot increase,
which means
p
?
2
2
it is maximized when kuT k = 0, for which it equals (1 + 7 c)(2 ? kuR k + 2 1 ?
?kuR k kuR k),
and setting kuR k to equal the x in Fact 7, we see that this expression is at most (1+7 c)(2+?).
Write the projection matrix P output by the streaming algorithm as U U T , where U is d ? k with
orthonormal columns ui (so R? R = P in the notation of Section 1). Applying Lemma 6 and Fact 7
to each of the columns ui , we show in the full version:
kBP k2F
?
k
X
?
(1 + 7 c)((2 + ?)k ?
kuiT k2 ).
(2)
i=1
Using the matrix Pythagorean theorem, we thus have,
kB ? BP k2F
= kBk2F ? kBP k2F
k
X
?
kuiT k2 ) using kBk2F = 2ck/ + k
? 2ck/ + k ? (1 + 7 c)((2 + ?)k ?
i=1
?
k
?
? X
2ck/ + k ? (1 + 7 c)(2 + ?)k + (1 + 7 c)
kuiT k2 .
(3)
i=1
We now argue that kB ? BP k2F cannot be too large if Alice and Bob succeed in solving f . First, we
?k of rank-k and bound kB ? B
?k k2 .
need to upper bound kB ? Bk k2F . To do so, we create a matrix B
F
?k will be 0 on the rows in B1 . We can group the rows of B2 into k pairs so that each pair
Matrix B
has the form ei + v i , ei , where i ? [ck/] and v i is a unit vector supported on [d] \ [ck/]. We let
Yi be the optimal (in Frobenius norm) rank-1 approximation to the matrix [ei + v i ; ei ]. By direct
?k then
computation 1 the maximum squared singular value of this matrix is 2 + ?. Our matrix B
?
consists of a single Yi for each pair in B2 . Observe that Bk has rank at most k and
?k k2F ? 2ck/ + k ? (2 + ?)k,
kB ? Bk k2F ? kB ? B
1
For an online SVD calculator, see http://www.bluebit.gr/matrix-calculator/
5
Therefore, if Bob succeeds in solving f on input B, then,
kB ? BP k2F
? (1 + )(2ck/ + k ? (2 + ?)k) ? 2ck/ + k ? (2 + ?)k + 2ck.
(4)
Comparing (3) and (4), we arrive at, conditioned on E:
k
X
i=1
kuiT k2 ?
?
1
? ? (7 c(2 + ?)k + 2ck) ? c1 k,
1+7 c
(5)
where c1 > 0 is a constant that can be made arbitrarily small by making c > 0 an arbitrarily small.
? +U
? , where the vectors in U
? are supported
Since P is a projector, kBP kF = kBU kF . Write U = U
?
on T , and the vectors in U are supported on [d] \ T . We have,
?
? k2F ? kBk22 c1 k ? (1 + 7 c)(2 + ?)c1 k ? c2 k,
kB U
? kF ? kBk2 kU
? kF and (5), the second inequality uses that event
where the first inequality uses kB U
E occurs, and the third inequality holds for a constant c2 > 0 that can be made arbitrarily small by
making the constant c > 0 arbitrarily small.
Combining with (4) and using the triangle inequality,
? kF ? kBP kF ? kB U
? kF using the triangle inequality
kB U
p
? k2F
? kBP kF ? c2 k using our bound on kB U
q
p
=
kBk2F ? kB ? BP k2F ? c2 k by the matrix Pythagorean theorem
p
p
?
(2 + ?)k ? 2ck ? c2 k by (4)
p
?
(2 + ?)k ? c3 k,
(6)
where c3 > 0 is a constant that can be made arbitrarily small for c > 0 an arbitrarily small constant
(note that c2 > 0 also becomes arbitrarily small as c > 0 becomes arbitrarily small). Hence,
? k2 ? k ? c4 k for a
? k2 ? (2 + ?)k ? c3 k, and together with Corollary 8, that implies kU
kB U
F
F
constant c4 that can be made arbitrarily small by making c > 0 arbitrarily small.
?,
? k2 . Consider any column u
? k2 is almost as large as kB U
? of U
Our next goal is to show that kB2 U
F
F
and write it as u
?S + u
?R . Hence,
kB u
?k2
= kRT u
?R k2 + kB2 u
?k2 using B1 u
? = RT u
?R
?
kRT u
?R k2 + k?
uS + RS u
?R k2 + k?
uS k2 by definition of the components
= kR?
uR k2 + 2k?
uS k2 + 2?
uTS RS u
?R using the Pythagorean theorem
?
2
? 1 + 7 c + k?
uS k + 2k?
uS kkRS u
?R k
?
using kR?
uR k2 ? (1 + 7 c)k?
uR k2 and k?
uR k2 + k?
uS k2 ? 1
(also using Cauchy-Schwarz to bound the other term).
?
Suppose kRS u
?R k = ? k?
uR k for a value 0 ? ? ? 1 + 7 c. Then
p
?
kB u
?k2 ? 1 + 7 c + k?
uS k2 + 2? k?
uS k 1 ? k?
uS k2 .
We thus have,
kB u
?k2
?
?
?
p
?
1 + 7 c + (1 ? ? )k?
uS k2 + ? (k?
uS k2 + 2k?
uS k 1 ? k?
uS k2 )
?
1 + 7 c + (1 ? ? ) + ? (1 + ?) by Fact 7
?
2 + ? ? + 7 c,
(7)
?
and hence, letting ?1 , . . . , ?k denote the corresponding values of ? for the k columns of U , we have
k
X
?
? k2F ? (2 + 7 c)k + ?
?i .
kB U
(8)
i=1
Comparing the square of (6) with (8), we have
k
X
i=1
?i
? k ? c5 k,
6
(9)
where c5 > 0 is a constant that can be made arbitrarily small by making c > 0 an arbitrarily small
? k2 ? k ? c4 k as shown above, while since kRs u
constant. Now, kU
?R k = ?i k?
uR k if u
?R is the i-th
F
?
column of U , by (9) we have
?R k2F ? (1 ? c6 )k
kRS U
(10)
for a constant c6 that can be made arbitrarily small by making c > 0 an arbitarily small constant.
?R k2 ? (1 + 7?c)k since event E occurs, and kRU
?R k2 = kRT U
?R k2 + kRS U
?R k2 since
Now kRU
F
F
F
F
the rows of R are the concatenation of rows of RS and RT , so combining with (10), we arrive at
?R k2
kRT U
F
? c7 k,
(11)
for a constant c7 > 0 that can be made arbitrarily small by making c > 0 arbitrarily small.
Combining the square of (6) with (11), we thus have
? k2F
kB2 U
? k2F ? kB1 U
? k2F = kB U
? k2F ? kRT U
?R k2F ? (2 + ?)k ? c3 k ? c7 k
= kB U
? (2 + ?)k ? c8 k,
(12)
where the constant c8 > 0 can be made arbitrarily small by making c > 0 arbitrarily small.
By the triangle inequality,
? kF ? kB2 U
? kF ? ((2 + ?)k ? c8 k)1/2 ? (c2 k)1/2 .
kB2 U kF ? kB2 U
(13)
Hence,
kB2 ? B2 P kF
q
kB2 k2F ? kB2 U k2F Matrix Pythagorean, kB2 U kF = kB2 P kF
q
? kF ? kB2 U
? kF )2 Triangle Inequality
?
kB2 k2F ? (kB2 U
q
?
3k ? (((2 + ?)k ? c8 k)1/2 ? (c2 k)1/2 )2 Using (13),kB2 k2F = 3k,(14)
=
(15)
or equivalently, kB2 ? B2 P k2F ? 3k ? ((2 + ?)k ? c8 k) ? (c2 k) + 2k(((2 + ?) ? c8 )c2 )1/2 ?
(1 ? ?)k + c8 k + 2k(((2 + ?) ? c8 )c2 )1/2 ? (1 ? ?)k + c9 k for a constant c9 > 0 that can be made
arbitrarily small by making the constant c > 0 small enough. This intuitively says that P provides a
good low rank approximation for the matrix B2 . Notice that by (14),
kB2 P k2F
= kB2 k2F ? kB2 ? B2 P k2F ? 3k ? (1 ? ?)k ? c9 k ? (2 + ?)k ? c9 k.
(16)
Now B2 is a 2k ? d matrix and we can partition its rows into k pairs of rows of the form Z` =
(ei` +Ri` , ei` ), for ` = 1, . . . , k. Here we abuse notation and think of Ri` as a d-dimensional vector,
its first ck/ coordinates set to 0. Each such pair of rows is a rank-2 matrix, which we abuse notation
and call Z`T . By direct computation2 Z`T has squared maximum singular value 2 + ?. We would
like to argue that the projection of P onto the row span of most Z` has length very close to 1. To
this end, for each Z` consider the orthonormal basis V`T of right singular vectors for its row space
T
T
(which is span(ei` , Ri` )). We let v`,1
, v`,2
be these two right singular vectors with corresponding
singular values ?1 and ?2 (which will be the same for all `, see below). We are interested in the
Pk
quantity ? = `=1 kV`T P k2F which intuitively measures how much of P gets projected onto the
row spaces of the Z`T . The following lemma and corollary are shown in the full version.
Lemma 9. Conditioned on event E, ? ? [k ? c10 k, k + c10 k], where c10 > 0 is a constant that can
be made arbitrarily small by making c > 0 arbitrarily small.
The following corollary is shown in the full version.
?
Corollary 10. Conditioned on event E, for a 1? c9 + 2c10 fraction of ` ? [k], kV`T P k2F ? 1+c11 ,
and for a 99/100 fraction of ` ? [k], we have kV`T P k2F ? 1 ? c11 , where c11 > 0 is a constant that
can be made arbitrarily small by making the constant c > 0 arbitrarily small.
2
We again used the calculator at http://www.bluebit.gr/matrix-calculator/
7
Recall that Bob holds i = i` for a random ` ? [k]. It follows (conditioned on E) by a union bound
that with probability at least 49/50, kV`T P k2F ? [1 ? c11 , 1 + c11 ], which we call the event F and
condition on. We also condition on event G that kZ`T P k2F ? (2+?)?c12 , for a constant c12 > 0 that
can be made arbitrarily small by making c > 0 an arbitrarily small constant. Combining the first part
of Corollary 10 together with (16), event G holds with probability at least 99.5/100, provided c > 0
is a sufficiently small constant. By a union bound it follows that E, F, and G occur simultaneously
with probability at least 49/51.
T
T
As kZ`T P k2F = ?12 kv`,1
P k2 + ?22 kv`,2
P k2 , with ?12 = 2 + ? and ?12 = 1 ? ?, events E, F, and G
T
2
imply that kv`,1 P k ? 1 ? c13 , where c13 > 0 is a constant that can be made arbitrarily small by
T
making the constant c > 0 arbitrarily small. Observe that kv`,1
P k2 = hv`,1 , zi2 , where z is a unit
vector in the direction of the projection of v`,1 onto P .
By the Pythagorean theorem, kv`,1 ? hv`,1 , zizk2 = 1 ? hv`,1 , zi2 , and so
kv`,1 ? hv`,1 , zizk2 ? c14 ,
(17)
for a constant c14 > 0 that can be made arbitrarily small by making c > 0 arbitrarily small.
We thus have Z`T P = ?1 hv`,1 , ziu`,1 z T + ?2 hv`,2 , wiu`,2 wT , where w is a unit vector in the
direction of the projection of of v`,2 onto P , and u`,1 , u`,2 are the left singular vectors of Z`T . Since
F occurs, we have that |hv`,2 , wi| ? c11 , where c11 > 0 is a constant that can be made arbitrarily
small by making the constant c > 0 arbitrarily small. It follows now by (17) that
t
kZ`T P ? ?1 u`,1 v`,1
k2F ? c15 ,
(18)
where c15 > 0 is a constant that can be made arbitrarily small by making the constant c > 0
arbitrarily small.
By direct calculation3 , u`,1 = ?.851ei` ? .526Ri` and v`,1 = ?.851ei` ? .526Ri` . It follows that
kZ`T P ? (2 + ?)[.724ei` + .448Ri` ; .448ei` + .277Ri` ]k2F ? c15 . Since ei` is the second row of
Z`T , it follows that keTi` P ? (2 + ?)(.448ei` + .277Ri` )k2 ? c15 .
Observe that Bob has ei` and P , and can therefore compute eTi` P . Moreover, as c15 > 0 can be made
arbitrarily small by making the constant c > 0 arbitrarily small, it follows that a 1 ? c16 fraction of
the signs of coordinates of eTi` P , restricted to coordinates in [d] \ [ck/], must agree with those of
(2 + ?).277Ri` , which in turn agree with those of Ri` . Here c16 > 0 is a constant that can be made
arbitrarily small by making the constant c > 0 arbitrarily small. Hence, in particular, the sign of the
j-th coordinate of Ri` , which Bob needs to output, agrees with that of the j-th coordinate of eTi` P
with probability at least 1 ? c16 . Call this event H.
By a union bound over the occurrence of events E, F, G, and H, and the streaming algorithm succeeding (which occurs with probability 3/4), it follows that Bob succeeds in solving Index with
probability at least 49/51 ? 1/4 ? c16 > 2/3, as required. This completes the proof.
3
Conclusion
We have shown an ?(dk/) bit lower bound for streaming algorithms in the row-update model for
outputting a k ? d matrix R with kA ? AR? RkF ? (1 + )kA ? Ak kF , thus showing that the
algorithm of [9] is optimal up to the word size. The next natural goal would be to obtain multi-pass
lower bounds, which seem quite challenging. Such lower bound techniques may also be useful for
showing the optimality of a constant-round O(sdk/) + (sk/)O(1) communication protocol in [12]
for low-rank approximation in the distributed communication model.
Acknowledgments. I would like to thank Edo Liberty and Jeff Phillips for many useful discusions
and detailed comments on this work (thanks to Jeff for the figure!). I would also like to thank
the XDATA program of the Defense Advanced Research Projects Agency (DARPA), administered
through Air Force Research Laboratory contract FA8750-12-C0323 for supporting this work.
3
Using the online calculator in earlier footnotes.
8
References
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[3] M. Brand. Incremental singular value decomposition of uncertain data with missing values. In
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page 739, 2001.
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9
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4,869 | 5,408 | Tight convex relaxations for sparse matrix
factorization
Emile Richard
Electrical Engineering
Stanford University
Guillaume Obozinski
Universit?e Paris-Est
Ecole des Ponts - ParisTech
Jean-Philippe Vert
MINES ParisTech
Institut Curie
Abstract
Based on a new atomic norm, we propose a new convex formulation for sparse
matrix factorization problems in which the number of non-zero elements of the
factors is assumed fixed and known. The formulation counts sparse PCA with
multiple factors, subspace clustering and low-rank sparse bilinear regression as
potential applications. We compute slow rates and an upper bound on the statistical dimension [1] of the suggested norm for rank 1 matrices, showing that
its statistical dimension is an order of magnitude smaller than the usual `1 -norm,
trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we
propose an active set algorithm leveraging the structure of the convex problem to
solve it and show promising numerical results.
1
Introduction
A range of machine learning problems such as link prediction in graphs containing community structure [16], phase retrieval [5], subspace clustering [18] or dictionary learning [12] amount to solve
sparse matrix factorization problems, i.e., to infer a low-rank matrix that can be factorized as the
product of two sparse matrices with few columns (left factor) and few rows (right factor). Such
a factorization allows more efficient storage, faster computation, more interpretable solutions and
especially leads to more accurate estimates in many situations. In the case of interaction networks,
for example, this is related to the assumption that the network is organized as a collection of highly
connected communities which can overlap. More generally, considering sparse low-rank matrices
combines two natural forms of sparsity, in the spectrum and in the support, which can be motivated
by the need to explain systems behaviors by a superposition of latent processes which only involve
a few parameters. Landmark applications of sparse matrix factorization are sparse principal components analysis (SPCA) [8, 21] or sparse canonical correlation analysis (SCCA)[19], which are
widely used to analyze high-dimensional data such as genomic data.
In this paper, we propose new convex formulations for the estimation of sparse low-rank matrices. In
particular, we assume that the matrix of interest should be factorized as the sum of rank one factors
that are the product of column and row vectors with respectively k and q non zero-entries, where k
and q are known. We first introduce below the (k, q)-rank of a matrix as the minimum number of
left and right factors, having respectively k and q non-zeros, required to reconstruct a matrix. This
index is a more involved complexity measure for matrices than the rank in that it conditions on the
number of non-zero elements of the left and right factors of the matrix. Based on this index, we
propose a new atomic norm for matrices [7] by considering its convex hull restricted to the unit ball
of the operator norm, resulting in convex surrogates to low (k, q)-rank matrix estimation problem.
We analyze the statistical dimension of the new norm and compare it to that of linear combinations
of the `1 and trace norms. In the vector case, our atomic norm actually reduces to k-support norm
introduced by [2] and our analysis shows that its statistical power is not better than that of the `1 1
norm. By contrast, in the matrix case, the statistical dimension of our norm is at least one order of
magnitude better than combinations of the `1 -norm and the trace norm.
However, while in the vector case the computation remains feasible in polynomial time, the norm we
introduce for matrices can not be evaluated in polynomial time. We propose algorithmic schemes
to approximately learn with the new norm. The same norm and meta-algorithms can be used as
a regularizer in supervised problems such as multitask learning or quadratic regression and phase
retrieval, highlighting the fact that our algorithmic contribution does not consist in providing more
efficient solutions to the rank-1 SPCA problem, but to combine atoms found by the rank-1 solvers
in a principled way.
2
Tight convex relaxations of sparse factorization constraints
In this section we propose a new matrix norm allowing to formulate various sparse matrix factorization problems as convex optimization problems. We start by defining the (k, q)-rank of a matrix
in section 2.1, a useful generalization of the rank which also quantifies the sparseness of a matrix
factorization. We then introduce in section 2.2 the (k, q)-trace norm, an atomic norm defined as the
convex relaxations of the (k, q)-rank over the operator norm ball. We discuss further properties and
potential applications of this norm used as a regularizer in section 2.3.
2.1
The (k, q)-rank of a matrix
The rank of a matrix Z ? Rm1 ?m2 is theP
minimum number of rank-1 matrices needed to express Z
r
as a linear combination of the form Z = i=1 ai b>
i . The following definition generalizes this rank
to incorporate conditions on the sparseness of the rank-1 elements:
m1 ?m2
Definition 1 ((k, q)-sparse decomposition and (k, q)-rank) For a matrix
, we call
Pr Z ? R
(k, q)-sparse decomposition of Z any decomposition of the form Z = i=1 ci ai b>
i where ai (resp.
bi ) are unit vectors with at most k (resp. q) non-zero elements, and with minimal r, which we call
the (k, q)-rank of Z.
The (k, q)-rank and (k, q)-sparse decomposition of Z can equivalently be defined as the optimal
value and a solution of the optimization problem:
min kck0
s.t.
Z=
?
X
ci ai b>
i ,
m2
1
(ai , bi , ci ) ? Am
k ? A q ? R+ ,
(1)
i=1
where for any 1 ? j ? n, Anj = {a ? Rn | kak0 ? j, kak2 = 1}. Since Ani ? Anj when i ? j, we
have for any k and q rank(Z) ? (k, q)-rank(Z) ? kZk0 . The (k, q)-rank is useful to formalize
problems such as sparse matrix factorization, which can be defined as approximating the solution of
a matrix valued problem by a matrix having low (k, q)-rank. For instance the standard rank-1 SPCA
problem consists in finding the symmetric matrix with (k, k)-rank equal to 1 and providing the best
approximation of the sample covariance matrix [21].
2.2
A convex relaxation for the (k, q)-rank
The (k, q)-rank is a discrete, nonconvex index, like the rank or the cardinality, leading to computational difficulties if one wants to learn matrices with small (k, q)-rank. We propose a convex
relaxation of the (k, q)-rank aimed at mitigating these difficulties. For that purpose, we consider an
atomic norm [7] that provides a convex relaxation of the (k, q)-trace norm, just like the `1 norm and
the trace norm are convex relaxations of the `0 semi-norm and the rank, respectively. An atomic
norm is a convex function defined based on a small set of elements called atoms which constitute
a basis on which an object of interest can be sparsely decomposed. The function (a norm if the set
is centrally symmetric) is defined as the gauge of the convex hull of atoms. In other terms, its unit
ball or level-set of value 1 is formed by the convex envelope of atoms. In case of atoms of interest,
namely rank-1 factors of given sparsities k and q, we define
m2
1
Definition 2 ((k, q)-trace norm) Let Ak,q be a set of atoms Ak,q = ab> : a ? Am
.
k , b ? Aq
For a matrix Z ? Rm1 ?m2 , the (k, q)-trace norm ?k,q (Z) is the atomic norm induced by Ak,q , i.e.,
2
n X
?k,q (Z) = inf
A?Ak,q
o
cA A, cA ? 0, ?A ? Ak,q .
X
cA : Z =
(2)
A?Ak,q
In words, Ak,q is the set of matrices A ? Rm1 ?m2 such that (k, q)-rank(A) = 1 and kAkop = 1.
The next lemma provides an explicit formulation for the (k, q)-trace norm and its dual:
Lemma 1 For any Z, K ? Rm1 ?m2 , and denoting Gkm = {I ? [[1, m]] : |I| = k}, we have
n
o
X
X
?k,q (Z) = inf
kZ (I,J) k? : Z =
Z (I,J) , supp(Z (I,J) ) ? I ? J ,
m1
m
?Gq 2
and
2.3
(3)
(I,J)
(I,J)?Gk
??k,q (K) = max kKI,J kop : I ? Gkm1 , J ? Gqm2 .
Learning matrices with sparse factors
In this section, we briefly discuss how the (k, q)-trace norm norm can be used to formulate various
problems involving the estimation of sparse low-rank matrices. A way to learn a matrix Z with
low empirical risk L(Z) and with low (k, q)-rank is to use ?k,q as a regularizer and minimize an
objective of the form
min
L(Z) + ??k,q (Z).
(4)
m ?m
Z?R
1
2
A number of problems can be formulated as variants of (4).
Bilinear regression. In bilinear regression, given two inputs x ? Rm1 and x0 ? Rm2 one observes as output a noisy version of y = x> Zx0 . Assuming that Z has low (k, q)-rank means that
the noiseless response is a sum of a small number of terms, each involving only a small number
of features from either of the input vectors. To estimate within such a model from observations
(xi , x0i , yi )i=1,...,n one can consider the following formulation, in which ` is a convex loss :
X
0
min
` x>
(5)
i Zxi , yi + ??k,q (Z) .
Z?Rm1 ?m2
i
Subspace clustering. In subspace clustering, one assumes that the data can be clustered in such a
way that the points in each cluster belong to a low dimensional space. If we have a design matrix
X ? Rn?p with each row corresponding to an observation, then the previous assumption means
that if X (j) ? Rnj ?p is a matrix formed by the rows of cluster j, there exist a low rank matrix
Z (j) ? Rnj ?nj such that Z (j) X (j) = X (j) . This means that there exists a block-diagonal matrix Z
such that ZX = X and with low-rank diagonal blocks. This idea, exploited recently by [18] implies
that Z is a sum of low rank sparse matrices; and this property still holds if the clustering is unknown.
We therefore suggest that if all subspaces are of dimension k, Z may be estimated via
min ?k,k (Z) s.t. ZX = X .
Z?Rn?n
Sparse PCA. One possible formulation of sparse PCA with multiple factors is the problem of ap? n by a low-rank matrix with sparse factors. This
proximation of an empirical covariance matrix ?
suggests to formulate sparse PCA as follows:
? n ? ZkF : (k, k)-rank(Z) ? r and Z 0 ,
min k?
(6)
Z
where q is the maximum number of non-zero coefficients allowed in each principal direction. By
contrast to sequential approaches that estimate the principal components one-by-one [11], this formulation requires to find simultaneously a set of complementary factors. If we require the decomposition of Z to be a sum of positive semi-definite (k, k)-sparse rank one factors (which is a stronger
assumption than assuming that Z is p.s.d.), the positivity constraint on Z is no longer necessary and
a natural convex relaxation for (6) using another atomic norm (in fact only a gauge here) is
min
Z?Rm?m
? n ? Zk2F + ??k, (Z) ,
k?
where ?k, is the gauge of the set of atoms Ak, := {aa> , a ? Am
k }.
3
(7)
3
Performance of the (k, q)-trace norm for denoising
In this section, we consider the problem of denoising a low-rank matrix Z ? ? Rm1 ?m2 with sparse
factors corrupted by additive Gaussian noise, that is noisy observations Y ? Rm1 ?m2 of the form
Y = Z ? + ?G , where ? > 0 and G is a random matrix with i.i.d. N (0, 1) entries. For a convex
penalty ? : Rm1 ?m2 ? R, we consider, for any ? > 0, the estimator
1
Z??? = arg min kZ ? Y k2F + ??(Z) .
(8)
Z 2
The following result is a straightforward generalization to any norm ? of the so-called slow rates
that are well know for the `1 norms and other norms such as the trace-norm (see e.g. [10]).
Lemma 2 If ? ? ??? (G)
then
?
Z?? ? Z ?
2 ? 4??(Z ? ) .
F
To derive an upper bound in estimation error from these inequalities, and to keep the argument as
simple as possible we consider the oracle1 estimate Z??Oracle equal to Z??? where ? = ??? (G). From
Lemma 2 we immediately get
(9)
E
Z??Oracle ? Z ? k2F ? 4? ?(Z ? ) E ??(G) .
This upper bound can be computed
for Z ? = ab> ? Ak,q for different norms. In particular, for
?
?
>
?(Z ), we have kab k1 ? kq and ?k,q (ab> ) = kab> k? = 1 which lead to the corollary:
Corollary 1 When Z ? = ab> ? Ak,q is an atom, the expected errors of the oracle estimators
Z??Oracle
, Z?1Oracle and Z??Oracle using respectively the (k, q)-trace norm, the `1 norm and the trace norm
k,q
are upper bounded as follows:
r
r
m1
m2
? 2
E kZ??Oracle
?
Z
k
?
8
?
k
log
q
log
+
2k
+
+
2q
,
F
k,q
k
q
p
p
(10)
E kZ?1Oracle ? Z ? k2F ? 2?kZ ? k1 2 log(m1 m2 ) ? 2? 2kq log(m1 m2 ) ,
?
?
E kZ??Oracle ? Z ? k2F ? 2?( m1 + m2 ) .
When the smallest p
entry in absolute value of a or b is close to 0, then the expected error is smaller for
Oracle
?
, reaching ? 2 log(m1 m2 ) on e1 e>
Z1
1 while not changing for the two other norms. But under
the?assumption that the smallest nonzero entries in absolute value of a and b are lower bounded by
c/ kq with c a constant, the upper bound on the rates obtained for the (k, q)-trace norm is at least
an order of magnitude larger than for the other norms. We report the order of magnitude of these
upper bounds in Table 1 for m1 =?
m2 = m and k = q and assuming that nonzeros coefficients are
lower bounded in magnitude by c/ kq.
Obviously the comparison of upper bounds is not enough to conclude to the superiority of
(k, q)-trace norm and, admittedly, the problem of denoising considered here is a special instance
of linear regression in which the design matrix is the identity, and, since this is a case in which the
design is trivially incoherent, it is possible to obtain fast rates for decomposable norms such as the
`1 or trace norm [13]; however, the slow rates obtained are the same if instead of Y a linear transformation of Z with incoherent design is observed, or when the signal to recover is only weakly
sparse, which is not the case for the fast rates. Moreover, Lemma 2 applies to matrices of any rank
and Corollary 1 generalizes to rank greater than 1. We present in the next section more sophisticated
results, based on bounds on the so-called statistical dimension of different norms [1].
4
A bound on the statistical dimension of the (k, q)-trace norm
The squared Gaussian width [7, and ref. therein] and the statistical dimension introduced recently
by Amelunxen et al. [1], provide quantified estimation guarantees. The two quantities are equal
1
We call it oracle estimate because the choice of ? depends on the unknown noise level. Virtually identical bounds (up to constants) holding with large probability could be derived for the non-oracle estimator by
controlling the deviations of ?? (G) from its expectation.
4
up to an additive term smaller than 1 and we thus present results only in terms of the statistical
dimension. The sample complexity of exact recovery and robust recovery are characterized by this
quantity [7]. It is also equal to the signal to noise ratio necessary for denoising [6] and demixing
[1] (see supplementary section 3). The statistical dimension is defined as follows: if T? (A) is the
tangent cone of a matrix norm ? : Rm1 ?m2 ? R+ at A, then, the statistical dimension of T? (A) is
h
2 i
S(Z, ?) := E
?T? (Z) (G)
F ,
where G ? Rm1 ?m2 is a random matrix with i.i.d. standard normal entries and ?T? (Z) (G) is the
orthogonal projection of G onto the cone T? (Z). In this section, we compute an upper bound on
the statistical dimension of ?k,q at an atoms A of Ak,q , which we will denote by S(A, ?k,q ), and
compare it to results known for linear combinations of the `1 and the trace norm of the form ?? with
?
?? (Z) := ? kZk1 + (1 ? ?)kZk? ,
(11)
kq
which are norms that have been used in the literature to infer sparse low-rank matrices [17]. The
ability to recover the support of a sparse vector typically depends on the size of its smallest non-zero
coefficient. For the recovery of a sparse rank 1 matrix, this motivates the following definition
?? ? [0, 1], ?Z ? Rm1 ?m2 ,
Definition 3 Let A = ab> ? Ak,q with I0 = supp(a) and J0 = supp(b). Denote a2min = min a2i and
i?I0
b2min = min b2j . We define the strength ?(a, b) ? (0, 1] as ?(a, b) := (k a2min ) ? (q b2min ).
j?J0
?
?
The strength of an atom takes the maximal value of 1 when |ai | = 1/ k, i ? I and |bj | = 1/ q, j ?
J where I and J are the supports of a and b. On the contrary, its strength is close to 0 as soon as one
of its nonzero entries is close to zero. We can now present our main result: a bound on the statistical
dimension of ?k,q on Ak,q .
Proposition 1 For A = ab> ? Ak,q with strength ? = ?(a, b), there exist universal constants
c1 , c2 , independent of m1 , m2 , k, q such that
c2
c1
S(A, ?k,q ) ? 2 (k + q) + (k + q) log(m1 ? m2 ) .
?
?
Our proof, presented in the appendix, follows the scheme proposed in [7] and used for the trace
norm and `1 norm. However, ?k,q is not decomposable and requires some work to obtain precise
upper bounds on various quantities.
Note first that S must be larger than the number of degrees of freedoms of elements of Ak,q which
is k + q ? 1. So the bound could not possibly be improved beyond logarithmic factors, besides
the logarithmic dependence on the dimension of the overall space is expected. To appreciate the
result, it should be compared with the statistical dimension for the `1 -norm which scales as the
product of the size of the support with the logarithm of the dimension of the ambient space, that is
as kq log(m1 m2 ). Using Landau notation, we report in Table 1 the upper and lower bounds known
for the statistical dimension of other norms in the case where k = q and m1 = m2 = m. The
rates are known exactly up to constants for the `1 and the trace norm (see e.g. [1]). Of particular
relevance is the comparison with norms of the form ?? which have been introduced with the aim
of improving over both the `1 -norm and the trace norm and have been the object of a significant
literature [17, 15, 9]. Using theorem 3.2 in [15], we prove in appendix 4 a lower bound on the
statistical dimension of ?? of order kq ? (m1 + m2 ) which holds for all values of ?, and which show
that, up to logarithmic factors, ?k,q is an order of magnitude smaller in term of k ? q.
In the right column of Table 1 we also report results in the vector case, that is, when m2 = q = 1. In
fact, in that case, ?k,1 is exactly the k-support norm proposed by [2]. But the statistical dimension
of that norm and the `1 norm is the same as it is known that the rate k log kp cannot be improved
([4]). So, perhaps surprisingly, there improvement in the matrix case but not in the vector case.
5
Algorithm
In this section, we present a working set algorithm that attempts to solve problem (4). Injecting the
variational form (3) of ?k,q in (4) and eliminating the variable Z from the optimization problem
5
Matrix norm
(k, q)-trace
`1
trace-norm
`1 + trace-n.
S
O(k log m)
?(k 2 log km2 )
?(m)
? k2 ? m
?(Z ? )E ?? (G)
1/2
(k log m
k)
2
(k log m)1/2
m1/2
1/2
O m ? (k 2 log m)1/2
Vector norm
k-support
`1
`2
elastic net
S
?(k log kp )
?(k log kp )
p
?(k log kp )
Table 1: Scaling of the statistical dimension S and of the upper bound ?(Z ? ) E?? (G) in estimation
error (slow-rates) of different matrix norms for elements of Ak,q with strength (see Definition
? 3)
lower bounded by a constant (or equivalently with nonzero coefficient lower bounded by c/ kq
for c a constant). Leftmost columns: scalings for matrices with k = q, m = m1 = m2 ; rightmost
columns: scalings for vectors with m1 = p and m2 = q = 1. We use the notations ? and ? with
f = ?(g) meaning g = O(f ) and f = ?(g) to mean that both g = O(f ) and f = O(g).
using Z =
min
Z (IJ) ?Rm1 m2
Z (IJ) , one obtains that, when S = Gkm1 ? Gqm2 , problem (4) is equivalent to
X
X
L
Z (IJ) + ?
kZ (IJ) k? , s.t. Supp(Z (IJ) ) ? I ? J, (I, J) ? S. (PS )
P
(I,J)?S
(I,J)?S
(I,J)?S
At the optimum of (4) however, most of the variables Z (IJ) are equal to zero, and the solution is
the same as the solution obtained from (PS ) in which S is reduced to the set of non-zero matrices
Z (IJ) obtained at optimality, that are often called the active components. We thus propose to solve
problem (4) using a so-called working set algorithm which solves a sequence of problems of the form
(PS ) for a growing sequence of working sets S, so as to keep a small number of non-zero matrices
Z (IJ) throughout. Problem (PS ) is solved easily using approximate block coordinate descent on
the (Z (IJ) )(I,J)?S [3, Chap. 4] , which consists in iterating proximal operators of the trace norm
on blocks I ? J. The principle of the working set algorithm is to solve problem (PS ) for the
current working set S and to check whether a new component should be added. It can be shown
that a component with support I ? J should be added if and only if k[?L(Z)]IJ kop > ? for the
current value of Z. If such a component is found, the corresponding (I, J) pair is added in S and
problem (PS ) is solved again. Given that for any component in S, we have k[?L(Z)]IJ kop ? ? at
the optimum of (PS ), the algorithm terminates if ??k,q (?L(Z)) ? ?.
m2
1
The main difficulty is that ??k,q (K) = max{a> Kb | a ? Am
k , b ? Aq }, which is NP-hard to
compute, since it reduces in particular to rank 1 sparse PCA when k = q and K is p.s.d.. This
implies that determining when the algorithm should stop and which new component to add is hard.
However, a significant amount of research has been carried out on sparse PCA recently, and we
thus propose to leverage some of the recently proposed relaxations and heuristics to solve this rank
1 sparse PCA problem (see [8, 20] and references therein). In particular, the Truncated Power
iteration (TPI) algorithm of [20] can easily be modified to compute ??k,q which corresponds to a
generalization of the rank 1 sparse PCA in which in general a 6= b and k 6= q.
In our numerical experiments, we used a variant of Truncated Power Iteration with multiple restarts,
keeping track of the highest found variance. It should be noted that under RIP conditions on the
matrix, [20] shows that the solution returned by TPI is guaranteed to solve the rank 1 sparse PCA
problem. Also, even if TPI finds a pair (I, J) which is suboptimal, adding it in S does not hurt as
the algorithm might determine subsequently that it is not necessary. However TPI might fail to find
some of the components violating the optimality conditions and terminate the algorithm early.
The proposed algorithm cannot be guaranteed to solve (4) if ??k,q is not computed exactly, but it
exploits as much as possible the structure of the convex optimization problem to find a candidate
solution. A similar active set algorithm can be designed to solve problems regularized by ?k, .
Formulations regularized by the trace norm require to compute its proximal operator, and thus to
compute an SVD. However, even when m1 , m2 are large, solving PS involves the computation of
trace norms of matrices of size only k ? q and so the SVDs that need to be computed are fairly
small. This means that the computational bottleneck of the algorithm is clearly in finding candidate
supports. It has been proved [20] that, under some conditions, the problem can be solved in linear
time. Multiple restarts allow to find good candidate supports in practice.
6
6
5
10
700
600
10
3
10
NMSE
500
Trace
?k,q
3
10
400
90 % overlap
Trace
?k,q
5
10
4
10
10
3
10
300
2
2
10
2
10
1
2
10
3
10
k
k
1
100
1
10
10
(k,q)?rank
200
k
10
0
10
l1
No overlap
4
NMSE
4
10
l1
800
Trace
?k,q
5
10
6
10
(k,k)?rank = 1
NMSE
900
l1
NMSE
6
10
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
10
0
10
1
1
10
2
10
3
10
10
0
10
1
10
2
10
Figure 1: Estimates of the statistical dimensions of the `1 , trace and ?k,q norms at a matrix Z ?
R1000?1000 in different setting. From left to right: (a): Z is an atom in Aek,k for different values of
k. (b) Z is a sum of r atoms in Aek,k with non-overlapping support, with k = 10 and varying r,
(c) Z is a sum of 3 atoms in Aek,k with non-overlapping support, for varying k. (d) Z is a sum of 3
atoms in Aek,k with overlapping support, for varying k.
6
Numerical experiments
In this section we report experimental results to assess the performance of sparse low-rank matrix
estimation using different techniques. We start in Section 6.1 with simulations that confirm and
illustrate the theoretical results on statistical dimension of ?k,q and assess how they generalize to
matrices with (k, q)-rank larger than 1. In Section 6.2 we compare several techniques for sparse
PCA on simulated data.
6.1
Empirical estimates of the statistical dimension.
In order to numerically estimate the statistical dimension S(Z, ?) of a regularizer ? at a matrix
Z, we add to Z a random Gaussian noise matrix and observe Y = Z + ?G where G has normal
i.i.d. entries following N (0, 1). We then denoise Y to form an estimate Z? of Z. For small ?, the
normalized mean-squared error (NMSE) defined as NMSE(?) := EkZ? ?Zk2F /? 2 is a good estimate
of the statistical dimension, since [14] show that S(Z, ?) = lim??0 NMSE(?) . Numerically, we
therefore estimate S(Z, ?) with the empirical NMSE(?) for ? = 10?4 , averaged over 20 replicates.
We consider square matrices with m1 = m2 = 1000, and estimate the statistical dimension of ?k,q ,
the `1 and the trace norms at different matrices Z. The constrained denoiser has a simple closed-form
for the `1 and the trace norm. For ?k,q , it can be obtained by a sequence of proximal projections
? has the correct value ?k,q (Z). Since the noise is small,
with different parameters ? until ?k,q (Z)
we found that it was sufficient and faster to perform a (k, q)-SVD of Y by computing a proximal of
?k,q with a small ?, and then apply the `1 constrained denoiser to the set of (k, q)-sparse singular
values.
We first estimate the statistical dimensions of the three norms at an atom Z in Aek,q for different
?
values of k = q, where Aek,q = {ab> ? Ak,q | kab> k? = 1/ kq} is the set of elements of
Ak,q with nonzero entries of constant magnitude . Figure 1.a shows the results, which confirm
the theoretical bounds summarized in Table 1. The statistical dimension of the trace norm does
not depend on k, while that of the `1 norm increases almost quadratically with k and that of ?k,q
increases linearly with k. The linear versus quadratic dependence of the statistical dimension on
k are reflected by the slopes of the curves in the log-log plot in Figure 1.a. As expected, ?k,q
interpolates between the `1 norm (for k = 1) and the trace norm (for k = m1 ), and outperforms both
norms for intermediate values of k. This experiments therefore confirms that our upper bound (1) on
S(Z, ?k,q ) captures the correct order in k, although the constants can certainly be much improved,
and that our algorithm manages, in this simple setting, to correctly approximate the solution of the
convex minimization problem.
Second, we estimate the statistical dimension of ?k,q on matrices with (k, q)-rank larger than 1,
a setting for which we proved no theoretical result. Figure 1.b shows the numerical estimate of
S(Z, ?k,q ) for matrices Z which are sums of r atoms in Aek,k with non-overlapping support, for
k = 10 and varying r. We observe that the increase in statistical dimension is roughly linear in
the (k, q)-rank. For a fixed (k, q)-rank of 3, Figures 1.c and 1.d compare the estimated statistical
dimensions of the three regularizers on matrices Z which are sums of 3 atoms in Aek,k with re7
Sample covariance
4.20 ? 0.02
Trace
0.98 ? 0.01
`1
2.07 ? 0.01
Trace + `1
0.96 ? 0.01
Sequential
0.93 ? 0.08
?k,
0.59 ? 0.03
Table 2: Relative error of covariance estimation with different methods.
spectively non-overlapping or overlapping supports. The shapes of the different curves are overall
similar to the rank 1 case, although the performance of ?k,q degrades when the supports of atoms
overlap. In both cases, ?k,q consistently outperforms the two other norms. Overall these experiments suggest that the statistical dimension of ?k,q at a linear combination of r atoms increases as
Cr (k log m1 + q log m2 ) where the coefficient C increases with the overlap among the supports of
the atoms.
6.2
Comparison of algorithms for sparse PCA
In this section we compare the performance of different algorithms in estimating a sparsely factored
covariance matrix that we denote ?? . The observed sample consists of n i.i.d. random vectors generated according to N (0, ?? + ? 2 Idp ), where (k, k)-rank(?? ) = 3. The matrix ?? is formed by
>
adding 3 blocks of rank 1, ?? = a1 a>
a3 a>
1 + a2 a2 +?
3 , having all the same sparsity kai k0 = k = 10,
3 ? 3 overlaps and nonzero entries equal to 1/ k. The noise level ? = 0.8 is set in order to make
the signal to noise ratio below the level ? = 1 where a spectral gap appears and makes the spectral
baseline (penalizing the trace of the PSD matrix) work. In our experiments the number of variables
is p = 200 and n = 80 points are observed. To estimate the true covariance matrix from the noisy
? n = 1 Pn xi x> , and given as input
observation, first the sample covariance matrix is formed as ?
i
i=1
n
? The methods we compared are the following:
to various algorithms which provide a new estimate ?.
? n as the estimate of the covariance.
? Sample covariance. Output ?
? n elementwise.
? `1 penalty. Soft-threshold ?
? n k2 + ? Tr Z .
? Trace penalty on the PSD cone. minZ0 12 kZ ? ?
F
? n k2 + ??? (Z).
? Trace + `1 penalty. minZ0 12 kZ ? ?
F
? n k2 + ??k, (Z) , with ?k, the gauge associated with Ak,
? ?k, penalty. minZ?Rp?p 12 kZ ? ?
F
introduced in Section 2.3.
? Sequential sparse PCA. This is the standard way of estimating multiple sparse principal components which consists of solving the problem for a single component at each step t = 1 . . . r, and
deflate to switch to the next (t + 1)st component. The deflation step used in this algorithm is the
>
orthogonal projection Zt+1 = (Idp ? ut u>
t ) Zt (Idp ? ut ut ) . The tuning parameters for this approach are the sparsity level k and the number of principal components r. The hyperparameters
were chosen by leaving one portion of the train data off (validation) and selecting the parameter
which allows to build an estimator approximating the best the validation set?s empirical covariance.
We assumed the true value of k known in advance for all algorithms.
? ? ?? kF /k?? kF over 10 runs of our experiments in Table 2. The
We report the relative errors k?
results indicate that sparse PCA methods, whether based on ?k, or the sequential method with
deflation steps, outperform spectral and `1 baselines, and that penalizing ?k, is superior to the
sequential approach. This was to be expected since our algorithm minimizes a loss function close to
the error measure used, whereas the sequential scheme does not optimize a well-defined objective.
7
Conclusion
We formulated the problem of matrix factorization with sparse factors of known sparsity as the
minimization of an index, the (k, q)-rank which tight convex relaxation is the (k, q)-trace norm
regularizer. This penalty is proved to have near optimal statistical performance. Despite theoretical
computational hardness in the worst-case scenario, exploiting the convex geometry of the problem
allowed us to build an efficient algorithm to minimize it. Future work will consist of relaxing the
constraint on the blocks size, and exploring applications such as finding small comminuties in large
random graph background.
Acknowlegments This project was partially funded by Agence Nationale de la Recherche grant
ANR-13-MONU-005-10 (CHORUS project) and by ERC grant SMAC-ERC-280032.
8
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[21] H. Zou, T. Hastie, and R. Tibshirani. Sparse principal component analysis. Journal of computational and graphical statistics, 15(2):265?286, 2006.
9
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4,870 | 5,409 | Robust Tensor Decomposition with Gross Corruption
Huan Gui? Jiawei Han
Department of Computer Science
University of Illinois
at Urbana-Champaign
Urbana, IL 61801
{huangui2,hanj}@illinois.edu
Quanquan Gu?
Department of Operations Research
and Financial Engineering
Princeton University
Princeton, NJ 08544
[email protected]
Abstract
In this paper, we study the statistical performance of robust tensor decomposition
with gross corruption. The observations are noisy realization of the superposition
of a low-rank tensor W ? and an entrywise sparse corruption tensor V ? . Unlike
conventional noise with bounded variance in previous convex tensor decomposition
analysis, the magnitude of the gross corruption can be arbitrary large. We show
that under certain conditions, the true low-rank tensor as well as the sparse corruption tensor can be recovered simultaneously. Our theory yields nonasymptotic
Frobenius-norm estimation error bounds for each tensor separately. We show
through numerical experiments that our theory can precisely predict the scaling
behavior in practice.
1
Introduction
Tensor data analysis have witnessed increasing applications in machine learning, data mining and
computer vision. For example, an ensemble of face images can be modeled as a tensor, whose mode
corresponds to pixels, subjects, illumination and viewpoint [23]. Traditional tensor decomposition
methods such as Tucker decomposition and CANDECOMP/PARAFAC(CP) decomposition [14, 13]
aim to factorize an input tensor into a number of low-rank factors. However, they are prone to local
optima because they are solving essentially non-convex optimization problems. In order to address
this problem, [15] [20] extended the trace norm of matrices [19] to tensors, and generalized convex
matrix completion [8] [7] and matrix decomposition [6] to convex tensor completion/decomposition.
For example, the goal of tensor decomposition aims to accurately estimate a low-rank tensor W ?
Rn1 ?...?nK from the noisy observation tensor Y ? Rn1 ?...?nK that is contaminated by dense
noises, i.e., Y = W ? + E, where W ? ? Rn1 ?...?nK is a low-rank tensor, E ? Rn1 ?...?nK is a
noise tensor whose entries are i.i.d. Gaussian noise with zero mean and bounded variance ? 2 , i.e.,
Ei1 ,...,iK ? N (0, ? 2 ). [22] [21] analyzed the statistical performance of convex tensor decomposition
under different extensions of trace norm. They showed that, under certain conditions, the estimation
error scales with the rank of the true tensor W ? . Furthermore, they demonstrated that given a noisy
tensor, the true low-rank tensor can be recovered under restricted strong convexity assumption [18].
However, all these algorithms [15] [20] and theoretical results [22] [21] reply on the assumption that
the observation noise has a bounded variance ? 2 . Without this assumption, we are not able to identify
c could be very far from the true tensor
the rank of W ? , and therefore the estimated low-rank tensor W
W ?.
On the other hand, in many practical applications such as face recognition and image/video denoising,
a portion of the observation tensor Y might be contaminated by gross error due to illumination,
occlusion or pepper/salt noise. This scenario is not covered by finite variance noise assumption,
therefore new mathematical models are demanded to address this problem. This motivates us to study
?
Equal Contribution
1
convex tensor decomposition with gross corruption. It is clear that if all the entries of a tensor are
corrupted by large error, there is no hope to recover the underlying low-rank tensor. To overcome
this problem, one common assumption is that the gross corruption is sparse. Under this assumption,
together with previous low-rank assumption, we formalize the noisy linear observation model as
follows:
Y = W ? + V ? + E,
(1)
where W ? ? Rn1 ?...?nK is a low-rank tensor, V ? ? Rn1 ?...?nK is a sparse corruption tensor, where
the locations of nonzero entries are unknown and the magnitudes of the nonzero entries can be
arbitrarily large, and E ? Rn1 ?...?nK is a noise tensor whose entries are i.i.d. Gaussian noise with
zero mean and bounded variance ? 2 , and thus dense. Our goal is to recover the low-rank tensor W ? ,
as well as the sparse corruption tensor V ? . Note that in some applications, the corruption tensor is of
independent interest and needs to be recovered.
Given the observation model in (1), and the low-rank as well as sparse assumptions on W ? and E ?
respectively, we propose the following convex minimization to estimate the unknown low-rank tensor
W ? and the sparse corruption tensor E ? simultaneously:
2
arg min |||Y ? W ? V|||F + ?M |||W|||S1 + ?M |||V|||1 ,
W,V
(2)
where |||?|||S1 is tensor Schatten-1 norm [22], |||?|||1 is entry-wise `1 norm of tensors, and ?M and ?M
are positive regularization parameters. We call this optimization Robust Tensor Decomposition, which
can been seen as a generalization of convex tensor decomposition in [15] [20] [22]. The regularization
associated with the E encourages sparsity on the corruption tensor, where parameter ?M controls the
sparsity level. In this paper, we focus on the following questions: under what conditions for the size
of the tensor, the rank of the tensor, and the fraction (sparsity level) of the corruption so that: (i) (2) is
able to recover W ? and V ? with small estimator error? (ii) (2) is able to recover the exact rank of
W ? and the support of V ? ? We will present nonasymptotic error bounds to answer these questions.
Experiments on synthetic datasets validate our theoretical results.
The rest of this paper is arranged as follows. Related work is discussed in Section 2. Section 3
introduces the background and notations. Section 4 presents the main results. Section 5 provides
an ADMM algorithm to solve the problem, followed by the numerical experiments in Section 6.
Section 7 concludes this work with remarks.
2
Related Work
The problem of recovering the data under gross error has gained many attentions recently in matrix
decomposition. A large body of work have been proposed and analyzed statistically. For example,
[9] considered the problem of recovering an unknown low-rank and an unknown sparse matrix, given
the sum of the two matrices. [5] proposed a similar problem, namely robust principal component
analysis (RPCA), which studies the problem of recovering the low-rank and sparse matrices by
solving a convex program. [10] studied multi-task regression which decomposes the coefficient
matrix into two matrices, and imposes different group sparse regularization on two matrices. [25]
considered more general case, where the parameter matrix could be the superposition of more than
two matrices with different structurally constraints. Our paper extends [5] from two perspective: we
extend the problem from matrices to high-order tensors, and consider the additional noise setting.
We notice that [16] extended RPCA to tensors, which aims to recover the low-rank and sparse
tensors by solving a constrained convex program. However, our formulation departs from [16] in
that we consider not only the sparse corruption, but also the dense noise. We also note that low-rank
noisy matrix completion [17] and robust matrix decomposition [1] [12] have been studied in in
the high dimensional setting as well. Our model can be seen as the high-order extension of robust
matrix decomposition. This extension is nontrivial, because the treatment of the tensor trace norm
(Schatten-1 norm) is more complicated. More importantly, for the robust matrix decomposition
problem considered [1], only the sum of error bound of two matrices (low-rank matrix and the sparse
corruption matrix) can be obtained under the assumption of restricted strongly convexity. In contrast,
under a different condition, our analysis provides error bound for each tensor component (low-rank
tensor and the sparse corruption tensor) separately, making our results more appealing in practice
and of independent theoretical interest. Since the problem in [1] is a special case of our problem, our
2
technical tool can be directly applied to their problem and yields new error bounds on the low-rank
matrix as well as the sparse corruption matrix separately.
3
Notation and Background
Before proceeding, we define our notation and state assumptions that will appear in various parts of
the analysis. For more details about tensor algebra, please refer to [14].
Scalars are denoted by lower case letters (a, b, . . .), vectors by bold lower case letters (a, b, . . .),
matrices by bold upper case letters (A, B, . . .), and high-order tensors by calligraphic upper case
letters (A, B, . . .). A tensor is a higher order generalization of a vector (first order tensor) and a matrix
(second order tensor). From a multi-linear algebra view, tensor is a multi-linear mapping over a set of
vector spaces. The order of tensor A ? Rn1 ?...?n2 ?...?nK is K, where nk is the dimensionality of
the k-th order. Elements of A are denoted as Ai1 ...ik ...in , 1 ? ik ? nk . We denote the number of
QK
elements in A by N = k=1 nk .
The mode-k vectors of a K order tensor A are the nk dimensional vectors obtained from A by
varying index ik while keeping the other indices fixed. The mode-k vectors are the column vectors
of mode-k flattening matrix A(k) ? Rnk ?(n1 ...nk?1 nk+1 ...nK ) that results by mode-k flattening the
tensor A. For example, matrix column vectors are referred to as mode-1 vectors and matrix row
vectors are referred to as mode-2 vectors.
n1 ...n2 ...nK
The
, is defined as hA, Bi =
P scalar
P product of two tensors A, B ? R
.
.
.
A
B
=
vec(A)vec(B),
where
vec(?)
is
a
vectorization. The Frobenius
i
...i
i
...i
1
1
K
K
i1
iK
p
norm of a tensor A is |||A|||F = hA, Ai.
There are multiple ways to define tensor rank. In this paper, following [22], we define the rank of
a tensor based on the mode-k rank of a tensor. More specifically, the mode-k rank of a tensor X ,
denoted by rankk (X), is the rank of the mode-k unfolding X(k) (note that X(k) is a matrix, so its
rank is well-defined). Based on mode-k rank, we define the rank of tensor X as r(X ) = (r1 , . . . , rk )
if the mode-k rank is rk for k = 1, . . . , K. Note that the mode-k rank can be computed in polynomial
time, because it boils down to computing a matrix rank, whereas computing tensor rank [14] is NP
complete.
Similarly, we extend the trace norm (a.k.a. nuclear norm) of matrices [19] to tensors. The overlapped
PK
1
Schatten-1 norm is defined as |||X |||S1 = K
X is the mode-k unfolding
k=1 kX(k) kS1 , where
Pr (k)
of X , and k ? kS1 is the Schatten-1 norm for a matrix, kXkS1 = j=1 ?j (X), where ?j (X) is the
j-th largest singular value of X. The dual norm of the Schatten-1 norm is Schatten-? norm (a.k.a.,
spectral norm) as kXkS? = maxj=1,...,r ?j (X).
By H?older?s inequality, we have |hW, Xi| ? kWkS1 kXkS? . It is easy to prove a similar result for
the overlapped Schatten-1 norm and its dual norm. We have the following H?older-like inequality [22]:
|hW, X i| ? |||W|||S1 |||X |||S ? ? |||W|||S1 |||X |||mean ,
1
where |||X |||mean :=
1
K
PK
k=1
(3)
kX(k) kS? .
Pn
Pn
Moreover, we define `1 -norm and `? -norm for tensors that |||X |||1 = i11=1 . . . iKK=1 |Xi1 ,...,iK |,
|||X |||? = max1?i1 ?n1 . . . max1?iK ?nK |Xi1 ,...,iK |. By H?older?s inequality, we have |hW, X i| ?
|||W|||1 |||X |||? , and the following inequality relates the overlapped Schatten-1 norm with the Frobenius norm,
|||X |||S1 ?
K
X
?
rk |||X |||F .
(4)
k=1
Let W ? ? Rn1 ?...?nK be the low-rank tensor that we wish to recover. We assume that W ? is
?
of rank (r1 , . . . , rK ). Thus, for each k, we have W(k)
= Uk Sk Vk> , where Uk ? Rnk ?rk and
?
Vk ? Rrk ?nk are orthogonal matrices, which consist of left and right singular vectors of W(k)
,
rk ?rk
n1 ?...?nK
Sk ? R
is a diagonal matrix whose diagonal elements are singular values. Let ? ? R
3
be an arbitrary tensor, we define the mode-k orthogonal complement ?00k of its mode-k unfolding
?
?(k) ? Rnk ?N\k with respect to the true low-rank tensor W ? as follows
>
?00k = (Ink ? Uk U>
?\k ? Vk Vk ).
k )?(k) (IN
?0k
(5)
?00k
In addition
= ?(k) ?
is the component which has overlapped row/column space with the
?
unfolding of the true tensor W(k)
. Note that the decomposition ?(k) = ?0k + ?00k is defined for
each mode.
In [18], the concept of decomposibility and a large class of decomposable norms are discussed
at length. Of particular relevance to us is the decomposability of the Schatten-1 norm and `1 norm. We have the following equality, i.e., mode-k decomposibility of the Schatten-1 norm that
?
?
+ ?00k kS1 = kW(k)
kS1 + k?00k kS1 , k = 1, . . . , K. To note that the decomposibility is defined
kW(k)
on each mode. It is also easy to check the decomposibility of the `1 -norm.
Let V ? ? Rn1 ?...?nK be the gross corruption tensor that we wish to recover. We assume the the
?
gross
corruption is sparse, in that the cardinality s = |supp(V
)| of its support, S = supp(V ? ) =
?
(i1 , i2 , . . . , iK ) ? [n1 ] ? . . . ? [nK ]|Vi1 ,...,iK 6= 0 . This assumption leads to the inequality
?
between the `1 norm and the Forbenius norm that |||V ? |||1 ? s |||V ? |||F . Moreover, we have
?
?
n1 ?...?nK
|||V |||1 = |||VS |||1 . For any D ? R
, we have |||D|||1 = |||DS |||1 + |||DS c |||1 .
4
Main Results
To get a deep theoretical insight into the recovery property of robust tensor decomposition, we will
now present a set of estimation error bounds. Unlike the analysis in [1], where only the summation
of the estimation errors on the low-rank matrix and gross corruption matrix are analyzed, we aim
at obtaining the estimation error bounds on each tensor (the low-rank tensor and corrupted tensor)
separately. All the proofs can be found in the longer version of this paper.
Instead of considering the observation model in 1, we consider the following more general observation
model
yi = hW ? , Xi i + hV ? , Xi i + i , i = 1, . . . , M,
(6)
where Xi can be seen as an observation operator, and i ?s are i.i.d. zero mean Gaussian noise with
variance ? 2 . Our goal is to estimate an unknown rank (r1 , . . . , rk ) of tensor W ? ? Rn1 ?...?nK ,
as well as the unknown support of tensor V ? , from observations yi , i = 1, . . . , M . We propose
the following convex minimization to estimate the unknown low-rank tensor W ? and the sparse
corruption tensor V ? simultaneously, with composite regularizers on W and V as follows:
c V)
b = arg min 1 ky ? X(W + V)k22 + ?M |||W||| + ?M |||V||| ,
(W,
1
S1
W,V 2M
(7)
where y = (y1 , . . . , yM )> is the collection of observations, X(W) is the linear observation model
that X(W) = [hW, X1 i, . . . , hW, XM i]> . Note that (2) is a special case of (7), where the linear
operator the identity tensor, we have yi as observation of each element in the summation of tensors
W ? + V ?.
? >
We also define y? = (y1? , . . . , yM
) , where yi? = hW ? + V ? , Xi i, is the true evaluation. Due to the
noise of observation model, we have y = y? + . In addition, we define the adjoint operator of X as
PM
X? : RM ? Rn1 ?...?nK that X? () = i=1 i Xi .
4.1
Deterministic Bounds
c?W ?
This section is devoted to obtain the deterministic bound of the residual low-rank tensor ? = W
?
b
and residual corruption tensor D = V ? V separately, which makes our analysis unique.
c ? W ? and D = V
b ? V ? , obtained
We begin with a key technical lemma on residual tensors ? = W
from the convex problem in (7).
c and V
b be the solution of minimization problem (7) with ?M ? 2 |||X? ()|||
Lemma 1. Let W
mean /M ,
?
?M ? 2 |||X ()|||? /M , we have
4
1. rank(?0k ) ? 2rk .
2. There exist ?1 ? 3 and ?2 ? 3, such that
|||DS c |||1 ? ?2 |||DS |||1 .
PK
k=1
k?00k kS1 ? ?1
PK
k=1
k?0k kS1 and
c and V,
b as well as the decomposibility of
The lemma can be obtained by utilizing the optimality of W
Schatten-1 norm and `1 -norm of tensors.
Also, we obtain the key property of the optimal solution of (7), presented in the following theorem.
c and V
b be the solution of minimization problem (7) with ?M ?
Theorem 1. Let W
?
2 |||X ()|||mean /M , ?M ? 2 |||X? ()|||? /M , we have
K
1
3?M X
3?M
kX(? + D)k22 ?
k?0k kS1 +
|||DS |||1 .
2M
2K
2
(8)
k=1
Theorem 1 provides a deterministic prediction error bound for model (7). This is a very general
result, and can be applied to any linear operator X, including the robust tensor decomposition case
that we are particularly interested in this paper. It also covers, for example, tensor regression, tensor
compressive sensing, to mention a few.
Furthermore, we impose an assumption on the linear operator and the residual low-rank tensor and
residue sparse corruption tensor, which generalized the restricted eigenvalue assumption [2] [10].
PK
PK
Assumption 1. Defining ? = {(?, D)| k=1 k?00k kS1 ? ?1 k=1 k?0k kS1 , |||DS c |||1 ?
?2 |||DS |||1 }, we assume there exist positive scalars ?1 , ?2 that
kX(? + D)k2
kX(? + D)k2
> 0, ?2 = min ?
> 0.
?1 = min ?
?,D??
?,D??
M |||?|||F
M |||D|||F
Note that Assumption 1 is also related to restricted strong convexity assumption, which is proposed
in [18] to analyze the statistical properties of general M-estimators in the high dimensional setting.
Combing the results in Theorem 1 and Assumption 1, we have the following theorem, which
summarizes our main result.
cV
b be an optimal solution of (7), and take the regularization parameters ?M ?
Theorem 2. Let W,
2 |||X? ()|||mean /M , ?M ? 2 |||X? ()|||? /M . Then the following results hold:
?
? !
K
3
1 X ?M 2rk
?M s
c
?
+
,
(9)
W ? W ?
?1 K
?1
?2
F
k=1
?
? !
K
1 X ?M 2rk
?M s
3
b
?
.
(10)
+
V ? V ?
?2 K
?1
?2
F
k=1
Theorem 2 provides us with the error bounds of each tensor separately. Specifically, these bounds not
only measure how well our decomposition model can approximate the observation model defined
in (6), but also measure how well the decomposition of the true low-rank tensor and gross corruption
tensor is. When s = 0, our theoretical results reduce to that proposed in [22], which is a special case
of our problem, i.e., noisy low-rank tensor decomposition without corruption.
On the other hand, the results obtained in Theorem 2 are very appealing both practically and
theoretically. From the perspective of applications, this result is quite useful as it helps us to better
understand the behavior of each tensor separately. From the theoretical point of view, this result is
novel, and is incomparable with previous results [1][17] or simple generalization of previous results.
c and V,
b it is unclear whether the rank
Though Theorem 2 has provided estimation error bounds of W
?
?
of W and the support of V can be exactly recovered. We show that under some assumptions about
the true tensors, both of them can be exactly recovered.
Corollary 1. Under the same conditions of Theorem 2, if the following condition holds:
PK ?
?
? !
K
X
6(1
+
?
)
2r
1
?
2r
?
s
1
k
k
M
M
?
k=1
?rk (W(k)
)>
+
,
(11)
?1 M K
K
?1
?2
k=1
5
?
?
where ?rk (W(k)
) is the rk -th largest singular value of W(k)
, then
PK ?
?
? !
K
X
3(1
+
?
)
2r
1
?
2r
?
s
1
k
M
k
M
k=1
c (k) ) >
rbk = arg max ?r (W
+
r
?1 M K
K
?1
?2
k=1
?
recovers the rank of W(k)
for all k.
Furthermore, if the following condition holds:
min
i1 ,...,iK
then
Sb =
|Vi?1 ,...,iK |
?
6(1 + ?2 ) s
>
?2 M
?
? !
K
1 X ?M 2rk
?M s
,
+
K
?1
?2
(12)
k=1
?
3(1 + ?2 ) s
b
(i1 , i2 , . . . , iK ) : Vi1 ,...,iK >
?2 M
?
? !
K
1 X ?M 2rk
?M s
+
K
?1
?2
k=1
recovers the true support of V ? .
Corollary 1, basically states that, under the assumption that the singular values of the low-rank tensor
W ? , and the entry values of corruption tensor V ? are above the noise level (e.g., (11) and (12)), we
can recover the rank and the support successfully.
4.2
Noisy Tensor Decomposition
Now we are going back to study robust tensor decomposition with corruption in (2), which is a special
case of (7), where the linear operator is identity tensor. As the linear operator X is a vectorization such
that M = N , and kX(? + D)k
? 2 = |||? + D|||F . In addition, it is easy to show that Assumption 1
holds with ?1 = ?2 = O(1/ N ). It remains to bound |||X? ()|||mean and |||X? ()|||? , as shown in
the following lemma [1] [24].
Lemma 2. Suppose that X : Rn1 ?????nK ? RN is a vectorization of a tensor. Then we have with
?\k )) ? 1/N that
probability at least 1 ? 2 exp(?C(nk + N
K
q
? X ?
?
?
|||X ()|||mean ?
nk + N\k ,
K
k=1
p
|||X? ()|||? ? 4? log N ,
where C is a universal constant.
With Theorem 2 and Lemma 2, we immediately have the following estimation error bounds for robust
tensor decomposition.
is a vectorization
of a tensor. Then for the
Theorem 3. Suppose that X : Rn1 ?????nK ? RN q
?
PK
?
?
regularization constants ?N ? 2?
nk + N\k /(N K), ?N > 8? log N /N , with
k=1
?\k )) ? 1/N , any solution of (2) have the following error
probability at least 1 ? 2 exp(?C(nk + N
bound:
q
PK ?
!
?
?\k ?2rk
K
?
n
+
N
X
k
k=1
6
1
4? s log N
c
?
+
,
W ? W ?
?1 K
?1 N K
?2 N
F
k=1
q
?
PK ?
!
?
?
K ?
n
+
N
2rk
X
k
\k
k=1
6
1
4? s log N
b
?
+
.
V ? V ?
?2 K
?1 N K
?2 N
F
k=1
c
In the special case that n1 = . . . = nK = n and r1 = . . . = rK = r, we have W
? W ? =
F
?
?
?
?
b
O ? rnK?1 + ? Ks log n and V
? V ? = O ? rnK?1 + ? Ks log n , which matches
F
the error bound of robust matrix decomposition [1] when K = 2.
Note that the high probability support and rank recovery guarantee for the special case of tensor
decomposition follows immediately from Corollary 1. Due to the space limit, we omit the result here.
6
5
Algorithm
In this section, we present an algorithm to solve (2). Since (2) is a special case of (7), we consider
the more general problem (7). It is easy to show that (7) is equivalent to the following problem with
auxiliary variables ?, ?:
min
W,V,Y,Z
K
K
1
?M X
?M X
ky ? x> (w + v)k22 +
|||?k |||S1 +
|||?k |||1 ,
2M
K
K
k=1
k=1
subject to Pk w = ?k , Pk v = ?k ,
PM
where x, w, v, ?k , ?k are the vectorizations of i=1 Xi , W, V, ?k , ?k respectively, and Pk is the
transformation matrix that change the order of rows and columns so that Pk w = ?k .
The augmented Lagrangian (AL) function of the above minimization problem with respect to the
primal variables (W t , V t ) is given as follows:
K
K
K
L? (W, V, {?k }K
k=1 , {?k }k=1 , {?k }k=1 , {?k }k=1 )
K
K
?M M X
1
?M M X
= ky ? x> (w + v)k22 +
|||?k |||S1 +
|||?k |||1
2
K
K
k=1
k=1
!
X
X
1
1
2
>
2
>
(?k (Pk v ? ?k ) + kPk v ? ?k k2 ) ,
+?
(?k (Pk w ? ?k ) + kPk w ? ?k k2 ) +
2
2
k
k
t
t
where ? , ? are Lagrangian multiplier vectors, and ? > 0 is a penalty parameter.
We then apply the algorithm of Alternating Direction Method of Multipliers (ADMM)
[3, 20] to solve the above optimization problem.
Starting from initial points
0 K
0 K
0 K
(w0 , v0 , {?0k }K
,
{?
}
,
{?
}
,
{?
}
),
ADMM
performs
the following updates
k=1
k k=1
k k=1
k k=1
iteratively:
!
K
X
t
t
wt+1 = (x> y ? x> xvt ) + ?
P>
k (?k ? ?k ) / (1 + ?K) ,
k=1
v
t+1
=
>
>
(x y ? x xw
t+1
)+?
K
X
!
t
P>
k (?k
?
?kt )
/ (1 + ?K) ,
k=1
t+1
?t+1
= proxtr
+ ?tk ),
?M (Pk w
k
?K
?t+1
= ?t+1
+ (Pk wt+1 ? ?kt+1 )
k
k
1
?t+1
= prox`?M
(Pk vt+1 + ?kt )
k
k = 1, . . . , K,
?K
?kt+1 = ?kt+1 + (Pk vt+1 ? ?t+1
k )
k = 1, . . . , K,
`1
where proxtr
? (?) is the soft-thresholding operator for trace norm, and prox? (?) is the soft-thresholding
operator for `1 norm [4, 11]. The stopping criterion is that all the partial (sub)gradients are (near)
zero, under which condition we obtain the saddle point of the augmented Lagrangian function. Since
(7) is strictly convex, the saddle point is the global optima for the primal problem.
6
Experiments
In this section, we conduct numerical experiments to confirm our analysis in previous sections. The
experiments are conducted under the setting of robust noisy tensor decomposition.
We follow the procedure described in [22] for the experimental part. We randomly generate low-rank
tensors of dimensions n(1) = (50, 50, 20) ( results are shown in Figure 1(a, b, c)) and n(2) =
(100, 100, 50)( results are shown in Figure 1(d, e, f)) for various rank (r1 , r2 , ..., rk ). Given a specific
rank, we first generated the ?core tensor? with elements r1 ? . . . ? rK from the standard normal
distribution, and then multiplied each mode of the core tensor with an orthonormal factor randomly
drawn from the Haar measure. For the gross corruption, we randomly generated the sparsity of
the corruption matrix s, and then randomly selected s elements in which we put values randomly
generated from uniform distribution. The additive independent Gaussian noise with variance ? 2
7
Nr = 4.0
3
2
1
0
mean error of low?rank tensor
15
20
25
30
35
40
Ns
|||?|||F
M
?5
x 10
N = 2.9
r
11
10
Nr = 4.0
9
Nr = 4.9
8
7
6
5
4
10
15
20
25
Ns
|||?|||F
M
5
30
35
against Ns of size n(2) . (e)
7
Ns = 25
4
6
Ns = 35
5
3
4
3
2
2
1
1
0
3
3.5
4
4.5
5
0
5.5
0
1
2
3
Nr
|||?|||F
M
against Ns of size n(1) . (b)
12
?6
x 10
8
Ns = 17
4
5
6
7
?6
x 10
?1
against Nr of size n(1) .
(c) ?1 against ?2 of size n(1) .
?5
x 10
9
?6
x 10
Ns = 15.8
4
8
Ns = 22.4
3.5
7
6
Ns = 31.6
3
2.5
?2
10
(a)
(d)
Nr = 5.4
4
?4
x 10
6
?2
5
mean error of low?rank tensor
Nr = 2.9
mean error of low?rank tensor
mean error of low?rank tensor
?4
x 10
6
5
2
4
1.5
3
1
2
1
1.5
2
2.5
3
3.5
0.5
0.5
4
Nr
|||?|||F
M
against Nr of size n(2) .
1
1.5
2
2.5
?1
3
3.5
4
?6
x 10
(f) ?1 against ?2 of size n(2) .
Figure 1: Results of robust noisy tensor decomposition with corruption, under different sizes.
was added to the observations of elements. We use the alternating direction method of multipliers
(ADMM) to solve the minimization problem (2). The whole experiments were repeated 50 times and
the averaged results are reported.
?
?
The results are shown in Figure 1, where Nr = K
Ns = s. In Figure 1(a, d), we first fix
k=1
rk /K , and
c
Nr at different values, and then draw the value of W ? W ? /N against Ns . Similarly, in Figure 1(b,
F
c ? W ? /N against Nr . In Figure 1(c, f), we
e), we first fix Ns at different values, and then draw W
F
c ? W ? /N scales linearly
study the values of ?1 and ?2 at various settings. We can see that W
F
with both Ns and Nr . Similar scalings of Vb ? V ? /N can be observed, hence we omit them due
F
to space limitation. We can also observe from
Figure 1(c, f) that,
under
various settings, ?1 ? ?2 ,
c
this finding is consistent with the fact that W ? W ? /N ? Vb ? V ? /N . All these results are
F
F
consistent with each other, validating our theoretical analysis.
P
7
Conclusions
In this paper, we analyzed the statistical performance of robust noisy tensor decomposition with
corruption. Our goal is to recover a pair of tensors, based on observing a noisy contaminated version
of their sum. It is based on solving a convex optimization with composite regularizations of Schatten-1
norm and `1 norm defined on tensors. We provided a general nonasymptotic estimator error bounds on
the underly low-rank tensor and sparse corruption tensor. Furthermore, the error bound we obtained
in this paper is new, and non-comparable with previous theoretical analysis.
Acknowledgement
We would like to thank the anonymous reviewers for their helpful comments. Research was sponsored
in part by the Army Research Lab, under Cooperative Agreement No. W911NF-09-2-0053 (NSCTA),
the Army Research Office under Cooperative Agreement No. W911NF-13-1-0193, National Science
Foundation IIS-1017362, IIS-1320617, and IIS-1354329, HDTRA1-10-1-0120, and MIAS, a DHSIDS Center for Multimodal Information Access and Synthesis at UIUC.
8
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arXiv preprint
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4,871 | 541 | Network activity determines
spatio-temporal integration in single cells
Ojvind Bernander, Christof Koch *
Computation and Neural Systems Program,
California Institut.e of Technology,
Pasadena, Ca 91125, USA.
Rodney J. Douglas
Anatomical Neuropharmacology Unit,
Dept. Pharmacology,
Oxford, UK.
Abstract
Single nerve cells with static properties have traditionally been viewed
as the building blocks for networks that show emergent phenomena. In
contrast to this approach, we study here how the overall network activity
can control single cell parameters such as input resistance, as well as time
and space constants, parameters that are crucial for excitability and spatiotemporal integration. Using detailed computer simulations of neocortical
pyramidal cells, we show that the spontaneous background firing of the
network provides a means for setting these parameters. The mechanism
for this control is through the large conductance change of the membrane
that is induced by both non-NMDA and NMDA excitatory and inhibitory
synapses activated by the spontaneous background activity.
1
INTRODUCTION
Biological neurons display a complexity rarely heeded in abstract network models.
Dendritic trees allow for local interactions, attenuation, and delays. Voltage- and
*To whom all correspondence should be a.ddressed.
43
44
Bernander, Koch, and Douglas
time-dependent conductances can give rise to adaptation, burst-firing, and other
non-linear effects. The extent of temporal integration is determined by the time
constant, and spatial integration by the "leakiness" of the membrane. It is unclear
which cell properties are computationally significant and which are not relevant
for information processing, even though they may be important for the proper
functioning of the cell. However, it is crucial to understand the function of the
component cells in order to make relevant abstractions when modeling biological
systems. In this paper we study how the spontaneous background firing of the
network as a whole can strongly influence some of the basic integration properties
of single cells.
1.1
Controlling parameters via background synaptic activity
The input resistance, RJn, is defined as ~, where dV is the steady state voltage
change in response to a small current step of amplitude dI. RJn will vary throughout
the cell, and is typically much larger in a long, narrow dendrite than in the soma.
However, the somatic input resistance is more relevant to the spiking behavior of
the neuron, since spikes are initiated at or close to the soma, and hence Rin,.oma
(henceforth simply referred to as Rin) will tell us something of the sensitivity of the
cell to charge reaching the soma.
The time constant, Tm , for a passive membrane patch is Rm . em, the membrane
resistance times the membrane capacitance. For membranes containing voltagedependent non-linearities, exponentials are fitted to the step response and the
largest. time constant is taken to be the membrane time constant. A large time
constant implies that any injected charge leaks away very slowly, and hence the cell
has a longer "memory" of previous events.
The parameters discussed above (Rin, Tm) clearly have computational significance
and it would be convenient to be able to chanfe them dynamically. Both depend
directly on the membrane conductance G m = Jr.;' so any change in G m will change
the ?parameters. Traditionally, however, G m has been viewed as static, so these
parameters have also been considered static. How can we change G m dynamically?
In traditional models, G m has two components: active (time- and voltagedependent) conductances and a passive "leak" conductance. Synapses are modeled as conductance changes, but if only a few are activated, the cable structure
of the cell will hardly change at all. However, it is well known that neocortical
neurons spike spontaneously, in the absence of sensory stimuli, at rates from 0 to
10 Hz. Since neocortical neurons receive on the order of 5,000 to 15,000 excitatory
synapses (Larkman, 1991), this spontaneous firing is likely to add up to a large total
conductance (Holmes & Woody, 1989) . This synaptic conductance becomes crucial
if the non-synaptic conductance components are small. Recent evidence show indeed that the non-synaptic conductances are relatively small (when the cell is not
spiking) (Anderson et aI., 1990). Our model uses a leak Rm
100,000 kOcm 2 ,
instead of more conventional values in the range of 2,500-10,000 kOcm 2 ? These
two facts, high Rm and synaptic background activity, allow R in and Tm to change
by more than ten-fold, as described below in this paper.
=
Nerwork activity determines spatio-temporal integration in single cells
2
MODEL
A typical layer V pyramidal cell (fig. 2) in striate cortex was filled with HRP during in vivo experiments in the anesthetized, adult cat (Douglas et aI., 1991). The
3-D coordinates and diameters of the dendritic tree were measured by a computerassisted method and each branch was replaced by a single equivalent cylinder. This
morphological data was fed into a modified version of NEURON, an efficient single cell simulator developed by Hines (1989). The dendrites were passive, while the
soma contained seven active conductances, underlying spike generation, adaptation,
and slow onset for weak stimuli. The model included two sodium conductances (a
fast spiking current and a. fJlower non-inactivating current), one calcium conductance, and four potassium conductances (delayed rectifier, slow 'M' and 'A' type
currents, and a calcium-dependent current). The active conductances were modeled
using a Hodgkin-Huxley-like formalism.
The model used a total of 5,000 synapses. The synaptic conductance change
in time was modeled with an alpha function, get) = ~.,.,... e te- tlt .,.... 4,000
synapses were fast excitatory non-NMDA or AMPA-type (tped 1.5 msec, gpeaJ: =
0.5 nS, E retJ
0 mV), 500 were medium-slow inhibitory GABAA (tpe4k =
10 msec, gpeole 1.0 nS, E retJ -70 mV), and 500 were slow inhibitory GABAB
(tpeok 40 msec, gpeok 0.1 nS, E,.etJ -95 mV). The excitatory synapses were
less concentrated towards the soma, while the inhibitory ones were more so. For a
more detailed description of the model, see Bernander et al. (1991).
...=.
=
=
=
=
=
=
120r------------------------------,
~~-----------------------,
RIn, no~NMDA
Rin, no~MDA and NMDA
100
60
i
140
i
40
20
./
""
-
.....
.....
--- --- --
20
'"--
2
3
4
5
6
Background frequency (Hz)
7
2
,3
4
5
6
7
Background frequency (Hz)
Figure 1: Input resistance and time constant as a function of background
frequency. In (a), the solid line corresponds to the "standard" model with passive
dendrites, while the dashed line includes active NMDA synapses as described in the
text.
45
46
Bernander, Koch, and Douglas
3
3.1
RESULTS
R,n and
Tm
change with background frequency
Fig. 1 illustrates what happens to ~n and Tm when the synaptic background activities of all synaptic types are varied simultaneously. In the absence of any synaptic
input, ~n
110 Mn and Tm
80 msec. At 1 Hz background activity, on average 5 synaptic events are impinging on the cell every msec, contributing a total
of 24 nS to the somatic input conductance Gin. Because of the reversal potential
of the excitatory synapses (0 mV), the membrane potential throughout the cell is
pulled towards more depolarizing potentials, activating additional active currents.
Although these trends continue as f is increased, the largest change can be observed
between 0 and 2 Hz.
=
=
Figure 2: Spatial integration as a function of background frequency.
Each dendrite has been "stretched" so that its apparent length corresponds to its
electrotonic length. The synaptic background frequency was 0 Hz (left) and 2 Hz
(right). The scale bar corresponds to 1 A (length constant).
Activating synaptic input has two distinct effects: the conductance of the postsynaptic membrane increases and the membrane is depolarized. The system can,
at least in principle, independently control these two effects by differentially varying the spontaneous firing frequencies of excitatory versus inhibitory inputs. Thus,
increasing f selectively for the GABAB inhibition will further increase the membrane conductance but move the resting potential towards more hyperpolarizing
Network activity determines spatio-temporal integration in single cells
potentials.
Note that the 0 Hz ca?c corresponds to experiments made with in vitro slice preparations or culture. In this case incoming fibers have been cut off and the spontaneous
firing rate is very small. Careful studies have shown very large values for Rin and
Tm under these circumstances (e.g. Spruston &. Johnston, 1991). In vivo preparations, on the other hand, leave the cortical circuitry intact and much smaller values
of R,n and Tm are usually recorded.
3.2
Spatial integration
Varying synaptic background activity can have a significant impact on the electrotonic structure of the cell (fig. 2). We plot the electrotonic distance of any particular
point from the cell body, that is the sum of the electrotonic length's L, = Ej(lj/Aj)
=J
associated with each dendritic segment i, where Aj
~m.R~j is the electrotonic
length constant of compartment i, Ij its anatomical length and the sum is taken
over all compartments between the soma and compartment i.
=
=
Increasing the synaptic background activity from I
0 to f
2 Hz has the effect
of stretching the "distance" of any particular synapse t.o the soma by a factor of
about 3, on average. Thus, while a distal synapse has an associated L value of
about 2.6 at 2 Hz it shrinks to 1.2 if all network activity is shut off, while for a
synapse at the tip of a basal dendrite, L shrinks from 0.7 t.o 0.2. In fact, the EPSP
induced by a single excitatory synapse at that location goes from 39 to 151 J,lV, a
decrease of about 4. Thus, when the overall network activity is low, synapses in the
superficial layer of cortex could have a significant effect on somatic discharge, while
having only a weak modulatory effect on the soma if the overall network activity is
high. Note that basal dendrites, which receive a larger number of synapses, stretch
more than apical dendrites.
3.3
Temporal integration
That the synaptic background activity can also modify the temporal integration
behavior of the cell is demonstrated in fig. 3. At any part.icular background frequency I, we compute the minimal number of additional excitatory synapses (at
gpeal: = 0.5 nS) necessary to barely generate one action potential. These synapses
were chosen randomly from among all excitatory synapses throughout the cell. We
compare the case in which all synapses are activated simultaneously (solid line)
with the case in which the inputs arrive asynchronously, smeared out over 25 msec
(dashed line). If I = 0, it requires 115 synapses firing simultaneously to generate
a single action potential, while 145 are needed if the input is desynchronized. This
small difference between inputs arriving synchronized and at random is due to the
long integration period of the cell.
If the background activity increases to f = 1 Hz, 113 synchronized synaptic
inputs-spread out all over the cell-are sufficient to fire the cell. If, however,
the synaptic input is spread out over 25 msec, 202 synapses are now needed in
order to trigger a response from the cell. This is mainly due to the much smaller
value of Tm relative to the period over which the synaptic input is spread out. Note
47
48
Bernander, Koch, and Douglas
that the difference in number of simultaneous synaptic inputs needed to fire the
cell for f
0 compared to f = 1 is small (i.e. 113 vs. 115), in spite of the more
than five-fold decrease in somatic input resistance. The effect of the smaller size of
the individual EPSP at higher values of f is compensated for by the fact that the
resting potential of the cell has been shifted towards the firing threshold of the cell
(about -49 mY).
=
,~----------------------------------------------,
-:
800
Unsynchronized Input
Synchronized input
o
I
!
600
...=-
-
J
E
:;,
z
?OL-------'------~2------~3~----~4------~5~----~I~----~7
Background frequency (Hz)
Figure 3: Phase detection.
A variable number of excitatory synapses were fired superimposed onto a constant
background frequency of 1 Hz. They fired either simultaneously (solid line) or
spread out in time uniformly during a 25 msec interval (dashed line). The y axis
shows the minimum number of synapses necessary to cause the cell to fire.
3.4
NMDA synapses
Fast excitatory synaptic input in cortex is mediated by both AMPA or non-NMDA
as well as NMDA receptors (Miller et aI., 1989). As opposed to the AMPA synapse,
the NMDA conductance change depends not only on time but also on the postsynaptic voltage:
(1)
where '1'1 = 40 msec, '1'2 = 0.335 msec, '1 = 0.33 mM-t, [M g2+] - 1 mM,
r = 0.06 mV-1. During spontaneous background activity many inputs impinge
on the cell and we can time-average the equation above. We will then be left with
a purely voltage-dependent conductance.
We measured the somatic input resistance, Rin, by injecting a small current pulse in
the soma (fig. 4) in the standard model. All synapses fired at a 0.5 Hz background
frequency. Next we added 4,000 NMDA synapses in addition to the 4,000 non-
Network activity determines spatio-temporal integration in single cells
NMDA synapses, also at 0.5 Hz, and again injected a current pulse. The voltage
response is now larger by about 65%, corresponding to a smaller input conducta.nce,
even though we are adding the positive NMDA conductance. This seeming paradox
depends on two effects. First, the input conductance is, by definition, ~
G(V)+
(V - Ern), where G(V) is the conductance specified in eq. (1). For the
N DA synapse this derivative is negative below about -35 mV. Second, due to the
excitation the membrane voltage has drifted towards more depolarized values. This
will cause a change in the activation of the other voltage-dependent currents. Even
though the summed conductance of these active currents will be larger at the new
voltage, the derivative
will be smaller at that point. In other words, activation
of NMDA synapses gives a negative contribution to the input conductance, even
though more conductances have opened up.
=
di) .
'*"
Next we replaced 2,000 of the 4,000 non-NMDA synapses in the old model with
2,000 NMDA synapses and recomputed the input resistance as a function of synaptic background activity. The result is overlaid in figure 1a (dashed line). The curve
shifts toward larger values of Rin for most values of f. This shift varies between
50 % - 200 %. The cell is more excitable than before.
-60
>
E
E
>
-61
-62
-63
-64
-65
-66
0
400
200
t
600
800
1000
lmsecl
Figure 4: Negative input conductance from NMDA activation.
At times t
250 msec and t
750 msec a 0.05 nA current pulse was injected
at the soma and the somatic voltage response was recorded. At t = 500 msec,
one NMDA synapse was activated for each non-NMDA synapse, for a total of 8,000
excitatory synaptic inputs. The background frequency was 0.5 Hz for all synapses.
=
4
=
DISCUSSION
We have seen that parameters such as Rtn, 7'm, and L are not static, but can
vary over about one order of magnitude under network control. The potential
computational possibilities could be significant.
49
50
Bernander, Koch, and Douglas
For example, if a low-contrast stimulus is presented within the receptive field of
the cell, the synaptic input rate will be small and the signal-t~noise ratio (SNR)
low. In this case, to make the cell more sensitive to the inputs we might want to
increase R;n. This would automatically be achieved as the total network activation
is low. We can improve the SNR by integrating over a longer time period, i.e. by
increasing Tm. This would also be a consequence of the reduced network activity.
The converse argument can be made for high-contrast stimuli, associated with high
overall network activity and low R;n and Tm values.
Many cortical cells are tuned for various properties of the stimulus, such as orientation, direction, and binocular disparity. As the effective membrane conductance,
G m , changes, the tuning curves are expected to change. Depending on the exact
circuitry and implementation of the tuning properties, this change in background
frequency could take many forms. One example of phase-tuning was given above.
In this case the temporal tuning increases with background frequency.
Acknowledgements
This work was supported by the Office of Naval Research, the National Science
Foundation, the James McDonnell Foundation and the International Human Frontier Science Program Organization. Thanks to Tom Tromey for writing the graphic
software and to Mike Hines for providing us with NEURON.
References
P. Anderson, M. Raastad &, J. F. Storm. (1990) Excitatory synaptic integration in
hippocampal pyramids and dentate granule cells. Symp. Quant. Bioi. 55, Cold
Spring Harbor Press, pp. 81-86.
O.
Bernander, R. J. Douglas, K. A. C. Martin &, C. Koch. (1991) Synaptic
background activity influences spatiotemporal integration in single pyramidal cells.
P.N.A.S, USA 88: 11569-11573.
R. J. Douglas, K. A. C. Martin &, D. Whitteridge. (1991) An intracellular analysis
of the visual responses of neurones in cat visual cortex. J. Physiol. 440: 659-696.
M. Hines. (1989) A program for simulation of nerve equations with branching
geometries. Int. J. Biomed. Comput. 24: 55-68.
W. R. Holmes &, C. D. Woody. (1989) Effects of uniform and non-uniform synaptic activation-distributions on the cable properties of modeled cortical pyramidal
neurons. Brain Research 505: 12-22.
A. U. Larkman. (1991) Dendritic morphology of pyramidal neurones of the visual
cortex of the rat: III. Spine distributions. J. Compo Neurol. 306: 332-343.
K. D. Miller, B. Chapman &, M. P. Stryker. (1989) Responses of cells in cat visual
cortex depend on NMDA receptors. P.N.A.S. 86: 5183-5187.
N. Spruston &, D. Johnston. (1992) Perforated patch-clamp analysis of the passive
membrane properties of three classes of hippocampal neurons. J. Netlrophysiol., in
press.
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4,872 | 5,410 | PEWA: Patch-based Exponentially Weighted
Aggregation for image denoising
Charles Kervrann
Inria Rennes - Bretagne Atlantique
Serpico Project-Team
Campus Universitaire de Beaulieu, 35 042 Rennes Cedex, France
[email protected]
Abstract
Patch-based methods have been widely used for noise reduction in recent years.
In this paper, we propose a general statistical aggregation method which combines
image patches denoised with several commonly-used algorithms. We show that
weakly denoised versions of the input image obtained with standard methods, can
serve to compute an efficient patch-based aggregated estimator. In our approach,
we evaluate the Stein?s Unbiased Risk Estimator (SURE) of each denoised candidate image patch and use this information to compute the exponential weighted
aggregation (EWA) estimator. The aggregation method is flexible enough to combine any standard denoising algorithm and has an interpretation with Gibbs distribution. The denoising algorithm (PEWA) is based on a MCMC sampling and is
able to produce results that are comparable to the current state-of-the-art.
1
Introduction
Several methods have been proposed to solve the image denoising problem including anisotropic
diffusion [15], frequency-based methods [26], Bayesian and Markov Random Fields methods [20],
locally adaptive kernel-based methods [17] and sparse representation [10]. The objective is to estimate a clean image generally assumed to be corrupted with additive white Gaussian (AWG) noise.
In recent years, state-of-the-art results have been considerably improved and the theoretical limits of denoising algorithms are currently discussed in the literature [4, 14]. The most competitive
methods are mostly patch-based methods, such as BM3D [6], LSSC [16], EPLL [28], NL-Bayes
[12], inspired from the N(on)L(ocal)-means [2]. In the NL-means method, each patch is replaced
by a weighted mean of the most similar patches found in the noisy input image. BM3D combines
clustering of noisy patches, DCT-based transform and shrinkage operation to achieve the current
state-of-the-art results [6]. PLOW [5], S-PLE [24] and NL-Bayes [12], falling in the same category of the so-called internal methods, are able to produce very comparable results. Unlike BM3D,
covariances matrices of clustered noisy patches are empirically estimated to compute a Maximum
A Posteriori (MAP) or a Minimum-Mean-Squared-Error (MMSE) estimate. The aforementioned
algorithms need two iterations [6, 12, 18] and the performances are surprisingly very close to the
state-of-the-art in average while the motivation and the modeling frameworks are quite different. In
this paper, the proposed Patch-based Exponential Weighted Aggregation (PEWA) algorithm, requiring no patch clustering, achieves also the state-of-the-art results.
A second category of patch-based external methods (e.g. FoE [20], EPLL [28], MLP [3]) has been
also investigated. The principle is to approximate the noisy patches using a set of patches of an
external learned dictionary. The statistics of a noise-free training set of image patches, serve as
priors for denoising. EPLL computes a prior from a mixture of Gaussians trained with a database
of clean image patches [28]; denoising is then performed by maximizing the so-called Expected
Patch Log Likelihood (EPLL) criteria using an optimization algorithm. In this line of work, a multi1
layer perceptron (MLP) procedure exploiting a training set of noisy and noise-free patches was able
to achieve the state-of-the-art performance [3]. Nevertheless, the training procedure is dedicated
to handle a fixed noise level and the denoising method is not flexible enough, especially for real
applications when the signal-to-noise ratio is not known.
Recently, the similarity of patch pairs extracted from the input noisy image and from clean patch
dataset has been studied in [27]. The authors observed that more repetitions are found in the same
noisy image than in a clean image patch database of natural images; also, it is not necessary to
examine patches far from the current patch to find good matching. While the external methods
are attractive, computation is not always feasible since a very large collection of clean patches are
required to denoise all patches in the input image. Other authors have previously proposed to learn
a dictionary on the noisy image [10] or to combine internal and external information (LSSC) [16].
In this paper, we focus on internal methods since they are more flexible for real applications than
external methods. They are less computationally demanding and remain the most competitive.
Our approach consists in estimating an image patch from ?weakly? denoised image patches in the
input image. We consider the general problem of combining multiple basic estimators to achieve
an estimation accuracy not much worse than that of the ?best? single estimator in some sense. This
problem is important for practical applications because single estimators often do not perform as
well as their combinations. The most important and widely studied aggregation method that achieves
the optimal average risk is the Exponential Weighted Aggregation (EWA) algorithm [13, 7, 19].
Salmon & Le Pennec have already interpreted the NL-means as a special case of the EWA procedure
but the results of the extended version described in [21] were similar to [2].
Our estimator combination is then achieved through a two-step procedure, where multiple estimators
are first computed and are then combined in a second separate computing step. We shall see that
the proposed method can be thought as a boosting procedure [22] since the performance of the precomputed estimators involved in the first step are rather poor, both visually and in terms of peak
signal-to-noise ratio (PSNR). Our contributions are the following ones:
1. We show that ?weak? denoised versions of the input noisy images can be combined to get
a boosted estimator.
2. A spatial Bayesian prior and a Gibbs energy enable to select good candidate patches.
3. We propose a dedicated Monte Carlo Markov Chain (MCMC) sampling procedure to compute efficiently the PEWA estimator.
The experimental results are comparable to BM3D [6] and the method is implemented efficiently
since all patches can be processed independently.
2
Patch-based image representation and SURE estimation
Formally, we represent a n-dimensional image patch at location x ? X ? R2 as a vector f (x) ? Rn .
We define the observation patch v(x) ? Rn as: v(x) = f (x) + ?(x) where ?(x) ? N (0, ? 2 In?n )
represents the errors. We are interested in an estimator fb(x) of f (x) assumed to be independent of
f (x) that achieves a small L2 risk. We consider the Stein?s Unbiased Risk Estimator
R(fb(x)) = kv(x) ? fb(x)k2n ? n? 2
in the Mean Square Error sense such that E[R(fb(x))] = E[kf (x) ? fb(x)k2n ] (E denotes the mathematical expectation). SURE has been already investigated for image denoising using NL-means
[23, 9, 22, 24] and for image deconvolution in [25].
3
Aggregation by exponential weights
Assume a family {f? (x), ? ? ?} of functions such that the mapping ? ? f? (x) is measurable
and ? = {1, ? ? ? , M }. Functions f? (x) can be viewed as some pre-computed estimators of f (x) or
?weak? denoisers independent of observations v(x), and considered as frozen in the following. The
set of M estimators is assumed to be very large, that is composed of several hundreds of thousands
2
of candidates. In this paper, we consider aggregates that are weighted averages of the functions in
the set {f? (x), ? ? ?} with some data-dependent weights:
fb(x) =
M
X
w? (x) f? (x) such that w? (x) ? 0 and
?=1
M
X
w? (x) = 1.
(1)
?=1
As suggested in [19], we can associate two probability measures w(x) = {w1 (x), ? ? ? , wM (x)} and
?(x) = {?1 (x), ? ? ? , ?M (x)} on {1, ? ? ? , M } and we define the Kullback-Leibler divergence as:
DKL (w(x), ?(x)) =
M
X
w? (x) log
?=1
w? (x)
?? (x)
.
(2)
The exponential weights are obtained as the solution of the following optimization problem:
(M
)
X
b
w(x)
= arg min
w? (x)?(R(f? (x))) + ? DKL (w(x), ?(x)) subject to (1)
w(x)?RM
(3)
?=1
where ? > 0 and ?(z) is a function of the following form ?(z) = |z|. From the Karush-KuhnTucker conditions, the unique closed-form solution is
exp(??(R(f? (x)))/?) ?? (x)
,
w? (x) = PM
0
0
?0 =1 exp(??(R(f? (x)))/?) ?? (x)
(4)
where ? can be interpreted as a ?temperature? parameter. This estimator satisfies oracle inequalities
of the following form [7]:
(M
)
X
b
E[R(f (x))] ?
min
w? (x)?(R(f? (x))) + ? DKL (w(x), ?(x)) .
(5)
w(x)?RM
?=1
The role of the distribution ? is to put a prior weight on the functions in the set. When there is no
preference, the uniform prior is a common choice but other choices are possible (see [7]).
In the proposed approach, we define the set of estimators as the set of patches taken in denoised
versions of the input image v. The next question is to develop a method to efficiently compute the
sum in (1) since the collection can be very large. For a typical image of N = 512 ? 512 pixels,
we could potentially consider M = L ? N pre-computed estimators if we apply L denoisers to the
input image v.
4
PEWA: Patch-based EWA estimator
Suppose that we are given a large collection of M competing estimators. These basis estimators
can be chosen arbitrarily among the researchers favorite denoising algorithm: Gaussian, Bilateral,
Wiener, Discrete Cosine Transform or other transform-based filterings. Let us emphasize here that
the number of basic estimators M is not expected to grow and is typically very large (M is chosen
on the order of several hundreds of thousands). In addition, the essential idea is that these basic
estimators only slightly improve the PSNR values of a few dBs.
Let us consider u` , ` = 1, ? ? ? , L denoised versions of v. A given pre-computed patch estimator
f? (x) is then a n-dimensional patch taken in the denoised image u` at any location y ? X , in the
spirit of the NL-means algorithm which considers only the noisy input patches for denoising. The
proposed estimator is then more general since a set of denoised patches at a given location are used.
Our estimator is then of the following form if we choose ?(z) = |z|:
fb(x) =
L
1 X X ?|R(u` (y))|/?
e
?` (y) u` (y),
Z(x)
`=1 y?X
Z(x) =
L X
X
0
e?|R(u`0 (y ))|/? ?`0 (y) (6)
`0 =1 y 0 ?X
where Z(x) is a normalization constant. Instead of considering a uniform prior over the set of
denoised patches taken in the whole image, it is appropriate to encourage patches located in the
3
neighborhood of x [27]. This can be achieved by introducing a spatial Gaussian prior G? (z) ?
2
2
e?z /(2? ) in the definition as
fbPEWA (x)
=
L
1 X X ?|R(u` (y))|/?
e
G? (x ? y) u` (y).
Z(x)
(7)
`=1 y?X
The Gaussian prior has a significant impact on the performance of the EWA estimator. Moreover, the
practical performance of the estimator strongly relies on an appropriate choice of ?. This important
question has been thoroughly discussed in [13] and ? = 4? 2 is motivated by the authors. Finally,
our patch-based EWA (PEWA) estimator can be written in terms of energies and Gibbs distributions
as:
fbPEWA (x)
=
L
1 X X ?E(u` (y))
e
u` (y),
Z(x)
Z(x) =
L X
X
0
e?E(u`0 (y )) ,
(8)
`0 =1 y 0 ?X
`=1 y?X
kx ? yk22
|kv(x) ? u` (y)k2n ? n? 2 |
+
.
2
4?
2? 2
The sums in (8) cannot be computed, especially when we consider a large collection of estimators.
In that sense, it differs from the NL-means methods [2, 11, 23, 9] which exploits patches generally
taken in a neighborhood of fixed size. Instead, we propose a Monte-Carlo sampling method to
approximately compute such an EWA when the number of aggregated estimators is large [1, 19].
E(u` (y))
4.1
=
Monte-Carlo simulations for computation
Because of the high dimensionality of the problem, we need efficient computational algorithms,
and therefore we suggest a stochastic approach to compute the PEWA estimator. Let us consider a random process (Fn (x))n?0 consisting in an initial noisy patch F0 (x) = v(x). The proposed Monte-Carlo procedure recommended to compute the estimator is based on the following
Metropolis-Hastings algorithm:
Draw a patch by considering a two-stage drawing procedure:
? draw uniformly a value ` in the set {1, 2, ? ? ? , L}.
? draw a pixel y = yc + ?, y ? X , with ? ? N (0, I2?2 ? 2 ) and yc is the position of the
current patch. At the initialization yc = x.
u` (y) if ? ? e??E(u` (y)),Fn (x))
Define Fn+1 (x) as: Fn+1 (x) =
(9)
Fn (x) otherwise
4
where ? is a random variable: ? ? U [0, 1] and ?E(u` (y), Fn (x)) = E(u` (y)) ? E(Fn (x)).
If we assume the Markov chain is ergodic, homogeneous, reductible, reversible and stationary, for
any F0 (x), we have almost surely
lim
T ?+?
T
X
1
Fn (x) ? fbPEWA (x)
T ? Tb
(10)
n=Tb
where T is the maximum number of samples of the Monte-Carlo procedure. It is also recommended
to introduce a burn-in phase to get a more satisfying estimator. Hence, the first Tb samples are
discarded in the average The Metropolis-Hastings rule allows reversibility and then stationarity of
the Markov chain. The chain is irreducible since it is possible to reach any patch in the set of possible
considered patches. The convergence is ensured when T tends to infinity. In practice, T is assumed
to be high to get a reasonable approximation of fbPEWA (x). In our implementation, we set T ? 1000
and Tb = 250 to produce fast and satisfying results. To improve convergence speed, we can use
several chains instead of only one [21].
In the Metropolis-Hastings dynamics, some patches are more frequently selected than others at a
given location. The number of occurrences of a particular candidate patch can be then evaluated. In
constant image areas, there is probably no preference for any one patch over any other and a low
number of candidate patches is expected along image contours and discontinuities.
4
4.2
Patch overlapping and iterations
The next step is to extend the PEWA procedure at every position of the entire image. To avoid
block effects at the patch boundaries, we overlap the patches. As a result, for the pixels lying
in the overlapping regions, we obtain multiple EWA estimates. These competing estimates must
be fused or aggregated into the single final estimate. The final aggregation can be performed by a
weighted average of the multiple EWA estimates as suggested in [21, 5, 22]. The simplest method of
aggregating such multiple estimates is to average them using equal weights. Such uniform averaging
provided the best results in our experiments and amounts to fusing n independent Markov chains.
The proposed implementation proceeds in two identical iterations. At the first iteration, the estimation is performed using several denoised versions of the noisy image. At the second iteration,
the first estimator is used as an additional denoised image in the procedure to improve locally the
estimation as in [6, 12]. The second iteration improves the PSNR values in the range of 0.2 to 0.5
dB as demonstrated by the experiments presented in the next section. Note that the first iteration is
able to produce very satisfying results for low and medium levels of noise. In practical imaging, we
use the method described in [11] to estimate the noise variance ? 2 for real-world noisy images.
5
Experimental results
We evaluated the PEWA algorithm on 25 natural images showing natural, man-made, indoor and
outdoor scenes (see Fig. 1). Each original image was corrupted with white Gaussian noise with zero
mean and variance ? 2 . In our experiments, the best results are obtained with n = 7 ? 7 patches
and L = 4 images ul denoised with DCT-based transform [26] ; we consider three different DCT
shrinkage thresholds: 1.25?, 1.5? and 1.75? to improve the PSNR of 1 to 6 db at most, depending
on ? and images (see Figs. 2-3). The fourth image is the noisy input image itself. We evaluated
the algorithm with a larger number L of denoised images and the quality drops by 0.1 db to 0.3 db,
which is visually imperceptible. Increasing L suggest also to considering more than 1000 samples
since the space of candidate patches is larger. The prior neighborhood size corresponds to a disk of
radius ? = 7 pixels but it can be smaller.
Performances of PEWA and other methods are quantified in terms of PSNR values for several noise
levels (see Tables 1-3). Table 1 reports the results obtained with PEWA on each individual image for
different values of standard deviation of noise. Table 2 compares the average PSNR values on these
25 images obtained by PEWA (after 1 and 2 iterations) and two state-of-the-art denoising methods
[6, 12]. We used the implementations provided by the authors: BM3D (http://www.cs.tut.fi/?foi/GCFBM3D/) and NL-Bayes (www.ipol.im). The best PSNR values are in bold and the results are quantitatively quite comparable except for very high levels of noise. We compared PEWA to the baseline
NL-means [2] and DCT [26] (using the implementation of www.ipol.im) since they form the core of
PEWA. The PSNR values increases of 1.5 db and 1.35 db on average over NL-means and DCT respectively. Finally, we compared the results to the recent S-PLE method which uses SURE to guide
the probabilistic patch-based filtering described in [24]. Figure 2 shows the denoising results on the
noisy Valdemossa (? = 15), Man (? = 20) and Castle (? = 25) images denoised with BM3D,
NL-Bayes and PEWA. Visual quality of methods is comparable.
Table 3 presents the denoising results with PEWA if the pre-computed estimators are obtained with
a Wiener filtering (spatial domain1 ) and DCT-based transform [26]. The results of PEWA with 5 ? 5
or 7 ? 7 patches are also given in Table 3, for one and two iterations. Note that NL-means can be
considered as a special case of the proposed method in which the original noisy patches constitute
the set of ?weak? estimators. The MCMC-based procedure can be then considered as an alternative
procedure to the usual implementation of NL-means to accelerate summation. Accordingly, in Table
3 we added a fair comparison (7?7 patches) with the implementation of NL-means algorithm (IPOL
(ipol.im)) which restricts the search of similar patches in a neighborhood of 21 ? 21 pixels. In these
experiments, ?PEWA basic? (1 iteration) produced better results especially for ? ? 10. Finally we
compared these results with the most popular and competitive methods on the same images. The
PSNR values are selected from publications cited in the literature. LSSC and BM3D are the most
var(v(x)) ? a` ? 2
u` (x) = mean(v(x)) + max 0,
? (v(x) ? mean(v(x))), where ` = {1, 2, 3} and
var(v(x))
a1 = 0.15, a2 = 0.20, a3 = 0.25.
1
5
cameraman
(256 ? 256)
peppers
(256 ? 256)
house
(256 ? 256)
Lena
(512 ? 512)
maya
(313 ? 473)
asia
(313 ? 473)
aircraft
(473 ? 313)
panther
(473 ? 313)
castle
(313 ? 473)
young man
(313 ? 473)
tiger
(473 ? 313)
man on wall picture
(473 ? 313)
barbara
(512 ? 512)
boat
(512 ? 512)
man
(512 ? 512)
couple
(512 ? 512)
hill
(512 ? 512)
alley
(192 ? 128)
computer
(704 ? 469)
dice
(704 ? 469)
flowers
(704 ? 469)
girl
(704 ? 469)
traffic
(704 ? 469)
trees
(192 ? 128)
valldemossa
(769 ? 338)
?
Figure 1: Set of 25 tested images. Top left: images from the BM3D website (cs.tut.fi/foi/GCFBM3D/) ; Bottom left: images from IPOL (ipol.im); Right: images from the Berkeley segmentation
database (eecs.berkeley.edu/Research/Projects/CS/ vision/bsds/).
performant but PEWA is able to produce better results on several piecewise smooth images while
BM3D is more appropriate for textured images.
In terms of computational complexity, denoising a 512 ? 512 grayscale image with an unoptimized
implementation of our method in C++ take about 2 mins (Intel Core i7 64-bit CPU 2.4 Ghz). Recently, PEWA has been implemented in parallel since every patch can be processed independently
and the computational times become a few seconds.
6
Conclusion
We presented a new general two-step denoising algorithm based on non-local image statistics and
patch repetition, that combines ideas from the popular NL-means [6] and BM3D algorithms [6] and
theoretical results from the statistical literature on Exponentially Weighted Aggregation [7, 21]. The
first step of PEWA involves the computation of denoised images obtained with a separate collection of multiple denoisers (Wiener, DCT... ) applied to the input image. In the second step, the
set of denoised image patches are selectively exploited to compute an aggregated estimator. We
showed that the estimator can be computed in reasonable time using a Monte-Carlo Markov Chain
(MCMC) sampling procedure. If we consider DCT-based transform [6] in the first step, the results
are comparable in average to the state-of-the-art results. The PEWA method generalizes the NLmeans algorithm in some sense but share also common features with BM3D (e.g. DCT transform,
two-stage collaborative filtering). tches, contrary to NL-Bayes and BM3D. For future work, waveletbased transform, multiple image patch sizes, robust statistics and sparse priors will be investigated
to improve the results of the flexible PEWA method.
6
noisy (PSNR = 24.61)
PEWA (PSNR = 29.25)
BM3D [6] (PSNR = 29.19)
NL-Bayes [12] (PSNR = 29.22)
Figure 2: Comparison of algorithms. Valldemossa image corrupted with white Gaussian noise (? =
15). The PSNR values of the three images denoised with DCT-based transform [26] are combined
with PEWA are 27.78, 27.04 and 26.26.)
noisy
(PSNR = 20.18)
PEWA
(PSNR = 29.49)
BM3D [6]
(PSNR = 29.36)
NL-Bayes [12]
(PSNR = 29.48)
noisy
(PSNR = 22.11)
PEWA
(PSNR = 30.50)
BM3D [6]
(PSNR = 30.59)
NL-Bayes [12]
(PSNR = 30.60)
Figure 3: Comparison of algorithms. First row: Castle image corrupted with white Gaussian noise
(? = 25). The PSNR values of the three images denoised with DCT-based transform [26] and
combined with PEWA are 25.77, 24.26 and 22.85. Second row: Man image corrupted with white
Gaussian noise (? = 20). The PSNR values of the three images denoised with DCT-based transform
[26] and combined with PEWA are 27.42, 26.00 and 24.67.
7
Cameraman
Peppers
House
Lena
Barbara
Boat
Man
Couple
Hill
Alley
Computer
Dice
Flowers
Girl
Traffic
Trees
Valldemossa
Aircraft
Asia
Castle
Man Picture
Maya
Panther
Tiger
Young man
Average
?=5
38.20
38.00
39.56
38.57
38.09
37.12
37.68
37.35
37.01
36.29
39.04
46.82
43.48
43.95
37.85
34.88
36.65
37.59
38.67
38.06
37.78
34.72
38.53
36.92
40.79
38.54
? = 10
34.23
34.68
36.40
35.78
34.73
33.75
33.93
33.91
33.52
32.20
35.13
43.87
39.67
41.22
33.54
29.93
31.79
34.62
34.46
34.13
33.58
29.64
33.91
32.85
37.36
34.75
? = 15
31.98
32.75
34.86
34.12
32.86
31.94
31.93
31.98
31.69
29.98
32.81
42.05
37.47
39.52
31.13
27.49
29.25
33.00
32.25
32.02
31.27
27.17
31.56
30.63
35.58
32.67
? = 20
30.60
31.40
33.72
32.90
31.43
30.64
30.50
30.57
30.50
28.54
31.23
40.58
35.90
38.27
29.58
25.86
27.59
31.75
30.73
30.56
29.73
25.42
30.02
29.13
34.30
31.26
? = 25
29.48
30.30
32.77
31.89
30.28
29.65
29.50
29.48
29.56
27.46
30.01
39.36
34.55
37.33
28.48
24.69
26.37
30.72
29.60
29.49
28.44
24.28
28.83
27.99
33.25
30.15
? = 50
26.25
26.69
29.29
28.83
26.58
26.64
26.67
26.02
26.92
24.13
26.38
35.33
30.81
34.14
25.50
21.78
23.18
27.68
26.63
26.15
24.65
22.85
25.59
24.63
29.59
26.95
? = 100
22.81
22.84
25.35
25.65
22.95
23.63
24.15
23.27
24.49
21.37
23.27
30.82
27.53
30.50
22.90
20.03
20.71
24.99
24.32
23.09
21.50
18.17
22.75
21.90
25.20
23.76
Table 1: Denoising results on the 25 tested images for several values of ?. The PSNR values are
averaged over 3 experiments corresponding to 3 different noise realizations.
PEWA 1
PEWA 2
BM3D [6]
NL-Bayes [12]
S-PLE [24]
NL-means [2]
DCT [26]
?=5
38.27
38.54
38.64
38.60
38.17
37.44
37.81
? = 10
34.39
34.75
34.78
34.75
34.38
33.35
33.57
? = 15
32.26
32.67
32.68
32.48
32.35
31.00
31.87
? = 20
30.76
31.26
31.25
31.22
30.67
30.16
29.95
? = 25
29.62
30.15
30.19
30.12
29.77
28.96
28.97
? = 50
26.00
26.95
26.97
26.90
26.46
25.53
25.91
? = 100
22.35
23.76
24.08
23.65
23.21
22.29
23.08
Table 2: Average of denoising results over the 25 tested images for several values of ?. The experiments with NL-Bayes [12], S-PLE[24], NL-means [2] and DCT [26] have been performed using the
using the implementation of IPOL (ipol.im). The best PSNR values are in bold.
Image
Peppers
House
Lena
Barbara
(256 ? 256)
(256 ? 256)
(512 ? 512)
(512 ? 512)
?
5.00 15.00 25.00 50.00
5.00 15.00 25.00 50.00
5.00 15.00 25.00 50.00
5.00 15.00 25.00 50.00
PEWA 1 (W) (5?5)
PEWA 2 (W) (5?5)
PEWA 1 (W) (7 ?7)
PEWA 2 (W) (7 ?7)
PEWA 1 (D) (5 ?5)
PEWA 2 (D) (5 ?5)
PEWA 1 (D) (7 ?7)
PEWA 2 (D) (7 ?7)
PEWA Basic (7?7)
NL-means [2] (7?7)
BM3D [6]
NL-Bayes [12]
ND-SAFIR [11]
K-SVD [10]
LSSC [16]
PLOW [5]
SOP [18]
36.69 30.58 27.50 22.85
37.45 32.20 29.72 26.09
36.72 30.60 27.60 22.82
37.34 32.34 30.11 26.53
37.70 32.45 29.83 26.01
37.95 32.80 30.20 26.66
37.71 32.43 29.87 26.00
38.00 32.75 30.30 26.69
36.88 31.34 29.47 26.02
36.77 30.93 28.76 24.24
38.12 32.70 30.16 26.68
38.09 32.26 29.79 26.10
37.34 32.13 29.73 25.29
37.80 32.23 29.81 26.24
38.18 32.82 30.21 26.62
37.69 31.82 29.53 26.32
37.63 32.40 30.01 26.75
37.89 31.88 28.55 23.49
38.98 34.27 32.13 28.35
37.90 31.90 28.59 23.52
39.00 34.57 32.51 29.04
39.28 34.23 31.79 27.72
39.46 34.74 31.67 29.15
39.27 34.26 31.79 27.71
39.56 34.83 32.77 29.29
37.88 34.13 32.14 28.25
37.75 32.36 31.11 27.54
39.83 34.94 32.86 29.69
39.39 33.77 31.36 27.62
37.62 34.08 32.22 28.67
39.33 34.19 31.97 28.01
39.93 35.35 33.15 30.04
39.52 34.72 32.70 29.08
38.76 34.35 32.54 29.64
37.27 31.43 28.30 23.45
38.05 33.40 31.11 27.80
37.26 31.45 28.33 23.45
38.00 33.65 31.56 28.40
38.46 33.72 31.33 27.59
38.57 33.96 31.81 28.43
38.45 33.72 31.25 27.62
38.58 34.12 31.89 28.83
37.39 33.26 31.20 27.92
36.65 32.00 30.45 27.32
38.72 34.27 32.08 29.05
38.75 33.51 31.16 27.62
37.91 33.70 31.73 28.38
38.63 33.76 31.35 27.85
38.69 34.15 31.87 28.87
38.66 33.90 31.92 28.32
38.31 33.84 31.80 28.96
36.39 30.18 29.31 22.71
37.13 31.94 29.47 25.58
36.40 30.18 27.32 22.71
37.00 32.10 30.00 26.20
37.71 32.20 29.55 25.58
38.03 32.70 30.03 26.01
37.70 32.30 29.84 26.20
38.09 32.86 30.28 26.58
36.80 31.89 29.76 25.83
36.79 30.65 28.99 25.63
38.31 33.11 30.72 27.23
38.38 32.47 30.02 26.45
37.12 31.80 29.24 24.09
38.08 32.33 29.54 25.43
38.48 33.00 30.47 27.06
37.98 21.17 30.20 26.29
37.74 32.65 30.37 27.35
Table 3: Comparison of several versions of PEWA (W (Wiener), D (DCT), Basic) and competitive
methods on a few standard images corrupted with white Gaussian noise. The best PSNR values are
in bold (PSNR values from publications cited in the literature).
8
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4,873 | 5,411 | A Multi-World Approach to Question Answering
about Real-World Scenes based on Uncertain Input
Mateusz Malinowski
Mario Fritz
Max Planck Institute for Informatics
Saarbr?ucken, Germany
{mmalinow,mfritz}@mpi-inf.mpg.de
Abstract
We propose a method for automatically answering questions about images by
bringing together recent advances from natural language processing and computer
vision. We combine discrete reasoning with uncertain predictions by a multiworld approach that represents uncertainty about the perceived world in a bayesian
framework. Our approach can handle human questions of high complexity about
realistic scenes and replies with range of answer like counts, object classes, instances and lists of them. The system is directly trained from question-answer
pairs. We establish a first benchmark for this task that can be seen as a modern
attempt at a visual turing test.
1
Introduction
As vision techniques like segmentation and object recognition begin to mature, there has been an
increasing interest in broadening the scope of research to full scene understanding. But what is
meant by ?understanding? of a scene and how do we measure the degree of ?understanding?? Most
often ?understanding? refers to a correct labeling of pixels, regions or bounding boxes in terms of
semantic annotations. All predictions made by such methods inevitably come with uncertainties
attached due to limitations in features or data or even inherent ambiguity of the visual input.
Equally strong progress has been made on the language side, where methods have been proposed
that can learn to answer questions solely from question-answer pairs [1]. These methods operate on
a set of facts given to the system, which is refered to as a world. Based on that knowledge the answer
is inferred by marginalizing over multiple interpretations of the question. However, the correctness
of the facts is a core assumption.
We like to unite those two research directions by addressing a question answering task based on realworld images. To combine the probabilistic output of state-of-the-art scene segmentation algorithms,
we propose a Bayesian formulation that marginalizes over multiple possible worlds that correspond
to different interpretations of the scene.
To date, we are lacking a substantial dataset that serves as a benchmark for question answering on
real-world images. Such a test has high demands on ?understanding? the visual input and tests a
whole chain of perception, language understanding and deduction. This very much relates to the
?AI-dream? of building a turing test for vision. While we are still not ready to test our vision system
on completely unconstrained settings that were envisioned in early days of AI, we argue that a
question-answering task on complex indoor scenes is a timely step in this direction.
Contributions: In this paper we combine automatic, semantic segmentations of real-world scenes
with symbolic reasoning about questions in a Bayesian framework by proposing a multi-world
approach for automatic question answering. We introduce a novel dataset of more than 12,000
1
question-answer pairs on RGBD images produced by humans, as a modern approach to a visual
turing test. We benchmark our approach on this new challenge and show the advantages of our
multi-world approach. Furthermore, we provide additional insights regarding the challenges that lie
ahead of us by factoring out sources of error from different components.
2
Related work
Semantic parsers: Our work is mainly inspired by [1] that learns the semantic representation for
the question answering task solely based on questions and answers in natural language. Although
the architecture learns the mapping from weak supervision, it achieves comparable results to the
semantic parsers that rely on manual annotations of logical forms ([2], [3]). In contrast to our work,
[1] has never used the semantic parser to connect the natural language to the perceived world.
Language and perception: Previous work [4, 5] has proposed models for the language grounding
problem with the goal of connecting the meaning of the natural language sentences to a perceived
world. Both methods use images as the representation of the physical world, but concentrate rather
on constrained domain with images consisting of very few objects. For instance [5] considers only
two mugs, monitor and table in their dataset, whereas [4] examines objects such as blocks, plastic
food, and building bricks. In contrast, our work focuses on a diverse collection of real-world indoor
RGBD images [6] - with many more objects in the scene and more complex spatial relationship
between them. Moreover, our paper considers complex questions - beyond the scope of [4] and [5]
- and reasoning across different images using only textual question-answer pairs for training. This
imposes additional challenges for the question-answering engines such as scalability of the semantic
parser, good scene representation, dealing with uncertainty in the language and perception, efficient
inference and spatial reasoning. Although others [7, 8] propose interesting alternatives for learning
the language binding, it is unclear if such approaches can be used to provide answers on questions.
Integrated systems that execute commands: Others [9, 10, 11, 12, 13] focus on the task of learning the representation of natural language in the restricted setting of executing commands. In such
scenario, the integrated systems execute commands given natural language input with the goal of using them in navigation. In our work, we aim for less restrictive scenario with the question-answering
system in the mind. For instance, the user may ask our architecture about counting and colors (?How
many green tables are in the image??), negations (?Which images do not have tables??) and superlatives (?What is the largest object in the image??).
Probabilistic databases: Similarly to [14] that reduces Named Entity Recognition problem into the
inference problem from probabilistic database, we sample multiple-worlds based on the uncertainty
introduced by the semantic segmentation algorithm that we apply to the visual input.
3
Method
Our method answers on questions based on images by combining natural language input with output
from visual scene analysis in a probabilistic framework as illustrated in Figure 1. In the single world
approach, we generate a single perceived world W based on segmentations - a unique interpretation
of a visual scene. In contrast, our multi-world approach integrates over many latent worlds W, and
hence taking different interpretations of the scene and question into account.
Single-world approach for question answering problem We build on recent progress on end-toend question answering systems that are solely trained on question-answer pairs (Q, A) [1]. Top part
of Figure 1 outlines how we build on [1] by modeling the logical forms associated with a question as
latent variable T given a single world W. More formally the task of predicting an answer A given
a question Q and a world W is performed by computing the following posterior which marginalizes
over the latent logical forms (semantic trees in [1]) T :
X
P (A|Q, W) :=
P (A|T , W)P (T |Q).
(1)
T
P (A|T , W) corresponds to denotation of a logical form T on the world W. In this setting,
the answer is unique given the logical form and the world: P (A|T , W) = 1[A ? ?W (T )]
with the evaluation function ?W , which evaluates a logical form on the world W. Following
[1] we use DCS Trees that yield the following recursive evaluation function ?W : ?W (T ) :=
2
single?
world
approach
Semantic
parsing
Q
Semantic
evaluation
T
A
logical
form
question
S
W
world
answer
sofa (1,brown, image 1, X,Y,Z)
table (1,brown, image 1,X,Y,Z)
wall (1,white, image 1, X,Y,Z)
Scene
analysis
bed (1, white, image 2 X,Y,Z)
chair (1,brown, image 4, X,Y,Z)
chair (2,brown, image 4, X,Y,Z)
chair (1,brown, image 5, X,Y,Z)
?
multi-world
approach
Semantic
parsing
Q
Semantic
evaluation
T
A
logical
form
question
W
S
latent
worlds
answer
segmentation
Figure 1: Overview of our approach to question answering with multiple latent worlds in contrast to single
world approach.
Td
j {v : v ? ?W (p), t ? ?W (Tj ), Rj (v, t)} where T := hp, (T1 , R1 ), (T2 , R2 ), ..., (Td , Rd )i is
the semantic tree with a predicate p associated with the current node, its subtrees T1 , T2 , ..., Td , and
relations Rj that define the relationship between the current node and a subtree Tj .
In the predictions, we use a log-linear distribution P (T |Q) ? exp(?T ?(Q, T )) over the logical
forms with a feature vector ? measuring compatibility between Q and T and parameters ? learnt
from training data. Every component ?j is the number of times that a specific feature template
occurs in (Q, T ). We use the same templates as [1]: string triggers a predicate, string is under a
relation, string is under a trace predicate, two predicates are linked via relation and a predicate has
a child. The model learns by alternating between searching over a restricted space of valid trees
and gradient descent updates of the model parameters ?. We use the Datalog inference engine to
produce the answers from the latent logical forms. The linguistic phenomena such as superlatives
and negations are handled by the logical forms and the inference engine. For a detailed exposition,
we refer the reader to [1].
Question answering on real-world images based on a perceived world Similar to [5], we
extend the work of [1] to operate now on what we call perceived world W. This still corresponds to the single world approach in our overview Figure 1. However our world is now populated with ?facts? derived from automatic, semantic image segmentations S. For this purpose, we
build the world by running a state-of-the-art semantic segmentation algorithm [15] over the images and collect the recognized information about objects such as object class, 3D position, and
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(b) Object?s coordinates.
Figure 2: Fig. 2a shows a few sampled worlds where only segments of the class ?person? are shown. In the
clock-wise order: original picture, most confident world, and three possible worlds (gray-scale values denote
the class confidence). Although, at first glance the most confident world seems to be a reasonable approach,
our experiments show opposite - we can benefit from imperfect but multiple worlds. Fig. 2b shows object?s
coordinates (original and Z, Y , X images in the clock-wise order), which better represent the spatial location
of the objects than the image coordinates.
3
auxiliary relations
spatial
Predicate
closeAbove(A, B)
closeLef tOf (A, B)
closeInF rontOf (A, B)
Xaux (A, B)
Zaux (A, B)
haux (A, B)
vaux (A, B)
daux (A, B)
lef tOf (A, B)
above(A, B)
inF rontOf (A, B)
on(A, B)
close(A, B)
Definition
above(A, B) and (Ymin (B) < Ymax (A) + )
lef tOf (A, B) and (Xmin (B) < Xmax (A) + )
inF rontOf (A, B) and (Zmin (B) < Zmax (A) + )
Xmean (A) < Xmax (B) and Xmin (B) < Xmean (A)
Zmean (A) < Zmax (B) and Zmin (B) < Zmean (A)
closeAbove(A, B) or closeBelow(A, B)
closeLef tOf (A, B) or closeRightOf (A, B)
closeInF rontOf (A, B) or closeBehind(A, B)
Xmean (A) < Xmean (B))
Ymean (A) < Ymean (B)
Zmean (A) < Zmean (B))
closeAbove(A, B) and Zaux (A, B) and Xaux (A, B)
haux (A, B) or vaux (A, B) or daux (A, B)
Table 1: Predicates defining spatial relations between A and B. Auxiliary relations define actual spatial relations. The Y axis points downwards, functions Xmax , Xmin , ... take appropriate values from the tuple
predicate, and is a ?small? amount. Symmetrical relations such as rightOf , below, behind, etc. can readily
be defined in terms of other relations (i.e. below(A, B) = above(B, A)).
color [16] (Figure 1 - middle part). Every object hypothesis is therefore represented as an n-tuple:
predicate(instance id, image id, color, spatial loc) where predicate ? {bag, bed, books, ...},
instance id is the object?s id, image id is id of the image containing the object, color is estimated color of the object [16], and spatial loc is the object?s position in the image. Latter is
represented as (Xmin , Xmax , Xmean , Ymin , Ymax , Ymean , Zmin , Zmax , Zmean ) and defines minimal, maximal, and mean location of the object along X, Y, Z axes. To obtain the coordinates we fit
axis parallel cuboids to the cropped 3d objects based on the semantic segmentation. Note that the
X, Y, Z coordinate system is aligned with direction of gravity [15]. As shown in Figure 2b, this is
a more meaningful representation of the object?s coordinates over simple image coordinates. The
complete schema will be documented together with the code release.
We realize that the skilled use of spatial relations is a complex task and grounding spatial relations
is a research thread on its own (e.g. [17], [18] and [19]). For our purposes, we focus on predefined
relations shown in Table 1, while the association of them as well as the object classes are still dealt
within the question answering architecture.
Multi-worlds approach for combining uncertain visual perception and symbolic reasoning
Up to now we have considered the output of the semantic segmentation as ?hard facts?, and hence
ignored uncertainty in the class labeling. Every such labeling of the segments corresponds to different interpretation of the scene - different perceived world. Drawing on ideas from probabilistic
databases [14], we propose a multi-world approach (Figure 1 - lower part) that marginalizes over
multiple possible worlds W - multiple interpretations of a visual scene - derived from the segmentation S. Therefore the posterior over the answer A given question Q and semantic segmentation S
of the image marginalizes over the latent worlds W and logical forms T :
XX
P (A | Q, S) =
P (A | W, T )P (W | S) P (T | Q)
(2)
W
T
The semantic segmentation of the image is a set of segments si with the associated probabilities
pij over the C object categories cj . More precisely S = {(s1 , L1 ), (s2 , L2 ), ..., (sk , Lk )} where
C
Li = {(cj , pij )}j=1 , P (si = cj ) = pij , and k is the number of segments of given image. Let
S?f = (s1 , cf (1) ), (s2 , cf (2) ), ..., (sk , cf (k) )) be an assignment of the categories into segments of
the image according to the binding function f ? F = {1, ..., C}{1,...,k} . With such notation, for
aQfixed binding function f , a world W is a set of tuples consistent with S?f , and define P (W |S) =
k
i p(i,f (i)) . Hence we have as many possible worlds as binding functions, that is C . Eq. 2 becomes
quickly intractable for k and C seen in practice, wherefore we use a sampling strategy that draws a
~ = (W1 , W2 , ..., WN ) from P (?|S) under an assumption that for each segment si
finite sample W
every object?s category cj is drawn independently according to pij . A few sampled perceived worlds
are shown in Figure 2a.
P
Regarding the computational efficiency, computing T P (A | Wi , T )P (T | Q) can be done independently for every Wi , and therefore in parallel without any need for synchronization. Since for
small N the computational costs of summing up computed probabilities is marginal, the overall cost
is about the same as single inference modulo parallelism. The presented multi-world approach to
question answering on real-world scenes is still an end-to-end architecture that is trained solely on
the question-answer pairs.
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Individual
Figure 3: NYU-Depth V2 dataset: image, Z axis, ground truth and predicted semantic segmentations.
Description
counting
counting and colors
room type
superlatives
counting and colors
negations type 1
negations type 2
negations type 3
Template
How many {object} are in {image id}?
How many {color} {object} are in {image id}?
Which type of the room is depicted in {image id}?
What is the largest {object} in {image id}?
How many {color} {object}?
Which images do not have {object}?
Which images are not {room type}?
Which images have {object} but do not have a {object}?
Example
How many cabinets are in image1?
How many gray cabinets are in image1?
Which type of the room is depicted in image1?
What is the largest object in image1?
How many black bags?
Which images do not have sofa?
Which images are not bedroom?
Which images have desk but do not have a lamp?
Table 2: Synthetic question-answer pairs. The questions can be about individual images or the sets of images.
Implementation and Scalability For worlds containing many facts and spatial relations the induction step becomes computationally demanding as it considers all pairs of the facts (we have
about 4 million predicates in the worst case). Therefore we use a batch-based approximation in such
situations. Every image induces a set of facts that we call a batch of facts. For every test image, we
find k nearest neighbors in the space of training batches with a boolean variant of TF.IDF to measure similarity [20]. This is equivalent to building a training world from k images with most similar
content to the perceived world of the test image. We use k = 3 and 25 worlds in our experiments.
Dataset and the source code can be found in our website 1 .
4
Experiments
4.1 DAtaset for QUestion Answering on Real-world images (DAQUAR)
Images and Semantic Segmentation Our new dataset for question answering is built on top of
the NYU-Depth V2 dataset [6]. NYU-Depth V2 contains 1449 RGBD images together with annotated semantic segmentations (Figure 3) where every pixel is labeled into some object class with a
confidence score. Originally 894 classes are considered. According to [15], we preprocess the data
to obtain canonical views of the scenes and use X, Y , Z coordinates from the depth sensor to define
spatial placement of the objects in 3D. To investigate the impact of uncertainty in the visual analysis
of the scenes, we also employ computer vision techniques for automatic semantic segmentation. We
use a state-of-the-art scene analysis method [15] which maps every pixel into 40 classes: 37 informative object classes as well as ?other structure?, ?other furniture? and ?other prop?. We ignore the
latter three. We use the same data split as [15]: 795 training and 654 test images. To use our spatial
representation on the image content, we fit 3d cuboids to the segmentations.
New dataset of questions and answers In the spirit of a visual turing test, we collect question
answer pairs from human annotators for the NYU dataset. In our work, we consider two types of the
annotations: synthetic and human. The synthetic question-answer pairs are automatically generated
question-answer pairs, which are based on the templates shown in Table 2. These templates are then
instantiated with facts from the database. To collect 12468 human question-answer pairs we ask 5
in-house participants to provide questions and answers. They were instructed to give valid answers
that are either basic colors [16], numbers or objects (894 categories) or sets of those. Besides the
answers, we don?t impose any constraints on the questions. We also don?t correct the questions as
we believe that the semantic parsers should be robust under the human errors. Finally, we use 6794
training and 5674 test question-answer pairs ? about 9 pairs per image on average (8.63, 8.75)2 .
1
https://www.d2.mpi-inf.mpg.de/visual-turing-challenge
Our notation (x, y) denotes mean x and trimean y. We use Tukey?s trimean 41 (Q1 + 2Q2 + Q3 ), where Qj
denotes the j-th quartile [21]. This measure combines the benefits of both median (robustness to the extremes)
and empirical mean (attention to the hinge values).
2
5
The database exhibit some biases showing humans tend to focus on a few prominent objects. For
instance we have more than 400 occurrences of table and chair in the answers. In average the
object?s category occurs (14.25, 4) times in training set and (22.48, 5.75) times in total. Figure 4
shows example question-answer pairs together with the corresponding image that illustrate some of
the challenges captured in this dataset.
Performance Measure While the quality of an answer that the system produces can be measured
in terms of accuracy w.r.t. the ground truth (correct/wrong), we propose, inspired from the work
on Fuzzy Sets [22], a soft measure based on the WUP score [23], which we call WUPS (WUP
Set) score. As the number of classes grows, the semantic boundaries between them are becoming
more fuzzy. For example, both concepts ?carton? and ?box? have similar meaning, or ?cup? and
?cup of coffee? are almost indifferent. Therefore we seek a metric that measures the quality of an
answer and penalizes naive solutions where the architecture outputs too many or too few answers.
PN
Standard Accuracy is defined as: N1 i=1 1{Ai = T i } ? 100 where Ai , T i are i-th answer and
ground-truth respectively. Since both the answers may include more than one object, it is beneficial
to represent them as sets of the objects T = {t1 , t2 , ...}. From this point of view we have for every
i ? {1, 2, ..., N }:
1{Ai = T i } = 1{Ai ? T i ? T i ? Ai } = min{1{Ai ? T i }, 1{T i ? Ai }}
Y
Y
Y
Y
= min{
1{a ? T i },
1{t ? Ai }} ? min{
?(a ? T i ),
?(t ? Ai )}
a?Ai
t?T i
a?Ai
(3)
(4)
t?T i
We use a soft equivalent of the intersection operator in Eq. 3, and a set membership measure ?,
with properties ?(x ? X) = 1 if x ? X, ?(x ? X) = maxy?X ?(x = y) and ?(x = y) ? [0, 1],
in Eq. 4 with equality whenever ? = 1. For ? we use a variant of Wu-Palmer similarity [23, 24].
WUP(a, b) calculates similarity based on the depth of two words a and b in the taxonomy[25, 26],
and define the WUPS score:
N
Y
Y
1 X
WUPS(A, T ) =
min{
maxi WUP(a, t),
max WUP(a, t)} ? 100
(5)
N i=1
t?T
a?Ai
i
i
a?A
t?T
Empirically, we have found that in our task a WUP score of around 0.9 is required for precise
answers. Therefore we have implemented down-weighting WUP(a, b) by one order of magnitude
(0.1 ? WUP) whenever WUP(a, b) < t for a threshold t. We plot a curve over thresholds t ranging
from 0 to 1 (Figure 5). Since ?WUPS at 0? refers to the most ?forgivable? measure without any downweighting and ?WUPS at 1.0? corresponds to plain accuracy. Figure 5 benchmarks architectures by
requiring answers with precision ranging from low to high. Here we show some examples of the pure
WUP score to give intuitions about the range: WUP(curtain, blinds) = 0.94, WUP(carton, box) =
0.94, WUP(stove, fire extinguisher) = 0.82.
4.2 Quantitative results
We perform a series of experiments to highlight particular challenges like uncertain segmentations, unknown true logical forms, some linguistic phenomena as well as show the advantages of
our proposed multi-world approach. In particular, we distinguish between experiments on synthetic question-answer pairs (SynthQA) based on templates and those collected by annotators (HumanQA), automatic scene segmentation (AutoSeg) with a computer vision algorithm [15] and human segmentations (HumanSeg) based on the ground-truth annotations in the NYU dataset as well
as single world (single) and multi-world (multi) approaches.
4.2.1 Synthetic question-answer pairs (SynthQA)
Based on human segmentations (HumanSeg, 37 classes) (1st and 2nd rows in Table 3) uses automatically generated questions (we use templates shown in Table 2) and human segmentations.
We have generated 20 training and 40 test question-answer pairs per template category, in total 140
training and 280 test pairs (as an exception negations type 1 and 2 have 10 training and 20 test examples each). This experiment shows how the architecture generalizes across similar type of questions
provided that we have human annotation of the image segments. We have further removed negations
of type 3 in the experiments as they have turned out to be particularly computationally demanding.
Performance increases hereby from 56% to 59.9% with about 80% training Accuracy. Since some
incorrect derivations give correct answers, the semantic parser learns wrong associations. Other difficulties stem from the limited training data and unseen object categories during training.
Based on automatic segmentations (AutoSeg, 37 classes, single) (3rd row in Table 3) tests the architecture based on uncertain facts obtained from automatic semantic segmentation [15] where the
6
most likely object labels are used to create a single world. Here, we are experiencing a severe drop
in performance from 59.9% to 11.25% by switching from human to automatic segmentation. Note
that there are only 37 classes available to us. This result suggests that the vision part is a serious
bottleneck of the whole architecture.
Based on automatic segmentations using multi-world approach (AutoSeg, 37 classes, multi)
(4th row in Table 3) shows the benefits of using our multiple worlds approach to predict the answer. Here we recover part of the lost performance by an explicit treatment of the uncertainty in the
segmentations. Performance increases from 11.25% to 13.75%.
4.3 Human question-answer pairs (HumanQA)
Based on human segmentations 894 classes (HumanSeg, 894 classes) (1st row in Table 4) switching to human generated question-answer pairs. The increase in complexity is twofold. First, the
human annotations exhibit more variations than the synthetic approach based on templates. Second,
the questions are typically longer and include more spatially related objects. Figure 4 shows a few
samples from our dataset that highlights challenges including complex and nested spatial reference
and use of reference frames. We yield an accuracy of 7.86% in this scenario. As argued above,
we also evaluate the experiments on the human data under the softer WUPS scores given different
thresholds (Table 4 and Figure 5). In order to put these numbers in perspective, we also show performance numbers for two simple methods: predicting the most popular answer yields 4.4% Accuracy,
and our untrained architecture gives 0.18% and 1.3% Accuracy and WUPS (at 0.9).
Based on human segmentations 37 classes (HumanSeg, 37 classes) (2nd row in Table 4) uses human segmentation and question-answer pairs. Since only 37 classes are supported by our automatic
segmentation algorithm, we run on a subset of the whole dataset. We choose the 25 test images
yielding a total of 286 question answer pairs for the following experiments. This yields 12.47% and
15.89% Accuracy and WUPS at 0.9 respectively.
Based on automatic segmentations (AutoSeg, 37 classes) (3rd row in Table 4) Switching from the
human segmentations to the automatic yields again a drop from 12.47% to 9.69% in Accuracy and
we observe a similar trend for the whole spectrum of the WUPS scores.
Based on automatic segmentations using multi-world approach (AutoSeg, 37 classes, multi)
(4th row in Table 4) Similar to the synthetic experiments our proposed multi-world approach yields
an improvement across all the measure that we investigate.
Human baseline (5th and 6th rows in Table 4 for 894 and 37 classes) shows human predictions on
our dataset. We ask independent annotators to provide answers on the questions we have collected.
They are instructed to answer with a number, basic colors [16], or objects (from 37 or 894 categories) or set of those. This performance gives a practical upper bound for the question-answering
algorithms with an accuracy of 60.27% for the 37 class case and 50.20% for the 894 class case.
We also ask to compare the answers of the AutoSeg single world approach with HumanSeg single
world and AutoSeg multi-worlds methods. We use a two-sided binomial test to check if difference
in preferences is statistically significant. As a result AutoSeg single world is the least preferred
method with the p-value below 0.01 in both cases. Hence the human preferences are aligned with
our accuracy measures in Table 4.
4.4 Qualitative results
We choose examples in Fig. 6 to illustrate different failure cases - including last example where all
methods fail. Since our multi-world approach generates different sets of facts about the perceived
worlds, we observe a trend towards a better representation of high level concepts like ?counting?
(leftmost the figure) as well as language associations. A substantial part of incorrect answers is
attributed to missing segments, e.g. no pillow detection in third example in Fig. 6.
5
Summary
We propose a system and a dataset for question answering about real-world scenes that is reminiscent
of a visual turing test. Despite the complexity in uncertain visual perception, language understanding
and program induction, our results indicate promising progress in this direction. We bring ideas
together from automatic scene analysis, semantic parsing with symbolic reasoning, and combine
them under a multi-world approach. As we have mature techniques in machine learning, computer
vision, natural language processing and deduction at our disposal, it seems timely to bring these
disciplines together on this open challenge.
7
QA: (What is behind the table?, window)!
Spatial relation like ?behind? are dependent
on the reference frame. Here the annotator
uses observer-centric view.!
The annotators are using different names to
call the same things. The names of the
brown object near the bed include ?night
stand?, ?stool?, and ?cabinet?.
QA: (what is beneath the candle holder,
decorative plate)!
Some annotators use variations on spatial
relations that are similar, e.g. ?beneath? is
closely related to ?below?.!
Some objects, like the table on the left of
image, are severely occluded or truncated.
Yet, the annotators refer to them in the
questions.
QA: (What is in front of toilet?, door)!
Here the ?open door? to the restroom is not
clearly visible, yet captured by the annotator.!
!
QA: (what is in front of the wall divider?,
cabinet)?
Annotators use additional properties to
clarify object references (i.e. wall divider).
Moreover, the perspective plays an
important role in these spatial relations
interpretations.
!
QA1:(How many doors are in the image?, 1)!QA: (How many drawers are there?, 8)!
QA2:(How many doors are in the image?, 5)!The annotators use their common-sense
knowledge for amodal completion. Here the
Different interpretation of ?door? results in
different counts: 1 door at the end of the hall ? annotator infers the 8th drawer from the
context
vs. 5 doors including lockers
QA: (what is behind the table?, sofa)!
Spatial relations exhibit different reference
frames. Some annotations use observercentric, others object-centric view!
QA: (how many lights are on?, 6)!
Moreover, some questions require detection
of states ?light on or off??
QA: (What is the shape of the green
chair?, horse shaped)!
In this example, an annotator refers to a
?horse shaped chair? which requires a quite
abstract reasoning about the shapes.!
QA1: (what is in front of the curtain behind
the armchair?, guitar)!
!
QA2: (what is in front of the curtain?,
guitar)!
!
Q: what is at the back side of the sofas?!
Annotators use wide range spatial relations,
such as ?backside? which is object-centric.
Spatial relations matter more in complex
environments where reference resolution
becomes more relevant. In cluttered scenes,
pragmatism starts playing a more important
role
QA: (What is the object on the counter in
the corner?, microwave)!
References like ?corner? are difficult to
resolve given current computer vision
models. Yet such scene features are
frequently used by humans.!
QA: (How many doors are open?, 1)!
Notion of states of object (like open) is not
well captured by current vision techniques.
Annotators use such attributes frequently
for disambiguation.!
QA: (Where is oven?, on the right side of
refrigerator)!
On some occasions, the annotators prefer to
use more complex responses. With spatial
relations, we can increase the answer?s
precision.!
Figure 4: Examples of human generated question-answer pairs illustrating the associated challenges. In the
descriptions we use following notation: ?A? - answer, ?Q? - question, ?QA? - question-answer pair. Last two
examples (bottom-right column) are from the extended dataset not used in our experiments.
?
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0.4
?
?
?
?
?
?
?
?
?
?
?
?
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?
?
?
?
?
?
?
?
0.0
0
0.1
0.2
0.3
?
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0.4
0.5
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?
0.8
0.9
1
synthetic question-answer pairs (SynthQA)
Segmentation
World(s)
# classes Accuracy
HumanSeg Single with Neg. 3
37
56.0%
HumanSeg
Single
37
59.5%
AutoSeg
Single
37
11.25%
AutoSeg
Multi
37
13.75%
?
?
?
?
HumanSeg, Single, 894
HumanSeg, Single, 37
AutoSeg, Single, 37
AutoSeg, Multi, 37
Human Baseline, 894
Human Baseline, 37
0.2
WUPS
0.6
0.8
HumanQA
?
0.6
0.7
?
?
?
Threshold
Figure 5: WUPS scores for different thresholds.
Table 3: Accuracy results for the experiments with synthetic question-answer pairs.
Human question-answer pairs (HumanQA)
Segmentation World(s) #classes Accuracy WUPS at 0.9
HumanSeg
Single
894
7.86%
11.86%
HumanSeg
Single
37
12.47%
16.49%
AutoSeg
Single
37
9.69%
14.73%
AutoSeg
Multi
37
12.73%
18.10%
Human Baseline
894
50.20%
50.82%
Human Baseline
37
60.27%
61.04%
WUPS at 0
38.79%
50.28%
48.57%
51.47%
67.27%
78.96%
Table 4: Accuracy and WUPS scores for the experiments with human question-answer pairs. We show WUPS
scores at two opposite sides of the WUPS spectrum.
Q: How many red chairs are there?!
H: ()!
M: 6!
C: blinds!
!
Q: How many chairs are at the table?!
H: wall?
M: 4!
C: chair
Q: What is on the right side of cabinet?!
H: picture?
M: bed!
C: bed
Q: What is on the wall?!
H: mirror!
M: bed!
C: picture
Q: What is the object on the chair?!
H: pillow!
M: floor, wall!
C: wall
Q: What is on the right side of the table?!
H: chair?
M: window, floor, wall!
C: floor
Q: What is behind the television?!
H: lamp?
M: brown, pink, purple!
C: picture
Q: What is in front of television?!
H: pillow!
M: chair!
C: picture
Figure 6: Questions and predicted answers. Notation: ?Q? - question, ?H? - architecture based on human
segmentation, ?M? - architecture with multiple worlds, ?C? - most confident architecture, ?()? - no answer. Red
color denotes correct answer.
8
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4,874 | 5,412 | Quantized Kernel Learning for Feature Matching
Danfeng Qin
ETH Z?urich
Xuanli Chen
TU Munich
Matthieu Guillaumin
ETH Z?urich
Luc Van Gool
ETH Z?urich
{qind, guillaumin, vangool}@vision.ee.ethz.ch, [email protected]
Abstract
Matching local visual features is a crucial problem in computer vision and its
accuracy greatly depends on the choice of similarity measure. As it is generally
very difficult to design by hand a similarity or a kernel perfectly adapted to the
data of interest, learning it automatically with as few assumptions as possible is
preferable. However, available techniques for kernel learning suffer from several
limitations, such as restrictive parametrization or scalability.
In this paper, we introduce a simple and flexible family of non-linear kernels
which we refer to as Quantized Kernels (QK). QKs are arbitrary kernels in the
index space of a data quantizer, i.e., piecewise constant similarities in the original feature space. Quantization allows to compress features and keep the learning
tractable. As a result, we obtain state-of-the-art matching performance on a standard benchmark dataset with just a few bits to represent each feature dimension.
QKs also have explicit non-linear, low-dimensional feature mappings that grant
access to Euclidean geometry for uncompressed features.
1
Introduction
Matching local visual features is a core problem in computer vision with a vast range of applications
such as image registration [28], image alignment and stitching [6] and structure-from-motion [1].
To cope with the geometric transformations and photometric distorsions that images exhibit, many
robust feature descriptors have been proposed. In particular, histograms of oriented gradients such
as SIFT [15] have proved successful in many of the above tasks. Despite these results, they are
inherently limited by their design choices. Hence, we have witnessed an increasing amount of work
focusing on automatically learning visual descriptors from data via discriminative embeddings [11,
4] or hyper-parameter optimization [5, 21, 23, 22].
A dual aspect of visual description is the measure of visual (dis-)similarity, which is responsible
for deciding whether a pair of features matches or not. In image registration, retrieval and 3D
reconstruction, for instance, nearest neighbor search builds on such measures to establish point
correspondences. Thus, the choice of similarity or kernel impacts the performance of a system as
much as the choice of visual features [2, 16, 18]. Designing a good similarity measure for matching
is difficult and commonly used kernels such as the linear, intersection, ?2 and RBF kernels are not
ideal as their inherent properties (e.g., stationarity, homogeneity) may not fit the data well.
Existing techniques for automatically learning similarity measures suffer from different limitations.
Metric learning approaches [25] learn to project the data to a lower-dimensional and more discriminative space where the Euclidean geometry can be used. However, these methods are inherently
linear. Multiple Kernel Learning (MKL) [3] is able to combine multiple base kernels in an optimal
way, but its complexity limits the amount of data that can be used and forces the user to pre-select
or design a small number of kernels that are likely to perform well. Additionally, the resulting kernel may not be easily represented in a reasonably small Euclidean space. This is problematic, as
many efficient algorithms (e.g. approximate nearest neighbor techniques) heavily rely on Euclidean
geometry and have non-intuitive behavior in higher dimensions.
1
In this paper, we introduce a simple yet powerful family of kernels, Quantized Kernels (QK), which
(a) model non-linearities and heterogeneities in the data, (b) lead to compact representations that
can be easily decompressed into a reasonably-sized Euclidean space and (c) are efficient to learn so
that large-scale data can be exploited. In essence, we build on the fact that vector quantizers project
data into a finite set of N elements, the index space, and on the simple observation that kernels on
finite sets are fully specified by the N?N Gram matrix of these elements (the kernel matrix), which
we propose to learn directly. Thus, QKs are piecewise constant but otherwise arbitrary, making
them very flexible. Since the learnt kernel matrices are positive semi-definite, we directly obtain the
corresponding explicit feature mappings and exploit their potential low-rankness.
In the remainder of the paper, we first further discuss related work (Sec. 2), then present QKs in detail
(Sec. 3). As important contributions, we show how to efficiently learn the quantizer and the kernel
matrix so as to maximize the matching performance (Sec. 3.2), using an exact linear-time inference
subroutine (Sec. 3.3), and devise practical techniques for users to incorporate knowledge about the
structure of the data (Sec. 3.4) and reduce the number of parameters of the system. Our experiments
in Sec. 4 show that our kernels yield state-of-the-art performance on a standard feature matching
benchmark and improve over kernels used in the literature for several descriptors, including one
based on metric learning. Our compressed features are very compact, using only 1 to 4 bits per
dimension of the original features. For instance, on SIFT descriptors, our QK yields about 10%
improvement on matching compared to the dot product, while compressing features by a factor 8.
2
Related work
Our work relates to a vast literature on kernel selection and tuning, descriptor, similarity, distance
and kernel learning. We present a selection of such works below.
Basic kernels and kernel tuning. A common approach for choosing a kernel is to pick one from
the literature: dot product, Gaussian RBF, intersection [16], ?2 , Hellinger, etc. These generic kernels
have been extensively studied [24] and have properties such as homogeneity or stationarity. These
properties may be inadequate for the data of interest and thus the kernels will not yield optimal
performance. Efficient yet approximate versions of such kernels [9, 20, 24] are similarly inadequate.
Descriptor learning. Early work on descriptor learning improved SIFT by exploring its parameter space [26]. Later, automatic parameter selection was proposed with a non-convex objective [5].
Recently, significant improvements in local description for matching have been obtained by optimizing feature encoding [4] and descriptor pooling [21, 23]. These works maximize the matching
performance directly via convex optimization [21] or boosting [23]. As we show in our experiments,
our approach improves matching even for such optimized descriptors.
Distance, similarity and kernel learning. Mahalanobis metrics (e.g., [25]) are probably the most
widely used family of (dis-)similarities in supervised settings. They extend the Euclidean metric
by accounting for correlations between input dimensions and are equivalent to projecting data to
a new, potentially smaller, Euclidean space. Learning the projection improves discrimination and
compresses feature vectors, but the projection is inherently linear.1 There are several attempts to
learn more powerful non-linear kernels from data. Multiple Kernel Learning (MKL) [3] operates
on a parametric family of kernels: it learns a convex combination of a few base kernels so as to
maximize classification accuracy. Recent advances now allow to combine thousands of kernels in
MKL [17] or exploit specialized families of kernels to derive faster algorithms [19]. In that work, the
authors combine binary base kernels based on randomized indicator functions but restricted them
to XNOR-like kernels. Our QK framework can also be seen as an efficient and robust MKL on
a specific family of binary base kernels. However, our binary base kernels originate from more
general quantizations: they correspond to their regions of constantness. As a consequence, the
resulting optimization problem is also more involves and thus calls for approximate solutions.
In parallel to MKL approaches, Non-Parametric Kernel Learning (NPKL) [10] has emerged as a
flexible kernel learning alternative. Without any assumption on the form of the kernel, these methods
aim at learning the Gram matrix of the data directly. The optimization problem is a semi-definite
program whose size is quadratic in the number of samples. Scalability is therefore an issue, and
approximation techniques must be used to compute the kernel on unobserved data. Like NPKL, we
learn the values of the kernel matrix directly. However, we do it in the index space instead of the
1
Metric learning can be kernelized, but then one has to choose the kernel.
2
original space. Hence, we restrict our family of kernels to piecewise constant ones2 , but, contrary to
NPKL, the complexity of the problems we solve does not grow with the number of data points but
with the refinement of the quantization and our kernels trivially generalize to unobserved inputs.
3
Quantized kernels
In this section, we present the framework of quantized kernels (QK). We start in Sec. 3.1 by defining
QKs and looking at some of their properties. We then present in Sec. 3.2 a general alternating
learning algorithm. A key step is to optimize the quantizer itself. We present in Sec. 3.3 our scheme
for quantization optimization for a single dimensional feature and how to generalize it to higher
dimensions in Sec. 3.4.
3.1
Definition and properties
D
D
Formally, quantized kernels QKD
N are the set of kernels kq on R ?R such that:
?q : RD 7? {1, . . . , N },
?K ? RN ?N 0,
?x, y ? RD ,
kq (x, y) = K(q(x), q(y)), (1)
where q is a quantization function which projects x ? RD to the finite index space {1, . . . , N },
and K 0 denotes that K is a positive semi-definite (PSD) matrix. As discussed above, quantized
kernels are an efficient parametrization of piecewise constant functions, where q defines the regions
of constantness. Moreover, the N ? N matrix K is unique for a given choice of kq , as it simply
accounts for the N (N+1)/2 possible values of the kernel and is the Gram matrix of the N elements
of the index space. We can also see q as a 1-of-N coding feature map ?q , such that:
kq (x, y) = K(q(x), q(y)) = ?q (x)> K?q (y).
(2)
The components of the matrix K fully parametrize the family of quantized kernels based on q, and
it is a PSD matrix if and only if kq is a PSD kernel. An explicit feature mapping of kq is easily
computed from the Cholesky decomposition of the PSD matrix K = P> P:
kq (x, y) = ?q (x)> K?q (y) = ?qP (x), ?qP (y) ,
(3)
where ?qP (x) = P?q (x). It is of particular interest to limit the rank N 0 ? N of K, and hence the
number of rows in P. In their compressed form, vectors require only log2 (N ) bits of memory for
0
storing q(x) and they can be decompressed in RN using P?q (x). Not only is this decompressed
vector smaller than one based on ?q , but it is also associated with the Euclidean geometry rather than
the kernel one. This allows the exploitation of the large literature of efficient methods specialized to
Euclidean spaces.
3.2 Learning quantized kernels
In this section, we describe a general alternating algorithm to learn a quantized kernel kq for feature
matching. This problem can be formulated as quadruple-wise constraints of the following form:
kq (x, y) > kq (u, v),
?(x, y) ? P,
?(u, v) ? N ,
(4)
where P denotes the set of positive feature pairs, and N is the negative one. The positive set contains
feature pairs that should be visually matched, while the negative pairs are mismatches.
We adopt a large-margin formulation of the above constraints using the trace-norm regularization
k ? k? on K, which is the tightest convex surrogate to low-rank regularization [8]. Using M training
pairs {(xj , yj )}j=1...M , we obtain the following optimization problem:
argmin
K0, q?QD
N
E(K, q) =
M
X
?
kKk? +
max 0, 1 ? lj ?q (xj )> K?q (yj ) ,
2
j=1
(5)
D
where QD
N denotes the set of quantizers q : R 7? {1, . . . , N }, the pair label lj ? {?1, 1} denotes
whether the feature pair (xj , yj ) is in N or P respectively. The parameter ? controls the trade-off
between the regularization and the empirical loss. Solving Eq. (5) directly is intractable. We thus
propose to alternate between the optimization of K and q. We describe the former below, and the
latter in the next section.
2
As any continuous function on an interval is the uniform limit of a series of piecewise constant functions,
this assumption does not inherently limit the flexibility of the family.
3
Optimizing K with fixed q. When fixing q in Eq. (5), the objective function becomes convex in
K but is not differentiable, so we resort to stochastic sub-gradient descent for optimization. Similar
to [21], we used Regularised Dual Averaging (RDA) [27] to optimize K iteratively. At iteration
t + 1, the kernel matrix Kt+1 is updated with the following rule:
?
t
Kt+1 = ? ?
Gt + ?I
?
(6)
Pt
where ? > 0 and Gt = 1t t0 =1 Gt0 is the rolling average of subgradients Gt0 of the loss computed
at step t0 from one sample pair. I is the identity matrix and ? is the projection onto the PSD cone.
3.3 Interval quantization optimization for a single dimension
To optimize an objective like Eq. (5) when K is fixed, we must consider how to design and
parametrize the elements of QD
N . In this work, we adopt interval quantizers, and in this section
we assume D = 1, i.e., restrict the study of quantization to R.
Interval quantizers. An interval quantizer q over R is defined by a set of N + 1 boundaries
bi ? R with b0 = ??, bN = ? and q(x) = i if and only if bi?1 < x ? bi . Importantly, interval
quantizers are monotonous, x ? y ? q(x) ? q(y), and boundaries bi can be set to any value between
maxq(x)=i x (included) and minq(x)=i+1 x (excluded). Therefore, Eq. (5) can be viewed as a data
labelling problem, where each value xj or yj takes a label in [1, N ], with a monotonicity constraint.
Thus, let us now consider the graph (V, E) where nodes V = {vt }t=1...2M represent the list of all
xj and yj in a sorted order and the edges E = {(vs , vt )} connect all pairs (xj , yj ). Then Eq. (5) with
fixed K is equivalent to the following discrete pairwise energy minimization problem:
argmin
q?[1,N ]2M
E 0 (q) =
X
Est (q(vs ), q(vt )) +
2M
X
Ct (q(vt?1 ), q(vt )),
(7)
t=2
(s,t)?E
where Est (q(vs ), q(vt )) = Ej (q(xj ), q(yj )) = max (0, 1 ? lj K(q(xj ), q(yj ))) and Ct is ? for
q(vt ) < q(vt?1 ) and 0 otherwise (i.e., it encodes the monotonicity of q in the sorted list of vt ).
The optimization of Eq. (7) is an NP-hard problem as the energies Est are arbitrary and the graph
does not have a bounded treewidth, in general. Hence, we iterate the individual optimization of each
of the boundaries using an exact linear-time algorithm, which we present below.
Exact linear-time optimization of a binary interval quantizer. We now consider solving equations of the form of Eq. (7) for the binary label case (N = 2). The main observation is that the
monotonicity constraint means that labels are 1 until a certain node t and then 2 from node t + 1,
and this switch can occur only once on the entire sequence, where vt ? b1 < vt+1 . This means
that there are only 2M +1 possible labellings and we can order them from (1, . . . , 1), (1, . . . , 1, 2)
to (2, . . . , 2). A na??ve algorithm consists in computing the 2M +1 energies explicitly. Since each
energy computation is linear in the number of edges, this results in a quadratic complexity overall.
A linear-time algorithm exist. It stems from the observation that the energies of two consecutive
labellings (e.g., switching the label of vt from 1 to 2) differ only by a constant number of terms:
E(q(vt?1 ) = 1, q(vt ) = 2, q(vt+1 ) = 2) = E(q(vt?1 ) = 1, q(vt ) = 1, q(vt+1 ) = 2)
+ Ct (1, 2) ? Ct (1, 1) + Ct+1 (2, 2) ? Ct+1 (1, 2) + Est (q(vs ), 2) ? Est (q(vs ), 1)
(8)
where, w.l.o.g., we have assumed (s, t) ? E. After finding the optimal labelling, i.e. finding the
label change (vt , vt+1 ), we set b1 = (vt +vt+1 )/2 to obtain the best possible generalization.
Finite spaces. When the input feature space has a finite number of different values (e.g., x ?
[1, T ]), then we can use linear-time sorting and merge all nodes with equal value in Eq. (7): this
results in considering at most T + 1 labellings, which is potentially much smaller than 2M + 1.
Extension to the multilabel case. Optimizing a single boundary bi of a multilabel interval quantization is essentially the same binary problem as above, where we limit the optimization to the
values currently assigned to i and i + 1 and keep the other assignments q fixed. We use unaries
Ej (q(xj ), q(yj )) or Ej (q(xj ), q(yj )) to model half-fixed pairs for xj or yj , respectively.
3.4 Learning higher dimensional quantized kernels
We now want to generalize interval quantizers to higher dimensions. This is readily feasible via
product quantization [13], using interval quantizers for each individual dimension.
4
Interval product quantization. An interval product quantizer q(x) : RD 7? {1, . . . , N } is of
the form q(x) = (q1 (x1 ), . . . , qD (xD )), where q1 , . . . , qD are interval quantizers with N1 , . . . , ND
QD
bins respectively, i.e., N = d=1 Nd . The learning algorithm devised above trivially generalizes
to interval product quantization by fixing all but one boundary of a single component quantizer qd .
However, learning K ? RN ? RN when N is very large becomes problematic: not only does RDA
scale unfavourably, but the lack of training data will eventually lead to severe overfitting. To address
these issues, we devise below variants of QKs that have practical advantages for robust learning.
Additive quantized kernels (AQK). We can drastically reduce the number of parameters by restricting product quantized kernels to additive ones, which consists in decomposing over dimensions:
kq (x, y) =
D
X
d=1
kqd (xd , yd ) =
D
X
?qd (xd )> Kd ?qd (yd ) = ?q (x)> K?q (y),
(9)
d=1
where qd ? Q1Nd , ?qd is the 1-of-Nd coding of dimension d, Kd is theP
Nd ? Nd Gram
P matrix of
dimension d, ?q is the concatenation of the D mappings ?qd , and K is a ( d Nd )?( d Nd ) blockdiagonal matrix of K
Q1 , . . . , KD . ThePbenefits of AQK are twofold. First, the explicit feature space is
reduced
N = d Nd to N 0 = d Nd . Second, the number of parameters to learn
P from
Pin K is now
only d Nd2 instead of N 2 . The compression ratio is unchanged since log2 (N ) = d log2 (Nd ).
To learn K in Eq. (9), we simply set the off-block-diagonal elements of Gt0 to zero in each iteration,
and iteratively update K as describe in Sec. 3.2. To optimize a product quantizer, we iterate the
optimization of each 1d quantizer qd following Sec. 3.3, while fixing qc for c 6= d. This leads to
using the following energy Ej for a pair (xj , yj ):
where ?j,d
Ej,d (qd (xj,d ), qd (yj,d )) = max (0, ?j,d ? lj Kd (qd (xj,d ), qd (yj,d ))) ,
P
= 1 ? lj c6=d Kc (qc (xc ), qc (yc )) acts as an adaptive margin.
(10)
Block quantized kernels (BQK). Although the additive assumption in AQK greatly reduces the
number of parameters, it is also very restrictive, as it assumes independent data dimensions. A
simple way to extend additive quantized kernels to model the inter-dependencies of dimensions is
to allow the off-diagonal elements of K in Eq. (9) to be nonzero. As a trade-off between a blockdiagonal (AQK) and a full matrix, in this work we also consider the grouping of the feature dimensions into B blocks, and only learn off-block-diagonal elements within each block, leading to Block
Quantized Kernels (BQK). In this way, assuming ?d Nd = n, the number of parameters in K is
B times smaller than for the full matrix. As a matter of fact, many features such as SIFT descriptors
exhibit block structure. SIFT is composed of a 4?4 grid of 8 orientation bins. Components within
the same spatial cell correlate more strongly than others and, thus, only modeling those jointly may
prove sufficient. The optimization of K and q are straightforwardly adapted from the AQK case.
Additional parameter sharing. Commonly, the different dimensions of a descriptor are generated by the same procedure and hence share similar properties. This results in block matrices
K1 , . . . , KD in AQK that are quite similar as well. We propose to exploit this observation and share
the kernel matrix for groups of dimensions, further reducing the number of parameters. Specifically,
we cluster dimensions based on their variances into G equally sized groups and use a single block
matrix for each group. During optimization,
dimensions sharingPthe same block matrix can conP
veniently be merged, i.e. ?q (x) = [ d s.t. Kd =K0 ?qd (xd ), . . . , d s.t. Kd =K0 ?qd (xd )], and then
1
G
K = diag(K01 , . . . , K0G ) is learnt following the procedure already described for AQK. Notably, the
quantizers themselves are not shared, so the kernel still adapts uniquely to every dimension of the
data, and the optimization of quantizers is not changed either. This parameter sharing strategy can
be readily applied to BQK as well.
4
Results
We now present our experimental results, starting with a description of our protocol. We then explore
parameters and properties of our kernels (optimization of quantizers, explicit feature maps). Finally,
we compare to the state-of-the-art in performance and compactness.
Dataset and evaluation protocol. We evaluate our method using the dataset of Brown et al. [5].
It contains three sets of patches extracted from Liberty, Notre Dame and Yosemite using the Difference of Gaussians (DoG) interest point detector. The patches are rectified with respect to the scale
5
Table 1: Impact of quantization optimization for different quantization
strategies
FPR @ 95% recall [%]
Optimized
21.68
25.70
14.29
FPR @ 95% recall [%]
Uniform
Adaptive
Adaptive+
Initial
24.84
25.99
14.62
20
18
16
14
2
6
10
14
#intervals
18
1
2
3
4
5
6
7
8
50
50
100
150
200
250
100 150 200 250
0.4
0.3
0.2
0.1
0
?0.1
?0.2
?0.3
?0.4
50
SIFT[15]
14
12
PR-proj[18]
10
8
1
SQ-4-DAISY[4]
2
3
#groups
4
Figure 1: Impact of N , the num- Figure 2: Impact of G, the number of quantization intervals
1 2 3 4 5 6 7 8
16
100
150
200
250
0.4
0.3
0.2
0.1
0
?0.1
?0.2
?0.3
?0.4
50
100
150
200
ber of dimension groups
250
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8
0.1
0.08
0.06
0.04
0.02
0
?0.02
?0.04
?0.06
?0.08
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: Our learned feature maps and additive quantized kernel of a single dimension. (a) shows the quantized kernel in index space, while (b) is in the original feature space for the first quantizer. (c,d) show the two
corresponding feature maps, and (e,f) the related rank-1 kernels.
and dominant orientation, and pairwise correspondences are computed using a multi-view stereo
algorithm. In our experiments, we use the standard evaluation protocol [5] and state-of-the-art descriptors: SIFT [15], PR-proj [21] and SQ-4-DAISY [4]. M =500k feature pairs are used for training
on each dataset, with as many positives as negatives. We report the false positive rate (FPR) at 95%
recall on the test set of 100k pairs. A challenge for this dataset is the bias in local patch appearance
for each set, so a key factor for performance is the ability to generalize and adapt across sets.
Below, in absence of other mention, AQKs are trained for SIFT on Yosemite and tested on Liberty.
Interval quantization and optimization. We first study the influence of initialization and optimization on the generalization ability of the interval quantizers. For initialization, we have used two
different schemes: a) Uniform quantization, i.e. the quantization with equal intervals; b) Adaptive
quantization, i.e. the quantization with intervals with equal number of samples. In both cases, it
allows to learn a first kernel matrix, and we can then iterate with boundary optimization (Sec. 3.3).
Typically, convergence is very fast (2-3 iterations) and takes less than 5 minutes in total (i.e., about
2s per feature dimension) with 1M nodes. We see in Table 1 that uniform binning outperforms the
adaptive one and that further optimization benefits the uniform case more. This may seem paradoxical at first, but this is due to the train/test bias problem: intervals with equal number of samples
are very different across sets, so refinements will not transfer well. Hence, following [7], we first
normalize the features with respect to their rank, separately for the training and test sets. We refer to
this process as Adaptive+. As Table 1 shows, not only does it bring a significant improvement, but
further optimization of the quantization boundaries is more beneficial than for the Adaptive case. In
the following, we thus adopt this strategy.
Number of quantization intervals. In Fig. 1, we show the impact of the number of intervals N
of the quantizer on the matching accuracy, using a single shared kernel submatrix (G = 1). This
number balances the flexibility of the model and its compression ratio. As we can see, using too few
intervals limits the performance of QK, and using too many eventually leads to overfitting. The best
performance for SIFT is obtained with between 8 and 16 intervals.
Explicit feature maps. Fig. 3a shows the additive quantized kernel learnt for SIFT with N = 8
and G = 1. Interestingly, the kernel has negative values far from the diagonal and positive values
near the diagonal. This is typical of stationary kernels: when both features have similar values,
they contribute more to the similarity. However, contrary to stationary kernels, diagonal elements
are far from being constant. There is a mode on small values and another one on large ones. The
second one is stronger: i.e., the co-occurrence of large values yields greater similarity. This is consistent with the voting nature of SIFT descriptors, where strong feature presences are both rarer and
more informative than their absences. The negative values far from the diagonal actually penalize
inconsistent observations, thus confirming existing results [12]. Looking at the values in the original space in Fig. 3b, we see that the quantizer has learnt that fine intervals are needed in the lower
6
Descriptor
Kernel
Dimensionality
Train on Yosemite
Train on Notredame
Notredame
Liberty
Yosemite
Liberty
Mean
SIFT[15]
SIFT[15]
SIFT[15]
SIFT[15]
SIFT[15]
Euclidean
?2
AQK(8)
AQK(8)
BQK(8)
128
128
128
256
256
24.02
17.65
10.72
9.26
8.05
31.34
22.84
16.90
14.48
13.31
27.96
23.50
10.72
10.16
9.88
31.34
22.84
16.85
14.43
13.16
28.66
21.71
13.80
12.08
11.10
SQ-4-DAISY [4]
SQ-4-DAISY [4]
SQ-4-DAISY [4]
SQ-4-DAISY [4]
Euclidean
?2
SQ [4]
AQK(8)
1360
1360
1360
?1813
10.08
10.61
8.42
4.96
16.90
16.25
15.58
9.41
10.47
12.19
9.25
5.60
16.90
16.25
15.58
9.77
13.58
13.82
12.21
7.43
Euclidean[21]
AQK(16)
<64
?102
7.11
5.41
14.82
10.90
10.54
7.65
12.88
10.54
11.34
8.63
PR-proj [21]
PR-proj [21]
Table 2: Performance of kernels on different datasets with different descriptors. AQK(N) denotes the additive
quantized kernel with N quantization intervals. Following [6], we report the False positive rate (%) at 95%
recall. The best results for each descriptor are in bold.
values, while larger ones are enough for larger values. This is consistent with previous observations
that the contribution of large values in SIFT should not grow proportionally [2, 18, 14].
In this experiment, the learnt kernel has rank 2. We show in Fig. 3c, 3d, 3e and 3f the corresponding
feature mappings and their associated rank 1 kernels. The map for the largest eigenvalue (Fig. 3c)
is monotonous but starts with negative values. This impacts dot product significantly, and accounts
for the above observation that negative similarities occur when inputs disagree. This rank 1 kernel
cannot allot enough contribution to similar mid-range values. This is compensated by the second
rank (Fig. 3f).
Number of groups. Fig. 2 now shows the influence of the number of groups G on performance,
for the three different descriptors (N = 8 for SIFT and SQ-4-DAISY, N = 16 for PR-proj). As for
intervals, using more groups adds flexibility to the model, but as less data is available to learn each
parameter, over-fitting will hurt performance. We choose G = 3 for the rest of the experiments.
Comparison to the state of the art. Table 2 reports the matching performance of different kernels
using different descriptors, for all sets, as well as the dimensionality of the corresponding explicit
feature maps. For all three descriptors and on all sets, our quantized kernels significantly and consistently outperform the best reported result in the literature. Indeed, AQK improves the mean error
rate at 95% recall from 28.66% to 12.08% for SIFT, from 13.58% to 7.43% for SQ-4-DAISY and
from 11.34% to 8.63% for PR-proj compared to the Euclidean distance, and about as much for the
?2 kernel. Note that PR-proj already integrates metric learning in its design ([21] thus recommends
using the Euclidean distance): as a consequence our experiments show that modelling non-linearities
can bring significant improvements. When comparing to sparse quantization (SQ) with hamming
distance as done in [4], the error is significantly reduced from 12.21% to 7.43%. This is a notable
achievement considering that [4] is the previous state of the art.
The SIFT descriptor has a grid block design which makes it particularly suited for the use of BQK.
Hence, we also evaluated our BQK variant for that descriptor. With BQK(8), we observed a relative
improvement of 8%, from 12.08% for AQK(8) to 11.1%.
We provide in Fig. 4 the ROC curves for the three descriptors when training on Yosemite and testing
on Notre Dame and Liberty. These figures show that the improvement in recall is consistent over the
full range of false positive rates. For further comparisons, our data and code are available online.3
Compactness of our kernels. In many applications of feature matching, the compactness of the
descriptor is important. In Table 3, we compare to other methods by grouping them according to
their memory footprint. As a reference, the best method reported in Table 2 (AQK(8) on SQ-4DAISY) uses 4080 bits per descriptor. As expected, error rates increase as fewer bits are used, the
original features being significantly altered. Notably, QKs consistently yield the best performance in
all groups. Even with a crude binary quantization of SQ-4-DAISY, our quantized kernel outperform
the state-of-the-art SQ of [4] by 3 to 4%. When considering the most compact encodings (? 64 bits),
our AQK(2) does not improve over BinBoost [22], a descriptor designed for extreme compactness, or
the product quantization (PQ [13]) encoding as used in [21]. This is because our current framework
does not yet allow for joint compression of multiple dimensions. Hence, it is unable to use less
3
See: http://www.vision.ee.ethz.ch/?qind/QuantizedKernel.html
7
BQK(8)
AQK(8)
AQK(2)
L2
80
75
5
10
15
20
25
False Positive Rate [%]
SIFT
90
85
BQK(8)
AQK(8)
AQK(2)
L2
80
75
5
10
15
90
85
80
AQK(8)
AQK(2)
SQ
75
5
20
25
False Positive Rate [%]
30
10
15
20
25
False Positive Rate [%]
90
85
80
AQK(8)
AQK(2)
SC
75
5
10
15
90
85
80
AQK(16)
AQK(4)
L2
75
5
10
20
25
False Positive Rate [%]
20
25
30
PR?proj
95
90
85
80
AQK(16)
AQK(4)
L2
75
70
0
30
15
False Positive Rate [%]
100
95
70
0
95
70
0
30
SQ-4-DAISY
100
95
70
0
95
70
0
30
PR?proj
100
True Positive Rate [%]
85
100
True Positive Rate [%]
True Positive Rate [%]
90
True Positive Rate [%]
True Positive Rate [%]
95
70
0
SQ-4-DAISY
100
True Positive Rate [%]
SIFT
100
5
10
15
20
25
False Positive Rate [%]
30
Figure 4: ROC curves when evaluating Notre Dame (top) and Liberty (bottom) from Yosemite
Descriptor
Encoding
Memory (bits)
Train on Yosemite
Train on Notredame
Notredame
Liberty
Yosemite
Liberty
Mean
SQ-4-DAISY [4]
SQ-4-DAISY [4]
SQ [4]
AQK(2)
1360
1360
8.42
5.86
15.58
10.81
9.25
6.36
15.58
10.94
12.21
8.49
SIFT[15]
PR-proj [21]
PR-proj [21]
AQK(8)
Bin [21]
AQK(16)
384
1024
<256
9.26
7.09
5.41
14.48
15.15
10.90
10.16
8.5
7.65
14.43
12.16
10.54
12.08
10.73
8.63
SIFT[15]
PR-proj [21]
PR-proj [21]
AQK(2)
Bin [21]
AQK(4)
128
128
<128
14.62
10.00
7.18
19.72
18.64
13.02
15.65
13.41
10.29
19.45
16.39
13.18
17.36
14.61
10.92
BinBoost[22]
PR-proj [21]
PR-proj [21]
PR-proj [21]
BinBoost[22]
AQK(2)
PQ [21]
PCA+AQK(4)
64
<64
64
64
14.54
14.80
12.91
10.74
21.67
20.59
20.15
17.46
18.97
19.38
19.32
14.44
20.49
22.24
17.97
17.60
18.92
19.26
17.59
15.06
Table 3: Performance comparison of different compact feature encoding. The number in the table is
reported as False positive rate (%) at 95% recall. The best results for each group are in bold.
than 1 bit per original dimension, and is not optimal in that case. To better understand the potential
benefits of decorrelating features and joint compression in future work, we pre-processed the data
with PCA, projecting to 32 dimensions and then using AQK(4). This simple procedure obtained
state-of-the-art performance with 15% error rate, now outperforming [22] and [21].
Although QKs yield very compact descriptors and achieve the best performance across many experimental setups, the computation of similarity values is slower than for competitors: in the binary
case, we double the complexity of hamming distance for the 2 ? 2 table look-up.
5
Conclusion
In this paper, we have introduced the simple yet powerful family of quantized kernels (QK), and
presented an efficient algorithm to learn its parameters, i.e. the kernel matrix and the quantization
boundaries. Despite their apparent simplicity, QKs have numerous advantages: they are very flexible, can model non-linearities in the data and provide explicit low-dimensional feature mappings
that grant access to the Euclidean geometry. Above all, they achieve state-of-the-art performance
on the main visual feature matching benchmark. We think that QKs have a lot of potential for further improvements. In future work, we want to explore new learning algorithms to obtain higher
compression ratios ? e.g. by jointly compressing feature dimensions ? and find the weight sharing
patterns that would further improve the matching performance automatically.
Acknowledgements
We gratefully thank the KIC-Climate project Modeling City Systems.
8
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9
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4,875 | 5,413 | Diverse Sequential Subset Selection for
Supervised Video Summarization
Boqing Gong?
Department of Computer Science
University of Southern California
Los Angeles, CA 90089
[email protected]
Wei-Lun Chao?
Department of Computer Science
University of Southern California
Los Angeles, CA 90089
[email protected]
Kristen Grauman
Department of Computer Science
University of Texas at Austin
Austin, TX 78701
[email protected]
Fei Sha
Department of Computer Science
University of Southern California
Los Angeles, CA 90089
[email protected]
Abstract
Video summarization is a challenging problem with great application potential.
Whereas prior approaches, largely unsupervised in nature, focus on sampling useful frames and assembling them as summaries, we consider video summarization
as a supervised subset selection problem. Our idea is to teach the system to learn
from human-created summaries how to select informative and diverse subsets, so
as to best meet evaluation metrics derived from human-perceived quality. To this
end, we propose the sequential determinantal point process (seqDPP), a probabilistic model for diverse sequential subset selection. Our novel seqDPP heeds the
inherent sequential structures in video data, thus overcoming the deficiency of the
standard DPP, which treats video frames as randomly permutable items. Meanwhile, seqDPP retains the power of modeling diverse subsets, essential for summarization. Our extensive results of summarizing videos from 3 datasets demonstrate
the superior performance of our method, compared to not only existing unsupervised methods but also naive applications of the standard DPP model.
1
Introduction
It is an impressive yet alarming fact that there is far more video being captured?by consumers, scientists, defense analysts, and others?than can ever be watched or browsed efficiently. For example,
144,000 hours of video are uploaded to YouTube daily; lifeloggers with wearable cameras amass
Gigabytes of video daily; 422,000 CCTV cameras perched around London survey happenings in
the city 24/7. With this explosion of video data comes an ever-pressing need to develop automatic
video summarization algorithms. By taking a long video as input and producing a short video (or
keyframe sequence) as output, video summarization has great potential to reign in raw video and
make it substantially more browseable and searchable.
Video summarization methods often pose the problem in terms of subset selection: among all the
frames (subshots) in the video, which key frames (subshots) should be kept in the output summary?
There is a rich literature in computer vision and multimedia developing a variety of ways to answer
this question [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Existing techniques explore a plethora of properties that a
good summary should capture, designing criteria that the algorithm should prioritize when deciding
?
Equal contribution
1
which subset of frames (or subshots) to select. These include representativeness (the frames should
depict the main contents of the videos) [1, 2, 10], diversity (they should not be redundant) [4, 11],
interestingness (they should have salient motion/appearance [2, 3, 6] or trackable objects [5, 12, 7]),
or importance (they should contain important objects that drive the visual narrative) [8, 9].
Despite valuable progress in developing the desirable properties of a summary, prior approaches are
impeded by their unsupervised nature. Typically the selection algorithm favors extracting content
that satisfies criteria like the above (diversity, importance, etc.), and performs some sort of frame
clustering to discover events. Unfortunately, this often requires some hand-crafting to combine the
criteria effectively. After all, the success of a summary ultimately depends on human perception.
Furthermore, due to the large number of possible subsets that could be selected, it is difficult to
directly optimize the criteria jointly on the selected frames as a subset; instead, sampling methods
that identify independently useful frames (or subshots) are common.
To address these limitations, we propose to consider video summarization as a supervised subset
selection problem. The main idea is to use examples of human-created summaries?together with
their original source videos?to teach the system how to select informative subsets. In doing so,
we can escape the hand-crafting often necessary for summarization, and instead directly optimize
the (learned) factors that best meet evaluation metrics derived from human-perceived quality. Furthermore, rather than independently select ?high scoring? frames, we aim to capture the interlocked
dependencies between a given frame and all others that could be chosen.
To this end, we propose the sequential determinantal point process (seqDPP), a new probabilistic
model for sequential and diverse subset selection. The determinantal point process (DPP) has recently emerged as a powerful method for selecting a diverse subset from a ?ground set? of items [13],
with applications including document summarization [14] and information retrieval [15]. However,
existing DPP techniques have a fatal modeling flaw if applied to video (or documents) for summarization: they fail to capture their inherent sequential nature. That is, a standard DPP treats the
inputs as bags of randomly permutable items agnostic to any temporal structure. Our novel seqDPP
overcomes this deficiency, making it possible to faithfully represent the temporal dependencies in
video data. At the same time, it lets us pose summarization as a supervised learning problem.
While learning how to summarize from examples sounds appealing, why should it be possible?
particularly if the input videos are expected to vary substantially in their subject matter?1 Unlike
more familiar supervised visual recognition tasks, where test data can be reasonably expected to
look like the training instances, a supervised approach to video summarization must be able to learn
generic properties that transcend the specific content of the training set. For example, the learner
can recover a ?meta-cue? for representativeness, if the input features record profiles of the similarity between a frame and its increasingly distant neighbor frames. Similarly, category-independent
cues about an object?s placement in the frame, the camera person?s active manipulation of viewpoint/zoom, etc., could play a role. In any such case, we can expect the learning algorithm to focus
on those meta-cues that are shared by the human-selected frames in the training set, even though the
subject matter of the videos may differ.
In short, our main contributions are: a novel learning model (seqDPP) for selecting diverse subsets
from a sequence, its application to video summarization (the model is applicable to other sequential
data as well), an extensive empirical study with three benchmark datasets, and a successful firststep/proof-of-concept towards using human-created video summaries for learning to select subsets.
The rest of the paper is organized as follows. In section 2, we review DPP and its application to
document summarization. In section 3, we describe our seqDPP method, followed by a discussion
of related work in section 4. We report results in section 5, then conclude in section 6.
2
Determinantal point process (DPP)
The DPP was first used to characterize the Pauli exclusion principle, which states that two identical particles cannot occupy the same quantum state simultaneously [16]. The notion of exclusion
has made DPP an appealing tool for modeling diversity in application such as document summarization [14, 13], or image search and ranking [17]. In what follows, we give a brief account on DPP and
how to apply it to document summarization where the goal is to generate a summary by selecting
1
After all, not all videos on YouTube are about cats.
2
several sentences from a long document [18, 19]. The selected sentences should be not only diverse
(i.e., different) to reduce the redundancy in the summary, but also representative of the document.
Background Let Y = {1, 2, ? ? ? , N} be a ground set of N items (eg., sentences). In its simplest
form, a DPP defines a discrete probability distribution over all the 2N subsets of Y. Let Y denote
the random variable of selecting a subset. Y is then distributed according to
det(Ly )
P (Y = y) =
(1)
det(L + I)
for y ? Y. The kernel L ? SN?N
is the DPP?s parameter and is constrained to be positive semidefi+
nite. I is the identity matrix. Ly is the principal minor (sub-matrix) with rows and columns selected
according to the indices in y. The determinant function det(?) gives rise to the interesting property
of pairwise repulsion. To see that, consider selecting a subset of two items i and j. We have
P (Y = {i, j}) ? Lii Ljj ? L2ij .
(2)
If the items i and j are the same, then P (Y = {i, j}) = 0 (because Lij = Lii = Ljj ). Namely,
identical items should not appear together in the same set. A more general case also holds: if i and
j are similar to each other, then the probability of observing i and j in a subset together is going to
be less than that of observing either one of them alone (see the excellent tutorial [13] for details).
The most diverse subset of Y is thus the one that attains the highest probability
y ? = arg maxy P (Y = y) = arg maxy det(Ly ),
(3)
?
where y results from MAP inference. This is a NP-hard combinatorial optimization problem.
However, there are several approaches to obtaining approximate solutions [13, 20].
Learning DPPs for document summarization Suppose we model selecting a subset of sentences
as a DPP over all sentences in a document. We are given a set of training samples in the form of
documents (i.e., ground sets) and the ground-truth summaries. How can we discover the underlying
parameter L so as to use it for generating summaries for new documents?
Note that the new documents will likely have sentences that have not been seen before in the training
samples. Thus, the kernel matrix L needs to be reparameterized in order to generalize to unseen
documents. [14] proposed a special reparameterization called quality/diversity decomposition:
1 T
T
Lij = qi ?i ?j qj , qi = exp
? xi ,
(4)
2
where ?i is the normalized TF-IDF vector of the sentence i so that ?Ti ?j computes the cosine angle
between two sentences. The ?quality? feature vector xi encodes the contextual information about i
and its representativeness of other items. In document summarization, xi are the sentence lengths,
positions of the sentences in the texts, and other meta cues. The parameter ? is then optimized with
maximum likelihood estimation (MLE) such that the target subsets have the highest probabilities
X
? ? = arg max?
log P (Y = yn? ; Ln (?)),
(5)
n
where Ln is the L matrix formulated using sentences in the n-th ground set, and yn? is the corresponding ground-truth summary.
Despite its success in document summarization [14], a direct application of DPP to video summarization is problematic. The DPP model is agnostic about the order of the items. For video (and to a
large degree, text data), it does not respect the inherent sequential structures. The second limitation
is that the quality-diversity decomposition, while cleverly leading to a convex optimization, limits
the power of modeling complex dependencies among items. Specifically, only the quality factor qi
is optimized on the training data. We develop new approaches to overcoming those limitations.
3
Approach
In what follows, we describe our approach for video summarization. Our approach contains three
components: (1) a preparatory yet crucial step that generates ground-truth summaries from multiple
human-created ones (section 3.1); (2) a new probabilistic model?the sequential determinantal point
process (seqDPP)?that models the process of sequentially selecting diverse subsets (section 3.2);
(3) a novel way of re-parameterizing seqDPP that enables learning more flexible and powerful representations for subset selection from standard visual and contextual features (section 3.3).
3
Figure 1: The agreement among human-created summaries is high, as is the agreement between the oracle
summary generated by our algorithm (cf. section 3.1) and human annotations.
3.1
Generating ground-truth summaries
The first challenge we need to address is what to provide to our learning algorithm as ground-truth
summaries. In many video datasets, each video is annotated (manually summarized) by multiple
human users. While the users were often well instructed on the annotation task, discrepancies are
expected due to many uncontrollable individual factors such as whether the person was attentive,
idiosyncratic viewing preferences, etc. There are some studies on how to evaluate automatically
generated summaries in the presence of multiple human-created annotations [21, 22, 23]. However,
for learning, our goal is to generate one single ground-truth or ?oracle? summary per video.
Our main idea is to synthesize the oracle summary that maximally agrees with all annotators. Our
hypothesis is that despite the discrepancies, those summaries nonetheless share the common traits of
reflecting the subject matters in the video. These commonalities, to be discovered by our synthesis
algorithm, will provide strong enough signals for our learning algorithm to be successful.
To begin with, we first describe a few metrics in quantifying the agreement in the simplest setting
where there are only two summaries. These metrics will also be used later in our empirical studies
to evaluate various summarization methods. Using those metrics, we then analyze the consistency
of human-created summaries in two video datasets to validate our hypothesis. Finally, we present
our algorithm for synthesizing one single oracle summary per video.
Evaluation metrics Given two video summaries A and B, we measure how much they are in
agreement by first matching their frames, as they might be of different lengths. Following [24], we
compute the pairwise distances between all frames across the two summaries. Two frames are then
?matched? if their visual difference is below some threshold; a frame is constrained to appear in
the matched pairs at most once. After the matching, we compute the following metrics (commonly
known as Precision, Recall and F-score):
PAB =
#matched frames
,
#frames in A
RAB =
#matched frames
,
#frames in B
FAB =
PAB ? RAB
.
0.5(PAB + RAB )
All of them lie between 0 and 1, and higher values indicate better agreement between A and B. Note
that these metrics are not symmetric ? if we swap A and B, the results will be different.
Our idea of examining the consistency among all summaries is to treat each summary in turn as if it
were the gold-standard (and assign it as B) while treating the other summaries as A?s. We report our
analysis of existing video datasets next.
Consistency in existing video databases We analyze video summaries in two video datasets: 50
videos from the Open Video Project (OVP) [25] and another 50 videos from Youtube [24]. Details
about these two video datasets are in section 5. We briefly point out that the two datasets have very
different subject matters and composition styles. Each of the 100 videos has 5 annotated summaries.
For each video, we compute the pairwise evaluation metrics in precision, recall, and F-score by
forming total 20 pairs of summaries from two different annotators. We then average them per video.
We plot how these averaged metrics distribute in Fig. 1. The plots show the number of videos
(out of 100) whose averaged metrics exceed certain thresholds, marked on the horizontal axes. For
example, more than 80% videos have an averaged F-score greater than 0.6, and 60% more than 0.7.
Note that there are many videos (?20) with averaged F-scores greater than 0.8, indicating that on
average, human-created summaries have a high degree of agreement. Note that the mean values of
the averaged metrics per video are also high.
4
Greedy algorithm for synthesizing an oracle summary Encouraged by our findings, we develop
a greedy algorithm for synthesizing one oracle summary per video, from multiple human-created
ones. This algorithm is adapted from a similar one for document summarization [14]. Specifically,
for each video, we initialize the oracle summary with the empty set y ? = ?. Iteratively, we then add
to y ? one frame i at a time from the video sequence
X
y ? ? y ? ? arg maxi
Fy? ?i,yu .
(6)
u
In words, the frame i is selected to maximally increase the F-score between the new oracle summary
and the human-created summaries yu . To avoid adding all frames in the video sequence, we stop
the greedy process as soon as there is no frame that can increase the F-score.
We measure the quality of the synthesized oracle summaries by computing their mean agreement
with the human annotations. The results are shown in Fig. 1 too. The quality is high: more than
90% of the oracle summaries agree well with other summaries, with an F-score greater than 0.6. In
what follows, we will treat the oracle summaries as ground-truth to inform our learning algorithms.
3.2
Sequential determinantal point processes (seqDPP)
The determinantal point process, as described in section 2, is a powerful tool for modeling diverse
subset selection. However, video frames are more than items in a set. In particular, in DPP, the
ground set is a bag ? items are randomly permutable such that the most diverse subset remains
unchanged. Translating this into video summarization, this modeling property essentially suggests
that we could randomly shuffle video frames and expect to get the same summary!
To address this serious deficiency, we propose sequential DPP, a new probabilistic model to introduce strong dependency structures between items. As a motivating example, consider a video
portraying the sequence of someone leaving home for school, coming back to home for lunch, leaving for market and coming back for dinner. If only visual appearance cues are available, a vanilla
DPP model will likely select only one frame from the home scene and repel other frames occurring
at the home. Our model, on the other hand, will recognize that the temporal span implies those
frames are still diverse despite their visual similarity. Thus, our modeling intuition is that diversity
should be a weaker prior for temporally distant frames but ought to act more strongly for closely
neighboring frames. We now explain how our seqDPP method implements this intuition.
Model definition Given a ground set (a long video sequence) Y, we partition it into T disjoint yet
ST
consecutive short segments t=1 Yt = Y. At time t, we introduce a subset selection variable Yt .
We impose a DPP over two neighboring segments where the ground set is Ut = Yt ? yt?1 , ie., the
union between the video segments and the selected subset in the immediate past. Let ?t denote the
L-matrix defined over the ground set Ut . The conditional distribution of Yt is thus given by,
det ?yt?1 ?yt
P (Yt = yt |Yt?1 = yt?1 ) =
.
(7)
det(?t + It )
As before, the subscript yt?1 ? yt selects the corresponding rows and columns from ?t . It is a
diagonal matrix, the same size as Ut . However, the elements corresponding to yt?1 are zeros and
the elements corresponding to Yt are 1 (see [13] for details). Readers who are familiar with DPP
might identify the conditional distribution is also a DPP, restricted to the ground set Yt .
The conditional probability is defined in such a way that at time t, the subset selected should be
diverse among Yt as well as be diverse from previously selected yt?1 . However, beyond those two
priors, the subset is not constrained by subsets selected in the distant past. Fig. 2 illustrates the idea
in graphical model notation. In particular, the joint distribution of all subsets is factorized
Y
P (Y1 = y1 , Y2 = y2 , ? ? ? , YT = yT ) = P (Y1 = y1 )
P (Yt = yt |Yt?1 = yt?1 ).
(8)
t=2
Inference and learning The MAP inference for the seqDPP model eq. (8) is as hard as the standard DPP model. Thus, we propose to use the following online inference, analogous to Bayesian
belief updates (for Kalman filtering):
y1? = arg maxy?Y1 P (Y1 = y)
y2? = arg maxy?Y2 P (Y2 = y|Y1 = y1? ) ? ? ?
?
yt? = arg maxy?Yt P (Yt = y|Yt?1 = yt?1
) ??????
5
Y1
Y2
Y3
Y1
Y2
Y3
Yt
???
Yt
YT
???
YT
Figure 2: Our sequential DPP for modeling sequential video data, drawn as a Bayesian network
Note that, at each step, the ground set could be quite small; thus an exhaustive search for the most
diverse subset is plausible. The parameter learning is similar to the standard DPP model. We
describe the details in the supplementary material.
3.3
Learning representations for diverse subset selection
As described previously, the kernel L of DPP hinges on the reparameterization with features of
the items that can generalize across different ground sets. The quality-diversity decomposition in
eq. (4), while elegantly leading to convex optimization, is severely limited in its power in modeling
complex items and dependencies among them. In particular, learning the subset selection rests solely
on learning the quality factor, as the diversity component remains handcrafted and fixed.
We overcome this deficiency with more flexible and powerful representations. Concretely, let fi
stand for the feature representation for item (frame) i, including both low-level visual cues and
meta-cues such as contextual information. We reparameterize the L matrix with fi in two ways.
Linear embeddings The simplest way is to linearly transform the original features
Lij = fiT W T W fj ,
(9)
where W is the transformation matrix.
Nonlinear hidden representation We use a one-hidden-layer neural network to infer a hidden
representation for fi
Lij = ziT W T W zj
where zi = tanh(U fi ),
(10)
where tanh(?) stands for the hyperbolic transfer function.
To learn the parameters W or U and W , we use maximum likelihood estimation (cf. eq. (5)), with
gradient-descent to optimize. Details are given in the supplementary material.
4
Related work
Space does not permit a thorough survey of video summarization methods. Broadly speaking, existing approaches develop a variety of selection criteria to prioritize frames for the output summary,
often combined with temporal segmentation. Prior work often aims to retain diverse and representative frames [2, 1, 10, 4, 11], and/or defines novel metrics for object and event saliency [3, 2, 6, 8].
When the camera is known to be stationary, background subtraction and object tracking are valuable
cues (e.g., [5]). Recent developments tackle summarization for dynamic cameras that are worn or
handheld [10, 8, 9] or design online algorithms to process streaming data [7].
Whereas existing methods are largely unsupervised, our idea to explicitly learn subset selection
from human-given summaries is novel. Some prior work includes supervised learning components
that are applied during selection (e.g., to generate learned region saliency metrics [8] or train classifiers for canonical viewpoints [10]), but they do not train/learn the subset selection procedure itself.
Our idea is also distinct from ?interactive? methods, which assume a human is in the loop to give
supervision/feedback on each individual test video [26, 27, 12].
Our focus on the determinantal point process as the building block is largely inspired by its appealing
property in modeling diversity in subset selection, as well as its success in search and ranking [17],
document summarization [14], news headline displaying [28], and pose estimation [29]. Applying
DPP to video summarization, however, is novel to the best of our knowledge.
Our seqDPP is closest in spirit to the recently proposed Markov DPP [28]. While both models enjoy
the Markov property by defining conditional probabilities depending only on the immediate past,
6
Table 1: Performance of various video summarization methods on OVP. Ours and its variants perform the best.
Unsupervised methods
F
P
R
DT
STIMO
VSUMM 1
VSUMM 2
[30]
57.6
67.7
53.2
[31]
63.4
60.3
72.2
[24]
70.3
70.6
75.8
[24]
68.2
73.1
69.1
+ Q/D
[14]
70.8?0.3
71.5?0.4
74.5?0.3
DPP
Supervised subset selection
Ours (seqDPP+)
Q/D
LINEAR
N . NETS
68.5?0.3 75.5?0.4 77.7?0.4
66.9?0.4 77.5?0.5 75.0?0.5
75.8?0.5 78.4?0.5 87.2?0.3
Table 2: Performance of our method with different representation learning
VSUMM 2
Youtube
Kodak
F
55.7
68.9
P
59.7
75.7
[24]
R
58.7
80.6
seqDPP+LINEAR
F
P
R
57.8?0.5 54.2?0.7 69.8?0.5
75.3?0.7 77.8?1.0 80.4?0.9
seqDPP+N . NETS
F
P
R
60.3?0.5 59.4?0.6 64.9?0.5
78.9?0.5 81.9?0.8 81.1?0.9
Markov DPP?s ground set is still the whole video sequence, whereas seqDPP can select diverse sets
from the present time. Thus, one potential drawback of applying Markov DPP is to select video
frames out of temporal order, thus failing to model the sequential nature of the data faithfully.
5
Experiments
We validate our approach of sequential determinantal point processes (seqDPP) for video summarization on several datasets, and obtain superior performance to competing methods.
5.1
Setup
Data We benchmark various methods on 3 video datasets: the Open Video Project (OVP), the
Youtube dataset [24], and the Kodak consumer video dataset [32]. They have 50, 392 , and 18 videos,
respectively. The first two have 5 human-created summaries per video and the last has one humancreated summary per video. Thus, for the first two datasets, we follow the algorithm described in
section 3.1 to create an oracle summary per video. We follow the same procedure as in [24] to
preprocess the video frames. We uniformly sample one frame per second and then apply two stages
of pruning to remove uninformative frames. Details are in the supplementary material.
Features Each frame is encoded with an `2-normalized 8192-dimensional Fisher vector ?i [33],
computed from SIFT features [34]. The Fisher vector represents well the visual appearance of the
video frame, and is hence used to compute the pairwise correlations of the frames in the qualitydiversity decomposition (cf. eq. (4)). We derive the quality features xi by measuring the representativeness of the frame. Specifically, we place a contextual window centered around the frame of
interest, and then compute its mean correlation (using the SIFT Fisher vector) to the other frames in
the window. By varying the size of the windows from 5 to 15, we obtain 12-dimensional contextual
features. We also add features computed from the frame saliency map [35]. To apply our method
for learning representations (cf. section 3.3), however, we do not make a distinction between the two
types, and instead compose a feature vector fi by concatenating xi and ?i . The dimension of our
linear transformed features W fi is 10, 40 and 100 for OVP, Youtube, and Kodak, respectively. For
the neural network, we use 50 hidden units and 50 output units.
Other details For each dataset, we randomly choose 80% of the videos for training and use the
remaining 20% for testing. We run 100 rounds of experiments and report the average performance,
which is evaluated by the aforementioned F-score, Precision, and Recall (cf. section 3.1). For
evaluation, we follow the standard procedure: for each video, we treat each human-created summary
as golden-standard and assess the quality of the summary output by our algorithm. We then average
over all human annotators to obtain the evaluation metrics for that video.
5.2
Results
We contrast our approach to several state-of-the-art methods for video summarization?which include several leading unsupervised methods?as well as the vanilla DPP model that has been successfully used for document summarization but does not model sequential structures. We compare
the methods in greater detail on the OVP dataset. Table 1 shows the results.
2
In total there are 50 Youtube videos. We keep 39 of them after excluding the cartoon videos.
7
Oracle
Youtube
(Video 99)
Sequential LINEAR
(F=70, P=60, R=88)
VSUMM1
(F=59, P=65, R=55)
User
Kodak
(Video 4)
Sequential LINEAR
(F=86, P=75, R=100)
VSUMM1
(F=50, P=100, R=33)
Figure 3: Exemplar video summaries by our seqDPP LINEAR vs.
VSUMM
summary [24].
Unsupervised or supervised? The four unsupervised methods are DT [30], STIMO [31],
VSUMM 1 [24], and VSUMM 2 with a postprocessing step to VSUMM 1 to improve the precision of
the results. We implement VSUMM ourselves using features described in the orignal paper and tune
its parameters to have the best test performance. All 4 methods use clustering-like procedures to
identify key frames as video summaries. Results of DT and STIMO are taken from their original
papers. They generally underperform VSUMM.
What is interesting is that the vanilla DPP does not outperform the unsupervised methods, despite
its success in other tasks. On the other end, our supervised method seqDPP, when coupled with the
linear or neural network representation learning, performs significantly better than all other methods.
We believe the improvement can be attributed to two factors working in concert: (1) modeling sequential structures of the video data, and (2) more flexible and powerful representation learning.
This is evidenced by the rather poor performance of seqDPP with the quality/diversity (Q/D) decomposition, where the representation of the items is severely limited such that modeling temporal
structures alone is simply insufficient.
Linear or nonlinear? Table 2 concentrates on comparing the effectiveness of these two types of
representation learning. The performances of VSUMM are provided for reference only. We see that
learning representations with neural networks generally outperforms the linear representations.
Qualitative results We present exemplar video summaries by different methods in Fig. 3. The
challenging Youtube video illustrates the advantage of sequential diverse subset selection. The visual
variance in the beginning of the video is far greater (due to close-shots of people) than that at the end
(zooming out). Thus the clustering-based VSUMM method is prone to select key frames from the first
half of the video, collapsing the latter part. In contrast, our seqDPP copes with time-varying diversity
very well. The Kodak video demonstrates again our method?s ability in attaining high recall when
users only make diverse selections locally but not globally. VSUMM fails to acknowledge temporally
distant frames can be diverse despite their visual similarities.
6
Conclusion
Our novel learning model seqDPP is a successful first step towards using human-created summaries
for learning to select subsets for the challenging video summarization problem. We just scratched
the surface of this fruit-bearing direction. We plan to investigate how to learn more powerful representations from low-level visual cues.
Acknowledgments B. G., W. C. and F. S. are partially supported by DARPA D11-AP00278, NSF IIS-1065243,
and ARO #W911NF-12-1-0241. K. G. is supported by ONR YIP Award N00014-12-1-0754 and gifts from
Intel and Google. B. G. and W. C. also acknowledge supports from USC Viterbi Doctoral Fellowship and USC
Annenberg Graduate Fellowship. We are grateful to Jiebo Luo for providing the Kodak dataset [32].
8
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9
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4,876 | 5,414 | Grouping-Based Low-Rank Trajectory Completion
and 3D Reconstruction
Marta Salas
Universidad de Zaragoza,
Zaragoza, Spain
[email protected]
Katerina Fragkiadaki
EECS, University of California,
Berkeley, CA 94720
[email protected]
Jitendra Malik
EECS, University of California,
Berkeley, CA 94720
[email protected]
Pablo Arbel?aez
Universidad de los Andes,
Bogot?a, Colombia
[email protected]
Abstract
Extracting 3D shape of deforming objects in monocular videos, a task known as
non-rigid structure-from-motion (NRSfM), has so far been studied only on synthetic datasets and controlled environments. Typically, the objects to reconstruct
are pre-segmented, they exhibit limited rotations and occlusions, or full-length
trajectories are assumed. In order to integrate NRSfM into current video analysis pipelines, one needs to consider as input realistic -thus incomplete- tracking,
and perform spatio-temporal grouping to segment the objects from their surroundings. Furthermore, NRSfM needs to be robust to noise in both segmentation and
tracking, e.g., drifting, segmentation ?leaking?, optical flow ?bleeding? etc. In
this paper, we make a first attempt towards this goal, and propose a method that
combines dense optical flow tracking, motion trajectory clustering and NRSfM
for 3D reconstruction of objects in videos. For each trajectory cluster, we compute multiple reconstructions by minimizing the reprojection error and the rank
of the 3D shape under different rank bounds of the trajectory matrix. We show
that dense 3D shape is extracted and trajectories are completed across occlusions
and low textured regions, even under mild relative motion between the object and
the camera. We achieve competitive results on a public NRSfM benchmark while
using fixed parameters across all sequences and handling incomplete trajectories,
in contrast to existing approaches. We further test our approach on popular video
segmentation datasets. To the best of our knowledge, our method is the first to
extract dense object models from realistic videos, such as those found in Youtube
or Hollywood movies, without object-specific priors.
1
Introduction
Structure-from-motion is the ability to perceive the 3D shape of objects solely from motion cues. It
is considered the earliest form of depth perception in primates, and is believed to be used by animals
that lack stereopsis, such as insects and fish [1].
In computer vision, non-rigid structure-from-motion (NRSfM) is the extraction of a time-varying
3D point cloud from its 2D point trajectories. The problem is under-constrained since many 3D
time-varying shapes and camera poses give rise to the same 2D image projections. To tackle this
ambiguity, early work of Bregler et al. [2] assumes the per frame 3D shapes lie in a low dimensional
subspace. They recover the 3D shape basis and coefficients, along with camera rotations, using
a 3K factorization of the 2D trajectory matrix, where K the dimension of the shape subspace,
1
Video sequence Trajectory clustering
Missing entries
3D Shape
Depth
NRSfM
Figure 1: Overview. Given a monocular video, we cluster dense flow trajectories using 2D motion
similarities. Each trajectory cluster results in an incomplete trajectory matrix that is the input to our
NRSfM algorithm. Present and missing trajectory entries for the chosen frames are shown in green
and red respectively. The color of the points in the rightmost column represents depth values (red is
close, blue is far). Notice the completion of the occluded trajectories on the belly dancer, that reside
beyond the image border.
extending the rank 3 factorization method for rigid SfM of Tomasi and Kanade [3]. Akhter et al.[4]
observe that the 3D point trajectories admit a similar low-rank decomposition: they can be written
as linear combinations over a 3D trajectory basis. This essentially reflects that 3D (and 2D) point
trajectories are temporally smooth. Temporal smoothness is directly imposed using differentials
over the 3D shape matrix in Dai et al. [5]. Further, rather than recovering the shape or trajectory
basis and coefficients, the authors propose a direct rank minimization of the 3D shape matrix, and
show superior reconstruction results.
Despite such progress, NRSfM has been so far demonstrated only on a limited number of synthetic
or lab acquired video sequences. Factors that limit the application of current approaches to realworld scenarios include:
(i) Missing trajectory data. The aforementioned state-of-the-art NRSfM algorithms assume complete trajectories. This is an unrealistic assumption under object rotations, deformations or occlusions. Work of Torresani et al. [6] relaxes the full-length trajectory assumption. They impose a
Gaussian prior over the 3D shape and use probabilistic PCA within a linear dynamical system for
extracting 3D deformation modes and camera poses; however, their method is sensitive to initialization and degrades with the amount of missing data. Gotardo and Martinez [7] combine the shape
and trajectory low-rank decompositions and can handle missing data; their method is one of our
baselines in Section 3. Park et al. [8] use static background structure to estimate camera poses
and handle missing data using a linear formulation over a predefined trajectory basis. Simon at al.
[9] consider a probabilistic formulation of the bilinear basis model of Akhter et al. [10] over the
non-rigid 3D shape deformations. This results in a matrix normal distribution for the time varying
3D shape with a Kronecker structured covariance matrix over the column and row covariances that
describe shape and temporal correlations respectively. Our work makes no assumptions regarding
temporal smoothness, in contrast to [8, 7, 9].
(ii) Requirement of accurate video segmentation. The low-rank priors typically used in NRSfM
require the object to be segmented from its surroundings. Work of [11] is the only approach that
attempts to combine video segmentation and reconstruction, rather than considering pre-segmented
objects. The authors projectively reconstruct small trajectory clusters assuming they capture rigidly
moving object parts. Reconstruction results are shown in three videos only, making it hard to judge
the success of this locally rigid model.
This paper aims at closing the gap between theory and application in object-agnostic NRSfM from
realistic monocular videos. We build upon recent advances in tracking, video segmentation and
low-rank matrix completion to extract 3D shapes of objects in videos under rigid and non-rigid
motion. We assume a scaled orthographic camera model, as standard in the literature [12, 13], and
low-rank object-independent shape priors for the moving objects. Our goal is a richer representation
of the video segments in terms of rotations and 3D deformations, and temporal completion of their
trajectories through occlusion gaps or tracking failures.
2
An overview of our approach is presented in Figure 1. Given a video sequence, we compute dense
point trajectories and cluster them using 2D motion similarities. For each trajectory cluster, we
first complete the 2D trajectory matrix using standard low-rank matrix completion. We then recover the camera poses through a rank 3 truncation of the trajectory matrix and Euclidean upgrade.
Last, keeping the camera poses fixed, we minimize the reprojection error of the observed trajectory
entries along with the nuclear norm of the 3D shape. A byproduct of affine NRSfM is trajectory
completion. The recovered 3D time-varying shape is backprojected in the image and the resulting
2D trajectories are completed through deformations, occlusions or other tracking ambiguities, such
as lack of texture. In summary, our contributions are:
(i) Joint study of motion segmentation and structure-from-motion. We use as input to reconstruction
dense trajectories from optical flow linking [14], as opposed to a) sparse corner trajectories [15],
used in previous NRSfM works [4, 5], or b) subspace trajectories of [16, 17], that are full-length
but cannot tolerate object occlusions. Reconstruction needs to be robust to segmentation mistakes.
Motion trajectory clusters are inevitably polluted with ?bleeding? trajectories that, although reside
on the background, they anchor on occluding contours. We use morphological operations to discard
such trajectories that do not belong to the shape subspace and confuse reconstruction.
(ii) Multiple hypothesis 3D reconstruction through trajectory matrix completion under various rank
bounds, for tackling the rank ambiguity.
(iii) We show that, under high trajectory density, rank 3 factorization of the trajectory matrix, as
opposed to 3K, is sufficient to recover the camera rotations in NRSfM. This allows the use of an
easy, well-studied Euclidean upgrade for the camera rotations, similar to the one proposed for rigid
SfM [3].
We present competitive results of our method on the recently proposed NRSfM benchmark of [17],
under a fixed set of parameters and while handling incomplete trajectories, in contrast to existing approaches. Further, we present extensive reconstruction results in videos from two popular video segmentation benchmarks, VSB100 [18] and Moseg [19], that contain videos from Hollywood movies
and Youtube. To the best of our knowledge, we are the first to show dense non-rigid reconstructions
of objects from real videos, without employing object-specific shape priors [10, 20]. Our code is
available at: www.eecs.berkeley.edu/?katef/nrsfm.
2
2.1
Low-rank 3D video reconstruction
Video segmentation by multiscale trajectory clustering
Given a video sequence, we want to segment the moving objects in the scene. Brox and Malik
[19] propose spectral clustering of dense point trajectories from 2D motion similarities and achieve
state-of-the-art performance on video segmentation benchmarks. We extend their method to produce multiscale (rather than single scale) trajectory clustering to deal with segmentation ambiguities
caused by scale and motion variations of the objects in the video scene. Specifically, we first compute a spectral embedding from the top eigenvectors of the normalized trajectory affinity matrix. We
then obtain discrete trajectory clusterings using the discretization method of [21], while varying the
number of eigenvectors to be 10, 20, 30 and 40 in each video sequence.
Ideally, each point trajectory corresponds to a sequence of 2D projections of a 3D physical point.
However, each trajectory cluster is spatially surrounded by a thin layer of trajectories that reside
outside the true object mask and do not represent projections of 3D physical points. They are
the result of optical flow ?bleeding ? to untextured surroundings [22], and anchor themselves on
occluding contours of the object. Although ?bleeding? trajectories do not drift across objects, they
are a source of noise for reconstruction since they do not belong to the subspace spanned by the true
object trajectories. We discard them by computing an open operation (erosion followed by dilation)
and an additional erosion of the trajectory cluster mask in each frame.
2.2
Non-rigid structure-from-motion
Given a trajectory cluster that captures an object in space and time, let Xtk ? R3?1 denote the 3D
coordinate [X Y Z]T of the kth object point at the tth frame. We represent 3D object shape with a
3
matrix S that contains the time varying coordinates of K object surface points in F frames:
? ? 1
S1
X1
? ? ?
= ? ... ? = ? ...
?
S3F ?P
SF
XF
1
X12
???
?
X1P
.. ? .
. ?
XF
2
???
XF
P
For the special case of rigid objects, shape coordinates are constant and the shape matrix takes the
simplified form: S3?P = [X1 X2 ? ? ? XP ] .
We adopt a scaled orthographic camera model for reconstruction [3]. Under orthography, the
projection rays are perpendicular to the image plane and the projection equation takes the form:
x = RX + t, where x = [x y]T is the vector of 2D pixel coordinates, R2?3 is a scaled truncated
rotation matrix and t2?1 is the camera translation. Combining the projection equations for all object
points in all fames, we obtain:
? 1
? 1?
1
1 ?
x1
? ..
.
xF
1
???
..
.
???
x2
..
.
xF
2
xP
t
.. ?
? .. ? ? 1P T ,
=
R
?
S
+
.
.
xF
tF
P
where the camera pose matrix R takes the form:
? 1?
Rrigid
2F ?3
R
= ? ... ? ,
RF
R1
= ? ...
0
?
Rnonrigid
2F ?3F
0
..
.
0
???
???
???
(1)
0
.. ?
.
.
F
R
?
(2)
We subtract the camera translation tt from the pixel coordinates xt , t = 1 ? ? ? F , fixing the origin
of the coordinate system on the objects?s center of mass in each frame, and obtain the centered
trajectory matrix W2F ?P for which W = R ? S.
? denote an incomplete trajectory matrix of a cluster obtained from our multiscale trajectory
Let W
clustering. Let H ? {0, 1}2F ?P denote a binary matrix that indicates presence or absence of
? Given W,
? H, we solve for complete trajectories W, shape S and camera pose R
entries in W.
by minimizing the camera reprojection error and 3D shape rank under various rank bounds for the
trajectory matrix. Rather than minimizing the matrix rank which is intractable, we minimize the
matrix nuclear norm instead (denoted by k?k? ), that yields the best convex approximation for the
matrix rank over the unit ball of matrices. Let denote Hadamard product and k?kF denote the
Frobenius matrix norm. Our cost function reads:
NRSfM(K):
min .
? 2 + kW ? R ? Sk2 + 1K>1 ? ?kSv k?
kH (W ? W)k
F
F
subject to
Rank(W) ? 3K, ??t , s.t. Rt (Rt )T = ?t I2?2 , t = 1 ? ? ? F.
W,R,S
(3)
We compute multiple reconstructions with K ? {1 ? ? ? 9}. Sv denotes the re-arranged shape matrix
where each row contains the vectorized 3D shape in that frame:
? 1
?
1
1
1
1
1
X1
SvF ?3P
= ? ..
.
X1F
Y1
..
.
Y1F
Z1
..
.
Z1F
???
???
???
XP
..
.
XPF
YP
..
.
YPF
ZP
..
.
ZPF
? = [PX
PY
PZ ] (I3 ? S),
(4)
where PX , PY , PZ are appropriate row selection matrices. Dai et al. [5] observe that SvF ?3P has
lower rank than the original S3F ?P since it admits a K-rank decomposition, instead of 3K, assuming per frame 3D shapes span a K dimensional subspace. Though S facilitates the writing of
the projection equations, minimizing the rank of the re-arranged matrix Sv avoids spurious degrees
of freedom. Minimization of the nuclear norm of Sv is used only in the non-rigid case (K > 1).
In the rigid case, the shape does not change in time and Sv1?3P has rank 1 by construction. We
approximately solve Eq. 3 in three steps.
Low-rank trajectory matrix completion
rank bound constraint:
min .
W
subject to
We want to complete the 2D trajectory matrix under a
? 2
kH (W ? W)k
F
Rank(W) ? 3K.
4
(5)
Due to its intractability, the rank bound constraint is typically imposed by a factorization, W =
U V T , U2F ?r, VP ?r , for our case r = 3K. Work of [23] empirically shows that the following
regularized problem is less prone to local minima than its non-regularized counterpart (? = 0):
min .
W,U2F ?3K ,VP ?3K
? 2 + ? (kUk2 + kVk2 )
kH (W ? W)k
F
F
F
2
W = UVT .
subject to
(6)
We solve Eq. 6 using the method of Augmented Lagrange multipliers. We want to explicitly search
over different rank bounds for the trajectory matrix W as we vary K. We do not choose to minimize
the nuclear norm instead, despite being convex, since different weights for the nuclear term result in
matrices of different ranks, thus is harder to control explicitly the rank bound. Prior work [24, 23]
shows that the bilinear formulation of Eq. 6, despite being non-convex in comparison to the nuclear
? 2 + kWk? ), it returns the same optimum in cases r >= r?,
regularized objective (kH (W ? W)k
F
where r? denotes the rank obtained by the unconstrained minimization of the nuclear regularized
objective. We use the continuation strategy proposed in [23] over r to avoid local minima for r < r?:
starting from large values of r, we iteratively reduce it till the desired rank bound 3K is achieved.
For details, please see [23, 24].
Euclidean upgrade Given a complete trajectory matrix, minimization of the reprojection error
term of Eq. 3 under the orthonormality constraints is equivalent to a SfM or NRSfM problem in its
standard form, previously studied in the seminal works of [3, 2]:
min .
kW ? R ? Sk2F
subject to
??t , s.t. Rt (Rt )T = ?t I2?2 , t = 1 ? ? ? F.
R,S
(7)
For rigid objects, Tomasi and Kanade [3] recover the camera pose and shape matrix via SVD of W
? ? S.
? The factorization is not unique
truncated to rank 3: W = UDVT = (UD1/2 )(D1/2 VT ) = R
?1
? ?S
?=R
? ? GG S.
? We estimate G so that RG
? satisfies the
since for any invertible matrix G3?3 : R
orthonormality constraints:
orthogonality:
same norm:
? 2t?1 GGT R
? T = 0, t = 1 ? ? ? F
R
2t
? 2t?1 GGT R
?T
? 2t GGT R
? T , t = 1 ? ? ? F.
R
=
R
2t?1
2t
(8)
The constraints of Eq. 8 form an overdetermined homogeneous linear system with respect to the
elements of the gram matrix Q = GGT . We estimate Q using least-squares and factorize it using
SVD to obtain G up to an arbitrary scaling and rotation of its row space [25]. Then, the rigid object
?
shape is obtained by S3?P = G?1 S.
For non-rigid objects, a similar Euclidean upgrade of the rotation matrices has been attempted using
a rank 3K (rather than 3) decomposition of W [26]. In the non-rigid case, the corrective transformation G has size 3K ? 3K. Each column triplet 3K ? 3 is recovered independently since it contains
the rotation information from all frames. For a long time, an overlooked rank 3 constraint on the
Gram matrix Qk = GTk Gk spurred conjectures regarding the ambiguity of shape recovery under
non-rigid motion [26]. This lead researchers to introduce additional priors for further constraining
the problem, such as temporal smoothness [27]. Finally, the work of [4] showed that orthonormality
constraints are sufficient to recover a unique non-rigid 3D shape. Dai et al. [5] proposed a practical
algorithm for Euclidean upgrade using rank 3K decomposition of W that minimizes the nuclear
norm of Qk under the orthonormality constraints.
Surprisingly, we have found that in practice it is not necessary to go beyond rank 3 truncation of W
to obtain the rotation matrices in the case of dense NRSfM. The large majority of trajectories span
the rigid component of the object, and their information suffices to compute the objects? rotations.
This is not the case for synthetic NRSfM datasets, where the number of tracked points on the articulating links is similar to the points spanning the ?torso-like? component, as in the famous ?Dance?
sequence [12]. In Section 3, we show dense face reconstruction results while varying the truncating
rank ?r of W for the Euclidean upgrade step, and verify that ?r = 3 is more stable than ?r > 3 for
NRSfM of faces.
Rank regularized least-squares for 3D shape recovery In the non-rigid case, given the recovered
camera poses R, we minimize the reprojection error of the observed trajectory entries and 3D shape
5
incomplete trajectories
groundtruth 3D shape
frontal view
missing entries
ours
complete trajectories
ours: frontal view
ours
ours: frontal view
sequence 3
99 frames long
mild deform./rot.
sequence 2
10 frames long
abrupt deform./rot.
rotated
Figure 2: Qualitative results in the synthetic benchmark of [17]. High quality reconstructions are
obtained with oracle (full-length) trajectories for both abrupt and smooth motion. For incomplete
trajectories, in the 3rd column we show in red the missing and in green the present trajectory entries.
The reconstruction result for the 2nd video sequence that has 30% missing data, though worse, is
still recognizable.
nuclear norm:
min .
1
2 kH
v
? ? R ? S)k2 + ?kSv k?
(W
F
(9)
PY PZ ] (I3 ? S).
? to constrain the 3D shape estimation; howNotice that we consider only the observed entries in W
ever, information from the complete W has been used for extracting the rotation matrices R. We
solve the convex, non-smooth problem in Eq. 9 using the nuclear minimization algorithm proposed
in [28]. It generalizes the accelerated proximal gradient method of [29] from l1 regularized leastsquares on vectors to nuclear norm regularized least-squares on matrices. It has a better iteration
complexity than the Fixed Point Continuation (FPC) method of [30] and the Singular Value Thresholding (SVT) method [31].
S
subject to
S = [PX
Given camera pose R and shape S, we backproject to obtain complete centered trajectory matrix
W = R ? S. Though we can in principle iterate over the extraction of camera pose and 3D shape, we
observed benefits from such iteration only in the rigid case. This observation agrees with the results
of Marques and Costeira [32] for rigid SfM from incomplete trajectories.
3
Experiments
The only available dense NRSfM benchmark has been recently introduced in Garg et al. [17]. They
propose a dense NRSfM method that minimizes a robust discontinuity term over the recovered 3D
depth along with 3D shape rank. However, their method assumes as input full-length trajectories
obtained via the subspace flow tracking method of [16]. Unfortunately, the tracker of [16] can
tolerate only very mild out-of-plane rotations or occlusions, which is a serious limitation for tracking
in real videos. Our method does not impose the full-length trajectory requirement. Also, we show
that the robust discontinuity term in [17] may not be necessary for high quality reconstructions.
The benchmark contains four synthetic video sequences that depict a deforming face, and three real
sequences that depict a deforming back, face and heart, respectively. Only the synthetic sequences
have ground-truth 3D shapes available, since it is considerably more difficult to obtain ground-truth
for NRSfM in non-synthetic environments. Dense full-length ground-truth 2D trajectories are provided for all sequences. For evaluation, we use the code supplied with the benchmark, that performs
a pre-alignment step at each frame between St and StGT using Procrustes analysis. Reconstruction
performance is measured by mean RMS error across all frames, where the per frame RMS error of
kSt ?St kF
.
a shape St with respect to ground-truth shape StGT is defined as: kSt GT
kF
GT
Figure 2 presents our qualitative results and Table 1 compares our performance against previous
state-of-the-art NRSfM methods: Trajectory Basis (TB) [12], Metric Projections (MP) [33], Variational Reconstruction (VR) [17] and CSF [7]. For CSF, we were not able to complete the experiment
for sequences 3 and 4 due to the non-scalable nature of the algorithm. Next to the error of each
6
Figure 3: Reconstruction results in the ?Back?, ?Face? and ?Heart? sequences of [17]. We show
present and missing trajectory entries, per frame depth maps and retextured depth maps.
method we show in parentheses the rank used, that is, the rank that gave the best error. Our method
uses exactly the same parameters and K = 9 for all four sequences. Baseline VR [17] adapts the
weight for the nuclear norm of S for each sequence. This shows robustness of our method under
varying object deformations. ?r is the truncated rank of W used for the Euclidean upgrade step.
When ?r > 3, we use the Euclidean upgrade proposed in [5]. ?r = 3 gives the most stable face
reconstruction results.
Next, to imitate a more realistic setup, we introduce missing entries to the ground-truth 2D tracks by
?hiding? trajectory entries that are occluded due to face rotations. The occluded points are shown in
red in Figure 2 3rd column. From the ?incomplete trajectories? section of Table 1, we see that the
error increase for our method is small in comparison to the full-length trajectory case.
In the real ?Back?, ?Face? and ?Heart? sequences of the benchmark, the objects are pre-segmented.
We keep all trajectories that are at least five frames long. This results in 29.29%, 30.54% and
? We used K = 8 for all sequences.
52.71% missing data in the corresponding trajectory matrices W.
We show qualitative results in Figure 3. The present and missing entries are shown in green and
red, respectively. The missing points occupy either occluded regions, or regions with ambiguous
correspondence, e.g., under specularities in the Heart sequence.
Next, we test our method on reconstructing objects from videos of two popular video segmentation
datasets: VSB100 [18], that contains videos uploaded to Youtube, and Moseg [19], that contains
videos from Hollywood movies. Each video is between 19 and 121 frames long. For all videos
we use K ? {1 ? ? ? 5}. We keep all trajectories longer than five frames. This results in missing
data varying from 20% to 70% across videos, with an average of 45% missing trajectory entries.
We visualize reconstructions for the best trajectory clusters (the ones closest to the ground-truth
segmentations supplied with the datasets) in Figure 4.
Discussion Our 3D reconstruction results in real videos show that, under high trajectory density,
small object rotations suffice to create the depth perception. We also observe the tracking quality to
be crucial for reconstruction. Optical flow deteriorates as the spatial resolution decreases, and thus
high video resolution is currently important for our method. The most important failure cases for our
Seq.1 (10)
Seq.2 (10)
Seq.3 (99)
Seq.4 (99)
ground-truth full trajectories
TB [12] MP [33] VR [17] ours
ours
?r = 3 ?r = 6
18.38 (2) 19.44 (3) 4.01 (9) 5.16
6.69
7.47 (2) 4.87 (3) 3.45 (9) 3.71
5.20
4.50 (4) 5.13 (6) 2.60 (9) 2.81
2.88
6.61 (4) 5.81 (4) 2.81 (9) 3.19
3.08
ours
?r = 9
21.02
25.6
3.00
3.54
incomplete trajectories
ours ?r = 3
CSF
4.92 (8.93% occl)
9.44 (31.60% occl)
3.40 (14.07% occl)
5.53 ( 13.63% occl)
15.6
36.8
??
??
Table 1: Reconstruction results on the NRSfM benchmark of [17]. We show mean RMS error per
cent (%). Numbers for TB, MP and VR baselines are from [17]. In the first column, we show in
parentheses the number of frames. ?r is the rank of W used for the Euclidean upgrade. The last
two columns shows the performance of our algorithm and CSF baseline when occluded points in the
ground-truth tracks are hidden.
7
K=4
K=2
K=1
K=3
K=3
K=3
K=1
K=1
K=2
K=2
Figure 4: Reconstruction results on the VSB and Moseg video segmentation datasets. For each
example we show a) the trajectory cluster, b) the present and missing entries, and c) the depths of the
visible (as estimated from ray casting) points, where red and blue denote close and far respectively.
method are highly articulated objects, which violates the low-rank assumptions. 3D reconstruction
of articulated bodies is the focus of our current work.
4
Conclusion
We have presented a practical method for extracting dense 3D object models from monocular uncalibrated video without object-specific priors. Our method considers as input trajectory motion
clusters obtained from automatic video segmentation that contain large amounts of missing data
due to object occlusions and rotations. We have applied our NRSfM method on synthetic dense reconstruction benchmarks and on numerous videos from Youtube and Hollywood movies. We have
shown that a richer object representation is achievable from video under mild conditions of camera
motion and object deformation: small object rotations are sufficient to recover 3D shape. ?We see
because we move, we move because we see?, said Gibson in his ?Perception of the Visual World?
[34]. We believe this paper has made a step towards encompassing 3D perception from motion into
general video analysis.
Acknowledgments
The authors would like to thank Philipos Modrohai for useful discussions. M.S. acknowledges
funding from Direcci?on General de Investigaci?on of Spain under project DPI2012-32168 and the
Ministerio de Educaci?on (scholarship FPU-AP2010-2906).
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4,877 | 5,415 | Learning Mixtures of Submodular Functions for
Image Collection Summarization
Rishabh Iyer
Department of Electrical Engineering
University of Washington
[email protected]
Sebastian Tschiatschek
Department of Electrical Engineering
Graz University of Technology
[email protected]
Haochen Wei
LinkedIn & Department of Electrical Engineering
University of Washington
[email protected]
Jeff Bilmes
Department of Electrical Engineering
University of Washington
[email protected]
Abstract
We address the problem of image collection summarization by learning mixtures of
submodular functions. Submodularity is useful for this problem since it naturally
represents characteristics such as fidelity and diversity, desirable for any summary.
Several previously proposed image summarization scoring methodologies, in fact,
instinctively arrived at submodularity. We provide classes of submodular component functions (including some which are instantiated via a deep neural network)
over which mixtures may be learnt. We formulate the learning of such mixtures as a
supervised problem via large-margin structured prediction. As a loss function, and
for automatic summary scoring, we introduce a novel summary evaluation method
called V-ROUGE, and test both submodular and non-submodular optimization
(using the submodular-supermodular procedure) to learn a mixture of submodular
functions. Interestingly, using non-submodular optimization to learn submodular
functions provides the best results. We also provide a new data set consisting of
14 real-world image collections along with many human-generated ground truth
summaries collected using Amazon Mechanical Turk. We compare our method
with previous work on this problem and show that our learning approach outperforms all competitors on this new data set. This paper provides, to our knowledge,
the first systematic approach for quantifying the problem of image collection summarization, along with a new data set of image collections and human summaries.
1
Introduction
The number of photographs being uploaded online is growing at an unprecedented rate. A recent
estimate is that 500 million images are uploaded to the internet every day (just considering Flickr,
Facebook, Instagram and Snapchat), a figure which is expected to double every year [22]. Organizing
this vast amount of data is becoming an increasingly important problem. Moreover, the majority
of this data is in the form of personal image collections, and a natural problem is to summarize
such vast collections. For example, one may have a collection of images taken on a holiday trip,
and want to summarize and arrange this collection to send to a friend or family member or to post
on Facebook. Here the problem is to identify a subset of the images which concisely represents
all the diversity from the holiday trip. Another example is scene summarization [28], where one
wants to concisely represent a scene, like the Vatican or the Colosseum. This is relevant for creating
a visual summary of a particular interest point, where we want to identify a representative set of
views. Another application that is gaining importance is summarizing video collections [26, 13] in
order to enable efficient navigation of videos. This is particularly important in security applications,
where one wishes to quickly identify representative and salient images in massive amounts of video.
1
These problems are closely related and can be unified via the problem of finding the most representative subset of images from an entire image collection. We argue that many formulations of
this problem are naturally instances of submodular maximization, a statement supported by the fact
that a number of scoring functions previously investigated for image summarization are (apparently
unintentionally) submodular [30, 28, 5, 29, 8].
A set function f (?) is said to be submodular if for any element v and sets A ? B ? V \{v}, where
V represents the ground set of elements, f (A ? {v}) ? f (A) ? f (B ? {v}) ? f (B). This is
called the diminishing returns property and states, informally, that adding an element to a smaller
set increases the function value more than adding that element to a larger set. Submodular functions
naturally model notions of coverage and diversity in applications, and therefore, a number of machine
learning problems can be modeled as forms of submodular optimization [11, 20, 18]. In this paper,
we investigate structured prediction methods for learning weighted mixtures of submodular functions
for image collection summarization.
Related Work: Previous work on image summarization can broadly be categorized into (a)
clustering-based approaches, and (b) approaches which directly optimize certain scoring functions.
The clustering papers include [12, 8, 16]. For example, [12] proposes a hierarchical clustering-based
summarization approach, while [8] uses k-medoids-based clustering to generate summaries. Similarly [16] proposes top-down based clustering. A number of other methods attempt to directly
optimize certain scoring functions. For example, [28] focuses on scene summarization and poses an
objective capturing important summarization metrics such as likelihood, coverage, and orthogonality.
While they do not explicitly mention this, their objective function is in fact a submodular function.
Furthermore, they propose a greedy algorithm to optimize their objective. A similar approach was proposed by [30, 29], where a set cover function (which incidentally also is submodular) is used to model
coverage, and a minimum disparity formulation is used to model diversity. Interestingly, they optimize
their objective using the same greedy algorithm. Similarly, [15] models the problem of diverse image
retrieval via determinantal point processes (DPPs). DPPs are closely related to submodularity, and in
fact, the MAP inference problem is an instance of submodular maximization. Another approach for
image summarization was posed by [5], where they define an objective function using a graph-cut function, and attempt to solve it using a semidefinite relaxation. They unintentionally use an objective that
is submodular (and approximately monotone [18]) that can be optimized using the greedy algorithm.
Our Contributions: We introduce a family of submodular function components for image collection
summarization over which a convex mixture can be placed, and we propose a large margin formulation
for learning the mixture. We introduce a novel data set of fourteen personal image collections, along
with ground truth human summaries collected via Amazon mechanical Turk, and then subsequently
cleaned via methods described below. Moreover, in order to automatically evaluate the quality of
novel summaries, we introduce a recall-based evaluation metric, which we call V-ROUGE, to compare
automatically generated summaries to the human ones. We are inspired by ROUGE [17], a wellknown evaluation criterion for evaluating summaries in the document summarization community, but
we are unaware of any similar efforts in the computer vision community for image summarization. We
show evidence that V-ROUGE correlates well with human evaluation. Finally, we extensively validate
our approach on these data sets, and show that it outperforms previously explored methods developed
for similar problems. The resulting learnt objective, moreover, matches human summarization
performance on test data.
2
Image Collection Summarization
Summarization is a task that most humans perform intuitively. Broadly speaking, summarization is
the task of extracting information from a source that is both minimal and most important. The precise
meaning of most important (relevance) is typically subjective and thus will differ from individual
to individual and hence is difficult to precisely quantify. Nevertheless, we can identify two general
properties that characterize good image collection summarizes [19, 28]:
Fidelity: A summary should have good coverage, meaning that all of the distinct ?concepts? in
the collection have at least one representative in the summary. For example, a summary of a photo
collection containing both mountains and beaches should contain images of both scene types.
Diversity: Summaries should be as diverse as possible, i.e., summaries should not contain images
that are similar or identical to each other. Other words for this concept include diversity or dispersion.
In computer vision, this property has been referred to as orthogonality [28].
2
Note that [28] also includes the notion of ?likelihood,? where summary images should have high
similarity to many other images in the collection. We believe, however, that such likelihood is
covered by fidelity. I.e., a summary that only has images similar to many in the collection might miss
certain outlier, or minority, concepts in the collection, while a summary that has high fidelity should
include a representative image for every both majority and minority concept in the collection.Also,
the above properties could be made very high without imposing further size or budget constraints.
I.e., the goal of a summary is to find a small or within-budget subset having the above properties.
2.1
Problem Formulation
We cast the problem of image collection summarization as a subset selection problem: given a
collection of images I = (I1 , I2 , ? ? ? , I|V | ) represented by an index set V and given a budget c, we
aim to find a subset S ? V, |S| ? c, which best summarizes the collection. Though alternative
approaches are possible, we aim to solve this problem by learning a scoring function F : 2V ? R+ ,
such that high quality summaries are mapped to high scores and low quality summaries to low scores.
Then, image collection summarization can be performed by computing:
S ? ? argmaxS?V,|S|?c F (S).
(1)
?
For arbitrary set functions, computing S is intractable, but for monotone submodular functions
we rely on the classic result [25] that the greedy algorithm offers a constant-factor mathematical
quality guarantee. Computational tractability notwithstanding, submodular functions are natural for
measuring fidelity and diversity [19] as we argue in Section 4.
2.2
Evaluation Criteria: V-ROUGE
Before describing practical submodular functions for mixture components, we discuss a crucial element for both summarization evaluation and for the automated learning of mixtures: an objective evaluation criterion for judging the quality of summaries. Our criterion is constructed similar to the popular
ROUGE score used in multi-document summarization [17] and that correlates well with human perception. For document summarization, ROUGE (which in fact, is submodular [19, 20]) is defined as:
P
P
min (cw (A), cw (S))
w?W
P S?S P
rS (A) =
( , r(A) when S is clear from the context), (2)
w?W
S?S cw (S)
where S is a set of human-generated reference summaries, W is a set of features (n-grams), and where
cw (A) is the occurrence-count of w in summary A. We may extend r(?) to handle images by letting W
be a set of visual words, S a set of reference summaries, and cw (A) be the occurrence-counts of visual
word w in summary A. Visual words can for example be computed from SIFT-descriptors [21] as common in the popular bag-of-words framework in computer vision [31]. We call this V-ROUGE (visual
ROUGE). In our experiments, we use visual words extracted from color histograms, from super-pixels,
and also from OverFeat [27], a deep convolutional network ? details are given in Section 5.
3
Learning Framework
We construct our submodular scoring functions Fw (?)
convex combinations of non-negative
Pas
m
submodular
functions
f
,
f
,
.
.
.
,
f
,
i.e.
F
(S)
=
1 2
m
w
i=1 wi fi (S), where w = (w1 , . . . , wm ),
P
wi ? 0, i wi = 1. The functions fi are submodular components and assumed to be normalized:
i.e., fi (?) = 0, and fi (V ) = 1 for polymatroid functions and maxA?V fi (A) ? 1 for non-monotone
functions. This ensures that the components are compatible with each other. Obviously, the merit of
the scoring function Fw (?) depends on the selection of the components. In Section 4, we provide a
large number of natural component choices, mixtures over which span a large diversity of submodular
functions. Many of these component functions have appeared individually in past work and are
unified into a single framework in our approach.
Large-margin Structured Prediction: We optimize the weights w of the scoring function Fw (?)
in a large-margin structured prediction framework, i.e. the weights are optimized such that human
summaries S are separated from competitor summaries by a loss-dependent margin:
Fw (S) ? Fw (S 0 ) + `(S 0 ),
?S ? S, S 0 ? Y \ S,
(3)
where `(?) is the considered loss function, and where Y is a structured output space (for example Y
is the set of summaries that satisfy a certain budget c, i.e. Y = {S 0 ? V : |S 0 | ? c}). We assume
3
the loss to be normalized, 0 ? `(S 0 ) ? 1, ?S 0 ? V , to ensure mixture and loss are calibrated.
Equation (3) can be stated as Fw (S) ? maxS 0 ?Y [Fw (S 0 ) + `(S 0 )] , ?S ? S which is called lossaugmented inference. We introduce slack variables and minimize the regularized sum of slacks [20]:
X
?
0
0
min
max
[F
(S
)
+
`(S
)]
?
F
(S)
+ kwk22 ,
(4)
w
w
S 0 ?Y
2
w?0,kwk1 =1
S?S
where the non-negative orthant constraint, w ? 0, ensures that the final mixture is submodular. Note
a 2-norm regularizer is used on top of a 1-norm constraint kwk1 = 1 which we interpret as a prior to
encourage higher entropy, and thus more diverse mixture, distributions. Tractability depends on the
choice of the loss function. An obvious choice is `(S) = 1 ? r(S), which yields a non-submodular
optimization problem suitable for optimization methods such as [10] (and which we try in Section 7).
We also consider other loss functions that retain submodularity in loss augmented inference. For
now, assume that S? = maxS 0 ?Y [Fw (S 0 ) + `(S 0 )] can be estimated efficiently. The objective in (4)
can then be minimized using standard stochastic gradient descent methods, where the gradient for
sample S with respect to weight wi is given as
?
?
2
?
?
? ? fi (S) + ?wi .
Fw (S) + `(S) ? Fw (S) + kwk2 = fi (S)
(5)
?wi
2
Loss Functions: A natural loss function is `1?R (S) = 1 ? r(S) where r(S) = V-ROUGE(S).
Because r(S) is submodular, 1 ? r(S) is supermodular and hence maximizing Fw (S 0 ) + `(S 0 )
requires difference-of-submodular set function maximization [24] which is NP-hard [10]. We
also consider two alternative loss functions [20], complement V-ROUGE and surrogate V-ROUGE.
Complement V-ROUGE sets `c (S) = r(V \ S) and is still submodular but it is non-monotone.
`c (?) does have the necessary characteristics of a proper loss: summaries S+ with large V-ROUGE
score are mapped to small values and summaries S? with small V-ROUGE score are mapped to
large values. In particular, submodularity means r(S) + r(V \ S) ? r(V ) + r(?) = r(V ) or
r(V \ S) ? r(V ) ? r(S) = 1 ? r(S), so complement rouge is a submodular
upper
P
P bound of the ideal
supermodular loss. We define surrogate V-ROUGE as `surr (A) = Z1 S?S w?W c cw (A), where
S
WSc is the set of visual words that do not appear in reference summary S and Z is a normalization
constant. Here, a summary has a high loss if it contains many visual words that do not occur in
reference summaries and a low loss if it mainly contains visual words that occur in the reference
summaries. Surrogate V-ROUGE is not only monotone submodular, it is modular.
Loss augmented Inference: Depending on the loss function, different algorithms for performing
loss augmented inference, i.e. computation of the maximum in (4), must be used. When using the
surrogate loss lsurr (?), the mixture function together with the loss, i.e. fL (S) = Fw (S) + `(S), is
submodular and monotone. Hence, the greedy algorithm [25] can be used for maximization. This
algorithm is extremely simple to implement, and starting at S 0 = ?, iteratively chooses an element
j?
/ S t that maximizes fL (S t ? j), until the budget constraint is violated. While the complexity of
this simple procedure is O(n2 ) function evaluations, it can be significantly accelerated, thanks again
to submodularity [23], which in practice we find is almost linear time. When using complement rouge
`c (?) as the loss, fL (S) is still submodular but no longer monotone, so we utilize the randomized
greedy algorithm [2] (which is essentially a randomized variant of the greedy algorithm above, and
has approximation guarantees). Finally, when using loss 1-V-ROUGE, Fw (S) + `(S) is neither
submodular nor monotone and approximate maximization is intractable. However, we resort to well
motivated and scalable heuristics, such as the submodular-supermodular procedures that have shown
good performance in various applications [24, 10].
Runtime Inference:
Having learnt the weights for the mixture components, the resulting function
Pm
Fw (S) = i=1 wi fi (S) is monotone submodular, which can be optimized by the accelerated greedy
algorithm [23]. Thanks to submodularity, we can obtain near optimal solutions very efficiently.
4
Submodular Component Functions
In this section, we consider candidate submodular component functions to use in Fw (?). We consider
first functions capturing more of the notion of fidelity, and then next diversity, although the distinction
is not entirely crystal clear in these functions as some have aspects of both. Many of the components
are graph-based. We define a weighted graph G(V, E, s), with V representing a the full set of images
and E is every pair of elements in V . Each edge (i, j) ? E has weight si,j computed from the visual
features as described in Section 7. The weight si,j is a similarity score between images i and j.
4
4.1
Fidelity-like Functions
A function representing the fidelity of a subset to the whole is one that gets a large value when
the subset faithfully represents that whole. An intuitively reasonable property for such a function
is one that scores a summary highly if it is the case that the summary, as a whole, is similar to a
large majority of items in the set V . In this case, if a given summary A has a fidelity of f (A), then
any superset B ? A should, if anything, have higher fidelity, and thus it seems natural to use only
monotone non-decreasing functions as fidelity functions. Submodularity is also a natural property
since as more and more properties of an image collection are covered by a summary, the less chance
any given image not part of the summary would have in offering additional coverage ? in other
words, submodularity is a natural model for measuring the inherent redundancy in any summary.
Given this, we briefly describe some possible choices for coverage functions:
Facility Location. Given a summary S ? V , we can quantify coverage of the whole image collection
V by the similarity between i ? V and its
Pclosest image j ? S. Summing these similarities yields the
facility location function ffac.loc. (S) = i?V maxj?S si,j . The facility location function has been
used for scene summarization in [28] and as one of the components in [20].
Sum Coverage. Here, we compute the average similarity in S rather than the similarity of the best
element in S only. From the graph perspective
P
P (G) we sum over the weights of edges with at least
one vertex in S. Thus, fsum cov. (S) = i?V j?S si,j .
Thresholded sum/truncated graph cut This function has been used in document summarization [20] and is similar to the sum coverage function except
that instead of summing over all elements
P
in S, we threshold the inner sum. Define ?i (S) = j?S si,j , i.e. informally, ?i (S) conveys how
much of image i is covered by S. In order to keep an element i P
from being overly covered by S as the
cause of the objective getting large, we define fthresh.sum (S) = i?V min(?i (S), ? ?i (V )), which is
both monotone and submodular [20]. Under budget constraints, this function avoids summaries that
over-cover any images.
Feature functions. Consider a bag-of-words image model where for i ? V , bi = (bi,w )w?W
is a bag-of-words representation of image i indexed by the set of visual words W (cf. Section 5).
We can then define
function [14], defined using the visual words, as follows:
P a feature
P coverage
ffeat.cov. (S) = w?W g
b
, where g(?) is a monotone non-decreasing concave function.
i,w
i?I
This class is both monotone and submodular, and an added benefit of scalability, since it does not
require computation of a O(n2 ) similarity matrix like the graph-based functions proposed above.
4.2
Diversity
Diversity is another trait of a good summary, and there are a number of ways to quantify it. In this
case, while submodularity is still quite natural, monotonicity sometimes is not.
Penalty based diversity/dispersion Given P
a set S,
Pwe penalize similarity within S by summing
over all pairs as follows: fdissim. (S) = ? i?S j?S,j>i si,j [28] (a variant, also submodular,
P
takes the form ? i,j?S si,j [19]). These functions are submodular, and monotone decreasing, so
when added to other functions can yield non-monotone submodular functions. Such functions have
occurred before in document summarization [19], as a dispersion function [1], and even for scene
summarization [28] (in this last case, the submodularity property was not explicitly mentioned).
Diversity reward based on clusters. As in [20], we define a cluster based function rewarding
diversity. Given clusters P1 , P2 , ? ? ? , Pk obtained by some clustering algorithm. We define diversity
Pk
reward functions fdiv.reward (S) = j=1 g(S ? Pj ), where g(?) is a monotone submodular function
so that fdiv.reward (?) is also monotone and submodular. Given a budget, fdiv.reward (S) is maximized
by selecting S as diverse, over different clusters, as possible because of diminishing credit when
repeatedly choosing an item in a cluster.
5
Visual Words for Evaluation
V-ROUGE (see Section 2.2) depends on a visual ?bag-of-words? vocabulary, and to construct a visual
vocabulary, multitude choices exists. Common choices include SIFT descriptors [21], color descriptors [34], raw image patches [7], etc. For encoding, vector quantization (histogram encoding) [4],
sparse coding [35], kernel codebook encoding [4], etc. can all be used. For the construction of our
5
V-ROUGE metric, we computed three lexical types and used their union as our visual vocabulary. The
different types are intended to capture information about the images at different scales of abstraction.
Color histogram. The goal here is to capture overall image information via color information. We
follow the method proposed in [34]: Firstly, we extract the most frequent colors in RGB color space
from the images in an image collection using 10 ? 10 pixel patches. Secondly, these frequent colors
are clustered by k-means into 128 clusters, resulting in 128 cluster centers. Finally, we quantize the
most frequent colors in every 10 ? 10 pixel image patch using nearest neighbor vector quantization.
For every image, the resulting bag-of-colors is normalized to unit `1 -norm.
Super pixels. Here, we wish to capture information about small objects or image regions that are
identified by segmentation. Images are first segmented using the quick shift algorithm implemented
in VLFeat [33]. For every segment, dense SIFT descriptors are computed and clustered into 200
clusters. Then, a patch-wise intermediate bag of words bpatch is computed by vector quantization
and the RGB color histogram of the corresponding patch cpatch is appended to that set of words.
This results in intermediate features ?patch = [bpatch , cpatch ]. These intermediate features are again
clustered into 200 clusters. Finally, the intermediate features are vector-quantized according to their
`1 -distance. This final bag-of-words representation is normalized to unit `1 -norm.
Deep convolutional neural network. Our deep neural network based words are meant to capture
high-level information from the images. We use OverFeat [27], i.e. an image recognizer and feature
extractor based on a convolutional neural network for extracting medium to high level image features.
A sliding window is moved across an input picture such that every image is divided into 10 ? 10
blocks (using a 50% overlap) and the pixels within the window are presented to OverFeat as input.
The activations on layer 17 are taken as intermediate features ?k and clustered by k-means into 300
clusters. Then, each ?k is encoded by kernel codebook encoding [4]. For every image, the resulting
bag-of-words representation is normalized to the unit `1 -norm.
6
Data Collection
Dataset. One major contribution of our paper is our new data set which we plan soon to publicly
release. Our data set consists of 14 image collections, each comprising 100 images. The image
collections are typical real world personal image collections as they, for the most part, were taken
during holiday trips. For each collection, human-generated summaries were collected using Amazon
mechanical Turk. Workers were asked to select a subset of 10 images from an image collection such
that it summarizes the collection in the best possible way.1 In contrast to previous work on movie
summarization [13], Turkers were not tested for their ability to produce high quality summaries.
Every Turker was rewarded 10 US cents for every summary.
Pruning of poor human-generated summaries. The summaries collected using Amazon
mechanical Turk differ drastically in quality. For example, some of the collected summaries have low
quality because they do not represent an image collection properly, e.g. they consist only of pictures
of the same people but no pictures showing, say, architecture. Even though we went through several
distinct iterations of summary collection via Amazon Turk, improving the quality of our instructions
each time, it was impossible to ensure that all individuals produced meaningful summaries. Such
low quality summaries can drastically degrade performance of the learning algorithm. We thus
developed a strategy to automatically prune away bad summaries, where ?bad? is defined as the
worst V-ROUGE score relative to a current set of human summaries. The strategy is depicted in
Algorithm 1. Each pruning step removes the worst human summary, and then creates a new instance
of V-ROUGE using the updated pruned summaries. Pruning proceeds as long as a significant fraction
(greater than a desired ?p-value?) of null-hypothesis summarizes (generated uniformly at random)
scores better than the worst human summary. We chose a significant value of p = 0.10.
7
Experiments
To validate our approach, we learned mixtures of submodular functions with 594 component
functions using the data set described in Section 6. In this data set, all human generated reference
summaries are size 10, and we evaluated performance of our learnt mixtures also by producing size
10 summaries. The component functions were the monotone submodular functions described in
1
We did not provide explicit instructions on precisely how to summarize an image collection and instead
only asked that they choose a representative subset. We relied on their high-level intuitive understanding that the
gestalt of the image collection should be preserved in the summary.
6
Algorithm 1 Algorithm for pruning poor human-generated summaries.
Require: Confidence level p, human summaries S, number of random summaries N
Sample N uniformly at random size-10 image sets, to be used as summaries R = (R1 , . . . , RN )
Instantiate
PV-ROUGE-score rS (?) instantiated with summaries S
1
o ? |R|
R?R 1{rS (R)>minS?S rS (S)} // fraction of random summaries better than worst human
while o > p do
S ? S \ (argminS?S rS (S))
Re-instantiate V-ROUGE score rS (?) using updated pruned human summaries S.
Recompute overlap o as above, but with updated V-ROUGE score.
end while
return Pruned human summaries S
Figure 1: Three example 10?10 image collections from our new data set.
Section 4 using features described in Section 5. For weight optimization, we used AdaGrad [6], an
adaptive subgradient method allowing for informative gradient-based learning. We do 20 passes
through the samples in the collection.
We considered two types of experiments: 1) cheating experiments to verify that our proposed mixture
components can effectively learn good scoring functions; and 2) a 14-fold cross-validation experiment
to test our approach in real- world scenarios. In the cheating experiments, training and testing is
performed on the same image collection, and this is repeated 14 times. By contrast, for our 14-fold
cross-validation experiments, training is performed on 13 out of 14 image collections and testing is
performed on the held out summary, again repeating this 14 times. In both experiment types, since
our learnt functions are always monotone submodular, we compute summaries S ? of size 10 that
approximately maximize the scoring functions using the greedy algorithm. For these summaries,
we compute the V-ROUGE score r(S ? ). For easy score interpretation, we normalize it according to
sc(S ? ) = (r(S ? ) ? R)/(H ? R), where R is the average V-ROUGE score of random summaries
(computed from 1000 summaries) and where H is the average V-ROUGE score of the collected final
pruned human summaries. The result sc(S ? ) is smaller than zero if S ? scores worse than the average
random summary and larger than one if it scores better than the average human summary.
The best cheating results are shown as Cheat in Table 1, learnt using 1-V-ROUGE as a loss. The
results in column Min are computed by constrainedly minimizing V-ROUGE via the methods of [11],
and the results in column Max are computed by maximizing V-ROUGE using the greedy algorithm.
Therefore, the Max column is an approximate upper bound on our achievable performance. Clearly,
we are able to learn good scoring functions, as on average we significantly exceed average human
performance, i.e., we achieve an average score of 1.42 while the average human score is 1.00.
Results for cross-validation experiments are presented in Table 1. In the columns Our Methods
we present the performance of our mixtures learnt using the proposed loss functions described in
Section 3. We also present a set of baseline comparisons, using similarity scores computed via a
histogram intersection [32] method over the visual words used in the construction of V-ROUGE. We
present baseline results for the following schemes:
the facility location objective ffac.loc. (S) alone;
the facility location objective mixed with a ?-weighted penalty, i.e. ffac.loc. (S) + ?fdissim. (S);
Maximal marginal relevance [3], using ? to tradeoff between relevance and diversity;
Graphcut mixed with a ?-weighted penalty, similar to FLpen but where graphcut is used in
place of facility location;
kM K-Medoids clustering [9, Algorithm 14.2]. Initial cluster centers were selected uniformly at
random. As a dissimilarity score between images i and j, we used 1 ? si,j . Clustering was
run 20 times, and we used the cluster centers of the best clustering as the summary.
FL
FLpen
MMR
GCpen
7
In each of the above cases where a ? weight is used, we take for each image collection the ? ?
{0, 0.1, 0.2, . . . , 0.9, 1.0} that produced a submodular function that when maximized produced the
best average V-ROUGE score on the 13 training image sets. This approach, therefore, selects the best
baseline possible when performing a grid-search on the training sets. Note that both ?-dependent
functions, i.e. FLpen and GCpen , are non-monotone submodular. Therefore, we used the randomized
greedy algorithm [2] for maximization which has a mathematical guarantee (we ran the algorithm 10
times and used the best result).
Table 1 shows that using 1-V-ROUGE as a loss significantly outperforms the other methods. Furthermore, the performance is on average better than human performance, i.e. we achieve an average score
of 1.13 while the average human score is 1.00. This indicates that we can efficiently learn scoring
functions suitable for image collection summarization. For the other two losses, i.e. surrogate and
complement V-ROUGE, performance is significantly worse. Thus, in this case it seems advantageous
to use the proper (supermodular) loss and heuristic optimization (the submodular-supermodular
procedure [24, 10]) for loss-augmented inference during training, compared to using an approximate
(submodular or modular) loss in combination with an optimization algorithm for loss-augmented
inference with strong guarantees. This could, however, perhaps be circumvented by constructing a
more accurate strictly submodular surrogate loss but we leave this to future work.
Table 1: Cross-Validation Experiments (see text for details). Average human performance is 1.00,
average random performance is 0.00. For each image collection, the best result achieved by any of
Our Methods and by any of the Baseline Methods is highlighted in bold.
Limits
Our Methods
Baseline Methods
8
No.
Min
Max
Cheat
`1?R
`c
`surr
FL
FLpen
MMR
GCpen
kM
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Avg.
-2.55
-2.06
-2.07
-3.20
-1.65
-2.83
-2.44
-1.66
-2.32
-1.46
-1.55
-1.74
-0.94
-1.46
-2.00
2.78
2.22
2.24
2.04
1.92
2.40
2.07
2.04
2.59
2.34
1.85
2.39
1.72
1.75
2.17
1.71
1.38
1.64
1.42
1.60
1.81
1.07
1.45
1.73
1.39
1.22
1.57
0.77
1.07
1.42
1.51
1.27
1.46
1.04
1.11
1.47
1.07
1.13
1.21
1.06
0.95
1.11
0.32
1.08
1.13
0.87
1.26
0.95
0.81
1.06
0.65
0.96
0.96
1.13
0.78
0.92
0.58
0.53
0.97
0.89
-0.36
0.44
0.23
-0.18
0.58
0.27
0.15
0.07
0.51
0.14
-0.08
0.12
0.14
0.77
0.20
1.45
0.18
0.47
0.71
0.96
1.26
0.93
0.62
0.81
1.58
0.43
0.78
0.02
0.23
0.75
0.82
0.58
0.94
1.01
0.93
1.16
0.70
0.38
0.94
0.99
0.56
0.54
-0.06
0.14
0.69
-0.51
0.65
0.85
0.51
0.95
-0.08
-0.33
0.57
0.09
-0.26
-0.29
0.02
0.52
0.22
0.21
1.06
0.21
-0.53
-0.02
-1.28
0.20
-0.84
-1.27
-0.59
0.07
0.05
-0.01
-0.04
-0.80
-0.27
1.23
0.89
0.52
1.32
0.70
1.05
0.97
0.91
0.38
0.73
0.26
0.63
0.02
0.29
0.71
Conclusions and Future Work
We have considered the task of automated summarization of image collections. A new data set
together with many human generated ground truth summaries was presented and a novel automated
evaluation metric called V-ROUGE was introduced. Based on large-margin structured prediction,
and either submodular or non-submodular optimization, we proposed a method for learning scoring
functions for image collection summarization and demonstrated its empirical effectiveness. In future
work, we would like to scale our methods to much larger image collections. A key step in this
direction is to consider low complexity and highly scalable classes of submodular functions. Another
challenge for larger image collections is how to collect ground truth, as it would be difficult for a
human to summarize a collection of, say, 10,000 images.
Acknowledgments: This material is based upon work supported by the National Science Foundation
under Grant No. (IIS-1162606), the Austrian Science Fund under Grant No. (P25244-N15), a Google
and a Microsoft award, and by the Intel Science and Technology Center for Pervasive Computing.
Rishabh Iyer is also supported by a Microsoft Research Fellowship award.
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4,878 | 5,416 | Deep Learning Face Representation by Joint
Identification-Verification
Yi Sun1
Yuheng Chen2
Xiaogang Wang3,4
Xiaoou Tang1,4
Department of Information Engineering, The Chinese University of Hong Kong
2
SenseTime Group
3
Department of Electronic Engineering, The Chinese University of Hong Kong
4
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences
1
[email protected] [email protected]
[email protected] [email protected]
Abstract
The key challenge of face recognition is to develop effective feature representations for reducing intra-personal variations while enlarging inter-personal
differences. In this paper, we show that it can be well solved with deep learning
and using both face identification and verification signals as supervision. The
Deep IDentification-verification features (DeepID2) are learned with carefully
designed deep convolutional networks. The face identification task increases the
inter-personal variations by drawing DeepID2 features extracted from different
identities apart, while the face verification task reduces the intra-personal
variations by pulling DeepID2 features extracted from the same identity together,
both of which are essential to face recognition. The learned DeepID2 features
can be well generalized to new identities unseen in the training data. On the
challenging LFW dataset [11], 99.15% face verification accuracy is achieved.
Compared with the best previous deep learning result [20] on LFW, the error rate
has been significantly reduced by 67%.
1
Introduction
Faces of the same identity could look much different when presented in different poses, illuminations, expressions, ages, and occlusions. Such variations within the same identity could overwhelm
the variations due to identity differences and make face recognition challenging, especially in
unconstrained conditions. Therefore, reducing the intra-personal variations while enlarging the
inter-personal differences is a central topic in face recognition. It can be traced back to early
subspace face recognition methods such as LDA [1], Bayesian face [16], and unified subspace
[22, 23]. For example, LDA approximates inter- and intra-personal face variations by using two
scatter matrices and finds the projection directions to maximize the ratio between them. More recent
studies have also targeted the same goal, either explicitly or implicitly. For example, metric learning
[6, 9, 14] maps faces to some feature representation such that faces of the same identity are close
to each other while those of different identities stay apart. However, these models are much limited
by their linear nature or shallow structures, while inter- and intra-personal variations are complex,
highly nonlinear, and observed in high-dimensional image space.
In this work, we show that deep learning provides much more powerful tools to handle the two types
of variations. Thanks to its deep architecture and large learning capacity, effective features for face
recognition can be learned through hierarchical nonlinear mappings. We argue that it is essential
to learn such features by using two supervisory signals simultaneously, i.e. the face identification
and verification signals, and the learned features are referred to as Deep IDentification-verification
features (DeepID2). Identification is to classify an input image into a large number of identity
1
classes, while verification is to classify a pair of images as belonging to the same identity or not
(i.e. binary classification). In the training stage, given an input face image with the identification
signal, its DeepID2 features are extracted in the top hidden layer of the learned hierarchical nonlinear
feature representation, and then mapped to one of a large number of identities through another
function g(DeepID2). In the testing stage, the learned DeepID2 features can be generalized to other
tasks (such as face verification) and new identities unseen in the training data. The identification
supervisory signal tends to pull apart the DeepID2 features of different identities since they have to
be classified into different classes. Therefore, the learned features would have rich identity-related
or inter-personal variations. However, the identification signal has a relatively weak constraint on
DeepID2 features extracted from the same identity, since dissimilar DeepID2 features could be
mapped to the same identity through function g(?). This leads to problems when DeepID2 features
are generalized to new tasks and new identities in test where g is not applicable anymore. We solve
this by using an additional face verification signal, which requires that every two DeepID2 feature
vectors extracted from the same identity are close to each other while those extracted from different
identities are kept away. The strong per-element constraint on DeepID2 features can effectively
reduce the intra-personal variations. On the other hand, using the verification signal alone (i.e. only
distinguishing a pair of DeepID2 feature vectors at a time) is not as effective in extracting identityrelated features as using the identification signal (i.e. distinguishing thousands of identities at a
time). Therefore, the two supervisory signals emphasize different aspects in feature learning and
should be employed together.
To characterize faces from different aspects, complementary DeepID2 features are extracted from
various face regions and resolutions, and are concatenated to form the final feature representation
after PCA dimension reduction. Since the learned DeepID2 features are diverse among different
identities while consistent within the same identity, it makes the following face recognition easier.
Using the learned feature representation and a recently proposed face verification model [3], we
achieved the highest 99.15% face verification accuracy on the challenging and extensively studied
LFW dataset [11]. This is the first time that a machine provided with only the face region achieves an
accuracy on par with the 99.20% accuracy of human to whom the entire LFW face image including
the face region and large background area are presented to verify.
In recent years, a great deal of efforts have been made for face recognition with deep learning
[5, 10, 18, 26, 8, 21, 20, 27]. Among the deep learning works, [5, 18, 8] learned features or
deep metrics with the verification signal, while DeepFace [21] and our previous work DeepID
[20] learned features with the identification signal and achieved accuracies around 97.45% on
LFW. Our approach significantly improves the state-of-the-art. The idea of jointly solving the
classification and verification tasks was applied to general object recognition [15], with the focus on
improving classification accuracy on fixed object classes instead of hidden feature representations.
Our work targets on learning features which can be well generalized to new classes (identities) and
the verification task.
2
Identification-verification guided deep feature learning
We learn features with variations of deep convolutional neural networks (deep ConvNets) [12].
The convolution and pooling operations in deep ConvNets are specially designed to extract visual
features hierarchically, from local low-level features to global high-level ones. Our deep ConvNets
take similar structures as in [20]. It contains four convolutional layers, with local weight sharing
[10] in the third and fourth convolutional layers. The ConvNet extracts a 160-dimensional DeepID2
feature vector at its last layer (DeepID2 layer) of the feature extraction cascade. The DeepID2
layer to be learned are fully-connected to both the third and fourth convolutional layers. We use
rectified linear units (ReLU) [17] for neurons in the convolutional layers and the DeepID2 layer.
An illustration of the ConvNet structure used to extract DeepID2 features is shown in Fig. 1 given
an RGB input of size 55 ? 47. When the size of the input region changes, the map sizes in the
following layers will change accordingly. The DeepID2 feature extraction process is denoted as
f = Conv(x, ?c ), where Conv(?) is the feature extraction function defined by the ConvNet, x is the
input face patch, f is the extracted DeepID2 feature vector, and ?c denotes ConvNet parameters to
be learned.
2
Figure 1: The ConvNet structure for DeepID2 feature extraction.
DeepID2 features are learned with two supervisory signals. The first is face identification signal,
which classifies each face image into one of n (e.g., n = 8192) different identities. Identification is
achieved by following the DeepID2 layer with an n-way softmax layer, which outputs a probability
distribution over the n classes. The network is trained to minimize the cross-entropy loss, which we
call the identification loss. It is denoted as
Ident(f, t, ?id ) = ?
n
X
pi log p?i = ? log p?t ,
(1)
i=1
where f is the DeepID2 feature vector, t is the target class, and ?id denotes the softmax layer
parameters. pi is the target probability distribution, where pi = 0 for all i except pt = 1
for the target class t. p?i is the predicted probability distribution. To correctly classify all
the classes simultaneously, the DeepID2 layer must form discriminative identity-related features
(i.e. features with large inter-personal variations). The second is face verification signal, which
encourages DeepID2 features extracted from faces of the same identity to be similar. The verification
signal directly regularize DeepID2 features and can effectively reduce the intra-personal variations.
Commonly used constraints include the L1/L2 norm and cosine similarity. We adopt the following
loss function based on the L2 norm, which was originally proposed by Hadsell et al.[7] for
dimensionality reduction,
(
Verif(fi , fj , yij , ?ve ) =
1
2
1
2
2
kfi ? fj k2
2
max 0, m ? kfi ? fj k2
if yij = 1
,
if yij = ?1
(2)
where fi and fj are DeepID2 feature vectors extracted from the two face images in comparison.
yij = 1 means that fi and fj are from the same identity. In this case, it minimizes the L2 distance
between the two DeepID2 feature vectors. yij = ?1 means different identities, and Eq. (2) requires
the distance larger than a margin m. ?ve = {m} is the parameter to be learned in the verification loss
function. Loss functions based on the L1 norm could have similar formulations [15]. The cosine
similarity was used in [17] as
Verif(fi , fj , yij , ?ve ) =
1
2
(yij ? ?(wd + b)) ,
2
(3)
f ?f
where d = kfi ki2 kfjj k2 is the cosine similarity between DeepID2 feature vectors, ?ve = {w, b} are
learnable scaling and shifting parameters, ? is the sigmoid function, and yij is the binary target of
whether the two compared face images belong to the same identity. All the three loss functions are
evaluated and compared in our experiments.
Our goal is to learn the parameters ?c in the feature extraction function Conv(?), while ?id and ?ve are
only parameters introduced to propagate the identification and verification signals during training.
In the testing stage, only ?c is used for feature extraction. The parameters are updated by stochastic
gradient descent. The identification and verification gradients are weighted by a hyperparameter ?.
Our learning algorithm is summarized in Tab. 1. The margin m in Eq. (2) is a special case, which
cannot be updated by gradient descent since this will collapse it to zero. Instead, m is fixed and
updated every N training pairs (N ? 200, 000 in our experiments) such that it is the threshold of
3
Table 1: The DeepID2 feature learning algorithm.
input: training set ? = {(xi , li )}, initialized parameters ?c , ?id , and ?ve , hyperparameter ?, learning rate ?(t), t ? 0
while not converge do
t ? t + 1 sample two training samples (xi , li ) and (xj , lj ) from ?
fi = Conv(xi , ?c ) and fj = Conv(xj , ?c )
? Ident(fj ,lj ,?id )
(fi ,li ,?id )
??id = ? Ident??
+
??id
id
? Verif(fi ,fj ,yij ,?ve )
??ve = ? ?
,
where
yij = 1 if li = lj , and yij = ?1 otherwise.
??ve
? Verif(fi ,fj ,yij ,?ve )
? Ident(fi ,li ,?id )
?fi =
+??
?fi
?fi
? Verif(fi ,fj ,yij ,?ve )
? Ident(fj ,lj ,?id )
+
?
?
?fj =
?fj
?fj
? Conv(xj ,?c )
? Conv(xi ,?c )
+ ?fj ?
??c = ?fi ?
??c
??c
update ?id = ?id ? ?(t) ? ??id , ?ve = ?ve ? ?(t) ? ??ve , and ?c = ?c ? ?(t) ? ??c .
end while
output ?c
Figure 2: Patches selected for feature extraction. The Joint Bayesian [3] face verification accuracy
(%) using features extracted from each individual patch is shown below.
the feature distances kfi ? fj k to minimize the verification error of the previous N training pairs.
Updating m is not included in Tab. 1 for simplicity.
3
Face Verification
To evaluate the feature learning algorithm described in Sec. 2, DeepID2 features are embedded into
the conventional face verification pipeline of face alignment, feature extraction, and face verification.
We first use the recently proposed SDM algorithm [24] to detect 21 facial landmarks. Then the face
images are globally aligned by similarity transformation according to the detected landmarks. We
cropped 400 face patches, which vary in positions, scales, color channels, and horizontal flipping,
according to the globally aligned faces and the position of the facial landmarks. Accordingly,
400 DeepID2 feature vectors are extracted by a total of 200 deep ConvNets, each of which is
trained to extract two 160-dimensional DeepID2 feature vectors on one particular face patch and
its horizontally flipped counterpart, respectively, of each face.
To reduce the redundancy among the large number of DeepID2 features and make our system
practical, we use the forward-backward greedy algorithm [25] to select a small number of effective
and complementary DeepID2 feature vectors (25 in our experiment), which saves most of the feature
extraction time during test. Fig. 2 shows all the selected 25 patches, from which 25 160-dimensional
DeepID2 feature vectors are extracted and are concatenated to a 4000-dimensional DeepID2 feature
vector. The 4000-dimensional vector is further compressed to 180 dimensions by PCA for face
verification. We learned the Joint Bayesian model [3] for face verification based on the extracted
DeepID2 features. Joint Bayesian has been successfully used to model the joint probability of two
faces being the same or different persons [3, 4].
4
4
Experiments
We report face verification results on the LFW dataset [11], which is the de facto standard test set
for face verification in unconstrained conditions. It contains 13, 233 face images of 5749 identities
collected from the Internet. For comparison purposes, algorithms typically report the mean face
verification accuracy and the ROC curve on 6000 given face pairs in LFW. Though being sound
as a test set, it is inadequate for training, since the majority of identities in LFW have only one
face image. Therefore, we rely on a larger outside dataset for training, as did by all recent highperformance face verification algorithms [4, 2, 21, 20, 13]. In particular, we use the CelebFaces+
dataset [20] for training, which contains 202, 599 face images of 10, 177 identities (celebrities)
collected from the Internet. People in CelebFaces+ and LFW are mutually exclusive. DeepID2
features are learned from the face images of 8192 identities randomly sampled from CelebFaces+
(referred to as CelebFaces+A), while the remaining face images of 1985 identities (referred to as
CelebFaces+B) are used for the following feature selection and learning the face verification models
(Joint Bayesian). When learning DeepID2 features on CelebFaces+A, CelebFaces+B is used as
a validation set to decide the learning rate, training epochs, and hyperparameter ?. After that,
CelebFaces+B is separated into a training set of 1485 identities and a validation set of 500 identities
for feature selection. Finally, we train the Joint Bayesian model on the entire CelebFaces+B data
and test on LFW using the selected DeepID2 features. We first evaluate various aspect of feature
learning from Sec. 4.1 to Sec. 4.3 by using a single deep ConvNet to extract DeepID2 features
from the entire face region. Then the final system is constructed and compared with existing best
performing methods in Sec. 4.4.
4.1
Balancing the identification and verification signals
We investigates the interactions of identification and verification signals on feature learning, by
varying ? from 0 to +?. At ? = 0, the verification signal vanishes and only the identification signal
takes effect. When ? increases, the verification signal gradually dominates the training process. At
the other extreme of ? ? +?, only the verification signal remains. The L2 norm verification loss
in Eq. (2) is used for training. Figure 3 shows the face verification accuracy on the test set by
comparing the learned DeepID2 features with L2 norm and the Joint Bayesian model, respectively.
It clearly shows that neither the identification nor the verification signal is the optimal one to learn
features. Instead, effective features come from the appropriate combination of the two.
This phenomenon can be explained from the view of inter- and intra-personal variations, which
could be approximated by LDA. According to LDA, the inter-personal scatter matrix is Sinter =
Pc
>
xi ? x
?) (?
xi ? x
?) , where x
?i is the mean feature of the i-th identity, x
? is the mean of the
i=1 ni ? (?
entire dataset, and ni is the number of face images of the i-th identity. The intra-personal scatter
Pc P
>
matrix is Sintra =
?i ) (x ? x
?i ) , where Di is the set of features of the i-th
i=1
x?Di (x ? x
identity, x
?i is the corresponding mean, and c is the number of different identities. The inter- and
intra-personal variances are the eigenvalues of the corresponding scatter matrices, and are shown in
Fig. 5. The corresponding eigenvectors represent different variation patterns. Both the magnitude
and diversity of feature variances matter in recognition. If all the feature variances concentrate on a
small number of eigenvectors, it indicates the diversity of intra- or inter-personal variations is low.
The features are learned with ? = 0, 0.05, and +?, respectively. The feature variances of each
given ? are normalized by the corresponding mean feature variance.
When only the identification signal is used (? = 0), the learned features contain both diverse
inter- and intra-personal variations, as shown by the long tails of the red curves in both figures.
While diverse inter-personal variations help to distinguish different identities, large and diverse
intra-personal variations are disturbing factors and make face verification difficult. When both the
identification and verification signals are used with appropriate weighting (? = 0.05), the diversity
of the inter-personal variations keeps unchanged while the variations in a few main directions
become even larger, as shown by the green curve in the left compared to the red one. At the
same time, the intra-personal variations decrease in both the diversity and magnitude, as shown
by the green curve in the right. Therefore, both the inter- and intra-personal variations changes in
a direction that makes face verification easier. When ? further increases towards infinity, both the
inter- and intra-personal variations collapse to the variations in only a few main directions, since
without the identification signal, diverse features cannot be formed. With low diversity on inter5
Figure 3: Face verification accuracy by varying Figure 4: Face verification accuracy of DeepID2
the weighting parameter ?. ? is plotted in log features learned by both the the face identification
scale.
and verification signals, where the number of
training identities (shown in log scale) used for
face identification varies. The result may be
further improved with more than 8192 identities.
Figure 5: Spectrum of eigenvalues of the inter- and intra-personal scatter matrices. Best viewed in
color.
personal variations, distinguishing different identities becomes difficult. Therefore the performance
degrades significantly.
Figure 6 shows the first two PCA dimensions of features learned with ? = 0, 0.05, and +?,
respectively. These features come from the six identities with the largest numbers of face images in
LFW, and are marked by different colors. The figure further verifies our observations. When ? = 0
(left), different clusters are mixed together due to the large intra-personal variations, although the
cluster centers are actually different. When ? increases to 0.05 (middle), intra-personal variations
are significantly reduced and the clusters become distinguishable. When ? further increases towards
infinity (right), although the intra-personal variations further decrease, the cluster centers also begin
to collapse and some clusters become significantly overlapped (as the red, blue, and cyan clusters in
Fig. 6 right), making it hard to distinguish again.
4.2
Rich identity information improves feature learning
We investigate how would the identity information contained in the identification supervisory signal
influence the learned features. In particular, we experiment with an exponentially increasing number
of identities used for identification during training from 32 to 8192, while the verification signal is
generated from all the 8192 training identities all the time. Fig. 4 shows how the verification
accuracies of the learned DeepID2 features (derived from the L2 norm and Joint Bayesian) vary
on the test set with the number of identities used in the identification signal. It shows that
6
Figure 6: The first two PCA dimensions of DeepID2 features extracted from six identities in LFW.
Table 2: Comparison of different verification signals.
verification signal
L2
L2+
L2-
L1
cosine none
L2 norm (%)
Joint Bayesian (%)
94.95
95.12
94.43
94.87
86.23
92.98
92.92
94.13
87.07
93.38
86.43
92.73
identifying a large number (e.g., 8192) of identities is key to learning effective DeepID2 feature
representation. This observation is consistent with those in Sec. 4.1. The increasing number of
identities provides richer identity information and helps to form DeepID2 features with diverse interpersonal variations, making the class centers of different identities more distinguishable.
4.3
Investigating the verification signals
As shown in Sec. 4.1, the verification signal with moderate intensity mainly takes the effect of
reducing the intra-personal variations. To further verify this, we compare our L2 norm verification
signal on all the sample pairs with those only constrain either the positive or negative sample pairs,
denoted as L2+ and L2-, respectively. That is, the L2+ only decreases the distances between
DeepID2 features of the same identity, while L2- only increases the distances between DeepID2
features of different identities if they are smaller than the margin. The face verification accuracies
of the learned DeepID2 features on the test set, measured by the L2 norm and Joint Bayesian
respectively, are shown in Table 2. It also compares with the L1 norm and cosine verification signals,
as well as no verification signal (none). The identification signal is the same (classifying the 8192
identities) for all the comparisons.
DeepID2 features learned with the L2+ verification signal are only slightly worse than those learned
with L2. In contrast, the L2- verification signal helps little in feature learning and gives almost
the same result as no verification signal is used. This is a strong evidence that the effect of the
verification signal is mainly reducing the intra-personal variations. Another observation is that the
face verification accuracy improves in general whenever the verification signal is added in addition
to the identification signal. However, the L2 norm is better than the other compared verification
metrics. This may be due to that all the other constraints are weaker than L2 and less effective in
reducing the intra-personal variations. For example, the cosine similarity only constrains the angle,
but not the magnitude.
4.4
Final system and comparison with other methods
Before learning Joint Bayesian, DeepID2 features are first projected to 180 dimensions by PCA.
After PCA, the Joint Bayesian model is trained on the entire CelebFaces+B data and tested on the
6000 given face pairs in LFW, where the log-likelihood ratio given by Joint Bayesian is compared
to a threshold optimized on the training data for face verification. Tab. 3 shows the face verification
accuracy with an increasing number of face patches to extract DeepID2 features, as well as the time
used to extract those DeepID2 features from each face with a single Titan GPU. We achieve 98.97%
accuracy with all the 25 selected face patches. The feature extraction process is also efficient and
takes only 35 ms for each face image. The face verification accuracy of each individual face patch
is provided in Fig. 2. The short DeepID2 signature is extremely efficient for face identification and
face image search when matching a query image with a large number of candidates.
7
Table 3: Face verification accuracy with DeepID2 features extracted from an increasing number of
face patches.
# patches
1
2
4
8
16
25
accuracy (%)
time (ms)
95.43
1.7
97.28
3.4
97.75
6.1
98.55
11
98.93
23
98.97
35
Table 4: Accuracy comparison with the previous best results on LFW.
method
accuracy (%)
High-dim LBP [4]
TL Joint Bayesian [2]
DeepFace [21]
DeepID [20]
GaussianFace [13]
DeepID2
95.17 ? 1.13
96.33 ? 1.08
97.35 ? 0.25
97.45 ? 0.26
98.52 ? 0.66
99.15 ? 0.13
Figure 7: ROC comparison with the previous best results on LFW. Best viewed in color.
To further exploit the rich pool of DeepID2 features extracted from the large number of patches, we
repeat the feature selection algorithm for another six times, each time choosing DeepID2 features
from the patches that have not been selected by previous feature selection steps. Then we learn
the Joint Bayesian model on each of the seven groups of selected features, respectively. We fuse the
seven Joint Bayesian scores on each pair of compared faces by further learning an SVM. In this way,
we achieve an even higher 99.15% face verification accuracy. The accuracy and ROC comparison
with previous state-of-the-art methods on LFW are shown in Tab. 4 and Fig. 7, respectively. We
achieve the best results and improve previous results with a large margin.
5
Conclusion
This paper have shown that the effect of the face identification and verification supervisory signals
on deep feature representation coincide with the two aspects of constructing ideal features for face
recognition, i.e., increasing inter-personal variations and reducing intra-personal variations, and the
combination of the two supervisory signals lead to significantly better features than either one of
them. When embedding the learned features to the traditional face verification pipeline, we achieved
an extremely effective system with 99.15% face verification accuracy on LFW. The arXiv report of
this paper was published in June 2014 [19].
8
References
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9
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4,879 | 5,417 | Fast Training of Pose Detectors in the Fourier Domain
Jo?ao F. Henriques
Pedro Martins
Rui Caseiro
Jorge Batista
Institute of Systems and Robotics
University of Coimbra
{henriques,pedromartins,ruicaseiro,batista}@isr.uc.pt
Abstract
In many datasets, the samples are related by a known image transformation, such
as rotation, or a repeatable non-rigid deformation. This applies to both datasets
with the same objects under different viewpoints, and datasets augmented with
virtual samples. Such datasets possess a high degree of redundancy, because
geometrically-induced transformations should preserve intrinsic properties of the
objects. Likewise, ensembles of classifiers used for pose estimation should also
share many characteristics, since they are related by a geometric transformation.
By assuming that this transformation is norm-preserving and cyclic, we propose a
closed-form solution in the Fourier domain that can eliminate most redundancies.
It can leverage off-the-shelf solvers with no modification (e.g. libsvm), and train
several pose classifiers simultaneously at no extra cost. Our experiments show that
training a sliding-window object detector and pose estimator can be sped up by orders of magnitude, for transformations as diverse as planar rotation, the walking
motion of pedestrians, and out-of-plane rotations of cars.
1
Introduction
To cope with the rich variety of transformations in natural images, recognition systems require a
representative sample of possible variations. Some of those variations must be learned from data
(e.g. non-rigid deformations), while others can be virtually generated (e.g. translation or rotation).
Recently, there has been a renewed interest in augmenting datasets with virtual samples, both in the
context of supervised [23, 17] and unsupervised learning [6]. This augmentation has the benefits of
regularizing high-capacity classifiers [6], while learning the natural invariances of the visual world.
Some kinds of virtual samples can actually make learning easier ? for example, with horizontallyflipped virtual samples [7, 4, 17], half of the weights of the template in the Dalal-Triggs detector [4]
become redundant by horizontal symmetry. A number of very recent works [14, 13, 8, 1] have shown
that cyclically translated virtual samples also constrain learning problems, which allows impressive
gains in computational efficiency. The core of this technique relies on approximately diagonalizing
the data matrix with the Discrete Fourier Transform (DFT).
In this work, we show that the ?Fourier trick? is not unique to cyclic translation, but can be generalized to other cyclic transformations. Our model captures a wide range of useful image transformations, yet retains the ability to accelerate training with the DFT. As it is only implicit, we can
accelerate training in both datasets of virtual samples and natural datasets with pose annotations.
Also due to the geometrically-induced structure of the training data, our algorithm can obtain several
transformed pose classifiers simultaneously. Some of the best object detection and pose estimation
systems currently learn classifiers for different poses independently [10, 7, 19], and we show how
joint learning of these classifiers can dramatically reduce training times.
1
(a)
(b)
(c)
Figure 1: (a) The horizontal translation of a 6 ? 6 image, by 1 pixel, can be achieved by a 36 ? 36 permutation
matrix P that reorders elements appropriately (depicted is the reordering of 2 pixels). (b) Rotation by a fixed
angle, with linearly-interpolated pixels, requires a more general matrix Q. By studying its influence on a dataset
of rotated samples, we show how to accelerate learning in the Fourier domain. Our model can also deal with
other transformations, including non-rigid. (c) Example HOG template (a car from the Google Earth dataset)
at 4 rotations learned by our model. Positive weights are on the first and third column, others are negative.
1.1
Contributions
Our contributions are as follows: 1) We generalize a previous successful model for translation
[14, 13] to other transformations, and analyze the properties of datasets with many transformed
images; 2) We present closed-form solutions that fully exploit the known structure of these datasets,
for Ridge Regression and Support Vector Regression, based on the DFT and off-the-shelf solvers;
3) With the same computational cost, we show how to train multiple classifiers for different poses
simultaneously; 4) Since our formulas do not require explicitly estimating or knowing the transformation, we demonstrate applicability to both datasets of virtual samples and structured datasets with
pose annotations. We achieve performance comparable to naive algorithms on 3 widely different
tasks, while being several orders of magnitude faster.
1.2
Related work
There is a vast body of works on image transformations and invariances, of which we can only mention a few. Much of the earlier computer vision literature focused on finding viewpoint-invariant
patterns [22]. They were based on image or scene-space coordinates, on which geometric transformations can be applied directly, however they do not apply to modern appearance-based representations. To relate complex transformations with appearance descriptors, a classic approach is to use
tangent vectors [3, 26, 16], which represent a first-order approximation. However, the desire for
more expressiveness has motivated the search for more general models.
Recent works have begun to approximate transformations as matrix-vector products, and try to estimate the transformation matrix explicitly. Tamaki et al. [27] do so for blur and affine transformations
in the context of LDA, while Miao et al. [21] approximate affine transformations with an E-M algorithm, based on a Lie group formulation. They estimate a basis for the transformation operator or the
transformed images, which is a hard analytical/inference problem in itself. The involved matrices
are extremely large for moderately-sized images, necessitating dimensionality reduction techniques
such as PCA, which may be suboptimal.
Several works focus on rotation alone [25, 18, 28, 2], most of them speeding up computations using
Fourier analysis, but they all explicitly estimate a reduced basis on which to project the data. Another
approach is to learn a transformation from data, using more parsimonious factored or deep models
[20]. In contrast, our method generalizes to other transformations and avoids a potentially costly
transformation model or basis estimation.
2
The cyclic orthogonal model for image transformations
Consider the m ? 1 vector x, obtained by vectorizing an image, i.e. stacking its elements into a
vector. The particular order does not matter, as long as it is consistent. The image may be a 32
dimensional array that contains multiple channels, such as RGB, or the values of a densely-sampled
image descriptor.
We wish to quickly train a classifier or regressor with transformed versions of sample images, to
make it robust to those transformations. The model we will use is an m ? m orthogonal matrix Q,
which will represent an incremental transformation of an image as Qx (for example, a small translation or rotation, see Fig. 1-a and 1-b). We can traverse different poses w.r.t. that transformation,
p ? Z, by repeated application of Q with a matrix power, Qp x.
In order for the number of poses to be finite, we must require the transformation to be cyclic, Qs =
Q0 = I, with some period s. This allows us to store all versions of x transformed to different poses
as the rows of an s ? m matrix,
?
T ?
Q0 x
T ?
?
? Q1 x
?
?
(1)
CQ (x) = ?
..
?
?
?
?
.
T
Qs?1 x
Due to Q being cyclic, any pose p ? Z can be found in the row (p mod s) + 1. Note that the first
row of CQ (x) contains the untransformed image x, since Q0 is the identity I. For the purposes of
training a classifier, CQ (x) can be seen as a data matrix, with one sample per row.
Although conceptually simple, we will show through experiments that this model can accurately
capture a variety of natural transformations (Section 5.2). More importantly, we will show that Q
never has to be created explicitly. The algorithms we develop will be entirely data-driven, using
an implicit description of Q from a structured dataset, either composed of virtual samples (e.g., by
image rotation), or natural samples (e.g. using pose annotations).
2.1
Image translation as a special case
A particular case of Q, and indeed what inspired the generalization that we propose, is the s ? s
cyclic shift matrix
T
0s?1
1
P =
,
(2)
Is?1 0s?1
where 0s?1 is an (s?1)?1 vector of zeros. This matrix cyclically permutes the elements of a vector
x as (x1 , x2 , x3 , . . . , xs ) ? (xs , x1 , x2 , . . . , xs?1 ). If x is a one-dimensional horizontal image, with
a single channel, then it is translated to the right by one pixel. An illustration is shown in Fig. 1-a.
By exploiting its relationship with the Discrete Fourier Transform (DFT), the cyclic shift model has
been used to accelerate a variety of learning algorithms in computer vision [14, 13, 15, 8, 1], with
suitable extensions to 2D and multiple channels.
2.2
Circulant matrices and the Discrete Fourier Transform
The basis for this optimization is the fact that the data matrix CP (x), or C(x) for short, formed by
all cyclic shifts of a sample image x, is circulant [5]. All circulant matrices are diagonalized by the
DFT, which can be expressed as the eigendecomposition
C(x) = U diag (F(x)) U H ,
(3)
where .H is the Hermitian transpose (i.e., transposition and complex-conjugation), F(x) denotes the
DFT of a vector x, and U is the unitary DFT basis. The constant matrix U can be used to compute
the DFT of any vector, since it satisfies U x = ?1s F(x). This is possible due to the linearity of the
DFT, though in practice the Fast Fourier Transform (FFT) algorithm is used instead. Note that U
is symmetric, U T = U , and unitary, U H = U ?1 . When working in Fourier-space, Eq. 3 shows
that circulant matrices in a learning problem become diagonal, which drastically reduces the needed
computations. For multiple channels or more images, they may become block-diagonal, but the
principles remain the same [13].
3
An important open question was whether the same diagonalization trick can be applied to image
transformations other than translation. We will show that this is true, using the model from Eq. 1.
3
Fast training with transformations of a single image
We will now focus on the main derivations of our paper, which allow us to quickly train a classifier
with virtual samples generated from an image x by repeated application of the transformation Q.
This section assumes only a single image x is given for training, which makes the presentation
simpler and we hope will give valuable insight into the core of the technique. Section 4 will expand
it to full generality, with training sets of an arbitrary number of images, all transformed by Q.
The first step is to show that some aspect of the data is diagonalizable by the DFT, which we do in
the following theorem.
Theorem 1. Given an orthogonal cyclic matrix Q, i.e. satisfying QT = Q?1 and Qs = Q0 , then
the s ? m matrix X = CQ (x) (from Eq. 1) verifies the following:
? The data matrix X and the uncentered covariance matrix X H X are not circulant in general,
unless Q = P (from Eq. 2).
? The Gram matrix G = XX H is always circulant.
Proof. See Appendix A.1.
Theorem 1 implies that the learning problem in its original form is not diagonalizable by the DFT
basis. However, the same diagonalization is possible for the dual problem, defined by the Gram
matrix G.
Because G is circulant, it has only s degrees of freedom and is fully specified by its first row g [11],
G = C(g). By direct computation from Eq. 1, we can verify that the elements of the first row g
are given by gp = xT Qp?1 x. One interpretation is that g contains the auto-correlation of x through
pose-space, i.e., the inner-product of x with itself as the transformation Q is applied repeatedly.
3.1
Dual Ridge Regression
For now we will restrict our attention to Ridge Regression (RR), since it has the appealing property
of having a solution in closed form, which we can easily manipulate. Section 4.1 will show how
to extend these results to Support Vector Regression. The goal of RR is to find the linear function
P
2
2
f (x) = wT x that minimizes a regularized squared error: i (f (xi ) ? yi ) + ? kwk .
Since we have s samples in the data matrix under consideration (Eq. 1), there are s dual variables,
?1
stored in a vector ?. The RR solution is given by ? = (G + ?I) y [24], where G = XX H is the
s ? s Gram matrix, y is the vector of s labels (one per pose), and ? is the regularization parameter.
The dual form of RR is usually associated with non-linear kernels [24], but since this is not our case
we can compute the explicit primal solution with w = X T ?, yielding
?1
w = X T (G + ?I)
y.
(4)
Applying the circulant eigendecomposition (Eq. 3) to G, and substituting it in Eq. 4,
?1
?1
w = X T U diag (?
g) U H + ?U U H
y = X T U (diag (?
g + ?)) U H y,
(5)
? = F (g), and similarly y
? = F (y). Since inversion of a
where we introduce the shorthand g
diagonal matrix can be done element-wise, and its multiplication by the vector U H y amounts to an
element-wise product, we obtain
?
y
,
(6)
w = X T F ?1
?+?
g
where F ?1 denotes the inverse DFT, and the division is taken element-wise. This formula allows us
to replace a costly matrix inversion with fast DFT and element-wise operations. We also do not need
to compute and store the full G, as the auto-correlation vector g suffices. As we will see in the next
section, there is a simple modification to Eq. 6 that turns out to be very useful for pose estimation.
4
3.2
Training several components simultaneously
A relatively straightforward way to estimate the object pose in an input image x is to train a classifier
for each pose (which we call components), evaluate all of them and take the maximum, i.e.
fpose (x) = arg max wpT x.
(7)
p
This can also be used as the basis for a pose-invariant classifier, by replacing argmax with max
[10]. Of course, training one component per pose can quickly become expensive. However, we can
exploit the fact that these training problems become tightly related when the training set contains
transformed images.
Recall that y specifies the labels for a training set of s transformed images, one label per pose.
Without any loss of generality, suppose that the label is 1 for a given pose t and 0 for all others, i.e.
y contains a single peak at element t. Then by shifting the peak with P p y, we will train a classifier
for pose t + p. In this manner we can train classifiers for all poses simply by varying the labels P p y,
with p = 0, . . . , s ? 1.
Based on Eq. 6, we can concatenate the solutions for all s components into a single m ? s matrix,
W =
w0
???
ws?1
?1
= X T (G + ?I)
?1
T
= X (G + ?I)
P 0y
???
P s?1 y
T
C (y) .
(8)
(9)
Diagonalization yields
??
y
F (X) ,
(10)
W =F
diag
?+?
g
where .? denotes complex-conjugation. Since their arguments are matrices, the DFT/IDFT operations here work along each column. The product of F (X) by the diagonal matrix simply amounts
to multiplying each of its rows by a scalar factor, which is inexpensive. Eq. 10 has nearly the same
computational cost as Eq. 6, which trains a single classifier.
T
4
?1
Transformation of multiple images
The training method described in the previous section would find little applicability for modern
recognition tasks if it remained limited to transformations of a single image. Naturally, we would
like to use
n images xi . We now
have a dataset of ns samples, which can be divided into n sample
groups Qp?1 xi |p = 1, . . . , s , each containing the transformed versions of one image.
This case becomes somewhat complicated by the fact that the data matrix X now has three dimensions ? the m features, the n sample groups, and the s poses of each sample group. In this m ? n ? s
array, each column vector (along the first dimension) is defined as
X?ip = Qp?1 xi ,
i = 1, . . . , n; p = 1, . . . , s,
(11)
where we have used ? to denote a one-dimensional slice of the three-dimensional array X.1 A twodimensional slice will be denoted by X??p , which yields a m ? n matrix, one for each p = 1, . . . , s.
Through a series of block-diagonalizations and reorderings, we can show (Appendix A.2-A.5) that
the solution W , of size m ? s, describing all s components (similarly to Eq. 10), is obtained with
?1 ?
? ?p = X
? ??p (?
W
g??p + ?I) Y??p
,
p = 1, . . . , s,
(12)
? is the DFT
where a hat ? over an array denotes the DFT along the dimension that has size s (e.g. X
of X along the third dimension), Yip specifies the label of the sample with pose p in group i, and g
is the n ? n ? s array with elements
1
For reference, our slice notation ? works the same way as the slice notation : in Matlab or NumPy.
5
T
gijp = xTi Qp?1 xj = X?i1
X?jp ,
i, j = 1, . . . , n; p = 1, . . . , s.
(13)
It may come as a surprise that, after all these changes, Eq. 12 still essentially looks like a dual Ridge
Regression (RR) problem (compare it to Eq. 4). Eq. 12 can be interpreted as splitting the original
problem into s smaller problems, one for each Fourier frequency, which are independent and can be
solved in parallel. A Matlab implementation is given in Appendix B.2
4.1
Support Vector Regression
Given that we can decompose such a large RR problem into s smaller RR problems, by applying
the DFT and slicing operators (Eq. 12), it is natural to ask whether the same can be done with
other algorithms. Leveraging a recent result [13], where this was done for image translation, the
same steps can be repeated for the dual formulation of other algorithms, such as Support Vector
Regression (SVR). Although RR can deal with complex data, SVR requires an extension to the
complex domain, which we show in Appendix A.6. We give a Matlab implementation in Appendix
B, which can use any off-the-shelf SVR solver without modification.
4.2
Efficiency
Naively training one detector per pose would require solving s large ns?ns systems (either with RR
or SVR). In contrast, our method learns jointly all detectors using s much smaller n?n subproblems.
The computational savings can be several orders of magnitude for large s. Our experiments seem to
validate this conclusion, even in relatively large recognition tasks (Section 6).
5
Orthogonal transformations in practice
Until now, we avoided the question of how to compute a transformation model Q. This may seem
like a computational burden, not to mention a hard estimation problem ? for example, what is the
cyclic orthogonal matrix Q that models planar rotations with period s? Inspecting Eq. 12-13,
however, reveals that we do not need to form Q explicitly, but can work with just a data matrix X
of transformed images. From there on, we exploit the knowledge that this data was obtained from
some matrix Q, and that is enough to allow fast training in the Fourier domain. This allows a great
deal of flexibility in implementation.
5.1
Virtual transformations
One way to obtain a structured data matrix X is with virtual samples. From the original dataset of n
samples, we can generate ns virtual samples using a standard image operator (e.g. planar rotation).
However, we should keep in mind that the accuracy of the proposed method will be affected by how
much the image operator resembles a pure cyclic orthogonal transformation.
Linearity. Many common image transformations, such as rotation or scale, are implemented by
nearest-neighbor or bilinear interpolation. For a fixed amount of rotation or scale, these functions
are linear functions in the input pixels, i.e. each output pixel is a fixed linear combination of some
of the input pixels. As such, they fulfill the linearity requirement.
Orthogonality. For an operator to be orthogonal, it must preserve the L2 norm of its inputs. At the
expense of introducing some non-linearity, we simply renormalize each virtual sample to have the
same norm as the original sample, which seems to work well in practice (Section 6).
Cyclicity. We conducted some experiments with planar rotation on satellite imagery (Section 6.1)
? rotation by 360/s degrees is cyclic with period s. In the future, we plan to experiment with noncyclic operators (similar to how cyclic translation is used to approximate image translation [14]).
2
The supplemental material is available at: www.isr.uc.pt/?henriques/transformations/
6
Figure 2: Example detections and estimated poses in 3 different settings. We can accelerate training with
(a) planar rotations (Google Earth), (b) non-rigid deformations in walking pedestrians (TUD-Campus/TUDCrossing), and (c) out-of-plane rotations (KITTI). Best viewed in color.
5.2
Natural transformations
Another interesting possibility is to use pose annotations to create a structured data matrix. This
data-driven approach allows us to consider more complicated transformations than those associated
with virtual samples. Given s views of n objects under different poses, we can build the m ? n ? s
data matrix X and use the same methodology as before. In Section 6 we describe experiments with
the walk cycle of pedestrians, and out-of-plane rotations of cars in street scenes. These transformations are cyclic, though highly non-linear, and we use the same renormalization as in Section 5.1.
5.3
Negative samples
One subtle aspect is how to obtain a structured data matrix from negative samples. This is simple for
virtual transformations, but not for natural transformations. For example, with planar rotation we
can easily generate rotated negative samples with arbitrary poses. However, the same operation with
walk cycles of pedestrians is not defined. How do we advance the walk cycle of a non-pedestrian? As
a pragmatic solution, we consider that negative samples are unaffected by natural transformations,
so a negative sample is constant for all s poses. Because the DFT of a constant signal is 0, except for
the DC value (the first frequency), we can ignore untransformed negative samples in all subproblems
for p 6= 1 (Eq. 12). This simple observation can result in significant computational savings.
6
Experiments
To demonstrate the generality of the proposed model, we conducted object detection and pose estimation experiments on 3 widely different settings, which will be described shortly. We implemented
a detector based on Histogram of Oriented Gradients (HOG) templates [4] with multiple components
[7]. This framework forms the basis on which several recent advances in object detection are built
[19, 10, 7]. The baseline algorithm independently trains s classifiers (components), one per pose, enabling pose-invariant object detection and pose prediction (Eq. 7). Components are then calibrated,
as usual for detectors with multiple components [7, 19]. The proposed method does not require any
ad-hoc calibration, since the components are jointly trained and related by the orthogonal matrix Q,
which preserves their L2 norm.
For the performance evaluation, ground truth objects are assigned to hypothesis by the widely used
Pascal criterion of bounding box overlap [7]. We then measure average precision (AP) and pose error (as epose /s, where epose is the discretized pose difference, taking wrap-around into account). We
tested two variants of each method, trained with both RR and SVR. Although parallelization is trivial, we report timings for single-core implementations, which more accurately reflect the total CPU
load. As noted in previous work [13], detectors trained with SVR have very similar performance to
those trained with Support Vector Machines.
6.1
Planar rotation in satellite images (Google Earth)
Our first test will be on a car detection task on satellite imagery [12], which has been used in several
works that deal with planar rotation [25, 18]. We annotated the orientations of 697 objects over half
the 30 images of the dataset. The first 7 annotated images were used for training, and the remaining
8 for validation. We created a structured data matrix X by augmenting each sample with 30 virtual
7
Google Earth
Time (s) AP
Pose
Fourier SVR
4.5
73.0
9.4
training RR
3.7
71.4
10.0
SVR 130.7
73.2
9.8
Standard
RR
399.3
72.7
10.3
TUD Campus/Crossing
KITTI
Time (s) AP
Pose Time (s) AP
0.1
81.5
9.3
15.0
53.5
0.08
82.2
8.9
15.5
53.4
40.5
80.2
9.5
454.2
56.5
45.8
81.6
9.4
229.6
54.5
Pose
14.9
15.0
13.8
14.0
Table 1: Results for pose detectors trained with Support Vector Regression (SVR) and Ridge Regression (RR).
We report training time, Average Precision (AP) and pose error (both in percentage).
samples, using 12? rotations. A visualization of trained weights is shown in Fig. 1-c and Appendix
B. Experimental results are presented in Table 1. Recall that our primary goal is to demonstrate
faster training, not to improve detection performance, which is reflected in the results. Nevertheless,
the two proposed fast Fourier algorithms are 29 to 107? faster than the baseline algorithms.
6.2
Walk cycle of pedestrians (TUD-Campus and TUD-Crossing)
We can consider a walking pedestrian to undergo a cyclic non-rigid deformation, with each period
corresponding to one step. Because this transformation is time-dependent, we can learn it from
video data. We used TUD-Campus for training and TUD-Crossing for testing (see Fig. 2). We
annotated a key pose in all 272 frames, so that the images of a pedestrian between two key poses
represent a whole walk cycle. Sampling 10 images per walk cycle (corresponding to 10 poses), we
obtained 10 sample groups for training, for a total of 100 samples.
From Table 1, the proposed algorithms seem to slightly outperform the baseline, showing that these
non-rigid deformations can be accurately accounted for. However, they are over 2 orders of magnitude faster. In addition to the speed benefits observed in Section 6.1, another factor at play is that
for natural transformations we can ignore the negative samples in s ? 1 of the subproblems (Section
5.3), whereas the baseline algorithms must consider them when training each of the s components.
6.3
Out-of-plane rotations of cars in street scenes (KITTI)
For our final experiment, we will attempt to demonstrate that the speed advantage of our method
still holds for difficult out-of-plane rotations. We chose the very recent KITTI benchmark [9], which
includes an object detection set of 7481 images of street scenes. The facing angle of cars (along the
vertical axis) is provided, which we bin into 15 discrete poses. We performed an 80-20% train-test
split of the images, considering cars of ?moderate? difficulty [9], and obtained 73 sample groups for
training with 15 poses each (for a total of 1095 samples).
Table 1 shows that the proposed method achieves competitive performance, but with a dramatically
lower computational cost. The results agree with the intuition that out-of-plane rotations strain the
assumptions of linearity and orthogonality, since they result in large deformations of the object.
Nevertheless, the ability to learn a useful model under such adverse conditions shows great promise.
7
Conclusions and future work
In this work, we derived new closed-form formulas to quickly train several pose classifiers at once,
and take advantage of the structure in datasets with pose annotation or virtual samples. Our implicit
transformation model seems to be surprisingly expressive, and in future work we would like to
experiment with other transformations, including non-cyclic. Other interesting directions include
larger-scale variants and the composition of multiple transformations.
Acknowledgements. The authors would like to thank Jo?ao Carreira for valuable discussions. They also acknowledge support by the FCT project PTDC/EEA-CRO/122812/2010, grants
SFRH/BD75459/2010, SFRH/BD74152/2010, and SFRH/BPD/90200/2012.
8
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4,880 | 5,418 | LSDA: Large Scale Detection through Adaptation
Judy Hoffman , Sergio Guadarrama , Eric Tzeng , Ronghang Hu? , Jeff Donahue ,
EECS, UC Berkeley, ? EE, Tsinghua University
{jhoffman, sguada, tzeng, jdonahue}@eecs.berkeley.edu
[email protected]
Ross Girshick , Trevor Darrell , Kate Saenko4
EECS, UC Berkeley, 4 CS, UMass Lowell
{rbg, trevor}@eecs.berkeley.edu, [email protected]
Abstract
A major challenge in scaling object detection is the difficulty of obtaining labeled
images for large numbers of categories. Recently, deep convolutional neural networks (CNNs) have emerged as clear winners on object classification benchmarks,
in part due to training with 1.2M+ labeled classification images. Unfortunately,
only a small fraction of those labels are available for the detection task. It is much
cheaper and easier to collect large quantities of image-level labels from search engines than it is to collect detection data and label it with precise bounding boxes.
In this paper, we propose Large Scale Detection through Adaptation (LSDA), an
algorithm which learns the difference between the two tasks and transfers this
knowledge to classifiers for categories without bounding box annotated data, turning them into detectors. Our method has the potential to enable detection for the
tens of thousands of categories that lack bounding box annotations, yet have plenty
of classification data. Evaluation on the ImageNet LSVRC-2013 detection challenge demonstrates the efficacy of our approach. This algorithm enables us to
produce a >7.6K detector by using available classification data from leaf nodes in
the ImageNet tree. We additionally demonstrate how to modify our architecture
to produce a fast detector (running at 2fps for the 7.6K detector). Models and
software are available at lsda.berkeleyvision.org.
1
Introduction
Both classification and detection are key visual recognition challenges, though historically very
different architectures have been deployed for each. Recently, the R-CNN model [1] showed how
to adapt an ImageNet classifier into a detector, but required bounding box data for all categories.
We ask, is there something generic in the transformation from classification to detection that can be
learned on a subset of categories and then transferred to other classifiers?
One of the fundamental challenges in training object detection systems is the need to collect a
large of amount of images with bounding box annotations. The introduction of detection challenge
datasets, such as PASCAL VOC [2], have propelled progress by providing the research community
a dataset with enough fully annotated images to train competitive models although only for 20
classes. Even though the more recent ImageNet detection challenge dataset [3] has extended the set
of annotated images, it only contains data for 200 categories. As we look forward towards the goal
of scaling our systems to human-level category detection, it becomes impractical to collect a large
quantity of bounding box labels for tens or hundreds of thousands of categories.
?
This work was supported in part by DARPA?s MSEE and SMISC programs, by NSF awards IIS-1427425,
and IIS-1212798, IIS-1116411, and by support from Toyota.
1
W
DET
Wdog
?
DET
apple
W
W
WCLASSIFY
cat
DET
Wcat
CLASSIFY
apple
ICLASSIFY
dog
I
DET
apple
dog
apple
CLASSIFY
dog
cat
ICLASSIFY
Detectors
Classifiers
IDET
Figure 1: The core idea is that we can learn detectors (weights) from labeled classification data (left),
for a wide range of classes. For some of these classes (top) we also have detection labels (right), and
can learn detectors. But what can we do about the classes with classification data but no detection
data (bottom)? Can we learn something from the paired relationships for the classes for which we
have both classifiers and detectors, and transfer that to the classifier at the bottom to make it into a
detector?
In contrast, image-level annotation is comparatively easy to acquire. The prevalence of image tags
allows search engines to quickly produce a set of images that have some correspondence to any
particular category. ImageNet [3], for example, has made use of these search results in combination
with manual outlier detection to produce a large classification dataset comprised of over 20,000
categories. While this data can be effectively used to train object classifier models, it lacks the
supervised annotations needed to train state-of-the-art detectors.
In this work, we propose Large Scale Detection through Adaptation (LSDA), an algorithm that
learns to transform an image classifier into an object detector. To accomplish this goal, we use
supervised convolutional neural networks (CNNs), which have recently been shown to perform well
both for image classification [4] and object detection [1, 5]. We cast the task as a domain adaptation
problem, considering the data used to train classifiers (images with category labels) as our source
domain, and the data used to train detectors (images with bounding boxes and category labels) as our
target domain. We then seek to find a general transformation from the source domain to the target
domain, that can be applied to any image classifier to adapt it into a object detector (see Figure 1).
Girshick et al. (R-CNN) [1] demonstrated that adaptation, in the form of fine-tuning, is very important for transferring deep features from classification to detection and partially inspired our approach.
However, the R-CNN algorithm uses classification data only to pre-train a deep network and then
requires a large number of bounding boxes to train each detection category.
Our LSDA algorithm uses image classification data to train strong classifiers and requires detection
bounding box labeled data for only a small subset of the final detection categories and much less
time. It uses the classes labeled with both classification and detection labels to learn a transformation
of the classification network into a detection network. It then applies this transformation to adapt
classifiers for categories without any bounding box annotated data into detectors.
Our experiments on the ImageNet detection task show significant improvement (+50% relative
mAP) over a baseline of just using raw classifier weights on object proposal regions. One can
adapt any ImageNet-trained classifier into a detector using our approach, whether or not there are
corresponding detection labels for that class.
2
Related Work
Recently, Multiple Instance Learning (MIL) has been used for training detectors using weak labels,
i.e. images with category labels but not bounding box labels. The MIL paradigm estimates latent
labels of examples in positive training bags, where each positive bag is known to contain at least one
positive example. Ali et al. [6] constructs positive bags from all object proposal regions in a weakly
labeled image that is known to contain the object, and uses a version of MIL to learn an object
detector. A similar method [7] learns detectors from PASCAL VOC images without bounding box
2
"
fcA"
det"
fc6"
det"
fc7"
?B"
det"
layers"175"
Input"image"
Region"
Proposals"
Warped""
region"
cat:"0.90"
cat?"yes"
dog:"0.45"
dog?"no"
adapt"
fcB"
LSDA"Net"
background"
background:"0.25"
Produce""
Predic=ons"
Figure 2: Detection with the LSDA network. Given an image, extract region proposals, reshape the
regions to fit into the network size and finally produce detection scores per category for the region.
Layers with red dots/fill indicate they have been modified/learned during fine-tuning with available
bounding box annotated data.
labels. MIL-based methods are a promising approach that is complimentary to ours. They have not
yet been evaluated on the large-scale ImageNet detection challenge to allow for direct comparison.
Deep convolutional neural networks (CNNs) have emerged as state of the art on popular object
classification benchmarks (ILSVRC, MNIST) [4]. In fact, ?deep features? extracted from CNNs
trained on the object classification task are also state of the art on other tasks, e.g., subcategory
classification, scene classification, domain adaptation [8] and even image matching [9]. Unlike the
previously dominant features (SIFT [10], HOG [11]), deep CNN features can be learned for each
specific task, but only if sufficient labeled training data are available. R-CNN [1] showed that finetuning deep features on a large amount of bounding box labeled data significantly improves detection
performance.
Domain adaptation methods aim to reduce dataset bias caused by a difference in the statistical distributions between training and test domains. In this paper, we treat the transformation of classifiers
into detectors as a domain adaptation task. Many approaches have been proposed for classifier
adaptation; e.g., feature space transformations [12], model adaptation approaches [13, 14] and joint
feature and model adaptation [15, 16]. However, even the joint learning models are not able to modify the feature extraction process and so are limited to shallow adaptation techniques. Additionally,
these methods only adapt between visual domains, keeping the task fixed, while we adapt both from
a large visual domain to a smaller visual domain and from a classification task to a detection task.
Several supervised domain adaptation models have been proposed for object detection. Given a
detector trained on a source domain, they adjust its parameters on labeled target domain data. These
include variants for linear support vector machines [17, 18, 19], as well as adaptive latent SVMs [20]
and adaptive exemplar SVM [21]. A related recent method [22] proposes a fast adaptation technique
based on Linear Discriminant Analysis. These methods require labeled detection data for all object
categories, both in the source and target domains, which is absent in our scenario. To our knowledge,
ours is the first method to adapt to held-out categories that have no detection data.
3
Large Scale Detection through Adaptation (LSDA)
We propose Large Scale Detection through Adaptation (LSDA), an algorithm for adapting classifiers
to detectors. With our algorithm, we are able to produce a detection network for all categories of
interest, whether or not bounding boxes are available at training time (see Figure 2).
Suppose we have K categories we want to detect, but we only have bounding box annotations for m
categories. We will refer to the set of categories with bounding box annotations as B = {1, ...m},
and the set of categories without bounding box annotations as set A = {m, ..., K}. In practice,
we will likely have m K, as is the case in the ImageNet dataset. We assume availability of
classification data (image-level labels) for all K categories and will use that data to initialize our
network.
3
LSDA transforms image classifiers into object detectors using three key insights:
1. Recognizing background is an important step in adapting a classifier into a detector
2. Category invariant information can be transferred between the classifier and detector feature representations
3. There may be category specific differences between a classifier and a detector
We will next demonstrate how our method accomplishes each of these insights as we describe the
training of LSDA.
3.1
Training LSDA: Category Invariant Adaptation
For our convolutional neural network, we adopt the architecture of Krizhevsky et al. [4], which
achieved state-of-the-art performance on the ImageNet ILSVRC2012 classification challenge. Since
this network requires a large amount of data and time to train its approximately 60 million parameters, we start by pre-training the CNN trained on the ILSVRC2012 classification dataset, which
contains 1.2 million classification-labeled images of 1000 categories. Pre-training on this dataset
has been shown to be a very effective technique [8, 5, 1], both in terms of performance and in terms
of limiting the amount of in-domain labeled data needed to successfully tune the network. Next, we
replace the last weight layer (1000 linear classifiers) with K linear classifiers, one for each category
in our task. This weight layer is randomly initialized and then we fine-tune the whole network on
our classification data. At this point, we have a network that can take an image or a region proposal
as input, and produce a set of scores for each of the K categories. We find that even using the net
trained on classification data in this way produces a strong baseline (see Section 4).
We next transform our classification network into a detection network. We do this by fine-tuning
layers 1-7 using the available labeled detection data for categories in set B. Following the Regionsbased CNN (R-CNN) [1] algorithm, we collect positive bounding boxes for each category in set B
as well as a set of background boxes using a region proposal algorithm, such as selective search [23].
We use each labeled region as a fine-tuning input to the CNN after padding and warping it to the
CNN?s input size. Note that the R-CNN fine-tuning algorithm requires bounding box annotated data
for all categories and so can not directly be applied to train all K detectors. Fine-tuning transforms
all network weights (except for the linear classifiers for set A) and produces a softmax detector for
categories in set B, which includes a weight vector for the new background class.
Layers 1-7 are shared between all categories in set B and we find empirically that fine-tuning induces
a generic, category invariant transformation of the classification network into a detection network.
That is, even though fine-tuning sees no detection data for categories in set A, the network transforms in a way that automatically makes the original set A image classifiers much more effective
at detection (see Figure 3). Fine-tuning for detection also learns a background weight vector that
encodes a generic ?background? category. This background model is important for modeling the
task shift from image classification, which does not include background distractors, to detection,
which is dominated by background patches.
3.2
Training LSDA: Category Specific Adaptation
Finally, we learn a category specific transformation that will change the classifier model parameters
into the detector model parameters that operate on the detection feature representation. The category
specific output layer (f c8) is comprised of f cA, f cB, ?B, and f c ? BG. For categories in set B,
this transformation can be learned through directly fine-tuning the category specific parameters f cB
(Figure 2). This is equivalent to fixing f cB and learning a new layer, zero initialized, ?B, with
equivalent loss to f cB , and adding together the outputs of ?B and f cB .
Let us define the weights of the output layer of the original classification network as W c , and the
weights of the output layer of the adapted detection network as W d . We know that for a category
i ? B, the final detection weights should be computed as Wid = Wic + ?Bi . However, since
there is no detection data for categories in A, we can not directly learn a corresponding ?A layer
during fine-tuning. Instead, we can approximate the fine-tuning that would have occurred to f cA
had detection data been available. We do this by finding the nearest neighbors categories in set B
for each category in set A and applying the average change. Here we define nearest neighbors as
4
those categories with the nearest (minimal Euclidean distance) `2 -normalized f c8 parameters in the
classification network. This corresponds to the classification model being most similar and hence,
we assume, the detection model should be most similar. We denote the k th nearest neighbor in set
B of category j ? A as NB (j, k), then we compute the final output detection weights for categories
in set A as:
?j ? A : Wjd
=
Wjc +
k
1X
?BNB (j,i)
k i=1
(1)
Thus, we adapt the category specific parameters even without bounding boxes for categories in set
A. In the next section we experiment with various values of k, including taking the full average:
k = |B|.
3.3
Detection with LSDA
At test time we use our network to extract K + 1 scores per region proposal in an image (similar to
the R-CNN [1] pipeline). One for each category and an additional score for the background category.
Finally, for a given region, the score for category i is computed by combining the per category score
with the background score: scorei ? scorebackground .
In contrast to the R-CNN [1] model which trains SVMs on the extracted features from layer 7
and bounding box regression on the extracted features from layer 5, we directly use the final score
vector to produce the prediction scores without either of the retraining steps. This choice results in a
small performance loss, but offers the flexibility of being able to directly combine the classification
portion of the network that has no detection labeled data, and reduces the training time from 3 days
to roughly 5.5 hours.
4
Experiments
To demonstrate the effectiveness of our approach we present quantitative results on the ILSVRC2013
detection dataset. The dataset offers a 200-category detection challenge. The training set has ?400K
annotated images and on average 1.534 object classes per image. The validation set has 20K annotated images with ?50K annotated objects. We simulate having access to classification labels for
all 200 categories and having detection annotations for only the first 100 categories (alphabetically
sorted).
4.1
Experiment Setup & Implementation Details
We start by separating our data into classification and detection sets for training and a validation
set for testing. Since the ILSVRC2013 training set has on average fewer objects per image than
the validation set, we use this data as our classification data. To balance the categories we use
?1000 images per class (200,000 total images). Note: for classification data we only have access
to a single image-level annotation that gives a category label. In effect, since the training set may
contain multiple objects, this single full-image label is a weak annotation, even compared to other
classification training data sets. Next, we split the ILSVRC2013 validation set in half as [1] did,
producing two sets: val1 and val2. To construct our detection training set, we take the images
with bounding box labels from val1 for only the first 100 categories (? 5000 images). Since the
validation set is relatively small, we augment our detection set with 1000 bounding box annotated
images per category from the ILSVRC2013 training set (following the protocol of [1]). Finally we
use the second half of the ILSVRC2013 validation set (val2) for our evaluation.
We implemented our CNN architectures and execute all fine-tuning using the open source software
package Caffe [24] and have made our model definitions weights publicly available.
4.2
Quantitative Analysis on Held-out Categories
We evaluate the importance of each component of our algorithm through an ablation study. As
a baseline we consider training the network with only the classification data (no adaptation) and
applying the network to the region proposals. The summary of the importance of our three adaptation
components is shown in Figure 3. Our full LSDA model achieves a 50% relative mAP boost over
5
Detection
Adaptation Layers
Output Layer
Adaptation
mAP Trained
100 Categories
mAP Held-out
100 Categories
mAP All
200 Categories
No Adapt (Classification Network)
fcbgrnd
fcbgrnd ,fc6
fcbgrnd ,fc7
fcbgrnd ,fcB
fcbgrnd ,fc6 ,fc7
fcbgrnd ,fc6 ,fc7 ,fcB
fcbgrnd ,layers1-7,fcB
-
12.63
14.93
24.72
23.41
18.04
25.78
26.33
27.81
10.31
12.22
13.72
14.57
11.74
14.20
14.42
15.85
11.90
13.60
19.20
19.00
14.90
20.00
20.40
21.83
fcbgrnd ,layers1-7,fcB
fcbgrnd ,layers1-7,fcB
fcbgrnd ,layers1-7,fcB
28.12
27.95
27.91
15.97
16.15
15.96
22.05
22.05
21.94
29.72
26.25
28.00
Avg NN (k=5)
Avg NN (k=10)
Avg NN (k=100)
Oracle: Full Detection Network
Table 1: Ablation study for the components of LSDA. We consider removing different pieces of our
algorithm to determine which pieces are essential. We consider training with the first 100 (alphabetically) categories of the ILSVRC2013 detection validation set (on val1) and report mean average
precision (mAP) over the 100 trained on and 100 held out categories (on val2). We find the best
improvement is from fine-tuning all layers and using category specific adaptation.
the classification only network. The most important step of our algorithm proved to be adapting
the feature representation, while the least important was adapting the category specific parameter.
This fits with our intuition that the main benefit of our approach is to transfer category invariant
information from categories with known bounding box annotation to those without the bounding
box annotations.
In Table 1, we present a more detailed analysis of the
different adaptation techniques we could use to train the
network. We find that the best category invariant adaptation approach is to learn the background category layer
and adapt all convolutional and fully connected layers,
bringing mAP on the held-out categories from 10.31% up
to 15.85%. Additionally, using output layer adaptation
(k = 10) further improves performance, bringing mAP
to 16.15% on the held-out categories (statistically significant at p = 0.017 using a paired sample t-test [25]). The
last row shows the performance achievable by our detection network if it had access to detection data for all 200
categories, and serves as a performance upper bound.1
LSDA
16.15
LSDA (bg+ft)
15.85
LSDA (bg only)
12.2
Classification Net
10.31
0
5
10
15
20
Figure 3: Comparison (mAP%) of our
We find that one of the biggest reasons our algorithm im- full system (LSDA) on categories with
proves is from reducing localization error. For example, no bounding boxes at training time.
in Figure 4, we show that while the classification only
trained net tends to focus on the most discriminative part
of an object (ex: face of an animal) after our adaptation, we learn to localize the whole object (ex:
entire body of the animal).
4.3
Error Analysis on Held Out Categories
We next present an analysis of the types of errors that our system (LSDA) makes on the held out
object categories. First, in Figure 5, we consider three types of false positive errors: Loc (localization errors), BG (confusion with background), and Oth (other error types, which is essentially
1
To achieve R-CNN performance requires additionally learning SVMs on the activations of layer 7 and
bounding box regression on the activations of layer 5. Each of these steps adds between 1-2mAP at high
computation cost and using the SVMs removes the adaptation capacity of the system.
6
Figure 4: We show example detections on held out categories, for which we have no detection
training data, where our adapted network (LSDA) (shown with green box) correctly localizes and
labels the object of interest, while the classification network baseline (shown in red) incorrectly
localizes the object. This demonstrates that our algorithm learns to adapt the classifier into a detector
which is sensitive to localization and background rejection.
correctly localizing an object, but misclassifying it). After separating all false positives into one of
these three error types we visually show the percentage of errors found in each type as you look at
the top scoring 25-3200 false positives.2 We consider the baseline of starting with the classification
only network and show the false positive breakdown in Figure 5(b). Note that the majority of false
positive errors are confusion with background and localization errors. In contrast, after adapting
the network using LSDA we find that the errors found in the top false positives are far less due to
localization and background confusion (see Figure 5(c)). Arguably one of the biggest differences between classification and detection is the ability to accurately localize objects and reject background.
Therefore, we show that our method successfully adapts the classification parameters to be more
suitable for detection.
In Figure 5(a) we show examples of the top scoring Oth error types for LSDA on the held-out
categories. This means the detector localizes an incorrect object type. For example, the motorcycle
detector localized and mislabeled bicycle and the lemon detector localized and mislabeled an orange.
In general, we noticed that many of the top false positives from the Oth error type were confusion
with very similar categories.
4.4
Large Scale Detection
To showcase the capabilities of our technique we produced a 7604 category detector. The first
categories correspond to the 200 categories from the ILSVRC2013 challenge dataset which have
bounding box labeled data available. The other 7404 categories correspond to leaf nodes in the
ImageNet database and are trained using the available full image labeled classification data. We
trained a full detection network using the 200 fully annotated categories and trained the other 7404
last layer nodes using only the classification data. Since we lack bounding box annotated data for
the majority of the categories we show example top detections in Figure 6. The results are filtered
using non-max suppression across categories to only show the highest scoring categories.
The main contribution of our algorithm is the adaptation technique for modifying a convolutional
neural network for detection. However, the choice of network and how the net is used at test time
both effect the detection time computation. We have therefore also implemented and released a
version of our algorithm running with fast region proposals [27] on a spatial pyramid pooling network [28], reducing our detection time down to half a second per image (from 4s per image) with
nearly the same performance. We hope that this will allow the use of our 7.6K model on large data
sources such as videos. We have released the 7.6K model and code to run detection (both the way
presented in this paper and our faster version) at lsda.berkeleyvision.org.
2
We modified the analysis software made available by Hoeim et al. [26] to work on ILSVRC-2013 detection
7
mushroom
microphone
motorcycle
microphone (sim): ov=0.00 1?r=?3.00
lemon
laptop
nail
miniskirt
1?r=?6.00
mushroom
(sim): ov=0.00 1?r=?8.00
miniskirt (sim): ov=0.00 motorcycle
1?r=?1.00 (sim): ov=0.00
nail (sim): ov=0.00 1?r=?4.00 laptop (sim): ov=0.00 1?r=?3.00 lemon (sim): ov=0.00 1?r=?5.00
(a) Example Top Scoring False Positives: LSDA correctly localizes but incorrectly labels object
Held?out Categories
Held?out Categories
100
Loc
Oth
BG
80
percentage of each type
percentage of each type
100
60
40
20
0
25
50
60
40
20
0
25
100 200 400 800 1600 3200
total false positives
(b) Classification Network
Loc
Oth
BG
80
50
100 200 400 800 1600 3200
total false positives
(c) LSDA Network
Figure 5: We examine the top scoring false positives from LSDA. Many of our top scoring false
positives come from confusion with other categories (a). (b-c) Comparison of error type breakdown
on the categories which have no training bounding boxes available (held-out categories). After
adapting the network using our algorithm (LSDA), the percentage of false positive errors due to
localization and background confusion is reduced (c) as compared to directly using the classification
network in a detection framework (b).
whippet: 2.0
dog: 4.1
taillight: 0.9
American bison: 7.0
wheel
and axle: 1.0
car: 6.0
sofa: 8.0
Figure 6: Example top detections from our 7604 category detector. Detections from the 200 categories that have bounding box training data available are shown in blue. Detections from the
remaining 7404 categories for which only classification training data is available are shown in red.
5
Conclusion
We have presented an algorithm that is capable of transforming a classifier into a detector. We
use CNN models to train both a classification and a detection network. Our multi-stage algorithm
uses corresponding classification and detection data to learn the change from a classification CNN
network to a detection CNN network, and applies that difference to future classifiers for which there
is no available detection data.
We show quantitatively that without seeing any bounding box annotated data, we can increase performance of a classification network by 50% relative improvement using our adaptation algorithm.
Given the significant improvement on the held out categories, our algorithm has the potential to
enable detection of tens of thousands of categories. All that would be needed is to train a classification layer for the new categories and use our fine-tuned detection model along with our output layer
adaptation techniques to update the classification parameters directly.
Our approach significantly reduces the overhead of producing a high quality detector. We hope that
in doing so we will be able to minimize the gap between having strong large-scale classifiers and
strong large-scale detectors. There is still a large gap to reach oracle (known bounding box labels)
performance. For future work we would like to explore multiple instance learning techniques to
discover and mine patches for the categories that lack bounding box data.
8
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9
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4,881 | 5,419 | Local Decorrelation for Improved Pedestrian Detection
Woonhyun Nam?
StradVision, Inc.
[email protected]
Piotr Doll?ar
Microsoft Research
Joon Hee Han
POSTECH, Republic of Korea
[email protected]
[email protected]
Abstract
Even with the advent of more sophisticated, data-hungry methods, boosted decision trees remain extraordinarily successful for fast rigid object detection, achieving top accuracy on numerous datasets. While effective, most boosted detectors
use decision trees with orthogonal (single feature) splits, and the topology of the
resulting decision boundary may not be well matched to the natural topology of
the data. Given highly correlated data, decision trees with oblique (multiple feature) splits can be effective. Use of oblique splits, however, comes at considerable
computational expense. Inspired by recent work on discriminative decorrelation
of HOG features, we instead propose an efficient feature transform that removes
correlations in local neighborhoods. The result is an overcomplete but locally
decorrelated representation ideally suited for use with orthogonal decision trees.
In fact, orthogonal trees with our locally decorrelated features outperform oblique
trees trained over the original features at a fraction of the computational cost. The
overall improvement in accuracy is dramatic: on the Caltech Pedestrian Dataset,
we reduce false positives nearly tenfold over the previous state-of-the-art.
1
Introduction
In recent years object detectors have undergone an impressive transformation [11, 32, 14]. Nevertheless, boosted detectors remain extraordinarily successful for fast detection of quasi-rigid objects.
Such detectors were first proposed by Viola and Jones in their landmark work on efficient sliding
window detection that made face detection practical and commercially viable [35]. This initial architecture remains largely intact today: boosting [31, 12] is used to train and combine decision trees
and a cascade is employed to allow for fast rejection of negative samples. Details, however, have
evolved considerably; in particular, significant progress has been made on the feature representation [6, 9, 2] and cascade architecture [3, 8]. Recent boosted detectors [1, 7] achieve state-of-the-art
accuracy on modern benchmarks [10, 22] while retaining computational efficiency.
While boosted detectors have evolved considerably over the past decade, decision trees with orthogonal (single feature) splits ? also known as axis-aligned decision trees ? remain popular and predominant. A possible explanation for the persistence of orthogonal splits is their efficiency: oblique
(multiple feature) splits incur considerable computational cost during both training and detection.
Nevertheless, oblique trees can hold considerable advantages. In particular, Menze et al. [23] recently demonstrated that oblique trees used in conjunction with random forests are quite effective
given high dimensional data with heavily correlated features.
To achieve similar advantages while avoiding the computational expense of oblique trees, we instead
take inspiration from recent work by Hariharan et al. [15] and propose to decorrelate features prior to
applying orthogonal trees. To do so we introduce an efficient feature transform that removes correlations in local image neighborhoods (as opposed to decorrelating features globally as in [15]). The
result is an overcomplete but locally decorrelated representation that is ideally suited for use with
orthogonal trees. In fact, orthogonal trees with our locally decorrelated features require estimation
of fewer parameters and actually outperform oblique trees trained over the original features.
?
This research was performed while W.N. was a postdoctoral researcher at POSTECH.
1
Figure 1: A comparison of boosting of orthogonal and oblique trees on highly correlated data while
varying the number (T ) and depth (D) of the trees. Observe that orthogonal trees generalize poorly
as the topology of the decision boundary is not well aligned to the natural topology of the data.
We evaluate boosted decision tree learning with decorrelated features in the context of pedestrian
detection. As our baseline we utilize the aggregated channel features (ACF) detector [7], a popular,
top-performing detector for which source code is available online. Coupled with use of deeper trees
and a denser sampling of the data, the improvement obtained using our locally decorrelated channel
features (LDCF) is substantial. While in the past year the use of deep learning [25], motion features
[27], and multi-resolution models [36] has brought down log-average miss rate (MR) to under 40%
on the Caltech Pedestrian Dataset [10], LDCF reduces MR to under 25%. This translates to a nearly
tenfold reduction in false positives over the (very recent) state-of-the-art.
The paper is organized as follows. In ?2 we review orthogonal and oblique trees and demonstrate
that orthogonal trees trained on decorrelated data may be equally or more effective as oblique trees
trained on the original data. We introduce the baseline in ?3 and in ?4 show that use of oblique
trees improves results but at considerable computational expense. Next, in ?5, we demonstrate that
orthogonal trees trained with locally decorrelated features are efficient and effective. Experiments
and results are presented in ?6. We begin by briefly reviewing related work next.
1.1
Related Work
Pedestrian Detection: Recent work in pedestrian detection includes use of deformable part models
and their extensions [11, 36, 26], convolutional nets and deep learning [33, 37, 25], and approaches
that focus on optimization and learning [20, 18, 34]. Boosted detectors are also widely used. In
particular, the channel features detectors [9, 1, 2, 7] are a family of conceptually straightforward and
efficient detectors based on boosted decision trees computed over multiple feature channels such as
color, gradient magnitude, gradient orientation and others. Current top results on the INRIA [6] and
Caltech [10] Pedestrian Datasets include instances of the channel features detector with additional
mid-level edge features [19] and motion features [27], respectively.
Oblique Decision Trees: Typically, decision trees are trained with orthogonal (single feature) splits;
however, the extension to oblique (multiple feature) splits is fairly intuitive and well known, see
e.g. [24]. In fact, Breiman?s foundational work on random forests [5] experimented with oblique
trees. Recently there has been renewed interest in random forests with oblique splits [23, 30] and
Marin et al. [20] even applied such a technique to pedestrian detection. Likewise, while typically
orthogonal trees are used with boosting [12], oblique trees can easily be used instead. The contribution of this work is not the straightforward coupling of oblique trees with boosting, rather, we
propose a local decorrelation transform that eliminates the necessity of oblique splits altogether.
Decorrelation: Decorrelation is a common pre-processing step for classification [17, 15]. In recent
work, Hariharan et al. [15] proposed an efficient scheme for estimating covariances between HOG
features [6] with the goal of replacing linear SVMs with LDA and thus allowing for fast training.
Hariharan et al. demonstrated that the global covariance matrix for a detection window can be estimated efficiently as the covariance between two features should depend only on their relative offset.
Inspired by [15], we likewise exploit the stationarity of natural image statistics, but instead propose
to estimate a local covariance matrix shared across all image patches. Next, rather than applying
global decorrelation, which would be computationally prohibitive for sliding window detection with
a nonlinear classifier1 , we instead propose to apply an efficient local decorrelation transform. The
result is an overcomplete representation well suited for use with orthogonal trees.
1
Global decorrelation coupled with a linear classifier is efficient as the two linear operations can be merged.
2
Figure 2: A comparison of boosting with orthogonal decision trees (T = 5) on transformed data.
Orthogonal trees with both decorrelated and PCA-whitened features show improved generalization
while ZCA-whitening is ineffective. Decorrelating the features is critical, while scaling is not.
2
Boosted Decision Trees with Correlated Data
Boosting is a simple yet powerful tool for classification and can model complex non-linear functions
[31, 12]. The general idea is to train and combine a number of weak learners into a more powerful
strong classifier. Decision trees are frequently used as the weak learner in conjunction with boosting,
and in particular orthogonal decision trees, that is trees in which every split is a threshold on a single
feature, are especially popular due to their speed and simplicity [35, 7, 1].
The representational power obtained by boosting orthogonal trees is not limited by use of orthogonal
splits; however, the number and depth of the trees necessary to fit the data may be large. This can
lead to complex decision boundaries and poor generalization, especially given highly correlated
features. Figure 1(a)-(c) shows the result of boosted orthogonal trees on correlated data. Observe
that the orthogonal trees generalize poorly even as we vary the number and depth of the trees.
Decision trees with oblique splits can more effectively model data with correlated features as the
topology of the resulting classifier can better match the natural topology of the data [23]. In oblique
trees, every split is based on a linear projection of the data z = w| x followed by thresholding.
The projection w can be sparse (and orthogonal splits are a special case with kwk0 = 1). While
in principle numerous approaches can be used to obtain w, in practice linear discriminant analysis
(LDA) is a natural choice for obtaining discriminative splits efficiently [16]. LDA aims to minimize
within-class scatter while maximizing between-class scatter. w is computed from class-conditional
mean vectors ?+ and ?? and a class-independent covariance matrix ? as follows:
w = ??1 (?+ ? ?? ).
(1)
The covariance may be degenerate if the amount or underlying dimension of the data is low; in this
case LDA can be regularized by using (1 ? )? + I in place of ?. In Figure 1(d) we apply boosted
oblique trees trained with LDA on the same data as before. Observe the resulting decision boundary
better matches the underlying data distribution and shows improved generalization.
The connection between whitening and LDA is well known [15]. Specifically, LDA simplifies to a
trivial classification rule on whitened data (data whose covariance is the identity). Let ? = Q?Q|
be the eigendecomposition1 of ? where
Q is an orthogonal matrix and ? is a diagonal matrix of
1
eigenvalues. W = Q?? 2 Q| = ?? 2 is known as a whitening matrix because the covariance of
x0 = Wx is the identity matrix. Given whitened data and means, LDA can be interpreted as
|
|
learning the trivial projection w0 = ?0+ ? ?0? = W?+ ? W?? since w0 x0 = w0 Wx = w| x.
Can whitening or a related transform likewise simplify learning of boosted decision trees?
Using standard terminology
[17], we define the following
related transforms: decorrelation (Q| ),
? 21 |
? 21 |
PCA-whitening (? Q ), and ZCA-whitening (Q? Q ). Figure 2 shows the result of boosting
orthogonal trees on the variously transformed features, using the same data as before. Observe that
with decorrelated and PCA-whitened features orthogonal trees show improved generalization. In
fact, as each split is invariant to scaling of individual features, orthogonal trees with PCA-whitened
and decorrelated features give identical results. Decorrelating the features is critical, while scaling
is not. The intuition is clear: each split operates on a single feature, which is most effective if
the features are decorrelated. Interestingly, the standard ZCA-whitened transform used by LDA is
ineffective: while the resulting features are not technically correlated, due to the additional rotation
by Q each resulting feature is a linear combination of features obtained by PCA-whitening.
3
3
Baseline Detector (ACF)
We next briefly review our baseline detector and evaluation benchmark. This will allow us to apply
the ideas from ?2 to object detection in subsequent sections. In this work we utilize the channel
features detectors [9, 7, 1, 2], a family of conceptually straightforward and efficient detectors for
which variants have been utilized for diverse tasks such as pedestrian detection [10], sign recognition
[22] and edge detection [19]. Specifically, for our experiments we focus on pedestrian detection and
employ the aggregate channel features (ACF) variant [7] for which code is available online2 .
Given an input image, ACF computes several feature channels, where each channel is a per-pixel
feature map such that output pixels are computed from corresponding patches of input pixels (thus
preserving image layout). We use the same channels as [7]: normalized gradient magnitude (1 channel), histogram of oriented gradients (6 channels), and LUV color channels (3 channels), for a total
of 10 channels. We downsample the channels by 2x and features are single pixel lookups in the
aggregated channels. Thus, given a h ? w detection window, there are h/2 ? w/2 ? 10 candidate
features (channel pixel lookups). We use RealBoost [12] with multiple rounds of bootstrapping to
train and combine 2048 depth-3 decision trees over these features to distinguish object from background. Soft-cascades [3] and an efficient multiscale sliding-window approach are employed. Our
baseline uses slightly altered parameters from [7] (RealBoost, deeper trees, and less downsampling);
this increases model capacity and benefits our final approach as we report in detail in ?6.
Current practice is to use the INRIA Pedestrian Dataset [6] for parameter tuning, with the test set
serving as a validation set, see e.g. [20, 2, 9]. We utilize this dataset in much the same way and report
full results on the more challenging Caltech Pedestrian Dataset [10]. Following the methodology
of [10], we summarize performance using the log-average miss rate (MR) between 10?2 and 100
false positives per image. We repeat all experiments 10 times and report the mean MR and standard
error for every result. Due to the use of a log-log scale, even small improvements in (log-average)
MR correspond to large reductions in false-positives. On INRIA, our (slightly modified) baseline
version of ACF scores at 17.3% MR compared to 17.0% MR for the model reported in [7].
4
Detection with Oblique Splits (ACF-LDA)
In this section we modify the ACF detector to enable oblique splits and report the resulting gains.
Recall that given input x, at each split of an oblique decision tree we need to compute z = w| x for
some projection w and threshold the result. For our baseline pedestrian detector, we use 128 ? 64
windows where each window is represented by a feature vector x of size 128/2 ? 64/2 ? 10 = 20480
(see ?3). Given the high dimensionality of the input x coupled with the use of thousands of trees in
a typical boosted classifier, for efficiency w must be sparse.
Local w: We opt to use w?s that correspond to local m?m blocks of pixels. In other words, we treat
x as a h/2 ? w/2 ? 10 tensor and allow w to operate over any m ? m ? 1 patch in a single channel
of x. Doing so holds multiple advantages. Most importantly, each pixel has strongest correlations
to spatially nearby pixels [15]. Since oblique splits are expected to help most when features are
strongly correlated, operating over local neighborhoods is a natural choice. In addition, using local
w allows for faster lookups due to the locality of adjacent pixels in memory.
Complexity: First, let us consider the complexity of training the oblique splits. Let d = h/2?w/2 be
the window size of a single channel. The number of patches per channel in x is about d, thus naively
training a single split means applying LDA d times ? once per patch ? and keeping w with lowest
error. Instead of computing d independent matrices ? per channel, for efficiency, we compute ?, a
d ? d covariance matrix for the entire window, and reconstruct individual m2 ? m2 ??s by fetching
appropriate entries from ?. A similar trick can be used for the ??s. Computing ? is O(nd2 ) given
n training examples (and could be made faster by omitting unnecessary elements). Inverting each
?, the bottleneck of computing Eq. (1), is O(dm6 ) but independent of n and thus fairly small as
n m. Finally computing z = w| x over all n training examples and d projections is O(ndm2 ).
Given the high complexity of each step, a naive brute-force approach for training is infeasible.
Speedup: While the weights over training examples change at every boosting iteration and after
every tree split, in practice we find it is unnecessary to recompute the projections that frequently.
Table 1, rows 2-4, shows the results of ACF with oblique splits, updated every T boosting iterations
2
http://vision.ucsd.edu/?pdollar/toolbox/doc/
4
Shared ?
T
Miss Rate
Training
-
-
17.3 ? .33
4.93m
ACF-LDA-4
ACF-LDA-16
ACF-LDA-?
No
No
No
4
16
?
14.9 ? .37
15.1 ? .28
17.0 ? .22
303.57m
78.11m
5.82m
ACF-LDA? -4
ACF-LDA? -16
ACF-LDA? -?
Yes
Yes
Yes
4
16
?
14.7 ? .29
15.1 ? .12
16.4 ? .17
194.26m
51.19m
5.79m
LDCF
Yes
-
13.7 ? .15
6.04m
ACF
Table 1: A comparison of boosted trees with orthogonal and oblique splits.
(denoted by ACF-LDA-T ). While more frequent updates improve accuracy, ACF-LDA-16 has negligibly higher MR than ACF-LDA-4 but a nearly fourfold reduction in training time (timed using 12
cores). Training the brute force version of ACF-LDA, updated at every iteration and each tree split
(7 interior nodes per depth-3 tree) would have taken about 5 ? 4 ? 7 = 140 hours. For these results we
used regularization of = .1 and patch size of m = 5 (effect of varying m is explored in ?6).
Shared ?: The crux and computational bottleneck of ACF-LDA is the computation and application
of a separate covariance ? at each local neighborhood. In recent work on training linear object
detectors using LDA, Hariharan et al. [15] exploited the observation that the statistics of natural
images are translationally invariant and therefore the covariance between two features should depend
only on their relative offset. Furthermore, as positives are rare, [15] showed that the covariances can
be precomputed using natural images. Inspired by these observations, we propose to use a single,
fixed covariance ? shared across all local image neighborhoods. We precompute one ? per channel
and do not allow it to vary spatially or with boosting iteration. Table 1, rows 5-7, shows the results
of ACF with oblique splits using fixed ?, denoted by ACF-LDA? . As before, the ??s and resulting
w are updated every T iterations. As expected, training time is reduced relative to ACF-LDA.
Surprisingly, however, accuracy improves as well, presumably due to the implicit regularization
effect of using a fixed ?. This is a powerful result we will exploit further.
Summary: ACF with local oblique splits and a single shared ? (ACF-LDA? -4) achieves 14.7% MR
compared to 17.3% MR for ACF with orthogonal splits. The 2.6% improvement in log-average MR
corresponds to a nearly twofold reduction in false positives but comes at considerable computational
cost. In the next section, we propose an alternative, more efficient approach for exploiting the use of
a single shared ? capturing correlations in local neighborhoods.
5
Locally Decorrelated Channel Features (LDCF)
We now have all the necessary ingredients to introduce our approach. We have made the following
observations: (1) oblique splits learned with LDA over local m ? m patches improve results over
orthogonal splits, (2) a single covariance matrix ? can be shared across all patches per channel, and
(3) orthogonal trees with decorrelated features can potentially be used in place of oblique trees. This
suggests the following approach: for every m ? m patch p in x, we can create a decorrelated representation by computing Q| p, where Q?Q| is the eigendecomposition of ? as before, followed by
use of orthogonal trees. However, such an approach is computationally expensive.
First, due to use of overlapping patches, computing Q| p for every overlapping patch results in an
overcomplete representation with a factor m2 increase in feature dimensionality. To reduce dimensionality, we only utilize the top k eigenvectors in Q, resulting in k < m2 features per pixel. The
intuition is that the top eigenvectors capture the salient neighborhood structure. Our experiments
in ?6 confirm this: using as few as k = 4 eigenvectors per channel for patches of size m = 5 is
sufficient. As our second speedup, we observe that the projection Q| p can be computed by a series
of k convolutions between a channel image and each m ? m filter reshaped from its corresponding
eigenvector (column of Q). This is possible because the covariance matrix ? is shared across all
patches per channel and hence the derived Q is likewise spatially invariant. Decorrelating all 10
channels in an entire feature pyramid for a 640 ? 480 image takes about .5 seconds.
5
Figure 3: Top-left: autocorrelation for each channel. Bottom-left: learned decorrelation filters.
Right: visualization of original and decorrelated channels averaged over positive training examples.
In summary, we modify ACF by taking the original 10 channels and applying k = 4 decorrelating
(linear) filters per channel. The result is a set of 40 locally decorrelated channel features (LDCF).
To further increase efficiency, we downsample the decorrelated channels by a factor of 2x which has
negligible impact on accuracy but reduces feature dimension to the original value. Given the new
locally decorrelated channels, all other steps of ACF training and testing are identical. The extra
implementation effort is likewise minimal: given the decorrelation filters, a few lines of code suffice
to convert ACF into LDCF. To further improve clarity, all source code for LDCF will be released.
Results of the LDCF detector on the INRIA dataset are given in the last row of Table 1. The
LCDF detector (which uses orthogonal splits) improves accuracy over ACF with oblique splits by
an additional 1% MR. Training time is significantly faster, and indeed, is only ?1 minute longer
than for the original ACF detector. More detailed experiments and results are reported in ?6. We
conclude by (1) describing the estimation of ? for each channel, (2) showing various visualizations,
and (3) discussing the filters themselves and connections to known filters.
Estimating ?: We can estimate a spatially constant ? for each channel using any large collection of natural images. ? for each channel is represented by a spatial autocorrelation function
?(x,y),(x+?x,y+?y) = C(?x, ?y). Given a collection of natural images, we first estimate a separate autocorrelation function for each image and then average the results. Naive computation of the
final function is O(np2 ) but the Wiener-Khinchin theorem reduces the complexity to O(np log p)
via the FFT [4], where n and p denote the number of images and pixels per image, respectively.
Visualization: Fig. 3, top-left, illustrates the estimated autocorrelations for each channel. Nearby
features are highly correlated and oriented gradients are spatially correlated along their orientation
due to curvilinear continuity [15]. Fig. 3, bottom-left, shows the decorrelation filters for each channel obtained by reshaping the largest eigenvectors of ?. The largest eigenvectors are smoothing
filters while the smaller ones resemble increasingly higher-frequency filters. The corresponding
eigenvalues decay rapidly and in practice we use the top k = 4 filters. Observe that the decorrelation filters for oriented gradients are aligned to their orientation. Finally, Fig. 3, right, shows original
and decorrelated channels averaged over positive training examples.
Discussion: Our decorrelation filters are closely related to sinusoidal, DCT basis, and Gaussian
derivative filters. Spatial interactions in natural images are often well-described by Markov models [13] and first-order stationary Markov processes are known to have sinusoidal KLT bases [29].
In particular, for the LUV color channels, our filters are similar to the discrete cosine transform
(DCT) bases that are often used to approximate the KLT. For oriented gradients, however, the decorrelation filters are no longer well modeled by the DCT bases (note also that our filters are applied
densely whereas the DCT typically uses block processing). Alternatively, we can interpret our filters
as Gaussian derivative filters. Assume that the autocorrelation is modeled by a squared-exponential
function C(?x) = exp(??x2 /2l2 ), which is fairly reasonable given the estimation results in Fig. 3.
In 1D, the k th largest eigenfunction of such an autocorrelation function is a k ? 1 order Gaussian
derivative filter [28]. It is straightforward to extend the result to an anisotropic multivariate case in
which case the eigenfunctions are Gaussian directional derivative filters similar to our filters.
6
Figure 4: (a-b) Use of k = 4 local decorrelation filters of size m = 5 gives optimal performance.
(c) Increasing tree depth while simultaneously enlarging the quantity of data available for training
can have a large impact on accuracy (blue stars indicate optimal depth at each sampling interval).
description
1.
2.
3.
4.
5.
6.
7.
ACF
LDCF small ?
LDCF random
LDCF LUV only
LDCF grad only
LDCF constant
LDCF
# channels
(modified) baseline
decorrelation w k smallest filters
filtering w k random filters
decorrelation of LUV channels only
decorrelation of grad channels only
decorrelation w constant filters
proposed approach
10
10k
10k
3k + 7
3 + 7k
10k
10k
miss rate
17.3 ? .33
61.7 ? .28
15.6 ? .26
16.2 ? .37
14.9 ? .29
14.2 ? .34
13.7 ? .15
Table 2: Locally decorrelated channels compared to alternate filtering strategies. See text.
6
Experiments
In this section, we demonstrate the effectiveness of locally decorrelated channel features (LDCF) in
the context of pedestrian detection. We: (1) study the effect of parameter settings, (2) test variations
of our approach, and finally (3) compare our results with the state-of-the-art.
Parameters: LDCF has two parameters: the count and size of the decorrelation filters. Fig. 4(a) and
(b) show the results of LDCF on the INRIA dataset while varying the filter count (k) and size (m),
respectively. Use of k = 4 decorrelation filters of size m = 5 improves performance up to ?4% MR
compared to ACF. Inclusion of additional higher-frequency filters or use of larger filters can cause
performance degradation. For all remaining experiments we fix k = 4 and m = 5.
Variations: We test variants of LDCF and report results on INRIA in Table 2. LDCF (row 7) outperforms all variants, including the baseline (1). Filtering the channels with the smallest k eigenvectors
(2) or k random filters (3) performs worse. Local decorrelation of only the color channels (4) or
only the gradient channels (5) is inferior to decorrelation of all channels. Finally, we test constant
decorrelation filters obtained from the intensity channel L that resemble the first k DCT basis filters.
Use of unique filters per channel outperforms use of constant filters across all channels (6).
Model Capacity: Use of locally decorrelated features implicitly allows for richer, more effective
splitting functions, increasing modeling capacity and generalization ability. Inspired by their success, we explore additional strategies for augmenting model capacity. For the following experiments,
we rely solely on the training set of the Caltech Pedestrian Dataset [10]. Of the 71 minute long training videos (?128k images), we use every fourth video as validation data and the rest for training. On
the validation set, LDCF outperforms ACF by a considerable margin, reducing MR from 46.2% to
41.7%. We first augment model capacity by increasing the number of trees twofold (to 4096) and the
sampled negatives fivefold (to 50k). Surprisingly, doing so reduces MR by an additional 4%. Next,
we experiment with increasing maximum tree depth while simultaneously enlarging the amount of
data available for training. Typically, every 30th image in the Caltech dataset is used for training and
testing. Instead, Figure 4(c) shows validation performance of LDCF with different tree depths while
varying the training data sampling interval. The impact of maximum depth on performance is quite
large. At a dense sampling interval of every 4th frame, use of depth-5 trees (up from depth-2 for the
original approach) improves performance by an additional 5% to 32.6% MR. Note that consistent
with the generalization bounds of boosting [31], use of deeper trees requires more data.
7
1
1
72% VJ
46% HOG
21% pAUCBoost
20% FisherBoost
20% LatSvm?V2
20% ConvNet
19% CrossTalk
17% ACF
16% VeryFast
15% RandForest
14% LDCF
14% Franken
14% Roerei
13% SketchTokens
.64
.50
.40
miss rate
.30
.20
.80
.64
.50
.40
.30
miss rate
.80
.10
.10
.05
.05
?2
10
?1
10
0
10
1
95% VJ
68% HOG
48% DBN?Mut
46% MF+Motion+2Ped
46% MOCO
45% MultiSDP
44% ACF?Caltech
43% MultiResC+2Ped
41% MT?DPM
39% JointDeep
38% MT?DPM+Context
37% ACF+SDt
30% ACF?Caltech+
25% LDCF
.20
?3
10
10
false positives per image
?2
10
?1
10
0
10
1
10
false positives per image
(a) INRIA Pedestrian Dataset
(b) Caltech Pedestrian Dataset
Figure 5: A comparison of our LDCF detector with state-of-the-art pedestrian detectors.
INRIA Results: In Figure 5(a) we compare LDCF with state-of-the-art detectors on INRIA [6]
using benchmark code maintained by [10]. Since the INRIA dataset is oft-used as a validation set,
including in this work, we include these results for completeness only. LDCF is essentially tied for
second place with Roerei [2] and Franken [21] and outperformed by ?1% MR by SketchTokens [19].
These approaches all belong to the family of channel features detectors, and as the improvements
proposed in this work are orthogonal, the methods could potentially be combined.
Caltech Results: We present our main result on the Caltech Pedestrian Dataset [10], see Fig. 5(b),
generated using the official evaluation code available online3 . The Caltech dataset has become the
standard for evaluating pedestrian detectors and the latest methods based on deep learning (JointDeep) [25], multi-resolution models (MT-DPM) [36] and motion features (ACF+SDt) [27] achieve
under 40% log-average MR. For a complete comparison, we first present results for an augmented
capacity ACF model which uses more (4096) and deeper (depth-5) trees trained with RealBoost
using dense sampling of the training data (every 4th image). See preceding note on model capacity
for details and motivation. This augmented model (ACF-Caltech+) achieves 29.8% MR, a considerable nearly 10% MR gain over previous methods, including the baseline version of ACF (ACFCaltech). With identical parameters, locally decorrelated channel features (LDCF) further reduce
error to 24.9% MR with substantial gains at higher recall. Overall, this is a massive improvement
and represents a nearly 10x reduction in false positives over the previous state-of-the-art.
7
Conclusion
In this work we have presented a simple, principled approach for improving boosted object detectors.
Our core observation was that effective but expensive oblique splits in decision trees can be replaced
by orthogonal splits over locally decorrelated data. Moreover, due to the stationary statistics of
image features, the local decorrelation can be performed efficiently via convolution with a fixed
filter bank precomputed from natural images. Our approach is general, simple and fast.
Our method showed dramatic improvement over previous state-of-the-art. While some of the gain
was from increasing model capacity, use of local decorrelation gave a clear and significant boost.
Overall, we reduced false-positives tenfold on Caltech. Such large gains are fairly rare.
In the present work we did not decorrelate features across channels (decorrelation was applied independently per channel). This is a clear future direction. Testing local decorrelation in the context of
other classifiers (e.g. convolutional nets or linear classifiers as in [15]) would also be interesting.
While the proposed locally decorrelated channel features (LDCF) require only modest modification
to existing code, we will release all source code used in this work to ease reproducibility.
3
http://www.vision.caltech.edu/Image_Datasets/CaltechPedestrians/
8
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9
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4,882 | 542 | Learning Global Direct Inverse Kinematics
Kenneth Kreutz-Delgado t
Electrical & Computer Eng.
UC San Diego
La Jolla, CA 92093-0407
David DeMers?
Computer Science & Eng.
UC San Diego
La Jolla, CA 92093-0114
Abstract
We introduce and demonstrate a bootstrap method for construction of an inverse function for the robot kinematic mapping using only sample configurationspace/workspace data. Unsupervised learning (clustering) techniques are used on
pre-image neighborhoods in order to learn to partition the configuration space
into subsets over which the kinematic mapping is invertible. Supervised leaming is then used separately on each of the partitions to approximate the inverse
function. The ill-posed inverse kinematics function is thereby regularized, and
a globa1 inverse kinematics solution for the wristless Puma manipulator is developed.
1 INTRODUCTION
The robot forward kinematics function is a continuous mapping
f : C ~ en -
w ~ Xm
which maps a set of n joint parameters from the configuration space, C, to the mdimensiona1 task space, W. If m S n, the robot has redundant degrees-of-freedom
(dof's). In general, control objectives such as the positioning and orienting of the endeffector are specified with respect to task space co-ordinates; however, the manipulator is
typica1ly controlled only in the configuration space. Therefore, it is important to be able
to find some 0 E C such that f(O) is a particular target va1ue
E W. This is the inverse
kinematics problem.
xo
? e-mail: [email protected]
t e-mail: [email protected]
589
590
DeMers and Kreutz-Delgado
The inverse kinematics problem is ill-posed. If there are redundant doCs then the problem is
locally ill-posed, because the solution is non-unique and consists of a non-trivial manifold 1
in C. With or without redundant dof's, the problem is generally globally ill-posed because
of the existence of a finite set of solution branches - there will typically be multiple
configurations which result in the same task space location. Thus computation of a direct
inverse is problematic due to the many-to--one nature (and therefore non-invertibility) of
the map I .
The inverse problem can be solved explicitly, that is, in closed form, for only certain kinds
of manipulators. E.g. six dof elbow manipulators with separable wrist (where the first three
joints are used for positioning and the last three have a common origin and are used for
orientation), such as the Puma 560, are solvable, see (Craig, 86). The alternative to a closed
form solution is a numerical solution, usually either using the inverse of the Jacobian, which
is a Newton-style approach, or by using gradient descent (also a Jacobian-based method).
These methods are iterative and require expensive Jacobian or gradient computation at each
step, thus they are not well-suited for real-time control.
Neural networks can be used to find an inverse by implementing either direct inverse
modeling (estimating the explicit function 1-1) or differential methods. Implementations
of the direct inverse approach typically fail due to the non-linearity of the solution sef,
or resolve this problem by restriction to a single solution a priori. However, such a prior
restriction of the solutions may not be possible or acceptable in all circumstances, and may
drastically reduce the dexterity and manipulability of the arm.
The differential approaches either find only the nearest local solution, or resolve the multiplicity of solutions at training time, as with Jordan's forward modeling (Jordan & Rumelhart,
1990) or the approach of (Nguyen & Patel, 1990). We seek to regularize the mapping in
such a way that all possible solutions are available at run-time, and can be computed
efficiently as a direct constant-time inverse rather than approximated by slower iterative
differential methods. To achieve the fast run-time solution, a significant cost in training
time must be paid; however, it is not unreasonable to invest resources in off-line learning
in order to attain on-line advantages. Thus we wish to gain the run-time computational
efficiency of a direct inverse solution while also achieving the benefits of the differential
approaches.
This paper introduces a method for performing global regularization; that is, identifying
the complete, finite set of solutions to the inverse kinematics problem for a non-redundant
manipulator. This will provide the ability to choose a particular solution at run time.
Resolving redundancy is beyond the scope of this paper, however, preliminary work on
a method which may be integrated with the work presented here is shown in (DeMers
& Kreutz-Delgado, 1991). In the remainder of this paper it will be assumed that the
manipulator does not have redundant dof's. It will also be assumed that all of the joints are
revolute, thus the configuration space is a subset of the n-torus, Tn.
IGenerically of dimensionality equal to n - m.
Zrhe target values are assumed to be in the range of I, i E W = I(C), so the existence of a
solution is not an issue in this paper.
3Training a network to minimize mean squared error with multiple target values for the same
input value results in a "learned" response of the average of the targets. Since the targets lie on a
number of non-linear manifolds (for the redundant case) or consist of a finite number of points (for
the non-redundant case), the average of multiple targets will typically not be a correct target.
Learning Global Direct Inverse Kinematics
2 TOPOLOGY OF THE KINEMATICS FUNCTION
The kinematics mapping is continuous and smooth and, generically, neighborhoods in
configuration space map to neighborhoods in the task space4 ? The configuration space,
C, is made up of a finite number of disjoint regions or partitions, separated by n - 1
dimensional surfaces where the Jacobian loses rank (called critical surfaces), see (Burdick,
1988, Burdick, 1991).
Let
I : Tn -- Rn
be the kinematic mapping. Then
k
W
= I(C) = UIi (Cd
i=l
=
where Ii is the restriction of I to Ci , Ii : Ci
en /1 -- Rn and the factor space
locally diffeomorphic to Rn. The Ci are each a connected region such that
VOECi ,
en /1 is
det(J(O));tO
where J is the Jacobian of I, J == d(J!. Define Wi as I(Cd. Generically, Ii is one-to-one
and onto open neighborhoods of Wi 5 , thus by the inverse function theorem
:3 gi(X)
=I
j-
1 :
Wi -- Ci, such that I
0
gi(X)
= X,
Vx E Wi
In the general case, with redundant dof's, the kinematics over a single configuration-space
region can be viewed as a fiber bundle, where the fibers are homeomorphic to Tn-m.
The base space is the reachable workspace (the image of Ci under j). Solution branch
resolution can be done by identifying distinct connected open coordinate neighborhoods of
the configuration space which cover the workspace. Redundancy resolution can be done by
a consistent parameterization of the fibers within each neighborhood. In the case at hand,
without redundant dof's, the "fibers" are singleton sets and no resolution is needed.
In the remainder of this paper, we will use input/output data to identify the individual
regions, Ci, of a non-redundant manipulator, over which the mapping Ii : Ci -- Wi is
invertible. The input/output data will then be partitioned modulo the configuration regions
Ci, and each li- 1 approximated individually.
3 SAMPLING APPROACH
If the manipulator can be measured and a large sample of (0, i) pairs taken, stored such
that the samples can be searched efficiently, a rough estimate of the inverse solutions at
a particular target point io may be obtained by finding all of the 0 points whose image lies
within some ( of
The pre-image of this (-ball will generically consist of several distinct
(distorted) balls in the configuration space. If the sampling is adequate then there will be
one such ball for each of the inverse solution branches. If each of the the points in each ball
is given a label for the solution branch, the labeled data may then be used for supervised
x
xo.
4This property fails when the manipulator is in a singular configuration, at which the Jacobian,
deft loses rank.
sSince it is generically true that J is non-singular.
591
592
DeMers and Kreutz-Delgado
learning of a classifier of solution branches in the configuration space. In this way we will
have "bootstrapped" our way to the development of a solution branch classifier.
Taking advantage of the continuous nature of the forward mapping, note that if io is slightly
perturbed by a "jump" to a neighboring target point then the pre-image balls will also be
perturbed. We can assign labels to the new data consistent with labels already assigned
to the previous data, by computing the distances between the new, unlabeled balls and the
previously labeled balls. Continuing in this fashion, io traces a path through the entire
workspace and solution branch labels may be given to all points in C which map to within
f of one of the selected i points along the sweep.
This procedure results in a significant and representative proportion of the data now being
labeled as to solution branch. Thus we now have labeled data (if, i, B(
where
8(
{I, ... , k} indicates which of the k solution branches, Ci , the point is in. We
can now construct a classifier using supervised learning to compute the branches B( e) for
a given B. Once an estimate of B( 0) is developed, we may use it to classify large amounts
of (if, i) data, and partition the data into k sets, one for each of the solution branches, Ci.
en,
e
e) =
4
RESOLUTION OF SOLUTION BRANCHES
We applied the above to the wristIess Puma 560, a 3-R manipulator for end-effector
positioning in R3. We took 40,000 samples of (if, i) points, and examined all points within
lOcm of selected target values ii. The ii formed a grid of 90 locations in the workspace.
3,062 of the samples fell within 10 cm of one of the ii. The configuration space points for
each target ii were clustered into four groups, corresponding to the four possible solution
branches of the wristless Puma 560. About 3% of the points were clustered into the wrong
group, based on the labeling scheme used. These 3,062 points were then used as training
patterns for a feedforward neural network classifier. A point was classified into the group
associated with the output unit of the neural network with maximum activation. The output
values were normalized to sum to 1.0. The network was tested on 50,000 new, previously
unseen (if, i) pairs, and correctly classified more than 98% of them.
All of the erroneous classifications were for points near the critical surfaces. Therefore the
activation levels of the output units can be used to estimate closeness to a critical surface.
Examining the test data and assigning all 0 points for which no output unit has activation
greater than or equal to 0.8 to the "near-a-singularity" class, the remaining points were
100% correctly classified.
Figure 1 shows the true critical manifold separating the regions of configuration space, and
the estimated manifold consisting of points from the test set where the maximum activation
of output units of the trained neural network is less than 0.8. The configuration space is a
subset of the 3-torus, which is shown here "sliced" along three generators and represented
as a cube. Because the Puma 560 has physiCal limits on the range of motion of its joints, the
regions shown are in fact six distinct regions, and there is no wraparound in any direction.
This classifier network is our candidate for an estimate of B( e). With it, the samples can
be separated into groups corresponding to the domains of each of the Ii, thus regularizing
into k 6 one-to-one invertible pieces6 ?
=
6 Although
there are only four inverse solutions for any i. If there were no joint limits, then the
Learning Global Direcr Inverse Kinemarics
Joint 2
Joint 2
Figure 1: The analytically derived critical surfaces, along with J ,000 points/or which no
unit 0/ the neural network classifier has greater than 0.8 activation.
5 DIRECT INVERSE SOLUTIONS
The classifier neural network can now be used to partition the data into four groups, one for
each of the branches, Ci . For each of these data sets, we train a feedforward network to learn
the mapping in the inverse direction. The target vectors were represented as vectors of the
sine of the half-angle (a measure motivated by the quatemion representation of orientation).
MSE under 0.001 were achieved for each of the four. This looks like a very small error,
however, this error is somewhat misleading. The configuration space error is measured in
units which are difficult to interpret. More important is the error in the workspace when
the solution computed is used in the forward kinematics mapping to position the ann. Over
a test set of 4,000 points, the average positioning error was 5.2 cm over the 92 cm radius
workspace.
We have as yet made no attempts to optimize the network or training for the direct inverse;
the thrust of our work is in achieving the regularization. It is clear that substantially better
performance can be developed, for example, by following (Ritter, et al., 1989), and we
expect end-effector positioning errors of less than 1% to be easily achievable.
6
DISCUSSION
We have shown that by exploiting the topological property of continuity of the kinematic
mapping for a non-redundant 3-dof robot we can determine all of the solution regions of the
inverse kinematic mapping. We have mapped out the configuration space critical surfaces
and thus discovered an important topological property of the mapping, corresponding to
an important physical property of the manipulator, by unsupervised learning. We can
boostrap from the original input/output data, unlabeled as to solution branch, and construct
an accurate classifier for the entire configuration space. The data can thereby be partitioned
into sets which are individually one-t()-{)ne and invenible, and the inverse mapping can
be directly approximated for each. Thus a large learning-time investment results in a fast
run-time direct inverse kinematics solution.
cube shown would be a true 3-torus, with opposite faces identified. Thus the small pieces in the
corners would be part of the larger regions by wraparound in the Joint 2 direction.
593
594
DeMers and Kreutz-Delgado
We need a thorough sampling of the configuration space in order to ensure that enough
points will fall within each f-ball, thus the data requirements are clearly exponential in the
number of degrees of freedom of the manipulator. Even with efficient storage and retrieval
in geometric data structures, such as a k-d tree, high dimensional systems may not be
tractable by our methods.
Fortunately practical and useful robotic systems of six and seven degrees of freedom
should be amenable to this method, especially if separable into positioning and orienting
subsystems.
Acknowledgements
This work was supported in part by NSF Presidential Young Investigator award IRI9057631 and a NASA/Netrologic grant. The first author would like to thank NIPS for
providing student travel grants. We thank Gary Cottrell for his many helpful comments and
enthusiastic discussions.
References
Joel Burdick (1991), "A Classification of 3R Regional Manipulator Singularities and Geometries", Proc. 19911?E? Inti. Con! Robotics & Automation, Sacramento.
Joel Burdick (1988), "Kinematics and Design of Redundant Robot Manipulators", Stanford
Ph.D. Thesis, Dept. of Mechanical Engineering.
John Craig (1986), Introduction to Robotics, Addison-Wesley.
David DeMers & Kenneth Kreutz-Delgado (1991), "Learning Global Topological Properties of Robot Kinematic Mappings for Neural Network-Based Configuration Control", in
Bekey, ed. Proc. USC Workshop on Neural Networks in Robotics, (to appear).
Michael I. Jordan (1988), "Supervised Learning and Systems with Excess Degrees of
Freedom", COINS Technical Report 88-27, University of Massachusetts at Amherst.
Michael!. Jordan & David E. Rumelhart (1990), "Forward Models: Supervised Learning
with a Distal Teacher". Submitted to Cognitive Science.
L. Nguyen & R.V. Patel (1990), "A Neural Network Based Strategy for the Inverse Kinematics Problem in Robotics", in Jamshidi and Saif, eds., Robotics and Manufacturing:
recent Trends in Research, Education and Applications, vol. 3, pp. 995-1000 (ASME
Press).
Helge J. Ritter, Thomas M. Martinetz, & Klaus J. Schulten (1989), ''Topology-Conserving
Maps for Learning Visuo-Motor-Coordination", Neural Networks, Vol. 2, pp. 159-168.
| 542 |@word achievable:1 proportion:1 open:2 seek:1 eng:2 paid:1 thereby:2 delgado:6 configuration:21 bootstrapped:1 activation:5 assigning:1 yet:1 must:1 john:1 cottrell:1 numerical:1 partition:5 thrust:1 burdick:4 motor:1 half:1 selected:2 parameterization:1 location:2 along:3 direct:10 differential:4 consists:1 manipulability:1 introduce:1 enthusiastic:1 globally:1 resolve:2 elbow:1 estimating:1 linearity:1 kind:1 cm:3 substantially:1 developed:3 finding:1 thorough:1 classifier:8 wrong:1 control:3 unit:6 grant:2 appear:1 engineering:1 local:1 limit:2 io:3 path:1 examined:1 co:1 range:2 unique:1 practical:1 wrist:1 investment:1 bootstrap:1 procedure:1 attain:1 puma:5 pre:3 onto:1 unlabeled:2 subsystem:1 storage:1 restriction:3 optimize:1 map:5 resolution:4 identifying:2 sacramento:1 regularize:1 his:1 deft:1 coordinate:1 diego:2 construction:1 target:12 modulo:1 origin:1 trend:1 rumelhart:2 expensive:1 approximated:3 labeled:4 electrical:1 solved:1 region:10 connected:2 sef:1 trained:1 efficiency:1 easily:1 joint:8 represented:2 fiber:4 train:1 separated:2 distinct:3 fast:2 labeling:1 klaus:1 neighborhood:6 dof:7 whose:1 posed:4 larger:1 stanford:1 tested:1 presidential:1 ability:1 gi:2 unseen:1 advantage:2 took:1 remainder:2 neighboring:1 achieve:1 conserving:1 jamshidi:1 invest:1 exploiting:1 requirement:1 measured:2 nearest:1 c:1 direction:3 radius:1 correct:1 vx:1 implementing:1 education:1 require:1 assign:1 clustered:2 preliminary:1 singularity:2 mapping:15 scope:1 proc:2 travel:1 label:4 coordination:1 individually:2 rough:1 clearly:1 rather:1 derived:1 rank:2 indicates:1 diffeomorphic:1 helpful:1 typically:3 integrated:1 entire:2 issue:1 classification:2 ill:4 orientation:2 priori:1 development:1 uc:2 cube:2 equal:2 construct:2 once:1 sampling:3 look:1 unsupervised:2 report:1 individual:1 usc:1 geometry:1 consisting:1 bekey:1 attempt:1 freedom:4 kinematic:6 joel:2 introduces:1 generically:4 bundle:1 amenable:1 accurate:1 tree:1 continuing:1 effector:2 classify:1 modeling:2 cover:1 cost:1 subset:3 examining:1 stored:1 perturbed:2 teacher:1 amherst:1 workspace:7 ritter:2 off:1 invertible:3 michael:2 squared:1 thesis:1 choose:1 corner:1 cognitive:1 style:1 li:1 singleton:1 student:1 invertibility:1 automation:1 explicitly:1 piece:1 sine:1 closed:2 minimize:1 formed:1 efficiently:2 identify:1 craig:2 classified:3 submitted:1 ed:2 pp:2 associated:1 visuo:1 con:1 demers:7 gain:1 massachusetts:1 dimensionality:1 nasa:1 wesley:1 supervised:5 response:1 done:2 hand:1 continuity:1 manipulator:14 orienting:2 normalized:1 true:3 regularization:2 assigned:1 analytically:1 distal:1 asme:1 complete:1 demonstrate:1 tn:3 motion:1 image:5 dexterity:1 common:1 physical:2 interpret:1 significant:2 grid:1 reachable:1 robot:6 surface:6 base:1 recent:1 jolla:2 certain:1 greater:2 somewhat:1 fortunately:1 determine:1 redundant:12 ii:9 branch:15 resolving:1 multiple:3 smooth:1 technical:1 positioning:6 retrieval:1 award:1 controlled:1 circumstance:1 achieved:1 robotics:5 separately:1 singular:2 regional:1 fell:1 comment:1 martinetz:1 jordan:4 near:2 feedforward:2 enough:1 topology:2 opposite:1 identified:1 reduce:1 space4:1 det:1 six:3 motivated:1 adequate:1 generally:1 useful:1 clear:1 amount:1 locally:2 ph:1 problematic:1 nsf:1 estimated:1 disjoint:1 correctly:2 vol:2 group:5 redundancy:2 four:5 achieving:2 kenneth:2 sum:1 run:5 inverse:29 angle:1 distorted:1 doc:1 acceptable:1 topological:3 uii:1 performing:1 separable:2 ball:8 slightly:1 wi:5 partitioned:2 helge:1 multiplicity:1 xo:2 inti:1 taken:1 resource:1 previously:2 kinematics:15 fail:1 r3:1 needed:1 addison:1 tractable:1 end:2 available:1 unreasonable:1 alternative:1 coin:1 slower:1 existence:2 original:1 thomas:1 clustering:1 remaining:1 ensure:1 newton:1 homeomorphic:1 especially:1 sweep:1 objective:1 already:1 strategy:1 gradient:2 distance:1 thank:2 mapped:1 separating:1 seven:1 mail:2 manifold:4 trivial:1 providing:1 difficult:1 trace:1 implementation:1 design:1 finite:4 descent:1 rn:3 ucsd:2 discovered:1 wraparound:2 ordinate:1 david:3 pair:2 mechanical:1 specified:1 learned:1 nip:1 able:1 beyond:1 usually:1 pattern:1 xm:1 critical:6 regularized:1 solvable:1 arm:1 scheme:1 misleading:1 ne:1 prior:1 geometric:1 acknowledgement:1 revolute:1 expect:1 generator:1 degree:4 consistent:2 cd:2 supported:1 last:1 drastically:1 fall:1 taking:1 face:1 benefit:1 forward:5 made:2 jump:1 san:2 author:1 nguyen:2 excess:1 approximate:1 patel:2 global:5 robotic:1 kreutz:7 assumed:3 continuous:3 iterative:2 learn:2 nature:2 ca:2 mse:1 domain:1 sliced:1 representative:1 en:4 fashion:1 fails:1 position:1 schulten:1 explicit:1 wish:1 torus:3 exponential:1 lie:2 candidate:1 jacobian:6 young:1 theorem:1 erroneous:1 closeness:1 consist:2 workshop:1 ci:11 suited:1 gary:1 loses:2 viewed:1 ann:1 leaming:1 manufacturing:1 called:1 ece:1 la:2 searched:1 investigator:1 dept:1 regularizing:1 |
4,883 | 5,420 | Do Convnets Learn Correspondence?
Jonathan Long
Ning Zhang
Trevor Darrell
University of California ? Berkeley
{jonlong, nzhang, trevor}@cs.berkeley.edu
Abstract
Convolutional neural nets (convnets) trained from massive labeled datasets [1]
have substantially improved the state-of-the-art in image classification [2] and object detection [3]. However, visual understanding requires establishing correspondence on a finer level than object category. Given their large pooling regions and
training from whole-image labels, it is not clear that convnets derive their success
from an accurate correspondence model which could be used for precise localization. In this paper, we study the effectiveness of convnet activation features for
tasks requiring correspondence. We present evidence that convnet features localize at a much finer scale than their receptive field sizes, that they can be used to
perform intraclass aligment as well as conventional hand-engineered features, and
that they outperform conventional features in keypoint prediction on objects from
PASCAL VOC 2011 [4].
1
Introduction
Recent advances in convolutional neural nets [2] dramatically improved the state-of-the-art in image
classification. Despite the magnitude of these results, many doubted [5] that the resulting features
had the spatial specificity necessary for localization; after all, whole image classification can rely
on context cues and overly large pooling regions to get the job done. For coarse localization, such
doubts were alleviated by record breaking results extending the same features to detection on PASCAL [3].
Now, the same questions loom on a finer scale. Are the modern convnets that excel at classification
and detection also able to find precise correspondences between object parts? Or do large receptive
fields mean that correspondence is effectively pooled away, making this a task better suited for
hand-engineered features?
In this paper, we provide evidence that convnet features perform at least as well as conventional
ones, even in the regime of point-to-point correspondence, and we show considerable performance
improvement in certain settings, including category-level keypoint prediction.
1.1
Related work
Image alignment Image alignment is a key step in many computer vision tasks, including face
verification, motion analysis, stereo matching, and object recognition. Alignment results in correspondence across different images by removing intraclass variability and canonicalizing pose.
Alignment methods exist on a supervision spectrum from requiring manually labeled fiducial points
or landmarks, to requiring class labels, to fully unsupervised joint alignment and clustering models.
Congealing [6] is an unsupervised joint alignment method based on an entropy objective. Deep
congealing [7] builds on this idea by replacing hand-engineered features with unsupervised feature
learning from multiple resolutions. Inspired by optical flow, SIFT flow [8] matches densely sampled
SIFT features for correspondence and has been applied to motion prediction and motion transfer. In
Section 3, we apply SIFT flow using deep features for aligning different instances of the same class.
1
Keypoint localization Semantic parts carry important information for object recognition, object
detection, and pose estimation. In particular, fine-grained categorization, the subject of many recent
works, depends strongly on part localization [9, 10]. Large pose and appearance variation across
examples make part localization for generic object categories a challenging task.
Most of the existing works on part localization or keypoint prediction focus on either facial landmark
localization [11] or human pose estimation. Human pose estimation has been approached using tree
structured methods to model the spatial relationships between parts [12, 13, 14], and also using
poselets [15] as an intermediate step to localize human keypoints [16, 17]. Tree structured models
and poselets may struggle when applied to generic objects with large articulated deformations and
wide shape variance.
Deep learning Convolutional neural networks have gained much recent attention due to their success in image classification [2]. Convnets trained with backpropagation were initially succesful in
digit recognition [18] and OCR [19]. The feature representations learned from large data sets have
been found to generalize well to other image classification tasks [20] and even to object detection
[3, 21]. Recently, Toshev et al. [22] trained a cascade of regression-based convnets for human pose
estimation and Jain et al. [23] combine a weak spatial model with deep learning methods.
The latter work trains multiple small, independent convnets on 64 ? 64 patches for binary bodypart detection. In contrast, we employ a powerful pretained ImageNet model that shares mid-elvel
feature representations among all parts in Section 5.
Several recent works have attempted to analyze and explain this overwhelming success. Zeiler and
Fergus [24] provide several heuristic visualizations suggesting coarse localization ability. Szegedy
et al. [25] show counterintuitive properties of the convnet representation, and suggest that individual
feature channels may not be more semantically meaningful than other bases in feature space. A
concurrent work [26] compares convnet features with SIFT in a standard descriptor matching task.
This work illuminates and extends that comparison by providing visual analysis and by moving
beyond single instance matching to intraclass correspondence and keypoint prediction.
1.2
Preliminaries
We perform experiments using a network architecture almost identical1 to that popularized by
Krizhevsky et al. [2] and trained for classification using the 1.2 million images of the ILSVRC
2012 challenge dataset [1]. All experiments are implemented using caffe [27], and our network
is the publicly available caffe reference model. We use the activations of each layer as features,
referred to as convn, pooln, or fcn for the nth convolutional, pooling, or fully connected layer,
respectively. We will use the term receptive field, abbreviated rf, to refer to the set of input pixels
that are path-connected to a particular unit in the convnet.
2
Feature visualization
In this section and Figures 1 and 2, we provide a
novel visual investigation of the effective pooling regions of convnet features.
Table 1: Convnet receptive field sizes and strides,
for an input of size 227 ? 227.
In Figure 1, we perform a nonparametric reconlayer
rf size
stride
struction of images from features in the spirit
conv1 11 ? 11
4?4
of HOGgles [28]. Rather than paired dictionary
conv2 51 ? 51
8?8
learning, however, we simply replace patches
conv3 99 ? 99
16 ? 16
with averages of their top-k nearest neighbors
conv4 131 ? 131 16 ? 16
in a convnet feature space. To do so, we first
conv5 163 ? 163 16 ? 16
compute all features at a particular layer, repool5 195 ? 195 32 ? 32
sulting in an 2d grid of feature vectors. We associate each feature vector with a patch in the
original image at the center of the corresponding receptive field and with size equal to the receptive
field stride. (Note that the strides of the receptive fields are much smaller than the receptive fields
1
Ours reverses the order of the response normalization and pooling layers.
2
conv4
conv5
uniform rf
5 neighbors
1 neighbor
5 neighbors
1 neighbor
conv3
Figure 1: Even though they have large receptive fields, convnet features carry local information at
a finer scale. Upper left: given an input image, we replaced 16 ? 16 patches with averages over
1 or 5 nearest neighbor patches, computed using convnet features centered at those patches. The
yellow square illustrates one input patch, and the black squares show the corresponding rfs for the
three layers shown. Right: Notice that the features retrieve reasonable matches for the centers of
their receptive fields, even though those rfs extend over large regions of the source image. In the
?uniform rf? column, we show the best that could be expected if convnet features discarded all
spatial information within their rfs, by choosing input patches uniformly at random from conv3sized neighborhoods. (Best viewed electronically.)
themselves, which overlap. Refer to Table 1 above for specific numbers.) We replace each such
patch with an average over k nearest neighbor patches using a database of features densely computed on the images of PASCAL VOC 2011. Our database contains at least one million patches for
every layer. Features are matched by cosine similarity.
Even though the feature rfs cover large regions of the source images, the specific resemblance of
the resulting images shows that information is not spread uniformly throughout those regions. Notable features (e.g., the tires of the bicycle and the facial features of the cat) are replaced in their
corresponding locations. Also note that replacement appears to become more semantic and less
visually specific as the layer deepens: the eyes and nose of the cat get replaced with differently colored or shaped eyes and noses, and the fur gets replaced with various animal furs, with the diversity
increasing with layer number.
Figure 2 gives a feature-centric rather than image-centric view of feature locality. For each column,
we first pick a random seed feature vector (computed from a PASCAL image), and find k nearest
neighbor features, again by cosine similarity. Instead of averaging only the centers, we average
the entire receptive fields of the neighbors. The resulting images show that similar features tend to
respond to similar colors specifically in the centers of their receptive fields.
3
conv4
conv5
500 nbrs 50 nbrs 5 nbrs
conv3
Figure 2: Similar convnet features tend to have similar receptive field centers. Starting from a
randomly selected seed patch occupying one rf in conv3, 4, or 5, we find the nearest k neighbor
features computed on a database of natural images, and average together the corresponding receptive
fields. The contrast of each image has been expanded after averaging. (Note that since each layer
is computed with a stride of 16, there is an upper bound on the quality of alignment that can be
witnessed here.)
3
Intraclass alignment
We conjecture that category learning implicitly aligns instances by pooling over a discriminative
mid-level representation. If this is true, then such features should be useful for post-hoc alignment
in a similar fashion to conventional features. To test this, we use convnet features for the task of
aligning different instances of the same class. We approach this difficult task in the style of SIFT
flow [8]: we retrieve near neighbors using a coarse similarity measure, and then compute dense
correspondences on which we impose an MRF smoothness prior which finally allows all images to
be warped into alignment.
Nearest neighbors are computed using fc7 features. Since we are specifically testing the quality of
alignment, we use the same nearest neighbors for convnet or conventional features, and we compute
both types of features at the same locations, the grid of convnet rf centers in the response to a single
image.
Alignment is determined by solving an MRF formulated on this grid of feature locations. Let p be a
point on this grid, let fs (p) be the feature vector of the source image at that point, and let ft (p) be the
feature vector of the target image at that point. For each feature grid location p of the source image,
there is a vector w(p) giving the displacement of the corresponding feature in the target image. We
use the energy function
X
X
kfs (p) ? ft (p + w(p))k2 + ?
kw(p) ? w(q)k22 ,
E(w) =
p
(p,q)?E
where E are the edges of a 4-neighborhood graph and ? is the regularization parameter. Optimization is performed using belief propagation, with the techniques suggested in [29]. Message passing
is performed efficiently using the squared Euclidean distance transform [30]. (Unlike the L1 regularization originally used by SIFT flow [8], this formulation maintains rotational invariance of w.)
Based on its performance in the next section, we use conv4 as our convnet feature, and SIFT with
descriptor radius 20 as our conventional feature. From validation experiments, we set ? = 3 ? 10?3
for both conv4 and SIFT features (which have a similar scale).
Given the alignment field w, we warp target to source using bivariate spline interpolation (implemented in SciPy [31]). Figure 3 gives examples of alignment quality for a few different seed images,
using both SIFT and convnet features. We show five warped nearest neighbors as well as keypoints
transferred from those neighbors.
We quantitatively assess the alignment by measuring the accuracy of predicted keypoints. To obtain
good predictions, we warp 25 nearest neighbors for each target image, and order them from smallest
to greatest deformation energy (we found this method to outperform ordering using the data term).
We take the predicted keypoints to be the median points (coordinate-wise) of the top five aligned
keypoints according to this ordering.
We assess correctness using mean PCK [32]. We consider a ground truth keypoint to be correctly
predicted if the prediction lies within a Euclidean distance of ? times the maximum of the bounding
4
five nearest neighbors
SIFT flow
conv4 flow
SIFT flow
conv4 flow
target image
Figure 3: Convnet features can bring different instances of the same class into good alignment at
least as well (on average) as traditional features. For each target image (left column), we show
warped versions of five nearest neighbor images aligned with conv4 flow (first row), and warped
versions aligned with SIFT flow [8] (second row). Keypoints from the warped images are shown
copied to the target image. The cat shows a case where convnet features perform better, while the
bicycle shows a case where SIFT features perform better. (Note that each instance is warped to a
square bounding box before alignment. Best viewed in color.)
Table 2: Keypoint transfer accuracy using convnet flow, SIFT flow, and simple copying from nearest
neighbors. Accuracy (PCK) is shown per category using ? = 0.1 (see text) and means are also
shown for the stricter values ? = 0.05 and 0.025. On average, convnet flow performs as well as
SIFT flow, and performs a bit better for stricter tolerances.
aero bike bird
conv4 flow 28.2 34.1 20.4
SIFT flow 27.6 30.8 19.9
NN transfer 18.3 24.8 14.5
boat
17.1
17.5
15.4
bttl
50.6
49.4
48.1
bus
36.7
36.4
27.6
car
20.9
20.7
16.0
mean
conv4 flow
SIFT flow
NN transfer
cat
19.6
16.0
11.1
chair
15.7
16.1
12.0
cow
25.4
25.0
16.8
? = 0.1
24.9
24.7
19.9
table
12.7
16.1
15.7
dog
18.7
16.3
12.7
? = 0.05
11.8
10.9
7.8
horse
25.9
27.7
20.2
mbike
23.1
28.3
18.5
prsn
21.4
20.2
18.7
plant
40.2
36.4
33.4
sheep
21.1
20.5
14.0
sofa
14.5
17.2
15.5
train
18.3
19.9
14.6
tv
33.3
32.9
30.0
mean
24.9
24.7
19.9
? = 0.025
4.08
3.55
2.35
box width and height, picking some ? ? [0, 1]. We compute the overall accuracy for each type of
keypoint, and report the average over keypoint types. We do not penalize predicted keypoints that
are not visible in the target image.
Results are given in Table 2. We show per category results using ? = 0.1, and mean results for
? = 0.1, 0.05, and 0.025. Indeed, convnet learned features are at least as capable as SIFT at
alignment, and better than might have been expected given the size of their receptive fields.
4
Keypoint classification
In this section, we specifically address the ability of convnet features to understand semantic information at the scale of parts. As an initial test, we consider the task of keypoint classification:
given an image and the coordinates of a keypoint on that image, can we train a classifier to label the
keypoint?
5
Table 3: Keypoint classification accuracies, in percent, on the twenty categories of PASCAL 2011
val, trained with SIFT or convnet features. The best SIFT and convnet scores are bolded in each
category.
aero
SIFT 10 36
(radius) 20 37
40 35
80 33
160 27
conv 1 16
(layer) 2 37
3 42
4 44
5 44
bike
42
50
54
43
36
14
43
50
53
51
bird
36
39
37
37
34
15
40
46
49
49
boat
32
35
41
42
38
19
35
41
42
41
bttl
67
74
76
75
72
20
69
76
78
77
bus
64
67
68
66
59
29
63
69
70
68
car
40
47
47
42
35
15
38
46
45
44
cat
37
40
37
30
25
22
44
52
55
53
chair
33
36
39
43
39
16
35
39
41
39
cow
37
43
40
36
30
17
40
45
48
45
table
60
68
69
70
67
29
61
64
68
63
dog
34
38
36
31
27
17
38
47
51
50
horse mbike prsn
39
38
29
42
48
33
42
49
32
36
51
27
32
46
25
14
16
15
40
44
34
48
52
40
51
53
41
49
52
39
(a)
(a) cat left eye
plant sheep sofa
63
37 42
70
44 52
69
39 52
70
35 49
70
29 48
33
18 12
65
39 41
74
46 50
76
49 52
73
47 47
train
64
68
74
69
66
27
63
71
73
71
tv
75
77
78
77
76
29
72
77
76
75
mean
45
50
51
48
44
20
47
54
56
54
(b)
Figure 5: Cross validation scores for cat
keypoint classification as a function of
the SVM parameter C. In (a), we plot
mean accuracy against C for five different convnet features; in (b) we plot
the same for SIFT features of different
sizes. We use C = 10?6 for all experiments in Table 3.
(b) cat nose
Figure 4: Convnet features show fine
localization ability, even beyond their
stride and in cases where SIFT features
do not perform as well. Each plot is
a 2D histogram of the locations of the
maximum responses of a classifer in a
21 by 21 pixel rectangle taken around a
ground truth keypoint.
For this task we use keypoint data [15] on the twenty classes of PASCAL VOC 2011 [4]. We extract
features at each keypoint using SIFT [33] and using the column of each convnet layer whose rf
center lies closest to the keypoint. (Note that the SIFT features will be more precisely placed as a
result of this approximation.) We trained one-vs-all linear SVMs on the train set using SIFT at five
different radii and each of the five convolutional layer activations as features (in general, we found
pooling and normalization layers to have lower performance). We set the SVM parameter C = 10?6
for all experiments based on five-fold cross validation on the training set (see Figure 5).
Table 3 gives the resulting accuracies on the val set. We find features from convnet layers consistently perform at least as well as and often better than SIFT at this task, with the highest performance
coming from layers conv4 and conv5. Note that we are specifically testing convnet features trained
only for classification; the same net could be expected to achieve even higher performance if trained
for this task.
Finally, we study the precise location understanding of our classifiers by computing their responses
with a single-pixel stride around ground truth keypoint locations. For two example keypoints (cat
left eye and nose), we histogram the locations of the maximum responses within a 21 pixel by 21
pixel rectangle around the keypoint, shown in Figure 4. We do not include maximum responses
that lie on the boundary of this rectangle. While the SIFT classifiers do not seem to be sensitive
to the precise locations of the keypoints, in many cases the convnet ones seem to be capable of
localization finer than their strides, not just their receptive field sizes. This observation motivates
our final experiments to consider detection-based localization performance.
6
5
Keypoint prediction
We have seen that despite their large receptive field sizes, convnets work as well as the handengineered feature SIFT for alignment and slightly better than SIFT for keypoint classification.
Keypoint prediction provides a natural follow-up test. As in Section 3, we use keypoint annotations
from PASCAL VOC 2011, and we assume a ground truth bounding box.
Inspired in part by [3, 34, 23], we train sliding window part detectors to predict keypoint locations
independently. R-CNN [3] and OverFeat [34] have both demonstrated the effectiveness of deep convolutional networks on the generic object detection task. However, neither of them have investigated
the application of CNNs for keypoint prediction.2 R-CNN starts from bottom-up region proposal
[35], which tends to overlook the signal from small parts. OverFeat, on the other hand, combines
convnets trained for classification and for regression and runs in multi-scale sliding window fashion.
We rescale each bounding box to 500 ? 500 and compute conv5 (with a stride of 16 pixels). Each
cell of conv5 contains one 256-dimensional descriptor. We concatenate conv5 descriptors from a
local region of 3 ? 3 cells, giving an overall receptive field size of 195 ? 195 and feature dimension
of 2304. For each keypoint, we train a linear SVM with hard negative mining. We consider the ten
closest features to each ground truth keypoint as positive examples, and all the features whose rfs
do not contain the keypoint as negative examples. We also train using dense SIFT descriptors for
comparison. We compute SIFT on a grid of stride eight and bin size of eight using VLFeat [36]. For
SIFT, we consider features within twice the bin size from the ground truth keypoint to be positives,
while samples that are at least four times the bin size away are negatives.
We augment our SVM detectors with a spherical Gaussian prior over candidate locations constructed
by nearest neighbor matching. The mean of each Gaussian is taken to be the location of the keypoint
in the nearest neighbor in the training set found using cosine similarity on pool5 features, and we
use a fixed standard deviation of 22 pixels. Let s(Xi ) be the output score of our local detector for
keypoint Xi , and let p(Xi ) be the prior score. We combine these to yield a final score f (Xi ) =
s(Xi )1?? p(Xi )? , where ? ? [0, 1] is a tradeoff parameter. In our experiments, we set ? = 0.1 by
cross validation. At test time, we predict the keypoint location as the highest scoring candidate over
all feature locations.
We evaluate the predicted keypoints using the measure PCK introduced in Section 3, taking ? = 0.1.
A predicted keypoint is defined as correct if the distance between it and the ground truth keypoint is
less than ? ? max(h, w) where h and w are the height and width of the bounding box. The results
using conv5 and SIFT with and without the prior are shown in Table 4. From the table, we can see
that local part detectors trained on the conv5 feature outperform SIFT by a large margin and that the
prior information is helpful in both cases. To our knowledge, these are the first keypoint prediction
results reported on this dataset. We show example results from five different categories in Figure
6. Each set consists of rescaled bounding box images with ground truth keypoint annotations and
predicted keypoints using SIFT and conv5 features, where each color corresponds to one keypoint.
As the figure shows, conv5 outperforms SIFT, often managing satisfactory outputs despite the
challenge of this task. A small offset can be noticed for some keypoints like eyes and noses, likely
due to the limited stride of our scanning windows. A final regression or finer stride could mitigate
this issue.
6
Conclusion
Through visualization, alignment, and keypoint prediction, we have studied the ability of the intermediate features implicitly learned in a state-of-the-art convnet classifier to understand specific,
local correspondence. Despite their large receptive fields and weak label training, we have found in
all cases that convnet features are at least as useful (and sometimes considerably more useful) than
conventional ones for extracting local visual information.
Acknowledgements This work was supported in part by DARPA?s MSEE and SMISC programs, by NSF
awards IIS-1427425, IIS-1212798, and IIS-1116411, and by support from Toyota.
2
But see works cited in Section 1.1 regarding keypoint localization.
7
Table 4: Keypoint prediction results on PASCAL VOC 2011. The numbers give average accuracy
of keypoint prediction using the criterion described in Section 3, PCK with ? = 0.1.
SIFT
SIFT+prior
conv5
conv5+prior
aero
17.9
33.5
38.5
50.9
Groundtruth
bike
16.5
36.9
37.6
48.8
bird
15.3
22.7
29.6
35.1
boat
15.6
23.1
25.3
32.5
bttl
25.7
44.0
54.5
66.1
SIFT+prior
bus
21.7
42.6
52.1
62.0
car
22.0
39.3
28.6
45.7
cat
12.6
22.1
31.5
34.2
chair
11.3
18.5
8.9
21.4
cow
7.6
23.5
30.5
41.1
table
6.5
11.2
24.1
27.2
dog
12.5
20.6
23.7
29.3
horse
18.3
32.2
35.8
46.8
Groundtruth
conv5+prior
mbike
15.1
33.9
29.9
45.6
prsn
15.9
26.7
39.3
47.1
plant
21.3
30.6
38.2
42.5
sheep
14.7
25.7
30.5
38.8
SIFT+prior
sofa
15.1
26.5
24.5
37.6
train
9.2
21.9
41.5
50.7
tv
19.9
32.4
42.0
45.6
mean
15.7
28.4
33.3
42.5
conv5+prior
Figure 6: Examples of keypoint prediction on five classes of the PASCAL dataset: aeroplane, cat,
cow, potted plant, and horse. Each keypoint is associated with one color. The first column is the
ground truth annotation, the second column is the prediction result of SIFT+prior and the third
column is conv5+prior. (Best viewed in color).
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4,884 | 5,421 | Deep Learning for Real-Time Atari Game Play
Using Offline Monte-Carlo Tree Search Planning
Satinder Singh
Computer Science and Eng.
University of Michigan
[email protected]
Xiaoxiao Guo
Computer Science and Eng.
University of Michigan
[email protected]
Honglak Lee
Computer Science and Eng.
University of Michigan
[email protected]
Richard Lewis
Department of Psychology
University of Michigan
[email protected]
Xiaoshi Wang
Computer Science and Eng.
University of Michigan
[email protected]
Abstract
The combination of modern Reinforcement Learning and Deep Learning approaches holds the promise of making significant progress on challenging applications requiring both rich perception and policy-selection. The Arcade Learning
Environment (ALE) provides a set of Atari games that represent a useful benchmark set of such applications. A recent breakthrough in combining model-free
reinforcement learning with deep learning, called DQN, achieves the best realtime agents thus far. Planning-based approaches achieve far higher scores than the
best model-free approaches, but they exploit information that is not available to
human players, and they are orders of magnitude slower than needed for real-time
play. Our main goal in this work is to build a better real-time Atari game playing
agent than DQN. The central idea is to use the slow planning-based agents to provide training data for a deep-learning architecture capable of real-time play. We
proposed new agents based on this idea and show that they outperform DQN.
1
Introduction
Many real-world Reinforcement Learning (RL) problems combine the challenges of closed-loop
action (or policy) selection with the already significant challenges of high-dimensional perception
(shared with many Supervised Learning problems). RL has made substantial progress on theory
and algorithms for policy selection (the distinguishing problem of RL), but these contributions have
not directly addressed problems of perception. Deep learning (DL) approaches have made remarkable progress on the perception problem (e.g., [11, 17]) but do not directly address policy selection.
RL and DL methods share the aim of generality, in that they both intend to minimize or eliminate
domain-specific engineering, while providing ?off-the-shelf? performance that competes with or exceeds systems that exploit control heuristics and hand-coded features. Combining modern RL and
DL approaches therefore offers the potential for general methods that address challenging applications requiring both rich perception and policy-selection.
The Arcade Learning Environment (ALE) is a relatively new and widely accessible class of benchmark RL problems that provide a particularly challenging combination of policy selection and perception. ALE includes an emulator and a large number of Atari 2600 (a 1970s?80s home video
console) games. The complexity and diversity of the games?both in terms of perceptual challenges
in mapping pixels to useful features for control and in terms of the control policies needed?make
1
ALE a useful set of benchmark RL problems, especially for evaluating general methods intended to
achieve success without hand-engineered features.
Since the introduction of ALE, there have been a number of attempts to build general-purpose Atari
game playing agents. The departure point for this paper is a recent and significant breakthrough [16]
that combines RL and DL to build agents for multiple Atari Games. It achieved the best machineagent real-time game play to date (in some games close to or better than human-level play), does not
require feature engineering, and indeed reuses the same perception architecture and RL algorithm
across all the games. We believe that continued progress on the ALE environment that preserves
these advantages will extend to broad advances in other domains with significant perception and
policy selection challenges. Thus, our immediate goal in the work reported here is to build even
better performing general-purpose Atari Game playing agents. We achieve this by introducing new
methods for combining RL and DL that use slow, off-line Monte Carlo tree search planning methods
to generate training data for a deep-learned classifier capable of state-of-the-art real-time play.
2
Brief background on RL and DL and challenges of perception
RL and more broadly decision-theoretic planning has a suite of methods that address the challenge of
selecting/learning good policies, including value function approximation, policy search, and MonteCarlo Tree Search [9, 10] (MCTS). These methods have different strengths and weaknesses and there
is increasing understanding of how to match them to different types of RL-environments. Indeed,
an accumulating number of applications attest to this success. But it is still not the case that there
are reasonably off-the-shelf approaches to solving complex RL problems of interest to Artificial
Intelligence (AI) such as the games in ALE. One reason for this is that despite major advances there
hasn?t been an off-the-shelf approach to significant perception problems. The perception problem
itself has two components: 1) the sensors at any time step do not capture all the information in
the history of observations, leading to partial observability, and 2) the sensors provide very highdimensional observations that introduce computational and sample-complexity challenges for policy
selection.
One way to handle the perception challenges when a model of the RL environment is available is
to avoid the perception problem entirely by eschewing the building of an explicit policy and instead
using repeated incremental planning via MCTS methods such as UCT [10] (discussed below). Either
when a model is not available, or when an explicit representation of the policy is required, the usual
approach to applied RL success has been to use expert-developed task-specific features of a short
history of observations in combination with function approximation methods and some trial-anderror on the part of the application developer (on small enough problems this can be augmented with
some automated feature selection methods). Eliminating the dependence of applied RL success on
engineered features motivates our interest in combining RL and DL (though see [20] for early work
in this direction).
Over the past decades, deep learning (see [3, 19] for a survey) has emerged as a powerful technique
for learning feature representations from data (again, this is in a stark contrast to the conventional
way of hand-crafting features by domain experts). For example, DL has achieved state-of-the-art results in image classification [11, 4], speech recognition [15, 17, 6], and activity recognition [12, 8].
In DL, features are learned in a compositional hierarchy. Specifically, low-level features are learned
to encode low-level statistical dependencies (e.g., ?edges? in images), and higher-level features encode higher-order dependencies of the lower-level features (e.g., ?object parts?) [14]. In particular,
for data that has strong spatial or temporal dependencies, convolutional neural networks [13] have
been shown to learn invariant high-level features that are informative for supervised tasks. Such
convolutional neural networks were used in the recent successful combination of DL and RL for
Atari Game playing [16] that forms the departure point of our work. We describe this work in more
detail below.
3
Existing Work on Atari Games and a Performance Gap
While the games in ALE are simpler than many modern games, they still pose significant challenges
to human players. In RL terms, for a human player these games are Partially-Observable Markov
Decision Processes (POMDPs). The true state of each game at any given point is captured by the
2
contents of the limited random-access memory (RAM). A human player does not observe the state
and instead perceives the game screen (frame) which is a 2D array of 7-bit pixels, 160 pixels wide by
210 pixels high. The action space available to the player depends on the game but maximally consists
of the 18 discrete actions defined by the joystick controller. The next state is a deterministic function
of the previous state and the player?s action choice. Stochasticity in these games is limited to the
choice of the initial state of the game (which can include a random number seed stored in RAM).
So even though the state transitions are deterministic, the transitions from history of observations
and actions to next observation can be stochastic (because of the stochastic initial hidden state). The
immediate reward at any given step is defined by the game and made available by the ALE; it is
usually a function of the current frame or the difference between current and previous frames. When
running in real-time, the simulator generates 60 frames per second. All the games we consider
terminate in a finite number of time-steps (and so are episodic). The goal in these games is to select
an optimal policy, i.e., to select actions in such a way so as to maximize the expected value of the
cumulative sum of rewards until termination.
Model-Free RL Agents for Atari Games. Here we discuss work that does not access the state
in the games and thus solves the game as a POMDP. In principle one could learn a state representation and infer an associated MDP model using frame-observation and action trajectories, but
these games are so complex that this is rarely done. Instead, partial observability is dealt with by
hand-engineering features of short histories of frames observed so far and model-free RL methods
are used to learn good policies as a function of those feature representations. For example, the paper that introduced ALE [1], used SARSA with several different hand-engineered features sets. The
contingency awareness approach [4] improved performance of the SARSA algorithm by augmenting
the feature sets with a learned representation of the parts of the screen that are under the agent?s control. The sketch-based approach [2] further improves performance by using the tug-of-war sketch
features. HyperNEAT-GGP [7] introduces an evolutionary policy search based Atari game player.
Most recently Deep Q-Network (hereafter DQN) [16] uses a modified version of Q-Learning with a
convolutional neural network (CNN) with three hidden layers for function approximation. This last
approach is the state of the art in this class of methods for Atari games and is the basis for our work;
we present the relevant details in Section 5. It does not use hand-engineered features but instead
provides the last four raw frames as input (four instead of one to alleviate partial observability).
Planning Agents for Atari Games based on UCT. These approaches access the state of the game
from the emulator and hence face a deterministic MDP (other than the random choice of initial state).
They incrementally plan the action to take in the current state using UCT, an algorithm widely used
for games. UCT has three parameters, the number of trajectories, the maximum-depth (uniform for
each trajectory), and a exploration parameter (a scalar set to 1 in all our experiments). In general,
the larger the trajectory & depth parameters are, the slower UCT is but the better it is. UCT uses the
emulator as a model to simulate trajectories as follows. Suppose it is generating the k th trajectory
and the current node is at depth d and the current state is s. It computes a score for each possible
action a in state-depth pair (s, d) as the sum of two terms, an exploitation term that is the MonteCarlo average of the discounted sum of rewards obtained from experiences
p with state-depth pair
(s, d) in the previous k ? 1 trajectories, and an exploration term that is log (n(s, d))/n(s, a, d)
where n(s, a, d) and n(s, d) are the number of experiences of action a with state-depth pair (s, d)
and with state-depth pair (s, d) respectively in the previous k ? 1 trajectories. UCT selects the
action to simulate in order to extend the trajectory greedily with respect to this summed score. Once
the input-parameter number of trajectories are generated each to maximum depth, UCT returns the
exploitation term for each action at the root node (which is the current state it is planning an action
for) as its estimate of the utility of taking that action in the current state of the game. UCT has the
nice theoretical property that the number of simulation steps (number of trajectories ? maximumdepth) needed to ensure any bound on the loss of following the UCT-based policy is independent of
the size of the state space; this result expresses the fact that the use of UCT avoids the perception
problem, but at the cost of requiring substantial computation for every time step of action selection
because it never builds an explicit policy.
Performance Gap & our Opportunity. The opportunity for this paper arises from the following
observations. The model-free RL agents for Atari games are fast (indeed faster than real-time, e.g.,
the CNN-based approach from our paper takes 10?4 seconds to select an action on our computer)
while the UCT-based planning agents are several orders of magnitude slower (much slower than
real-time, e.g., they take seconds to select an action on the same computer). On the other hand,
3
the performance of UCT-based planning agents is much better than the performance of model-free
RL agents (this will be evident in our results below). Our goal is to develop methods that retain
the DL advantage of not needing hand crafted features and the online real-time play ability of the
model-free RL agents by exploiting data generated by UCT-planning agents.
4
Methods for Combining UCT-based RL with DL
We first describe the baseline UCT agent, and then three agents that instantiate different methods of
combining the UCT agent with DL. Recall that in keeping with the goal of building general-purpose
methods as in the DQN work we impose the constraint of reusing the same input representations,
the same function approximation architecture, and the same planning method for all the games.
4.1
Baseline UCT agent that provides training data
This agent requires no training. It does, however, require specification of its two parameters, the
number of trajectories and the maximum-depth. Recall that our proposed new agents will all use
data from this UCT-agent to train a CNN-based policy and so it is reasonable that the resulting
performance of our proposed agents will be worse than that of the UCT-agent. Therefore, in our
experiments we set these two parameters large enough to ensure that they outscore the published
DQN scores, but not so large that they make our computational experiments unreasonably slow.
Specifically, we elected to use 300 as maximum-depth and 10000 as number of trajectories for all
games but two. Pong turns out to be a much simpler game and we could reduce the number of
trajectories to 500, and Enduro turned out to have more distal rewards than the other games and so
we used a maximum-depth of 400. As will be evident from the results in Section 5 this allowed the
UCT agent to significantly outperform DQN in all games but Pong in which DQN already performs
perfectly. We emphasize that the UCT agent does not meet our goal of real-time play. For example,
to play a game just 800 times with the UCT agent (we do this to collect training data for our agent?s
below) takes a few days on a recent multicore computer for each game.
4.2
Our three methods and their corresponding agents
Method 1: UCTtoRegression (for UCT to CNN via Regression). The key idea is to use the action
values computed by the UCT-agent to train a regression-based CNN. The following is done for each
game. Collect 800 UCT-agent runs by playing the game 800 times from start to finish using the UCT
agent above. Build a dataset (table) from these runs as follows. Map the last four frames of each state
along each trajectory into the action-values of all the actions as computed by UCT. This training data
is used to train the CNN via regression (see below for CNN details). The UCTtoRegression-agent
uses the CNN learned by this training procedure to select actions during evaluation.
Method 2: UCTtoClassification (for UCT to CNN via Classification). The key idea is to use the
action choice computed by the UCT-agent (selected greedily from action-values) to train a classifierbased CNN. The following is done for each game. Collect 800 UCT-agent runs as above. These runs
yield a table in which the rows correspond to the last four frames at each state along each trajectory
and the single column is the choice of action that is best according to the UCT-agent at that state of
the trajectory. This training data is used to train the CNN via multinomial classification (see below
for CNN details). The UCTtoClassification-agent uses the CNN-classifier learned by this training
procedure to select actions during evaluation.
One potential issue with the above two agents is that the training data?s input distribution is generated
by the UCT-agent while during testing the UCTtoRegression and UCTtoClassification agents will
perform differently from the UCT-agent and thus could experience an input distribution quite difference from that of the UCT-agent?s. This could limit the testing performance of the UCTtoRegression
and UCTtoClassification agents. Thus, it might be desirable to somehow bias the distribution over
inputs to those likely to be encountered by these agents; this observation motivates our next method.
Method 3: UCTtoClassification-Interleaved (for UCT to CNN via Classification-Interleaved).
The key idea is to focus UCT planning on that part of the state space experienced by the (partially
trained) CNN player. The method accomplishes this by interleaving training and data collection as
4
84
84
fully-connectedlayer (max(0,x))
conv-layer
(tanh)
conv-layer
(tanh)
20
4
20
9
9
16
32
fully-connectedlayer (linear)
256
Figure 1: The CNN architecture from DQN [6] that we adopt in our agents. See text for details.
follows1 . Collect 200 UCT-agent runs as above; these will obviously have the same input distribution
concern raised above. The data from these runs is used to train the CNN via multinomial classification just as in the UCTtoClassification-agent?s method (we do not do this for the UCTtoRegressionagent because as we show below it performs worse than the UCTtoClassification-agent). The trained
CNN is then used to decide action choices in collecting a further 200 runs (though 5% of the time
a random action is chosen to ensure some exploration). At each state of the game along each trajectory, UCT is asked to compute its choice of action and the original data set is augmented with
the last four frames for each state as the rows and the column as UCT?s action choice. This 400
trajectory dataset?s input distribution is now potentially different from that of the UCT-agent. This
dataset is used to train the CNN again via multinomial classification. This interleaved procedure is
repeated until there are a total of 800 runs worth of data in the dataset for the final round of training
of the CNN. The UCTtoClassification-Interleaved agent uses the final CNN-classifier learned by this
training procedure to select actions during testing.
In order to focus our empirical evaluation on the contribution of the non-DL part of our three new
agents, we reused exactly the same convolutional neural network architecture as used in the DQN
work (we describe this architecture in brief detail below). The DQN work modified the reward
functions for some of the games (by saturating them at +1 and ?1) while we use unmodified reward
functions (these only play a role in the UCT-agent components of our methods and not in the CNN
component). We also follow DQN?s frame-skipping techniques: the agent sees and selects actions
on every k th frame instead of every frame (k = 3 for Space Invaders and k = 4 for all other games),
and the latest chosen-action is repeated on subsequently-skipped frames.
4.3
Details of Data Preprocessing and CNN Architecture
Preprocessing (identical to DQN to the best of our understanding). Raw Atari game frames are
160 ? 210 pixel images with a 128-color palette. We convert the RGB representation to gray-scale
and crop an 160 ? 160 region of the image that captures the playing area, and then the cropped
image is down-sampled to 84 ? 84 in order to reuse DQN?s CNN architecture. This procedure is
applied to the last 4 frames associated with a state and stacked to produce a 84 ? 84 ? 4 preprocessed
input representation for each state. We subtracted the pixel-level means and scale the inputs to lie in
the range [-1, 1]. We shuffle the training data to break the strong correlations between consecutive
samples, which therefore reduces the variance of the updates.
CNN Architecture. We use the same deep neural network architecture as DQN [16] for our agents.
As depicted in Figure 1, our network consists of three hidden layers. The input to the neural network
is an 84 ? 84 ? 4 image produced by the preprocessing procedure above. The first hidden layer
convolves 16, 8 ? 8, filters with stride 4 with the input image and applies a rectifier nonlinearity
(tanh). The second hidden layer convolves 32, 4 ? 4, filters with stride 2 again followed by a
rectifier nonlinearity (tanh). The final hidden layer is fully connected and consists of 256 rectifier
(max) units. In the multi-regression-based agent (UCTtoRegression), the output layer is a fully
connected linear layer with a single output for each valid action. In the classification-based agents
(UCTtoClassification, UCTtoClassification-Interleaved), a softmax (instead of linear) function is
applied to the final output layer. We refer the reader to the DQN paper for further detail.
1
Our UCTtoClassification-Interleaved method is a special case of DAgger [18] (in the use of a CNNclassifier and in the use of specific choices of parameters ?1 = 1, and for i > 1, ?i = 0). As a small
point of difference, we note that our emphasis in this paper was in the use of CNNs to avoid the use of handcrafted domain specific features, while the empirical work for DAgger did not have the same emphasis and so
used handcrafted features.
5
Table 1: Performance (game scores) of the four real-time game playing agents, where UCR is short for UCTtoRegression, UCC is short for UCTtoClassification, and UCC-I is short for UCTtoClassification-Interleaved.
Agent
B.Rider
Breakout
Enduro
Pong
Q*bert
Seaquest
S.Invaders
DQN
-best
4092
5184
168
225
470
661
20
21
1952
4500
1705
1740
581
1075
UCC
-best
-greedy
5342 (20)
10514
5676
175(5.63)
351
269
558(14)
942
692
19(0.3)
21
21
11574(44)
29725
19890
2273(23)
5100
2760
672(5.3)
1200
680
UCC-I
-best
-greedy
5388(4.6)
10732
5702
215(6.69)
413
380
601(11)
1026
741
19(0.14)
21
21
13189(35.3)
29900
20025
2701(6.09)
6100
2995
670(4.24)
910
692
UCR
2405(12)
143(6.7)
566(10.2)
19(0.3)
12755(40.7)
1024 (13.8)
441(8.1)
Table 2: Performance (game scores) of the off-line UCT game playing agent.
5
Agent
B.Rider
Breakout
Enduro
Pong
Q*bert
Seaquest
S.Invaders
UCT
7233
406
788
21
18850
3257
2354
Experimental Results
First we present our main performance results and then present some visualizations to help understand the performance of our agents. In Table 1 we compare and contrast the performance of the
four real-time game playing agents, three of which (UCTtoRegression, UCTtoClassification, and
UCTtoClassification-Interleaved) we implemented and evaluated; the performance of the DQN was
obtained from [16].
The columns correspond to the seven games named in the header, and the rows correspond to different assessments of the four agents. Throughout the numbers in parentheses are standard-errors.
The DQN row reports the average performance (game score) of the DQN agent (a random action is
chosen 5% of the time during testing). The DQN-best row is the best performance of the DQN over
all the attempts at each game incorporated in the row corresponding to DQN. Comparing the performance of the UCTtoClassification and UCTtoRegression agents (both use 5% exploration), we see
that the UCTtoClassification agent either competes well with or significantly outperforms the UCTtoRegression agent. More importantly the UCTtoClassification agent outperforms the DQN agent
in all games but Pong (in which both agents do nearly perfectly because the maximum score in this
game is 21). In some games (B.Rider, Enduro, Q*Bert, Sequest and S.Invaders) the percentageperformance gain of UCTtoClassification over DQN is quite large. Similar gains are obtained in the
comparison of UCTtoClassification-best to DQN-best.
We used 5% exploration in our agents to match what the DQN agent does, but it is not clear why
one should consider random action selection during testing. In any case, the effect of this randomness in action-selection will differ across games (based, e.g., on whether a wrong action can
be terminal). Thus, we also present results for the UCTtoClassification-greedy agent in which we
don?t do any exploration. As seen by comparing the rows corresponding to UCTtoClassification
and UCTtoClassification-greedy, the latter agent always outperforms the former and in four games
(Breakout, Enduro, Q*Bert, and Seaquest) achieves further large-percentage improvements.
Table 2 gives the performance of our non-realtime UCT agent (again, with 5% exploration). As
discussed above we selected UCT-agent?s parameters to ensure that this agent outperforms the DQN
agent allowing room for our agents to perform in the middle.
Finally, recall that the UCTtoClassification-Interleaved agent was designed so that its input distribution during training is more likely to match its input distribution during evaluation and we hypothesized that this would improve performance relative to UCTtoClassification. Indeed, in all games but
B. Rider, Pong and S.Invaders in which the two agents perform similarly, UCTtoClassificationInterleaved significantly outperforms UCTtoClassification. The same holds when comparing
6
frame: t-3
t-2
t-1
t
?submarine?
?diver?
?enemy?
?enemy+diver?
Figure 2: Visualization of the first-layer features learned from Seaquest. (Left) visualization of four first-layer
filters; each filter covers four frames, showing the spatio-temporal template. (Middle) a captured screen. (Right)
gray-scale version of the input screen which is fed into the CNN. Four filters were color-coded and visualized
as dotted bounding boxes at the locations where they get activated. This figure is best viewed in color.
UCTtoClassification-Interleaved-best and UCTtoClassification-best as well as UCTtoClassificationInterleaved-greedy and UCTtoClassification-greedy.
In a further preliminary exploration of the effectiveness of the UCTtoClassification-Interleaved
in exploiting additional computational resources for generating UCT runs, on the game Enduro
we compared UCTtoClassification and UCTtoClassification-Interleaved where we allowed each of
them twice the number of UCT runs used in producing the Table 1 above, i.e., 1600 runs while
keeping a batch size of 200. The performance of UCTtoClassification improves from 558 to 581
while the performance of UCTtoClassification-Interleaved improves from 601 to 670, i.e., the interleaved method improved more in absolute and percentage terms as we increased the amount of
training data. This is encouraging and is further confirmation of the hypothesis that motivated the
interleaved method, because the interleaved input distribution would be even more like that of the
final agent with the larger data set.
Learned Features from Convolutional Layers. We
provide visualizations of the learned filters in order to
gain insights on what the CNN learns. Specifically, we
apply the ?optimal stimuli? method [5] to visualize the
features CNN learned after training. The method picks
the input image patches that generate the greatest responsive after convolution with the trained filters. We
select 8*8*4 input patches to visualize the first convolutional layer features and 20*20*4 to visualize the second convolutional layer filters. Note that these patch
sizes correspond to receptive field sizes of the learned
features in each layer.
In Figure 2, we show four first-layer filters of the CNN
trained from Seaquest for UCTtoClassification-agent.
Specifically, each filter covers four frames of 8*8 pixels, which can be viewed as a spatio-temporal tem- Figure 3: Visualization of the second-layer
plate that captures specific patterns and their temporal features learned from Seaquest.
changes. We also show an example screen capture and
visualize where the filters get activated in the gray-scale version of the image (which is the actual
input to the CNN model). The visualization suggests that the first-layer filters capture ?object-part?
patterns and their temporal movements.
Figure 3 visualizes the four second-layer features via the optimal stimulus method, where each row
corresponds to a filter. We can see that the second-layer features capture bigger spatial patterns
(often covering beyond the size of individual objects), while encoding interactions between objects,
such as two enemies moving together, and submarine moving along a direction. Overall, these
qualitative results suggest that the CNN learns relevant patterns useful for game playing.
7
Step 69: FIRE
Step 70: DOWN+FIRE
Step 74:DOWN+FIRE
Step 75:RIGHT+FIRE
Step 76:RIGHT+FIRE
Step 78: RIGHT+FIRE
Step 79:DOWN+FIRE
Figure 4: A visualization of the UCTtoClassification agent?s policy as it kills an enemy agent.
Visualization of Learned Policy. Here we present visualizations of the policy learned by the UCTtoClassification agent with the aim of illustrating both what it does well and what it does not.
Figure 4 shows the policy learned by UCTtoClassification to destroy nearby enemies. The CNN
changes the action from ?Fire? to ?Down+Fire? at time step 70 when the enemies first show up at
the right columns of the screen, which will move the submarine to the same horizontal position of the
closest enemy. At time step 75, the submarine is at the horizontal position of the closest enemy and
the action changes to ?Right+Fire?. The ?Right+Fire? action is repeated until the enemy is destroyed
at time step 79. At time step 79, the predicted action is changed to ?Down+Fire? again to move the
submarine to the horizontal position of the next closest enemy. This shows the UCTtoClassification
agent?s ability to deal with delayed reward as it learns to take a sequence of unrewarded actions
before it obtains any reward when it finally destroys an enemy.
Figure 4 also shows a shortcoming in the UCTtoClassification agent?s policy, namely it does not
purposefully take actions to save a diver (saving a diver can lead to a large reward). For example, at
time step 69, even though there are two divers below and to the right of the submarine (our agent), the
learned policy does not move the submarine downward. This phenomenon was observed frequently.
The reason for this shortcoming is that it can take a large number of time steps to capture 6 divers
and bring them to surface (bringing fewer divers to the surface does not yield a reward); this takes
longer than the planning depth of UCT. Thus, it is UCT that does not purposefully save divers and
thus the training data collected via UCT reflects that defect which is then also present in the play of
the UCTtoClassification (and UCTtoClassification-Interleaved) agent.
6
Conclusion
UCT-based planning agents are unrealistic for Atari game play in at least two ways. First, to play
the game they require access to the state of the game which is unavailable to human players, and
second they are orders of magnitude slower than realtime. On the other hand, by slowing the game
down enough to allow UCT to play leads to the highest scores on the games they have been tried on.
Indeed, by allowing UCT more and more time (and thus allowing for larger number of trajectories
and larger maximum-depth) between moves one can presumably raise the score more and more. We
identified a gap between the UCT-based planning agents performance and the best realtime player
DQN?s performance and developed new agents to partially fill this gap. Our main applied result is
that at the time of the writing of this paper we have the best realtime Atari game playing agents
on the same 7 games that were used to evaluate DQN. Indeed, in most of the 7 games our best
agent beats DQN significantly. Another result is that at least in our experiments training the CNN to
learn a classifier that maps game observations to actions was better than training the CNN to learn
a regression function that maps game observations to action-values (we intend to do further work
to confirm how general this result is on ALE). Finally, we hypothesized that the difference in input
distribution between the UCT agent that generates the training data and the input distribution experienced by our learned agents would diminish performance. The UCTtoClassification-Interleaved
agent we developed to deal with this issue indeed performed better than the UCTtoClassification
agent indirectly confirming our hypothesis and solving the underlying issue.
Acknowledgments. This work was supported in part by NSF grant IIS-1148668. Any opinions,
findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.
8
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9
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4,885 | 5,422 | On the Number of Linear Regions of
Deep Neural Networks
Guido Mont?ufar
Max Planck Institute for Mathematics in the Sciences
[email protected]
Razvan Pascanu
Universit?e de Montr?eal
[email protected]
Yoshua Bengio
Universit?e de Montr?eal, CIFAR Fellow
[email protected]
Kyunghyun Cho
Universit?e de Montr?eal
[email protected]
Abstract
We study the complexity of functions computable by deep feedforward neural networks
with piecewise linear activations in terms of the symmetries and the number of linear
regions that they have. Deep networks are able to sequentially map portions of each
layer?s input-space to the same output. In this way, deep models compute functions
that react equally to complicated patterns of different inputs. The compositional
structure of these functions enables them to re-use pieces of computation exponentially
often in terms of the network?s depth. This paper investigates the complexity of such
compositional maps and contributes new theoretical results regarding the advantage
of depth for neural networks with piecewise linear activation functions. In particular,
our analysis is not specific to a single family of models, and as an example, we employ
it for rectifier and maxout networks. We improve complexity bounds from pre-existing
work and investigate the behavior of units in higher layers.
Keywords: Deep learning, neural network, input space partition, rectifier, maxout
1 Introduction
Artificial neural networks with several hidden layers, called deep neural networks, have become popular
due to their unprecedented success in a variety of machine learning tasks (see, e.g., Krizhevsky et al.
2012, Ciresan et al. 2012, Goodfellow et al. 2013, Hinton et al. 2012). In view of this empirical evidence,
deep neural networks are becoming increasingly favored over shallow networks (i.e., with a single layer
of hidden units), and are often implemented with more than five layers. At the time being, however, the
theory of deep networks still poses many questions. Recently, Delalleau and Bengio (2011) showed that
a shallow network requires exponentially many more sum-product hidden units1 than a deep sum-product
network in order to compute certain families of polynomials. We are interested in extending this kind
of analysis to more popular neural networks, such as those with maxout and rectifier units.
There is a wealth of literature discussing approximation, estimation, and complexity of artificial neural
networks (see, e.g., Anthony and Bartlett 1999). A well-known result states that a feedforward neural
network with a single, huge, hidden layer is a universal approximator of Borel measurable functions (see
Hornik et al. 1989, Cybenko 1989). Other works have investigated universal approximation of probability
distributions by deep belief networks (Le Roux and Bengio 2010, Mont?ufar and Ay 2011), as well as
their approximation properties (Mont?ufar 2014, Krause et al. 2013).
These previous theoretical results, however, do not trivially apply to the types of deep neural networks
that have seen success in recent years. Conventional neural networks often employ either hidden units
1
A single sum-product hidden layer summarizes a layer of product units followed by a layer of sum units.
1
Figure 1: Binary classification using a shallow model with 20 hidden units (solid line) and a deep model
with two layers of 10 units each (dashed line). The right panel shows a close-up of the left panel. Filled
markers indicate errors made by the shallow model.
with a bounded smooth activation function, or Boolean hidden units. On the other hand, recently it has
become more common to use piecewise linear functions, such as the rectifier activation g(a) = max{0, a}
(Glorot et al. 2011, Nair and Hinton 2010) or the maxout activation g(a1, . . . , ak ) = max{a1, . . . , ak }
(Goodfellow et al. 2013). The practical success of deep neural networks with piecewise linear units calls
for the theoretical analysis specific for this type of neural networks.
In this respect, Pascanu et al. (2013) reported a theoretical result on the complexity of functions computable
by deep feedforward networks with rectifier units. They showed that, in the asymptotic limit of many
hidden layers, deep networks are able to separate their input space into exponentially more linear response
regions than their shallow counterparts, despite using the same number of computational units.
Building on the ideas from Pascanu et al. (2013), we develop a general framework for analyzing deep
models with piecewise linear activations. We describe how the intermediary layers of these models
are able to map several pieces of their inputs into the same output. The layer-wise composition of the
functions computed in this way re-uses low-level computations exponentially often as the number of
layers increases. This key property enables deep networks to compute highly complex and structured
functions. We underpin this idea by estimating the number of linear regions of functions computable by
two important types of piecewise linear networks: with rectifier units and with maxout units. Our results
for the complexity of deep rectifier networks yield a significant improvement over the previous results
on rectifier networks mentioned above, showing a favorable behavior of deep over shallow networks even
with a moderate number of hidden layers. Furthermore, our analysis of deep rectifier and maxout networks
provides a platform to study a broad variety of related networks, such as convolutional networks.
The number of linear regions of the functions that can be computed by a given model is a measure of the
model?s flexibility. An example of this is given in Fig. 1, which compares the learned decision boundary of a
single-layer and a two-layer model with the same number of hidden units (see details in the Supplementary
Material). This illustrates the advantage of depth; the deep model captures the desired boundary more
accurately, approximating it with a larger number of linear pieces. As noted earlier, deep networks are able
to identify an exponential number of input neighborhoods by mapping them to a common output of some
intermediary hidden layer. The computations carried out on the activations of this intermediary layer are
replicated many times, once in each of the identified neighborhoods. This allows the networks to compute
very complex looking functions even when they are defined with relatively few parameters. The number
of parameters is an upper bound for the dimension of the set of functions computable by a network, and
a small number of parameters means that the class of computable functions has a low dimension. The
set of functions computable by a deep feedforward piecewise linear network, although low dimensional,
achieves exponential complexity by re-using and composing features from layer to layer.
2 Feedforward Neural Networks and their Compositional Properties
In this section we discuss the ability of deep feedforward networks to re-map their input-space to create
complex symmetries by using only relatively few computational units. The key observation of our analysis
is that each layer of a deep model is able to map different regions of its input to a common output. This
leads to a compositional structure, where computations on higher layers are effectively replicated in all
input regions that produced the same output at a given layer. The capacity to replicate computations over
the input-space grows exponentially with the number of network layers. Before expanding these ideas, we
introduce basic definitions needed in the rest of the paper. At the end of this section, we give an intuitive
perspective for reasoning about the replicative capacity of deep models.
2
2.1 Definitions
A feedforward neural network is a composition of layers of computational units which defines a function
F : Rn0 ? Rout of the form
F (x; ?) = fout ? gL ? fL ? ? ? ? ? g1 ? f1(x),
(1)
where fl is a linear preactivation function and gl is a nonlinear activation function. The parameter ? is
composed of input weight matrices Wl ? Rk?nl ?nl?1 and bias vectors bl ? Rk?nl for each layer l ? [L].
>
The output of the l-th layer is a vector xl = [xl,1, . . . , xl,nl ] of activations xl,i of the units i ? [nl ] in
that layer. This is computed from the activations of the preceding layer by xl = gl (fl (xl?1)). Given the
activations xl?1 of the units in the (l ? 1)-th layer, the preactivation of layer l is given by
fl (xl?1) = Wlxl?1 + bl ,
>
where fl = [fl,1, . . . , fl,nl ] is an array composed of nl preactivation vectors fl,i ? Rk , and the activation
of the i-th unit in the l-th layer is given by
xl,i = gl,i(fl,i(xl?1)).
We will abbreviate gl ? fl by hl . When the layer index l is clear, we will drop the corresponding subscript.
We are interested in piecewise linear activations, and will consider the following two important types.
? Rectifier unit:
gi(fi) = max {0, fi}, where fi ? R and k = 1.
? Rank-k maxout unit: gi(fi) = max{fi,1, . . . , fi,k }, where fi = [fi,1, . . . , fi,k ] ? Rk .
The structure of the network refers to the way its units are arranged. It is specified by the number n0 of
input dimensions, the number of layers L, and the number of units or width nl of each layer.
We will classify the functions computed by different network structures, for different choices of parameters,
in terms of their number of linear regions. A linear region of a piecewise linear function F : Rn0 ? Rm
is a maximal connected subset of the input-space Rn0 , on which F is linear. For the functions that we
consider, each linear region has full dimension, n0.
2.2 Shallow Neural Networks
Rectifier units have two types of behavior; they can be either constant 0 or linear, depending on their
inputs. The boundary between these two behaviors is given by a hyperplane, and the collection of all
the hyperplanes coming from all units in a rectifier layer forms a hyperplane arrangement. In general,
if the activation function g : R ? R has a distinguished (i.e., irregular) behavior at zero (e.g., an inflection
point or non-linearity), then the function Rn0 ? Rn1 ; x ?
7 g(Wx + b) has a distinguished behavior at
all inputs from any of the hyperplanes Hi := {x ? Rn0 : Wi,:x + bi = 0} for i ? [n1]. The hyperplanes
capturing this distinguished behavior also form a hyperplane arrangement (see, e.g., Pascanu et al. 2013).
The hyperplanes in the arrangement split the input-space into several regions. Formally, a region of a
hyperplane arrangement {H1, . . . , Hn1 } is a connected component of the complement Rn0 \ (?iHi),
i.e., a set of points delimited by these hyperplanes (possibly open towards infinity). The number of regions
of an arrangement can be given in terms of a characteristic function of the arrangement, as P
shown in a
n0
n1
well-known result by Zaslavsky (1975). An arrangement of n1 hyperplanes in Rn0 has at most j=0
j
regions. Furthermore, this number of regions is attained if and only if the hyperplanes are in general
position. This implies that the maximal number of linear
of functions computed by a shallow
Pn0 regions
n1
rectifier network with n0 inputs and n1 hidden units is j=0
j (see Pascanu et al. 2013; Proposition 5).
2.3 Deep Neural Networks
We start by defining the identification of input neighborhoods mentioned in the introduction more formally:
Definition 1. A map F identifies two neighborhoods S and T of its input domain if it maps them to a common subset F (S) = F (T ) of its output domain. In this case we also say that S and T are identified by F .
Example 2. The four quadrants of 2-D Euclidean space are regions that are identified by the absolute
>
value function g : R2 ? R2; (x1, x2) ?
7 [|x1|, |x2|] .
3
3.
2. Fold along the
horizontal axis
1. Fold along the
vertical axis
(a)
Input Space
First Layer Space
S40 S10
S30 S20
S10 S40
S20 S30
S40 S10
S4 S1
S3 S2
S20
S10
S30
S40
S30 S20
S30
S40
S20
S10
Second Layer
Space
(b)
(c)
Figure 2: (a) Space folding of 2-D Euclidean space along the two coordinate axes. (b) An illustration of
how the top-level partitioning (on the right) is replicated to the original input space (left). (c) Identification
of regions across the layers of a deep model.
The computation carried out by the l-th layer of a feedforward network on a set of activations from the
(l ? 1)-th layer is effectively carried out for all regions of the input space that lead to the same activations
of the (l ? 1)-th layer. One can choose the input weights and biases of a given layer in such a way that
the computed function behaves most interestingly on those activation values of the preceding layer which
have the largest number of preimages in the input space, thus replicating the interesting computation many
times in the input space and generating an overall complicated-looking function.
For any given choice of the network parameters, each hidden layer l computes a function hl = gl ? fl on
the output activations of the preceding layer. We consider the function Fl : Rn0 ? Rnl ; Fl := hl ? ? ? ? ? h1
that computes the activations of the l-th hidden layer. We denote the image of Fl by Sl ? Rnl , i.e., the
set of (vector valued) activations reachable by the l-th layer for all possible inputs. Given a subset R ? Sl ,
? 1, . . . , R
? k ? Sl?1 that are mapped by hl onto R; that is, subsets
we denote by PRl the set of subsets R
? 1) = ? ? ? = hl (R
? k ) = R. See Fig. 2 for an illustration.
that satisfy hl (R
The number of separate input-space neighborhoods that are mapped to a common neighborhood
R ? Sl ? Rnl can be given recursively as
X
NRl =
NRl?1
NR0 = 1, for each region R ? Rn0 .
(2)
0 ,
l
R0 ?PR
For example, PR1 is the set of all disjoint input-space neighborhoods whose image by the function
computed by the first layer, h1 : x ?
7 g(Wx + b), equals R ? S1 ? Rn1 .
The recursive formula (2) counts the number of identified sets by moving along the branches of a tree
rooted at the set R of the j-th layer?s output-space (see Fig. 2 (c)). Based on these observations, we can
estimate the maximal number of linear regions as follows.
Lemma 3. The maximal number of linear regionsP
of the functions computed by an L-layer neural network
with piecewise linear activations is at least N = R?P L NRL?1, where NRL?1 is defined by Eq. (2), and
P L is a set of neighborhoods in distinct linear regions of the function computed by the last hidden layer.
Here, the idea to construct a function with many linear regions is to use the first L ? 1 hidden layers to
identify many input-space neighborhoods, mapping all of them to the activation neighborhoods P L of
the (L ? 1)-th hidden layer, each of which belongs to a distinct linear region of the last hidden layer. We
will follow this strategy in Secs. 3 and 4, where we analyze rectifier and maxout networks in detail.
2.4 Identification of Inputs as Space Foldings
In this section, we discuss an intuition behind Lemma 3 in terms of space folding. A map F that identifies
two subsets S and S 0 can be considered as an operator that folds its domain in such a way that the two
4
Figure 3: Space folding of 2-D space in a non-trivial way. Note how the folding can potentially identify
symmetries in the boundary that it needs to learn.
subsets S and S 0 coincide and are mapped to the same output. For instance, the absolute value function
g : R2 ? R2 from Example 2 folds its domain twice (once along each coordinate axis), as illustrated
in Fig. 2 (a). This folding identifies the four quadrants of 2-D Euclidean space. By composing such
operations, the same kind of map can be applied again to the output, in order to re-fold the first folding.
Each hidden layer of a deep neural network can be associated with a folding operator. Each hidden layer
folds the space of activations of the previous layer. In turn, a deep neural network effectively folds its
input-space recursively, starting with the first layer. The consequence of this recursive folding is that
any function computed on the final folded space will apply to all the collapsed subsets identified by the
map corresponding to the succession of foldings. This means that in a deep model any partitioning of
the last layer?s image-space is replicated in all input-space regions which are identified by the succession
of foldings. Fig. 2 (b) offers an illustration of this replication property.
Space foldings are not restricted to foldings along coordinate axes and they do not have to preserve lengths.
Instead, the space is folded depending on the orientations and shifts encoded in the input weights W and
biases b and on the nonlinear activation function used at each hidden layer. In particular, this means that the
sizes and orientations of identified input-space regions may differ from each other. See Fig. 3. In the case
of activation functions which are not piece-wise linear, the folding operations may be even more complex.
2.5 Stability to Perturbation
Our bounds on the complexity attainable by deep models (Secs. 3 and 4) are based on suitable choices
of the network weights. However, this does not mean that the indicated complexity is only attainable
in singular cases. The parametrization of the functions computed by a neural network is continuous.
More precisely, the map ? : RN ? C(Rn0 ; RnL ); ? ?
7 F? , which maps input weights and biases
n0
? = {Wi, bi}L
? RnL computed by the network, is continuous.
i=1 to the continuous functions F? : R
Our analysis considers the number of linear regions of the functions F? . By definition, each linear region
contains an open neighborhood of the input-space Rn0 . Given any function F? with a finite number
of linear regions, there is an > 0 such that for each -perturbation of the parameter ?, the resulting
function F?+ has at least as many linear regions as F? . The linear regions of F? are preserved under
small perturbations of the parameters, because they have a finite volume.
If we define a probability density on the space of parameters, what is the probability of the event that
the function represented by the network has a given number of linear regions? By the above discussion,
the probability of getting a number of regions at least as large as the number resulting from any particular
choice of parameters (for a uniform measure within a bounded domain) is nonzero, even though it may be
very small. This is because there exists an epsilon-ball of non-zero volume around that particular choice of
parameters, for which at least the same number of linear regions is attained. For example, shallow rectifier
networks generically attain the maximal number of regions, even if in close vicinity of any parameter
choice there may be parameters corresponding to functions with very few regions.
For future work it would be interesting to study the partitions of parameter space RN into pieces where
the resulting functions partition their input-spaces into isomorphic linear regions, and to investigate how
many of these pieces of parameter space correspond to functions with a given number of linear regions.
2.6 Empirical Evaluation of Folding in Rectifier MLPs
We empirically examined the behavior of a trained MLP to see if it folds the input-space in the way described
above. First, we note that tracing the activation of each hidden unit in this model gives a piecewise linear
map Rn0 ? R (from inputs to activation values of that unit). Hence, we can analyze the behavior of each
5
h2
?
h(x)
3
2
1
h1
0
1
h3
h1 ? h2 + h3
x
2
h1 ? h2
Figure 4: Folding of the real line into equal-length segments by a sum of rectifiers.
unit by visualizing the different weight matrices corresponding to the different linear pieces of this map. The
weight matrix of one piece of this map can be found by tracking the linear piece used in each intermediary
layer, starting from an input example. This visualization technique, a byproduct of our theoretical analysis,
is similar to the one proposed by Zeiler and Fergus (2013), but is motivated by a different perspective.
After computing the activations of an intermediary hidden unit for each training example, we can, for
instance, inspect two examples that result in similar levels of activation for a hidden unit. With the linear
maps of the hidden unit corresponding to the two examples we perturb one of the examples until it results
in exactly the same activation. These two inputs then can be safely considered as points in two regions
identified by the hidden unit. In the Supplementary Material we provide details and examples of this
visualization technique. We also show inputs identified by a deep MLP.
3 Deep Rectifier Networks
In this section we analyze deep neural networks with rectifier units, based on the general observations
from Sec. 2. We improve upon the results by Pascanu et al. (2013), with a tighter lower-bound on the
maximal number of linear regions of functions computable by deep rectifier networks. First, let us note the
following upper-bound, which follows directly from the fact that each linear region of a rectifier network
corresponds to a pattern of hidden units being active:
Proposition 4. The maximal number of linear regions of the functions computed by any rectifier network
with a total of N hidden units is bounded from above by 2N .
3.1 Illustration of the Construction
Consider a layer of n rectifiers with n0 input variables, where n ? n0. We partition the set of rectifier
units into n0 (non-overlapping) subsets of cardinality p = b n/n0 c and ignore the remainder units. Consider
the units in the j-th subset. We can choose their input weights and biases such that
h1(x) = max {0, wx} ,
h2(x) = max {0, 2wx ? 1} ,
h3(x) = max {0, 2wx ? 2} ,
..
.
hp(x) = max {0, 2wx ? (p ? 1)} ,
where w is a row vector with j-th entry equal to 1 and all other entries set to 0. The product wx selects
the j-th coordinate of x. Adding these rectifiers with alternating signs, we obtain following scalar function:
?j (x) = 1, ?1, 1, . . . , (?1)p?1 [h1(x), h2(x), h3(x), . . . , hp(x)]> .
h
(3)
?j acts only on the j-th input coordinate, we may redefine it to take a scalar input, namely the
Since h
j-th coordinate of x. This function has p linear regions given by the intervals (??, 0], [0, 1], [1, 2],
?j onto the interval (0, 1), as
. . . , [p ? 1, ?). Each of these intervals has a subset that is mapped by h
?
illustrated in Fig. 4. The function hj identifies the input-space strips with j-th coordinate xj restricted to
?=
the intervals (0, 1), (1, 2), . . . , (p ? 1, p). Consider now all the n0 subsets of rectifiers and the function h
>
?
?
?
h1, h2, . . . , hp . This function is locally symmetric about each hyperplane with a fixed j-th coordinate
6
equal to xj = 1, . . . , xj = p ? 1 (vertical lines in Fig. 4), for all j = 1, . . . , n0. Note the periodic pattern
? identifies a total of pn0 hypercubes delimited by these hyperplanes.
that emerges. In fact, the function h
? arises from h by composition with a linear function (alternating sums). This linear
Now, note that h
? as
function can be effectively absorbed in the preactivation function of the next layer. Hence we can treat h
being the function computed by the current layer. Computations by deeper layers, as functions of the unit
hypercube output of this rectifier layer, are replicated on each of the pn0 identified input-space hypercubes.
3.2 Formal Result
We can generalize the construction described above to the case of a deep rectifier network with n0 inputs
and L hidden layers of widths ni ? n0 for all i ? [L]. We obtain the following lower bound for the
maximal number of linear regions of deep rectifier networks:
Theorem 5. The maximal number of linear regions of the functions computed by a neural network with
n0 input units and L hidden layers, with ni ? n0 rectifiers at the i-th layer, is lower bounded by
! n
L?1
0
Y ni n0 X
nL
.
n
j
0
j=0
i=1
The next corollary gives an expression for the asymptotic behavior of these bounds. Assuming that
n0 = O(1) and ni = n for all i ? 1, the number of regions of a single layer model with Ln hidden units
behaves as O(Ln0 nn0 ) (see Pascanu et al. 2013; Proposition 10). For a deep model, Theorem 5 implies:
Corollary 6. A rectifier neural network with n0 input
units and L hidden layers of width n ? n0 can
(L?1)n0 n0
n
compute functions that have ? ( /n0 )
n
linear regions.
Thus we see that the number of linear regions of deep models grows exponentially in L and polynomially
in n, which is much faster than that
with nL hidden units. Our result is a significant
of shallow models
L?1 n0
improvement over the bound ? ( n/n0 )
n
obtained by Pascanu et al. (2013). In particular, our
result demonstrates that even for small values of L and n, deep rectifier models are able to produce
substantially more linear regions than shallow rectifier models. Additionally, using the same strategy
as Pascanu et al. (2013), our result can be reformulated in terms of the number of linear regions per
parameter. This results in a similar behavior, with deep models being exponentially more efficient than
shallow models (see the Supplementary Material).
4 Deep Maxout Networks
A maxout network is a feedforward network with layers defined as follows:
Definition 7. A rank-k maxout layer with n input and m output units is defined by a preactivation function
of the form f : Rn ? Rm?k ; f(x) = Wx+b, with input and bias weights W ? Rm?k?n, b ? Rm?k , and
activations of the form gj (z) = max{z(j?1)k+1, . . . , zjk } for all j ? [m]. The layer computes a function
?
?
max{f1(x), . . . , fk (x)}
?
?
..
g ? f : Rn ? Rm; x ?
7 ?
(4)
?.
.
max{f(m?1)k+1(x), . . . , fmk (x)}
Since the maximum of two convex functions is convex, maxout units and maxout layers compute convex
functions. The maximum of a collection of functions is called their upper envelope. We can view the graph
of each linear function fi : Rn ? R as a supporting hyperplane of a convex set in (n + 1)-dimensional
space. In particular, if each fi, i ? [k] is the unique maximizer fi = max{fi0 : i0 ? [k]} at some input
neighborhood, then the number of linear regions of the upper envelope g1 ? f = max{fi : i ? [k]} is
exactly k. This shows that the maximal number of linear regions of a maxout unit is equal to its rank.
The linear regions of the maxout layer are the intersections of the linear regions of the individual maxout
units. In order to obtain the number of linear regions for the layer, we need to describe the structure of
the linear regions of each maxout unit, and study their possible intersections. Voronoi diagrams can be
7
lifted to upper envelopes of linear functions, and hence they describe input-space partitions generated
by maxout units. Now, how many regions do we obtain by intersecting the regions of m Voronoi diagrams
with k regions each? Computing the intersections of Voronoi diagrams is not easy, in general. A trivial
upper bound for the number of linear regions is km, which corresponds to the case where all intersections
of regions of different units are different from each other. We will give a better bound in Proposition 8.
Now, for the purpose of computing lower bounds, here it will be sufficient to consider certain well-behaved
special cases. One simple example is the division of input-space by k?1 parallel hyperplanes. If m ? n, we
can consider the arrangement of hyperplanes Hi = {x ? Rn : xj = i} for i = 1, . . . , k ? 1, for each maxout unit j ? [m]. In this case, the number of regions is km. If m > n, the same arguments yield kn regions.
Proposition 8. The maximal number of regions of a single layer maxout network with n inputs and m
Pn
2
outputs of rank k is lower bounded by kmin{n,m} and upper bounded by min{ j=0 k jm , km}.
Now we take a look at the deep maxout model. Note that a rank-2 maxout layer can be simulated by a
rectifier layer with twice as many units. Then, by the results from the last section, a rank-2 maxout network
with L ? 1 hidden layers of width n = n0 can identify 2n0 (L?1) input-space regions, and, in turn, it can
compute functions with 2n0 (L?1)2n0 = 2n0 L linear regions. For the rank-k case, we note that a rank-k
maxout unit can identify k cones from its input-domain, whereby each cone is a neighborhood of the
positive half-ray {rWi ? Rn : r ? R+} corresponding to the gradient Wi of the linear function fi for
all i ? [k]. Elaborating this observation, we obtain:
Theorem 9. A maxout network with L layers of width n0 and rank k can compute functions with at least
kL?1kn0 linear regions.
Theorem 9 and Proposition 8 show that deep maxout networks can compute functions with a number of
linear regions that grows exponentially with the number of layers, and exponentially faster than the maximal
number of regions of shallow models with the same number of units. Similarly to the rectifier model, this
exponential behavior can also be established with respect to the number of network parameters. We note
that although certain functions that can be computed by maxout layers can also be computed by rectifier
layers, the rectifier construction from last section leads to functions that are not computable by maxout
networks (except in the rank-2 case). The proof of Theorem 9 is based on the same general arguments
from Sec. 2, but uses a different construction than Theorem 5 (details in the Supplementary Material).
5 Conclusions and Outlook
We studied the complexity of functions computable by deep feedforward neural networks in terms of their
number of linear regions. We specifically focused on deep neural networks having piecewise linear hidden
units which have been found to provide superior performance in many machine learning applications
recently. We discussed the idea that each layer of a deep model is able to identify pieces of its input in
such a way that the composition of layers identifies an exponential number of input regions. This results
in exponentially replicating the complexity of the functions computed in the higher layers. The functions
computed in this way by deep models are complicated, but still they have an intrinsic rigidity caused by
the replications, which may help deep models generalize to unseen samples better than shallow models.
This framework is applicable to any neural network that has a piecewise linear activation function. For
example, if we consider a convolutional network with rectifier units, as the one used in (Krizhevsky et al.
2012), we can see that the convolution followed by max pooling at each layer identifies all patches of the
input within a pooling region. This will let such a deep convolutional neural network recursively identify
patches of the images of lower layers, resulting in exponentially many linear regions of the input space.
The structure of the linear regions depends on the type of units, e.g., hyperplane arrangements for shallow
rectifier vs. Voronoi diagrams for shallow maxout networks. The pros and cons of each type of constraint
will likely depend on the task and are not easily quantifiable at this point. As for the number of regions,
in both maxout and rectifier networks we obtain an exponential increase with depth. However, our bounds
are not conclusive about which model is more powerful in this respect. This is an interesting question
that would be worth investigating in more detail.
The parameter space of a given network is partitioned into the regions where the resulting functions have
corresponding linear regions. The combinatorics of such structures is in general hard to compute, even for
simple hyperplane arrangements. One interesting question for future analysis is whether many regions of the
parameter space of a given network correspond to functions which have a given number of linear regions.
8
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4,886 | 5,423 | Generative Adversarial Nets
Ian J. Goodfellow?, Jean Pouget-Abadie?, Mehdi Mirza, Bing Xu, David Warde-Farley,
Sherjil Ozair?, Aaron Courville, Yoshua Bengio?
D?epartement d?informatique et de recherche op?erationnelle
Universit?e de Montr?eal
Montr?eal, QC H3C 3J7
Abstract
We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G
that captures the data distribution, and a discriminative model D that estimates
the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This
framework corresponds to a minimax two-player game. In the space of arbitrary
functions G and D, a unique solution exists, with G recovering the training data
distribution and D equal to 21 everywhere. In the case where G and D are defined
by multilayer perceptrons, the entire system can be trained with backpropagation.
There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate
the potential of the framework through qualitative and quantitative evaluation of
the generated samples.
1
Introduction
The promise of deep learning is to discover rich, hierarchical models [2] that represent probability
distributions over the kinds of data encountered in artificial intelligence applications, such as natural
images, audio waveforms containing speech, and symbols in natural language corpora. So far, the
most striking successes in deep learning have involved discriminative models, usually those that
map a high-dimensional, rich sensory input to a class label [14, 20]. These striking successes have
primarily been based on the backpropagation and dropout algorithms, using piecewise linear units
[17, 8, 9] which have a particularly well-behaved gradient . Deep generative models have had less
of an impact, due to the difficulty of approximating many intractable probabilistic computations that
arise in maximum likelihood estimation and related strategies, and due to difficulty of leveraging
the benefits of piecewise linear units in the generative context. We propose a new generative model
estimation procedure that sidesteps these difficulties. 1
In the proposed adversarial nets framework, the generative model is pitted against an adversary: a
discriminative model that learns to determine whether a sample is from the model distribution or the
data distribution. The generative model can be thought of as analogous to a team of counterfeiters,
trying to produce fake currency and use it without detection, while the discriminative model is
analogous to the police, trying to detect the counterfeit currency. Competition in this game drives
both teams to improve their methods until the counterfeits are indistiguishable from the genuine
articles.
?
Ian Goodfellow is now a research scientist at Google, but did this work earlier as a UdeM student
Jean Pouget-Abadie did this work while visiting Universit?e de Montr?eal from Ecole Polytechnique.
?
Sherjil Ozair is visiting Universit?e de Montr?eal from Indian Institute of Technology Delhi
?
Yoshua Bengio is a CIFAR Senior Fellow.
1
All code and hyperparameters available at http://www.github.com/goodfeli/adversarial
?
1
This framework can yield specific training algorithms for many kinds of model and optimization
algorithm. In this article, we explore the special case when the generative model generates samples
by passing random noise through a multilayer perceptron, and the discriminative model is also a
multilayer perceptron. We refer to this special case as adversarial nets. In this case, we can train
both models using only the highly successful backpropagation and dropout algorithms [16] and
sample from the generative model using only forward propagation. No approximate inference or
Markov chains are necessary.
2
Related work
Until recently, most work on deep generative models focused on models that provided a parametric
specification of a probability distribution function. The model can then be trained by maximizing the log likelihood. In this family of model, perhaps the most succesful is the deep Boltzmann
machine [25]. Such models generally have intractable likelihood functions and therefore require
numerous approximations to the likelihood gradient. These difficulties motivated the development
of ?generative machines??models that do not explicitly represent the likelihood, yet are able to generate samples from the desired distribution. Generative stochastic networks [4] are an example of
a generative machine that can be trained with exact backpropagation rather than the numerous approximations required for Boltzmann machines. This work extends the idea of a generative machine
by eliminating the Markov chains used in generative stochastic networks.
Our work backpropagates derivatives through generative processes by using the observation that
lim ?x E?N (0,?2 I) f (x + ) = ?x f (x).
??0
We were unaware at the time we developed this work that Kingma and Welling [18] and Rezende
et al. [23] had developed more general stochastic backpropagation rules, allowing one to backpropagate through Gaussian distributions with finite variance, and to backpropagate to the covariance
parameter as well as the mean. These backpropagation rules could allow one to learn the conditional variance of the generator, which we treated as a hyperparameter in this work. Kingma and
Welling [18] and Rezende et al. [23] use stochastic backpropagation to train variational autoencoders (VAEs). Like generative adversarial networks, variational autoencoders pair a differentiable
generator network with a second neural network. Unlike generative adversarial networks, the second network in a VAE is a recognition model that performs approximate inference. GANs require
differentiation through the visible units, and thus cannot model discrete data, while VAEs require
differentiation through the hidden units, and thus cannot have discrete latent variables. Other VAElike approaches exist [12, 22] but are less closely related to our method.
Previous work has also taken the approach of using a discriminative criterion to train a generative
model [29, 13]. These approaches use criteria that are intractable for deep generative models. These
methods are difficult even to approximate for deep models because they involve ratios of probabilities which cannot be approximated using variational approximations that lower bound the probability. Noise-contrastive estimation (NCE) [13] involves training a generative model by learning the
weights that make the model useful for discriminating data from a fixed noise distribution. Using a
previously trained model as the noise distribution allows training a sequence of models of increasing
quality. This can be seen as an informal competition mechanism similar in spirit to the formal competition used in the adversarial networks game. The key limitation of NCE is that its ?discriminator?
is defined by the ratio of the probability densities of the noise distribution and the model distribution,
and thus requires the ability to evaluate and backpropagate through both densities.
Some previous work has used the general concept of having two neural networks compete. The most
relevant work is predictability minimization [26]. In predictability minimization, each hidden unit
in a neural network is trained to be different from the output of a second network, which predicts
the value of that hidden unit given the value of all of the other hidden units. This work differs from
predictability minimization in three important ways: 1) in this work, the competition between the
networks is the sole training criterion, and is sufficient on its own to train the network. Predictability
minimization is only a regularizer that encourages the hidden units of a neural network to be statistically independent while they accomplish some other task; it is not a primary training criterion.
2) The nature of the competition is different. In predictability minimization, two networks? outputs
are compared, with one network trying to make the outputs similar and the other trying to make the
2
outputs different. The output in question is a single scalar. In GANs, one network produces a rich,
high dimensional vector that is used as the input to another network, and attempts to choose an input
that the other network does not know how to process. 3) The specification of the learning process
is different. Predictability minimization is described as an optimization problem with an objective
function to be minimized, and learning approaches the minimum of the objective function. GANs
are based on a minimax game rather than an optimization problem, and have a value function that
one agent seeks to maximize and the other seeks to minimize. The game terminates at a saddle point
that is a minimum with respect to one player?s strategy and a maximum with respect to the other
player?s strategy.
Generative adversarial networks has been sometimes confused with the related concept of ?adversarial examples? [28]. Adversarial examples are examples found by using gradient-based optimization
directly on the input to a classification network, in order to find examples that are similar to the
data yet misclassified. This is different from the present work because adversarial examples are
not a mechanism for training a generative model. Instead, adversarial examples are primarily an
analysis tool for showing that neural networks behave in intriguing ways, often confidently classifying two images differently with high confidence even though the difference between them is
imperceptible to a human observer. The existence of such adversarial examples does suggest that
generative adversarial network training could be inefficient, because they show that it is possible to
make modern discriminative networks confidently recognize a class without emulating any of the
human-perceptible attributes of that class.
3
Adversarial nets
The adversarial modeling framework is most straightforward to apply when the models are both
multilayer perceptrons. To learn the generator?s distribution pg over data x, we define a prior on
input noise variables pz (z), then represent a mapping to data space as G(z; ?g ), where G is a
differentiable function represented by a multilayer perceptron with parameters ?g . We also define a
second multilayer perceptron D(x; ?d ) that outputs a single scalar. D(x) represents the probability
that x came from the data rather than pg . We train D to maximize the probability of assigning the
correct label to both training examples and samples from G. We simultaneously train G to minimize
log(1 ? D(G(z))). In other words, D and G play the following two-player minimax game with
value function V (G, D):
min max V (D, G) = Ex?pdata (x) [log D(x)] + Ez?pz (z) [log(1 ? D(G(z)))].
(1)
G
D
In the next section, we present a theoretical analysis of adversarial nets, essentially showing that
the training criterion allows one to recover the data generating distribution as G and D are given
enough capacity, i.e., in the non-parametric limit. See Figure 1 for a less formal, more pedagogical
explanation of the approach. In practice, we must implement the game using an iterative, numerical
approach. Optimizing D to completion in the inner loop of training is computationally prohibitive,
and on finite datasets would result in overfitting. Instead, we alternate between k steps of optimizing
D and one step of optimizing G. This results in D being maintained near its optimal solution, so
long as G changes slowly enough. The procedure is formally presented in Algorithm 1.
In practice, equation 1 may not provide sufficient gradient for G to learn well. Early in learning,
when G is poor, D can reject samples with high confidence because they are clearly different from
the training data. In this case, log(1 ? D(G(z))) saturates. Rather than training G to minimize
log(1 ? D(G(z))) we can train G to maximize log D(G(z)). This objective function results in the
same fixed point of the dynamics of G and D but provides much stronger gradients early in learning.
4
Theoretical Results
The generator G implicitly defines a probability distribution pg as the distribution of the samples
G(z) obtained when z ? pz . Therefore, we would like Algorithm 1 to converge to a good estimator
of pdata , if given enough capacity and training time. The results of this section are done in a nonparametric setting, e.g. we represent a model with infinite capacity by studying convergence in the
space of probability density functions.
We will show in section 4.1 that this minimax game has a global optimum for pg = pdata . We will
then show in section 4.2 that Algorithm 1 optimizes Eq 1, thus obtaining the desired result.
3
...
x
XXX
z
Z Z Z
(a)
(b)
(c)
(d)
Figure 1: Generative adversarial nets are trained by simultaneously updating the discriminative distribution
(D, blue, dashed line) so that it discriminates between samples from the data generating distribution (black,
dotted line) px from those of the generative distribution pg (G) (green, solid line). The lower horizontal line is
the domain from which z is sampled, in this case uniformly. The horizontal line above is part of the domain
of x. The upward arrows show how the mapping x = G(z) imposes the non-uniform distribution pg on
transformed samples. G contracts in regions of high density and expands in regions of low density of pg . (a)
Consider an adversarial pair near convergence: pg is similar to pdata and D is a partially accurate classifier.
(b) In the inner loop of the algorithm D is trained to discriminate samples from data, converging to D? (x) =
pdata (x)
. (c) After an update to G, gradient of D has guided G(z) to flow to regions that are more likely
pdata (x)+pg (x)
to be classified as data. (d) After several steps of training, if G and D have enough capacity, they will reach a
point at which both cannot improve because pg = pdata . The discriminator is unable to differentiate between
the two distributions, i.e. D(x) = 12 .
Algorithm 1 Minibatch stochastic gradient descent training of generative adversarial nets. The number of
steps to apply to the discriminator, k, is a hyperparameter. We used k = 1, the least expensive option, in our
experiments.
for number of training iterations do
for k steps do
? Sample minibatch of m noise samples {z (1) , . . . , z (m) } from noise prior pg (z).
? Sample minibatch of m examples {x(1) , . . . , x(m) } from data generating distribution
pdata (x).
? Update the discriminator by ascending its stochastic gradient:
m
? ?d
i
1 Xh
log D x(i) + log 1 ? D G z (i)
.
m i=1
end for
? Sample minibatch of m noise samples {z (1) , . . . , z (m) } from noise prior pg (z).
? Update the generator by descending its stochastic gradient:
m
?? g
1 X
log 1 ? D G z (i)
.
m i=1
end for
The gradient-based updates can use any standard gradient-based learning rule. We used momentum in our experiments.
4.1
Global Optimality of pg = pdata
We first consider the optimal discriminator D for any given generator G.
Proposition 1. For G fixed, the optimal discriminator D is
?
DG
(x) =
pdata (x)
pdata (x) + pg (x)
4
(2)
Proof. The training criterion for the discriminator D, given any generator G, is to maximize the
quantity V (G, D)
Z
Z
V (G, D) = pdata (x) log(D(x))dx + pz (z) log(1 ? D(g(z)))dz
z
Zx
= pdata (x) log(D(x)) + pg (x) log(1 ? D(x))dx
(3)
x
For any (a, b) ? R2 \ {0, 0}, the function y ? a log(y) + b log(1 ? y) achieves its maximum in
a
[0, 1] at a+b
. The discriminator does not need to be defined outside of Supp(pdata ) ? Supp(pg ),
concluding the proof.
Note that the training objective for D can be interpreted as maximizing the log-likelihood for estimating the conditional probability P (Y = y|x), where Y indicates whether x comes from pdata
(with y = 1) or from pg (with y = 0). The minimax game in Eq. 1 can now be reformulated as:
C(G) = max V (G, D)
D
?
?
=Ex?pdata [log DG
(x)] + Ez?pz [log(1 ? DG
(G(z)))]
(4)
?
?
=Ex?pdata [log DG (x)] + Ex?pg [log(1 ? DG
(x))]
pg (x)
pdata (x)
=Ex?pdata log
+ Ex?pg log
Pdata (x) + pg (x)
pdata (x) + pg (x)
Theorem 1. The global minimum of the virtual training criterion C(G) is achieved if and only if
pg = pdata . At that point, C(G) achieves the value ? log 4.
?
?
Proof. For pg = pdata , DG
(x) = 12 , (consider Eq. 2). Hence, by inspecting Eq. 4 at DG
(x) = 21 , we
1
1
find C(G) = log 2 + log 2 = ? log 4. To see that this is the best possible value of C(G), reached
only for pg = pdata , observe that
Ex?pdata [? log 2] + Ex?pg [? log 2] = ? log 4
?
, G), we obtain:
and that by subtracting this expression from C(G) = V (DG
pdata + pg
pdata + pg
C(G) = ? log(4) + KL pdata
+
KL
p
(5)
g
2
2
where KL is the Kullback?Leibler divergence. We recognize in the previous expression the Jensen?
Shannon divergence between the model?s distribution and the data generating process:
C(G) = ? log(4) + 2 ? JSD (pdata kpg )
(6)
Since the Jensen?Shannon divergence between two distributions is always non-negative, and zero
iff they are equal, we have shown that C ? = ? log(4) is the global minimum of C(G) and that the
only solution is pg = pdata , i.e., the generative model perfectly replicating the data distribution.
4.2
Convergence of Algorithm 1
Proposition 2. If G and D have enough capacity, and at each step of Algorithm 1, the discriminator
is allowed to reach its optimum given G, and pg is updated so as to improve the criterion
?
?
Ex?pdata [log DG
(x)] + Ex?pg [log(1 ? DG
(x))]
then pg converges to pdata
Proof. Consider V (G, D) = U (pg , D) as a function of pg as done in the above criterion. Note
that U (pg , D) is convex in pg . The subderivatives of a supremum of convex functions include the
derivative of the function at the point where the maximum is attained. In other words, if f (x) =
sup??A f? (x) and f? (x) is convex in x for every ?, then ?f? (x) ? ?f if ? = arg sup??A f? (x).
This is equivalent to computing a gradient descent update for pg at the optimal D given the corresponding G. supD U (pg , D) is convex in pg with a unique global optima as proven in Thm 1,
therefore with sufficiently small updates of pg , pg converges to px , concluding the proof.
In practice, adversarial nets represent a limited family of pg distributions via the function G(z; ?g ),
and we optimize ?g rather than pg itself, so the proofs do not apply. However, the excellent performance of multilayer perceptrons in practice suggests that they are a reasonable model to use despite
their lack of theoretical guarantees.
5
Model
DBN [3]
Stacked CAE [3]
Deep GSN [5]
Adversarial nets
MNIST
138 ? 2
121 ? 1.6
214 ? 1.1
225 ? 2
TFD
1909 ? 66
2110 ? 50
1890 ? 29
2057 ? 26
Table 1: Parzen window-based log-likelihood estimates. The reported numbers on MNIST are the mean loglikelihood of samples on test set, with the standard error of the mean computed across examples. On TFD, we
computed the standard error across folds of the dataset, with a different ? chosen using the validation set of
each fold. On TFD, ? was cross validated on each fold and mean log-likelihood on each fold were computed.
For MNIST we compare against other models of the real-valued (rather than binary) version of dataset.
5
Experiments
We trained adversarial nets an a range of datasets including MNIST[21], the Toronto Face Database
(TFD) [27], and CIFAR-10 [19]. The generator nets used a mixture of rectifier linear activations [17,
8] and sigmoid activations, while the discriminator net used maxout [9] activations. Dropout [16]
was applied in training the discriminator net. While our theoretical framework permits the use of
dropout and other noise at intermediate layers of the generator, we used noise as the input to only
the bottommost layer of the generator network.
We estimate probability of the test set data under pg by fitting a Gaussian Parzen window to the
samples generated with G and reporting the log-likelihood under this distribution. The ? parameter
of the Gaussians was obtained by cross validation on the validation set. This procedure was introduced in Breuleux et al. [7] and used for various generative models for which the exact likelihood
is not tractable [24, 3, 4]. Results are reported in Table 1. This method of estimating the likelihood
has somewhat high variance and does not perform well in high dimensional spaces but it is the best
method available to our knowledge. Advances in generative models that can sample but not estimate
likelihood directly motivate further research into how to evaluate such models. In Figures 2 and 3
we show samples drawn from the generator net after training. While we make no claim that these
samples are better than samples generated by existing methods, we believe that these samples are at
least competitive with the better generative models in the literature and highlight the potential of the
adversarial framework.
6
Advantages and disadvantages
This new framework comes with advantages and disadvantages relative to previous modeling frameworks. The disadvantages are primarily that there is no explicit representation of pg (x), and that D
must be synchronized well with G during training (in particular, G must not be trained too much
without updating D, in order to avoid ?the Helvetica scenario? in which G collapses too many values
of z to the same value of x to have enough diversity to model pdata ), much as the negative chains of a
Boltzmann machine must be kept up to date between learning steps. The advantages are that Markov
chains are never needed, only backprop is used to obtain gradients, no inference is needed during
learning, and a wide variety of functions can be incorporated into the model. Table 2 summarizes
the comparison of generative adversarial nets with other generative modeling approaches.
The aforementioned advantages are primarily computational. Adversarial models may also gain
some statistical advantage from the generator network not being updated directly with data examples, but only with gradients flowing through the discriminator. This means that components of the
input are not copied directly into the generator?s parameters. Another advantage of adversarial networks is that they can represent very sharp, even degenerate distributions, while methods based on
Markov chains require that the distribution be somewhat blurry in order for the chains to be able to
mix between modes.
7
Conclusions and future work
This framework admits many straightforward extensions:
6
a)
b)
c)
d)
Figure 2: Visualization of samples from the model. Rightmost column shows the nearest training example of
the neighboring sample, in order to demonstrate that the model has not memorized the training set. Samples
are fair random draws, not cherry-picked. Unlike most other visualizations of deep generative models, these
images show actual samples from the model distributions, not conditional means given samples of hidden units.
Moreover, these samples are uncorrelated because the sampling process does not depend on Markov chain
mixing. a) MNIST b) TFD c) CIFAR-10 (fully connected model) d) CIFAR-10 (convolutional discriminator
and ?deconvolutional? generator)
Figure 3: Digits obtained by linearly interpolating between coordinates in z space of the full model.
1. A conditional generative model p(x | c) can be obtained by adding c as input to both G and D.
2. Learned approximate inference can be performed by training an auxiliary network to predict z
given x. This is similar to the inference net trained by the wake-sleep algorithm [15] but with
the advantage that the inference net may be trained for a fixed generator net after the generator
net has finished training.
3. One can approximately model all conditionals p(xS | x6S ) where S is a subset of the indices
of x by training a family of conditional models that share parameters. Essentially, one can use
adversarial nets to implement a stochastic extension of the deterministic MP-DBM [10].
4. Semi-supervised learning: features from the discriminator or inference net could improve performance of classifiers when limited labeled data is available.
5. Efficiency improvements: training could be accelerated greatly by devising better methods for
coordinating G and D or determining better distributions to sample z from during training.
This paper has demonstrated the viability of the adversarial modeling framework, suggesting that
these research directions could prove useful.
7
Deep directed
graphical models
Deep undirected
graphical models
Inference needed
during training.
MCMC needed to
approximate
partition function
gradient.
Generative
autoencoders
Adversarial models
Enforced tradeoff
between mixing
and power of
reconstruction
generation
Synchronizing the
discriminator with
the generator.
Helvetica.
Learned
approximate
inference
Training
Inference needed
during training.
Inference
Learned
approximate
inference
Variational
inference
MCMC-based
inference
Sampling
No difficulties
Requires Markov
chain
Evaluating p(x)
Intractable, may be
approximated with
AIS
Intractable, may be
approximated with
AIS
Requires Markov
chain
Not explicitly
represented, may be
approximated with
Parzen density
estimation
Model design
Models need to be
designed to work
with the desired
inference scheme
? some inference
schemes support
similar model
families as GANs
Careful design
needed to ensure
multiple properties
Any differentiable
function is
theoretically
permitted
No difficulties
Not explicitly
represented, may be
approximated with
Parzen density
estimation
Any differentiable
function is
theoretically
permitted
Table 2: Challenges in generative modeling: a summary of the difficulties encountered by different approaches
to deep generative modeling for each of the major operations involving a model.
Acknowledgments
We would like to acknowledge Patrice Marcotte, Olivier Delalleau, Kyunghyun Cho, Guillaume
Alain and Jason Yosinski for helpful discussions. Yann Dauphin shared his Parzen window evaluation code with us. We would like to thank the developers of Pylearn2 [11] and Theano [6, 1],
particularly Fr?ed?eric Bastien who rushed a Theano feature specifically to benefit this project. Arnaud Bergeron provided much-needed support with LATEX typesetting. We would also like to thank
CIFAR, and Canada Research Chairs for funding, and Compute Canada, and Calcul Qu?ebec for
providing computational resources. Ian Goodfellow is supported by the 2013 Google Fellowship in
Deep Learning. Finally, we would like to thank Les Trois Brasseurs for stimulating our creativity.
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4,887 | 5,424 | Deep Symmetry Networks
Robert Gens
Pedro Domingos
Department of Computer Science and Engineering
University of Washington
Seattle, WA 98195-2350, U.S.A.
{rcg,pedrod}@cs.washington.edu
Abstract
The chief difficulty in object recognition is that objects? classes are obscured by
a large number of extraneous sources of variability, such as pose and part deformation. These sources of variation can be represented by symmetry groups,
sets of composable transformations that preserve object identity. Convolutional
neural networks (convnets) achieve a degree of translational invariance by computing feature maps over the translation group, but cannot handle other groups.
As a result, these groups? effects have to be approximated by small translations,
which often requires augmenting datasets and leads to high sample complexity.
In this paper, we introduce deep symmetry networks (symnets), a generalization
of convnets that forms feature maps over arbitrary symmetry groups. Symnets
use kernel-based interpolation to tractably tie parameters and pool over symmetry
spaces of any dimension. Like convnets, they are trained with backpropagation.
The composition of feature transformations through the layers of a symnet provides a new approach to deep learning. Experiments on NORB and MNIST-rot
show that symnets over the affine group greatly reduce sample complexity relative
to convnets by better capturing the symmetries in the data.
1
Introduction
Object recognition is a central problem in vision. What makes it challenging are all the nuisance
factors such as pose, lighting, part deformation, and occlusion. It has been shown that if we could
remove these factors, recognition would be much easier [2, 17]. Convolutional neural networks
(convnets), the current state-of-the-art method for object recognition, capture only one type of invariance (translation); the rest have to be approximated via it and standard features. In practice,
the best networks require enormous datasets which are further expanded by affine transformations
[7, 13] yet are sensitive to imperceptible image perturbations [23]. We propose deep symmetry networks, a generalization of convnets based on symmetry group theory [20] that makes it possible to
capture a broad variety of invariances, and correspondingly improves generalization.
A symmetry group is a set of transformations that preserve the identity of an object and obey the
group axioms. Most of the visual nuisance factors are symmetry groups themselves, and by incorporating them into our model we are able to reduce the sample complexity of learning from data
transformed by these groups. Deep symmetry networks (symnets) form feature maps over any symmetry group, rather than just the translation group. A feature map in a deep symmetry network
is defined analogously to convnets as a filter that is applied at all points in the symmetry space.
Each layer in our general architecture is constructed by applying every symmetry in the group to
the input, computing features on the transformed input, and pooling over neighborhoods. The entire
architecture is then trained by backpropagation. In this paper, we instantiate the architecture with the
affine group, resulting in deep affine networks. In addition to translation, the affine group includes
rotation, scaling and shear. The affine group of the two-dimensional plane is six-dimensional (i.e.,
an affine transformation can be represented by a point in 6D affine space). The key challenge with
1
extending convnets to affine spaces is that it is intractable to explicitly represent and compute with
a high-dimensional feature map. We address this by approximating the map using kernel functions,
which not only interpolate but also control pooling in the feature maps. Compared to convnets,
this architecture substantially reduces sample complexity on image datasets involving 2D and 3D
transformations.
We share with other researchers the hypothesis that explanatory factors cannot be disentangled unless they are represented in an appropriate symmetry space [4, 11]. Our adaptation of a representation to work in symmetry space is similar in some respects to the use of tangent distance in
nearest-neighbor classifiers [22]. Symnets, however, are deep networks that compute features in
symmetry space at every level. Whereas the tangent distance approximation is only locally accurate,
symnet feature maps can represent large displacements in symmetry space. There are other deep
networks that reinterpret the invariance of convolutional networks. Scattering networks [6] are cascades of wavelet decompositions designed to be invariant to particular Lie groups, where translation
and rotation invariance have been demonstrated so far. The M-theory of Anselmi et al. [2] constructs features invariant to a symmetry group by using statistics of dot products with group orbits.
We differ from these networks in that we model multiple symmetries jointly in each layer, we do not
completely pool out a symmetry, and we discriminatively train our entire architecture. The first two
differences are important because objects and their subparts may have relative flexibility but not total
invariance along certain dimensions of symmetry space. For example, a leg of a person can be seen
in some but not all combinations of rotation and scale relative to the torso. Without discriminative
training, scattering networks and M-theory are limited to representing features whose invariances
may be inappropriate for a target concept because they are fixed ahead of time, either by the wavelet
hierarchy of the former or unsupervised training of the latter. The discriminative training of symnets
yields features with task-oriented invariance to their sub-features. In the context of digit recognition
this might mean learning the concept of a ?0? with more rotation invariance than a ?6?, which would
incur loss if it had positive weights in the region of symmetry space where a ?9? would also fire.
Much of the vision literature is devoted to features that reduce or remove the effects of certain symmetry groups, e.g., [18, 17]. Each feature by itself is not discriminative for object recognition, so
structure is modeled separately, usually with a representation that does not generalize to novel viewpoints (e.g., bags-of-features) or with a rigid alignment algorithm that cannot represent uncertainty
over geometry (e.g. [9, 19]). Compared to symnets, these features are not learned, have invariance
limited to a small set of symmetries, and destroy information that could be used to model object
sub-structure. Like deformable part models [10], symnets can model and penalize relative transformations that compose up the hierarchy, but can also capture additional symmetries.
Symmetry group theory has made a limited number of appearances in machine learning [8]. A few
applications are discussed by Kondor [12], and they are also used in determinantal point processes
[14]. Methods for learning transformations from examples [24, 11] could potentially benefit from
being embedded in a deep symmetry network. Symmetries in graphical models [21] lead to effective
lifted probabilistic inference algorithms. Deep symmetry networks may be applicable to these and
other areas.
In this paper, we first review symmetry group theory and its relation to sample complexity. We then
describe symnets and their affine instance, and develop new methods to scale to high-dimensional
symmetry spaces. Experiments on NORB and MNIST-rot show that affine symnets can reduce by a
large factor the amount of data required to achieve a given accuracy level.
2
Symmetry Group Theory
A symmetry of an object is a transformation that leaves certain properties of that object intact [20].
A group is a set S with an operator ? on it with the four properties of closure, associativity, an
identity element, and an inverse element. A symmetry group is a type of group where the group
elements are functions and the operator is function composition. A simple geometric example is
the symmetry group of a square, which consists of four reflections and {0, 1, 2, 3} multiples of 90degree rotations. These transformations can be composed together to yield one of the original eight
symmetries. The identity element is the 0-degree rotation. Each symmetry has a corresponding
inverse element. Composition of these symmetries is associative.
2
Lie groups are continuous symmetry groups whose elements form a smooth differentiable manifold.
For example, the symmetries of a circle include reflections and rotations about the center. The affine
group is a set of transformations that preserves collinearity and parallel lines. The Euclidean group
is a subgroup of the affine group that preserves distances, and includes the set of rigid body motions
(translations and rotations) in three-dimensional space.
The elements of a symmetry group can be represented as matrices. In this form, function composition can be performed via matrix multiplication. The transformation P followed by Q (also denoted
Q ? P) is computed as R = QP. In this paper we treat the transformation matrix P as a point
in D-dimensional space, where D depends on the particular representation of the symmetry group
(e.g., D = 6 for affine transformations in the plane).
A generating set of a group is a subset of the group such that any group element can be expressed
through combinations of generating set elements and their inverses. For example, a generating set
of the translation symmetry group is {x ? x + , y ? y + } for infinitesimal . We define the
k-neighborhood of element f in group S under generating set G as the subset of S that can be
expressed as f composed with elements of G or their inverses at most k times. With the previous
example, the k-neighborhood of a translation vector f would take the shape of a diamond centered
at f in the xy-plane.
The orbit of an object x is the set of objects obtained by applying each element of a symmetry group
to x. Formally, a symmetry group S acting on a set of objects X defines an orbit for each x ? X:
Ox = {s ? x : s ? S}. For example, the orbit of an image I(u) whose points are transformed by the
rotation symmetry group s ? I(u) = I(s?1 ? u) is the set of images resulting from all rotations of
that image. If two orbits share an element, they are the same
S orbit. In this way, a symmetry group
S partitions the set of objects into unique orbits X = a Oa . If a data distribution D(x, y) has
the property that all the elements of an orbit share the same label y, S imposes a constraint on the
hypothesis class of a learner, effectively lowering its VC-dimension and sample complexity [1].
3
Deep Symmetry Networks
Deep symmetry networks represent rich compositional structure that incorporates invariance to highdimensional symmetries. The ideas behind these networks are applicable to any symmetry group, be
it rigid-body transformations in 3D or permutation groups over strings. The architecture of a symnet
consists of several layers of feature maps. Like convnets, these feature maps benefit from weight
tying and pooling, and the whole network is trained with backpropagation. The maps and the filters
they apply are in the dimension D of the chosen symmetry group S.
A deep symmetry network has L layers l ? {1, ..., L} each with Il features and corresponding
feature maps. A feature is the dot-product of a set of weights with a corresponding set of values from
a local region of a lower layer followed by a nonlinearity. A feature map represents the application
of a filter at all points in symmetry space. A feature at point P is computed from the feature maps
of the lower layer at points in the k-neighborhood of P. As P moves in the symmetry space of a
feature map, so does its neighborhood of inputs in the lower layer. Feature map i of layer l is denoted
M [l, i] : RD ? R, a scalar function of the D-dimensional symmetry space. Given a generating set
G ? S, the points in the k-neighborhood of the identity element are stored in an array T[ ]. Each
filter i of layer l defines a weight vector w[l, i, j] for each point T[j] in the k-neighborhood. The
vector w[l, i, j] is the size of Il?1 , the number of features in the underlying layer. For example, a
feature in an affine symnet that detects a person would have positive weight for an arm sub-feature in
the region of the k-neighborhood that would transform the arm relative to the person (e.g., smaller,
rotated, and translated relative to the torso). The value of feature map i in layer l at point P is
the dot-product of weights and underlying feature values in the neighborhood of P followed by a
nonlinearity:
M [l, i](P) =
v(P, l, i) =
? (v(P, l, i))
P|T|
x(P0 ) =
w[l, i, j] ? x(P ? T[j])
+
S(M [l ? 1, 0])(P0 )
...
S(M [l ? 1, Il?1 ])(P0 )
j
*
3
(1)
(2)
(3)
Layer l
Feature map i
Layer l-1
Pooled feature maps 0,1,2
Layer l-1
Feature maps 0,1,2
Kernels
Figure 1: The evaluation of point P in map M [l, i]. The elements of the k-neighborhood of P are
computed P ? T[j]. Each point in the neighborhood is evaluated in the pooled feature maps of the
lower layer l ? 1. The pooled maps are computed with kernels on the underlying feature maps. The
dashed line intersects the points in the pooled map whose values form x(P ? T[j]) in Equation 3; it
also intersects the contours of kernels used to compute those pooled values. The value of the feature
is the sum of the dot-products w[l, i, j] ? x(P ? T[j]) over all j, followed by a nonlinearity.
where ? is the nonlinearity (e.g., tanh(x) or max(x, 0)), v(P, l, i) is the dot product, P ? T[j] represents element j in the k-neighborhood of P, and x(P0 ) is the vector of values from the underlying
pooled maps at point P0 . This definition is a generalization of feature maps in convnets1 . Similarly,
the same filter weights w[l, i, j] are tied across all points P in feature map M [l, i]. The evaluation
of a point in a feature map is visualized in Figure 1.
Feature maps M [l, i] are pooledR via kernel convolution to become S(M [l, i]). In the case of
sum-pooling, S(M [l, i])(P) = M [l, i](P ? Q)K(Q) dQ; for max-pooling, S(M [l, i])(P) =
maxQ M [l, i](P ? Q)K(Q). The kernel K(Q) is also a scalar function of the D-dimensional symmetry space. In the previous example of a person feature, the arm feature map could be pooled over
a wide range of rotations but narrow range of translations and scales so that the person feature allows
for moveable but not unrealistic arms. Each filter can specify the kernels it uses to pool lower layers,
but for the sake of brevity and analogy to convnets we assume that the feature maps of a layer are
pooled by the same kernel. Note that convnets discretize these operations, subsample the pooled
map, and use a uniform kernel K(Q) = 1{kQk? < r}.
As with convnets, the values of points in a symnet feature map are used by higher symnet layers,
layers of fully connected hidden units, and ultimately softmax classification. Hidden units take the
familiar form o = ?(Wx + b), with input x, output o, weight
P matrix W, and bias b. The log-loss
of the softmax L on an instance is ?wi ? x ? bi + log ( c exp (wc ? x + bc )), where Y = i is
the true label, wc and bc are the weight vector and bias for class c, and the summation is over the
classes. The input image is treated as a feature map (or maps, if color or stereo) with values in the
translation symmetry space.
Deep symmetry networks are trained with backpropagation and are amenable to the same best practices as convnets. Though feature maps are defined as continuous, in practice the maps and their
gradients are evaluated on a finite set of points P ? M [l, i]. We provide the partial derivative of the
loss L with respect to a weight vector.
?L
=
?w[l, i, j]
?M [l, i](P)
=
?w[l, i, j]
?M [l,i](P)
?L
P?M [l,i] ?M [l,i](P) ?w[l,i,j]
(4)
? 0 (v(P, l, i)) x(P ? T[j])
(5)
P
1
The neighborhood that defines a square filter in convnets is the reference point translated by up to k times
in x and k times in y.
4
A
B1
B2
B3
B4
B5
A
B5
C2
B1
A
B2
C2
B3
C1
C2
C3
C4
B2
B4
B3
C3
C4
B1
B2
B3
B4
B5
C1
C3
B5
A
C2
C1
C4
B1
B1
B1
A
B2
C1
C1
B3
C1
C2
C3
C4
B1
B4
C1
B5
C3
B4
C4
Figure 2: The feature hierarchy of a three-layer deep affine net is visualized with and without
pooling. From top to bottom, the layers (A,B,C) contain one, five, and four feature maps, each corresponding to a labeled part of the cartoon figure. Each horizontal line represents a six-dimensional
affine feature map, and bold circles denote six-dimensional points in the map. The dashed lines
represent the affine transformation from a feature to the location of one of its filter points. For clarity, only a subset of filter points are shown. Left: Without pooling, the hierarchy represents a rigid
affine transformation among all maps. Another point on feature map A is visualized in grey. Right:
Feature maps B1 and C1 are pooled with a kernel that gives those features flexibility in rotation.
The partial derivative of the loss L with respect to the value of a point in a lower layer is
?L
=
?M [l ? 1, i](P)
?M [l, i0 ](P0 )
=
?M [l ? 1, i](P)
?M [l,i0 ](P0 )
?L
P0 ?M [l,i0 ] ?M [l,i0 ](P0 ) ?M [l?1,i](P)
PIl P
i0
? 0 (v(P0 , l, i0 ))
where the gradient of the pooled feature map
P|T|
j
0
[l?1,i])(P ?T[j])
w[l, i0 , j][i] ?S(M?M
[l?1,i](P)
?S(M [l,i])(P)
?M [l,i](Q)
(6)
(7)
equals K(P ? Q) for sum-pooling.
None of this treatment depends explicitly on the dimensionality of the space except for the kernel
and transformation composition which have polynomial dependence on D. In the next section we
apply this architecture to the affine group in 2D, but it could also be applied to the affine group in
3D or any other symmetry group.
4
Deep Affine Networks
We instantiate a deep symmetry network with the
affine symmetry group in the plane. The affine
symmetry group contains transformations capable of rotating, scaling, shearing, and translating
two-dimensional points. The transformation is described by six coordinates:
0
x
a b
x
e
=
+
y0
c d
y
f
1
0
This means that each of the feature maps M [l, i]
?1
and elements T[j] of the k-neighborhood is represented in six dimensions. The identity transformation is a = d = 1, b = c = e = f = 0. The generating
?1
0
1
set of the affine symmetry group contains six el- Figure 3: The six transformations in the generements, each of which is obtained by adding to ating set of the affine group applied to a square
one of the six coordinates in the identity transform. (exaggerated = 0.2, identity is black square).
This generating set is visualized in Figure 3.
A deep affine network can represent a rich part hierarchy where each weight of a feature modulates
the response to a subpart at a point in the affine neighborhood. The geometry of a deep affine network
is best understood by tracing a point on a feature map through its filter point transforms into lower
layers. Figure 2 visualizes this structure without and with pooling on the left and right sides of
the diagram, respectively. Without pooling, the feature hierarchy defines a rigid affine relationship
between the point of evaluation on a map and the location of its sub-features. In contrast, a pooled
value on a sub-feature map is computed from a neighborhood defined by the kernel of points in
affine space; this can represent model flexibility along certain dimensions of affine space.
5
5
Scaling to High-Dimensional Symmetry Spaces
It would be intractable to explicitly represent the high-dimensional feature maps of symnets. Even a
subsampled grid becomes unwieldy at modest dimensions (e.g., a grid in affine space with ten steps
per axis has 106 points). Instead, each feature map is evaluated at N control points. The control
points are local maxima of the feature in symmetry space, found by Gauss-Newton optimization,
each initialized from a prior. This can be seen as a form of non-maximum suppression. Since the
goal is recognition, there is no need to approximate the many points in symmetry space where the
feature is not present. The map is then interpolated with kernel functions; the shape of the function
also controls pooling.
5.1 Transformation Optimization
Convnets max-pool a neighborhood of translation space by exhaustive evaluation of feature locations. There are a number of algorithms that solve for a maximal feature location in symmetry space
but they are not efficient when the feature weights are frequently adjusted [9, 19]. We adopt an
iterative approach that dovetails with the definition of our features.
If a symnet is based on a Lie group, gradient based optimization can be used to find a point P?
that locally maximizes the feature value (Equation 1) initialized at point P. In our experiments
with deep affine nets, we follow the forward compositional (FC) warp [3] to align filters with the
image. An extension of Lucas-Kanade, FC solves for an image alignment. We adapt this procedure
P|T|
to our filters and weight vectors: min?P j kw[l, i, j] ? x(P ? ?P ? T[j])k2 . We run an FC
alignment for each of the N control points in feature map M [l, i], each initialized from a prior.
P|T|
Assuming j kx(P ? ?P ? T[j])k2 is constant, this procedure locally maximizes the dot product
between the filter and the map in Equation 2. Each iteration of FC takes a Gauss-Newton step to
solve for a transformation of the neighborhood of the feature in the underlying map ?P, which is
then composed with the control point: P ? P ? ?P.
5.2 Kernels
Given a set of N local optima O?
=
{(P1 , v1 ), . . . , (PN , vN )} in D-dimensional feature
map M [l, i], we use kernel-based interpolation to compute
a pooled map S(M [l, i]). The kernel performs three
functions: penalizing relative locations of sub-features
in symmetry space (cf. [10]), interpolating the map, and
pooling a region of the map. These roles could be split
into separate filter-specific kernels that are then convolved
appropriately. The choice of these kernels will vary
with the application. In our experiments, we lump these
functions into a single kernel for a layer. We use a Gaussian
T ?1
kernel K(Q) = e?q ? q where q is the D-dimensional
vector representation of Q and the D?D covariance matrix
? controls the shape and extent of the kernel. Several
instances of this kernel are shown in Figure 4. Max-pooling
produced the best results on our tests.
6
Figure 4: Contours of three 6D Gaussian kernels visualized on a surface
in affine space. Points are visualized
by an oriented square transformed by
the affine transformation at that point.
Each kernel has a different covariance
matrix ?.
Experiments
In our experiments we test the hypothesis that a deep network with access to a larger symmetry group
will generalize better from fewer examples, provided those symmetries are present in the data. In
particular, theory suggests that a symnet will have better sample complexity than another classifier
on a dataset if it is based on a symmetry group that generates variations present in that dataset [1].
We compare deep affine symnets to convnets on the MNIST-rot and NORB image classification
datasets, which finely sample their respective symmetry spaces such that learning curves measure
the amount of augmentation that would be required to achieve similar performance. On both datasets
affine symnets achieve a substantial reduction in sample complexity. This is particularly remarkable
on NORB because its images are generated by a symmetry space in 3D. Symnet running time was
within an order of magnitude of convnets, and could be greatly optimized.
6.1 MNIST-rot
MNIST-rot [15] consists of 28x28 pixel greyscale images: 104 for training, 2 ? 103 for validation,
and 5 ? 104 for testing. The images are sampled from the MNIST digit recognition dataset and each
6
Figure 5: Impact of training set size on MNIST-rot test performance for architectures that use either
one convolutional layer or one affine symnet layer.
is rotated by a random angle in the uniform distribution [0, 2?]. With transformations that apply
to the whole image, MNIST-rot is a good testbed for comparing the performance of a single affine
layer to a single convnet layer.
We modified the Theano [5] implementation of convolutional networks so that the network consisted
of a single layer of convolution and maxpooling followed by a hidden layer of 500 units and then
softmax classification. The affine net layer was directly substituted for the convolutional layer. The
control points of the affine net were initialized at uniformly random positions with rotations oriented
around the image center, and each control point was locally optimized with four iterations of GaussNewton updates. The filter points of the affine net were arranged in a square grid. Both the affine
net and the convnet compute a dot-product and use the sigmoid nonlinearity. Both networks were
trained with 50 epochs of mini-batch gradient descent with momentum, and test results are reported
on the network with lowest error on the validation set2 . The convnet did best with small 5 ? 5
filters and the symnet with large 20 ? 20 filters. This is not surprising because the convnet must
approximate the large rotations of the dataset with translations of small patches. The affine net can
pool directly in this space of rotations with large filters.
Learning curves for the two networks are presented in Figure 5. We observe that the affine symnet
roughly halves the error of the convnet. With small sample sizes, the symnet achieves an accuracy
for which the convnet requires about eight times as many samples.
6.2 NORB
MNIST-rot is a synthetic dataset with symmetries that are not necessarily representative of real
images. The NORB dataset [16] contains 2 ? 108 ? 108 pixel stereoscopic images of 50 toys in five
categories: quadrupeds, human figures, airplanes, trucks, and cars. Five of the ten instances of each
category are reserved for the test set. Each toy is photographed on a turntable from an exhaustive
set of angles and lighting conditions. Each image is then perturbed by a random translation shift,
planar rotation, luminance change, contrast change, scaling, distractor object, and natural image
background. A sixth blank category containing just the distractor and background is also used. As
in other papers, we downsample the images to 2?48?48. To compensate for the effect of distractors
in smaller training sets, we also train and test on a version of the dataset that is centrally-cropped to
2 ? 24 ? 24. We report results for whichever version had lower validation error. In our experiments
we train on a variable subset of the first training fold, using the first 2 ? 103 images of the second
fold for validation. Our results use both of the testing folds.
We compare architectures that use two convolutional layers or two affine ones, which performed
better than single-layer ones. As with the MNIST-rot experiments, the symnet and convnet layers
are followed by a layer of 500 hidden units and softmax classification. The symnet control points in
the first layer were arranged in three concentric rings in translation space, with 8 points spaced across
rotation (200 total points). Control points in the second layer were fixed at the center of translation
2
Grid search over learning rate {.1, .2}, mini-batch size {10, 50, 100}, filter size {5, 10, 15, 20, 25}, number of filters {20, 50, 80}, pooling size (convnet) {2, 3, 4}, and number of control points (symnet) {5, 10, 20}.
7
Figure 6: Impact of training set size on NORB test performance for architectures with two convolutional or affine symnet layers followed by a fully connected layer and then softmax classification.
space arranged over 8 rotations and up to 2 vertical scalings (16 total points) to approximate the
effects of elevation change. Control points were not iteratively optimized due to the small size of
object parts in downsampled images. The filter points of the first layer of the affine net were arranged
in a square grid. The second layer filter points were arranged in a circle in translation space at a 3
or 4 pixel radius, with 8 filter points evenly spaced across rotation at each translation. We report the
test results of the networks with lowest validation error on a range of hyperparameters3 .
The learning curves for convnets and affine symnets are shown in Figure 6. Even though the primary
variability in NORB is due to rigid 3D transformations, we find that our affine networks still have
an advantage over convnets. A 3D rotation can be locally approximated with 2D scales, shears, and
rotations. The affine net can represent these transformations and so it benefited from larger filter
patches. The translation approximation of the convnet is unable to properly align larger features to
the true symmetries, and so it performed better with smaller filters. The convnet requires about four
times as much data to reach the accuracy of the symnet with the smallest training set. Larger filters
capture more structure than smaller ones, allowing symnets to generalize better than convnets, and
effectively giving each symnet layer the power of more than one convnet layer.
The left side of the graph may be more indicative of the types of gains symnets may have over
convnets in more realistic datasets that do not have thousands of images of identical 3D shapes.
With the ability to apply more realistic transformations to sub-parts, symnets may also be better able
to reuse substructure on datasets with many interrelated or fine-grained categories. Since symnets
are a clean generalization of convnets, they should benefit from the learning, regularization, and
efficiency techniques used by state-of-the-art networks [13].
7
Conclusion
Symmetry groups underlie the hardest challenges in computer vision. In this paper we introduced
deep symmetry networks, the first deep architecture that can compute features over any symmetry
group. It is a natural generalization of convolutional neural networks that uses kernel interpolation
and transformation optimization to address the difficulties in representing high-dimensional feature
maps. In experiments on two image datasets with 2D and 3D variability, affine symnets achieved
higher accuracy than convnets while using significantly less data.
Directions for future work include extending to other symmetry groups (e.g., lighting, 3D space),
modeling richer distortions, incorporating probabilistic inference, and scaling to larger datasets.
Acknowledgments
This research was partly funded by ARO grant W911NF-08-1-0242, ONR grants N00014-13-10720 and N00014-12-1-0312, and AFRL contract FA8750-13-2-0019. The views and conclusions
contained in this document are those of the authors and should not be interpreted as necessarily
representing the official policies, either expressed or implied, of ARO, ONR, AFRL, or the United
States Government.
3
Grid search over filter size in each layer {6, 9}, pooling size in each layer (convnet) {2, 3, 4}, first layer
control point translation spacing (symnet) {2, 3}, momentum {0, 0.5, 0.9}, others as in MNIST-rot.
8
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9
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4,888 | 5,425 | A Multiplicative Model for Learning Distributed
Text-Based Attribute Representations
Ryan Kiros, Richard S. Zemel, Ruslan Salakhutdinov
University of Toronto
Canadian Institute for Advanced Research
{rkiros, zemel, rsalakhu}@cs.toronto.edu
Abstract
In this paper we propose a general framework for learning distributed representations of attributes: characteristics of text whose representations can be jointly
learned with word embeddings. Attributes can correspond to a wide variety of
concepts, such as document indicators (to learn sentence vectors), language indicators (to learn distributed language representations), meta-data and side information (such as the age, gender and industry of a blogger) or representations of
authors. We describe a third-order model where word context and attribute vectors
interact multiplicatively to predict the next word in a sequence. This leads to the
notion of conditional word similarity: how meanings of words change when conditioned on different attributes. We perform several experimental tasks including
sentiment classification, cross-lingual document classification, and blog authorship attribution. We also qualitatively evaluate conditional word neighbours and
attribute-conditioned text generation.
1
Introduction
Distributed word representations have enjoyed success in several NLP tasks [1, 2]. More recently,
the use of distributed representations have been extended to model concepts beyond the word level,
such as sentences, phrases and paragraphs [3, 4, 5, 6], entities and relationships [7, 8] and embeddings of semantic categories [9, 10].
In this paper we propose a general framework for learning distributed representations of attributes:
characteristics of text whose representations can be jointly learned with word embeddings. The use
of the word attribute in this context is general. Table 1 illustrates several of the experiments we
perform along with the corresponding notion of attribute. For example, an attribute can represent
an indicator of the current sentence or language being processed. This allows us to learn sentence
and language vectors, similar to the proposed model of [6]. Attributes can also correspond to side
information, or metadata associated with text. For instance, a collection of blogs may come with
information about the age, gender or industry of the author. This allows us to learn vectors that can
capture similarities across metadata based on the associated body of text. The goal of this work
is to show that our notion of attribute vectors can achieve strong performance on a wide variety of
NLP related tasks. In particular, we demonstrate strong quantitative performance on three highly
diverse tasks: sentiment classification, cross-lingual document classification, and blog authorship
attribution.
To capture these kinds of interactions between attributes and text, we propose to use a third-order
model where attribute vectors act as gating units to a word embedding tensor. That is, words are
represented as a tensor consisting of several prototype vectors. Given an attribute vector, a word
embedding matrix can be computed as a linear combination of word prototypes weighted by the
attribute representation. During training, attribute vectors reside in a separate lookup table which
can be jointly learned along with word features and the model parameters. This type of three-way
1
Table 1: Summary of tasks and attribute types used in our experiments. The first three are quantitative while the second three are qualitative.
Task
Sentiment Classification
Cross-Lingual Classification
Authorship Attribution
Conditional Text Generation
Structured Text Generation
Conditional Word Similarity
Dataset
Sentiment Treebank
RCV1/RCV2
Blog Corpus
Gutenberg Corpus
Gutenberg Corpus
Blogs & Europarl
Attribute type
Sentence Vector
Language Vector
Author Metadata
Book Vector
Part of Speech Tags
Author Metadata / Language
interaction can be embedded into a neural language model, where the three-way interaction consists
of the previous context, the attribute and the score (or distribution) of the next word after the context.
Using a word embedding tensor gives rise to the notion of conditional word similarity. More specifically, the neighbours of word embeddings can change depending on which attribute is being conditioned on. For example, the word ?joy? when conditioned on an author with the industry attribute
?religion? appears near ?rapture? and ?god? but near ?delight? and ?comfort? when conditioned on
an author with the industry attribute ?science?. Another way of thinking of our model would be
the language analogue of [11]. They used a factored conditional restricted Boltzmann machine for
modelling motion style defined by real or continuous valued style variables. When our factorization
is embedded into a neural language model, it allows us to generate text conditioned on different
attributes in the same manner as [11] could generate motions from different styles. As we show in
our experiments, if attributes are represented by different books, samples generated from the model
learn to capture associated writing styles from the author. Furthermore, we demonstrate a strong
performance gain for authorship attribution when conditional word representations are used.
Multiplicative interactions have also been previously incorporated into neural language models. [12]
introduced a multiplicative model where images are used for gating word representations. Our
framework can be seen as a generalization of [12] and in the context of their work an attribute would
correspond to a fixed representation of an image. [13] introduced a multiplicative recurrent neural
network for generating text at the character level. In their model, the character at the current timestep
is used to gate the network?s recurrent matrix. This led to a substantial improvement in the ability to
generate text at the character level as opposed to a non-multiplicative recurrent network.
2
Methods
In this section we describe the proposed models. We first review the log-bilinear neural language
model of [14] as it forms the basis for much of our work. Next, we describe a word embedding
tensor and show how it can be factored and introduced into a multiplicative neural language model.
This is concluded by detailing how our attribute vectors are learned.
2.1
Log-bilinear neural language models
The log-bilinear language model (LBL) [14] is a deterministic model that may be viewed as a feedforward neural network with a single linear hidden layer. Each word w in the vocabulary is represented as a K-dimensional real-valued vector rw ? RK . Let R denote the V ? K matrix of
word representation vectors where V is the vocabulary size. Let (w1 , . . . wn?1 ) be a tuple of n ? 1
words where n ? 1 is the context size. The LBL model makes a linear prediction of the next word
representation as
n?1
X
?
r=
C(i) rwi ,
(1)
i=1
(i)
where C , i = 1, . . . , n ? 1 are K ? K context parameter matrices. Thus, ?
r is the predicted
representation of rwn . The conditional probability P (wn = i|w1:n?1 ) of wn given w1 , . . . , wn?1 is
exp(?
rT ri + bi )
P (wn = i|w1:n?1 ) = PV
,
rT rj + bj )
j=1 exp(?
where b ? RV is a bias vector. Learning can be done using backpropagation.
2
(2)
(a) NLM
(b) Multiplicative NLM
(c) Multiplicative NLM with language switch
Figure 1: Three different formulations for predicting the next word in a neural language model. Left:
A standard neural language model (NLM). Middle: The context and attribute vectors interact via
a multiplicative interaction. Right: When words are unshared across attributes, a one-hot attribute
vector gates the factors-to-vocabulary matrix.
2.2
A word embedding tensor
Traditionally, word representation matrices are represented as a matrix R ? RV ?K , such as in
the case of the log-bilinear model. Throughout this work, we instead represent words as a tensor
T ? RV ?K?D where D corresponds to the number of tensor slices. Given an attribute vector
PD
x ? RD , we can compute attribute-gated word representations as T x = i=1 xi T (i) i.e. word
representations with respect to x are computed as a linear combination of slices weighted by each
component xi of x.
It is often unnecessary to use a fully unfactored tensor. Following [15, 16], we re-represent T in
terms of three matrices Wf k ? RF ?K , Wf d ? RF ?D and Wf v ? RF ?V , such that
T x = (Wf v )> ? diag(Wf d x) ? Wf k ,
(3)
where diag(?) denotes the matrix with its argument on the diagonal. These matrices are parametrized
by a pre-chosen number of factors F .
2.3
Multiplicative neural language models
We now show how to embed our word representation tensor T into the log-bilinear neural language
model. Let E = (Wf k )> Wf v denote a ?folded? K ? V matrix of word embeddings. Given the
context w1 , . . . , wn?1 , the predicted next word representation ?
r is given by
?
r=
n?1
X
C(i) E(:, wi ),
(4)
i=1
where E(:, wi ) denotes the column of E for the word representation of wi and C(i) , i = 1, . . . , n?1
are K ? K context matrices. Given a predicted next word representation ?
r, the factor outputs are
f = (Wf k?
r) ? (Wf d x),
(5)
where ? is a component-wise product. The conditional probability P (wn = i|w1:n?1 , x) of wn
given w1 , . . . , wn?1 and x can be written as
exp (Wf v (:, i))> f + bi
P (wn = i|w1:n?1 , x) = PV
.
fv
>
j=1 exp (W (:, j)) f + bj
Here, Wf v (:, i) denotes the column of Wf v corresponding to word i. In contrast to the log-bilinear
model, the matrix of word representations R from before is replaced with the factored tensor T , as
shown in Fig. 1.
2.4
Unshared vocabularies across attributes
Our formulation for T assumes that word representations are shared across all attributes. In some
cases, words may only be specific to certain attributes and not others. An example of this is crosslingual modelling, where it is necessary to have language specific vocabularies. As a running example, consider the case where each attribute corresponds to a language representation vector. Let
3
Table 2: Samples generated from the model when conditioning on various attributes. For the last
example, we condition on the average of the two vectors (symbol <#> corresponds to a number).
Attribute
Bible
Caesar
1
2
(Bible +
Caesar)
Sample
<#> : <#> for thus i enquired unto thee , saying , the lord had not come unto
him . <#> : <#> when i see them shall see me greater am that under the name
of the king on israel .
to tell vs pindarus : shortly pray , now hence , a word . comes hither , and
let vs exclaim once by him fear till loved against caesar . till you are now which
have kept what proper deed there is an ant ? for caesar not wise cassi
let our spring tiger as with less ; for tucking great fellowes at ghosts of broth .
industrious time with golden glory employments . <#> : <#> but are far in men
soft from bones , assur too , set and blood of smelling , and there they cost ,
i learned : love no guile his word downe the mystery of possession
x denote the attribute vector for language ` and x0 for language `0 (e.g. English and French). We
can then compute language-specific word representations T ` by breaking up our decomposition into
language dependent and independent components (see Fig. 1c):
T ` = (W`f v )> ? diag(Wf d x) ? Wf k ,
(6)
where (W`f v )> is a V` ? F language specific matrix. The matrices Wf d and Wf k do not depend
on the language or the vocabulary, whereas (W`f v )> is language specific. Moreover, since each language may have a different sized vocabulary, we use V` to denote the vocabulary size of language `.
Observe that this model has an interesting property in that it allows us to share statistical strength
across word representations of different languages. In particular, we show in our experiments how
we can improve cross-lingual classification performance between English and German when a large
amount of parallel data exists between English and French and only a small amount of parallel data
exists between English and German.
2.5
Learning attribute representations
We now discuss how to learn representation vectors x. Recall that when training neural language
models, the word representations of w1 , . . . , wn?1 are updated by backpropagating through the
word embedding matrix. We can think of this as being a linear layer, where the input to this layer
is a one-hot vector with the i-th position active for word wi . Then multiplying this vector by the
embedding matrix results in the word vector for wi . Thus the columns of the word representations
matrix consisting of words from w1 , . . . , wn?1 will have non-zero gradients with respect to the loss.
This allows us to consistently modify the word representations throughout training.
We construct attribute representations in a similar way. Suppose that L is an attribute lookup table,
where x = f (L(:, x)) and f is an optional non-linearity. We often use a rectifier non-linearity in
order to keep x sparse and positive, which we found made training much more stable. Initially, the
entries of L are generated randomly. During training, we treat L in the same way as the word embedding matrix. This way of learning language representations allows us to measure how ?similar?
attributes are as opposed to using a one-hot encoding of attributes for which no such similarity could
be computed.
In some cases, attributes that are available during training may not also be available at test time.
An example of this is when attributes are used as sentence indicators for learning representations
of sentences. To accommodate for this, we use an inference step similar to that proposed by [6].
That is, at test time all the network parameters are fixed and stochastic gradient descent is used for
inferring the representation of an unseen attribute vector.
3
Experiments
In this section we describe our experimental evaluation and results. Throughout this section we refer
to our model as Attribute Tensor Decomposition (ATD). All models are trained using stochastic gradient descent with an exponential learning rate decay and linear (per epoch) increase in momentum.
We first demonstrate initial qualitative results to get a sense of the tasks our model can perform. For
these, we use the small project Gutenberg corpus which consists of 18 books, some of which have
the same author. We first trained a multiplicative neural language model with a context size of 5,
4
Table 3: A modified version of the game Mad Libs. Given an initialization, the model is to generate
the next 5 words according to the part-of-speech sequence (note that these are not hard constraints).
[DT, NN, IN, DT, JJ]
the meaning of life is...
the cure of the bad
the truth of the good
a penny for the fourth
the globe of those modern
all man upon the same
[TO, VB, VBD, JJS, NNS]
my greatest accomplishment is...
to keep sold most wishes
to make manned most magnificent
to keep wounded best nations
to be allowed best arguments
to be mentioned most people
[PRP, NN, ?,? , JJ, NN]
i could not live without...
his regard , willing tenderness
her french , serious friend
her father , good voice
her heart , likely beauty
her sister , such character
Table 4: Classification accuracies on various tasks. Left: Sentiment classification on the treebank dataset. Competing methods include the Neural Bag of words (NBoW) [5], Recursive Network (RNN) [17], Matrix-Vector Recursive Network (MV-RNN) [18], Recursive Tensor Network
(RTNN) [3], Dynamic Convolutional Network (DCNN) [5] and Paragraph Vector (PV) [6]. Right:
Cross-lingual classification on RCV2. Methods include statistical machine translation (SMT), IMatrix [19], Bag-of-words autoencoders (BAE-*) [20] and BiCVM, BiCVM+ [21]. The use of ?+?
on cross-lingual tasks indicate the use of a third language (French) for learning embeddings.
Method
SVM
BiNB
NBoW
RNN
MVRNN
RTNN
DCNN
PV
ATD
Fine-grained
40.7%
41.9%
42.4%
43.2%
44.4%
45.7%
48.5%
48.7%
45.9%
Positive / Negative
79.4%
83.1%
80.5%
82.4%
82.9%
85.4%
86.8%
87.8%
83.3%
Method
SMT
I-Matrix
BAE-cr
BAE-tree
BiCVM
BiCVM+
BAE-corr
ATD
ATD+
EN ? DE
68.1%
77.6%
78.2%
80.2%
83.7%
86.2%
91.8%
80.8%
83.4%
DE ? EN
67.4%
71.1%
63.6%
68.2%
71.4%
76.9%
72.8%
71.8%
72.9%
where each attribute is represented as a book. This results in 18 learned attribute vectors, one for
each book. After training, we can condition on a book vector and generate samples from the model.
Table 2 illustrates some the generated samples. Our model learns to capture the ?style? associated
with different books. Furthermore, by conditioning on the average of book representations, the
model can generate reasonable samples that represent a hybrid of both attributes, even though such
attribute combinations were not observed during training.
Next, we computed POS sequences from sentences that occur in the training corpus. We trained
a multiplicative neural language model with a context size of 5 to predict the next word from its
context, given knowledge of the POS tag for the next word. That is, we model P (wn = i|w1:n?1 , x)
where x denotes the POS tag for word wn . After training, we gave the model an initial input and
a POS sequence and proceeded to generate samples. Table 3 shows some results for this task.
Interestingly, the model can generate rather funny and poetic completions to the initial context.
3.1
Sentiment classification
Our first quantitative experiments are performed on the sentiment treebank of [3]. A common challenge for sentiment classification tasks is that the global sentiment of a sentence need not correspond
to local sentiments exhibited in sub-phrases of the sentence. To address this issue, [3] collected annotations from the movie reviews corpus of [22] of all subphrases extracted from a sentence parser.
By incorporating local sentiment into their recursive architectures, [3] was able to obtain significant
performance gains with recursive networks over bag of words baselines.
We follow the same experimental procedure proposed by [3] for which evaluation is reported on
two tasks: fine-grained classification of categories {very negative, negative, neutral, positive, very
positive } and binary classification {positive, negative }. We extracted all subphrases of sentences
that occur in the training set and used these to train a multiplicative neural language model. Here,
each attribute is represented as a sentence vector, as in [6]. In order to compute subphrases for
unseen sentences, we apply an inference procedure similar to [6], where the weights of the network
are frozen and gradient descent is used to infer representations for each unseen vector. We trained a
logistic regression classifier using all training subphrases in the training set. At test time, we infer a
representation for a new sentence which is used for making a review prediction. We used a context
5
size of 8, 100 dimensional word vectors initialized from [2] and 100 dimensional sentence vectors
initialized by averaging vectors of words from the corresponding sentence.
Table 4, left panel, illustrates our results on this task in comparison to all other proposed approaches.
Our results are on par with the highest performing recursive network on the fine-grained task and
outperforms all bag-of-words baselines and recursive networks with the exception of the RTNN on
the binary task. Our method is outperformed by the two recently proposed approaches of [5] (a
convolutional network trained on sentences) and Paragraph Vector [6].
3.2
Cross-lingual document classification
We follow the experimental procedure of [19], for which several existing baselines are available to
compare our results. The experiment proceeds as follows. We first use the Europarl corpus [23] for
inducing word representations across languages. Let S be a sentence with words w in language `
and let x be the corresponding language vector. Let
X
X
v` (S) =
T ` (:, w) =
(W`f v (:, w))> ? diag(Wf d x) ? Wf k
(7)
w?S
w?S
denote the sentence representation of S, defined as the sum of language conditioned word representations for each w ? S. Equivalently we define a sentence representation for the translation S 0 of S
denoted as v`0 (S 0 ). We then optimize the following ranking objective:
XX
2
2
2
minimize
max 0, ? +
v` (S) ? v`0 (S 0 )
2 ?
v` (S) ? v`0 (Ck )
2 + ?
?
2
?
S
k
subject to the constraints that each sentence vector has unit norm. Each Ck is a constrastive (nontranslation) sentence of S and ? denotes all model parameters. This type of cross-language ranking
loss was first used by [21] but without the norm constraint which we found significantly improved
the stability of training. The Europarl corpus contains roughly 2 million parallel sentence pairs
between English and German as well as English and French, for which we induce 40 dimensional
word representations. Evaluation is then performed on English and German sections of the Reuters
RCV1/RCV2 corpora. Note that these documents are not parallel. The Reuters dataset contains
multiple labels for each document. Following [19], we only consider documents which have been
assigned to one of the top 4 categories in the label hierarchy. These are CCAT (Corporate/Industrial),
ECAT (Economics), GCAT (Government/Social) and MCAT (Markets). There are a total of 34,000
English documents and 42,753 German documents with vocabulary sizes of 43614 English words
and 50,110 German words. We consider both training on English and evaluating on German and
vice versa. To represent a document, we sum over the word representations of words in that document followed by a unit-ball projection. Following [19] we use an averaged perceptron classifier.
Classification accuracy is then evaluated on a held-out test set in the other language. We used a
monolingual validation set for tuning the margin ?, which was set to ? = 1. Five contrastive terms
were used per example which were randomly assigned per epoch.
Table 4, right panel, shows our results compared to all proposed methods thus far. We are competitive with the current state-of-the-art approaches, being outperformed only by BiCVM+ [21] and
BAE-corr [20] on EN ? DE. The BAE-corr method combines both a reconstruction term and a
correlation regularizer to match sentences, while our method does not consider reconstruction. We
also performed experimentation on a low resource task, where we assume the same conditions as
above with the exception that we only use 10,000 parallel sentence pairs between English and German while still incorporating all English and French parallel sentences. For this task, we compare
against a separation baseline, which is the same as our model but with no parameter sharing across
languages (and thus resembles [21]). Here we achieve 74.7% and 69.7% accuracies (EN?DE and
DE?EN) while the separation baseline obtains 63.8% and 67.1%. This indicates that parameter sharing across languages can be useful when only a small amount of parallel data is available.
Figure 2 further shows t-SNE embeddings of English-German word pairs.1
Another interesting consideration is whether or not the learned language vectors can capture any
interesting properties of various languages. To look into this, we trained a multiplicative neural
language model simultaneously on 5 languages: English, French, German, Czech and Slovak. To
our knowledge, this is the most languages word representations have been jointly learned on. We
1
We note that Germany and Deutschland are nearest neighbours in the original space.
6
(a) Months
(b) Countries
5
4
3
2
1
0.3
0.2
0.1
0.0
? 0.1
uncondit ioned ATD
? 0.2
0
(a) Correlation matrix
uncondit ioned ATD
LBL
condit ioned ATD
6
Inferred at t ribut es difference
Im provem ent over init ial m odel
Figure 2: t-SNE embeddings of English-German word pairs learned from Europarl.
5
10
25
50
100
# Docum ent s (t housands)
382
5
10
25
50
100
# Docum ent s (t housands)
382
(b) Effect of conditional embeddings (c) Effect of inferring attribute vectors
Figure 3: Results on the Blog classification corpus. For the middle and right plots, each pair of same
coloured bars corresponds to the non-inclusion or inclusion of inferred attribute vectors, respectively.
computed a correlation matrix from the language vectors, illustrated in Fig. 3a. Interestingly, we
observe high correlation between Czech and Slovak representations, indicating that the model may
have learned some notion of lexical similarity. That being said, additional experimentation for future
work is necessary to better understand the similarities exhibited through language vectors.
3.3
Blog authorship attribution
For our final task, we use the Blog corpus of [24] which contains 681,288 blog posts from 19,320
authors. For our experiments, we break the corpus into two separate datasets: one containing the
1000 most prolific authors (most blog posts) and the other containing all the rest. Each author comes
with an attribute tag corresponding to a tuple (age, gender, industry) indicating the age range of the
author (10s, 20s or 30s), whether the author is male or female, and what industry the author works
in. Note that industry does not necessary correspond to the topic of blog posts. We use the dataset
of non-prolific authors to train a multiplicative language model conditioned on an attribute tuple
of which there are 234 unique tuples in total. We used 100 dimensional word vectors initialized
from [2], 100 dimensional attribute vectors with random initialization and a context size of 5. A
1000-way classification task is then performed on the prolific author subset and evaluation is done
using 10-fold cross-validation. Our initial experimentation with baselines found that tf-idf performs
well on this dataset (45.9% accuracy). Thus, we consider how much we can improve on the tf-idf
baseline by augmenting word and attribute features.
For the first experiment, we determine the effect conditional word embeddings have on classification
performance, assuming attributes are available at test time. For this, we compute two embedding
matrices from a trained ATD model, one without and with attribute knowledge:
unconditioned ATD :
conditioned ATD :
(Wf v )> Wf k
(8)
(Wf v )> ? diag(Wf d x) ? Wf k .
(9)
We represent a blog post as the sum of word vectors projected to unit norm and augment these with
tf-idf features. As an additional baseline we include a log-bilinear language model [14]. 2 Figure
3b illustrates the results from which we observe that conditioned word embeddings are significantly
more discriminative over word embeddings computed without knowledge of attribute vectors.
2
The log-bilinear model has no concept of attributes.
7
Table 5: Results from a conditional word similarity task using Blog attributes and language vectors.
Query,A,B
school
f/10/student
m/20/tech
journal
f/10/student
m/30/adv.
create
f/30/arts
f/30/internet
joy
m/30/religion
m/20/science
cool
m/10/student
f/10/student
Common
work
church
college
diary
blog
webpage
build
develop
maintain
happiness
sadness
pain
nice
funny
awesome
Unique to A
choir
prom
skool
project
book
yearbook
provide
acquire
generate
rapture
god
heartbreak
beautiful
amazing
neat
Unique to B
therapy
tech
job
zine
app
referral
compile
follow
analyse
delight
comfort
soul
sexy
hott
lame
English
january
june
october
market
markets
internal
war
weapons
global
said
stated
told
two
two-thirds
both
French
janvier
decembre
juin
marche
marches
interne
guerre
terrorisme
mondaile
dit
disait
declare
deux
deuxieme
seconde
German
januar
dezember
juni
markt
binnenmarktes
marktes
krieg
globale
krieges
sagte
gesagt
sagten
zwei
beiden
zweier
For the second experiment, we determine the effect of inferring attribute vectors at test time if they
are not assumed to be available. To do this, we train a logistic regression classifier within each fold
for predicting attributes. We compute an inferred vector by averaging each of the attribute vectors
weighted by the log-probabilities of the classifier. In Fig. 3c we plot the difference in performance
when an inferred vector is augmented vs. when it is not. These results show consistent, albeit small
improvement gains when attribute vectors are inferred at test time.
To get a better sense of the attribute features learned from the model, the supplementary material
contains a t-SNE embedding of the learned attribute vectors. Interestingly, the model learns features
which largely isolate the vectors of all teenage bloggers independent of gender and topic.
3.4
Conditional word similarity
One of the key properties of our tensor formulation is the notion of conditional word similarity,
namely how neighbours of word representations change depending on the attributes that are conditioned on. In order to explore the effects of this, we performed two qualitative comparisons: one
using blog attribute vectors and the other with language vectors. These results are illustrated in
Table 5. For the first comparison on the left, we chose two attributes from the blog corpus and a
query word. We identify each of these attribute pairs as A and B. Next, we computed a ranked list of
the nearest neighbours (by cosine similarity) of words conditioned on each attribute and identified
the top 15 words in each. Out of these 15 words, we display the top 3 words which are common
to both ranked lists, as well as 3 words that are unique to a specific attribute. Our results illustrate
that the model can capture distinctive notions of word similarities depending on which attributes
are being conditioned. On the right of Table 5, we chose a query word in English (italicized) and
computed the nearest neighbours when conditioned on each language vector. This results in neighbours that are either direct translations of the query word or words that are semantically similar. The
supplementary material includes additional examples with nearest neighbours of collocations.
4
Conclusion
There are several future directions from which this work can be extended. One application area
of interest is in learning representations of authors from papers they choose to review as a way of
improving automating reviewer-paper matching [25]. Since authors contribute to different research
topics, it might be more useful to instead consider a mixture of attribute vectors that can allow for
distinctive representations of the same author across research areas. Another interesting application
is learning representations of graphs. Recently, [26] proposed an approach for learning embeddings
of nodes in social networks. Introducing network indicator vectors could allow us to potentially
learn representations of full graphs. Finally, it would be interesting to train a multiplicative neural
language model simultaneously across dozens of languages.
Acknowledgments
We would also like to thank the anonymous reviewers for their valuable comments and suggestions.
This work was supported by NSERC, Google, Samsung, and ONR Grant N00014-14-1-0232.
8
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4,889 | 5,426 | Sparse Polynomial Learning and Graph Sketching
Murat Kocaoglu1? , Karthikeyan Shanmugam1? , Alexandros G.Dimakis1? , Adam Klivans2?
1
Department of Electrical and Computer Engineering, 2 Department of Computer Science
The University of Texas at Austin, USA
?
[email protected], ? [email protected]
?
[email protected], ? [email protected]
Abstract
Let f : {?1, 1}n ? R be a polynomial with at most s non-zero real coefficients.
We give an algorithm for exactly reconstructing f given random examples from
the uniform distribution on {?1, 1}n that runs in time polynomial in n and 2s
and succeeds if the function satisfies the unique sign property: there is one output
value which corresponds to a unique set of values of the participating parities. This
sufficient condition is satisfied when every coefficient of f is perturbed by a small
random noise, or satisfied with high probability when s parity functions are chosen
randomly or when all the coefficients are positive. Learning sparse polynomials
over the Boolean domain in time polynomial in n and 2s is considered notoriously
hard in the worst-case. Our result shows that the problem is tractable for almost
all sparse polynomials.
Then, we show an application of this result to hypergraph sketching which is the
problem of learning a sparse (both in the number of hyperedges and the size of
the hyperedges) hypergraph from uniformly drawn random cuts. We also provide
experimental results on a real world dataset.
1
Introduction
Learning sparse polynomials over the Boolean domain is one of the fundamental problems from
computational learning theory and has been studied extensively over the last twenty-five years [1?
6]. In almost all cases, known algorithms for learning or interpolating sparse polynomials require
query access to the unknown polynomial. An outstanding open problem is to find an algorithm
for learning s-sparse polynomials with respect to the uniform distribution on {?1, 1}n that runs in
time polynomial in n and g(s) (where g is any fixed function independent of n) and requires only
randomly chosen examples to succeed. In particular, such an algorithm would imply a breakthrough
result for the problem of learning k-juntas (functions that depend on only k n input variables; it
is not known how to learn ?(1)-juntas in polynomial time).
We present an algorithm and a set of natural conditions such that any sparse polynomial f satisfying these conditions can be learned from random examples in time polynomial in n and 2s . In
particular, any f whose coefficients have been subjected to a small perturbation (smoothed analysis
setting) satisfies these conditions (for example, if a Gaussian with arbitrarily small variance is added
independently to each coefficient, f satisfies these conditions with probability 1). We state our main
result here:
Theorem 1. Let f be an s-sparse function that satisfies at least one of the following properties:
a) (smoothed analysis setting)The coefficients {ci }si=1 are in general position or all of them are
perturbed by a small random noise. b) The s parity functions are linearly independent. c) All the
coefficients are positive. Then we learn f with high probability in time poly(n, 2s ).
1
We note that smoothed-analysis, pioneered in [7], has now become a common alternative for problems that seem intractable in the worst-case.
Our algorithm also succeeds in the presence of noise:
Theorem 2. Let f = f1 + f2 be a polynomial such that f1 and f2 depend on mutually disjoint set
of parity functions. f1 is s-sparse and the values of f1 are ?well separated?. Further, kf2 k1 ? ?,
(i.e., f is approximately sparse). If observations are corrupted by additive noise bounded by , then
there exists an algorithm which takes + ? as an input, that gives g in time polynomial in n and 2s
such that kf ? gk2 ? O(? + ) with high probability.
The treatment of the noisy case, i.e., the formal statement of this theorem, the corresponding algorithm, and the related proofs are relegated to the supplementary material. All these results are
based on what we call as the unique sign property: If there is one value that f takes which uniquely
specifies the signs of the parity functions involved, then the function is efficiently learnable. Note
that our results cannot be used for learning juntas or other Boolean-valued sparse polynomials, since
the unique sign property does not hold in these settings.
We show that this property holds for the complement of the cut function on a hypergraph (no. of
hyperedges ? cut value). This fact can be used to learn the cut complement function and eventually
infer the structure of a sparse hypergraph from random cuts. Sparsity implies that the number of
hyperedges and the size of each hyperedge is of constant size. Hypergraphs can be used to represent
relations in many real world data sets. For example, one can represent the relation between the books
and the readers (users) on the Amazon dataset with a hypergraph. Book titles and Amazon users
can be mapped to nodes and hyperedges, respectively ([8]). Then a node belongs to a hyperedge, if
the corresponding book is read by the user represented by that hyperedge. When such graphs evolve
over time (and space), the difference graph filtered by time and space is often sparse. To locate
and learn the few hyperedges from random cuts in such difference graphs constitutes hypergraph
sketching. We test our algorithms on hypergraphs generated from the dataset that contain the time
stamped record of messages between Yahoo! messenger users marked with the user locations (zip
codes).
1.1
Approach and Related Work
The problem of recovering the sparsest solution of a set of underdetermined linear equations has received significant recent attention in the context of compressed sensing [9?11]. In compressed sensing, one tries to recover an unknown sparse vector using few linear observations (measurements),
possibly in the presence of noise.
The recent papers [12,13] are of particular relevance to us since they establish a connection between
learning sparse polynomials and compressed sensing. The authors show that the problem of learning
a sparse polynomial is equivalent to recovering the unknown sparse coefficient vector using linear
measurements. By applying techniques from compressed sensing theory, namely Restricted Isometry Property (see [12]) and incoherence (see [13]), the authors independently established results for
reconstructing sparse polynomials using convex optimization. The results have near-optimal sample
complexity. However, the running time of these algorithms is exponential in the underlying dimension, n. This is because the measurement matrix of the equivalent compressed sensing problem
requires one column for every possible non-zero monomial.
In this paper, we show how to solve this problem in time polynomial in n and 2s under the assumption of unique sign property on the sparse polynomial. Our key contribution is a novel identification
procedure that can reduce the list of potentially non-zero coefficients from the naive bound of 2n to
2s when the function has this property.
On the theoretical side, there has been interesting recent work of [14] that approximately learns
sparse polynomial functions when the underlying domain is Gaussian. Their results do not seem to
translate to the Boolean domain. We also note the work of [15] that gives an algorithm for learning
sparse Boolean functions with respect to a randomly chosen product distribution on {?1, 1}n . Their
work does not apply to the uniform distribution on {?1, 1}n .
On the practical side, we give an application of the theory to the problem of hypergraph sketching.
We generalize a prior work [12] that applied the compressed sensing approach discussed before to
2
graph sketching on evolving social network graphs. In our algorithm, while the sample complexity
requirements are higher, the time complexity is greatly reduced in comparison. We test our algorithms on a real dataset and show that the algorithm is able to scale well on sparse hypergraphs
created out of Yahoo! messenger dataset by filtering through time and location stamps.
2
Definitions
Consider a real-valued function over the Boolean hypercube f : {?1, 1}n ? R. Given a sequence
of labeled samples of the form hf (x), xi, where x is sampled from the uniform distribution U over
the hypercube {?1, 1}n , we are interested in an efficient algorithm that learns the function f with
high probability. Through Fourier expansion, f can be written as a linear combination of monomials:
X
f (x) =
cS ?S (x), ? x ? {?1, 1}n
(1)
S?[n]
where [n] is the set of integers from 1 to n, ?S (x) =
Q
xi and cS ? R. Let c be the vector of
i?S
coefficients cS . A monomial ?S (x) is also called a parity function. More background on Boolean
functions and the Fourier expansion can be found in [16].
In this work, we restrict ourselves to sparse polynomials f with sparsity s in the Fourier domain, i.e.,
f is a linear combination of unknown parity functions ?S1 (x), ?S2 (x), . . . ?Ss (x) with s unknown
real coefficients given by {cSi }si=1 such that cSi 6= 0, ?1 ? i ? s; all other coefficients are 0. Let
the subsets corresponding to the s parity functions form a family of sets I = {Si }si=1 . Finding I is
equivalent to finding the s parity functions.
Note: In certain places, where the context makes it clear, we slightly abuse the notation such that
the set Si identifying a specific parity function is replaced by just the index i. The coefficients may
be denoted simply by ci and the parity functions by ?i (?).
Let F2 denote the binary field. Every parity function ?i (?) can be represented by a vector pi ? Fn?1
.
2
The j-th entry pi (j) in the vector pi is 1, if j ? Si and is 0 otherwise.
Definition 1. A set of s parity functions {?i (?)}si=1 are said to be linearly independent if the corresponding set of vectors {pi }si=1 are linearly independent over F2 .
Similarly, they are said to have rank r if the dimension of the subspace spanned by {pi }si=1 is r.
Definition 2. The coefficients {ci }si=1 are said to be in general position if for all possible set of
s
P
values bi ? {0, 1, ?1}, ? 1 ? i ? s, with at least one nonzero bi ,
ci bi 6= 0
i=1
Definition 3. The coefficients {ci }si=1 are said to be ?-separated
if for all possible set of values
s
P
bi ? {0, 1, ?1}, ? 1 ? i ? s with at least one nonzero bi , ci bi > ?.
i=1
Definition 4. A sign pattern is a distinct vector of signs a = [?1 (?) , ?2 (?) , . . . ?s (?))] ?
{?1, 1}1?s assumed by the set of s parity functions.
Since this work involves switching representations between the real and the binary field, we define
a function q that does the switch.
Definition 5. q : {?1, 1}a?b ? F2a?b is a function that converts a sign matrix X to a matrix Y
over F2 such that Yij = q(Xij ) = 1 ? F2 , if Xij = ?1 and Yij = q(Xij ) = 0 ? F2 , if Xij = 1.
Clearly, it has an inverse function q ?1 such that q ?1 (Y) = X.
We also present some definitions to deal with the case when the polynomial f is not exactly s-sparse
and observations are noisy. Let 2[n] denote the power set of [n].
n
Definition 6. A polynomial f : {?1,
P 1} ? R is called approximately (s, ?)-sparse if there exists
[n]
I ? 2 with |I| = s such that
|cS | < ?, where {cS } are the Fourier coefficients as in (1).
S?I c
In other words, the sum of the absolute values of all the coefficients except the ones corresponding
to I are rather small.
3
3
Problem Setting
Suppose m labeled samples hf (x) , xim
drawn from the uniform distribution U on the Boolean
i=1 are
n
hypercube. For any B ? 2[n] , let cB ? R2 ?1 be the vector of real coefficients such that cB (S) =
n
cS , ?S ? B and cB (S) = 0, ?S ?
/ B. Let A ? Rm?2 be such that every row of A corresponds
to one random input sample x ? U . Let x also denote the row index and S ? [n] denote the
column index of A. A(x, S) = ?S (x). Let AS denote the sub matrix formed by the columns
corresponding to the subsets in S. Let I be the set consisting of the s parity functions of interest
in both the sparse and the approximately sparse cases. A sparse representation of an approximately
(s, ?)-sparse function f is fI = A(x) cI , where cI is as defined above.
We review the compressed sensing framework used in [12] and [13]. Specifically, for the remainder
of the paper, we rely on [13] as a point of reference. We review their framework and explain how
we use it to obtain our results, particularly for the noisy case.
n
Let y ? Rm and ?S ? R2 , such that ?S = 0, ?S ? S c . Note that, here S is a subset of the power
set 2[n] . Now, consider the following convex program for noisy compressed sensing in this setting:
r
1
mink?S k1 subject to
kA?S ? yk2 ? .
(2)
m
Let ?Sopt be an optimum for the program (2). Note that only the columns of A in S are used in the
program. The convex program runs in time poly (m, |S|). The incoherence property of the matrix
A in [13] implies the following.
Theorem 3. ( [13]) For any family of subsets I ? 2[n] such that |I| = s, m = 4096ns2 and
c1 = 4, c2 = 8, for any feasible point ?S of program 2, we have:
n 1/4
k?I c T S k1
(3)
k?S ? ?Sopt k2 ? c1 + c2
m
with probability at least 1 ? O 41n
When S is set to the power set 2[n] , = 0 and y is the vector of observed values for an s-sparse
polynomial, the s-sparse vector cI is a feasible point to program (2). By Theorem 3, the program
recovers the sparse vector cI and hence learns the function. The only caveat is that the complexity
is exponential in n.
The main idea behind our algorithms for noiseless and noisy sparse function learning is to ?capture?
the actual s-sparse set I of interest in a small set S : |S| = O (2s ) of coefficients by a separate
algorithm that runs in time poly(n, 2s ). Using the restricted set of coefficients S, we search for the
sparse solution under the noisy and noiseless cases using program (2).
Lemma 1. Given an algorithm that runs in time poly(n, 2s ) and generates a set of parities S such
that |S| = O (2s ) , I ? S with |I| = s, program (2) with S and m = 4096ns2 random samples
as
inputs runs in time poly(n, 2s ) and learns the correct function with probability 1 ? O 41n .
Unique Sign Pattern Property: The key property that lets us find a small S efficiently is the
unique sign pattern property. Observe that an s-sparse function can produce at most 2s different real
values. If the maximum value obtained always corresponds to a unique pattern of signs of parities,
by looking only at the random samples x corresponding to the subsequent O(n) occurrences of this
maximum value, we show that all the parity functions needed to learn f are captured in a small set
of size 2s+1 (see Lemma 2 and its proof). The unique sign property again plays an important role,
along with Theorem 3 with more technicalities added, in the noisy case, which we visit in Section 2
of the supplementary material.
In the next section, we provide an algorithm to generate the bounded set S for the noiseless case for
an s-sparse function f and provide guarantees for the algorithm formally.
4
Algorithm and Guarantees: Noiseless case
Let I be the family of s subsets {Si }si=1 each corresponding to the s parity functions ?Si (?) in an
s-sparse function f . In this section, we provide an algorithm, named LearnBool, that finds a small
4
subset S of the power set 2[n] that contains elements of I first and then uses program (2) with S.
We show that the algorithm learns f in time poly (n, 2s ) from uniformly randomly drawn labeled
samples from the Boolean hypercube with high probability under some natural conditions.
Recall that if the function is such that f (x) attains its maximum value only if
[?1 (x), ?2 (x) . . . ?s (x)] = amax ? {?1, 1}s for some unique sign pattern amax , then the function
is said to possess the unique sign property. Now we state the main technical lemma for the unique
sign property.
Lemma 2. If an s-sparse function f has the unique
sign property then, in Algorithm 1, S is such
that I ? S, |S| ? 2s+1 with probability 1 ? O n1 and runs in time poly(n, 2s ).
Proof. See the supplementary material.
The proof of the above lemma involves showing that the random matrix Ymax (see Algorithm 1) has
rank at least n ? s, leading to at most 2s solutions for each equation in (4). The feasible solutions
can be obtained by Gaussian elimination in the binary field.
Theorem 4. Let f be an s-sparse function that satisfies at least one of the following properties:
(a) The coefficients {ci }si=1 are in general position.
(b) The s parity functions are linearly independent.
(c) All the coefficients are positive.
Given labeled samples,
Algorithm 1 learns f exactly (or vopt = c) in time poly (n, 2s ) with proba
1
bility 1 ? O n .
Proof. See the supplementary material.
Smoothed Analysis Setting: Perturbing ci ?s with Gaussian random variables of standard deviation
? > 0 or by random variables drawn from any set of reasonable continuous distributions ensures
that the perturbed function satisfies property (a) with probability 1.
Random Parity Functions: When ci ?s are arbitrary and the set of s parity functions are drawn uniformly randomly from 2[n] , then property (b) holds with high probability if s is a constant.
1
Input: Sparsity parameter s, m1 = 2n2s random labeled samples {hf (xi ) , xi i}m
i=1 .
max
Pick samples {xij }nj=1
corresponding to the maximum value of f observed in all the m samples.
Stack all xij row wise into a matrix Xmax of dimensions nmax ? n.
Initialise S = ?. Let Ymax = q (Xmax ).
Find all feasible solutions p ? Fn?1
such that:
2
1nmax ?1 = Ymax p or 0nmax ?1 = Ymax p
(4)
F2n?1 .
Collect all feasible solutions p to either of the above equations in the set P ?
S = {{j ? [n] : p(j) = 1}|p ? P }.
Using m = 4096ns2 more samples (number of rows of A is m corresponding to these new
samples), solve:
?Sopt = mink?S k1 such that A?S = y,
(5)
where y is the vector of m observed values.
Set vopt = ?Sopt .
Output: vopt .
Algorithm 1: LearnBool
5
A Sparse Polynomial Learning Application: Hypergraph Sketching
Hypergraphs can be used to model the relations in real world data sets (e.g., books read by users in
Amazon). We show that the cut functions on hypergraphs satisfy the unique sign property. Learning a cut function of a sparse hypergraph from random cuts is a special case of learning a sparse
5
polynomial from samples drawn uniformly from the Boolean hypercube. To track the evolution of
large hypergraphs over a small time interval, it is enough to learn the cut function of the difference
graph which is often sparse. This is called the graph sketching problem. Previously, graph sketching
was applied to social network evolution [12]. We generalize this to hypergraphs showing that they
satisfy the unique sign property, which enable faster algorithms, and provide experimental results
on real data sets.
5.1
Graph Sketching
A hypergraph G = (V, E) is a set of vertices V along with a set E of subsets of V called the
hyperedges. The size of a hyperedge is the number of variables that the hyperedge connects. Let d
be the maximum hyperedge size of graph G. Let |V | = n and |E| = s.
A random cut S ? V is aT
set of vertices
T selected uniformly at random. Define the value of the cut S
to be c(S) = |{e ? E : e S 6= ?, e V ? S 6= ?}|. Graph sketching is the problem of identifying
the graph structure from random queries that evaluate the value of a random cut, where s n
(sparse setting). Hypergraphs naturally specify relations among a set of objects through hyperedges.
For example, Amazon users can form the set E and Amazon books can form the set V . Each user
may read a subset of books which represents the hyperedge. Learning the hypergraph corresponds
to identifying the sets of books bought by each user. For more examples of hypergraphs in real data
sets, we refer the reader to [8]. Such hypergraphs evolve over time. The difference graph between
two consecutive time instants is expected to be sparse (number of edges s and maximum hyperedge
size d are small). We are interested in learning such hypergraphs from random cut queries.
For simplicity and convenience, we consider the cut complement query, i.e., c?cut, which returns
s ? c(S). One can easily represent the c?cut query with a sparse polynomial as follows: Let node
i correspond to variable xi ? {?1, +1}. A random cut involves choosing xi uniformly randomly
from {?1, +1}. The variables assigned to +1 belong to the random cut S. The value is given by
the polynomial
?
?
!
X Y (1 + xi ) Y (1 ? xi )
X 1 ? X
Y
?
fc?cut (x) =
+
=
(1 +
xi )? . (6)
?
|I|?1
2
2
2
J ?I,
I?E i?I
i?I
I?E
i?J
|J |is even
Hence, the c?cut function is a sparse polynomial where the sparsity is at most s2d?1 . The variables
corresponding to the nodes that belong to some hyperedge appear in the polynomial. We call these
the relevant variables and the number of relevant variables is denoted by k. Note that, in our sparse
setting k ? sd. We note that for a hypergraph with no singleton hyperedge, given the c?cut function,
it is easy to recover the hyper edges from (6). Therefore, we focus on learning the c?cut function to
sketch the hypergraph.
When G is a graph with edges (of cardinality 2), the compressed sensing approach (using program
2) using the cut (or c?cut) values as measurements is shown to be very efficient in [12] in terms
of the sample complexity, i.e., the required number of queries. The run time is efficient because
total number of candidate parities is O(n2 ). However when we consider hypergraphs, i.e., when
d is a large constant, the compressed sensing approach cannot scale computationally (poly(nd )
runtime). Here, based on the theory developed, we give a faster algorithm based on the unique
sign property with sample complexity m1 = O(2k d log n + 22d+1 s2 (log n + k)) and run time of
O(m1 2k , n2 log n)).
We observe that the c?cut polynomial satisfies the unique sign property. From (6), it is evident
that the polynomial has only positive coefficients. Therefore, by Theorem 4, algorithm LearnBool
succeeds. The maximum value of the c?cut function is the number of edges. Notice that the
maximum value is definitely observed in two configurations of the relevant variables: If either all
relevant variables are +1 or all are ?1. Therefore, the maximum value is observed in every 2k?1 ?
2sd samples. Thus, a direct application of LearnBool yields poly(n, 2k?1 ) time complexity, which
improves the O(nd ) bound for small s and d.
Improving further, we provide a more efficient algorithm tailored for the hypergraph sketching problem, which makes use of the unique sign property and some other properties of the cut function.
Algorithm LearnGraph (Algorithm 4) is provided in the supplementary material.
6
4
10
Error Probability vs. ?
Runtime of LearnGraph vs. standard compressed sensing
3
0.25
Prob. of Error
Runtime (seconds)
10
2
10
1
0
0
0.2
0.15
10
10
Setting 1
Setting 3
Setting 2
Setting 4
LearnGraph
Comp. Sensing
200
400
600
No. of variables, n
800
0.1
1000
(a) Runtime vs. # of variables, d = 3 and s = 1.
1
2
3
4
5
6
7
? (# of samples/n)
8
9
10
(b) Probability of error vs. ?.
Figure 1: Performance figures comparing LearnGraph and Compressed Sensing approach.
Theorem 5. Algorithm 4 exactly learns the c?cut function with probability 1 ? O( n1 )with sample
complexity m1 = O(2k d log n + 22d+1 s2 (log n + k)) and time complexity O(2k m1 + n2 d log n)) .
Proof. See the supplementary material.
5.2
Yahoo! Messenger User Communication Pattern Dataset
We performed simulations using MATLAB on an Intel(R) Xeon(R) quad-core 3.6 GHz machine
with 16 GB RAM and 10M cache. We run our algorithm on the Yahoo! Messenger User Communication Pattern Dataset [17]. This dataset contains the timestamped user communication data, i.e.,
information about a large number of messages sent over Yahoo! Messenger, for a duration of 28
days.
Dataset: Each row represents a message. The first two columns show the day and time (time
stamp) of the message respectively. The third and fifth columns show the ID of the transmitting and
receiving users, respectively. The fourth column shows the zipcode (spatial stamp) from which this
particular message is transmitted. The sixth column shows if the transmitter was in the contact list
of the reciver user (y) or not (n). If a transmitter sends the same receiver more than one message
from the same zipcode, only the first message is shown in the dataset. In total, there are 100000
unique users and 5649 unique zipcodes.
We form a hypergraph from the dataset as follows: The transmitting users form the hyperedges and
the receiving users form the nodes of the hypergraph. A hyperedge connects a set T of users if
there is a transmitting user that sends a message to all the users in T . In any given time interval ?t
(short time interval) and small set of locations ?x specified by the number of zip codes, there are
few users who transmit (s) and they transmit to very few users (d). The complete set of nodes in the
hypergraph (n) is taken to be those receiving users who are active during m consecutive intervals
of length ?t and in a set of ?x zipcodes. This gives rise to a sparse graph. We identify the active
set of transmitting users (hyperedges) and their corresponding receivers (nodes in these hyperedges)
during a short time interval ?t and a randomly selected space interval (?x, i.e., zip codes) from a
large pool of receivers (nodes) that are observed during m intervals of length ?t. Details of ?t, m
and ?x chosen for experiments are given in Table 1. We note that n is in the order of 1000 usually.
Remark: Our task is to learn the c?cut function from the random queries, i.e., random +/-1 assignment of variables and corresponding c?cut values. The generated sparse graph contains only
hyperedges that have more than 1 node. Other hyperedges (transmitting users) with just one node in
the sparse hypergraph are not taken into account. This is because a singleton hyperedge i is always
counted in the c?cut function thereby effectively its presence is masked. First, we identify the relevant variables that participate in the sparse graph. After identifying this set of candidates, correlating
the corresponding candidate parities with the function output yields the Fourier coefficient of that
parity (see Algorithm 4).
7
Table 1: Runtime for different graphs. LG: LearnGraph, CS: Compressed sensing based alg.
(b) Runtime for d = 4 and s = 3 graph.
(a) Runtime for d = 4 and s = 1 graph.
HnH
Alg. H
LG
CS
88
159
288
556
1221
1.96
265.63
2.13
-
2.23
-
2.79
-
4.94
-
n
HH
Alg. H
LG
CS
52
104
246
412
1399
1.91
39.89
2.08
> 10823
2.08
-
2.30
-
4.98
-
(c) Simulation parameters for Fig. 1b
5.2.1
Setting No.
Interval
# of Int.
n
max(d)
max(s)
Zip. Set Size
Setting 1
Setting 2
Setting 3
Setting 4
5 min.
20 sec.
10 min.
2 min.
20
200
10
50
6822
5730
6822
6822
10
22
11
30
19
4
13
21
20
200
2
50
Performance Comparison with Compressed Sensing Approach
First, we compare the runtime of our implementation LearnGraph with the compressed sensing
based algorithm from [12]. Both algorithms correctly identify the relevant variables in all the considered range of parameters. The last step of finding the corresponding Fourier coefficients is omitted
and can be easily implemented (Algorithm 4) without significantly affecting the running time. As
can be seen in Tables 1a, 1b and Fig. 1a, LearnGraph scales well to graphs on thousands of nodes.
On the contrary, the compressed sensing approach must handle a measurement matrix of size O(nd ),
which becomes prohibitively large on graphs involving more than a few hundred nodes.
5.2.2
Error Performance of LearnGraph
Error probability (probability that the correct c?cut function is not recovered) versus the number
of samples used is plotted for four different experimental settings of ?t, ?x and m in Fig. 1b. For
each time interval, the error probability is calculated by averaging the number of errors among 100
different trials. For each value of ? (number of samples), the error probability is averaged over time
intervals to illustrate the error performance. We only keep the intervals for which the graph filtered
with the considered zipcodes contains at least one user with more than one neighbor. We find that
for the first 3 settings, the error probability decreases with more samples. For the fourth setting, d
and s are very large and hence a large number of samples are required. For that reason, the error
probability does not improve significantly. The probability of error can be reduced by repeating the
experiment multiple times and taking a majority, at the cost of significantly more samples. Our plot
shows only the probability of error without such a majority amplification.
6
Conclusions
We presented a novel algorithm for learning sparse polynomials by random samples on the Boolean
hypercube. While the general problem of learning all sparse polynomials is notoriously hard, we
show that almost all sparse polynomials can be efficiently learned using our algorithm. This is
because our unique sign property holds for randomly perturbed coefficients, in addition to several
other natural settings. As an application, we show that graph and hypergraph sketching lead to sparse
polynomial learning problems that always satisfy the unique sign property. This allows us to obtain
efficient reconstruction algorthms that outperform the previous state of the art for these problems.
An important open problem is to achieve the sample complexity of [12] while keeping the computational complexity polynomial in n.
Acknowledgments
M.K, K.S. and A.D. acknowledge the support of NSF via CCF 1422549, 1344364, 1344179 and
DARPA STTR and a ARO YIP award.
8
References
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J. Comput., vol. 22, no. 6, 1993, pp. 1331?1348.
[2] Y. Mansour, ?Randomized interpolation and approximation of sparse polynomials,? in SIAM
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1995, pp. 357?368.
[3] R. Schapire and R. Sellie, ?Learning sparse multivariate polynomials over a field with queries
and counterexamples,? in JCSS: Journal of Computer and System Sciences, vol. 52, 1996.
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[6] A. Akavia, ?Deterministic sparse Fourier approximation via fooling arithmetic progressions,?
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[7] D. Spielman and S. Teng, ?Smoothed analysis of algorithms: Why the simplex algorithm
usually takes polynomial time,? in JACM: Journal of the ACM, vol. 51, 2004.
?
[8] P. Li, ?Relational learning with hypergraphs,? Ph.D. dissertation, Ecole
Polytechnique F?ed?erale
de Lausanne, 2013.
[9] E. J. Cand`es, J. Romberg, and T. Tao, ?Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,? Information Theory, IEEE Transactions
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[10] E. J. Cand`es and T. Tao, ?Decoding by linear programming,? Information Theory, IEEE Transactions on, vol. 51, no. 12, pp. 4203?4215, 2005.
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[12] P. Stobbe and A. Krause, ?Learning Fourier sparse set functions,? in Proceedings of the International Conference on Artificial Intelligence and Statistics, 2012, pp. 1125?1133.
[13] S. Negahban and D. Shah, ?Learning sparse boolean polynomials,? in Proceedings of the Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on.
IEEE, 2012, pp. 2032?2036.
[14] A. Andoni, R. Panigrahy, G. Valiant, and L. Zhang, ?Learning sparse polynomial functions,?
in Proceedings of SODA, 2014.
[15] A. T. Kalai, A. Samorodnitsky, and S.-H. Teng, ?Learning and smoothed analysis,? in Proceedings of FOCS. IEEE Computer Society, 2009, pp. 395?404.
[16] R. O?Donnell, Analysis of Boolean Functions.
Cambridge University Press, 2014.
[17] Yahoo, ?Yahoo! webscope dataset ydata-ymessenger-user-communication-pattern-v1 0,? http:
//research.yahoo.com/Academic Relations.
9
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4,890 | 5,427 | A Residual Bootstrap for High-Dimensional
Regression with Near Low-Rank Designs
Miles E. Lopes
Department of Statistics
University of California, Berkeley
Berkeley, CA 94720
[email protected]
Abstract
We study the residual bootstrap (RB) method in the context of high-dimensional
linear regression. Specifically, we analyze the distributional approximation of linear contrasts c> (?b? ? ?), where ?b? is a ridge-regression estimator. When regression coefficients are estimated via least squares, classical results show that RB
consistently approximates the laws of contrasts, provided that p n, where the
design matrix is of size n ? p. Up to now, relatively little work has considered
how additional structure in the linear model may extend the validity of RB to the
setting where p/n 1. In this setting, we propose a version of RB that resamples
residuals obtained from ridge regression. Our main structural assumption on the
design matrix is that it is nearly low rank ? in the sense that its singular values
decay according to a power-law profile. Under a few extra technical assumptions, we derive a simple criterion for ensuring that RB consistently approximates
the law of a given contrast. We then specialize this result to study confidence
intervals for mean response values Xi> ?, where Xi> is the ith row of the design. More precisely, we show that conditionally on a Gaussian design with near
low-rank structure, RB simultaneously approximates all of the laws Xi> (?b? ? ?),
i = 1, . . . , n. This result is also notable as it imposes no sparsity assumptions on
?. Furthermore, since our consistency results are formulated in terms of the Mallows (Kantorovich) metric, the existence of a limiting distribution is not required.
1
Introduction
Until recently, much of the emphasis in the theory of high-dimensional statistics has been on ?first
order? problems, such as estimation and prediction. As the understanding of these problems has
become more complete, attention has begun to shift increasingly towards ?second order? problems,
dealing with hypothesis tests, confidence intervals, and uncertainty quantification [1?6]. In this
direction, much less is understood about the effects of structure, regularization, and dimensionality
? leaving many questions open. One collection of such questions that has attracted growing interest
deals with the operating characteristics of the bootstrap in high dimensions [7?9] . Due to the fact
that bootstrap is among the most widely used tools for approximating the sampling distributions of
test statistics and estimators, there is much practical value in understanding what factors allow for
the bootstrap to succeed in the high-dimensional regime.
The regression model and linear contrasts. In this paper, we focus our attention on highdimensional linear regression, and our aim is to know when the residual bootstrap (RB) method
consistently approximates the laws of linear contrasts. (A review of RB is given in Section 2.)
1
To specify the model, suppose that we observe a response vector Y ? Rn , generated according to
Y = X? + ?,
n?p
(1)
p
where X ? R
is the observed design matrix, ? ? R is an unknown vector of coefficients, and
the error variables ? = (?1 , . . . , ?n ) are drawn i.i.d. from an unknown distribution F0 , with mean
0 and unknown variance ? 2 < ?. As is conventional in high-dimensional statistics, we assume
the model (1) is embedded in a sequence of models indexed by n. Hence, we allow X, ?, and p to
vary implicitly with n. We will leave p/n unconstrained until Section 3.3, where we will assume
p/n 1 in Theorem 3, and then in Section 3.4, we will assume further that p/n is bounded strictly
between 0 and 1. The distribution F0 is fixed with respect to n, and none of our results require F0
to have more than four moments.
Although we are primarily interested in cases where the design matrix X is deterministic, we will
also study the performance of the bootstrap conditionally on a Gaussian design. For this reason,
we will use the symbol E[. . . |X] even when the design is non-random so that confusion does not
arise in relating different sections of the paper. Likewise, the symbol E[. . . ] refers to unconditional
expectation over all sources of randomness. Whenever the design is random, we will assume X ?
? ?,
denoting the distribution of X by PX , and the distribution of ? by P? .
Within the context of the regression, we will be focused on linear contrasts c> (?b ??), where c ? Rp
is a fixed vector and ?b ? Rp is an estimate of ?. The importance of contrasts arises from the fact
that they unify many questions about a linear model. For instance, testing the significance of the ith
coefficient ?i may be addressed by choosing c to be the standard basis vector c> = e>
i . Another
important problem is quantifying the uncertainty of point predictions, which may be addressed by
choosing c> = Xi> , i.e. the ith row of the design matrix. In this case, an approximation to the law
of the contrast leads to a confidence interval for the mean response value E[Yi ] = Xi> ?. Further
applications of contrasts occur in the broad topic of ANOVA [10].
Intuition for structure and regularization in RB. The following two paragraphs explain the core
conceptual aspects of the paper. To understand the role of regularization in applying RB to highdimensional regression, it is helpful to think of RB in terms of two ideas. First, if ?bLS denotes the
ordinary least squares estimator, then it is a simple but important fact that contrasts can be written
as c> (?bLS ? ?) = a> ? where a>:= c> (X > X)?1 X > . Hence, if it were possible to sample directly
from F0 , then the law of any such contrast could be easily determined. Since F0 is unknown, the
second key idea is to use the residuals of some estimator ?b as a proxy for samples from F0 . When
p n, the least-squares residuals are a good proxy [11, 12]. However, it is well-known that leastsquares tends to overfit when p/n 1. When ?bLS fits ?too well?, this means that its residuals are
b
?too small?, and hence they give a poor proxy for F0 . Therefore, by using a regularized estimator ?,
b
overfitting can be avoided, and the residuals of ? may offer a better way of obtaining ?approximate
samples? from F0 .
The form of regularized regression we will focus on is ridge regression:
?b? := (X > X + ?Ip?p )?1 X > Y,
(2)
where ? > 0 is a user-specificed regularization parameter. As will be seen in Sections 3.2 and 3.3,
the residuals obtained from ridge regression lead to a particularly good approximation of F0 when
the design matrix X is nearly low-rank, in the sense that most of its singular values are close to
0. In essence, this condition is a form of sparsity, since it implies that the rows of X nearly lie
in a low-dimensional subspace of Rp . However, this type of structural condition has a significant
advantage over the the more well-studied assumption that ? is sparse. Namely, the assumption that
X is nearly low-rank can be inspected directly in practice ? whereas sparsity in ? is typically
unverifiable. In fact, our results will impose no conditions on ?, other than that k?k2 remains
bounded as (n, p) ? ?. Finally, it is worth noting that the occurrence of near low-rank design
matrices is actually very common in applications, and is often referred to as collinearity [13, ch.
17].
Contributions and outline. The primary contribution of this paper is a complement to the work of
Bickel and Freedman [12] (hereafter B&F 1983) ? who showed that in general, the RB method fails
2
to approximate the laws of least-squares contrasts c> (?bLS ? ?) when p/n 1. Instead, we develop
an alternative set of results, proving that even when p/n 1, RB can successfully approximate the
laws of ?ridged contrasts? c> (?b? ? ?) for many choices of c ? Rp , provided that the design matrix
X is nearly low rank. A particularly interesting consequence of our work is that RB successfully
approximates the law c> (?b? ? ?) for a certain choice of c that was shown in B&F 1983 to ?break?
RB when applied to least-squares. Specifically, such a c can be chosen as one of the rows of X with
a high leverage score (see Section 4). This example corresponds to the practical problem of setting
confidence intervals for mean response values E[Yi ] = Xi> ?. (See [12, p. 41], as well as Lemma 2
and Theorem 4 in Section 3.4). Lastly, from a technical point of view, a third notable aspect of our
results is that they are formulated in terms of the Mallows-`2 metric, which frees us from having to
impose conditions that force a limiting distribution to exist.
Apart from B&F 1983, the most closely related works we are aware of are the recent papers [7]
and [8], which also consider RB in the high-dimensional setting. However, these works focus on
role of sparsity in ? and do not make use of low-rank structure in the design, whereas our work deals
only with structure in the design and imposes no sparsity assumptions on ?.
The remainder of the paper is organized as follows. In Section 2, we formulate the problem of
approximating the laws of contrasts, and describe our proposed methodology for RB based on ridge
regression. Then, in Section 3 we state several results that lay the groundwork for Theorem 4, which
shows that that RB can successfully approximate all of the laws L(Xi> (?b? ? ?)|X), i = 1, . . . , n,
conditionally on a Gaussian design. Due to space constraints, all proofs are deferred to material that
will appear in a separate work.
Notation and conventions. If U and V are random variables, then L(U |V ) denotes the law of U ,
conditionally on V . If an and bn are two sequences of real numbers, then the notation an . bn
means that there is an absolute constant ?0 > 0 and an integer n0 ? 1 such that an ? ?0 bn for all
n ? n0 . The notation an bn means that an . bn and bn . an . For a square matrix A ? Rk?k
whose eigenvalues are real, we denote them by ?min (A) = ?k (A) ? ? ? ? ? ?1 (A) = ?max (A).
2
Problem setup and methodology
Problem setup. For any c ? Rp , it is clear that conditionally on X, the law of c> (?b? ? ?) is
completely determined by F0 , and hence it makes sense to use the notation
?
?
(3)
?? (F0 ; c) := L c> (?b? ? ?)?
?X .
The problem we aim to solve is to approximate the distribution ?? (F0 ; c) for suitable choices of c.
Review of the residual bootstrap (RB) procedure. We briefly explain the steps involved in the
residual bootstrap procedure, applied to the ridge estimator ?b? of ?. To proceed somewhat indirectly,
consider the following ?bias-variance? decomposition of ?? (F0 ; c), conditionally on X,
?
?
?? (F0 ; c) = L c> ?b? ? E[?b? |X] ?
(4)
?X + c> E[?b? |X] ? ? .
|
{z
}
|
{z
}
=: ?? (F0 ;c)
=: bias(?? (F0 ;c))
Note that the distribution ?(F0 ; c) has mean zero, and so that the second term on the right side is
the bias of ?? (F0 ; c) as an estimator of ?? (F0 ; c). Furthermore, the distribution ?? (F0 ; c) may be
viewed as the ?variance component? of ?? (F0 ; c). We will be interested in situations where the
regularization parameter ? may be chosen small enough so that the bias component is small. In this
case, one has ?? (F0 ; c) ? ?? (F0 ; c), and then it is enough to find an approximation to the law
?? (F0 ; c), which is unknown. To this end, a simple manipulation of c> (?b? ? E[?b? ]) leads to
?
?
?? (F0 ; c) = L(c> (X > X + ?Ip?p )?1 X > ??
(5)
?X).
Now, to approximate ?? (F0 ; c), let Fb be any centered estimate of F0 . (Typically, Fb is obtained by
using the centered residuals of some estimator of ?, but this is not necessary in general.) Also, let
?? = (??1 , . . . , ??n ) ? Rn be an i.i.d. sample from Fb. Then, replacing ? with ?? in line (5) yields
?
?
?? (Fb; c) = L(c> (X > X + ?Ip?p )?1 X > ?? ?
(6)
?X).
3
At this point, we define the (random) measure ?? (Fb; c) to be the RB approximation to ?? (F0 ; c).
Hence, it is clear that the RB approximation is simply a ?plug-in rule?.
A two-stage approach. An important feature of the procedure just described is that we are free
to use any centered estimator Fb of F0 . This fact offers substantial flexibility in approximating
?? (F0 ; c). One way of exploiting this flexibility is to consider a two-stage approach, where a ?pilot?
ridge estimator ?b% is used to first compute residuals whose centered empirical distribution function
is Fb% , say. Then, in the second stage, the distribution Fb% is used to approximate ?? (F0 ; c) via the
relation (6). To be more detailed, if (b
e1 (%), . . . , ebn (%)) = eb(%) := Y ? X ?b% are the residuals of ?b% ,
b
then we define F% to be the distribution that places mass 1/n at each of the values ebi (%) ? e?(%) with
Pn
e?(%) := n1 i=1 ebi (%). Here, it is important to note that the value % is chosen to optimize Fb% as an
approximation to F0 . By contrast, the choice of ? depends on the relative importance of width and
coverage probability for confidence intervals based on ?? (Fb% ; c). Theorems 1, 3, and 4 will offer
some guidance in selecting % and ?.
Resampling algorithm. To summarize the discussion above, if B is user-specified number of
bootstrap replicates, our proposed method for approximating ?? (F0 ; c) is given below.
1. Select ? and %, and compute the residuals eb(%) = Y ? X ?b% .
2. Compute the centered distribution function Fb% , putting mass 1/n at each ebi (%) ? e?(%).
3. For j = 1, . . . , B:
? Draw a vector ?? ? Rn of n i.i.d. samples from Fb% .
? Compute zj := c> (X > X + ?Ip?p )?1 X > ?? .
4. Return the empirical distribution of z1 , . . . , zB .
Clearly, as B ? ?, the empirical distribution of z1 , . . . , zB converges weakly to ?? (Fb% ; c), with
probability 1. As is conventional, our theoretical analysis in the next section will ignore Monte Carlo
issues, and address only the performance of ?? (Fb% ; c) as an approximation to ?? (F0 ; c).
3
Main results
The following metric will be central to our theoretical results, and has been a standard tool in the
analysis of the bootstrap, beginning with the work of Bickel and Freedman [14].
The Mallows (Kantorovich) metric. For two random vectors U and V in a Euclidean space, the
Mallows-`2 metric is defined by
n h
i
o
d22 (L(U ), L(V )) := inf E kU ? V k22 : (U, V ) ? ?
(7)
???
where the infimum is over the class ? of joint distributions ? whose marginals are L(U ) and L(V ).
It is worth noting that convergence in d2 is strictly stronger than weak convergence, since it also
requires convergence of second moments. Additional details may be found in the paper [14].
3.1
A bias-variance decomposition for bootstrap approximation
To give some notation for analyzing the bias-variance decomposition of ?? (F0 ; c) in line (4), we
define the following quantities based upon the ridge estimator ?b? . Namely, the variance is
v? = v? (X; c) := var(?? (F0 ; c)|X) = ? 2 kc> (X > X + ?Ip?p )?1 X > k22 .
To express the bias of ?? (F0 ; c), we define the vector ?(X) ? Rp according to
?(X) := ? ? E[?b? ] = Ip?p ? (X > X + ?Ip?p )?1 X > X ?,
4
(8)
and then put
b2? = b2? (X; c) := bias2 (?? (F0 ; c)) = (c> ?(X))2 .
(9)
We will sometimes omit the arguments of v? and b2? to lighten notation. Note that v? (X; c) does not
depend on ?, and b2? (X; c) only depends on ? through ?(X).
The following result gives a regularized and high-dimensional extension of some lemmas in Freedman?s early work [11] on RB for least squares. The result does not require any structural conditions
on the design matrix, or on the true parameter ?.
Theorem 1 (consistency criterion). Suppose X ? Rn?p is fixed. Let Fb be any estimator of F0 , and
let c ? Rp be any vector such that v? = v? (X; c) 6= 0. Then with P? -probability 1, the following
inequality holds for every n ? 1, and every ? > 0,
b2
d22 ?1v? ?? (F0 ; c), ?1v? ?? (Fb; c) ? ?12 d22 (F0 , Fb) + v?? .
(10)
?
Remarks. Observe that the normalization 1/ v? ensures that the bound is non-trivial, since the
?
distribution ?? (F0 ; c)/ v? has variance equal to 1 for all n (and hence does not become degenerate
for large n). To consider the choice of ?, it is simple to verify that the ratio b2? /v? decreases monotonically as ? decreases. Note also that as ? becomes small, the variance v? becomes large, and
likewise, confidence intervals based on ?? (Fb; c) become wider. In other words, there is a trade-off
between the width of the confidence interval and the size of the bound (10).
Sufficient conditions for consistency of RB. An important practical aspect of Theorem 1 is that
for any given contrast c, the variance v? (X; c) can be easily estimated, since it only requires an
estimate of ? 2 , which can be obtained from Fb. Consequently, whenever theoretical bounds on
d22 (F0 , Fb) and b2? (X; c) are available, the right side of line (10) can be controlled. In this way,
Theorem 1 offers a simple route for guaranteeing that RB is consistent. In Sections 3.2 and 3.3 to
follow, we derive a bound on E[d22 (F0 , Fb)|X] in the case where Fb is chosen to be Fb% . Later on in
Section 3.4, we study RB consistency in the context of prediction with a Gaussian design, and there
we derive high probability bounds on both v? (X; c) and b2? (X; c) where c is a particular row of X.
3.2
A link between bootstrap consistency and MSPE
If ?b is an estimator of ?, its mean-squared prediction error (MSPE), conditionally on X, is defined
as
?
?
mspe(?b |X) := n1 E kX(?b ? ?)k22 ?
(11)
?X .
The previous subsection showed that in-law approximation of contrasts is closely tied to the approximation of F0 . We now take a second step of showing that if the centered residuals of an estimator
?b are used to approximate F0 , then the quality of this approximation can be bounded naturally in
terms of mspe(?b |X). This result applies to any estimator ?b computed from the observations (1).
Theorem 2. Suppose X ? Rn?p is fixed. Let ?b be any estimator of ?, and let Fb be the empirical
b Also, let Fn denote the empirical distribution of n i.i.d.
distribution of the centered residuals of ?.
samples from F0 . Then for every n ? 1,
?
2
?
?
E d22 (Fb, F0 )?X
? 2 mspe(?b |X) + 2 E[d22 (Fn , F0 )] + 2?n .
(12)
Remarks. As we will see in the next section, the MSPE of ridge regression can be bounded in a
sharp way when the design matrix is approximately low rank, and there we will analyze mspe(?b% |X)
for the pilot estimator. Consequently, when near low-rank structure is available, the only remaining
issue in controlling the right side of line (12) is to bound the quantity E[d22 (Fn , F0 )|X]. The very
recent work of Bobkov and Ledoux [15] provides an in-depth study of this question, and they derive
a variety bounds under different tail conditions on F0 . We summarize one of their results below.
Lemma 1 (Bobkov and Ledoux, 2014). If F0 has a finite fourth moment, then
E[d22 (Fn , F0 )] . log(n)n?1/2 .
5
(13)
Remarks. The fact that the squared distance is bounded at the rate of log(n)n?1/2 is an indication that d2 is a rather strong metric on distributions. For a detailed discussion of this result, see
Corollaries 7.17 and 7.18 in the paper [15]. Although it is possible to obtain faster rates when more
b
stringent tail conditions are placed on F0 , we will only need a fourth moment, since the mspe(?|X)
term in Theorem 2 will often have a slower rate than log(n)n?1/2 , as discussed in the next section.
3.3
Consistency of ridge regression in MSPE for near low rank designs
In this subsection, we show that when the tuning parameter % is set at a suitable rate, the pilot ridge
estimator ?b% is consistent in MSPE when the design matrix is near low-rank ? even when p/n is
large, and without any sparsity constraints on ?. We now state some assumptions.
A1. There is a number ? > 0, and absolute constants ?1 , ?2 > 0, such that
b ? ?2 i??
?1 i?? ? ?i (?)
for all i = 1, . . . , n ? p.
A2. There are absolute constants ?, ? > 0, such that for every n ? 1,
%
n
= n?? and
?
n
= n?? .
A3. The vector ? ? Rp satisfies k?k2 . 1.
Due to Theorem 2, the following bound shows that the residuals of ?b% may be used to extract a
consistent approximation to F0 . Two other notable features of the bound are that it is non-asymptotic
and dimension-free.
Theorem 3. Suppose that X ? Rn?p is fixed and that Assumptions 1?3 hold, with p/n 1. Assume
1
?
1
further that ? is chosen as ? = 2?
3 when ? ? (0, 2 ), and ? = ?+1 when ? > 2 . Then,
(
2?
if ? ? (0, 12 ),
n? 3
?
(14)
mspe(?b% |X) .
? ?+1
n
if ? > 21 .
Also, both bounds in (14) are tight in the sense that ? can be chosen so that ?b% attains either rate.
b are observable, they may be used to estimate ? and guide
Remarks. Since the eigenvalues ?i (?)
the selection of %/n = n?? . However, from a practical point of view, we found it easier to select %
via cross-validation in numerical experiments, rather than via an estimate of ?.
A link with Pinsker?s Theorem. In the particular case when F0 is a centered Gaussian distribution, the ?prediction problem? of estimating X? is very similar to estimating the mean parameters of
a Gaussian sequence model, with error measured in the `2 norm. In the alternative sequence-model
format, the decay condition on the eigenvalues of n1 X > X translates into an ellipsoid constraint on
the mean parameter sequence [16, 17]. For this reason, Theorem 3 may be viewed as ?regression
version? of `2 error bounds for the sequence model under an ellipsoid constraint (cf. Pinsker?s Theorem, [16, 17]). Due to the fact that the latter problem has a very well developed literature, there
may be various ?neighboring results? elsewhere. Nevertheless, we could not find a direct reference
for our stated MSPE bound in the current setup. For the purposes of our work in this paper, the more
important point to take away from Theorem 3 is that it can be coupled with Theorem 2 for proving
consistency of RB.
3.4
Confidence intervals for mean responses, conditionally on a Gaussian design
In this section, we consider the situation where the design matrix X has rows Xi> ? Rp drawn
i.i.d. from a multivariate normal distribution N (0, ?), with X ?
? ?. (The covariance matrix ? may
vary with n.) Conditionally on a realization of X, we analyze the RB approximation of the laws
?? (F0 ; Xi ) = L(Xi> (?b? ? ?)|X). As discussed in Section 1, this corresponds to the problem of
setting confidence intervals for the mean responses E[Yi ] = Xi> ?. Assuming that the population
eigenvalues ?i (?) obey a decay condition, we show below in Theorem 4 that RB succeeds with high
PX -probability. Moreover, this consistency statement holds for all of the laws ?? (F0 ; Xi ) simultaneously. That is, among the n distinct laws ?? (F0 ; Xi ), i = 1, . . . , n, even the worst bootstrap
approximation is still consistent. We now state some population-level assumptions.
6
A4. The operator norm of ? ? Rp?p satisfies k?kop . 1.
Next, we impose a decay condition on the eigenvalues of ?. This condition also ensures that ? is
invertible for each fixed p ? even though the bottom eigenvalue may become arbitrarily small as p
becomes large. It is important to notice that we now use ? for the decay exponent of the population
eigenvalues, whereas we used ? when describing the sample eigenvalues in the previous section.
A5. There is a number ? > 0, and absolute constants k1 , k2 > 0, such that for all i = 1, . . . , p,
k1 i?? ? ?i (?) ? k2 i?? .
A6. There are absolute constants k3 , k4 ? (0, 1) such that for all n ? 3, we have the bounds
k3 ? np ? k4 and p ? n ? 2.
The following lemma collects most of the effort needed in proving our final result in Theorem 4.
Here it is also helpful to recall the notation ?/n = n?? and %/n = n?? from Assumption 2.
Lemma 2. Suppose that the matrix X ? Rn?p has rows Xi> drawn i.i.d. from N (0, ?), and that
Assumptions 2?6 hold. Furthermore, assume that ? chosen so that 0 < ? < min{?, 1}. Then, the
statements below are true.
(i) (bias inequality)
Fix any ? > 0. Then, there is an absolute constant ?0 > 0, such that for all large n, the following
event holds with PX -probability at least 1 ? n?? ? ne?n/16 ,
max b2? (X; Xi ) ? ?0 ? n?? ? (? + 1) log(n + 2).
1?i?n
(15)
(ii) (variance inequality)
There are absolute constants ?1 , ?2 > 0 such that for all large n, the following event holds with
?
PX -probability at least 1 ? 4n exp(??1 n ? ),
1
1?i?n v? (X;Xi )
max
?
? ?2 n1? ? .
(16)
(iii) (mspe inequalities)
?
when
Suppose that ? is chosen as ? = 2?/3 when ? ? (0, 21 ), and that ? is chosen as ? = 1+?
1
? > 2 . Then, there are absolute constants ?3 , ?4 , ?5 , ?6 > 0 such that for all large n,
(
2?
with PX -probability at least 1 ? exp(??3 n2?4?/3 ), if ? ? (0, 12 )
?4 n? 3
b
mspe(?% |X) ?
?
2
? ?+1
?6 n
with PX -probability at least 1 ? exp(??5 n 1+? ),
if ? > 21 .
Remarks. Note that the two rates in part (iii) coincide as ? approaches 1/2. At a conceptual level,
the entire lemma may be explained in relatively simple terms. Viewing the quantities mspe(?b% |X),
b2? (X; Xi ) and v? (X; Xi ) as functionals of a Gaussian matrix, the proof involves deriving concentration bounds for each of them. Indeed, this is plausible given that these quantities are smooth
functionals of X. However, the difficulty of the proof arises from the fact that they are also highly
non-linear functionals of X. We now combine Lemmas 1 and 2 with Theorems 1 and 2 to show that
all of the laws ?? (F0 ; Xi ) can be simultaneously approximated via our two-stage RB method.
Theorem 4. Suppose that F0 has a finite fourth moment, Assumptions 2?6 hold, and ? is chosen
?
so that 1+?
< ? < min{?, 1}. Also suppose that ? is chosen as ? = 2?/3 when ? ? (0, 12 ), and
?
? = ?+1
when ? > 12 . Then, there is a sequence of positive numbers ?n with limn?? ?n = 0, such
that the event
?
h
i
?
?
E max d22 ?1v? ?? (F0 ; Xi ), ?1v? ?? (Fb% ; Xi ) ?
X
? ?n
(17)
?
?
1?i?n
has PX -probability tending to 1 as n ? ?.
Remark. Lemma 2 gives explicit bounds on the numbers ?n , as well as the probabilities of the
corresponding events, but we have stated the result in this way for the sake of readability.
7
4
Simulations
In four different settings of n, p, and the decay parameter ?, we compared the nominal 90% confidence intervals (CIs) of four methods: ?oracle?, ?ridge?, ?normal?, and ?OLS?, to be described
below. In each setting, we generated N1 := 100 random designs X with i.i.d. rows drawn from
N (0, ?), where ?j (?) = j ?? , j = 1, . . . , p, and the eigenvectors of ? were drawn randomly by
setting them to be the Q factor in a QR decomposition of a standard p ? p Gaussian matrix. Then,
for each realization of X, we generated N2 := 1000 realizations of Y according to the model (1),
where ? = 1/k1k2 ? Rp , and F0 is the centered t distribution on 5 degrees of freedom, rescaled to
have standard deviation ? = 0.1. For each X, and each corresponding Y , we considered the problem of setting a 90% CI for the mean response value Xi>? ?, where Xi>? is the row with the highest
leverage score, i.e. i? = argmax1?i?n Hii and H := X(X > X)?1 X > . This problem was shown in
B&F 1983 to be a case where the standard RB method based on least-squares fails when p/n 1.
Below, we refer to this method as ?OLS?.
To describe the other three methods, ?ridge? refers to the interval [Xi>? ?b? ? qb0.95 , Xi>? ?b? ? qb0.05 ],
where qb? is the ?% quantile of the numbers z1 , . . . , zB computed in the proposed algorithm in
Section 2, with B = 1000 and c> = Xi>? . To choose the parameters ? and % for a given X and Y ,
we first computed rb as the value that optimized the MSPE error of a ridge estimator ?br with respect
to 5-fold cross validation; i.e. cross validation was performed for every distinct pair (X, Y ). We then
put % = 5b
r and ? = 0.1b
r, as we found the prefactors 5 and 0.1 to work adequately across various
settings. (Optimizing % with respect to MSPE is motivated by Theorems 1, 2, and 3. Also, choosing ?
to be somewhat smaller than % conforms with the constraints on ? and ? in Theorem 4.) The method
?normal? refers to the CI based on the normal approximation L(Xi>? (?b? ??)|X) ? N (0, ?b2 ), where
?b2 = ?
b2 kXi>? (X > X +?Ip?p )?1 X > k22 , ? = 0.1b
r, and ?
b2 is the usual unbiased estimate of ? 2 based
on OLS residuals. The ?oracle? method refers to the interval [Xi>? ?b? ? q?0.95 , Xi>? ?b? ? q?0.05 ], with
? = 0.1b
r, and q?? being the empirical ?% quantile of Xi> (?b? ? ?) over all 1000 realizations of Y
based on a given X. (This accounts for the randomness in ? = 0.1b
r.)
Within a given setting of the triplet (n, p, ?), we refer to the ?coverage? of a method as the fraction of
the N1 ?N2 = 105 instances where the method?s CI contained the parameter Xi>? ?. Also, we refer to
?width? as the average width of a method?s intervals over all of the 105 instances. The four settings of
(n, p, ?) correspond to moderate/high dimension and moderate/fast decay of the eigenvalues ?i (?).
Even in the moderate case of p/n = 0.45, the results show that the OLS intervals are too narrow
and have coverage noticeably less than 90%. As expected, this effect becomes more pronounced
when p/n = 0.95. The ridge and normal intervals perform reasonably well across settings, with
both performing much better than OLS. However, it should be emphasized that our study of RB
is motivated by the desire to gain insight into the behavior of the bootstrap in high dimensions
? rather than trying to outperform particular methods. In future work, we plan to investigate the
relative merits of the ridge and normal intervals in greater detail.
Table 1: Comparison of nominal 90% confidence intervals
setting 1
n = 100, p = 45, ? = 0.5
setting 2
n = 100, p = 95, ? = 0.5
setting 3
n = 100, p = 45, ? = 1
setting 4
n = 100, p = 95, ? = 1
width
coverage
width
coverage
width
coverage
width
coverage
oracle
0.21
0.90
0.22
0.90
0.20
0.90
0.21
0.90
ridge
0.20
0.87
0.26
0.88
0.21
0.90
0.26
0.92
normal
0.23
0.91
0.26
0.88
0.22
0.91
0.23
0.87
OLS
0.16
0.81
0.06
0.42
0.16
0.81
0.06
0.42
Acknowledgements. MEL thanks Prof. Peter J. Bickel for many helpful discussions, and gratefully acknowledges the DOE CSGF under grant DE-FG02-97ER25308, as well as the NSF-GRFP.
8
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9
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4,891 | 5,428 | Fast and Robust Least Squares Estimation in
Corrupted Linear Models
Brian McWilliams?
Gabriel Krummenacher? Mario Lucic Joachim M. Buhmann
Department of Computer Science
ETH Z?urich, Switzerland
{mcbrian,gabriel.krummenacher,lucic,jbuhmann}@inf.ethz.ch
Abstract
Subsampling methods have been recently proposed to speed up least squares estimation in large scale settings. However, these algorithms are typically not robust
to outliers or corruptions in the observed covariates.
The concept of influence that was developed for regression diagnostics can be
used to detect such corrupted observations as shown in this paper. This property
of influence ? for which we also develop a randomized approximation ? motivates
our proposed subsampling algorithm for large scale corrupted linear regression
which limits the influence of data points since highly influential points contribute
most to the residual error. Under a general model of corrupted observations, we
show theoretically and empirically on a variety of simulated and real datasets that
our algorithm improves over the current state-of-the-art approximation schemes
for ordinary least squares.
1
Introduction
To improve scalability of the widely used ordinary least squares algorithm, a number of randomized
approximation algorithms have recently been proposed. These methods, based on subsampling the
dataset, reduce the computational time from O np2 to o(np2 )1 [14]. Most of these algorithms
are concerned with the classical fixed design setting or the case where the data is assumed to be
sampled i.i.d. typically from a sub-Gaussian distribution [7]. This is known to be an unrealistic
modelling assumption since real-world data are rarely well-behaved in the sense of the underlying
distributions.
We relax this limiting assumption by considering the setting where with some probability, the observed covariates are corrupted with additive noise. This scenario corresponds to a generalised
version of the classical problem of ?errors-in-variables? in regression analysis which has recently
been considered in the context of sparse estimation [12]. This corrupted observation model poses a
more realistic model of real data which may be subject to many different sources of measurement
noise or heterogeneity in the dataset.
A key consideration for sampling is to ensure that the points used for estimation are typical of the
full dataset. Typicality requires the sampling distribution to be robust against outliers and corrupted
points. In the i.i.d. sub-Gaussian setting, outliers are rare and can often easily be identified by
examining the statistical leverage scores of the datapoints.
Crucially, in the corrupted observation setting described in ?2, the concept of an outlying point
concerns the relationship between the observed predictors and the response. Now, leverage alone
cannot detect the presence of corruptions. Consequently, without using additional knowledge about
?
1
Authors contributed equally.
Informally: f (n) = o(g(n)) means f (n) grows more slowly than g(n).
1
the corrupted points, the OLS estimator (and its subsampled approximations) are biased. This also
rules out stochastic gradient descent (SGD) ? which is often used for large scale regression ? since
convex cost functions and regularizers which are typically used for noisy data are not robust with
respect to measurement corruptions.
This setting motivates our use of influence ? the effective impact of an individual datapoint exerts on
the overall estimate ? in order to detect and therefore avoid sampling corrupted points. We propose
an algorithm which is robust to corrupted observations and exhibits reduced bias compared with
other subsampling estimators.
Outline and Contributions. In ?2 we introduce our corrupted observation model before reviewing
the basic concepts of statistical leverage and influence in ?3. In ?4 we briefly review two subsampling
approaches to approximating least squares based on structured random projections and leverage
weighted importance sampling. Based on these ideas we present influence weighted subsampling
(IWS-LS), a novel randomized least squares algorithm based on subsampling points with small
influence in ?5.
In ?6 we analyse IWS-LS in the general setting where the observed predictors can be corrupted
with additive sub-Gaussian noise. Comparing the IWS-LS estimate with that of OLS and other
randomized least squares approaches we show a reduction in both bias and variance. It is important
to note that the simultaneous reduction in bias and variance is relative to OLS and randomized
approximations which are only unbiased in the non-corrupted setting. Our results rely on novel
finite sample characteristics of leverage and influence which we defer to ?SI.3. Additionally, in
?SI.4 we prove an estimation error bound for IWS-LS in the standard sub-Gaussian model.
Computing influence exactly is not practical in large-scale applications and so we propose two randomized approximation algorithms based on the randomized leverage approximation of [8]. Both
of these algorithms run in o(np2 ) time which improve scalability in large problems. Finally, in ?7
we present extensive experimental evaluation which compares the performance of our algorithms
against several randomized least squares methods on a variety of simulated and real datasets.
2
Statistical model
In this work we consider a variant of the standard linear model
(1)
y = X + ?,
where ? 2 R is a noise term independent of X 2 R
. However, rather than directly observing
X we instead observe Z where
Z = X + U W.
(2)
U = diag(u1 , . . . , un ) and ui is a Bernoulli random variable with probability ? of being 1.
W 2 Rn?p is a matrix of measurement corruptions. The rows of Z therefore are corrupted with
probability ? and not corrupted with probability (1 ?).
Definition 1 (Sub-gaussian matrix). A zero-mean matrix X is called sub-Gaussian with parameter
1
p
>
( n1 x2 , n1 ?x ) if (a) Each row x>
i 2 R is sampled independently and has E[xi xi ] = n ?x . (b) For
p
>
any unit vector v 2 R , v xi is a sub-Gaussian random variable with parameter at most p1p x .
n
n?p
We consider the specific instance of the linear corrupted observation model in Eqs. (1), (2) where
? X, W 2 Rn?p are sub-Gaussian with parameters ( n1 x2 , n1 ?x ) and ( n1
tively,
? ? 2 Rn is sub-Gaussian with parameters ( n1 ?2 , n1 ?2 In ),
2 1
w , n ?w )
respec-
and all are independent of each other.
The key challenge is that even when ? and the magnitude of the corruptions, w are relatively small,
the standard linear regression estimate is biased and can perform poorly (see ?6). Sampling methods
which are not sensitive to corruptions in the observations can perform even worse if they somehow
subsample a proportion rn > ?n of corrupted points. Furthermore, the corruptions may not be large
enough to be detected via leverage based techniques alone.
The model described in this section generalises the ?errors-in-variables? model from classical least
squares modelling. Recently, similar models have been studied in the high dimensional (p
n)
2
setting in [4?6, 12] in the context of robust sparse estimation. The ?low-dimensional? (n > p)
setting is investigated in [4], but the ?big data? setting (n
p) has not been considered so far.2
In the high-dimensional problem, knowledge of the corruption covariance, ?w [12], or the data
covariance ?x [5], is required to obtain a consistent estimate. This assumption may be unrealistic in
many settings. We aim to reduce the bias in our estimates without requiring knowledge of the true
covariance of the data or the corruptions, and instead sub-sample only non-corrupted points.
3
Diagnostics for linear regression
In practice, the sub-Gaussian linear model assumption is often violated either by heterogeneous
noise or by a corruption model as in ?2. In such scenarios, fitting a least squares model to the full
dataset is unwise since the outlying or corrupted points can have a large adverse effect on the model
fit. Regression diagnostics have been developed in the statistics literature to detect such points (see
e.g. [2] for a comprehensive overview). Recently, [14] proposed subsampling points for least squares
based on their leverage scores. Other recent works suggest related influence measures that identify
subspace [16] and multi-view [15] clusters in high dimensional data.
3.1
Statistical leverage
For the standard linear model in Eq. (1), the well known least squares solution is
b = arg min ky
X k2 = X > X
1
X> y.
(3)
The projection matrix I L with L := X(X> X) 1 X> specifies the subspace in which the residual
lies. The diagonal elements of the ?hat matrix? L, li := Lii , i = 1, . . . , n are the statistical leverage
scores of the ith sample. Leverage scores quantify to what extent a particular sample is an outlier
with respect to the distribution of X.
An equivalent definition from [14] which will be useful later concerns any matrix U 2 Rn?p which
spans the column space of X (for example, the matrix whose columns are the left singular vectors of
X). The statistical leverage scores of the rows of X are the squared row norms of U, i.e. li = kUi k2 .
Although the use of leverage can be motivated from the least squares solution in Eq. (3), the leverage scores do not take into account the relationship between the predictor variables and the response
variable y. Therefore, low-leverage points may have a weak predictive relationship with the response and vice-versa. In other words, it is possible for such points to be outliers with respect to the
conditional distribution P (y|X) but not the marginal distribution on X.
3.2
Influence
A concept that captures the predictive relationship between covariates and response is influence.
Influential points are those that might not be outliers in the geometric sense, but instead adversely
affect the estimated coefficients.
One way to assess the influence of a point is to compute the change in the learned model when
the point is removed from the estimation step. [2]. We can compute a leave-one-out least squares
estimator by straightforward application of the Sherman-Morrison-Woodbury formula (see Prop. 3
in ?SI.3):
where ei = yi
we have
2
3
b
i
= X> X
x>
i xi
1
X> y
b
x>
i yi =
? 1 x>
i ei
1 li
xi b OLS . Defining the influence3 , di as the change in expected mean squared error
?
di = b
b
i
?>
?
X> X b
b
i
?
=
e2i li
(1
li )
Unlike [5, 12] and others we do not consider sparsity in our solution since n
The expression we use is also called Cook?s distance [2].
3
2.
p.
Points with large values of di are those which, if added to the model, have the largest adverse effect
on the resulting estimate. Since influence only depends on the OLS residual error and the leverage
scores, it can be seen that the influence of every point can be computed at the cost of a least squares
fit. In the next section we will see how to approximate both quantities using random projections.
4
Fast randomized least squares algorithms
We briefly review two randomized approaches to least squares approximation: the importance
weighted subsampling approach of [9] and the dimensionality reduction approach [14]. The former proposes an importance sampling probability distribution according to which, a small number
of rows of X and y are drawn and used to compute the regression coefficients. If the sampling probabilities are proportional to the statistical leverages, the resulting estimator is close to the optimal
estimator [9]. We refer to this as LEV-LS.
The dimensionality reduction approach can be viewed as a random projection step followed by a
uniform subsampling. The class of Johnson-Lindenstrauss projections ? e.g. the SRHT ? has been
shown to approximately uniformize leverage scores in the projected space. Uniformly subsampling
the rows of the projected matrix proves to be equivalent to leverage weighted sampling on the original dataset [14]. We refer to this as SRHT-LS. It is analysed in the statistical setting by [7] who also
propose ULURU, a two step fitting procedure which aims to correct for the subsampling bias and
consequently converges to the OLS estimate at a rate independent of the number of subsamples [7].
Subsampled Randomized Hadamard Transform (SRHT) The SHRT consists of a preconditioning step after
q which nsubs rows of the new matrix are subsampled uniformly at random in the
n
following way nsubs
SHD ? X = ?X with the definitions [3]:
? S is a subsampling matrix.
? D is a diagonal matrix whose entries are drawn independently from { 1, 1}.
? H 2 Rn?n is a normalized Walsh-Hadamard matrix4 which is defined recursively as
?
?
Hn/2 Hn/2
+1 +1
Hn =
, H2 =
.
Hn/2
Hn/2
+1
1
We set H =
p1 Hn
n
so it has orthonormal columns.
As a result, the rows of the transformed matrix ?X have approximately uniform leverage scores.
(see [17] for detailed analysis of the SRHT). Due to the recursive nature of H, the cost of applying
the SRHT is O (pn log nsubs ) operations, where nsubs is the number of rows sampled from X [1].
The SRHT-LS algorithm solves b SRHT = arg min k?y ?X k2 which for an appropriate
2
? which satisfies
subsampling ratio, r = ?( ?p2 ) results in a residual error, e
(4)
k?
ek ? (1 + ?)kek
where e = y
X b OLS is the vector of OLS residual errors [14].
Randomized leverage computation Recently, a method based on random projections has been
proposed to approximate the leverage scores based on first reducing the dimensionality of the data
using the SRHT followed by computing the leverage scores using this low-dimensional approximation [8?10, 13].
The leverage approximation algorithm of [8] uses a SRHT, ?1 2 Rr1 ?n to first compute the approximate SVD of X,
>
?1 X = U?X ??X V?X
. Followed by a second SHRT ?2 2 Rp?r2 to compute an approximate
orthogonal basis for X
R
1
1
? = XR
= V?X ??X
2 Rp?p , U
1
?2 2 Rn?r2 .
(5)
4
For the Hadamard transform, n must be a power of two but other transforms exist (e.g. DCT, DFT) for
which similar theoretical guarantees hold and there is no restriction on n.
4
? ?li = kU
? i k2 .
The approximate leverage scores are now the squared row norms of U,
From [14] we derive the following result relating to randomized approximation of the leverage
?li ? (1 + ?l )li ,
(6)
where the approximation error, ?l depends on the choice of projection dimensions r1 and r2 .
The leverage weighted least squares (LEV-LS) algorithm samples rows of X and y with probability
proportional to li (or ?li in the approximate case) and performs least squares on this subsample. The
residual error resulting from the leverage weighted least squares is bounded by Eq. (4) implying
that LEV-LS and SRHT-LS are equivalent [14]. It is important to note that under the corrupted
observation model these approximations will be biased.
5
Influence weighted subsampling
In the corrupted observation model, OLS and therefore the random approximations to OLS described in ?4 obtain poor predictions. To remedy this, we propose influence weighted subsampling
(IWS-LS) which is described in Algorithm 1. IWS-LS subsamples
points according to the distriPn
bution, Pi = c/di where c is a normalizing constant so that i=1 Pi = 1. OLS is then estimated on
the subsampled points. The sampling procedure ensures that points with high influence are selected
infrequently and so the resulting estimate is less biased than the full OLS solution. Several approaches similar in spirit have previously been proposed based on identifying and down-weighting
the effect of highly influential observations [19].
Obviously, IWS-LS is impractical in the scenarios we consider since it requires the OLS residuals
and full leverage scores. However, we use this as a baseline and to simplify the analysis. In the next
section, we propose an approximate influence weighted subsampling algorithm which combines the
approximate leverage computation of [8] and the randomized least squares approach of [14].
Algorithm 1 Influence weighted subsampling
(IWS-LS).
Input: Data: Z, y
1: Solve b OLS = arg min ky Z k2
2: for i = 1 . . . n do
3:
ei = yi zi b OLS
>
1
4:
l i = z>
zi
i (Z Z)
5:
di = e2i li /(1 li )2
6: end for
? y
? ) of (Z, y) proportional to
7: Sample rows (Z,
8: Solve b IWS = arg min k?
y
Output: b IWS
? k2
Z
1
di
Algorithm 2 Residual weighted subsampling
(aRWS-LS)
Input: Data: Z, y
1: Solve b SRHT = arg min k? ? (y Z )k2
? = y Z b SRHT
2: Estimate residuals: e
?
? ) of (Z, y) proportional to
3: Sample rows (Z, y
1
e?2i
4: Solve b RW S = arg min k?
y
Output: b RW S
? k2
Z
Randomized approximation algorithms. Using the ideas from ?4 and ?4 we obtain the following
randomized approximation to the influence scores
d?i =
e?2i ?li
,
(1 ?li )2
(7)
where e?i is the ith residual error computed using the SRHT-LS estimator. Since the approximation errors of e?i and ?li are bounded (inequalities (4) and (6)), this suggests that our randomized
approximation to influence is close to the true influence.
Basic approximation. The first approximation algorithm is identical to Algorithm 1 except that
leverage and residuals are replaced by their randomized approximations as in Eq. (7). We refer to
this algorithm as Approximate influence weighted subsampling (aIWS-LS). Full details are given
in Algorithm 3 in ?SI.2.
5
Residual Weighted Sampling. Leverage scores are typically uniform [7, 13] for sub-Gaussian
data. Even in the corrupted setting, the difference in leverage scores between corrupted and noncorrupted points is small (see ?6). Therefore, the main contribution to the influence for each point
will originate from the residual error, e2i . Consequently, we propose sampling with probability
inversely proportional to the approximate residual, e?12 . The resulting algorithm Residual Weighted
i
Subsampling (aRWS-LS) is detailed in Algorithm 2. Although aRWS-LS is not guaranteed to be
a good approximation to IWS-LS, empirical results suggests that it works well in practise and is
faster to compute than aIWS-LS.
Computational complexity. Clearly, the computational complexity of IWS-LS is O np2 . The
computation complexity of aIWS-LS is O np log nsubs + npr2 + nsubs p2 , where the first term
is the cost of SRHT-LS, the second term is the cost of approximate leverage computation and the
last term solves OLS on the subsampled dataset. Here, r2 is the dimension of the random projection detailed in Eq. (5). The cost of aRWS-LS is O np log nsubs + np + nsubs p2 where
the first term is the cost of SRHT-LS, the second term is the cost of computing the residuals
e, and the last term solves OLS on the subsampled dataset. This computation can be reduced to
O np log nsubs + nsubs p2 . Therefore the cost of both aIWS-LS and aRWS-LS is o(np2 ).
6
Estimation error
In this section we will prove an upper bound on the estimation error of IWS-LS in the corrupted
model. First, we show that the OLS error consists of two additional variance terms that depend on the
size and proportion of the corruptions and an additional bias term. We then show that IWS-LS can
significantly reduce the relative variance and bias in this setting, so that it no longer depends on the
magnitude of the corruptions but only on their proportion. We compare these results to recent results
from [4, 12] suggesting that consistent estimation requires knowledge about ?w . More recently, [5]
show that incomplete knowledge about this quantity results in a biased estimator where the bias is
proportional to the uncertainty about ?w . We see that the form of our bound matches these results.
Inequalities are said to hold with high probability (w.h.p.) if the probability of failure is not more
than C1 exp( C2 log p) where C1 , C2 are positive constants that do not depend on the scaling quantities n, p, w . The symbol . means that we ignore constants that do not depend on these scaling
quantities. Proofs are provided in the supplement. Unless otherwise stated, k?k denotes the `2 norm
for vectors and the spectral norm for matrices.
Corrupted observation model. As a baseline, we first investigate the behaviour of the OLS estimator in the corrupted model.
2 2
Theorem 1 (A bound on k b OLS
k). If n & minx (?wx ) p log p then w.h.p.
!
r
p log p
1
2
2p
b
k OLS
k.
+ ? w pk k ?
(8)
? x+? ? w +?
w + w x k k
n
where 0 < ? min (?x ) + ? min (?w ).
Remark 1 (No corruptions case). Notice for a fixed w , taking lim?!0 or for a fixed ? taking
lim w !0 (i.e. there are no corruptions) the above error reduces to the least squares result (see for
example [4]).
p
Remark 2 (Variance and Bias). The first three terms inp
(8) scale with 1/n so as n ! 1, these
terms tend towards 0. The last term does not depend on 1/n and so for some non-zero ? the least
squares estimate will incur some bias depending on the fraction and magnitude of corruptions.
We are now ready to state our theorem characterising the mean squared error of the influence
weighted subsampling estimator.
2
2
Theorem 2 (Influence sampling in the corrupted model). For n & minx(?w?x ) p log p we have
!
?
?r
? ?
p log p
1
p
b
k IWS
k.
+ ?k k
+ ? pk k .
? x+
( w + 1)
nsubs
where 0 <
?
min (??x )
and ??x is the covariance of the influence weighted subsampled data.
6
(b) Leverage (0.1)
(a) Influence (1.1)
Figure 1: Comparison of the distribution of the influence and leverage for corrupted and noncorrupted points. The `1 distance between the histograms is shown in brackets.
Remark 3. Theorem 2 states that the influence weighted subsampling estimator removes p
the proportional dependance
of
the
error
on
so
the
additional
variance
terms
scale
as
O(?/
?
p/nsubs )
w
w
p
p
and O(? p/nsubs ). The relative contribution of the bias term is ? pk k compared with
2p
? w
pk k for the OLS or non-influence-based subsampling methods.
Comparison with fully corrupted setting. We note that the bound in Theorem 1 is similar to the
bound in [5] for an estimator where all data points are corrupted (i.e. ? = 1) and where incomplete
knowledge of the covariance matrix of the corruptions, ?w is used. The additional bias in the
estimator is proportional to the uncertainty in the estimate of ?w ? in Theorem 1 this corresponds to
2
w . Unbiased estimation is possible if ?w is known. See the Supplementary Information for further
discussion, where the relevant results from [5] are provided in Section SI.6.1 as Lemma 16.
7
Experimental results
We compare IWS-LS against the methods SRHT-LS [14], ULURU [7]. These competing methods
represent current state-of-the-art in fast randomized least squares. Since SRHT-LS is equivalent to
LEV-LS [9] the comparison will highlight the difference between importance sampling according
to the two difference types of regression diagnostic in the corrupted model. Similar to IWS-LS,
ULURU is also a two-step procedure where the first is equivalent to SRHT-LS. The second reduces
bias by subtracting the result of regressing onto the residual. The experiments with the corrupted
data model will demonstrate the difference in robustness of IWS-LS and ULURU to corruptions in
the observations. Note that we do not compare with SGD. Although SGD has excellent properties
for large-scale linear regression, we are not aware of a convex loss function which is robust to the
corruption model we propose.
We assess the empirical performance of our method compared with standard and state-of-the-art
randomized approaches to linear regression in several difference scenarios. We evaluate these methods on the basis of the estimation error: the `2 norm of the difference between the true weights and
the learned weights, k b
k. We present additional results for root mean squared prediction error
(RMSE) on the test set in ?SI.7.
For all the experiments on simulated data sets we use ntrain = 100, 000, ntest = 1000, p = 500.
For datasets of this size, computing exact leverage is impractical and so we report on results for
IWS-LS in ?SI.7. For aIWS-LS and aRWS-LS we used the same number of sub-samples to
approximate the leverage scores and residuals as for solving the regression. For aIWS-LS we set
r2 = p/2 (see Eq. (5)). The results are averaged over 100 runs.
Corrupted data. We investigate the corrupted data noise model described in Eqs. (1)-(2). We
show three scenarios where ? = {0.05, 0.1, 0.3}. X and W were sampled from independent, zeromean Gaussians with standard deviation x = 1 and w = 0.4 respectively. The true regression
coefficients, were sampled from a standard Gaussian. We added i.i.d. zero-mean Gaussian noise
with standard deviation e = 0.1.
Figure 1 shows the difference in distribution of influence and leverage between non-corrupted points
(top) and corrupted points (bottom) for a dataset with 30% corrupted points. The distribution of
leverage is very similar between the corrupted and non-corrupted points, as quantified by the `1
difference. This suggests that leverage alone cannot be used to identify corrupted points.
7
(a) 5% Corruptions
(b) 30% Corruptions
(c) Airline delay
Figure 2: Comparison of mean estimation error and standard deviation on two corrupted simulated
datasets and the airline delay dataset.
On the other hand, although there are some corrupted points with small influence, they typically
have a much larger influence than non-corrupted points. We give a theoretical explanation of this
phenomenon in ?SI.3 (remarks 4 and 5).
Figure 2(a) and (b) shows the estimation error and the mean squared prediction error for different
subsample sizes. In this setting, computing IWS-LS is impractical (due to the exact leverage computation) so we omit the results but we notice that aIWS-LS and aRWS-LS quickly improve over
the full least squares solution and the other randomized approximations in all simulation settings. In
all cases, influence based methods also achieve lower-variance estimates.
For 30% corruptions for a small number of samples ULURU outperforms the other subsampling
methods. However, as the number of samples increases, influence based methods start to outperform
OLS. Here, ULURU converges quickly to the OLS solution but is not able to overcome the bias
introduced by the corrupted datapoints. Results for 10% corruptions are shown in Figs. 5 and 6 and
we provide results on smaller corrupted datasets (to show the performance of IWS-LS) as well as
non-corrupted data simulated according to [13] in ?SI.7.
Airline delay dataset The dataset consists of details of all commercial flights in the USA over 20
years. Dataset along with visualisations available from http://stat-computing.org/dataexpo/2009/.
Selecting the first ntrain = 13, 000 US Airways flights from January 2000 (corresponding to approximately 1.5 weeks) our goal is to predict the delay time of the next ntest = 5, 000 US Airways
flights. The features in this dataset consist of a binary vector representing origin-destination pairs
and a real value representing distance (p = 170).
The dataset might be expected to violate the usual i.i.d. sub-Gaussian design assumption of standard
linear regression since the length of delays are often very different depending on the day. For
example, delays may be longer due to public holidays or on weekends. Of course, such regular
events could be accounted for in the modelling step, but some unpredictable outliers such as weather
delay may also occur. Results are presented in Figure 2(c), the RMSE is the error in predicted delay
time in minutes. Since the dataset is smaller, we can run IWS-LS to observe the accuracy of
aIWS-LS and aRWS-LS in comparison. For more than 3000 samples, these algorithm outperform
OLS and quickly approach IWS-LS. The result suggests that the corrupted observation model is a
good model for this dataset. Furthermore, ULURU is unable to achieve the full accuracy of the OLS
solution.
8
Conclusions
We have demonstrated theoretically and empirically under the generalised corrupted observation
model that influence weighted subsampling is able to significantly reduce both the bias and variance
compared with the OLS estimator and other randomized approximations which do not take influence
into account. Importantly our fast approximation, aRWS-LS performs similarly to IWS-LS. We
find ULURU quickly converges to the OLS estimate, although it is not able to overcome the bias
induced by the corrupted datapoints despite its two-step procedure. The performance of IWS-LS
relative to OLS in the airline delay problem suggests that the corrupted observation model is a more
realistic modelling scenario than the standard sub-Gaussian design model for some tasks. Software
is available at http://people.inf.ethz.ch/kgabriel/software.html.
Acknowledgements. We thank David Balduzzi, Cheng Soon Ong and the anonymous reviewers
for invaluable discussions, suggestions and comments.
8
References
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9
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4,892 | 5,429 | Fast Multivariate Spatio-temporal Analysis
via Low Rank Tensor Learning
Mohammad Taha Bahadori?
Dept. of Electrical Engineering
Univ. of Southern California
Los Angeles, CA 90089
[email protected]
Qi (Rose) Yu?
Dept. of Computer Science
Univ. of Southern California
Los Angeles, CA 90089
[email protected]
Yan Liu
Dept. of Computer Science
Univ. of Southern California
Los Angeles, CA 90089
[email protected]
Abstract
Accurate and efficient analysis of multivariate spatio-temporal data is critical in
climatology, geology, and sociology applications. Existing models usually assume
simple inter-dependence among variables, space, and time, and are computationally expensive. We propose a unified low rank tensor learning framework for multivariate spatio-temporal analysis, which can conveniently incorporate different
properties in spatio-temporal data, such as spatial clustering and shared structure
among variables. We demonstrate how the general framework can be applied to
cokriging and forecasting tasks, and develop an efficient greedy algorithm to solve
the resulting optimization problem with convergence guarantee. We conduct experiments on both synthetic datasets and real application datasets to demonstrate
that our method is not only significantly faster than existing methods but also
achieves lower estimation error.
1
Introduction
Spatio-temporal data provide unique information regarding ?where? and ?when?, which is essential
to answer many important questions in scientific studies from geology, climatology to sociology. In
the context of big data, we are confronted with a series of new challenges when analyzing spatiotemporal data because of the complex spatial and temporal dependencies involved.
A plethora of excellent work has been conducted to address the challenge and achieved successes to
a certain extent [8, 13]. Often times, geostatistical models use cross variogram and cross covariance
functions to describe the intrinsic dependency structure. However, the parametric form of cross
variogram and cross covariance functions impose strong assumptions on the spatial and temporal
correlation, which requires domain knowledge and manual work. Furthermore, parameter learning
of those statistical models is computationally expensive, making them infeasible for large-scale
applications.
Cokriging and forecasting are two central tasks in multivariate spatio-temporal analysis. Cokriging
utilizes the spatial correlations to predict the value of the variables for new locations. One widely
adopted method is multitask Gaussian process (MTGP) [4], which assumes a Gaussian process prior
over latent functions to directly induce correlations between tasks. However, for a cokriging task
with M variables of P locations for T time stamps, the time complexity of MTGP is O(M 3 P 3 T )
[4]. For forecasting, popular methods in multivariate time series analysis include vector autoregressive (VAR) models, autoregressive integrated moving average (ARIMA) models, and cointegration
models. An alternative method for spatio-temporal analysis is Bayesian hierarchical spatio-temporal
models with either separable and non-separable space-time covariance functions [6]. Rank reduced
?
Authors have equal contributions.
1
models have been proposed to capture the inter-dependency among variables [1]. However, very
few models can directly handle the correlations among variables, space and time simultaneously in
a scalable way. In this paper, we aim to address this problem by presenting a unified framework for
many spatio-temporal analysis tasks that are scalable for large-scale applications.
Tensor representation provides a convenient way to capture inter-dependencies along multiple dimensions. Therefore it is natural to represent the multivariate spatio-temporal data in tensor. Recent
advances in low rank learning have led to simple models that can capture the commonalities among
each mode of the tensor [15, 20]. Similar argument can be found in the literature of spatial data recovery [11], neuroimaging analysis [26], and multi-task learning [20]. Our work builds upon recent
advances in low rank tensor learning [15, 11, 26] and further considers the scenario where additional
side information of data is available. For example, in geo-spatial applications, apart from measurements of multiple variables, geographical information is available to infer location adjacency; in
social network applications, friendship network structure is collected to obtain preference similarity.
To utilize the side information, we can construct a Laplacian regularizer from the similarity matrices,
which favors locally smooth solutions.
We develop a fast greedy algorithm for learning low rank tensors based on the greedy structure
learning framework [2, 24, 21]. Greedy low rank tensor learning is efficient, as it does not require
full singular value decomposition of large matrices as opposed to other alternating direction methods
[11]. We also provide a bound on the difference between the loss function at our greedy solution
and the one at the globally optimal solution. Finally, we present experiment results on simulation
datasets as well as application datasets in climate and social network analysis, which show that our
algorithm is faster and achieves higher prediction accuracy than state-of-art approaches in cokriging
and forecasting tasks.
2
Tensor formulation for multivariate spatio-temporal analysis
The critical element in multivariate spatio-temporal analysis is an efficient way to incorporate the
spatial temporal correlations into modeling and automatically capture the shared structures across
variables, locations, and time. In this section, we present a unified low rank tensor learning framework that can perform various types of spatio-temporal analysis. We will use two important applications, i.e., cokriging and forecasting, to motivate and describe the framework.
2.1
Cokriging
In geostatistics, cokriging is the task of interpolating the data of one variable for unknown locations
by taking advantage of the observations of variables from known locations. For example, by making
use of the correlations between precipitation and temperature, we can obtain more precise estimate
of temperature in unknown locations than univariate kriging. Formally, denote the complete data
for P locations over T time stamps with M variables as X ? RP ?T ?M . We only observe the
measurements for a subset of locations ? ? {1, . . . , P } and their side information such as longitude
and latitude. Given the measurements X? and the side information, the goal is to estimate a tensor
W ? RP ?T ?M that satisfies W? = X? . Here X? represents the outcome of applying the index
operator I? to X:,:,m for all variables m = 1, . . . , M . The index operator I? is a diagonal matrix
whose entries are one for the locations included in ? and zero otherwise.
Two key consistency principles have been identified for effective cokriging [9, Chapter 6.2]: (1)
Global consistency: the data on the same structure are likely to be similar. (2) Local consistency: the
data in close locations are likely to be similar. The former principle is akin to the cluster assumption
in semi-supervised learning [25]. We incorporate these principles in a concise and computationally
efficient low-rank tensor learning framework.
To achieve global consistency, we constrain the tensor W to be low rank. The low rank assumption
is based on the belief that high correlations exist within variables, locations and time, which leads to
natural clustering of the data. Existing literature have explored the low rank structure among these
three dimensions separately, e.g., multi-task learning [19] for variable correlation, fixed rank kriging
[7] for spatial correlations. Low rankness assumes that the observed data can be described with a
few latent factors. It enforces the commonalities along three dimensions without an explicit form
for the shared structures in each dimension.
2
For local consistency, we construct a regularizer via the spatial Laplacian matrix. The Laplacian
matrix is defined as L = D ?
PA, where A is a kernel matrix constructed by pairwise similarity
and diagonal matrix Di,i =
j (Ai,j ). Similar ideas have been explored in matrix completion
[16]. In cokriging literature, the local consistency is enforced via the spatial covariance matrix. The
Bayesian models often impose the Gaussian process prior on the observations with the covariance
matrix K = Kv ? Kx where Kv is the covariance between variables and Kx is that for locations.
The Laplacian regularization term corresponds to the relational Gaussian process [5] where the
covariance matrix is approximated by the spatial Laplacian.
In summary, we can perform cokriging and find the value of tensor W by solving the following
optimization problem:
(
)
M
X
2
>
c
W = argmin kW? ? X? kF + ?
tr(W:,:,m LW:,:,m )
s.t.
rank(W) ? ?,
(1)
W
m=1
qP
2
where the Frobenius norm of a tensor A is defined as kAkF =
i,j,k Ai,j,k and ?, ? > 0
are the parameters that make tradeoff between the local and global consistency, respectively. The
low rank constraint finds the principal components of the tensor and reduces the complexity of
the model while the Laplacian regularizer clusters the data using the relational information among
the locations. By learning the right tradeoff between these two techniques, our method is able to
benefit from both. Due to the various definitions of tensor rank, we use rank as supposition for rank
complexity, which will be specified in later section.
2.2
Forecasting
Forecasting estimates the future value of multivariate time series given historical observations.
For ease of presentation, we use the classical VAR model with K lags and coefficient tensor
W ? RP ?KP ?M as an example. Using the matrix representation, the VAR(K) process defines
the following data generation process:
X:,t,m = W:,:,m Xt,m + E:,t,m , for m = 1, . . . , M and t = K + 1, . . . , T,
(2)
>
>
]> denotes the concatenation of K-lag historical data before
where Xt,m = [X:,t?1,m
, . . . , X:,t?K,m
time t. The noise tensor E is a multivariate Gaussian with zero mean and unit variance .
Existing multivariate regression methods designed to capture the complex correlations, such as
Tucker decomposition [20], are computationally expensive. A scalable solution requires a simpler
model that also efficiently accounts for the shared structures in variables, space, and time. Similar
global and local consistency principles still hold in forecasting. For global consistency, we can use
low rank constraint to capture the commonalities of the variables as well as the spatial correlations
on the model parameter tensor, as in [8]. For local consistency, we enforce the predicted value
for close locations to be similar via spatial Laplacian regularization. Thus, we can formulate the
forecasting task as the following optimization problem over the model coefficient tensor W:
(
)
M
X
2
>
c
b
b
b
W = argmin kX ? X kF + ?
tr(X:,:,m LX:,:,m )
s.t. rank(W) ? ?, Xb:,t,m = W:,:,m Xt,m
W
m=1
(3)
Though cokriging and forecasting are two different tasks, we can easily see that both formulations
follow the global and local consistency principles and can capture the inter-correlations from spatialtemporal data.
2.3
Unified Framework
We now show that both cokriging and forecasting can be formulated into the same tensor learning
framework. Let us rewrite the loss function in Eq. (1) and Eq. (3) in the form of multitask regression
and complete the quadratic form for the loss function. The cokriging task can be reformulated as
follows:
( M
)
X
> ?1
2
c
W = argmin
kW:,:,m H ? (H ) X?,m kF
s.t. rank(W) ? ?
(4)
W
m=1
3
where we define HH > = I? + ?L.1 For the forecasting problem, HH > = IP + ?L and we have:
( M
)
T
X X
c = argmin
W
kHW:,:,m Xt,m ? (H ?1 )X:,t,m k2F
s.t. rank(W) ? ?,
(5)
W
m=1 t=K+1
By slight change of notation (cf. Appendix D), we can easily see that the optimal solution of both
problems can be obtained by the following optimization problem with appropriate choice of tensors
Y and V:
( M
)
X
c = argmin
W
kW:,:,m Y:,:,m ? V:,:,m k2F
s.t. rank(W) ? ?.
(6)
W
m=1
After unifying the objective function, we note that tensor rank has different notions such as CP
rank, Tucker rank and mode n-rank [15, 11]. In this paper, we choose the mode-n rank, which is
computationally more tractable [11, 23]. The mode-n rank of a tensor W is the rank of its mode-n
unfolding W(n) .2 In particular, for a tensor W with N mode, we have the following definition:
mode-n rank(W) =
N
X
rank(W(n) ).
(7)
n=1
A common practice to solve this formulation with mode n-rank constraint is to relax the rank constraint to a convex nuclear norm constraint [11, 23]. However, those methods are computationally
expensive since they need full singular value decomposition of large matrices. In the next section,
we present a fast greedy algorithm to tackle the problem.
3
Fast greedy low rank tensor learning
To solve the non-convex problem in Eq. (6) and find its optimal solution, we propose a greedy
learning algorithm by successively adding rank-1 estimation of the mode-n unfolding. The main
idea of the algorithm is to unfold the tensor into a matrix, seek for its rank-1 approximation and
then fold back into a tensor with same dimensionality. We describe this algorithm in three steps:
(i) First, we show that we can learn rank-1 matrix estimations efficiently by solving a generalized
eigenvalue problem, (ii) We use the rank-1 matrix estimation to greedily solve the original tensor
rank constrained problem, and (iii) We propose an enhancement via orthogonal projections after
each greedy step.
Optimal rank-1 Matrix Learning The following lemma enables us to find such optimal rank-1
estimation of the matrices.
Lemma 1. Consider the following rank constrained problem:
n
o
2
b1 = argmin
A
kY ? AXkF ,
(8)
A:rank(A)=1
q?n
p?n
b1 can be written as
where Y ? R
,X ? R
, and A ? Rq?p . The optimal solution of A
>
b
b
b
b
A1 = uv , kb
vk2 = 1 where v is the dominant eigenvector of the following generalized eigenvalue
problem:
(XY > Y X > )v = ?(XX > )v
(9)
b
and u can be computed as
1
b.
b= >
Y X >v
(10)
u
b XX > v
b
v
Proof is deferred to Appendix A. Eq. (9) is a generalized eigenvalue problem whose dominant
eigenvector can be found efficiently [12]. If XX > is full rank, as assumed in Theorem 2, the
problem is simplified to a regular eigenvalue problem whose dominant eigenvector can be efficiently
computed.
1
We can use Cholesky decomposition to obtain H. In the rare cases that I? + ?L is not full rank, IP is
added where is a very small positive value.
2
The mode-n unfolding of a tensor is the matrix resulting from treating n as the first mode of the matrix,
and cyclically concatenating other modes. Tensor refolding is the reverse direction operation [15].
4
Algorithm 1 Greedy Low-rank Tensor Learning
1: Input: transformed data Y, V of M variables, stopping criteria ?
2: Output: N mode tensor W
3: Initialize W ? 0
4: repeat
5:
for n = 1 to N do
6:
Bn ? argmin L(refold(W(n) + B); Y, V)
B: rank(B)=1
7:
8:
9:
?n ? L(W; Y, V) ? L(refold(W(n) + Bn ); Y, V)
end for
n? ? argmax{?n }
n
10:
11:
12:
13:
if ?n? > ? then
W ? W + refold(Bn? , n? )
end if
W ? argminrow(A(1) )?row(W(1) ) L(A; Y, V)
# Optional Orthogonal Projection Step.
col(A(1) )?col(W(1) )
14: until ?n? < ?
Greedy Low n-rank Tensor Learning The optimal rank-1 matrix learning serves as a basic element in our greedy algorithm. Using Lemma 1, we can solve the problem in Eq. (6) in the Forward
Greedy Selection framework as follows: at each iteration of the greedy algorithm, it searches for the
mode that gives the largest decrease in the objective function. It does so by unfolding the tensor in
that mode and finding the best rank-1 estimation of the unfolded tensor. After finding the optimal
mode, it adds the rank-1 estimate in that mode to the current estimation of the tensor. Algorithm
PM
1 shows the details of this approach, where L(W; Y, V) = m=1 kW:,:,m Y:,:,m ? V:,:,m k2F . Note
that we can find the optimal rank-1 solution in only one of the modes, but it is enough to guarantee
the convergence of our greedy algorithm.
Theorem 2 bounds the difference between the loss function evaluated at each iteration of the greedy
algorithm and the one at the globally optimal solution.
>
Theorem 2. Suppose in Eq. (6) the matrices Y:,:,m
Y:,:,m for m = 1, . . . , M are positive definite.
The solution of Algo. 1 at its kth iteration step satisfies the following inequality:
L(Wk ; Y, V) ? L(W ? ; Y, V) ?
?
(kYk2 kW(1)
k? )2
(k + 1)
,
(11)
where W ? is the global minimizer of the problem in Eq. (6) and kYk2 is the largest singular value
of a block diagonal matrix created by placing the matrices Y(:, :, m) on its diagonal blocks.
The detailed proof is given in Appendix B. The key idea of the proof is that the amount of decrease
in the loss function by each step in the selected mode is not smaller than the amount of decrease if we
had selected the first mode. The theorem shows that we can obtain the same rate of convergence for
learning low rank tensors as achieved in [22] for learning low rank matrices. The greedy algorithm
in Algorithm 1 is also connected to mixture regularization in [23]: the mixture approach decomposes
the solution into a set of low rank structures while the greedy algorithm successively learns a set of
rank one components.
Greedy Algorithm with Orthogonal Projections It is well-known that the forward greedy algorithm may make steps in sub-optimal directions because of noise. A common solution to alleviate the
effect of noise is to make orthogonal projections after each greedy step [2, 21]. Thus, we enhance the
forward greedy algorithm by projecting the solution into the space spanned by the singular vectors
of its mode-1 unfolding. The greedy algorithm with orthogonal projections performs an extra step in
line 13 of Algorithm 1: It finds the top k singular vectors of the solution: [U, S, V ] ? svd(W(1) , k)
where k is the iteration number. Then it finds the best solution in the space spanned by U and V by
solving Sb ? minS L(U SV > , Y, V) which has a closed form solution. Finally, it reconstructs the
b > , 1). Note that the projection only needs to find top k singular vectors
solution: W ? refold(U SV
which can be computed efficiently for small values of k.
5
1
0.9
0.8
0.7
0.6
Forward
Orthogonal
ADMM
Trace
1000
15
Run Time (Sec)
1.1
20
Mixture Rank Complexity
Parameter Estimation RMSE
1200
Forward
Orthogonal
ADMM
Trace
MTL?L1
MTL?L21
MTL?Dirty
1.2
10
5
0
50
100
150
# of Samples
(a) RMSE
200
250
?5
0
600
400
200
0.5
0.4
0
800
Forward Greedy
Orthogonal Greedy
ADMM
50
100
150
# of Samples
(b) Rank
200
0 1
10
2
# of Variables
10
(c) Scalability
Figure 1: Tensor estimation performance comparison on the synthetic dataset over 10 random runs.
(a) parameter Estimation RMSE with training time series length, (b) Mixture Rank Complexity with
training time series length, (c) running time for one single round with respect to number of variables.
4
Experiments
We evaluate the efficacy of our algorithms on synthetic datasets and real-world application datasets.
4.1
Low rank tensor learning on synthetic data
For empirical evaluation, we compare our method with multitask learning (MTL) algorithms, which
also utilize the commonalities between different prediction tasks for better performance. We use the
following baselines: (1) Trace norm regularized MTL (Trace), which seeks the low rank structure
only on the task dimension; (2) Multilinear MTL [20], which adapts the convex relaxation of low
rank tensor learning solved with Alternating Direction Methods of Multiplier (ADMM) [10] and
Tucker decomposition to describe the low rankness in multiple dimensions; (3) MTL-L1 , MTL-L21
[19], and MTL-LDirty [14], which investigate joint sparsity of the tasks with Lp norm regularization.
For MTL-L1 , MTL-L21 [19] and MTL-LDirty , we use MALSAR Version 1.1 [27].
We construct a model coefficient tensor W of size 20 ? 20 ? 10 with CP rank equals to 1.
Then, we generate the observations Y and V according to multivariate regression model V:,:,m =
W:,:,m Y:,:,m + E:,:,m for m = 1, . . . , M , where E is tensor with zero mean Gaussian noise elements.
We split the synthesized data into training and testing time series and vary the length of the training
time series from 10 to 200. For each training length setting, we repeat the experiments for 10 times
and select the model parameters via 5-fold cross validation. We measure the prediction performance
via two criteria: parameter estimation accuracy and rank complexity. For accuracy, we calculate the
RMSE of the estimation versus the true model coefficient tensor. For rank complexity, we calculate
PN
the mixture rank complexity [23] as M RC = n1 n=1 rank(W(n) ).
The results are shown in Figure 1(a) and 1(b). We omit the Tucker decomposition as the results are
not comparable. We can clearly see that the proposed greedy algorithm with orthogonal projections
achieves the most accurate tensor estimation. In terms of rank complexity, we make two observations: (i) Given that the tensor CP rank is 1, greedy algorithm with orthogonal projections produces
the estimate with the lowest rank complexity. This can be attributed to the fact that the orthogonal
projections eliminate the redundant rank-1 components that fall in the same spanned space. (ii) The
rank complexity of the forward greedy algorithm increases as we enlarge the sample size. We believe that when there is a limited number of observations, most of the new rank-1 elements added
to the estimate are not accurate and the cross-validation steps prevent them from being added to the
model. However, as the sample size grows, the rank-1 estimates become more accurate and they are
preserved during the cross-validation.
To showcase the scalability of our algorithm, we vary the number of variables and generate a series
of tensor W ? R20?20?M for M from 10 to 100 and record the running time (in seconds) for three
tensor learning algorithms, i.e, forward greedy, greedy with orthogonal projections and ADMM. We
measure the run time on a machine with a 6-core 12-thread Intel Xenon 2.67GHz processor and
12GB memory. The results are shown in Figure 1(c). The running time of ADMM increase rapidly
with the data size while the greedy algorithm stays steady, which confirms the speedup advantage
of the greedy algorithm.
6
Table 1: Cokriging RMSE of 6 methods averaged over 10 runs. In each run, 10% of the locations
are assumed missing.
DATASET
USHCN
CCDS
Y ELP
F OURSQUARE
4.2
ADMM
0.8051
0.8292
0.7730
0.1373
F ORWARD
0.7594
0.5555
0.6993
0.1338
O RTHOGONAL
0.7210
0.4532
0.6958
0.1334
S IMPLE
0.8760
0.7634
NA
NA
O RDINARY
0.7803
0.7312
NA
NA
MTGP
1.0007
1.0296
NA
NA
Spatio-temporal analysis on real world data
We conduct cokriging and forecasting experiments on four real-world datasets:
USHCN The U.S. Historical Climatology Network Monthly (USHCN)3 dataset consists of
monthly climatological data of 108 stations spanning from year 1915 to 2000. It has three climate variables: (1) daily maximum, (2) minimum temperature averaged over month, and (3) total
monthly precipitation.
CCDS The Comprehensive Climate Dataset (CCDS)4 is a collection of climate records of North
America from [18]. The dataset was collected and pre-processed by five federal agencies. It contains
monthly observations of 17 variables such as Carbon dioxide and temperature spanning from 1990 to
2001. The observations were interpolated on a 2.5 ? 2.5 degree grid, with 125 observation locations.
Yelp The Yelp dataset5 contains the user rating records for 22 categories of businesses on Yelp
over ten years. The processed dataset includes the rating values (1-5) binned into 500 time intervals
and the corresponding social graph for 137 active users. The dataset is used for the spatio-temporal
recommendation task to predict the missing user ratings across all business categories.
Foursquare The Foursquare dataset [17] contains the users? check-in records in Pittsburgh area
from Feb 24 to May 23, 2012, categorized by different venue types such as Art & Entertainment,
College & University, and Food. The dataset records the number of check-ins by 121 users in each
of the 15 category of venues over 1200 time intervals, as well as their friendship network.
4.2.1
Cokriging
We compare the cokriging performance of our proposed method with the classical cokriging approaches including simple kriging and ordinary cokriging with nonbias condition [13] which are
applied to each variables separately. We further compare with multitask Gaussian process (MTGP)
[4] which also considers the correlation among variables. We also adapt ADMM for solving the
nuclear norm relaxed formulation of the cokriging formulation as a baseline (see Appendix C for
more details). For USHCN and CCDS, we construct a Laplacian matrix by calculating the pairwise
Haversine distance of locations. For Foursquare and Yelp, we construct the graph Laplacian from
the user friendship network.
For each dataset, we first normalize it by removing the trend and diving by the standard deviation.
Then we randomly pick 10% of locations (or users for Foursquare) and eliminate the measurements
of all variables over the whole time span. Then, we produce the estimates for all variables of each
timestamp. We repeat the procedure for 10 times and report the average prediction RMSE for all
timestamps and 10 random sets of missing locations. We use the MATLAB Kriging Toolbox6 for
the classical cokriging algorithms and the MTGP code provided by [4].
Table 1 shows the results for the cokriging task. The greedy algorithm with orthogonal projections is
significantly more accurate in all three datasets. The baseline cokriging methods can only handle the
two dimensional longitude and latitude information, thus are not applicable to the Foursquare and
Yelp dataset with additional friendship information. The superior performance of the greedy algorithm can be attributed to two of its properties: (1) It can obtain low rank models and achieve global
consistency; (2) It usually has lower estimation bias compared to nuclear norm relaxed methods.
3
http://www.ncdc.noaa.gov/oa/climate/research/ushcn
http://www-bcf.usc.edu/?liu32/data/NA-1990-2002-Monthly.csv
5
http://www.yelp.com/dataset_challenge
6
http://globec.whoi.edu/software/kriging/V3/english.html
4
7
Table 2: Forecasting RMSE for VAR process with 3 lags, trained with 90% of the time series.
DATASET
USHCN
CCDS
FSQ
T UCKER
0.8975
0.9438
0.1492
ADMM F ORWARD
0.9227 0.9171
0.8448 0.8810
0.1407 0.1241
O RTHO O RTHO NL
0.9069 0.9175
0.8325 0.8555
0.1223 0.1234
T RACE
0.9273
0.8632
0.1245
MTLl1
0.9528
0.9105
0.1495
MTLl21 MTLdirty
0.9543 0.9735
0.9171 1.0950
0.1495 0.1504
Table 3: Running time (in seconds) for cokriging and forecasting.
DATASET
ORTHO
ADMM
4.2.2
USHCN
93.03
791.25
C OKRIGING
CCDS
YELP
16.98
78.47
320.77 2928.37
FSQ
91.51
720.40
F ORECASTING
USHCN CCDS
FSQ
75.47
21.38
37.70
235.73
45.62
33.83
Forecasting
We present the empirical evaluation on the forecasting task by comparing with multitask regression
algorithms. We split the data along the temporal dimension into 90% training set and 10% testing
set. We choose VAR(3) model and during the training phase, we use 5-fold cross-validation.
As shown in Table 2, the greedy algorithm with orthogonal projections again achieves the best prediction accuracy. Different from the cokriging task, forecasting does not necessarily need the correlations of locations for prediction. One might raise the question as to whether the Laplacian regularizer helps. Therefore, we report the results for our formulation without Laplacian (ORTHONL)
for comparison. For efficiency, we report the running time (in seconds) in Table 3 for both tasks of
cokriging and forecasting. Compared with ADMM, which is a competitive baseline also capturing
the commonalities among variables, space, and time, our greedy algorithm is much faster for most
datasets.
As a qualitative study, we plot the map of most predictive regions analyzed by the greedy algorithm using CCDS dataset in Fig. 2. Based on the concept
of how informative the past values of the climate
measurements in a specific location are in predicting future values of other time series, we define the
aggregate strength of predictiveness of each region
PP PM
as w(t) = p=1 m=1 |Wp,t,m |. We can see that
two regions are identified as the most predictive regions: (1) The southwest region, which reflects the
impact of the Pacific ocean and (2) The southeast region, which frequently experiences relative sea level
rise, hurricanes, and storm surge in Gulf of Mexico.
Another interesting region lies in the center of Colorado, where the Rocky mountain valleys act as a
funnel for the winds from the west, providing locally
divergent wind patterns.
5
Figure 2: Map of most predictive regions
analyzed by the greedy algorithm using 17
variables of the CCDS dataset. Red color
means high predictiveness whereas blue denotes low predictiveness.
Conclusion
In this paper, we study the problem of multivariate spatio-temporal data analysis with an emphasis
on two tasks: cokriging and forecasting. We formulate the problem into a general low rank tensor
learning framework which captures both the global consistency and the local consistency principle.
We develop a fast and accurate greedy solver with theoretical guarantees for its convergence. We
validate the correctness and efficiency of our proposed method on both the synthetic dataset and realapplication datasets. For future work, we are interested in investigating different forms of shared
structure and extending the framework to capture non-linear correlations in the data.
Acknowledgment
We thank the anonymous reviewers for their helpful feedback and comments. The research was
sponsored by the NSF research grants IIS-1134990, IIS- 1254206 and Okawa Foundation Research
Award. The views and conclusions are those of the authors and should not be interpreted as representing the official policies of the funding agency, or the U.S. Government.
8
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4,893 | 543 | Recognition of Manipulated Objects
by Motor Learning
Hiroaki Gomi
Mitsuo Kawato
ATR Auditory and Visual Perception Research Laboratories,
Inui-dani, Sanpei-dani, Seika-cho, Soraku-gun, Kyoto 619-02, Japan
Abstract
We present two neural network controller learning schemes based on feedbackerror-learning and modular architecture for recognition and control of multiple
manipulated objects. In the first scheme, a Gating Network is trained to acquire
object-specific representations for recognition of a number of objects (or sets of
objects). In the second scheme, an Estimation Network is trained to acquire
function-specific, rather than object-specific, representations which directly estimate
physical parameters. Both recognition networks are trained to identify manipulated
objects using somatic and/or visual information. After learning, appropriate
motor commands for manipulation of each object are issued by the control
networks.
1 INTRODUCTION
Conventional feedforward neural-network controllers (Barto et aI., 1983; Psaltis et al.,
1987; Kawato et aI., 1987, 1990; Jordan, 1988; Katayama & Kawato, 1991) can not cope
with multiple or changeable manipulated objects or disturbances because they cannot
change immediately the control law corresponding to the object. In interaction with
manipulated objects or, in more general terms, in interaction with an environment which
contains unpredictable factor, feedback information is essential for control and object
recognition. From these considerations, Gomi & Kawato (1990) have examined the
adaptive feedback controller learning schemes using feedback-error-Iearning, from which
impedance control (Hogan, 1985) can be obtained automatically. However, in that scheme,
some higher system needs to supervise the setting of the appropriate mechanical impedance
for each manipulated object or environment.
In this paper, we introduce semi-feedforward control schemes using neural networks which
receive feedback and/or feedforward information for recognition of multiple manipulated
objects based on feedback-error-learning and modular network architecture. These schemes
have two advantages over previous ones as follows. (1) Learning is achieved without the
547
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Gomi and Kawato
exact target motor command vector, which is unavailable during supervised motor learning.
(2) Although somatic information alone was found to be sufficient to recognize objects,
object identification is predictive and more reliable when both somatic and visual information
are used.
2 RECOGNITION OF MANIPULATED OBJECTS
The most important issues in object manipulation are (l) how to recognize the manipulated
object and (2) how to achieve uniform performance for different objects. There are several
ways to acquire helpful information for recognizing manipulated objects. Visual information
and somatic information (performance by motion) are most informative for object recognition
for manipulation.
The physical characteristics useful for object manipulation such as mass, softness and
slipperiness, can not be predicted without the experience of manipulating similar objects.
In this respect, object recognition for manipulation should be learned through object
manipulation.
3 MODULAR ARCHITECTURE USING GATING NETWORK
Jacobs et al. (1990, 1991) and Nowlan & Hinton (1990, 1991) have proposed a competitive
modular network architecture which is applied to the task decomposition problem or
classification problems. Jacobs (1991) applied this network architecture to the multi-payload
robotics task in which each expert network controller is trained for each category of
manipulated objects in terms of the object's mass. In his scheme, the payload's identity is
fed to the gating network to select a suitable expert network which acts as a feedforward
controller.
We examined modular network architecture using feedback-e"or-learning for simultaneous
learning of object recognition and control task as shown in Fig.l.
M1Tt rU."nbmmOOi--._,",So
v
"'::" ,:f,
t:~~ ~~t~ t:~
Gatmg
Network ....+----.
Expert Network 1
Expert Network 2
t--=-.c::u-~
Expert Network 3
+
u~
u
Controlled
t -.......eo{4~"""""-"l~ object
...-_..
1--. . . .
Fig.1 Configuration of the modular architecture using Gating Network
for object manipulation based on feedback-error-learning
In this learning scheme, the quasi-target vector for combined output of expert networks is
employed instead of the exact target vector. This is because it is unlikely that the exact
target motor command vector can be provided in learning. The quasi-target vector of
feedforward motor command, u' is produced by :
(1)
U '- U + Ufo '
Recognition of Manipulated Objects by Motor Learning
Here, U denotes the previous final motor command and ufo denotes the feedback motor
command. Using this quasi-target vector, the gating and expert networks are trained to
maximize the log-likelihood function, In L, by using backpropagation.
In L
=In i
gje -IU'-u,r /2(1,2
(2)
j=!
Here, uj is the i th expert network output, (Ij is a variance scaling parameter of the i th
expert network and gj' the i th output of gating network, is calculated by
gj
e
S,
(3)
= - 1 1- ,
Le
sJ
j=!
where Sj denotes the weighted input received by the i th output unit. The total output of
the modular network is
11
uff
=~gjUj'
j=l
(4)
By maximizing Eq.2 using steepest ascent method, the gating network learns to choose
the expert network whose output is closest to the quasi-target command, and each expert
network is tuned correctly when it is chosen by the gating network. The desired trajectory
is fed to the expert networks so as to make them work as feedforward controllers.
4 SIMULATION OF OBJECT MANIPULATION
BY MODULAR ARCHITECTURE WITH GATING NETWORK
We show the advantage of the learning schemes presented above by simulation results
below. The configuration of the controlled object and manipulated object is shown in
Fig.2 in which M, B, K respectively denote the mass, viscosity and stiffness of the
coupled object (controlled- and manipulated-object). The manipulated object is changed
every epoch (l [sec]) while the coupled object is controlled to track the desired trajectory.
Fig.3 shows the selected object, the feedforward and feedback motor commands, and the
desired and actual trajectories before learning.
----------
l~__
~
a
x
-4--j
M
t:~
.24,------~1------r-----"--~~1
o
Fig.2 Configuration of the controlled
object and the manipulated object
5
20
time [??c]
Fig.3 Temporal patterns of the selected
object, the motor commands, the desired
and actual trajectories before learning
The desired trajectory, x d ' was produced by Ornstein-Uhlenbeck random process. As
shown in Fig.3, the error between the desired trajectory and the actual trajectory remained
because the feedback controller in which the gains were fixed, was employed in this
condition. (Physical characteristics of the objects used are listed in Fig.4a)
549
550
Gomi and Kawato
4.1 SOMATIC INFORMATION FOR GATING NETWORK
We call the actual trajectory vector, x, and the final motor command,
"somatic
infonnation". Somatic infonnation should be most useful for on-line (feedback) recognition
of the dynamical characteristics of manipulated objects. The latest four times data of
somatic information were used as the gating network inputs for identification of the
coupled object in this simulation. s ofEq.3 is expressed as:
s(t) = '1'1 (x(t), x(t -1), x(t - 2), x(t - 3), u(t), u(t -1), u(t - 2), u(t - 3?).
(5)
The dynamical characteristics of coupled objects are shown in Fig.4a. The object was
changed in every epoch (l [secD. The variance scaling parameter was (Jj = 0.8 and the
learning rates were 77g a,e =1. 0 x 10-3 and 77 expert i = 1. 0 x 10-5 ? The three-layered feedforward
neural network (input 16, hidden 30, output 3) was employed for the gating network and
the two-layered linear networks (input 3, output 1) were used for the expert networks.
Comparing the expert's weights after learning and the coupled object characteristics in
Fig.4a, we realize that expert networks No.1, No.2, No.3 obtained the inverse dynamics
of coupled objects y, (3, a, respectively. The time variation of object, the gating network
outputs, motor commands and trajectories after learning are shown in Fig.4b. The gating
network outputs for the objects responded correctly in the most of the time and the
feedback motor command, ufo' was almost zero. As a consequence of adaptation, the
actual trajectory almost perfectly corresponded with the desired trajectory.
b.
a.
-
'Y
Gating Net Outputs v.s. Objects
U ,
-- -- ---
relinal
chara::loristiicsl ,mago D-u::-;-""""::'::"::-;:;':-:=-r-"7.:""':...--i
M B K
a
1.0 2.0 8.0
none
f3
5.0 7.0 4.0
none
- 20 --\---'_ _---"'.-_ _----'-..L"--_ _
~~---.:,;
!1~actC:al
none
1~L_8.03.01
____.0 L:::Lll?[ili?~
....~.... =::O~
~d.::
L.
IS.
_
o
5
_
10
tlmo
15
20
[,.cl
Fig.4 Somatic information for gating network, a. Statistical analysis of the
correspondence of the expert networks with each object after learning (averaged
gating outputs), b. Temporal patterns of objects, gating outputs, motor commands
and trajectories after learning
4.2 VISUAL INFORMATION FOR GATING NETWORK
We usually assume the manipulated object's characteristics by using visual infonnation.
Visual information might be helpful for feedforward recognition. In this case, s of Eq.3 is
expressed as:
s(t) = 'l'2(V(t?) .
(6)
We used three visual cues corresponding to each coupled object in this simulation as
shown in Fig.5a. At each epoch in this simulation, one of three visual cues selected
randomly is randomly placed at one of four possible locations on a 4 x 4 retinal matrix.
Recognition of Manipulated Objects by Motor Learning
The visual cues of each object are different, but object ex and ex* have the same dynamical
characteristics as shown in Fig.5a. The gating network should identify the object and
select a suitable expert network for feedforward control by using this visual information.
The learning coefficients were O"j 0.7, 17gate 1. 0 X 10-3 , 17eXpert j = 1. 0 X 10-5 . The same
networks used in above experiment were used in this simulation.
=
=
After learning, the expert network No.2 acquired the inverse dynamics of object ex and ex * ,
and expert network No.3 accomplished this for object y. It is recognized from Fig.5b that
the gating network almost perfectly selected expert network No.2 for object ex and ex*,
and almost perfectly selected expert network No.3 for object y. Expert network No.1
which did not acquire inverse dynamics corresponding to any of the three objects, was not
selected in the test period after learning. The actual trajectory in the test period corresponded
almost perfectly to the desired trajectory.
b.
a.
Gating Net. Outputs V.s. Objects
------ - -
tlma [sac]
Fig. 5 Visual information for gating network, a. Statistical analysis of the
correspondence of the expert networks with each Object after learning (averaged
gating outputs), b. Temporal patterns of objects, gating outputs, motor commands
and trajectories after learning
4.3 SOMATIC & VISUAL INFORMATION FOR GATING NETWORK
We show here the simulation results by using both of somatic and visual information as
the gating network inputs. In this case, s ofEq.3 is represented as:
s(t)= 'l'3(x(t),?? ?,x(t-3),u(t),???,u(t-3),V(t)).
(7)
In this simulation, the object ex and ~* had different dynamical characteristics, but shared
same visual cue as listed in Fig.6a. Thus, to identify the coupled object one by one, it is
necessary for the gating network to utilize not only visual information but also somatic
information. The learning coefficients were O"j =1. 0, 17gale = 1. 0 X 10-3 and
17expert j = 1. 0 X 10-5 . The gating network had 32 input units, 50 hidden units, and 1 output
unit, and the expert networks were the same as in the above experiment.
After learning, expert networks No.1, No.2, No.3 acquired the inverse dynamics of
objects y, ~*, ex respectively. As shown in Fig.6b, the gating network identified the
object almost correctly.
551
552
Gomi and Kawato
--
b.
a.
-
- ---
Gating Net. Outputs v.s. Objects
Objac1
physical
charactonstics
M
B K
- 20~------~----~
____________
~
8
:2j~
1
LL__.J?~2Lill1iiliJill___? ? ?"l
-
actual
~
0
j
o
5
10
15
20
time [.ocJ
Fig. 6 Somatic & Visual information for gating network, a. Statistical analysis of the
correspondence of the expert networks with each object after learning (averaged gating
outputs), b. Temporal patterns of objects, gating outputs, motor commands and
trajectories after learning
4.4 UNKNOWN OBJECT RECOGNITION
BY USING SOMATIC INFORMATION
Fig.7b shows the responses for unknown objects whose physical characteristics were
slightly different from known objects (see Fig.7a and Fig.4a) in the case using somatic
information as the gating network inputs. Even if each tested object was not the same as
any of the known (learned) objects, the closest expert network was selected. (compare
Fig.4a and Fig.7a) During some period in the test phase, the feedback command increased
because of an inappropriate feedforward command.
-- --- - --- -----
b.
a.
Gating Net. Outputs v.s. Objects
object
physical
charactorisbCS
M
a'
rotinal II---..:-..---.-----..-:~_=_r____._;:_;;__t
Imago
B K
II--'--'----'--+---'--"---'---+--'---'--"-t
2.0 3.0 7.0
none
4.0 6 .0 5.0
none
20
__III
~
0
~
?2
....,..".~~,y 'i~~~k--~ ~
O~--....lIt,.------':'--i:..i,-_ _~--'-_~
9.0 2.0 2.0 none
tim. [secJ
Fig. 7 Unknown objects recognition by using Somatic information, a. Statistical
analysis of the correspondence of the expert networks with each object after learning
(averaged gating outputs), b. Temporal patterns of objects, gating outputs, motor
commands and trajectories after learning
Recognition of Manipulated Obj ects by Motor Learning
5 MODULAR ARCHITECTURE
USING ESTIMATION NETWORK
The previous modular architecture is competitive in the sense that expert networks
compete with each other to occupy its niche in the input space. We here propose a new
cooperative modular architecture where expert networks specified for different functions
cooperate to produce the required output. In this scheme, estimation networks are trained
to recognize physical characteristics of manipulated objects by using feedback information.
Using this method, an infinite number of manipulated objects in the limited domain can
be treated by using a small number of estimation networks. We applied this method to
recognizing the mass of the manipulated objects. (see Fig.8)
Fig.9a shows the output of the estimation network compared to actual masses. The
realized trajectory almost coincided with the desired trajectory as shown in Fig.9b. This
learning scheme can be applied not only to estimating mass but also to other physical
characteristics such as softness or slipperiness.
a.
~
8
6
~ 4
~
2
0'r--_"""T""'_""""''''-_-'--_--'
0.0
0.5
1.0
~me
b. _
~
2.0
I f\.~
desired traj9Ctory
2
1
actual traJecklry
.~ a -"'--__ ~ I
8.
1.5
(sec]
'..-1
-1
,1"\
\
\
-2
-3
o
j ,i.x
Fig. 8 Confaguration of the modular architecture using
mass estimation network for object manipulation by
feedback-error-Iearning
5
10
15
20
time (sec]
Fig. 9 a. Comparison of actual &
estimated mass, b. desired & actual
trajectory
6 DISCUSSION
In the first scheme, the internal models for object manipulation (in this case, inverse
dynamics) were represented not in terms of visual information but rather, of somatic
information (see 4.2). Although the current simulation is primitive, it indicates the very
important issue that functional internal-representations of objects (or environments),
rather than declarative ones, were acquired by motor learning.
The quasi-target motor command in the first scheme and the motor command error in the
second scheme are not always exactly correct in each time step because the proposed
learning schemes are based on the feedback-error-learning method. Thus, the learning rates
in the proposed schemes should be slower than those schemes in which exact target
commands are employed. In our preliminary simulation, it was about five times slower.
However, we emphasize that exact target motor commands are not available in supervised
motor learning.
The limited number of controlled objects which can be dealt with by the modular network
with a gating network is a considerable problem (Jacobs, 1991; Nowlan, 1990, 1991).
This problem depends on choosing an appropriate number of expert networks and value of
the variance scaling parameter, (J' . Once this is done, the expert networks can interpolate
553
554
Gomi and Kawato
the appropriate output for a number of unknown objects. Our second scheme provides a
more satisfactory solution to this problem.
On the other hand, one possible drawback of the second scheme is that it may be difficult
to estimate many physical parameters for complicated objects, even though the learning
scheme which directly estimates the physical parameters can handle any number of
objects.
We showed here basic examinations of two types of neural networks - a gating network
and a direct estimation network. Both networks use feedback and/or feedforward information
for recognition of multiple manipulated objects. In future. we will attempt to integrate
these two architectures in order to model tasks involving skilled motor coordination and
high level recognition.
Ack nowledgmen t
We would like to thank Drs. E. Yodogawa and K. Nakane of AlR Auditory and Visual
Perception Research Laboratories for their continuing encouragement. Supported by HFSP
Grant to M.K.
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4,894 | 5,430 | Provable Non-convex Robust PCA
Praneeth Netrapalli 1? U N Niranjan2?
1
Sujay Sanghavi3
Animashree Anandkumar2
Prateek Jain4
Microsoft Research, Cambridge MA. 2 The University of California at Irvine.
3
The University of Texas at Austin. 4 Microsoft Research, India.
Abstract
We propose a new method for robust PCA ? the task of recovering a low-rank matrix from sparse corruptions that are of unknown value and support. Our method
involves alternating between projecting appropriate residuals onto the set of lowrank matrices, and the set of sparse matrices; each projection is non-convex but
easy to compute. In spite of this non-convexity, we establish exact recovery of the
low-rank matrix, under the same conditions that are required by existing methods
(which are based on convex optimization).
For an m?n input matrix (m ? n), our
method has a running time of O r2 mn per iteration, and needs O (log(1/)) iterations to reach an accuracy of . This is close to the running times of simple PCA
via the power method, which requires O (rmn) per iteration, and O (log(1/)) iterations. In contrast, the existing methods
for robust PCA, which are based on
convex optimization, have O m2 n complexity per iteration, and take O (1/)
iterations, i.e., exponentially more iterations for the same accuracy.
Experiments on both synthetic and real data establishes the improved speed and
accuracy of our method over existing convex implementations.
Keywords:
tions.
1
Robust PCA, matrix decomposition, non-convex methods, alternating projec-
Introduction
Principal component analysis (PCA) is a common procedure for preprocessing and denoising, where
a low rank approximation to the input matrix (such as the covariance matrix) is carried out. Although
PCA is simple to implement via eigen-decomposition, it is sensitive to the presence of outliers,
since it attempts to ?force fit? the outliers to the low rank approximation. To overcome this, the
notion of robust PCA is employed, where the goal is to remove sparse corruptions from an input
matrix and obtain a low rank approximation. Robust PCA has been employed in a wide range
of applications, including background modeling [LHGT04], 3d reconstruction [MZYM11], robust
topic modeling [Shi13], and community detection [CSX12], and so on.
Concretely, robust PCA refers to the following problem: given an input matrix M = L? + S ? , the
goal is to decompose it into sparse S ? and low rank L? matrices. The seminal works of [CSPW11,
CLMW11] showed that this problem can be provably solved via convex relaxation methods, under
some natural conditions on the low rank and sparse components. While the theory is elegant, in
practice, convex techniques are expensive to run on a large scale and have poor convergence rates.
Concretely, for decomposing an m?n matrix, say with m ? n, the best specialized
implementations
(typically first-order methods) have a per-iteration complexity of O m2 n , and require O(1/)
number of iterations to achieve an error of . In contrast, the usual PCA, which carries out a rankr approximation of the input matrix, has O(rmn) complexity per iteration ? drastically smaller
?
Part of the work done while interning at Microsoft Research, India
1
when r is much smaller than m, n. Moreover, PCA requires exponentially fewer iterations for
convergence: an accuracy is achieved with only O (log(1/)) iterations (assuming constant gap in
singular values).
In this paper, we design a non-convex algorithm which is ?best of both the worlds? and bridges the
gap between (the usual) PCA and convex methods for robust PCA. Our method has low computational complexity similar to PCA (i.e. scaling costs and convergence rates), and at the same time,
has provable global convergence guarantees, similar to the convex methods. Proving global convergence for non-convex methods is an exciting recent development in machine learning. Non-convex
alternating minimization techniques have recently shown success in many settings such as matrix
completion [Kes12, JNS13, Har13], phase retrieval [NJS13], dictionary learning [AAJ+ 13], tensor
decompositions for unsupervised learning [AGH+ 12], and so on. Our current work on the analysis
of non-convex methods for robust PCA is an important addition to this growing list.
1.1
Summary of Contributions
We propose a simple intuitive algorithm for robust PCA with low per-iteration cost and a fast convergence rate. We prove tight guarantees for recovery of sparse and low rank components, which
match those for the convex methods. In the process, we derive novel matrix perturbation bounds,
when subject to sparse perturbations. Our experiments reveal significant gains in terms of speed-ups
over the convex relaxation techniques, especially as we scale the size of the input matrices.
Our method consists of simple alternating (non-convex) projections onto low-rank and sparse matrices. For an m?n matrix, our method has a running time of O(r2 mn log(1/)), where r is the rank of
the low rank component. Thus, our method has a linear convergence rate, i.e. it requires O(log(1/))
iterations to achieve an error of , where r is the rank of the low rank component L? . When the rank
r is small, this nearly matches the complexity of PCA, (which is O(rmn log(1/))).
We prove recovery of the sparse and low rank components under a set of requirements which are
tight and match those for the convex techniques (up to constant factors). In particular, under the
deterministic sparsity model, where each row and each column
of the sparse matrix S ? has at most
2
? fraction of non-zeros, we require that ? = O 1/(? r) , where ? is the incoherence factor (see
Section 3).
In addition to strong theoretical guarantees, in practice, our method enjoys significant advantages over the state-of-art solver for (1), viz., the inexact augmented Lagrange multiplier (IALM)
method [CLMW11]. Our method outperforms IALM in all instances, as we vary the sparsity levels,
incoherence, and rank, in terms of running time to achieve a fixed level of accuracy. In addition,
on a real dataset involving the standard task of foreground-background separation [CLMW11], our
method is significantly faster and provides visually better separation.
Overview of our techniques: Our proof technique involves establishing error contraction with
each projection onto the sets of low rank and sparse matrices. We first describe the proof ideas when
L? is rank one. The first projection step is a hard thresholding procedure on the input matrix M to
remove large entries and then we perform rank-1 projection of the residual to obtain L(1) . Standard
matrix perturbation results (such as Davis-Kahan) provide `2 error bounds between the singular
vectors of L(1) and L? . However, these bounds do not suffice for establishing the correctness of our
method. Since the next step in our method involves hard thresholding of the residual M ? L(1) ,
we require element-wise error bounds on our low rank estimate. Inspired by the approach of Erd?os
et al. [EKYY13], where they obtain similar element-wise bounds for the eigenvectors of sparse
Erd?os?R?enyi graphs, we derive these bounds by exploiting the fixed point characterization of the
eigenvectors1 . A Taylor?s series expansion reveals that the perturbation between the estimated and
the true eigenvectors consists of bounding the walks in a graph whose adjacency matrix corresponds
to (a subgraph of) the sparse component S ? . We then show that if the graph is sparse enough,
then this perturbation can be controlled, and thus, the next thresholding step results in further error
contraction. We use an induction argument to show that the sparse estimate is always contained in
the true support of S ? , and that there is an error contraction in each step. For the case, where L? has
rank r > 1, our algorithm proceeds in several stages, where we progressively compute higher rank
1
If the input matrix M is not symmetric, we embed it in a symmetric matrix and consider the eigenvectors
of the corresponding matrix.
2
projections which alternate with the hard thresholding steps. In stage k = [1, 2, . . . , r], we compute
rank-k projections, and show that after a sufficient number of alternating projections, we reduce the
error to the level of (k + 1)th singular value of L? , using similar arguments as in the rank-1 case.
We then proceed to performing rank-(k + 1) projections which alternate with hard thresholding.
This stage-wise procedure is needed for ill-conditioned matrices, since we cannot hope to recover
lower eigenvectors in the beginning when there are large perturbations. Thus, we establish global
convergence guarantees for our proposed non-convex robust PCA method.
1.2
Related Work
Guaranteed methods for robust PCA have received a lot of attention in the past few years, starting
from the seminal works of [CSPW11, CLMW11], where they showed recovery of an incoherent low
rank matrix L? through the following convex relaxation method:
Conv-RPCA :
min kLk? + ?kSk1 ,
s.t., M = L + S,
(1)
L,S
where kLk? denotes the nuclear norm of L (nuclear norm is the sum of singular values). A typical
solver for this convex program involves projection on to `1 and nuclear norm balls (which are convex
sets). Note that the convex method can be viewed as ?soft? thresholding in the standard and spectral
domains, while our method involves hard thresholding in these domains.
[CSPW11] and [CLMW11] consider two different models of sparsity for S ? . Chandrasekaran et
al. [CSPW11] consider a deterministic sparsity model, where each row and column of the m ? n
matrix, S, ?
has at
most ? fraction of non-zero entries. For guaranteed recovery, they require ? =
O 1/(?2 r n) , where ? is the incoherence level of L? , and r is its rank. Hsu et al. [HKZ11]
improve upon this result to obtain guarantees for an optimal sparsity level of ? = O 1/(?2 r) .
This matches the requirements of our non-convex method for exact recovery. Note that when the
rank r = O(1), this allows for a constant fraction of corrupted entries. Cand`es et al. [CLMW11]
consider a different model with random sparsity and additional incoherence constraints, viz.,
p they
?
require kU V > k? < ? r/n. Note that our assumption of incoherence, viz., kU (i) k < ? r/n,
only yields kU V > k? < ?2 r/n. The additional assumption enables [CLMW11] to prove exact
recovery with a constant fraction of corrupted entries, even when L? is nearly full-rank. We note that
removing the kU V > k? condition
? for robust PCA would imply solving the planted clique problem
when the clique size is less than n [Che13]. Thus, our recovery guarantees are tight upto constants
without these additional assumptions.
A number of works have considered modified models under the robust PCA framework,
e.g. [ANW12, XCS12]. For instance, Agarwal et al. [ANW12] relax the incoherence assumption to
a weaker ?diffusivity? assumption, which bounds the magnitude of the entries in the low rank part,
but incurs an additional approximation error. Xu et al.[XCS12] impose special sparsity structure
where a column can either be non-zero or fully zero.
In terms of state-of-art specialized solvers, [CLMW11] implements the in-exact augmented Lagrangian multipliers (IALM) method and provides guidelines for parameter tuning. Other related
methods such as multi-block alternating directions method of multipliers (ADMM) have also been
considered for robust PCA, e.g. [WHML13]. Recently, a multi-step multi-block stochastic ADMM
method was analyzed for this problem [SAJ14], and this requires 1/ iterations to achieve an error
of . In addition, the convergence rate is tight in terms of scaling with respect to problem size (m, n)
and sparsity and rank parameters, under random noise models.
There is only one other work which considers a non-convex method for robust PCA [KC12]. However, their result holds only for significantly more restrictive settings and does not cover the deterministic sparsity assumption that we study. Moreover, the projection step in their method can
have an arbitrarily large rank, so the running time is still O(m2 n), which is the same as the convex
methods. In contrast, we have an improved running time of O(r2 mn).
2
Algorithm
In this section, we present our algorithm for the robust PCA problem. The robust PCA problem can
be formulated as the following optimization problem: find L, S s.t. kM ? L ? SkF ? 2 and
2
is the desired reconstruction error
3
Figure 1: Illustration of alternating projections. The goal is to find a matrix L? which lies in the
intersection of two sets: L = { set of rank-r matrices} and SM = {M ? S, where S is a sparse
matrix}. Intuitively, our algorithm alternately projects onto the above two non-convex sets, while
appropriately relaxing the rank and the sparsity levels.
1. L lies in the set of low-rank matrices,
2. S lies in the set of sparse matrices.
A natural algorithm for the above problem is to iteratively project M ? L onto the set of sparse
matrices to update S, and then to project M ? S onto the set of low-rank matrices to update L. Alternatively, one can view the problem as that of finding a matrix L in the intersection of the following
two sets: a) L = { set of rank-r matrices}, b) SM = {M ?S, where S is a sparse matrix}. Note that
these projections can be done efficiently, even though the sets are non-convex. Hard thresholding
(HT) is employed for projections on to sparse matrices, and singular value decomposition (SVD) is
used for projections on to low rank matrices.
Rank-1 case: We first describe our algorithm for the special case when L? is rank 1. Our algorithm performs an initial hard thresholding to remove very large entries from input M . Note that if
we performed the projection on to rank-1 matrices without the initial hard thresholding, we would
not make any progress since it is subject to large perturbations. We alternate between computing
the rank-1 projection of M ? S, and performing hard thresholding on M ? L to remove entries
exceeding a certain threshold. This threshold is gradually decreased as the iterations proceed, and
the algorithm is run for a certain number of iterations (which depends on the desired reconstruction
error).
General rank case: When L? has rank r > 1, a naive extension of our algorithm consists of alternating projections on to rank-r matrices and sparse matrices. However, such a method has poor
performance on ill-conditioned matrices. This is because after the initial thresholding of the input
matrix M , the sparse corruptions in the residual are of the order of the top singular value (with the
choice of threshold as specified in the algorithm). When the lower singular values are much smaller,
the corresponding singular vectors are subject to relatively large perturbations and thus, we cannot
make progress in improving the reconstruction error. To alleviate the dependence on the condition
number, we propose an algorithm that proceeds in stages. In the k th stage, the algorithm alternates
between rank-k projections and hard thresholding for a certain number of iterations. We run the
algorithm for r stages, where r is the rank of L? . Intuitively, through this procedure, we recover
the lower singular values only after the input matrix is sufficiently denoised, i.e. sparse corruptions
at the desired level have been removed. Figure 1 shows a pictorial representation of the alternating
projections in different stages.
Parameters: As can be seen, the only real parameter to the algorithm is ?, used in thresholding,
which represents ?spikiness? of L? . That is if the user expects L? to be ?spiky? and the sparse part
to be heavily diffused, then higher value of ? can be provided. In our implementation, we found
that selecting
? aggressively helped speed up recovery of our algorithm. In particular, we selected
?
? = 1/ n.
Complexity: The complexity of each iteration within a single stage is O(kmn), since it involves
calculating the rank-k approximation3 of an m?n matrix (done e.g. via vanilla PCA). The number of
iterations in each stage is O (log (1/)) and there are at most r stages. Thus the overall complexity
2
of the entire algorithm
is then O(r mn log(1/)). This is drastically lower than the best known
2
bound of O m n/ on the number of iterations required by convex methods, and just a factor r
away from the complexity of vanilla PCA.
3
Note that we only require a rank-k approximation of the matrix rather than the actual singular vectors.
Thus, the computational complexity has no dependence on the gap between the singular values.
4
b S)
b = AltProj(M, , r, ?): Non-convex Alternating Projections based Robust PCA
Algorithm 1 (L,
1: Input: Matrix M ? Rm?n , convergence criterion , target rank r, thresholding parameter ?.
2: Pk (A) denotes the best rank-k approximation of matrix A. HT? (A) denotes hard-thresholding,
i.e. (HT? (A))ij = Aij if |Aij | ? ? and 0 otherwise.
3: Set initial threshold ?0 ? ??1 (M ).
4: L(0) = 0, S (0) = HT?0 (M ? L(0) )
5: for Stage k = 1 to r do
6:
for Iteration t = 0 to T = 10 log n?
M ? S (0)
2 / do
7:
Set threshold ? as
!
t
1
(t)
(t)
? = ? ?k+1 (M ? S ) +
?k (M ? S )
(2)
2
8:
L(t+1) = Pk (M ? S (t) )
9:
S (t+1) = HT? (M ? L(t+1) )
10:
end for
then
11:
if ??k+1 (L(t+1) ) < 2n
(T )
(T )
12:
Return: L , S
/* Return rank-k estimate if remaining part has small norm */
13:
else
14:
S (0) = S (T )
/* Continue to the next stage */
15:
end if
16: end for
17: Return: L(T ) , S (T )
3
Analysis
In this section, we present our main result on the correctness of AltProj. We assume the following
conditions:
(L1) Rank of L? is at most r.
(L2) L? is ?-incoherent, i.e., if L? = U ? ?? (V ? )> is the SVD of L? , then k(U ? )i k2 ?
?
?
?
? r,
m
th
?1 ? i ? m and k(V ? )i k2 ? ??nr , ?1 ? i ? n, where (U ? )i and (V ? )i denote the i
rows of U ? and V ? respectively.
(S1) Each row and column of S have at most ? fraction of non-zero entries such that ? ?
1
512?2 r .
Note that in general, it is not possible to have a unique recovery of low-rank and sparse components.
For example, if the input matrix M is both sparse and low rank, then there is no unique decomposition (e.g. M = e1 e>
1 ). The above conditions ensure uniqueness of the matrix decomposition
problem.
Additionally, we set the parameter ? in Algorithm 1 be set as ? =
4?2 r
?
.
mn
We now establish that our proposed algorithm recovers the low rank and sparse components under
the above conditions.
Theorem 1 (Noiseless Recovery). Under conditions (L1), (L2) and S ? , and choice of ? as above,
b and Sb of Algorithm 1 satisfy:
the outputs L
b
, and Supp Sb ? Supp (S ? ) .
L ? L?
? ,
Sb ? S ?
? ?
mn
F
?
Remark (tight recovery conditions): Our result is tight up to constants, in terms of allowable
sparsity level under the deterministic sparsity model. In other words, if we exceed the sparsity limit
imposed in S1, it is possible to construct instances where there is no unique decomposition4 . Our
4
For instance, consider the n ? n matrix which has r copies of the all ones matrix, each of size nr , placed
across the diagonal. We see that this matrix has rank r and is incoherent with parameter ? = 1. Note that
5
conditions L1, L2 and S1 also match the conditions required by the convex method for recovery, as
established in [HKZ11].
Remark (convergence rate): Our method has a linear rate of convergence, i.e. O(log(1/))
to achieve an error of , and hence we provide a strongly polynomial method for robust PCA. In
contrast, the best known bound for convex methods for robust PCA is O(1/) iterations to converge
to an -approximate solution.
Theorem 1 provides recovery guarantees assuming that L? is exactly rank-r. However, in several
real-world scenarios, L? can be nearly rank-r. Our algorithm can handle such situations, where
M = L? + N ? + S ? , with N ? being an additive noise. Theorem 1 is a special case of the following
theorem which provides recovery guarantees when N ? has small `? norm.
Theorem 2 (Noisy Recovery). Under conditions (L1), (L2) and S ? , and choice of ? as in Theo?
r (L )
b Sb of Algorithm 1 satisfy:
,the outputs L,
rem 1, when the noise kN ? k? ? ?100n
?
8 mn
b
2
?
?
?
?
?
+
2?
r
7
kN
k
+
L
?
L
kN
k
2
? ,
r
F
?
2?2 r
8 mn
b
?
?
?
?
?
?
+
7 kN k2 +
kN k? , and Supp Sb ? Supp (S ? ) .
S ? S
?
mn
mn
r
?
3.1
Proof Sketch
We now present the key steps in the proof of Theorem 1. A detailed proof is provided in the appendix.
Step I: Reduce to the symmetric case, while maintaining incoherence of L? and sparsity of S ? .
Using standard symmetrization arguments, we can reduce the problem to the symmetric case, where
all the matrices involved are symmetric. See appendix for details on this step.
Step II: Show decay in kL ? L? k? after projection onto the set of rank-k matrices. The
t-th iterate L(t+1) of the k-th stage is given by L(t+1) = Pk (L? + S ? ? S (t) ). Hence, L(t+1) is
obtained by using the top principal components of a perturbation of L? given by L? + (S ? ? S (t) ).
The key step in our analysis is to show that when an incoherent and low-rank L? is perturbed by a
sparse matrix S ? ? S (t) , then kL(t+1) ? L? k? is small and is much smaller than |S ? ? S (t) |? . The
following lemma formalizes the intuition; see the appendix for a detailed proof.
Lemma 1. Let L? , S ? be symmetric and satisfy the assumptions of Theorem 1 and let S (t) and L(t)
be the tth iterates of the k th stage of Algorithm 1. Let ?1? , . . . , ?n? be the eigenvalues of L? , s.t.,
|?1? | ? ? ? ? ? |?r? |. Then, the following holds:
!
t
1
2?2 r ?
(t+1)
?
?
?k+1 +
?L
?
|?k | ,
L
n
2
?
!
t
8?2 r ?
1
?
(t+1)
?
?k+1 +
|?k | , and Supp S (t+1) ? Supp (S ? ) .
S ? S
?
n
2
?
b and Sb of Algorithm 1 satisfy:
Moreover, the outputs L
b
L ? L?
? ,
Sb ? S ?
? , and Supp Sb ? Supp (S ? ) .
n
F
?
Step III: Show decay in kS ? S ? k? after projection onto the set of sparse matrices. We next
show that if kL(t+1) ? L? k? is much smaller than kS (t) ? S ? k? then the iterate S (t+1) also has
a much smaller error (w.r.t. S ? ) than S (t) . The above given lemma formally provides the error
bound.
Step IV: Recurse the argument. We have now reduced the `? norm of the sparse part by a factor of
half, while maintaining its sparsity. We can now go back to steps II and III and repeat the arguments
for subsequent iterations.
a fraction of ? = O (1/r) sparse perturbations suffice to erase one of these blocks making it impossible to
recover the matrix.
6
500
600
n?
700
800
IALM
2
400
200
500
600
n?
700
800
10
1
n = 2000, ? = 1, n ? = 3r
AltProj
IALM
1.5
AltProj
IALM
2
10
Time(s)
600
Time(s)
2
10
AltProj
IALM
n = 2000, r = 10, n ? = 100
n = 2000, r = 5, ? = 1
Max. Rank
Time(s)
n = 2000, r = 5, ? = 1
2
?
2.5
3
50
100
r
150
200
(a)
(b)
(c)
(d)
Figure 2: Comparison of AltProj and IALM on synthetic datasets. (a) Running time of AltProj and
IALM with varying ?. (b) Maximum rank of the intermediate iterates of IALM. (c) Running time
of AltProj and IALM with varying ?. (d) Running time of AltProj and IALM with varying r.
4
Experiments
We now present an empirical study of our AltProj method. The goal of this study is two-fold: a)
establish that our method indeed recovers the low-rank and sparse part exactly, without significant
parameter tuning, b) demonstrate that AltProj is significantly faster than Conv-RPCA (see (1)); we
solve Conv-RPCA using the IALM method [CLMW11], a state-of-the-art solver [LCM10]. We
implemented our method in Matlab and used a Matlab implementation of the IALM method by
[LCM10].
We consider both synthetic experiments and experiments on real data involving the problem of
foreground-background separation in a video. Each of our results for synthetic datasets is averaged
over 5 runs.
Parameter Setting: Our pseudo-code (Algorithm 1) prescribes the threshold ? in Step 4, which
depends on the knowledge of the singular values of the low rank component L? . Instead, in the
?S (t) )
?
experiments, we set the threshold at the (t + 1)-th step of k-th stage as ? = ??k+1 (M
. For
n
synthetic experiments, we employ the ? used for data generation, and for real-world datasets, we
tune ? through cross-validation. We found that the above thresholding provides exact recovery
while speeding up the computation significantly. We would
also like to note that [CLMW11] sets
?
the regularization parameter ? in Conv-RPCA (1) as 1/ n (assuming m ? n). However, we found
that for problems with ?
large incoherence such a parameter setting does not provide exact recovery.
Instead, we set ? = ?/ n in our experiments.
Synthetic datasets: Following the experimental setup of [CLMW11], the low-rank part L? = U V T
is generated using normally distributed U ? Rm?r , V ? Rn?r . Similarly, supp(S ? ) is generated
?
by sampling a uniformly random subset of [m] ?[n] with size kS ? k0 and each non-zero Sij
is drawn
?
?
i.i.d. from the uniform distribution over [r/(2 mn), r/ mn]. For increasing incoherence of L? ,
we randomly zero-out rows of U, V and then re-normalize them.
There are three key problem parameters for RPCA with a fixed matrix size: a) sparsity of S ? ,
b) incoherence of L? , c) rank of L? . We investigate performance of both AltProj and IALM by
varying each of the three parameters while fixing the others. In our plots (see Figure 2), we report
computational time required by each of the two methods for decomposing M into L + S up to a
relative error (kM ? L ? SkF /kM kF ) of 10?3 . Figure 2 shows that AltProj scales significantly
better than IALM for increasingly dense S ? . We attribute this observation to the fact that as kS ? k0
increases, the problem is ?harder? and the intermediate iterates of IALM have ranks significantly
larger than r. Our intuition is confirmed by Figure 2 (b), which shows that when density (?) of S ?
is 0.4 then the intermediate iterates of IALM can be of rank over 500 while the rank of L? is only
5. We observe a similar trend for the other parameters, i.e., AltProj scales significantly better than
IALM with increasing incoherence parameter ? (Figure 2 (c)) and increasing rank (Figure 2 (d)).
See Appendix C for additional plots.
Real-world datasets: Next, we apply our method to the problem of foreground-background (F-B)
separation in a video [LHGT04]. The observed matrix M is formed by vectorizing each frame and
stacking them column-wise. Intuitively, the background in a video is the static part and hence forms
a low-rank component while the foreground is a dynamic but sparse perturbation.
Here, we used two benchmark datasets named Escalator and Restaurant dataset. The Escalator
dataset has 3417 frames at a resolution of 160 ? 130. We first applied the standard PCA method for
extracting low-rank part. Figure 3 (b) shows the extracted background from the video. There are
7
(a)
(b)
(c)
(d)
Figure 3: Foreground-background separation in the Escalator video. (a): Original image frame. (b):
Best rank-10 approximation; time taken is 3.1s. (c): Low-rank frame obtained using AltProj; time
taken is 63.2s. (d): Low-rank frame obtained using IALM; time taken is 1688.9s.
(a)
(b)
(c)
(d)
Figure 4: Foreground-background separation in the Restaurant video. (a): Original frame from the
video. (b): Best rank-10 approximation (using PCA) of the original frame; 2.8s were required to
compute the solution (c): Low-rank part obtained using AltProj; computational time required by
AltProj was 34.9s. (d): Low-rank part obtained using IALM; 693.2s required by IALM to compute
the low-rank+sparse decomposition.
several artifacts (shadows of people near the escalator) that are not desirable. In contrast, both IALM
and AltProj obtain significantly better F-B separation (see Figure 3(c), (d)). Interestingly, AltProj removes the steps of the escalator which are moving and arguably are part of the dynamic foreground,
while IALM keeps the steps in the background part. Also, our method is significantly faster, i.e.,
our method, which takes 63.2s is about 26 times faster than IALM, which takes 1688.9s.
Restaurant dataset: Figure 4 shows the comparison of AltProj and IALM on a subset of the ?Restaurant? dataset where we consider the last 2055 frames at a resolution of 120?160. AltProj was around
19 times faster than IALM. Moreover, visually, the background extraction seems to be of better quality (for example, notice the blur near top corner counter in the IALM solution). Plot(b) shows the
PCA solution and that also suffers from a similar blur at the top corner of the image, while the
background frame extracted by AltProj does not have any noticeable artifacts.
5
Conclusion
In this work, we proposed a non-convex method for robust PCA, which consists of alternating projections on to low rank and sparse matrices. We established global convergence of our method under
conditions which match those for convex methods. At the same time, our method has much faster
running times, and has superior experimental performance. This work opens up a number of interesting questions for future investigation. While we match the convex methods, under the deterministic
sparsity model, studying the random sparsity model is of interest. Our noisy recovery results assume deterministic noise; improving the results under random noise needs to be investigated. There
are many decomposition problems beyond the robust PCA setting, e.g. structured sparsity models,
robust tensor PCA problem, and so on. It is interesting to see if we can establish global convergence
for non-convex methods in these settings.
Acknowledgements
AA and UN would like to acknowledge NSF grant CCF-1219234, ONR N00014-14-1-0665, and Microsoft faculty fellowship. SS would like to acknowledge NSF grants 1302435, 0954059, 1017525
and DTRA grant HDTRA1-13-1-0024. PJ would like to acknowledge Nikhil Srivastava and
Deeparnab Chakrabarty for several insightful discussions during the course of the project.
8
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9
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4,895 | 5,431 | Spectral Methods Meet EM: A Provably Optimal
Algorithm for Crowdsourcing
Yuchen Zhang?
Xi Chen]
Dengyong Zhou?
Michael I. Jordan?
?
University of California, Berkeley, Berkeley, CA 94720
{yuczhang,jordan}@berkeley.edu
]
New York University, New York, NY 10012
[email protected]
?
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052
[email protected]
Abstract
The Dawid-Skene estimator has been widely used for inferring the true labels
from the noisy labels provided by non-expert crowdsourcing workers. However,
since the estimator maximizes a non-convex log-likelihood function, it is hard to
theoretically justify its performance. In this paper, we propose a two-stage efficient algorithm for multi-class crowd labeling problems. The first stage uses the
spectral method to obtain an initial estimate of parameters. Then the second stage
refines the estimation by optimizing the objective function of the Dawid-Skene
estimator via the EM algorithm. We show that our algorithm achieves the optimal
convergence rate up to a logarithmic factor. We conduct extensive experiments on
synthetic and real datasets. Experimental results demonstrate that the proposed
algorithm is comparable to the most accurate empirical approach, while outperforming several other recently proposed methods.
1
Introduction
With the advent of online crowdsourcing services such as Amazon Mechanical Turk, crowdsourcing
has become an appealing way to collect labels for large-scale data. Although this approach has
virtues in terms of scalability and immediate availability, labels collected from the crowd can be of
low quality since crowdsourcing workers are often non-experts and can be unreliable. As a remedy,
most crowdsourcing services resort to labeling redundancy, collecting multiple labels from different
workers for each item. Such a strategy raises a fundamental problem in crowdsourcing: how to infer
true labels from noisy but redundant worker labels?
For labeling tasks with k different categories, Dawid and Skene [8] propose a maximum likelihood
approach based on the Expectation-Maximization (EM) algorithm. They assume that each worker
is associated with a k ? k confusion matrix, where the (l, c)-th entry represents the probability that
a randomly chosen item in class l is labeled as class c by the worker. The true labels and worker confusion matrices are jointly estimated by maximizing the likelihood of the observed worker
labels, where the unobserved true labels are treated as latent variables. Although this EM-based
approach has had empirical success [21, 20, 19, 26, 6, 25], there is as yet no theoretical guarantee
for its performance. A recent theoretical study [10] shows that the global optimal solutions of the
Dawid-Skene estimator can achieve minimax rates of convergence in a simplified scenario, where
the labeling task is binary and each worker has a single parameter to represent her labeling accuracy (referred to as a ?one-coin model? in what follows). However, since the likelihood function is
non-convex, this guarantee is not operational because the EM algorithm may get trapped in a local
optimum. Several alternative approaches have been developed that aim to circumvent the theoretical
deficiencies of the EM algorithm, still in the context of the one-coin model [14, 15, 11, 7]. Unfor1
tunately, they either fail to achieve the optimal rates or depend on restrictive assumptions which are
hard to justify in practice.
We propose a computationally efficient and provably optimal algorithm to simultaneously estimate
true labels and worker confusion matrices for multi-class labeling problems. Our approach is a
two-stage procedure, in which we first compute an initial estimate of worker confusion matrices
using the spectral method, and then in the second stage we turn to the EM algorithm. Under some
mild conditions, we show that this two-stage procedure achieves minimax rates of convergence up
to a logarithmic factor, even after only one iteration of EM. In particular, given any ? ? (0, 1),
we provide the bounds on the number of workers and the number of items so that our method can
correctly estimate labels for all items with probability at least 1 ? ?. We also establish a lower bound
to demonstrate the optimality of this approach. Further, we provide both upper and lower bounds for
estimating the confusion matrix of each worker and show that our algorithm achieves the optimal
accuracy.
This work not only provides an optimal algorithm for crowdsourcing but sheds light on understanding the general method of moments. Empirical studies show that when the spectral method is used
as an initialization for the EM algorithm, it outperforms EM with random initialization [18, 5]. This
work provides a concrete way to theoretically justify such observations. It is also known that starting
from a root-n consistent estimator obtained by the spectral method, one Newton-Raphson step leads
to an asymptotically optimal estimator [17]. However, obtaining a root-n consistent estimator and
performing a Newton-Raphson step can be demanding computationally. In contrast, our initialization doesn?t need to be root-n consistent, thus a small portion of data suffices to initialize. Moreover,
performing one iteration of EM is computationally more attractive and numerically more robust than
a Newton-Raphson step especially for high-dimensional problems.
2
Related Work
Many methods have been proposed to address the problem of estimating true labels in crowdsourcing
[23, 20, 22, 11, 19, 26, 7, 15, 14, 25]. The methods in [20, 11, 15, 19, 14, 7] are based on the
generative model proposed by Dawid and Skene [8]. In particular, Ghosh et al. [11] propose a
method based on Singular Value Decomposition (SVD) which addresses binary labeling problems
under the one-coin model. The analysis in [11] assumes that the labeling matrix is full, that is,
each worker labels all items. To relax this assumption, Dalvi et al. [7] propose another SVD-based
algorithm which explicitly considers the sparsity of the labeling matrix in both algorithm design
and theoretical analysis. Karger et al. propose an iterative algorithm for binary labeling problems
under the one-coin model [15] and extend it to multi-class labeling tasks by converting a k-class
problem into k ? 1 binary problems [14]. This line of work assumes that tasks are assigned to
workers according to a random regular graph, thus imposing specific constraints on the number
of workers and the number of items. In Section 5, we compare our theoretical results with that
of existing approaches [11, 7, 15, 14]. The methods in [20, 19, 6] incorporate Bayesian inference
into the Dawid-Skene estimator by assuming a prior over confusion matrices. Zhou et al. [26,
25] propose a minimax entropy principle for crowdsourcing which leads to an exponential family
model parameterized with worker ability and item difficulty. When all items have zero difficulty, the
exponential family model reduces to the generative model suggested by Dawid and Skene [8].
Our method for initializing the EM algorithm in crowdsourcing is inspired by recent work using
spectral methods to estimate latent variable models [3, 1, 4, 2, 5, 27, 12, 13]. The basic idea in this
line of work is to compute third-order empirical moments from the data and then to estimate parameters by computing a certain orthogonal decomposition of a tensor derived from the moments. Given
the special symmetric structure of the moments, the tensor factorization can be computed efficiently using the robust tensor power method [3]. A problem with this approach is that the estimation
error can have a poor dependence on the condition number of the second-order moment matrix and
thus empirically it sometimes performs worse than EM with multiple random initializations. Our
method, by contrast, requires only a rough initialization from the moment of moments; we show that
the estimation error does not depend on the condition number (see Theorem 2 (b)).
3
Problem Setup
Throughout this paper, [a] denotes the integer set {1, 2, . . . , a} and ?b (A) denotes the b-th largest
singular value of the matrix A. Suppose that there are m workers, n items and k classes. The true
2
Algorithm 1: Estimating confusion matrices
Input: integer k, observed labels zij ? Rk for i ? [m] and j ? [n].
bi ? Rk?k for i ? [m].
Output: confusion matrix estimates C
(1) Partition the workers into three disjoint and non-empty group G1 , G2 and G3 . Compute the
group aggregated labels Zgj by Eq. (1).
(2) For (a, b, c) ? {(2, 3, 1), (3, 1, 2), (1, 2, 3)}, compute the second and the third order moments
c2 ? Rk?k , M
c3 ? Rk?k?k by Eq. (2a)-(2d), then compute C
bc ? Rk?k and W
c ? Rk?k by
M
tensor decomposition:
b ? Rk?k (such that Q
bT M
c2 Q
b = I) using SVD.
(a) Compute whitening matrix Q
c3 (Q,
b Q,
b Q)
b
(b) Compute eigenvalue-eigenvector pairs {(b
?h , vbh )}kh=1 of the whitened tensor M
by using the robust tensor power method [3]. Then compute w
bh = ?
bh?2 and
b T )?1 (b
?
bh = (Q
?h vbh ).
bc by some ?
(c) For l = 1, . . . , k, set the l-th column of C
bh whose l-th coordinate has the
c by w
greatest component, then set the l-th diagonal entry of W
bh .
bi by Eq. (3).
(3) Compute C
label yj of item j ? [n] is assumed to be sampled from a probability distribution P[yj = l] = wl
Pk
where {wl : l ? [k]} are positive values satisfying l=1 wl = 1. Denote by a vector zij ? Rk
the label that worker i assigns to item j. When the assigned label is c, we write zij = ec , where ec
represents the c-th canonical basis vector in Rk in which the c-th entry is 1 and all other entries are
0. A worker may not label every item. Let ?i indicate the probability that worker i labels a randomly
chosen item. If item j is not labeled by worker i, we write zij = 0. Our goal is to estimate the true
labels {yj : j ? [n]} from the observed labels {zij : i ? [m], j ? [n]}.
In order to obtain an estimator, we need to make assumptions on the process of generating observed
labels. Following the work of Dawid and Skene [8], we assume that the probability that worker i
labels an item in class l as class c is independent of any particular chosen item, that is, it is a constant
over j ? [n]. Let us denote the constant probability by ?ilc . Let ?il = [?il1 ?il2 ? ? ? ?ilk ]T . The
matrix Ci = [?i1 ?i2 . . . ?ik ] ? Rk?k is called the confusion matrix of worker i. Besides estimating
the true labels, we also want to estimate the confusion matrix for each worker.
4
Our Algorithm
In this section, we present an algorithm to estimate confusion matrices and true labels. Our algorithm
consists of two stages. In the first stage, we compute an initial estimate of confusion matrices via
the method of moments. In the second stage, we perform the standard EM algorithm by taking the
result of the Stage 1 as an initialization.
4.1
Stage 1: Estimating Confusion Matrices
Partitioning the workers into three disjoint and non-empty groups G1 , G2 and G3 , the outline of
this stage is the following: we use the spectral method to estimate the averaged confusion matrices
for the three groups, then utilize this intermediate estimate to obtain the confusion matrix of each
individual worker. In particular, for g ? {1, 2, 3} and j ? [n], we calculate the averaged labeling
within each group by
1 X
Zgj :=
zij .
(1)
|Gg |
i?Gg
P
Denoting the aggregated confusion matrix columns by ?gl := E(Zgj |yj = l) = |G1g | i?Gg ?i ?il ,
our first step is to estimate Cg := [?g1 , ?g2 , . . . , ?gk ] and to estimate the distribution of true labels
3
W := diag(w1 , w2 , . . . , wk ). The following proposition shows that we can solve for Cg and W
from the moments of {Zgj }.
Proposition 1 (Anandkumar et al. [3]). Assume that the vectors {?g1 , ?g2 , . . . , ?gk } are linearly
independent for each g ? {1, 2, 3}. Let (a, b, c) be a permutation of {1, 2, 3}. Define
?1
Zaj ,
?1
Zbj ,
0
Zaj
:= E[Zcj ? Zbj ] (E[Zaj ? Zbj ])
0
Zbj
:= E[Zcj ? Zaj ] (E[Zbj ? Zaj ])
0
E[Zaj
0
Zbj
]
0
0
M2 :=
?
and M3 := E[Zaj
? Zbj
? Zcj ];
Pk
P
k
then we have M2 = l=1 wl ?cl ? ?cl and M3 = l=1 wl ?cl ? ?cl ? ?cl .
Since we only have finite samples, the expectations in Proposition 1 have to be approximated by
empirical moments. In particular, they are computed by averaging over indices j = 1, 2, . . . , n. For
each permutation (a, b, c) ? {(2, 3, 1), (3, 1, 2), (1, 2, 3)}, we compute
0
baj
Z
:=
0
bbj
Z
:=
n
1 X
n
n
1 X
n
j=1
n
1 X
n
Zcj ? Zbj
Zcj ? Zaj
n
?1
Zaj ,
(2a)
Zbj ,
(2b)
j=1
n
1 X
j=1
Zaj ? Zbj
Zbj ? Zaj
?1
j=1
n
X
0
0
c2 := 1
baj
bbj
M
Z
?Z
,
n j=1
(2c)
n
X
0
0
c3 := 1
baj
bbj
M
Z
?Z
? Zcj .
n j=1
(2d)
The statement of Proposition 1 suggests that we can recover the columns of Cc and the diagonal
c2 and M
c3 . This is implemented by the tensor facentries of W by operating on the moments M
torization method in Algorithm 1. In particular, the tensor factorization algorithm returns a set of
vectors {(b
?h , w
bh ) : h = 1, . . . , k}, where each (b
?h , w
bh ) estimates a particular column of Cc (for
some ?cl ) and a particular diagonal entry of W (for some wl ). It is important to note that the tensor
factorization algorithm doesn?t provide a one-to-one correspondence between the recovered column and the true columns of Cc . Thus, ?
b1 , . . . , ?
bk represents an arbitrary permutation of the true
columns.
To discover the index correspondence, we take each ?
bh and examine its greatest component. We
assume that within each group, the probability of assigning a correct label is always greater than
the probability of assigning any specific incorrect label. This assumption will be made precise
in the next section. As a consequence, if ?
bh corresponds to the l-th column of Cc , then its l-th
bc to
coordinate is expected to be greater than other coordinates. Thus, we set the l-th column of C
some vector ?
bh whose l-th coordinate has the greatest component (if there are multiple such vectors,
then randomly select one of them; if there is no such vector, then randomly select a ?
bh ). Then, we
c to the scalar w
set the l-th diagonal entry of W
bh associated with ?
bh . Note that by iterating over
b for c = 1, 2, 3 respectively. There will be
(a, b, c) ? {(2, 3, 1), (3, 1, 2), (1, 2, 3)}, we obtain C
c
c
three copies of W estimating the same matrix W ?we average them for the best accuracy.
In the second step, we estimate each individual confusion matrix Ci . The following proposition
shows that we can recover Ci from the moments of {zij }. See [24] for the proof.
Proposition 2. For any g ? {1, 2, 3} and any i ? Gg , let a ? {1, 2, 3}\{g} be one of the remaining
group index. Then
T
?i Ci W (Ca )T = E[zij Zaj
].
bi using the empirical approximation
Proposition 2 suggests a plug-in estimator for Ci . We compute C
T
b c
b
of E[zij Zaj ] and using the matrices Ca , Cb , W obtained in the first step. Concretely, we calculate
?
?
n
? 1 X
?1 ?
T
bi := normalize
c (C
ba )T
zij Zaj
W
,
(3)
C
? n
?
j=1
4
where the normalization operator rescales the matrix columns, making sure that each column sums
to one. The overall procedure for Stage 1 is summarized in Algorithm 1.
4.2
Stage 2: EM algorithm
The second stage is devoted to refining the initial estimate provided by Stage 1. The joint likelihood
of true label yj and observed labels zij , as a function of confusion matrices ?i , can be written as
L(?; y, z) :=
n Y
m Y
k
Y
(?iyj c )I(zij =ec ) .
j=1 i=1 c=1
By assuming
a uniform prior over y, we maximize the marginal log-likelihood function `(?) :=
P
log( y?[k]n L(?; y, z)). We refine the initial estimate of Stage 1 by maximizing the objective function, which is implemented by the Expectation Maximization (EM) algorithm. The EM algorithm
takes the values {b
?ilc } provided as output by Stage 1 as initialization, then executes the following
E-step and M-step for at least one round.
E-step Calculate the expected value of the log-likelihood function, with respect to the conditional
distribution of y given z under the current estimate of ?:
( k
!)
n
m Y
k
X
X
Y
I(zij =ec )
qbjl log
(?ilc )
,
Q(?) := Ey|zf,b? [log(L(?; y, z))] =
j=1
where
qbjl ? Pk
exp
l0 =1
i=1 c=1
l=1
Pm Pk
exp
?ilc )
i=1
c=1 I(zij = ec ) log(b
Pm Pk
?il0 c )
i=1
c=1 I(zij = ec ) log(b
for j ? [n], l ? [k].
(4)
M-step Find the estimate ?
b that maximizes the function Q(?):
Pn
bjl I(zij = ec )
j=1 q
for i ? [m], l ? [k], c ? [k].
?
bilc ? Pk Pn
bjl I(zij = ec0 )
c0 =1
j=1 q
(5)
In practice, we alternatively execute the updates (4) and (5), for one iteration or until convergence.
Each update increases the objective function `(?). Since `(?) is not concave, the EM update doesn?t
guarantee converging to the global maximum. It may converge to distinct local stationary points for
different initializations. Nevertheless, as we prove in the next section, it is guaranteed that the EM
algorithm will output statistically optimal estimates of true labels and worker confusion matrices if
it is initialized by Algorithm 1.
5
Convergence Analysis
To state our main theoretical results, we first need to introduce some notation and assumptions. Let
wmin := min{wl }kl=1 and ?min := min{?i }m
i=1
be the smallest portion of true labels and the most extreme sparsity level of workers. Our first
assumption assumes that both wmin and ?min are strictly positive, that is, every class and every
worker contributes to the dataset.
Our second assumption assumes that the confusion matrices for each of the three groups, namely
C1 , C2 and C3 , are nonsingular. As a consequence, if we define matrices Sab and tensors Tabc for
any a, b, c ? {1, 2, 3} as
Sab :=
k
X
wl ?al ? ?bl = Ca W (Cb )T
and
l=1
Tabc :=
k
X
wl ?al ? ?bl ? ?cl ,
l=1
then there will be a positive scalar ?L such that ?k (Sab ) ? ?L > 0.
Our third assumption assumes that within each group, the average probability of assigning a correct
label is always higher than the average probability of assigning any incorrect label. To make this
5
statement rigorous, we define a quantity
? := min
min min {?gll ? ?glc }
g?{1,2,3} l?[k] c?[k]\{l}
indicating the smallest gap between diagonal entries and non-diagonal entries in the same confusion
matrix column. The assumption requires ? being strictly positive. Note that this assumption is
group-based, thus does not assume the accuracy of any individual worker.
Finally, we introduce a quantity that measures the average ability ofP
workers in identifying distinct
labels. For two discrete distributions P and Q, let DKL (P, Q) := i P (i) log(P (i)/Q(i)) represent the KL-divergence between P and Q. Since each column of the confusion matrix represents a
discrete distribution, we can define the following quantity:
m
1 X
D = min0
?i DKL (?il , ?il0 ) .
(6)
l6=l m
i=1
The quantity D lower bounds the averaged KL-divergence between two columns. If D is strictly
positive, it means that every pair of labels can be distinguished by at least one subset of workers. As
the last assumption, we assume that D is strictly positive.
The following two theorems characterize the performance of our algorithm. We split the convergence analysis into two parts. Theorem 1 characterizes the performance of Algorithm 1, providing
sufficient conditions for achieving an arbitrarily accurate initialization. We provide the proof of
Theorem 1 in the long version of this paper [24].
n
o
Theorem 1. For any scalar ? > 0 and any scalar satisfying ? min ?min36?k
,
2
, if the
wmin ?L
number of items n satisfies
5
k log((k + m)/?)
n=?
,
2 w 2 ? 13
2 ?min
min L
then the confusion matrices returned by Algorithm 1 are bounded as
bi ? Ci k? ?
kC
for all i ? [m],
with probability at least 1 ? ?. Here, k ? k? denotes the element-wise `? -norm of a matrix.
Theorem 2 characterizes the error rate in Stage 2. It states that when a sufficiently accurate
initialization is taken, the updates (4) and (5) refine the estimates ?
b and yb to the optimal accuracy.
See the long version of this paper [24] for the proof.
Theorem 2. Assume that there is a positive scalar ? such that ?ilc ? ? for all (i, l, c) ? [m] ? [k]2 .
bi are initialized in a manner such that
For any scalar ? > 0, if confusion matrices C
bi ? Ci k? ? ? := min ? , ?D
kC
for all i ? [m],
(7)
2 16
and the number of workers m and the number of items n satisfy
log(mk/?)
log(1/?) log(kn/?) + log(mn)
and n = ?
,
m=?
?min wmin ?2
D
then, for ?
b and qb obtained by iterating (4) and (5) (for at least one round), with probability at least
1 ? ?,
(a) Letting ybj = arg maxl?[k] qbjl , we have that ybj = yj holds for all j ? [n].
(b) kb
?il ? ?il k22 ?
48 log(2mk/?)
?i wl n
holds for all (i, l) ? [m] ? [k].
In Theorem 2, the assumption that all confusion matrix entries are lower bounded by ? > 0 is
somewhat restrictive. For datasets violating this assumption, we enforce positive confusion matrix
entries by adding random noise: Given any observed label zij , we replace it by a random label in
{1, ..., k} with probability k?. In this modified model, every entry of the confusion matrix is lower
bounded by ?, so that Theorem 2 holds. The random noise makes the constant D smaller than its
original value, but the change is minor for small ?.
6
Dataset name
Bird
RTE
TREC
Dog
Web
# classes
2
2
2
4
5
# items
108
800
19,033
807
2,665
# workers
39
164
762
52
177
# worker labels
4,212
8,000
88,385
7,354
15,567
Table 1: Summary of datasets used in the real data experiment.
To see the consequence of the convergence analysis, we take error rate in Theorem 1 equal to the
constant ? defined in Theorem 2. Then we combine the statements of the two theorems. This shows
that if we choose the number
n such that
of workers m andthe number of items
5
k
1
e
e
and n = ?
;
(8)
m=?
2 w 2 ? 13 min{?2 , (?D)2 }
D
?min
min L
that is, if both m and n are lower bounded by a problem-specific constant and logarithmic terms,
then with high probability, the predictor yb will be perfectly accurate, and the estimator ?
b will be
e
bounded as kb
?il ? ?il k22 ? O(1/(?
w
n)).
To
show
the
optimality
of
this
convergence
rate, we
i l
present the following minimax lower bounds. Again, see [24] for the proof.
Theorem 3. There are universal constants c1 > 0 and c2 > 0 such that:
(a) For any {?ilc }, {?i } and any number of items n, if the number of workers m ? 1/(4D), then
n
i
hX
I(b
yj 6= yj ) {?ilc }, {?i }, y = v ? c1 n.
inf sup E
y
b v?[k]n
j=1
(b) For any {wl }, {?i }, any worker-item pair (m, n) and any pair of indices (i, l) ? [m] ? [k], we
have
h
i
1
2
inf
sup E kb
?il ? ?il k2 {wl }, {?i } ? c2 min 1,
.
?
b ??Rm?k?k
?i w l n
In part (a) of Theorem 3, we see that the number of workers should be at least 1/(4D), otherwise
any predictor will make many mistakes. This lower bound matches our sufficient condition on the
number of workers m (see Eq. (8)). In part (b), we see that the best possible estimate for ?il has
?(1/(?i wl n)) mean-squared error. It verifies the optimality of our estimator ?
bil . It is worth noting
that the constraint on the number of items n (see Eq. (8)) might be improvable. In real datasets we
usually have n m so that the optimality for m is more important than for n.
It is worth contrasting our convergence rate with existing algorithms. Ghosh et al. [11] and Dalvi et
al. [7] proposed consistent estimators for the binary one-coin model. To attain an error rate ?, their
algorithms require m and n scaling with 1/? 2 , while our algorithm only requires m and n scaling
with log(1/?). Karger et al. [15, 14] proposed algorithms for both binary and multi-class problems.
Their algorithm assumes that workers are assigned by a random regular graph. Moreover, their
analysis assumes that the limit of number of items goes to infinity, or that the number of workers is
many times the number of items. Our algorithm no longer requires these assumptions.
We also compare our algorithm with the majority voting estimator, where the true label is simply
estimated by a majority vote among workers. Gao and Zhou [10] showed that if there are many
spammers and few experts, the majority voting estimator gives almost a random guess. In cone
to guarantee good performance. Since mD is the
trast, our algorithm only requires mD = ?(1)
aggregated KL-divergence, a small number of experts are sufficient to ensure it is large enough.
6
Experiments
In this section, we report the results of empirical studies comparing the algorithm we propose in
Section 4 (referred to as Opt-D&S) with a variety of existing methods which are also based on the
generative model of Dawid and Skene. Specifically, we compare to the Dawid & Skene estimator
7
0.35
Opt?D&S: 1st iteration
Opt?D&S: 50th iteration
MV?D&S: 1st iteration
MV?D&S: 50th iteration
0.2
Label prediction error
Label prediction error
0.2
Opt?D&S: 1st iteration
Opt?D&S: 50th iteration
MV?D&S: 1st iteration
MV?D&S: 50th iteration
0.21
0.18
0.16
0.14
0.12
0.1
Opt?D&S: 1st iteration
Opt?D&S: 50th iteration
MV?D&S: 1st iteration
MV?D&S: 50th iteration
0.3
Label prediction error
0.22
0.19
0.18
0.17
0.25
0.2
0.16
0.15
0.08
0.15
?6
10
?5
10
?4
10
?3
10
?2
10
?1
10
10
?6
?5
10
?4
10
?3
10
?2
10
10
?1
?6
10
?5
10
?4
10
?3
10
Threshold
Threshold
Threshold
(a) RTE
(b) Dog
(c) Web
?2
10
?1
10
Figure 1: Comparing MV-D&S and Opt-D&S with different thresholding parameter ?. The label
prediction error is plotted after the 1st EM update and after convergence.
Bird
RTE
TREC
Dog
Web
Opt-D&S
10.09
7.12
29.80
16.89
15.86
MV-D&S
11.11
7.12
30.02
16.66
15.74
Majority Voting
24.07
10.31
34.86
19.58
26.93
KOS
11.11
39.75
51.96
31.72
42.93
Ghosh-SVD
27.78
49.13
42.99
?
?
EigenRatio
27.78
9.00
43.96
?
?
Table 2: Error rate (%) in predicting true labels on real data.
initialized by majority voting (referred to as MV-D&S), the pure majority voting estimator, the
multi-class labeling algorithm proposed by Karger et al. [14] (referred to as KOS), the SVD-based
algorithm proposed by Ghosh et al. [11] (referred to as Ghost-SVD) and the ?Eigenvalues of Ratio?
algorithm proposed by Dalvi et al. [7] (referred to as EigenRatio). The evaluation is made on five
real datasets.
We compare the crowdsourcing algorithms on three binary tasks and two multi-class tasks. Binary
tasks include labeling bird species [22] (Bird dataset), recognizing textual entailment [21] (RTE
dataset) and assessing the quality of documents in the TREC 2011 crowdsourcing track [16] (TREC
dataset). Multi-class tasks include labeling the breed of dogs from ImageNet [9] (Dog dataset) and
judging the relevance of web search results [26] (Web dataset). The statistics for the five datasets
are summarized in Table 1. Since the Ghost-SVD algorithm and the EigenRatio algorithm work on
binary tasks, they are evaluated only on the Bird, RTE and TREC datasets. For the MV-D&S and
the Opt-D&S methods, we iterate their EM steps until convergence.
Since entries of the confusion matrix are positive, we find it helpful to incorporate this prior knowledge into the initialization stage of the Opt-D&S algorithm. In particular, when estimating the confusion matrix entries by Eq. (3), we add an extra checking step before the normalization, examining
if the matrix components are greater than or equal to a small threshold ?. For components that are
smaller than ?, they are reset to ?. The default choice of the thresholding parameter is ? = 10?6 .
Later, we will compare the Opt-D&S algorithm with respect to different choices of ?. It is important to note that this modification doesn?t change our theoretical result, since the thresholding is not
needed in case that the initialization error is bounded by Theorem 1.
Table 2 summarizes the performance of each method. The MV-D&S and the Opt-D&S algorithms
consistently outperform the other methods in predicting the true label of items. The KOS algorithm,
the Ghost-SVD algorithm and the EigenRatio algorithm yield poorer performance, presumably due
to the fact that they rely on idealized assumptions that are not met by the real data. In Figure 1, we
compare the Opt-D&S algorithm with respect to different thresholding parameters ? ? {10?i }6i=1 .
We plot results for three datasets (RET, Dog, Web), where the performance of MV-D&S is equal to or
slightly better than that of Opt-D&S. The plot shows that the performance of the Opt-D&S algorithm
is stable after convergence. But at the first EM iterate, the error rates are more sensitive to the choice
of ?. A proper choice of ? makes Opt-D&S outperform MV-D&S. The result suggests that a
proper initialization combined with one EM iterate is good enough for the purposes of prediction.
In practice, the best choice of ? can be obtained by cross validation.
8
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4,896 | 5,432 | Unsupervised Transcription of Piano Music
Taylor Berg-Kirkpatrick Jacob Andreas Dan Klein
Computer Science Division
University of California, Berkeley
{tberg,jda,klein}@cs.berkeley.edu
Abstract
We present a new probabilistic model for transcribing piano music from audio to
a symbolic form. Our model reflects the process by which discrete musical events
give rise to acoustic signals that are then superimposed to produce the observed
data. As a result, the inference procedure for our model naturally resolves the
source separation problem introduced by the the piano?s polyphony. In order to
adapt to the properties of a new instrument or acoustic environment being transcribed, we learn recording-specific spectral profiles and temporal envelopes in an
unsupervised fashion. Our system outperforms the best published approaches on
a standard piano transcription task, achieving a 10.6% relative gain in note onset
F1 on real piano audio.
1
Introduction
Automatic music transcription is the task of transcribing a musical audio signal into a symbolic representation (for example MIDI or sheet music). We focus on the task of transcribing piano music,
which is potentially useful for a variety of applications ranging from information retrieval to musicology. This task is extremely difficult for multiple reasons. First, even individual piano notes are
quite rich. A single note is not simply a fixed-duration sine wave at an appropriate frequency, but
rather a full spectrum of harmonics that rises and falls in intensity. These profiles vary from piano
to piano, and therefore must be learned in a recording-specific way. Second, piano music is generally polyphonic, i.e. multiple notes are played simultaneously. As a result, the harmonics of the
individual notes can and do collide. In fact, combinations of notes that exhibit ambiguous harmonic
collisions are particularly common in music, because consonances sound pleasing to listeners. This
polyphony creates a source-separation problem at the heart of the transcription task.
In our approach, we learn the timbral properties of the piano being transcribed (i.e. the spectral and
temporal shapes of each note) in an unsupervised fashion, directly from the input acoustic signal. We
present a new probabilistic model that describes the process by which discrete musical events give
rise to (separate) acoustic signals for each keyboard note, and the process by which these signals are
superimposed to produce the observed data. Inference over the latent variables in the model yields
transcriptions that satisfy an informative prior distribution on the discrete musical structure and at
the same time resolve the source-separation problem.
For the problem of unsupervised piano transcription where the test instrument is not seen during
training, the classic starting point is a non-negative factorization of the acoustic signal?s spectrogram. Most previous work improves on this baseline in one of two ways: either by better modeling
the discrete musical structure of the piece being transcribed [1, 2] or by better adapting to the timbral properties of the source instrument [3, 4]. Combining these two kinds of approaches has proven
challenging. The standard approach to modeling discrete musical structures?using hidden Markov
or semi-Markov models?relies on the availability of fast dynamic programs for inference. Here,
coupling these discrete models with timbral adaptation and source separation breaks the conditional
independence assumptions that the dynamic programs rely on. In order to avoid this inference
problem, past approaches typically defer detailed modeling of discrete structure or timbre to a postprocessing step [5, 6, 7].
1
Event Params
PLAY
?(n)
M (nr)
velocity
Note Events
REST
velocity
duration
time
Envelope Params
Note Activation
?
(n)
A
(nr)
time
time
Component Spectrogram
(n)
S (nr)
freq
freq
Spectral Params
time
N
freq
Spectrogram
X (r)
R
time
Figure 1: We transcribe a dataset consisting of R songs produced by a single piano with N notes. For each
keyboard note, n, and each song, r, we generate a sequence of musical events, M (nr) , parameterized by ?(n) .
Then, conditioned on M (nr) , we generate an activation time series, A(nr) , parameterized by ?(n) . Next,
conditioned on A(nr) , we generate a component spectrogram for note n in song r, S (nr) , parameterized by
? (n) . The observed total spectrogram for song r is produced by superimposing component spectrograms:
P
X (r) = n S (nr) .
We present the first approach that tackles these discrete and timbral modeling problems jointly. We
have two primary contributions: first, a new generative model that reflects the causal process underlying piano sound generation in an articulated way, starting with discrete musical structure; second,
a tractable approximation to the inference problem over transcriptions and timbral parameters. Our
approach achieves state-of-the-art results on the task of polyphonic piano music transcription. On
a standard dataset consisting of real piano audio data, annotated with ground-truth onsets, our approach outperforms the best published models for this task on multiple metrics, achieving a 10.6%
relative gain in our primary measure of note onset F1 .
2
Model
Our model is laid out in Figure 1. It has parallel random variables for each note on the piano keyboard. For now, we illustrate these variables for a single concrete note?say C] in the 4th octave?
and in Section 2.4 describe how the parallel components combine to produce the observed audio
signal. Consider a single song, divided into T time steps. The transcription will be I musical events
long. The component model for C] consists of three primary random variables:
M , a sequence of I symbolic musical events, analogous to the locations
and values of symbols along the C] staff line in sheet music,
A, a time series of T activations, analogous to the loudness of sound emitted by the C] piano string over time as it peaks and attenuates during each
event in M ,
S, a spectrogram of T frames, specifying the spectrum of frequencies over
time in the acoustic signal produced by the C] string.
2
M (nr)
Envelope Params
?(n)
Event State
E2
E1
E3
Truncate to duration Di
Scale to velocity Vi
Add noise
Duration
Velocity
D3
D2
D1
V1
V2
V3
A(nr)
Figure 2: Joint distribution on musical events, M (nr) , and activations, A(nr) , for note n in song r, conditioned
on event parameters, ?(n) , and envelope parameters, ?(n) . The dependence of Ei , Di , and Vi on n and r is
suppressed for simplicity.
The parameters that generate each of these random variables are depicted in Figure 1. First, musical
]
events, M , are generated from a distribution parameterized by ?(C ) , which specifies the probability
]
that the C key is played, how long it is likely to be held for (duration), and how hard it is likely to
be pressed (velocity). Next, the activation of the C] string over time, A, is generated conditioned on
]
M from a distribution parameterized by a vector, ?(C ) (see Figure 1), which specifies the shape of
the rise and fall of the string?s activation each time the note is played. Finally, the spectrogram, S,
]
is generated conditioned on A from a distribution parameterized by a vector, ? (C ) (see Figure 1),
which specifies the frequency distribution of sounds produced by the C] string. As depicted in
]
Figure 3, S is produced from the outer product of ? (C ) and A. The joint distribution for the note1 is:
]
]
]
]
P (S, A, M |? (C ) , ?(C ) , ?(C ) ) = P (M |?(C ) )
? P (A|M, ?
(C] )
]
[Event Model, Section 2.1]
)
? P (S|A, ? (C ) )
[Activation Model, Section 2.2]
[Spectrogram Model, Section 2.3]
In the next three sections we give detailed descriptions for each of the component distributions.
2.1
Event Model
Our symbolic representation of musical structure (see Figure 2) is similar to the MIDI format used by
musical synthesizers. M consists of a sequence of I random variables representing musical events
for the C] piano key: M = (M1 , M2 , . . . , MI ). Each event Mi , is a tuple consisting of a state,
Ei , which is either PLAY or REST, a duration Di , which is a length in time steps, and a velocity Vi ,
which specifies how hard the key was pressed (assuming Ei is PLAY).
The graphical model for the process that generates M is depicted in Figure 2. The sequence of
states, (E1 , E2 , . . . , EI ), is generated from a Markov model. The transition probabilities, ?TRANS ,
control how frequently the note is played (some notes are more frequent than others). An event?s
duration, Di , is generated conditioned on Ei from a distribution parameterized by ?DUR . The durations of PLAY events have a multinomial parameterization, while the durations of REST events are
distributed geometrically. An event?s velocity, Vi , is a real number on the unit interval and is generated conditioned on Ei from a distribution parameterized by ?VEL . If Ei = REST, deterministically
]
Vi = 0. The complete event parameters for keyboard note C] are ?(C ) = (?TRANS , ?DUR , ?VEL ).
1
For notational convenience, we suppress the C] superscripts on M , A, and S until Section 2.4.
3
2.2
Activation Model
In an actual piano, when a key is pressed, a hammer strikes a string and a sound with sharply
rising volume is emitted. The string continues to emit sound as long as the key remains depressed, but the volume decays since no new energy is being transferred. When the key is
released, a damper falls back onto the string, truncating the decay. Examples of this trajectory are depicted in Figure 1 in the graph of activation values. The graph depicts the note being played softly and held, and then being played more loudly, but held for a shorter time.
In our model, PLAY events represent hammer strikes on a piano string with raised damper,
while REST events represent the lowered damper. In our parameterization, the shape of the
rise and decay is shared by all PLAY events for a given note, regardless of their
?
duration and velocity. We call this shape an envelope and describe it using a posi]
tive vector of parameters. For our running example of C , this parameter vector is
]
?(C ) (depicted to the right).
The time series of activations for the C] string, A, is a positive vector of length T , where T denotes
the total length of the song in time steps. Let [A]t be the activation at time step t. As mentioned in
Section 2, A may be thought of as roughly representing the loudness of sound emitted by the piano
string as a function of time. The process that generates A is depicted in Figure 2. We generate A
conditioned on the musical events, M . Each musical event, Mi = (Ei , Di , Vi ), produces a segment
of activations, Ai , of length Di . For PLAY events, Ai will exhibit an increase in activation. For
REST events, the activation will remain low. The segments are appended together to make A. The
activation values in each segment are generated in a way loosely inspired by piano mechanics. Given
]
]
?(C ) , we generate the values in segment Ai as follows: ?(C ) is first truncated to duration Di , then
is scaled by the velocity of the strike, Vi , and, finally, is used to parameterize an activation noise
distribution which generates the segment Ai . Specifically, we add independent Gaussian noise to
]
each dimension after ?(C ) is truncated and scaled. In principle, this choice of noise distribution
gives a formally deficient model, since activations are positive, but in practice performs well and has
the benefit of making inference mathematically simple (see Section 3.1).
2.3
Component Spectrogram Model
Piano sounds have a harmonic structure; when played, each piano string primarily emits
energy at a fundamental frequency determined by the string?s length, but also at all
integer multiples of that frequency (called partials) with diminishing strength (see the depiction to the right). For example, the audio signal produced by the C] string will vary in
loudness, but its frequency distribution will remain mostly fixed. We call this frequency
distribution a spectral profile. In our parameterization, the spectral profile of C] is speci]
]
fied by a positive spectral parameter vector, ? (C ) (depicted to the right). ? (C ) is a vector
]
of length F , where [? (C ) ]f represents the weight of frequency bin f .
In our model, the audio signal produced by the C] string over the course of the song is represented
as a spectrogram, S, which is a positive matrix with F rows, one for each frequency bin, f , and T
columns, one for each time step, t (see Figures 1 and 3 for examples). We denote the magnitude
of frequency f at time step t as [S]f t . In order to generate the spectrogram (see Figure 3), we
first produce a matrix of intermediate values by taking the outer product of the spectral profile,
]
? (C ) , and the activations, A. These intermediate values are used to parameterize a spectrogram
noise distribution that generates S. Specifically, for each frequency bin f and each time step t, the
corresponding value of the spectrogram, [S]f t , is generated from a noise distribution parameterized
]
by [? (C ) ]f ? [A]t . In practice, the choice of noise distribution is very important. After examining
residuals resulting from fitting mean parameters to actual musical spectrograms, we experimented
with various noising assumptions, including multiplicative gamma noise, additive Gaussian noise,
log-normal noise, and Poisson noise. Poisson noise performed best. This is consistent with previous
findings in the literature, where non-negative matrix factorization using KL divergence (which has
a generative interpretation as maximum likelihood inference in a Poisson model [8]) is commonly
chosen [7, 2]. Under the Poisson noise assumption, the spectrogram is interpreted as a matrix of
(large) integer counts.
4
A(1r)
(N )
freq
...
freq
(1)
A(N r)
time
time
S (1r)
S (N r)
freq
X (r)
2.4
Figure 3:
Conditional distribution for song r on the observed
total spectrogram, X (r) , and
the component spectrograms for
each note, (S (1r) , . . . , S (N r) ),
given the activations for each
note, (A(1r) , . . . , A(N r) ), and
spectral parameters for each note,
(? (1) , . . . , ? (N ) ).
X (r) is the
superposition of the component
P
spectrograms: X (r) = n S (nr) .
time
Full Model
So far we have only looked at the component of the model corresponding to a single note?s contribution to a single song. Our full model describes the generation of a collection of many songs, from
a complete instrument with many notes. This full model is diagrammed in Figures 1 and 3. Let a
piano keyboard consist of N notes (on a standard piano, N is 88), where n indexes the particular
note. Each note, n, has its own set of musical event parameters, ?(n) , envelope parameters, ?(n) ,
and spectral parameters, ? (n) . Our corpus consists of R songs (?recordings?), where r indexes a
particular song. Each pair of note n and song r has it?s own musical events variable, M (nr) , activations variable, A(nr) , and component spectrogram S (nr) . The full spectrogram for song r, which is
the observed input, is denoted
as X (r) . Our model generates X (r) by superimposing the component
P (nr)
(r)
spectrograms: X = n S
. Going forward, we will need notation to group together variables
across all N notes: let ? = (?(1) , . . . , ?(N ) ), ? = (?(1) , . . . , ?(N ) ), and ? = (? (1) , . . . , ? (N ) ).
Also let M (r) = (M (1r) , . . . , M (N r) ), A(r) = (A(1r) , . . . , A(N r) ), and S (r) = (S (1r) , . . . , S (N r) ).
3
Learning and Inference
Our goal is to estimate the unobserved musical events for each song, M (r) , as well as the unknown
envelope and spectral parameters of the piano that generated the data, ? and ?. Our inference will
estimate both, though our output is only the musical events, which specify the final transcription.
Because MIDI sample banks (piano synthesizers) are readily available, we are able to provide the
system with samples from generic pianos (but not from the piano being transcribed). We also give
the model information about the distribution of notes in real musical scores by providing it with an
external corpus of symbolic music data. Specifically, the following information is available to the
model during inference: 1) the total spectrogram for each song, X (r) , which is the input, 2) the event
parameters, ?, which we estimate by collecting statistics on note occurrences in the external corpus
of symbolic music, and 3) truncated normal priors on the envelopes and spectral profiles, ? and ?,
which we extract from the MIDI samples.
? = (M (1) , . . . , M (R) ), A? = (A(1) , . . . , A(R) ), and S? = (S (1) , . . . , S (R) ), the tuples of event,
Let M
activation, and spectrogram variables across all notes n and songs r. We would like to compute
? , ?, and ?. However, marginalizing over the activations A? couples
the posterior distribution on M
the discrete musical structure with the superposition process of the component spectrograms in an
? , A,
? ?, and ? via iterated
intractable way. We instead approximate the joint MAP estimates of M
? Specifically, we
conditional modes [9], only marginalizing over the component spectrograms, S.
perform the following optimization via block-coordinate ascent:
"
#
Y X
max
P (X (r) , S (r) , A(r) , M (r) |?, ?, ?) ? P (?, ?)
? ,A,?,?
?
M
r
S (r)
? in Section 3.1,
The updates for each group of variables are described in the following sections: M
? in Section 3.2, A? in Section 3.3, and ? in Section 3.4.
5
3.1
Updating Events
? to maximize the objective while the other variables are held fixed. The joint disWe update M
? and A? is a hidden semi-Markov model [10]. Given the optimal velocity for each
tribution on M
segment of activations, the computation of the emission potentials for the semi-Markov dynamic
? can be performed exactly and efficiently. We let
program is straightforward and the update over M
the distribution of velocities for PLAY events be uniform. This choice, together with the choice of
Gaussian activation noise, yields a simple closed-form solution for the optimal setting of the velocity
(nr)
(nr)
variable Vi
. Let [?(n) ]j denote the jth value of the envelope vector ?(n) . Let [Ai ]j be the jth
(nr)
entry of the segment of A(nr) generated by event Mi . The velocity that maximizes the activation
segment?s probability is given by:
PDi(nr) (n)
(nr)
[?
]
?
[A
]
j
j
j=1
i
(nr)
Vi
=
PDi(nr) (n) 2
]j
j=1 [?
3.2
Updating Envelope Parameters
Given settings of the other variables, we update the envelope parameters, ?, to optimize the objec(nr)
tive. The truncated normal prior on ? admits a closed-form update. Let I(j, n, r) = {i : Di
? j},
(n)
the set of event indices for note n in song r with durations no longer than j time steps. Let [?0 ]j
(n)
be the location parameter for the prior on [? ]j , and let ? be the scale parameter (which is shared
across all n and j). The update for [?(n) ]j is given by:
P
P
(nr)
(nr)
(n)
[Ai ]j + 2?1 2 [?0 ]j
(n,r)
i?I(j,n,r) Vi
(n)
[? ]j =
P
P
(nr) 2
]j + 2?1 2
(n,r)
i?I(j,n,r) [Ai
3.3
Updating Activations
? we optimize the objective with respect to A,
? with the other
In order to update the activations, A,
variables held fixed. The choice of Poisson noise for generating each of the component spectrograms,P
S (nr) , means that the conditional distribution of the total spectrogram for each song,
(r)
(nr)
X
=
, with S (r) marginalized out, is also
nS
Poisson. Specifically, the distribution of
P
(r)
[X ]f t is Poisson with mean n [? (n) ]f ? [A(nr) ]t . Optimizing the probability of X (r) under
this conditional distribution with respect to A(r) corresponds to computing the supervised NMF
using KL divergence [7]. However, to perform the correct update in our model, we must also incorporate the distribution of A(r) , and so, instead of using the standard multiplicative NMF updates,
we use exponentiated gradient ascent [11] on the log objective. Let L denote the log objective, let
?
? (n, r, t) denote the velocity-scaled envelope value used to generate the activation value [A(nr) ]t ,
and let ? 2 denote the variance parameter for the Gaussian activation noise. The gradient of the log
objective with respect to [A(nr) ]t is:
"
#
X
[X (r) ]f t ? [? (n) ]f
1 (nr)
?L
(n)
P
=
?
[?
]
?
[A
]
?
?
?
(n,
r,
t)
f
t
0
0
(n ) ] ? [A(n r) ]
?2
?[A(nr) ]t
f
t
n0 [?
f
3.4
Updating Spectral Parameters
The update for the spectral parameters, ?, is similar to that of the activations. Like the activations, ?
is part of the parameterization of the Poisson distribution on each X (r) . We again use exponentiated
(n)
gradient ascent. Let [?0 ]f be the location parameter of the prior on [? (n) ]f , and let ? be the scale
parameter (which is shared across all n and f ). The gradient of the the log objective with respect to
[? (n) ]f is given by:
"
#
X
?L
[X (r) ]f t ? [A(nr) ]t
1 (n)
(n)
(nr)
P
=
?
[A
]
?
[?
]
?
[?
]
t
f
f
0
(n0 ) ] ? [A(n0 r) ]
?2
?[? (n) ]f
f
t
n0 [?
(r,t)
6
4
Experiments
Because polyphonic transcription is so challenging, much of the existing literature has either worked
with synthetic data [12] or assumed access to the test instrument during training [5, 6, 13, 7]. As
our ultimate goal is the transcription of arbitrary recordings from real, previously-unseen pianos, we
evaluate in an unsupervised setting, on recordings from an acoustic piano not observed in training.
Data We evaluate on the MIDI-Aligned Piano Sounds (MAPS) corpus [14]. This corpus includes
a collection of piano recordings from a variety of time periods and styles, performed by a human
player on an acoustic ?Disklavier? piano equipped with electromechanical sensors under the keys.
The sensors make it possible to transcribe directly into MIDI while the instrument is in use, providing
a ground-truth transcript to accompany the audio for the purpose of evaluation. In keeping with much
of the existing music transcription literature, we use the first 30 seconds of each of the 30 ENSTDkAm
recordings as a development set, and the first 30 seconds of each of the 30 ENSTDkCl recordings
as a test set. We also assume access to a collection of synthesized piano sounds for parameter
initialization, which we take from the MIDI portion of the MAPS corpus, and a large collection of
symbolic music data from the IMSLP library [15, 16], used to estimate the event parameters in our
model.
Preprocessing We represent the input audio as a magnitude spectrum short-time Fourier transform
with a 4096-frame window and a hop size of 512 frames, similar to the approach used by Weninger
et al. [7]. We temporally downsample the resulting spectrogram by a factor of 2, taking the maximum magnitude over collapsed bins. The input audio is recorded at 44.1 kHz and the resulting
spectrogram has 23ms frames.
Initialization and Learning We estimate initializers and priors for the spectral parameters, ?,
and envelope parameters, ?, by fitting isolated, synthesized, piano sounds. We collect these isolated
sounds from the MIDI portion of MAPS, and average the parameter values across several synthesized
pianos. We estimate the event parameters ? by counting note occurrences in the IMSLP data. At
decode time, to fit the spectral and envelope parameters and predict transcriptions, we run 5 iterations
of the block-coordinate ascent procedure described in Section 3.
Evaluation We report two standard measures of performance: an onset evaluation, in which a
predicted note is considered correct if it falls within 50ms of a note in the true transcription, and
a frame-level evaluation, in which each transcription is converted to a boolean matrix specifying
which notes are active at each time step, discretized to 10ms frames. Each entry is compared to
the corresponding entry in the true matrix. Frame-level evaluation is sensitive to offsets as well
as onsets, but does not capture the fact that note onsets have greater musical significance than do
offsets. As is standard, we report precision (P), recall (R), and F1 -measure (F1 ) for each of these
metrics.
4.1
Comparisons
We compare our system to three state-of-the-art unsupervised systems: the hidden semi-Markov
model described by Benetos and Weyde [2] and the spectrally-constrained factorization models described by Vincent et al. [3] and O?Hanlon and Plumbley [4]. To our knowledge, Benetos and Weyde
[2] report the best published onset results for this dataset, and O?Hanlon and Plumbley [4] report the
best frame-level results.
The literature also includes a number of supervised approaches to this task. In these approaches,
a model is trained on annotated recordings from a known instrument. While best performance
is achieved when testing on the same instrument used for training, these models can also achieve
reasonable performance when applied to new instruments. Thus, we also compare to a discriminative
baseline, a simplified reimplementation of a state-of-the-art supervised approach [7] which achieves
slightly better performance than the original on this task. This system only produces note onsets,
and therefore is not evaluated at a frame-level. We train the discriminative baseline on synthesized
audio with ground-truth MIDI annotations, and apply it directly to our test instrument, which the
system has never seen before.
7
System
Discriminative [7]
Benetos [2]
Vincent [3]
O?Hanlon [4]
This work
P
Onsets
R
F1
P
Frames
R
F1
76.8
62.7
48.6
78.1
65.1
76.8
73.0
74.7
70.4
68.6
69.0
58.3
76.4
79.6
73.4
69.1
63.6
72.8
80.7
68.0
70.7
73.2
74.4
Table 1: Unsupervised transcription results on the MAPS corpus. ?Onsets? columns show scores for identification (within ?50ms) of note start times. ?Frames? columns show scores for 10ms frame-level evaluation. Our
system achieves state-of-the-art results on both metrics.2
4.2
Results
Our model achieves the best published numbers on this task: as shown in Table 1, it achieves an
onset F1 of 76.4, which corresponds to a 10.6% relative gain over the onset F1 achieved by the
system of Vincent et al. [3], the top-performing unsupervised baseline on this metric. Surprisingly,
the discriminative baseline [7], which was not developed for the unsupervised task, outperforms
all the unsupervised baselines in terms of onset evaluation, achieving an F1 of 70.4. Evaluated on
frames, our system achieves an F1 of 74.4, corresponding to a more modest 1.6% relative gain over
the system of O?Hanlon and Plumbley [4], which is the best performing baseline on this metric.
The surprisingly competitive discriminative baseline shows that it is possible to achieve high onset
accuracy on this task without adapting to the test instrument. Thus, it is reasonable to ask how much
of the gain our model achieves is due to its ability to learn instrument timbre. If we skip the blockcoordinate ascent updates (Section 3) for the envelope and spectral parameters, and thus prevent our
system from adapting to the test instrument, onset F1 drops from 76.4 to 72.6. This result indicates
that learning instrument timbre does indeed help performance.
As a short example of our system?s behavior, Figure 4 shows our system?s output passed through a
commercially-available MIDI-to-sheet-music converter. This example was chosen because its onset
F1 of 75.5 and error types are broadly representative of the system?s performance on our data. The
resulting score has musically plausible errors.
Predicted score
Reference score
Figure 4: Result of passing our system?s prediction and the reference transcription MIDI through the GarageBand MIDI-to-sheet-music converter. This is a transcription of the first three bars of Schumann?s Hobgoblin.
A careful inspection of the system?s output suggests that a large fraction of errors are either off by
an octave (i.e. the frequency of the predicted note is half or double the correct frequency) or are
segmentation errors (in which a single key press is transcribed as several consecutive key presses).
While these are tricky errors to correct, they may also be relatively harmless for some applications
because they are not detrimental to musical perception: converting the transcriptions back to audio
using a synthesizer yields music that is qualitatively quite similar to the original recordings.
5
Conclusion
We have shown that combining unsupervised timbral adaptation with a detailed model of the generative relationship between piano sounds and their transcriptions can yield state-of-the-art performance. We hope that these results will motivate further joint approaches to unsupervised music
transcription. Paths forward include exploring more nuanced timbral parameterizations and developing more sophisticated models of discrete musical structure.
2
For consistency we re-ran all systems in this table with our own evaluation code (except for the system
of Benetos and Weyde [2], for which numbers are taken from the paper). For O?Hanlon and Plumbley [4]
scores are higher than the authors themselves report; this is due to an extra post-processing step suggested by
O?Hanlon in personal correspondence.
8
References
[1] Masahiro Nakano, Yasunori Ohishi, Hirokazu Kameoka, Ryo Mukai, and Kunio Kashino. Bayesian nonparametric music parser. In IEEE International Conference on Acoustics, Speech and Signal Processing,
2012.
[2] Emmanouil Benetos and Tillman Weyde. Explicit duration hidden markov models for multiple-instrument
polyphonic music transcription. In International Society for Information Music Retrieval, 2013.
[3] Emmanuel Vincent, Nancy Bertin, and Roland Badeau. Adaptive harmonic spectral decomposition for
multiple pitch estimation. IEEE Transactions on Audio, Speech, and Language Processing, 2010.
[4] Ken O?Hanlon and Mark D. Plumbley. Polyphonic piano transcription using non-negative matrix factorisation with group sparsity. In IEEE International Conference on Acoustics, Speech, and Signal Processing, 2014.
[5] Graham E. Poliner and Daniel P.W. Ellis. A discriminative model for polyphonic piano transcription.
EURASIP Journal on Advances in Signal Processing, 2007.
[6] R. Lienhart C. G. van de Boogaart. Note onset detection for the transcription of polyphonic piano music.
In Multimedia and Expo ICME. IEEE, 2009.
[7] Felix Weninger, Christian Kirst, Bjorn Schuller, and Hans-Joachim Bungartz. A discriminative approach
to polyphonic piano note transcription using supervised non-negative matrix factorization. In IEEE International Conference on Acoustics, Speech and Signal Processing, 2013.
[8] Paul H. Peeling, Ali Taylan Cemgil, and Simon J. Godsill. Generative spectrogram factorization models
for polyphonic piano transcription. IEEE Transactions on Audio, Speech, and Language Processing,
2010.
[9] Julian Besag. On the statistical analysis of dirty pictures. Journal of the Royal Statistical Society, 1986.
[10] Stephen Levinson. Continuously variable duration hidden Markov models for automatic speech recognition. Computer Speech & Language, 1986.
[11] Jyrki Kivinen and Manfred K. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Information and Computation, 1997.
[12] Matti P. Ryynanen and Anssi Klapuri. Polyphonic music transcription using note event modeling. In
IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, 2005.
[13] Sebastian B?ock and Markus Schedl. Polyphonic piano note transcription with recurrent neural networks.
In IEEE International Conference on Acoustics, Speech and Signal Processing, 2012.
[14] Valentin Emiya, Roland Badeau, and Bertrand David. Multipitch estimation of piano sounds using a
new probabilistic spectral smoothness principle. IEEE Transactions on Audio, Speech, and Language
Processing, 2010.
[15] The international music score library project, June 2014. URL http://imslp.org.
[16] Vladimir Viro. Peachnote: Music score search and analysis platform. In The International Society for
Music Information Retrieval, 2011.
9
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4,897 | 5,433 | Combinatorial Pure Exploration of
Multi-Armed Bandits
Shouyuan Chen1? Tian Lin2
Irwin King1
Michael R. Lyu1
Wei Chen3
2
3
The Chinese University of Hong Kong
Tsinghua University
Microsoft Research Asia
1
2
{sychen,king,lyu}@cse.cuhk.edu.hk
[email protected] 3 [email protected]
1
Abstract
We study the combinatorial pure exploration (CPE) problem in the stochastic multi-armed
bandit setting, where a learner explores a set of arms with the objective of identifying
the optimal member of a decision class, which is a collection of subsets of arms with
certain combinatorial structures such as size-K subsets, matchings, spanning trees or paths,
etc. The CPE problem represents a rich class of pure exploration tasks which covers not
only many existing models but also novel cases where the object of interest has a nontrivial combinatorial structure. In this paper, we provide a series of results for the general
CPE problem. We present general learning algorithms which work for all decision classes
that admit offline maximization oracles in both fixed confidence and fixed budget settings.
We prove problem-dependent upper bounds of our algorithms. Our analysis exploits the
combinatorial structures of the decision classes and introduces a new analytic tool. We also
establish a general problem-dependent lower bound for the CPE problem. Our results show
that the proposed algorithms achieve the optimal sample complexity (within logarithmic
factors) for many decision classes. In addition, applying our results back to the problems
of top-K arms identification and multiple bandit best arms identification, we recover the
best available upper bounds up to constant factors and partially resolve a conjecture on the
lower bounds.
1
Introduction
Multi-armed bandit (MAB) is a predominant model for characterizing the tradeoff between exploration and exploitation in decision-making problems. Although this is an intrinsic tradeoff in many
tasks, some application domains prefer a dedicated exploration procedure in which the goal is to
identify an optimal object among a collection of candidates and the reward or loss incurred during
exploration is irrelevant. In light of these applications, the related learning problem, called pure exploration in MABs, has received much attention. Recent advances in pure exploration MABs have
found potential applications in many domains including crowdsourcing, communication network
and online advertising.
In many of these application domains, a recurring problem is to identify the optimal object with
certain combinatorial structure. For example, a crowdsourcing application may want to find the best
assignment from workers to tasks such that overall productivity of workers is maximized. A network
routing system during the initialization phase may try to build a spanning tree that minimizes the
delay of links, or attempts to identify the shortest path between two sites. An online advertising
system may be interested in finding the best matching between ads and display slots. The literature
of pure exploration MAB problems lacks a framework that encompasses these kinds of problems
where the object of interest has a non-trivial combinatorial structure. Our paper contributes such
a framework which accounts for general combinatorial structures, and develops a series of results,
including algorithms, upper bounds and lower bounds for the framework.
In this paper, we formulate the combinatorial pure exploration (CPE) problem for stochastic multiarmed bandits. In the CPE problem, a learner has a fixed set of arms and each arm is associated with
an unknown reward distribution. The learner is also given a collection of sets of arms called decision
class, which corresponds to a collection of certain combinatorial structures. During the exploration
period, in each round the learner chooses an arm to play and observes a random reward sampled from
?
This work was done when the first two authors were interns at Microsoft Research Asia.
1
the associated distribution. The objective is when the exploration period ends, the learner outputs a
member of the decision class that she believes to be optimal, in the sense that the sum of expected
rewards of all arms in the output set is maximized among all members in the decision class.
The CPE framework represents a rich class of pure exploration problems. The conventional pure exploration problem in MAB, whose objective is to find the single best arm, clearly fits into this framework, in which the decision class is the collection of all singletons. This framework also naturally
encompasses several recent extensions, including the problem of finding the top K arms (henceforth
T OP K) [18, 19, 8, 20, 31] and the multi-bandit problem of finding the best arms simultaneously
from several disjoint sets of arms (henceforth MB) [12, 8]. Further, this framework covers many
more interesting cases where the decision classes correspond to collections of non-trivial combinatorial structures. For example, suppose that the arms represent the edges in a graph. Then a decision
class could be the set of all paths between two vertices, all spanning trees or all matchings of the
graph. And, in these cases, the objectives of CPE become identifying the optimal paths, spanning
trees and matchings through bandit explorations, respectively. To our knowledge, there are no results
available in the literature for these pure exploration tasks.
The CPE framework raises several interesting challenges to the design and analysis of pure exploration algorithms. One challenge is that, instead of solving each type of CPE task in an ad-hoc way,
one requires a unified algorithm and analysis that support different decision classes. Another challenge stems from the combinatorial nature of CPE, namely that the optimal set may contain some
arms with very small expected rewards (e.g., it is possible that a maximum matching contains the
edge with the smallest weight); hence, arms cannot be eliminated simply based on their own rewards in the learning algorithm or ignored in the analysis. This differs from many existing approach
of pure exploration MABs. Therefore, the design and analysis of algorithms for CPE demands novel
techniques which take both rewards and combinatorial structures into account.
Our results. In this paper, we propose two novel learning algorithms for general CPE problem: one
for the fixed confidence setting and one for the fixed budget setting. Both algorithms support a wide
range of decision classes in a unified way. In the fixed confidence setting, we present Combinatorial
Lower-Upper Confidence Bound (CLUCB) algorithm. The CLUCB algorithm does not need to know
the definition of the decision class, as long as it has access to the decision class through a maximization oracle. We upper bound the number of samples used by CLUCB. This sample complexity bound
depends on both the expected rewards and the structure of decision class. Our analysis relies on a
novel combinatorial construction called exchange class, which may be of independent interest for
other combinatorial optimization problems. Specializing our result to T OP K and MB, we recover
the best available sample complexity bounds [19, 13, 20] up to constant factors. While for other decision classes in general, our result establishes the first sample complexity upper bound. We further
show that CLUCB can be easily extended to the fixed budget setting and PAC learning setting and
we provide related theoretical guarantees in the supplementary material.
Moreover, we establish a problem-dependent sample complexity lower bound for the CPE problem.
Our lower bound shows that the sample complexity of the proposed CLUCB algorithm is optimal
(to within logarithmic factors) for many decision classes, including T OP K, MB and the decision
classes derived from matroids (e.g., spanning tree). Therefore our upper and lower bounds provide
a nearly full characterization of the sample complexity of these CPE problems. For more general
decision classes, our results show that the upper and lower bounds are within a relatively benign
factor. To the best of our knowledge, there are no problem-dependent lower bounds known for pure
exploration MABs besides the case of identifying the single best arm [24, 1]. We also notice that
our result resolves the conjecture of Bubeck et al. [8] on the problem-dependent sample complexity
lower bounds of T OP K and MB problems, for the cases of Gaussian reward distributions.
In the fixed budget setting, we present a parameter-free algorithm called Combinatorial Successive
Accept Reject (CSAR) algorithm. We prove a probability of error bound of the CSAR algorithm. This
bound can be shown to be equivalent to the sample complexity bound of CLUCB within logarithmic
factors, although the two algorithms are based on quite different techniques. Our analysis of CSAR
re-uses exchange classes as tools. This suggests that exchange classes may be useful for analyzing
similar problems. In addition, when applying the algorithm to back T OP K and MB, our bound
recovers the best known result in the fixed budget setting due to Bubeck et al. [8] up to constant
factors.
2
2
Problem Formulation
In this section, we formally define the CPE problem. Suppose that there are n arms and the arms
are numbered 1, 2, . . . , n. Assume that each arm e 2 [n] is associated with a reward distribution
T
'e . Let w = w(1), . . . , w(n) denote the vector of expected rewards, where each entry w(e) =
EX?'e [X] denotes the expected reward of arm e. Following standard assumptions of stochastic
MABs, we assume that all reward distributions have R-sub-Gaussian tails for some known constant
R > 0. Formally,
if X is a random
variable drawn from 'e for some e 2 [n], then, for all t 2 R,
?
?
one has E exp(tX tE[X]) ? exp(R2 t2 /2). It is known that the family of R-sub-Gaussian tail
distributions encompasses all distributions that are supported on [0, R] as well as many unbounded
distributions such as Gaussian distributions with variance R2 (see e.g., [27, 28]).
We define a decision class M ? 2[n] as a collection of sets of arms. Let M? = arg maxM 2M w(M )
denote the optimal member of the decision class M which maximizes the sum of expected rewards1 .
A learner?s objective is to identify M? from M by playing the following game with the stochastic
environment. At the beginning of the game, the decision class M is revealed to the learner while
the reward distributions {'e }e2[n] are unknown to her. Then, the learner plays the game over a
sequence of rounds; in each round t, she pulls an arm pt 2 [n] and observes a reward sampled
from the associated reward distribution 'pt . The game continues until certain stopping condition is
satisfied. After the game finishes, the learner need to output a set Out 2 M.
We consider two different stopping conditions of the game, which are known as fixed confidence
setting and fixed budget setting in the literature. In the fixed confidence setting, the learner can stop
the game at any round. She need to guarantee that Pr[Out = M? ] 1
for a given confidence
parameter . The learner?s performance is evaluated by her sample complexity, i.e., the number of
pulls used by the learner. In the fixed budget setting, the game stops after a fixed number T of rounds,
where T is given before the game starts. The learner tries to minimize the probability of error, which
is formally Pr[Out 6= M? ], within T rounds. In this setting, her performance is measured by the
probability of error.
3
Algorithm, Exchange Class and Sample Complexity
In this section, we present Combinatorial Lower-Upper Confidence Bound (CLUCB) algorithm, a
learning algorithm for the CPE problem in the fixed confidence setting, and analyze its sample complexity. En route to our sample complexity bound, we introduce the notions of exchange classes and
the widths of decision classes, which play an important role in the analysis and sample complexity
bound. Furthermore, the CLUCB algorithm can be extended to the fixed budget and PAC learning
settings, the discussion of which is included in the supplementary material (Appendix B).
Oracle. We allow the CLUCB algorithm to access a maximization oracle. A maximization oracle
takes a weight vector v 2 Rn as input and finds an optimal set from a given decision class M with
respect to the weight vector v. Formally, we call a function Oracle: Rn ! M a maximization oracle
for M if, for all v 2 Rn , we have Oracle(v) 2 arg maxM 2M v(M ). It is clear that a wide range
of decision classes admit such maximization oracles, including decision classes corresponding to
collections of matchings, paths or bases of matroids (see later for concrete examples). Besides the
access to the oracle, CLUCB does not need any additional knowledge of the decision class M.
Algorithm. Now we describe the details of CLUCB, as shown in Algorithm 1. During its execution,
the CLUCB algorithm maintains empirical mean w
?t (e) and confidence radius radt (e) for each arm
e 2 [n] and each round t. The construction of confidence radius ensures that |w(e) w
?t (e)| ?
radt (e) holds with high probability for each arm e 2 [n] and each round t > 0. CLUCB begins
with an initialization phase in which each arm is pulled once. Then, at round t
n, CLUCB uses
the following procedure to choose an arm to play. First, CLUCB calls the oracle which finds the
set Mt = Oracle(w
? t ). The set Mt is the ?best? set with respect to the empirical means w
? t . Then,
CLUCB explores possible refinements of Mt . In particular, CLUCB uses the confidence radius to
compute an adjusted expectation vector w
? t in the following way: for each arm e 2 Mt , w
?t (e) is
equal to to the lower confidence bound w
?t (e) = w
?t (e) radt (e); and for each arm e 62 Mt , w
?t (e) is
equal to the upper confidence bound w
?t (e) = w
?t (e) + radt (e). Intuitively, the adjusted expectation
vector w
? t penalizes arms belonging to the current set Mt and encourages exploring arms out of
P
We define v(S) , i2S v(i) for any vector v 2 Rn and any set S ? [n]. In addition, for convenience,
we will assume that M? is unique.
1
3
Algorithm 1 CLUCB: Combinatorial Lower-Upper Confidence Bound
Require: Confidence 2 (0, 1); Maximization oracle: Oracle(?) : Rn ! M
Initialize: Play each arm e 2 [n] once. Initialize empirical means w
? n and set Tn (e)
1 for all e.
1: for t = n, n + 1, . . . do
2:
Mt
Oracle(w
?t)
3:
Compute confidence radius radt (e) for all e 2 [n]
. radt (e) is defined later in Theorem 1
4:
for e = 1, . . . , n do
5:
if e 2 Mt then w
?t (e)
w
?t (e) radt (e)
6:
else w
?t (e)
w
?t (e) + radt (e)
?t
7:
M
Oracle(w
?t)
? t) = w
8:
if w
? t (M
?t (Mt ) then
9:
Out
Mt
10:
return Out
11:
pt
arg maxe2(M? t \Mt )[(Mt \M? t ) radt (e)
. break ties arbitrarily
12:
Pull arm pt and observe the reward
13:
Update empirical means w
? t+1 using the observed reward
14:
Update number of pulls: Tt+1 (pt )
Tt (pt ) + 1 and Tt+1 (e)
Tt (e) for all e 6= pt
Mt . CLUCB then calls the oracle using the adjusted expectation vector w
? t as input to compute a
? t = Oracle(w
? t) = w
refined set M
? t ). If w
? t (M
?t (Mt ) then CLUCB stops and returns Out = Mt .
? t and
Otherwise, CLUCB pulls the arm that belongs to the symmetric difference between Mt and M
has the largest confidence radius (intuitively the largest uncertainty). This ends the t-th round of
CLUCB. We note that CLUCB generalizes and unifies the ideas of several different fixed confidence
algorithms dedicated to the T OP K and MB problems in the literature [19, 13, 20].
3.1 Sample complexity
Now we establish a problem-dependent sample complexity bound of the CLUCB algorithm. To formally state our result, we need to introduce several notions.
Gap. We begin with defining a natural hardness measure of the CPE problem. For each arm e 2 [n],
we define its gap e as
?
w(M? ) maxM 2M:e2M w(M ) if e 62 M? ,
(1)
e =
w(M? ) maxM 2M:e62M w(M ) if e 2 M? ,
where we adopt the convention that the maximum value of an empty set is
hardness H as the sum of inverse squared gaps
X
2
H=
e .
1. We also define the
(2)
e2[n]
We see that, for each arm e 62 M? , the gap e represents the sub-optimality of the best set that
includes arm e; and, for each arm e 2 M? , the gap e is the sub-optimality of the best set that does
not include arm e. This naturally generalizes and unifies previous definitions of gaps [1, 12, 18, 8].
Exchange class and the width of a decision class. A notable challenge of our analysis stems from
the generality of CLUCB which, as we have seen, supports a wide range of decision classes M.
Indeed, previous algorithms for special cases including T OP K and MB require a separate analysis
for each individual type of problem. Such strategy is intractable for our setting and we need a unified
analysis for all decision classes. Our solution to this challenge is a novel combinatorial construction
called exchange class, which is used as a proxy for the structure of the decision class. Intuitively,
an exchange class B for a decision class M can be seen as a collection of ?patches? (borrowing
concepts from source code management) such that, for any two different sets M, M 0 2 M, one can
transform M to M 0 by applying a series of patches of B; and each application of a patch yields a
valid member of M. These patches are later used by our analysis to build gadgets that interpolate
between different members of the decision class and serve to bridge key quantities. Furthermore, the
maximum patch size of B will play an important role in our sample complexity bound.
Now we formally define the exchange class. We begin with the definition of exchange sets, which
formalize the aforementioned ?patches?. We define an exchange set b as an ordered pair of disjoint
sets b = (b+ , b ) where b+ \ b = ; and b+ , b ? [n]. Then, we define operator such that, for
any set M ? [n] and any exchange set b = (b+ , b ), we have M b , M \b [ b+ . Similarly, we
also define operator such that M b , M \b+ [ b .
4
We call a collection of exchange sets B an exchange class for M if B satisfies the following property.
For any M, M 0 2 M such that M 6= M 0 and for any e 2 (M \M 0 ), there exists an exchange set
(b+ , b ) 2 B which satisfies five constraints: (a) e 2 b , (b) b+ ? M 0 \M , (c) b ? M \M 0 , (d)
(M b) 2 M and (e) (M 0 b) 2 M.
Intuitively, constraints (b) and (c) resemble the concept of patches in the sense that b+ contains
only the ?new? elements from M 0 and b contains only the ?old? elements of M ; constraints (d)
and (e) allow one to transform M one step closer to M 0 by applying a patch b 2 B to yield (M
b) 2 M (and similarly for M 0 b). These transformations are the basic building blocks in our
analysis. Furthermore, as we will see later in our examples, for many decision classes, there are
exchange classes representing natural combinatorial structures, e.g., augmenting paths and cycles of
matchings.
In our analysis, the key quantity of exchange class is called width, which is defined as the size of the
largest exchange set as follows
width(B) =
max
(b+ ,b )2B
(3)
|b+ | + |b |.
Let Exchange(M) denote the family of all possible exchange classes for M. We define the width
of a decision class M as the width of the thinnest exchange class
width(M) =
min
B2Exchange(M)
(4)
width(B).
Sample complexity. Our main result of this section is a problem-dependent sample complexity
bound of the CLUCB algorithm which show that, with high probability, CLUCB returns the optimal
? width(M)2 H samples.
set M? and uses at most O
Theorem 1. Given any 2 (0, 1), any decision class M ? 2[n] and any expected rewards w 2 Rn .
Assume that the reward distribution 'e for each arm e 2 [n] has mean w(e) with
qan R-sub-Gaussian
3
tail. Let M? = arg maxM 2M w(M ) denote the optimal set. Set radt (e) = R 2 log 4nt /Tt (e)
for all t > 0 and e 2 [n]. Then, with probability at least 1
, the CLUCB algorithm (Algorithm 1)
returns the optimal set Out = M? and
T ? O R2 width(M)2 H log nR2 H/
,
(5)
where T denotes the number of samples used by Algorithm 1, H is defined in Eq. (2) and width(M)
is defined in Eq. (4).
3.2 Examples of decision classes
Now we investigate several concrete types of decision classes, which correspond to different CPE
tasks. We analyze the width of these decision classes and apply Theorem 1 to obtain the sample
complexity bounds. A detailed analysis and the constructions of exchange classes can be found in
the supplementary material (Appendix F). We begin with the problems of top-K arm identification
(T OP K) and multi-bandit best arms identification (MB).
Example 1 (T OP K and MB). For any K 2 [n], the problem of finding the top K arms with the
largest expected reward can be modeled by decision class MTOP K(K) = {M ? [n] | M = K}.
Let A = {A1 , . . . , Am } be a partition of [n]. The problem of identifying the best arms from each
group of arms A1 , . . . , Am can be modeled by decision class MMB(A) = {M ? [n] | 8i 2
[m], |M \ Ai | = 1}. Note that maximization oracles for these two decision classes are trivially the
functions of returning the top k arms or the best arms of each group.
Then we have width(MTOP K(K) ) ? 2 and width(MMB(A) ) ? 2 (see Fact 2 and 3 in the supplementary material) and therefore the sample complexity of CLUCB for solving T OP K and MB is
O H log(nH/ ) , which matches previous results in the fixed confidence setting [19, 13, 20] up to
constant factors.
Next we consider the problem of identifying the maximum matching and the problem of finding
the shortest path (by negating the rewards), in a setting where arms correspond to edges. For these
problems, Theorem 1 establishes the first known sample complexity bound.
5
Example 2 (Matchings and Paths). Let G(V, E) be a graph with n edges and assume there is a oneto-one mapping between edges E and arms [n]. Suppose that G is a bipartite graph. Let MM ATCH(G)
correspond to the set of all matchings in G. Then we have width(MM ATCH(G) ) ? |V | (In fact, we
construct an exchange class corresponding to the collection of augmenting cycles and augmenting
paths of G; see Fact 4).
Next suppose that G is a directed acyclic graph and let s, t 2 V be two vertices. Let MPATH(G,s,t)
correspond to the set of all paths from s to t. Then we have width(MPATH(G,s,t) ) ? |V | (In fact,
we construct an exchange class corresponding to the collection of disjoint pairs of paths; see
Fact 5). Therefore the sample complexity bounds of CLUCB for decision classes MM ATCH(G) and
MPATH(G,s,t) are O |V |2 H log(nH/ ) .
Last, we investigate the general problem of identifying the maximum-weight basis of a matroid.
Again, Theorem 1 is the first sample complexity upper bound for this type of pure exploration tasks.
Example 3 (Matroids). Let T = (E, I) be a finite matroid, where E is a set of size n (called
ground set) and I is a family of subsets of E (called independent sets) which satisfies the axioms of
matroids (see Footnote 3 in Appendix F). Assume that there is a one-to-one mapping between E and
[n]. Recall that a basis of matroid T is a maximal independent set. Let MM ATROID(T ) correspond
to the set of all bases of T . Then we have width(MM ATROID(T ) ) ? 2 (derived from strong basis
exchange property of matroids; see Fact 1) and the sample complexity of CLUCB for MM ATROID(T )
is O H log(nH/ ) .
The last example MM ATROID(T ) is a general type of decision class which encompasses many pure
exploration tasks including T OP K and MB as special cases, where T OP K corresponds to uniform
matroids of rank K and MB corresponds to partition matroids. It is easy to see that MM ATROID(T )
also covers the decision class that contains all spanning trees of a graph. On the other hand, it has
been established that matchings and paths cannot be formulated as matroids since they are matroid
intersections [26].
4
Lower Bound
In this section, we present a problem-dependent lower bound on the sample complexity of the CPE
problem. To state our results, we first define the notion of -correct algorithm as follows. For any
2 (0, 1), we call an algorithm A a -correct algorithm if, for any expected reward w 2 Rn , the
probability of error of A is at most , i.e., Pr[M? 6= Out] ? , where Out is the output of A.
We show that, for any decision class M and any expected rewards w, a -correct algorithm A must
use at least ? H log(1/ ) samples in expectation.
Theorem 2. Fix any decision class M ? 2[n] and any vector w 2 Rn . Suppose that, for each
arm e 2 [n], the reward distribution 'e is given by 'e = N (w(e), 1), where we let N (?, 2 )
denote Gaussian distribution with mean ? and variance 2 . Then, for any 2 (0, e 16 /4) and any
-correct algorithm A, we have
? ?
1
1
E[T ]
H log
,
(6)
16
4
where T denote the number of total samples used by algorithm A and H is defined in Eq. (2).
In Example 1 and Example 3, we have seen that the sample complexity of CLUCB is
O(H log(nH/ )) for pure exploration tasks including T OP K, MB and more generally the CPE
tasks with decision classes derived from matroids, i.e., MM ATROID(T ) (including spanning trees).
Hence, our upper and lower bound show that the CLUCB algorithm achieves the optimal sample
complexity within logarithmic factors for these pure exploration tasks. In addition, we remark that
Theorem 2 resolves the conjecture of Bubeck et al. [8] that the lower bounds of sample complexity
of T OP K and MB problems are ? H log(1/ ) , for the cases of Gaussian reward distributions.
On the other hand, for general decision classes with non-constant widths, we see that there is a gap of
2
?
?(width(M)
) between the upper bound Eq. (5) and the lower bound Eq. (6). Notice that we have
width(M) ? n for any decision class M and therefore the gap is relatively benign. Our lower bound
also suggests that the dependency on H of the sample complexity of CLUCB cannot be improved up
to logarithmic factors. Furthermore, we conjecture that the sample complexity lower bound might
inherently depend on the size of exchange sets. In the supplementary material (Appendix C.2), we
6
provide evidences on this conjecture which is a lower bound on the sample complexity of exploration
of the exchange sets.
5
Fixed Budget Algorithm
In this section, we present Combinatorial Successive Accept Reject (CSAR) algorithm, which is a
parameter-free learning algorithm for the CPE problem in the fixed budget setting. Then, we upper
bound the probability of error CSAR in terms of gaps and width(M).
Constrained oracle. The CSAR algorithm requires access to a constrained oracle, which is a function denoted as COracle : Rn ? 2[n] ? 2[n] ! M [ {?} and satisfies
(
arg maxM 2MA,B v(M ) if MA,B 6= ;
COracle(v, A, B) =
(7)
?
if MA,B = ;,
where we define MA,B = {M 2 M | A ? M, B \ M = ;} as the collection of feasible sets
and ? is a null symbol. Hence we see that COracle(v, A, B) returns an optimal set that includes all
elements of A while excluding all elements of B; and if there are no feasible sets, the constrained
oracle COracle(v, A, B) returns the null symbol ?. In the supplementary material (Appendix G),
we show that constrained oracles are equivalent to maximization oracles up to a transformation on
the weight vector. In addition, similar to CLUCB, CSAR does not need any additional knowledge of
M other than accesses to a constrained oracle for M.
Algorithm. The idea of the CSAR algorithm is as follows. The CSAR algorithm divides the budget
of T rounds into n phases. In the end of each phase, CSAR either accepts or rejects a single arm. If
an arm is accepted, then it is included into the final output. Conversely, if an arm is rejected, then it
is excluded from the final output. The arms that are neither accepted nor rejected are sampled for an
equal number of times in the next phase.
Now we describe the procedure of the CSAR algorithm for choosing an arm to accept/reject. Let
At denote the set of accepted arms before phase t and let Bt denote the set of rejected arms before
phase t. We call an arm e to be active if e 62 At [ Bt . In the beginning of phase t, CSAR samples
each active arm for T?t T?t 1 times, where the definition of T?t is given in Algorithm 2. Next,
CSAR calls the constrained oracle to compute an optimal set Mt with respect to the empirical means
w
? t , accepted arms At and rejected arms Bt , i.e., Mt = COracle(w
? t , At , Bt ). It is clear that the
output of COracle(w
? t , At , Bt ) is independent from the input w
?t (e) for any e 2 At [ Bt . Then, for
each active arm e, CSAR estimates the ?empirical gap? of e in the following way. If e 2 Mt , then
? t,e that does not include e, i.e., M
? t,e = COracle(w
CSAR computes an optimal set M
? t , At , Bt [
?
? t,e =
{e}). Conversely, if e 62 Mt , then CSAR computes an optimal Mt,e which includes e, i.e., M
?
COracle(w
? t , At [{e}, Bt ). Then, the empirical gap of e is calculated as w
?t (Mt ) w
?t (Mt,e ). Finally,
CSAR chooses the arm pt which has the largest empirical gap. If pt 2 Mt then pt is accepted,
otherwise pt is rejected. The pseudo-code CSAR is shown in Algorithm 2. We note that CSAR can
be considered as a generalization of the ideas of the two versions of SAR algorithm due to Bubeck
et al. [8], which are designed specifically for the T OP K and MB problems respectively.
5.1
Probability of error
In the following theorem, we bound the probability of error of the CSAR algorithm.
Theorem 3. Given any T > n, any decision class M ? 2[n] and any expected rewards w 2
Rn . Assume that the reward distribution 'e for each arm e 2 [n] has mean w(e) with an R-subGaussian tail. Let (1) , . . . , (n) be a permutation of 1 , . . . , n (defined in Eq. (1)) such that
2
(1) ? . . . . . . (n) . Define H2 , maxi2[n] i (i) . Then, the CSAR algorithm uses at most T
samples and outputs a solution Out 2 M [ {?} such that
?
?
(T n)
2
Pr[Out 6= M? ] ? n exp
,
(8)
?
18R2 log(n)
width(M)2 H2
Pn
?
where log(n)
,
i 1 , M? = arg max
w(M ) and width(M) is defined in Eq. (4).
i=1
M 2M
One can verify that H2 is equivalent to H up to a logarithmic factor: H2 ? H ? log(2n)H2 (see
[1]). Therefore, by setting the probability of error (the RHS of Eq. (8)) to a constant, one can see
2
?
that CSAR requires a budget of T = O(width(M)
H) samples. This is equivalent to the sample
complexity bound of CLUCB up to logarithmic factors. In addition, applying Theorem 3 back to
T OP K and MB, our bound matches the previous fixed budget algorithm due to Bubeck et al. [8].
7
Algorithm 2 CSAR: Combinatorial Successive Accept Reject
Require: Budget: T > 0; Constrained oracle: COracle : Rn ? 2[n] ? 2[n] ! M [ {?}.
P
1
?
1: Define log(n)
, n
i=1 i
?
2: T0
0, A1
;, B1
;
3: for t = 1,l. . . , n do
m
T n
4:
T?t
?
log(n)(n
t+1)
5:
Pull each arm e 2 [n]\(At [ Bt ) for T?t T?t 1 times
6:
Update the empirical means w
? t for each arm e 2 [n]\(At [ Bt )
. set w
?t (e) = 0, 8e 2 At [ Bt
7:
Mt
COracle(w
? t , At , Bt )
8:
if Mt = ? then
9:
fail: set Out
? and return Out
10:
for each e 2 [n]\(At [ Bt ) do
? t,e
11:
if e 2 Mt then M
COracle(w
? t , At , Bt [ {e})
?
12:
else Mt,e
COracle(w
? t , At [ {e}, Bt )
? t,e )
13:
pt
arg maxe2[n]\(At [Bt ) w
?t (Mt ) w
? t (M
. define w
?t (?) = 1; break ties arbitrarily
14:
if pt 2 Mt then
15:
At+1
At [ {pt }, Bt+1
Bt
16:
else
17:
At+1
At , Bt+1
Bt [ {pt }
18: Out
An+1
19: return Out
6
Related Work
The multi-armed bandit problem has been extensively studied in both stochastic and adversarial
settings [22, 3, 2]. We refer readers to [5] for a survey on recent advances. Many work in MABs focus
on minimizing the cumulative regret, which is an objective known to be fundamentally different
from the objective of pure exploration MABs [6]. Among these work, a recent line of research
considers a generalized setting called combinatorial bandits in which a set of arms (satisfying certain
combinatorial constraints) are played on each round [9, 17, 25, 7, 10, 14, 23, 21]. Note that the
objective of these work is to minimize the cumulative regret, which differs from ours.
In the literature of pure exploration MABs, the classical problem of identifying the single best arm
has been well-studied in both fixed confidence and fixed budget settings [24, 11, 6, 1, 13, 15, 16].
A flurry of recent work extend this classical problem to T OP K and MB problems and obtain algorithms with upper bounds [18, 12, 13, 19, 8, 20, 31] and worst-case lower bounds of T OP K [19, 31].
Our framework encompasses these two problems as special cases and covers a much larger class of
combinatorial pure exploration problems, which have not been addressed in current literature. Applying our results back to T OP K and MB, our upper bounds match best available problem-dependent
bounds up to constant factors [13, 19, 8] in both fixed confidence and fixed budget settings; and our
lower bound is the first proven problem-dependent lower bound for these two problems, which are
conjectured earlier by Bubeck et al. [8].
7
Conclusion
In this paper, we proposed a general framework called combinatorial pure exploration (CPE) that
can handle pure exploration tasks for many complex bandit problems with combinatorial constraints,
and have potential applications in various domains. We have shown a number of results for the
framework, including two novel learning algorithms, their related upper bounds and a novel lower
bound. The proposed algorithms support a wide range of decision classes in a unifying way and our
analysis introduced a novel tool called exchange class, which may be of independent interest. Our
upper and lower bounds characterize the complexity of the CPE problem: the sample complexity of
our algorithm is optimal (up to a logarithmic factor) for the decision classes derived from matroids
(including T OP K and MB), while for general decision classes, our upper and lower bounds are
within a relatively benign factor.
Acknowledgments. The work described in this paper was partially supported by the National Grand
Fundamental Research 973 Program of China (No. 2014CB340401 and No. 2014CB340405), the
Research Grants Council of the Hong Kong Special Administrative Region, China (Project No.
CUHK 413212 and CUHK 415113), and Microsoft Research Asia Regional Seed Fund in Big Data
Research (Grant No. FY13-RES-SPONSOR-036).
8
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9
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4,898 | 5,434 | From Stochastic Mixability to Fast Rates
Robert C. Williamson
Research School of Computer Science
Australian National University and NICTA
[email protected]
Nishant A. Mehta
Research School of Computer Science
Australian National University
[email protected]
Abstract
Empirical risk minimization (ERM) is a fundamental learning rule for statistical
learning problems where the data is generated according to some unknown distribution P and returns a hypothesis f chosen from a fixed class F with small loss
? `.
In the parametric setting, depending upon (`, F, P) ERM can have slow (1/ n)
or fast (1/n) rates of convergence of the excess risk as a function of the sample
size n. There exist several results that give sufficient conditions for fast rates in
terms of joint properties of `, F, and P, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there
is an analogous slow and fast rate phenomenon, and it is entirely characterized in
terms of the mixability of the loss ` (there being no role there for F or P). The
notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents
a direct proof of fast rates for ERM in terms of stochastic mixability of (`, F, P),
and in so doing provides new insight into the fast-rates phenomenon. The proof
exploits an old result of Kemperman on the solution to the general moment problem. We also show a partial converse that suggests a characterization of fast rates
for ERM in terms of stochastic mixability is possible.
1
Introduction
Recent years have unveiled central contact points between the areas of statistical and online learning.
These include Abernethy et al.?s [1] unified Bregman-divergence based analysis of online convex
optimization and statistical learning, the online-to-batch conversion of the exponentially weighted
average forecaster (a special case of the aggregating algorithm for mixable losses) which yields the
progressive mixture rule as can be seen e.g. from the work of Audibert [2], and most recently Van
Erven et al.?s [21] injection of the concept of mixability into the statistical learning space in the form
of stochastic mixability. It is this last connection that will be our departure point for this work.
Mixability is a fundamental property of a loss that characterizes when constant regret is possible in
the online learning game of prediction with expert advice [23]. Stochastic mixability is a natural
adaptation of mixability to the statistical learning setting; in fact, in the special case where the function class consists of all possible functions from the input space to the prediction space, stochastic
mixability is equivalent to mixability [21]. Just as Vovk and coworkers (see e.g. [24, 8]) have developed a rich convex geometric understanding of mixability, stochastic mixability can be understood
as a sort of effective convexity.
In this work, we study the O(1/n)-fast rate phenomenon in statistical learning from the perspective
of stochastic mixability. Our motivation is that stochastic mixability might characterize fast rates in
statistical learning. As a first step, Theorem 5 herein establishes via a rather direct argument that
stochastic mixability implies an exact oracle inequality (i.e. with leading constant 1) with a fast rate
for finite function classes, and Theorem 7 extends this result to VC-type classes. This result can be
understood as a new chapter in an evolving narrative that started with Lee et al.?s [13] seminal paper
1
showing fast rates for agnostic learning with squared loss over convex function classes, and that was
continued by Mendelson [18] who showed that fast rates are possible for p-losses (y, y?) 7? |y ? y?|p
over effectively convex function classes by passing through a Bernstein condition (defined in (12)).
We also show that when stochastic mixability does not hold in a certain sense (described in Section 5), then the risk minimizer is not unique in a bad way. This is precisely the situation at the
heart of the works of Mendelson [18] and Mendelson and Williamson [19], which show that having
non-unique minimizers is symptomatic of bad geometry of the learning problem. In such situations,
there are certain targets (i.e. output conditional distributions) close to the original target under which
empirical risk minimization learns (ERM) at a slow rate, where the guilty target depends on the sample size and the target sequence approaches the original target asymptotically. Even the best known
upper bounds have constants that blow up in the case of non-unique minimizers. Thus, whereas
stochastic mixability implies fast rates, a sort of converse is also true, where learning is hard in a
?neighborhood? of statistical learning problems for which stochastic mixability does not hold. In
addition, since a stochastically mixable problem?s function class looks convex from the perspective
of risk minimization, and since when stochastic mixability fails the function class looks non-convex
from the same perspective (it has multiple well-separated minimizers), stochastic mixability characterizes the effective convexity of the learning problem from the perspective of risk minimization.
Much of the recent work in obtaining faster learning rates in agnostic learning has taken place in settings where a Bernstein condition holds, including results based on local Rademacher complexities
[3, 10]. The Bernstein condition appears to have first been used by Bartlett and Mendelson [4] in
their analysis of ERM; this condition is subtly different from the margin condition of Mammen and
Tsybakov [15, 20], which has been used to obtain fast rates for classification. Lecu?e [12] pinpoints
that the difference between the two conditions is that the margin condition applies to the excess loss
relative to the best predictor (not necessarily in the model class) whereas the Bernstein condition
applies to the excess loss relative to the best predictor in the model class. Our approach in this work
is complementary to the approaches of previous works, coming from a different assumption that
forms a bridge to the online learning setting. Yet this assumption is related; the Bernstein condition
implies stochastic mixability under a bounded losses assumption [21]. Further understanding the
connection between the Bernstein condition and stochastic mixability is an ongoing effort.
?
Contributions. The core contribution of this work is to show a new path to the O(1/n)-fast
rate
in statistical learning. We are not aware of previous results that show fast rates from the stochastic
mixability assumption. Secondly, we establish intermediate learning rates that interpolate between
the fast and slow rate under a weaker notion of stochastic mixability. Finally, we show that in a
certain sense stochastic mixability characterizes the effective convexity of the statistical problem.
In the next section we formally define the statistical problem, review stochastic mixability, and
explain our high-level approach toward getting fast rates. This approach involves directly appealing
to the Cram?er-Chernoff method, from which nearly all known concentration inequalities arose in one
way or another. In Section 3, we frame the problem of computing a particular moment of a certain
excess loss random variable as a general moment problem. We sufficiently bound the optimal value
of the moment, which allows for a direct application of the Cram?er-Chernoff method. These results
easily imply a fast rates bound for finite classes that can be extended to parametric (VC-type) classes,
as shown in Section 4. We describe in Section 5 how stochastic mixability characterizes a certain
notion of convexity of the statistical learning problem. In Section 6, we extend the fast rates results to
classes that obey a notion we call weak stochastic mixability. Finally, Section 7 concludes this work
with connections to related topics in statistical learning theory and a discussion of open problems.
2
Stochastic mixability, Cram?er-Chernoff, and ERM
Let (`, F, P) be a statistical learning problem with ` : Y ? R ? R+ a nonnegative loss, F ? RX a
compact function class, and P a probability measure over X ? Y for input space X and output/target
space Y. Let Z be a random variable defined as Z = (X, Y ) ? P. We assume for all f ? F,
`(Y, f (X)) ? V almost surely (a.s.) for some constant V .
A probability measure P operates on functions and loss-composed functions as:
P `(?, f ) = E(X,Y )?P ` Y, f (X) .
P f = E(X,Y )?P f (X)
2
Similarly, an empirical measure Pn associated with an n-sample z, comprising n iid samples
(x1 , y1 ), . . . , (xn , yn ), operates on functions and loss-composed functions as:
n
n
1X
1X
Pn f =
f (xj )
Pn `(?, f ) =
` yj , f (xj ) .
n j=1
n j=1
Let f ? be any function for which P `(?, f ? ) = inf f ?F P `(?, f ). For each f ? F define the excess
risk random variable Zf := ` Y, f (X) ? ` Y, f ? (X) .
We frequently work with the following two subclasses. For any ? > 0, define the subclasses
F? := {f ? F : P Zf ? ?}
F? := {f ? F : P Zf ? ?} .
2.1
Stochastic mixability
For ? > 0, we say that (`, F, P) is ?-stochastically mixable if for all f ? F
log E exp(??Zf ) ? 0.
(1)
If ?-stochastic mixability holds for some ? > 0, then we say that (`, F, P) is stochastically mixable.
Throughout this paper it is assumed that the stochastic mixability condition holds, and we take ? ? to
be the largest ? such that ?-stochastic mixability holds. Condition (1) has a rich history, beginning
from the foundational thesis of Li [14] who studied the special case of ? ? = 1 in density estimation
with log loss from the perspective of information geometry. The connections that Li showed between
this condition and convexity were strengthened by Gr?unwald [6, 7] and Van Erven et al. [21].
2.2
Cram?er-Chernoff
The high-level strategy taken here is to show that with high probability ERM will not select a fixed
hypothesis function f with excess risk above na for some constant a > 0. For each hypothesis, this
guarantee will flow from the Cram?er-Chernoff method [5] by controlling the cumulant generating
function (CGF) of ?Zf in a particular way to yield exponential concentration. This control will be
possible because the ? ? -stochastic mixability condition implies that the CGF of ?Zf takes the value
0 at some ? ? ? ? , a fact later exploited by our key tool Theorem 3.
Let Z be a real-valued random variable. Applying Markov?s inequality to an exponentially transformed random variable yields that, for any ? ? 0 and t ? R
Pr(Z ? t) ? exp(??t + log E exp(?Z));
(2)
the inequality is non-trivial only if t > E Z and ? > 0.
2.3
Analysis of ERM
We consider the ERM estimator f?z := arg minf ?F Pn `(?, f ). That is, given an n-sample z, ERM
selects any f?z ? F minimizing the empirical risk Pn `(?, f ). We say ERM is ?-good when f?z ? F? .
In order to show that ERM is ?-good it is sufficient to show that for all f ? F \ F? we have
P Zf > 0. The goal is to show that with high probability ERM is ?-good, and we will do this by
showing that with high probability uniformly for all f ? F \ F? we have Pn Zf > t for some
slack t > 0 that will come in handy later.
For a real-valued random variable X, recall that the cumulant generating function of X is ? 7?
?X (?) := log E e?X ; we allow ?X (?) to be infinite for some ? > 0.
Theorem 1 (Cram?er-Chernoff Control on ERM). Let a > 0 and select f such that E Zf > 0.
Let t < E Zf . If there exists ? > 0 such that ??Zf (?) ? ? na , then
n
o
Pr Pn `(?, f ) ? Pn `(?, f ? ) + t ? exp(?a + ?t).
Pn
Proof. Let Zf,1 , . . . , Zf,n be iid copies of Zf , and define the sum Sf,n :=
j=1 ?Zf,j . Since
1
(?t) > E n Sf,n , then from (2) we have
X
n
1
1
Pr
Zf,j ? t = Pr
Sf,n ? ?t ? exp (?t + log E exp(?Sf,n ))
n j=1
n
n
= exp(?t) E exp(??Zf ) .
3
Making the replacement ??Zf (?) = log E exp(??Zf ) yields
1
log Pr
Sf,n ? ?t ? ?t + n??Zf (?).
n
By assumption, ??Zf (?) ? ? na , and so Pr{Pn Zf ? t} ? exp(?a + ?t) as desired.
This theorem will be applied by showing that for an excess loss random variable Zf taking values in
[?1, 1], if for some ? > 0 we have E exp(??Zf ) = 1 and if E Zf = na for some constant a (that can
and must depend on n), then ??Zf (?/2) ? ? c?a
n where c > 0 is a universal constant. This is the
nature of the next section. We then extend this result to random variables taking values in [?V, V ].
3
Semi-infinite linear programming and the general moment problem
The key subproblem now is to find, for each excess loss random variable Zf with mean na and
??Zf (?) = 0 (for some ? ? ? ? ), a pair of constants ?0 > 0 and c > 0 for which ??Zf (?0 ) ? ? ca
n.
Theorem 1 would then imply that ERM will prefer f ? over this particular f with high probability for
ca large enough. This subproblem is in fact an instance of the general moment problem, a problem
on which Kemperman [9] has conducted a very nice geometric study. We now describe this problem.
The general moment problem. Let P(A) be the space of probability measures over a measurable
space A = (A, S). For real-valued measurable functions h and (gj )j?[m] on a measurable space
A = (A, S), the general moment problem is
inf
EX?? h(X)
??P(A)
(3)
subject to EX?? gj (X) = yj , j ? {1, . . . , m}.
Let the vector-valued map g : A ? Rm be defined in terms of coordinate functions as (g(x))j =
gj (x), and let the vector y ? Rm be equal to (y1 , . . . , ym ).
Let D? ? Rm+1 be the set
m
X
?
?
m+1
D := d = (d0 , d1 , . . . , dm ) ? R
: h(x) ? d0 +
dj gj (x) for all x ? A .
(4)
j=1
Theorem 3 of [9] states that if y ? int conv g(A), the optimal value of problem (3) equals
m
X
?
?
dj yj : d = (d0 , d1 , . . . , dm ) ? D .
sup d0 +
(5)
j=1
Our instantiation. We choose A = [?1, 1], set m = 2 and define h, (gj )j?{1,2} , and y ? R2 as:
a
h(x) = ?e(?/2)x ,
g1 (x) = x,
g2 (x) = e?x ,
y1 = ? ,
y2 = 1,
n
for any ? > 0, a > 0, and n ? N. This yields the following instantiation of problem (3):
inf
??P([?1,1])
subject to
EX?? ?e(?/2)X
a
n
= 1.
(6a)
EX?? X = ?
(6b)
EX?? e?X
(6c)
Note that equation (5) from the general moment problem now instantiates to
n
o
a
sup d0 ? d1 + d2 : d? = (d0 , d1 , d2 ) ? D? ,
n
with D? equal to the set
n
o
d? = (d0 , d1 , d2 ) ? R3 : ?e(?/2)x ? d0 + d1 x + d2 e?x for all x ? [?1, 1] .
(7)
(8)
Applying Theorem 3 of [9] requires the condition y ? int conv g([?1, 1]). We first characterize
when y ? conv g([?1, 1]) holds and handle the int conv g([?1, 1]) version after Theorem 3.
4
Lemma 2 (Feasible Moments). The point y = ? na , 1 ? conv g([?1, 1]) if and only if
cosh(?) ? 1
a
e? + e?? ? 2
=
?
.
n
e? ? e??
sinh(?)
(9)
Proof. Let W denote the convex hull of g([?1, 1]). We need to see if ? na , 1 ? W . Note that W
is the convex set formed by starting with the graph of x 7? e?x on the domain [?1, 1], including the
line segment connecting this curve?s endpoints (?1, e?? ) to (1, e?x ), and including all of the points
below this line segment but above the aforementioned graph. That is, W is precisely the set
e? + e??
e? ? e??
W := (x, y) ? R2 : e?x ? y ?
+
x, ?x ? [?1, 1] .
2
2
It remains to check that 1 is sandwiched between the lower and upper bounds at x = ? na . Clearly
the lower bound holds. Simple algebra shows that the upper bound is equivalent to condition (9).
Note that if (9) does not hold, then the semi-infinite linear program (6) is infeasible; infeasibility in
turn implies that such an excess loss random variable cannot exist. Thus, we need not worry about
whether (9) holds; it holds for any excess loss random variable satisfying constraints (6b) and (6c).
The following theorem is a key technical result for using stochastic mixability to control the CGF.
The proof is long and can be found in Appendix A.
Theorem 3 (Stochastic Mixability Concentration). Let f be an element of F with Zf taking values in [?1, 1], n ? N, E Zf = na for some a > 0, and ??Zf (?) = 0 for some ? > 0. If
e? + e?? ? 2
a
<
,
n
e? ? e??
E e(?/2)(?Zf ) ? 1 ?
then
(10)
0.18(? ? 1)a
.
n
Note that since log(1 ? x) ? ?x when x < 1, we have ??Zf (?/2) ? ? 0.18(?n? 1)a .
In order to apply Theorem 3, we need (10) to hold, but only (9) is guaranteed to hold. The corner
case is if (9) holds with equality. However, observe that one can always approximate the random
variable X by a perturbed version X 0 which has nearly identical mean a0 ? a and a nearly identical
0
0
? 0 ? ? for which EX 0 ??0 e? X = 1, and yet the inequality in (9) is strict. Later, in the proof
of Theorem 5, for any random variable that required perturbation to satisfy the interior condition
(10), we implicitly apply the analysis to the perturbed version, show that ERM would not pick the
(slightly different) function corresponding to the perturbed version, and use the closeness of the two
functions to show that ERM also would not pick the original function.
We now present a necessary extension for the case of losses with range [0, V ], proved in Appendix A.
Lemma 4 (Bounded Losses). Let g1 (x) = x and y2 = 1 be common settings for the following two
problems. The instantiation of problem (3) with A = [?V, V ], h(x) = ?e(?/2)x , g2 (x) = e?x ,
and y1 = ? na has the same optimal value as the instantiation of problem (3) with A = [?1, 1],
h(x) = ?e(V ?/2)x , g2 (x) = e(V ?)x , and y1 = ? a/V
n .
4
Fast rates
We now show how the above results can be used to obtain an exact oracle inequality with a fast rate.
We first present a result for finite classes and then present a result for VC-type classes (classes with
logarithmic universal metric entropy).
?
Theorem 5 (Finite Classes Exact Oracle Inequality). Let (`, F,
P) be ? -stochastically mixable,
where |F| = N , ` is a nonnegative loss, and supf ?F ` Y, f (X) ? V a.s. for a constant V . Then
for all n ? 1, with probability at least 1 ? ?
n
o
6 max V, ?1? log 1? + log N
?
P `(?, f?z ) ? P `(?, f ) +
.
n
5
(?)
Proof. Let ?n = na for a constant a to be fixed later. For each ? > 0, let F?n ? F?n correspond
to those functions in F?n for which ? is the largest constant such that E exp(??Zf ) = 1. Let
hyper
F?
? F?n correspond to functions f in F?n for which lim??? E exp(??Zf ) < 1. Clearly,
n
S
(?)
hyper
F?n =
??[? ? ,?) F?n ? F?n . The excess loss random variables corresponding to elements
hyper
f ? F?
are ?hyper-concentrated? in the sense that they are infinitely stochastically mixable.
n
However, Lemma 10 in Appendix B shows that for each hyper-concentrated Zf , there exists another
excess loss random variable Zf0 with mean arbitrarily close to that of Zf , with E exp(??Zf0 ) = 1 for
some arbitrarily large but finite ?, and with Zf0 ? Zf with probability 1. The last property implies
that the empirical risk of Zf0 is no greater than that of Zf ; hence for each hyper-concentrated Zf it is
sufficient (from the perspective of ERM) to study a corresponding Zf0 . From now on, we implicitly
S
(?)
make this replacement in F?n itself, so that we now have F?n = ??[?? ,?) F?n .
(?)
Consider an arbitrary a > 0. For some fixed ? ? [? ? , ?) for which |F?n | > 0, consider
(?)
the subclass F?n . Individually for each such function, we will apply Theorem 1 as follows.
From Lemma 4, we have ??Zf (?/2) = ?? V1 Zf (V ?/2). From Theorem 3, the latter is at most
1)(a/V )
? 0.18(V ? ?
= ? (V0.18?a
n
? ? 1)n . Hence, Theorem 1 with t = 0 and the ? from the Theorem taken to be ?/2 implies that the probability of the event Pn `(?, f ) ? Pn `(?, f ? ) is at most
exp ?0.18 V ??? 1 a . Applying the union bound over all of F?n , we conclude that
0.18a
Pr {?f ? F?n : Pn `(?, f ) ? Pn `(?, f ? )} ? N exp ?? ?
.
V ?? ? 1
Since ERM selects hypotheses on their empirical risk, from inversion it holds that with probability at
6 max{V, ?1? }(log ?1 +log N )
.
least 1 ? ? ERM will not select any hypothesis with excess risk at least
n
Before presenting the result for VC-type classes, we require some definitions. For a pseudometric
space (G, d), for any ? > 0, let N (?, G, d) be the ?-covering number of (G, d); that is, N (?, G, d) is
the minimal number of balls of radius ? needed to cover G. We will further constrain the cover (the
set of centers of the balls) to be a subset of G (i.e. to be proper), thus ensuring that the stochastic
mixability assumption transfers to any (proper) cover of F. Note that the ?proper? requirement at
most doubles the constant K below, as shown by Vidyasagar [22, Lemma 2.1].
We now state a localization-based result that allows us to extend the result for finite classes to VCtype classes. Although the localization result can be obtained by combining standard techniques,1
we could not find this particular result in the literature. Below, an ?-net F? of a set F is a subset of
F such that F is contained in the union of the balls of radius ? with centers in F? .
Theorem 6. Let F be a separable function class whose functions have range bounded in [0, V ] and
for which, for a constant K ? 1, for each u ? (0, K] the L2 (P) covering numbers are bounded as
C
K
N (u, F, L2 (P)) ?
.
(11)
u
Suppose F? is a minimal ?-net for F in the L2 (P) norm, with ? = n1 . Denote by ? : F ? F? an
L2 (P)-metric projection from F to F? . Then, provided that ? ? 21 , with probability at most ? can
there exist f ? F such that
s
!
V
1
e
Pn f < Pn (?(f )) ?
1080C log(2Kn) + 90
log
C log(2Kn) + log
.
n
?
?
The proof is presented in Appendix C. We now present the fast rates result for VC-type classes.
The proof (in Appendix C) uses Theorem 6 and the proof of the Theorem 5. Below, we denote the
loss-composed version of a function class F as ` ? F := {`(?, f ) : f ? F}.
1
See e.g. the techniques of Massart and N?ed?elec [16] and equation (3.17) of Koltchinskii [11].
6
Theorem 7 (VC-Type Classes Exact Oracle Inequality). Let (`, F, P) be ? ? -stochastically mixable with ` ? F separable, where, for a constant K ? 1, for each ? ? (0, K] we have
C
N (` ? F, L2 (P), ?) ? K
, and supf ?F ` Y, f (X) ? V a.s. for a constant V ? 1. Then
?
for all n ? 5 and ? ? 12 , with probability at least 1 ? ?
o
n
?
?
C log(Kn) + log 2? ,
8 max V, ?1?
1
?
q
P `(?, fz ) ? P `(?, f ) + max
? 2V 1080C log(2Kn) + 90 log 2 C log(2Kn) + log
n
?
?
?
?
5
2e
?
?
+
1
.
n
Characterizing convexity from the perspective of risk minimization
In the following, when we say (`, F, P) has a unique minimizer we mean that any two minimizers
f1? , f2? of P `(?, f ) over F satisfy ` Y, f1? (X) = ` Y, f2? (X) a.s. We say the excess loss class
{`(?, f ) ? `(?, f ? ) : f ? F} satisfies a (?, B)-Bernstein condition with respect to P for some B > 0
and 0 < ? ? 1 if, for all f ? F:
2
?
P `(?, f ) ? `(?, f ? ) ? B P `(?, f ) ? `(?, f ? )
.
(12)
It already is known that the stochastic mixability condition guarantees that there is a unique minimizer [21]; this is a simple consequence of Jensen?s inequality. This leaves open the question: if
stochastic mixability does not hold, are there necessarily non-unique minimizers? We show that in
a certain sense this is indeed the case, in bad way: the set of minimizers will be a disconnected set.
For any ? > 0, define G? as the class G? := {f ? } ? f ? F : kf ? f ? kL1 (P) ? ? , where in case
there are multiple minimizers in F we arbitrarily select one of them as f ? . Since we assume that F
is compact and G? \ {f ? } is equal to F minus an open set homeomorphic to the unit L1 (P) ball,
G? \ {f ? } is also compact.
Theorem 8 (Non-Unique Minimizers). Suppose there exists some ? > 0 such that G? is not
stochastically mixable. Then there are minimizers
f1? , f2? ? F of P `(?, f ) over F such that it is
?
?
not the case that ` Y, f1 (X) = ` Y, f2 (X) a.s.
Proof. Select ? > 0 as in the theorem and some fixed ? > 0. Since G? is not ?-stochastically
mixable, there exists f? ? G? such that ??Zf? (?) > 0. Note that there exists ? 0 ? (0, ?) with
??Zf? (? 0 ) = 0; if not, lim??0
??Zf (?)???Zf
?
?
?
(0)
> 0 ? ?0?Zf? (0) > 0, so ?0?Zf? (0) = E(?Zf? )
implies that E Zf? < 0, a contradiction! From Lemma 2, E Zf? ?
0
cosh(? 0 )?1
sinh(? 0 ) ;
for ? 0 ? 0 the RHS
0
0
)?1
2 0
1
has upper bound ?2 since the derivative of ?2 ? cosh(?
sinh(? 0 ) is the nonnegative function 2 tanh (? /2)
0
0
)?1
and ?2 ? cosh(?
|?0 =0 = 0. Thus, E Zf? ? 0 as ? ? 0. As G? \ {f ? } is compact, we can take
sinh(? 0 )
a positive decreasing sequence (?j )j approaching 0, corresponding to a sequence (f?j )j ? G? \{f ? }
with limit point g ? ? G? \ {f ? } for which E Zg? = 0, and so there is a risk minimizer in G? \ {f ? }.
The implications of having non-unique risk minimizers. In the case of non-unique risk minimizers, Mendelson [17] showed that for p-losses (y, y?) 7? |y ? y?|p with p ? [2, ?) there is an
n-indexed sequence of probability measures (P(n) )n approaching the true probability measure as
n ? ? such that, for each n, ERM learns at a slow rate under sample size n when the true distribution is P(n) . This behavior is a consequence of the statistical learning problem?s poor geometry:
there are multiple minimizers and the set of minimizers is not even connected. Furthermore, in this
case, the best known fast rate upper bounds (see [18] and [19]) have a multiplicative constant that
approaches ? as the target probability measure approaches a probability measure for which there
are non-unique minimizers. The reason for the poor upper bounds in this case is that the constant B
in the Bernstein condition explodes, and the upper bounds rely upon the Bernstein condition.
6
Weak stochastic mixability
For some ? ? [0, 1], we say (`, F, P) is (?, ?0 )-weakly stochastically mixable if, for every ? > 0, for
all f ? {f ? } ? F? , the inequality log E exp(??? Zf ) ? 0 holds with ?? := ?0 ?1?? . This concept
was introduced by Van Erven et al. [21] without a name.
7
Suppose that some fixed function has excess risk a = ?. Then, roughly, with high probability
ERM does not make a mistake provided that a?a = n1 , i.e. when ? ? ?0 ?1?? = n1 and hence when
? = (?0 n)?1/(2??) . Modifying the proof of the finite classes result (Theorem 5) to consider all
functions in the subclass F?n for ?n = (?0 n)?1/(2??) yields the following corollary of Theorem 5.
Corollary 9. Let (`, F, P) be (?, ?0 )-weakly stochastically
mixable for some ? ? [0, 1], where
|F| = N , ` is a nonnegative loss, and supf ?F ` Y, f (X) ? V a.s. for a constant V . Then for any
n ? ?10 V (1??)/(2??) , with probability at least 1 ? ?
6 log 1? + log N
?
?
P `(?, fz ) ? P `(?, f ) +
.
(?0 n)1/(2??)
It is simple to show a similar result for VC-type classes; the ?-net can still be taken at the resolution
1
?1/(2??)
.
n , but we need only apply the analysis to the subclass of F with excess risk at least (?0 n)
7
Discussion
We have shown that stochastic mixability implies fast rates for VC-type classes, using a direct argument based on the Cram?er-Chernoff method and sufficient control of the optimal value of a certain
instance of the general moment problem. The approach is amenable to localization in that the analysis separately controls the probability of large deviations for individual elements of F. An important
open problem is to extend the results presented here for VC-type classes to results for nonparametric
classes with polynomial metric entropy, and moreover, to achieve rates similar to those obtained for
these classes under the Bernstein condition.
There are still some unanswered questions with regards to the connection between the Bernstein
condition and stochastic mixability. Van Erven et al. [21] showed that for bounded losses the Bernstein condition implies stochastic mixability. Therefore, when starting from a Bernstein condition,
Theorem 5 offers a different path to fast rates. An open problem is to settle the question of whether
the Bernstein condition and stochastic mixability are equivalent. Previous results [21] suggest that
the stochastic mixability does imply a Bernstein condition, but the proof was non-constructive, and
it relied upon a bounded losses assumption. It is well known (and easy to see) that both stochastic
mixability and the Bernstein condition hold only if there is a unique minimizer. Theorem 8 shows in
a certain sense that if stochastic mixability does not hold, then there cannot be a unique minimizer.
Is the same true when the Bernstein condition fails to hold? Regardless of whether stochastic mixability is equivalent to the Bernstein condition, the direct argument presented here and the connection
to classical mixability, which does characterize constant regret in the simpler non-stochastic setting,
motivates further study of stochastic mixability.
Finally, it would be of great interest to discard the bounded losses assumption. Ignoring the dependence of the metric entropy on the maximum possible loss, the upper bound on the loss V enters the
final bound through the difficulty of controlling the minimum value of u? (?1) when ? is large (see
the proof of Theorem 3). From extensive experiments with a grid-approximation linear program,
we have observed that the worst (CGF-wise) random variables for fixed negative mean and fixed
optimal stochastic mixability constant are those which place very little probability mass at ?V and
most of the probability mass at a small positive number that scales with the mean. These random
variables correspond to functions that with low probability beat f ? by a large (loss) margin but with
high probability have slightly higher loss than f ? . It would be useful to understand if this exotic
behavior is a real concern and, if not, find a simple, mild condition on the moments that rules it out.
Acknowledgments
RCW thanks Tim van Erven for the initial discussions around the Cram?er-Chernoff method during
his visit to Canberra in 2013 and for his gracious permission to proceed with the present paper
without him as an author, and both authors thank him for the further enormously helpful spotting
of a serious error in our original proof for fast rates for VC-type classes. This work was supported
by the Australian Research Council (NAM and RCW) and NICTA (RCW). NICTA is funded by the
Australian Government through the Department of Communications and the Australian Research
Council through the ICT Centre of Excellence program.
8
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9
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4,899 | 5,435 | Beyond Disagreement-based Agnostic Active
Learning
Chicheng Zhang
University of California, San Diego
9500 Gilman Drive, La Jolla, CA 92093
[email protected]
Kamalika Chaudhuri
University of California, San Diego
9500 Gilman Drive, La Jolla, CA 92093
[email protected]
Abstract
We study agnostic active learning, where the goal is to learn a classi?er in a prespeci?ed hypothesis class interactively with as few label queries as possible, while
making no assumptions on the true function generating the labels. The main algorithm for this problem is disagreement-based active learning, which has a high
label requirement. Thus a major challenge is to ?nd an algorithm which achieves
better label complexity, is consistent in an agnostic setting, and applies to general
classi?cation problems.
In this paper, we provide such an algorithm. Our solution is based on two novel
contributions; ?rst, a reduction from consistent active learning to con?dence-rated
prediction with guaranteed error, and second, a novel con?dence-rated predictor.
1
Introduction
In this paper, we study active learning of classi?ers in an agnostic setting, where no assumptions
are made on the true function that generates the labels. The learner has access to a large pool of
unlabelled examples, and can interactively request labels for a small subset of these; the goal is to
learn an accurate classi?er in a pre-speci?ed class with as few label queries as possible. Speci?cally,
we are given a hypothesis class H and a target ?, and our aim is to ?nd a binary classi?er in H
whose error is at most ? more than that of the best classi?er in H, while minimizing the number of
requested labels.
There has been a large body of previous work on active learning; see the surveys by [10, 28] for
overviews. The main challenge in active learning is ensuring consistency in the agnostic setting
while still maintaining low label complexity. In particular, a very natural approach to active learning
is to view it as a generalization of binary search [17, 9, 27]. While this strategy has been extended
to several different noise models [23, 27, 26], it is generally inconsistent in the agnostic case [11].
The primary algorithm for agnostic active learning is called disagreement-based active learning.
The main idea is as follows. A set Vk of possible risk minimizers is maintained with time, and the
label of an example x is queried if there exist two hypotheses h1 and h2 in Vk such that h1 (x) ?=
h2 (x). This algorithm is consistent in the agnostic setting [7, 2, 12, 18, 5, 19, 6, 24]; however, due
to the conservative label query policy, its label requirement is high. A line of work due to [3, 4, 1]
have provided algorithms that achieve better label complexity for linear classi?cation on the uniform
distribution over the unit sphere as well as log-concave distributions; however, their algorithms are
limited to these speci?c cases, and it is unclear how to apply them more generally.
Thus, a major challenge in the agnostic active learning literature has been to ?nd a general active
learning strategy that applies to any hypothesis class and data distribution, is consistent in the agnostic case, and has a better label requirement than disagreement based active learning. This has been
mentioned as an open problem by several works, such as [2, 10, 4].
1
In this paper, we provide such an algorithm. Our solution is based on two key contributions, which
may be of independent interest. The ?rst is a general connection between con?dence-rated predictors and active learning. A con?dence-rated predictor is one that is allowed to abstain from
prediction on occasion, and as a result, can guarantee a target prediction error. Given a con?dencerated predictor with guaranteed error, we show how to to construct an active label query algorithm
consistent in the agnostic setting. Our second key contribution is a novel con?dence-rated predictor
with guaranteed error that applies to any general classi?cation problem. We show that our predictor
is optimal in the realizable case, in the sense that it has the lowest abstention rate out of all predictors
guaranteeing a certain error. Moreover, we show how to extend our predictor to the agnostic setting.
Combining the label query algorithm with our novel con?dence-rated predictor, we get a general
active learning algorithm consistent in the agnostic setting. We provide a characterization of the label
complexity of our algorithm, and show that this is better than the bounds known for disagreementbased active learning in general. Finally, we show that for linear classi?cation with respect to the
uniform distribution and log-concave distributions, our bounds reduce to those of [3, 4].
2
2.1
Algorithm
The Setting
We study active learning for binary classi?cation. Examples belong to an instance space X , and
their labels lie in a label space Y = {?1, 1}; labelled examples are drawn from an underlying data
distribution D on X ? Y. We use DX to denote the marginal on D on X , and DY |X to denote the
conditional distribution on Y |X = x induced by D. Our algorithm has access to examples through
two oracles ? an example oracle U which returns an unlabelled example x ? X drawn from DX and
a labelling oracle O which returns the label y of an input x ? X drawn from DY |X .
Given a hypothesis class H of VC dimension d, the error of any h ? H with respect to a
data distribution ? over X ? Y is de?ned as err? (h) = P(x,y)?? (h(x) ?= y). We de?ne:
h? (?) = argminh?H err? (h), ? ? (?) = err? (h? (?)). For a set S, we abuse notation and use S
to also denote the uniform distribution over the elements of S. We de?ne P? (?) := P(x,y)?? (?),
E? (?) := E(x,y)?? (?).
Given access to examples from a data distribution D through an example oracle U and a labeling
? ? H such that with probability ? 1 ? ?, errD (h)
? ?
oracle O, we aim to provide a classi?er h
? ? (D) + ?, for some target values of ? and ?; this is achieved in an adaptive manner by making
as few queries to the labelling oracle O as possible. When ? ? (D) = 0, we are said to be in the
realizable case; in the more general agnostic case, we make no assumptions on the labels, and thus
? ? (D) can be positive.
Previous approaches to agnostic active learning have frequently used the notion of disagreements.
The disagreement between two hypotheses h1 and h2 with respect to a data distribution ? is
the fraction of examples according to ? to which h1 and h2 assign different labels; formally:
?? (h1 , h2 ) = P(x,y)?? (h1 (x) ?= h2 (x)). Observe that a data distribution ? induces a pseudometric ?? on the elements of H; this is called the disagreement metric. For any r and any h ? H,
de?ne B? (h, r) to be the disagreement ball of radius r around h with respect to the data distribution
?. Formally: B? (h, r) = {h? ? H : ?? (h, h? ) ? r}.
For notational simplicity, we assume that the hypothesis space is ?dense? with repsect to the data
distribution D, in the sense that ?r > 0, suph?BD (h? (D),r) ?D (h, h? (D)) = r. Our analysis will
still apply without the denseness assumption, but will be signi?cantly more messy. Finally, given a
set of hypotheses V ? H, the disagreement region of V is the set of all examples x such that there
exist two hypotheses h1 , h2 ? V for which h1 (x) ?= h2 (x).
This paper establishes a connection between active learning and con?dence-rated predictors with
guaranteed error. A con?dence-rated predictor is a prediction algorithm that is occasionally allowed to abstain from classi?cation. We will consider such predictors in the transductive setting.
Given a set V of candidate hypotheses, an error guarantee ?, and a set U of unlabelled examples,
a con?dence-rated predictor P either assigns a label or abstains from prediction on each unlabelled
2
x ? U . The labels are assigned with the guarantee that the expected disagreement1 between the
label assigned by P and any h ? V is ? ?. Speci?cally,
for all h ? V,
Px?U (h(x) ?= P (x), P (x) ?= 0) ? ?
(1)
This ensures that if some h? ? V is the true risk minimizer, then, the labels predicted by P on U do
not differ very much from those predicted by h? . The performance of a con?dence-rated predictor
which has a guarantee such as in Equation (1) is measured by its coverage, or the probability of
non-abstention Px?U (P (x) ?= 0); higher coverage implies better performance.
2.2
Main Algorithm
Our active learning algorithm proceeds in epochs, where the goal of epoch k is to achieve excess
generalization error ?k = ?2k0 ?k+1 , by querying a fresh batch of labels. The algorithm maintains a
candidate set Vk that is guaranteed to contain the true risk minimizer.
The critical decision at each epoch is how to select a subset of unlabelled examples whose labels
should be queried. We make this decision using a con?dence-rated predictor P . At epoch k, we run
P with candidate hypothesis set V = Vk and error guarantee ? = ?k /64. Whenever P abstains, we
query the label of the example. The number of labels mk queried is adjusted so that it is enough to
achieve excess generalization error ?k+1 .
An outline is described in Algorithm 1; we next discuss each individual component in detail.
Algorithm 1 Active Learning Algorithm: Outline
1: Inputs: Example oracle U , Labelling oracle O, hypothesis class H of VC dimension d,
con?dence-rated predictor P , target excess error ? and target con?dence ?.
2: Set k0 = ?log 1/??. Initialize candidate set V1 = H.
3: for k = 1, 2, ..k0 do
?
4:
Set ?k = ?2k0 ?k+1 , ?k = 2(k0 ?k+1)
2.
5:
Call U to generate a fresh unlabelled sample Uk = {zk,1 , ..., zk,nk } of size nk =
512 2
288
2
192( 512
?k ) (d ln 192( ?k ) + ln ?k ).
6:
Run con?dence-rated predictor P with inpuy V = Vk , U = Uk and error guarantee
? = ?k /64 to get abstention probabilities ?k,1 , . . . , ?k,nk on the examples in U
?kn. k These
?k,i .
probabilities induce a distribution ?k on Uk . Let ?k = Px?Uk (P (x) = 0) = n1k i=1
7:
if in the Realizable Case then
1536?k
k
8:
Let mk = 1536?
+ ln ?48k ). Draw mk i.i.d examples from ?k and query
?k (d ln
?k
O for labels of these examples to get a labelled data set Sk . Update Vk+1 using Sk :
Vk+1 := {h ? Vk : h(x) = y, for all (x, y) ? Sk }.
9:
else
10:
In the non-realizable case, use Algorithm 2 with inputs hypothesis set Vk , distribution
?k
?k , target excess error 8?
, target con?dence ?2k , and the labeling oracle O to get a new
k
hypothesis set Vk+1 .
? ? Vk +1 .
11: return an arbitrary h
0
Candidate Sets. At epoch k, we maintain a set Vk of candidate hypotheses guaranteed to contain
the true risk minimizer h? (D) (w.h.p). In the realizable case, we use a version space as our candidate
set. The version space with respect to a set S of labelled examples is the set of all h ? H such that
h(xi ) = yi for all (xi , yi ) ? S.
Lemma 1. Suppose we run Algorithm 1 in the realizable case with inputs example oracle U , labelling oracle O, hypothesis class H, con?dence-rated predictor P , target excess error ? and target
con?dence ?. Then, with probability 1, h? (D) ? Vk , for all k = 1, 2, . . . , k0 + 1.
In the non-realizable case, the version space is usually empty; we use instead a (1 ? ?)-con?dence
set for the true risk minimizer. Given a set S of n labelled examples, let C(S) ? H be a function of
1
where the expectation is with respect to the random choices made by P
3
S; C(S) is said to be a (1 ? ?)-con?dence set for the true risk minimizer if for all data distributions
? over X ? Y,
PS??n [h? (?) ? C(S)] ? 1 ? ?,
?
Recall that h (?) = argminh?H err? (h). In the non-realizable case, our candidate sets are (1 ? ?)con?dence sets for h? (D), for ? = ?. The precise setting of Vk is explained in Algorithm 2.
Lemma 2. Suppose we run Algorithm 1 in the non-realizable case with inputs example oracle U ,
labelling oracle O, hypothesis class H, con?dence-rated predictor P , target excess error ? and
target con?dence ?. Then with probability 1 ? ?, h? (D) ? Vk , for all k = 1, 2, . . . , k0 + 1.
Label Query. We next discuss our label query procedure ? which examples should we query labels
for, and how many labels should we query at each epoch?
Which Labels to Query? Our goal is to query the labels of the most informative examples. To
choose these examples while still maintaining consistency, we use a con?dence-rated predictor P
with guaranteed error. The inputs to the predictor are our candidate hypothesis set Vk which contains
(w.h.p) the true risk minimizer, a fresh set Uk of unlabelled examples, and an error guarantee ? =
?k /64. For notation simplicity, assume the elements in Uk are distinct. The output is a sequence of
abstention probabilities {?k,1 , ?k,2 , . . . , ?k,nk }, for each example in Uk . It induces a distribution ?k
over Uk , from which we independently draw examples for label queries.
How Many Labels to Query? The goal of epoch k is to achieve excess generalization error ?k .
2
?
To achieve this, passive learning requires O(d/?
k ) labelled examples in the realizable case, and
?
2
?
O(d(? (D) + ?k )/?k ) examples in the agnostic case. A key observation in this paper is that in
order to achieve excess generalization error ?k on D, it suf?ces to achieve a much larger excess
generalization error O(?k /?k ) on the data distribution induced by ?k and DY |X , where ?k is the
fraction of examples on which the con?dence-rated predictor abstains.
1536?k
k
In the realizable case, we achieve this by sampling mk = 1536?
+ ln ?48k ) i.i.d examples
?k (d ln
?k
from ?k , and querying their labels to get a labelled dataset Sk . Observe that as ?k is the abstention
probability of P with guaranteed error ? ?k /64, it is generally smaller than the measure of the
disagreement region of the version space; this key fact results in improved label complexity over
disagreement-based active learning. This sampling procedure has the following property:
Lemma 3. Suppose we run Algorithm 1 in the realizable case with inputs example oracle U , labelling oracle O, hypothesis class H, con?dence-rated predictor P , target excess error ? and target
con?dence ?. Then with probability 1 ? ?, for all k = 1, 2, . . . , k0 + 1, and for all h ? Vk ,
? returned at the end of the algorithm satis?es errD (h)
? ? ?.
errD (h) ? ?k . In particular, the h
The agnostic case has an added complication ? in practice, the value of ? ? is not known ahead of
time. Inspired by [24], we use a doubling procedure(stated in Algorithm 2) which adaptively ?nds
the number mk of labelled examples to be queried and queries them. The following two lemmas
illustrate its properties ? that it is consistent, and that it does not use too many label queries.
Lemma 4. Suppose we run Algorithm 2 with inputs hypothesis set V , example distribution ?,
? Let ?
? be the joint distribution on
labelling oracle O, target excess error ?? and target con?dence ?.
? such that on E,
?
?
? (1)
X ? Y induced by ? and DY |X . Then there exists an event E, P(E) ? 1 ? ?,
Algorithm 2 halts and (2) the set Vj0 has the following properties:
? ?
(2.1) If for h ? H, err?
?/2, then h ? Vj0 .
? (h) ? err?
? (h (?)) ? ?
? ?
?.
(2.2) On the other hand, if h ? Vj0 , then err?
? (h) ? err?
? (h (?)) ? ?
? happens, we say Algorithm 2 succeeds.
When event E
Lemma 5. Suppose we run Algorithm 2 with inputs hypothesis set V , example distribution ?,
? There exists some absolute constant
labelling oracle O, target excess error ?? and target con?dence ?.
? ?
?
). Thus
c1 > 0, such that on the event that Algorithm 2 succeeds, nj0 ? c1 ((d ln 1?? + ln 1?? ) ? (?)+?
??2
? ?
?j0
?
1
1 ? (?)+?
the total number of labels queried is j=1 nj ? 2nj0 ? 2c1 ((d ln ?? + ln ?? ) ??2 ).
2
? hides logarithmic factors
O(?)
4
A naive approach (see Algorithm 4 in the Appendix) which uses an additive VC bound gives a
? ??2 ); Algorithm 2 gives a better sample complexity.
sample complexity of O((d ln(1/?
?) + ln(1/?))?
The following lemma is a consequence of our label query procedure in the non-realizable case.
Lemma 6. Suppose we run Algorithm 1 in the non-realizable case with inputs example oracle U ,
labelling oracle O, hypothesis class H, con?dence-rated predictor P , target excess error ? and
target con?dence ?. Then with probability 1 ? ?, for all k = 1, 2, . . . , k0 + 1, and for all h ? Vk ,
? returned at the end of the algorithm satis?es
errD (h) ? errD (h? (D)) + ?k . In particular, the h
?
? ? errD (h (D)) + ?.
errD (h)
Algorithm 2 An Adaptive Algorithm for Label Query Given Target Excess Error
1: Inputs: Hypothesis set V of VC dimension d, Example distribution ?, Labeling oracle O,
?
target excess error ??, target con?dence ?.
2: for j = 1, 2, . . . do
3:
Draw nj = 2j i.i.d examples from ?; query their labels from O to get a labelled dataset
?
+ 1)).
Sj . Denote ??j := ?/(j(j
?
4:
Train an ERM classi?er hj ? V over Sj .
5:
De?ne the set Vj as follows:
?
?
?
? j ) + ?? + ?(nj , ??j ) + ?(nj , ??j )?S (h, h
?j )
Vj = h ? V : errSj (h) ? errSj (h
j
2
Where ?(n, ?) :=
16
ln 2en
n (2d?
d
6:
if suph?Vj (?(nj , ??j ) +
7:
j0 = j, break
8: return Vj0 .
2.3
+ ln 24
? ).
?
? j )) ?
?(nj , ?j )?S (h, h
j
??
6
then
Con?dence-Rated Predictor
Our active learning algorithm uses a con?dence-rated predictor with guaranteed error to make its
label query decisions. In this section, we provide a novel con?dence-rated predictor with guaranteed
error. This predictor has optimal coverage in the realizable case, and may be of independent interest.
The predictor P receives as input a set V ? H of hypotheses (which is likely to contain the true
risk minimizer), an error guarantee ?, and a set of U of unlabelled examples. We consider a soft
prediction algorithm; so, for each example in U , the predictor P outputs three probabilities that add
up to 1 ? the probability of predicting 1, ?1 and 0. This output is subject to the constraint that the
expected disagreement3 between the ?1 labels assigned by P and those assigned by any h ? V is
at most ?, and the goal is to maximize the coverage, or the expected fraction of non-abstentions.
Our key insight is that this problem can be written as a linear program, which is described in Algorithm 3. There are three variables, ?i , ?i and ?i , for each unlabelled zi ? U ; there are the probabilities with which we predict 1, ?1 and 0 on zi respectively. Constraint (2) ensures that the expected
disagreement between the label predicted and any h ? V is no more than ?, while the LP objective
maximizes the coverage under these constraints. Observe that the LP is always feasible. Although
the LP has in?nitely many constraints, the number of constraints in Equation (2) distinguishable by
Uk is at most (em/d)d , where d is the VC dimension of the hypothesis class H.
The performance of a con?dence-rated predictor is measured by its error and coverage. The error of
a con?dence-rated predictor is the probability with which it predicts the wrong label on an example,
while the coverage is its probability of non-abstention. We can show the following guarantee on the
performance of the predictor in Algorithm 3.
Theorem 1. In the realizable case, if the hypothesis set V is the version space with respect to
a training set, then Px?U (P (x) ?= h? (x), P (x) ?= 0) ? ?. In the non-realizable case, if the
hypothesis set V is an (1 ? ?)-con?dence set for the true risk minimizer h? , then, w.p ? 1 ? ?,
Px?U (P (x) ?= y, P (x) ?= 0) ? Px?U (h? (x) ?= y) + ?.
3
where the expectation is taken over the random choices made by P
5
Algorithm 3 Con?dence-rated Predictor
1: Inputs: hypothesis set V , unlabelled data U = {z1 , . . . , zm }, error bound ?.
2: Solve the linear program:
min
m
?
?i
i=1
subject to:
?i, ?i + ?i + ?i = 1
?
?
?h ? V,
?i +
i:h(zi )=1
i:h(zi )=?1
?i ? ?m
(2)
?i, ?i , ?i , ?i ? 0
3: For each zi ? U , output probabilities for predicting 1, ?1 and 0: ?i , ?i , and ?i .
In the realizable case, we can also show that our con?dence rated predictor has optimal coverage.
Observe that we cannot directly show optimality in the non-realizable case, as the performance
depends on the exact choice of the (1 ? ?)-con?dence set.
Theorem 2. In the realizable case, suppose that the hypothesis set V is the version space with
respect to a training set. If P ? is any con?dence rated predictor with error guarantee ?, and if P is
the predictor in Algorithm 3, then, the coverage of P is at least much as the coverage of P ? .
3
Performance Guarantees
An essential property of any active learning algorithm is consistency ? that it converges to the true
risk minimizer given enough labelled examples. We observe that our algorithm is consistent provided we use any con?dence-rated predictor P with guaranteed error as a subroutine. The consistency of our algorithm is a consequence of Lemmas 3 and 6 and is shown in Theorem 3.
Theorem 3 (Consistency). Suppose we run Algorithm 1 with inputs example oracle U , labelling
oracle O, hypothesis class H, con?dence-rated predictor P , target excess error ? and target
? returned by Algorithm 1 satis?es
con?dence ?. Then with probability 1 ? ?, the classi?er h
?
?
errD (h) ? errD (h (D)) ? ?.
We now establish a label complexity bound for our algorithm; however, this label complexity bound
applies only if we use the predictor described in Algorithm 3 as a subroutine.
For any hypothesis set V , data distribution D, and ?, de?ne ?D (V, ?) to be the minimum abstention probability of a con?dence-rated predictor which guarantees that the disagreement between its
predicted labels and any h ? V under DX is at most ?.
Formally, ?D (V, ?) = min{ED ?(x) : ED [I(h(x) = +1)?(x) + I(h(x) = ?1)?(x)] ?
? for all h ? V, ?(x) + ?(x) + ?(x) ? 1, ?(x), ?(x), ?(x) ? 0}. De?ne ?(r, ?) :=
?D (BD (h? , r), ?). The label complexity of our active learning algorithm can be stated as follows.
Theorem 4 (Label Complexity). Suppose we run Algorithm 1 with inputs example oracle U , labelling oracle O, hypothesis class H, con?dence-rated predictor P of Algorithm 3, target excess
error ? and target con?dence ?. Then there exist constants c3 , c4 > 0 such that with probability
1 ? ?:
(1) In the realizable case, the total number of labels queried by Algorithm 1 is at most:
c3
?log
1
?? ?
k=1
(d ln
?(?k , ?k /256)
?log(1/?)? ? k + 1 ?(?k , ?k /256)
))
+ ln(
?k
?
?k
(2) In the agnostic case, the total number of labels queried by Algorithm 1 is at most:
c4
1
?? ?
?log
k=1
(d ln
?(2? ? (D) + ?k , ?k /256)
?log(1/?)? ? k + 1 ?(2? ? (D) + ?k , ?k /256)
? ? (D)
+ln(
(1+
)
))
?k
?
?k
?k
6
Comparison. The label complexity of disagreement-based active learning is characterized in
terms of the disagreement coef?cient. Given a radius r, the disagreement coef?cent ?(r) is de?ned
as:
P(DIS(BD (h? , r? )))
,
?(r) = sup
r?
r ? ?r
where for any V ? H, DIS(V ) is the disagreement region of V . As P(DIS(BD (h? , r))) =
?
?(r, 0) [13], in our notation, ?(r) = supr? ?r ?(rr?,0) .
In the realizable case, the best known bound for label complexity of disagreement-based active
?
learning is O(?(?)
? ln(1/?) ? (d ln ?(?) + ln ln(1/?))) [20]4 . Our label complexity bound may be
simpli?ed to:
??
?
?
?
?
1
?(?
,
?
/256)
?(?
,
?
/256)
1
k
k
k
k
? ln ?
sup
,
+ ln ln
? d ln
sup
O
? k??log(1/?)?
?k
?k
?
k??log(1/?)?
which is essentially the bound of [20] with ?(?) replaced by supk??log(1/?)? ?(?k ,??kk/256) . As enforcing a lower error guarantee requires more abstention, ?(r, ?) is a decreasing function of ?; as a
result,
?(?k , ?k /256)
? ?(?),
sup
?k
k??log(1/?)?
and our label complexity bound is better.
?
2
?
?
In the agnostic case, [12] provides a label complexity bound of O(?(2?
(D)+?)?(d ? (D)
ln(1/?)+
?2
2
d ln (1/?))) for disagreement-based active-learning. In contrast, by Proposition 1 our label complexity is at most:
?
? ?
??
?
2
?(2?
(D)
+
?
,
?
/256)
(D)
?
k
k
2
?
? d
ln(1/?) + d ln (1/?)
O
sup
2? ? (D) + ?k
?2
k??log(1/?)?
Again, this is essentially the bound of [12] with ?(2? ? (D) + ?) replaced by the smaller quantity
?(2? ? (D) + ?k , ?k /256)
,
2? ? (D) + ?k
k??log(1/?)?
sup
[20] has provided a more re?ned analysis of disagreement-based active learning that gives a label
?
2
?
?
complexity of O(?(?
(D) + ?)( ? (D)
+ ln 1? )(d ln ?(? ? (D) + ?) + ln ln 1? )); observe that their
?2
?
dependence is still on ?(? (D) + ?). We leave a more re?ned label complexity analysis of our
algorithm for future work.
An important sub-case of learning from noisy data is learning under the Tsybakov noise conditions [30]. We defer the discussion into the Appendix.
3.1
Case Study: Linear Classi?cation under the Log-concave Distribution
We now consider learning linear classi?ers with respect to?log-concave data distribution on Rd . In
this case, for any r, the disagreement coef?cient ?(r) ? O( d ln(1/r)) [4]; however, for any ? > 0,
?(r,?)
? O(ln(r/?)) (see Lemma 14 in the Appendix), which is much smaller so long as ?/r is not
r
too small. This leads to the following label complexity bounds.
Corollary 1. Suppose DX is isotropic and log-concave on Rd , and H is the set of homogeneous linear classi?ers on Rd . Then Algorithm 1 with inputs example oracle U , labelling oracle O, hypothesis
class H, con?dence-rated predictor P of Algorithm 3, target excess error ? and target con?dence ?
satis?es the following properties. With probability 1 ? ?:
(1) In the realizable case, there exists some absolute constant c8 > 0 such that the total number of
labels queried is at most c8 ln 1? (d + ln ln 1? + ln 1? ).
4
? notation hides factors logarithmic in 1/?
Here the O(?)
7
(2) In the agnostic case, there exists some absolute constant c9 > 0 such that the total number of la?
2
?
?
?
+ ln 1? ) ln ?+? ? (D) (d ln ?+? ? (D) + ln 1? ) + ln 1? ln ?+? ? (D) ln ln 1? .
bels queried is at most c9 ( ? (D)
?2
(3) If (C0 , ?)-Tsybakov Noise condition holds for D with respect to H, then there exists some
constant c10 > 0 (that depends on C0 , ?) such that the total number of labels queried is at most
2
c10 ? ? ?2 ln 1? (d ln 1? + ln 1? ).
In the realizable case, our bound matches [4]. For disagreement-based
algorithms, the bound is
?
3
2 1
1
?
2
O(d ln ? (ln d + ln ln ? )), which is worse by a factor of O( d ln(1/?)). [4] does not address the
fully agnostic case directly; however, if ? ? (D) is known a-priori, then their algorithm can achieve
roughly the same label complexity as ours.
For the Tsybakov Noise Condition with ? > 1, [3, 4] provides a label complexity bound for
? ?2 ?2 ln2 1 (d + ln ln 1 )) with an algorithm that has a-priori knowledge of C0 and ?. We get
O(?
?
?
a slightly better bound. On the other hand, a disagreement based algorithm [20] gives a label
?
? 32 ln2 1 ? ?2 ?2 (ln d + ln ln 1 )). Again our bound is better by factor of ?( d)
complexity of O(d
?
?
over disagreement-based algorithms. For ? = 1, we can tighten our label complexity to get a
1
1
1
?
O(ln
? (d + ln ln ? + ln ? )) bound, which again matches [4], and is better than the ones provided by
? 32 ln2 1 (ln d + ln ln 1 )) [20].
disagreement-based algorithm ? O(d
?
?
4
Related Work
Active learning has seen a lot of progress over the past two decades, motivated by vast amounts of
unlabelled data and the high cost of annotation [28, 10, 20]. According to [10], the two main threads
of research are exploitation of cluster structure [31, 11], and ef?cient search in hypothesis space,
which is the setting of our work. We are given a hypothesis class H, and the goal is to ?nd an h ? H
that achieves a target excess generalization error, while minimizing the number of label queries.
Three main approaches have been studied in this setting. The ?rst and most natural one is generalized
binary search [17, 8, 9, 27], which was analyzed in the realizable case by [9] and in various limited
noise settings by [23, 27, 26]. While this approach has the advantage of low label complexity, it is
generally inconsistent in the fully agnostic setting [11]. The second approach, disagreement-based
active learning, is consistent in the agnostic PAC model. [7] provides the ?rst disagreement-based
algorithm for the realizable case. [2] provides an agnostic disagreement-based algorithm, which
is analyzed in [18] using the notion of disagreement coef?cient. [12] reduces disagreement-based
active learning to passive learning; [5] and [6] further extend this work to provide practical and ef?cient implementations. [19, 24] give algorithms that are adaptive to the Tsybakov Noise condition.
The third line of work [3, 4, 1], achieves a better label complexity than disagreement-based active
learning for linear classi?ers on the uniform distribution over unit sphere and logconcave distributions. However, a limitation is that their algorithm applies only to these speci?c settings, and it is
not apparent how to apply it generally.
Research on con?dence-rated prediction has been mostly focused on empirical work, with relatively
less theoretical development. Theoretical work on this topic includes KWIK learning [25], conformal prediction [29] and the weighted majority algorithm of [16]. The closest to our work is the recent
learning-theoretic treatment by [13, 14]. [13] addresses con?dence-rated prediction with guaranteed
error in the realizable case, and provides a predictor that abstains in the disagreement region of the
version space. This predictor achieves zero error, and coverage equal to the measure of the agreement region. [14] shows how to extend this algorithm to the non-realizable case and obtain zero
error with respect to the best hypothesis in H. Note that the predictors in [13, 14] generally achieve
less coverage than ours for the same error guarantee; in fact, if we plug them into our Algorithm 1,
then we recover the label complexity bounds of disagreement-based algorithms [12, 19, 24].
A formal connection between disagreement-based active learning in realizable case and perfect
con?dence-rated prediction (with a zero error guarantee) was established by [15]. Our work can
be seen as a step towards bridging these two areas, by demonstrating that active learning can be
further reduced to imperfect con?dence-rated prediction, with potentially higher label savings.
Acknowledgements. We thank NSF under IIS-1162581 for research support. We thank Sanjoy
Dasgupta and Yoav Freund for helpful discussions. CZ would like to thank Liwei Wang for introducing the problem of selective classi?cation to him.
8
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9
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